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The Millimeter-Wave Bolometric Interferometer: Data analysis, simulations and microwave instrumentation

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T h e M il l im e t e r - w a v e B o l o m e t r ic I n t e r f e r o m e t e r : D a ta A n a l y s is ,
S im u l a t io n s
and
M ic r o w a v e I n s t r u m e n t a t io n
by
SlDDHARTH S . M ALU
A dissertation subm itted in partial fulfillment of the
requirements for the degree of
D o c t o r o f P h il o s o p h y
( P h y s ic s )
at the
U n i v e r s i t y o f W is c o n s in - M a d is o n
2007
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
UMI Number: 3298524
Copyright 2007 by
Malu, Siddharth S.
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© Copyright by Siddharth S. Malu 2007
All Rights Reserved
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
C om m ittee’s Page. This page is not to be hand-written except for the
A dissertation entitled
THE MILLIMETER-WAVE BOLOMETRIC INTERPEROMETER:
DATA ANALYSIS, SIMULATIONS AND MICROWAVE INSTRUMENTATION
submitted to the Graduate School of the
University of Wisconsin-Madison
in partial fulfillment of the requirements for the
degree of Doctor of Philosophy
SIDDHARTH SAVYASACHI MALU
Date of Final Oral Examination: September 28, 2007
Com m ittee’s Page. This page is not to be hand-written except for the signatures
Month & Year Degree to be awarded:
December , 2007
May
A ugust
Approval Signatures of Dissertation Committee
Signature, Dean of Graduate School
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
The Millimeter-Wave Bolometric Interferometer:
Data Analysis, Simulations and Microwave Instrumentation
Siddharth S. Malu
Under the supervision of Professor Peter T. Timbie
At the University of W isconsin-M adison
A b stract
The following advances have occured in Cosmic Microwave Background (CMB) cosmology
in the past decade:
1. A system atic characterization of cosmological models.
2. Accurate M easurements of CMB tem perature power spectrum.
3. Detection of CMB polarization.
4. Appearance of large CMB datasets w ith new techniques for d a ta analysis.
Results from CMB theory, experiments and analysis have thus dom inated advances in cosmol­
ogy over the past few years, and are expected to do so with the upcoming experiments and
analysis techniques as well. The aforementioned results fit well within and are explained well
by the inflationary paradigm. However, current evidence for inflation is indirect. The next gen­
eration of CMB experiments will aim at providing the most direct evidence for the inflationary
paradigm through the detection of B-modes in CMB polarization. In this thesis, we describe the
design, construction and plans for implementation of a novel instrum ent, the Millimeter-Wave
Bolometric Interferometer (MBI), an interferometer designed to measure the power spectrum
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
11
of CMB polarization. We introduce novel m ethods in optics and d a ta analysis and discuss the
instrum ent.
MBI is designed for sensitive measurements of the polarization of the CMB with a 7°
field-of-view in the multipole range £=150-270. MBI combines the differencing capabilities of
an interferometer w ith the high sensitivity of bolometers in the W -band (75-110 GHz). We
introduce a novel beam combination scheme - the Fizeau system - th a t will be able to extract
both images and visibilities and provide spectral information. Gibbs sampling, an efficient and
complete Bayesian technique is described and applied to interferometry. Instrum entation and
analysis of d a ta from two components - a ferrite-based waveguide phase m odulator and an
overmoded circular waveguide system - is also discussed. A combination of these techniques,
especially the unique abilities of the Fizeau system and the com putational efficiency of Gibbs
sampling will make MBI a probe of CMB polarization signal and foregrounds w ith exquisite
sensitivity and control of system atic effects.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
A cknow ledgem ents
r^'Tl'M
<JWT*P=T
1
- Sanskrit. Translation: W hat I am dedicating to you, O Guru, O Lord, was never mine
- it was always yours.
Friend, philosopher and guide - th a t is w hat a G u r u is supposed to be. It is my pleasure
to have worked w ith an advisor who has turned out to be all of these, in every sense of the word.
Peter Timbie has been a pillar of support the entire tim e th a t I have been his student. Obviously,
I have learnt everything I know about laboratory techniques in Experim ental Cosmology from
him. He has, however, taught me much more th an th a t - to be patient when the first few
versions of anything do not work out, to keep my calm when everything th a t can possibly go
wrong does, b u t above all, to believe in myself - and th a t, at times when I had almost given
up.
Of course, one could describe those many dinners, picnics, and ’work-parties’ th a t were
a lot of fun, b u t it really is P eter’s dedication to students - teaching, training, and sometimes
even tolerating them - th a t makes him a true Guru.
Ben W andelt has been equally encouraging and supportive during the time th a t I have
worked w ith him.
Peter and Ben are together responsible for most of my knowledge and
achievements during the course of my thesis, and it w ith them in mind th a t I quote the Sanskrit
shloka above.
I have learned a great deal from members of the MBI team - Carolina helped me through
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
d a ta analysis, Jaiseung with programming, Andrei with instrum entation and instrum ent design.
It has been fun working with a wonderful team at UW-Madison - Peter H., A m anda and Emily,
my fellow graduate students. A special thanks to the undergraduates who worked w ith me, in
variuos projects - Steve Kaeppler, Seth Bruch, Eric Lopez and Lauren Levac. It was insipiring
to work w ith such a dedicated bunch of people.
I have been fortunate to have been guided by others quite like Peter and Ben throughout
my life, the first of them being my parents. It is one thing to guide and support, and quite
another to brave all the storm, ridicule APART from guiding and supporting me through all
the troubles I faced, because of the obviously wrong decisions I made in my life. It takes a huge
amount of strength to believe in someone when all they are doing is com m itting mistakes, and
repeating them over and over. I am proud to say th a t my parents were never found wanting,
and while I am sorry th a t I made them go through all th a t they did in the past ten years (which
had nothing to do with this thesis!), I am glad th at they taught me, along w ith Peter Timbie,
to believe in myself and the people close to me. They have been my base, my pillar of support,
without which I would barely become a tenth of w hat I have, far less achieve anything. They
changed their lives around my sister and me, ju st so we could have a stable childhood. They
stayed apart for long periods of tim e, so th a t we would not have to change cities or even schools
as my parents’ jobs took them from one place to another. Nor can I forget the contribution
of the rest of my family - my grandparents in particular, who had already filled up our home
with all sorts of books and supported us through difficult times, because they, like my parents,
believed in the value of a good education. It was my parents th at filled in us (my sister and
me) a sense of curiosity for the world/universe around us and the value and im portance of
perseverence in the face of all difficulty and disenchantm ent. This thesis is dedicated to them my father, Suman Malu, and my m other, Shashi Rani Malu. And to my sister, who, w ith her
great sense of hum our and wit kept me alive.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Going through my school years will produce a long list of people, all of them dedicated
teachers and great colleagues, but a few of them stand out in my memory. Ms. Suchita Bhengra,
for making even the dreariest parts of Chemistry come alive; Mr. Alan Cowell, for teaching
me the value of discipline and for making men out of us children; Mr. Donald M artin, for
patiently plowing through the derivations; Ms. Annamma, for kindling my interest in Biology;
and my friends Evanjan Banerjee, Rohit Sharma and Ravikirti for being constant support and
unwavering belief in my abilities, especially through two of the toughest years in my life; and
finally, Don Bosco Academy, Patna, which was my anchor for 12 years.
St. Stephen’s College, while elitist and exclusive, gave me the rare opportunity to learn
from Dr. Bhargava, Dr. Swaminathan, Dr. Phookun and Mr. B hatia - every one of them a
gem of a teacher. I owe my m athem atical physics background to Dr. Bhargava, who made
the subject so lively th a t I ended up extending one of the ideas he gave out in a lecture as
a full project! Working on this project with Dr. Bhargava and Dr. Phookun has been one
of the most rewarding experiences of my life - only now do I realize the full extent of their
dedication to the welfare and training of students and their patience. Yes, it would be fair
to say th a t I wouldn’t have the training or the courage to end up in Physics had it not been
for these two Gurus. They taught me to take my dreams more seriously than I thought was
possible. They also taught me to keep my feet firmly on ground, in order to be able to translate
those dreams into reality.
SSC also introduced me to some truly colourful characters th a t
have provided different shades of companionship and amusement - from Swam it’s unending
laugh-fest to Chako’s paranoia; Vivek’s overcautiousness and conscientiousness to Sum antra’s
pragmaticism; Vinayak and Vikram’s steely resolve to uncover the mysteries of Geek-land to
Advaith and P ranjal’s crazy ideas of fun.
Under Prof. Stone and Dr. Podsiadlowski’s guidance, I continued my training a t Oxford.
I thank Prof. Stone for his encoragement, particularly when I needed it during the dreary,
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
grey days. He drives his students and appreciates their qualities in a way th a t I have rarely
ever seen anyone do. Dr. Podisadlowski has an amazing knack for presenting anything in
theoretical physics and making it look simple. I am forever in debt of Jenny, my High Energy
Physics supervisor/tutor - she has to be the most enthusiastic and encouraging tu to r I have
come across. My classmates Rachel, Tom and the two Wills helped me get through the doom
of the Finals. Venkat and Prashant have been my pillars of support here in Madison through
my worst times, as also Yogini.
The author gratefully acknowledges support from Sigma-Xi through the Grants-In-Aid
of Research program , grant number G20063131556544060. The MBI program has been made
possible by the NASA ARPA grants. Lauren Levac was supported by the Bernice D urand Award
for her work w ith the MBI team in summer 2007. Prof. van der Weide in UW Engineering
very kindly allowed us to use his equipment for our tests.
This thesis has made extensive use of CMBFAST and HealPix packages, and the LAMBDA
website and tools.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
To m y fa m ily - m y first Gurus
sf^JT
3J^: wrerra" 'T T ^ r
W
H ^C I
lsft
in r : II
- Sanskrit couplet about th e Guru. Translation: Creation, sustenance and destruction
are but like child’s play to the Guru, who is th e supreme Lord, and to this Lord do I bow w ith
all my soul.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
v iii
C ontents
1
2
O verview
1
1.1
Thesis overview ....................................................................................................................
1
1.2
C o n trib u tio n s .......................................................................................................................
1
In tro d u ctio n
5
2.1
Hubble’s Law and FRWL C osm ology.............................................................................
6
2.2
Cosmodynamic c a lc u la tio n s ........................
2.3
3
10
2.2.1
P re lim in a rie s ....................................................................................................... . 10
2.2.2
Horizon size a t recombination
2.2.3
Age of the Universe
........................................................................ 11
.................................................................................................. 12
The C M B ..................................................................................................................................13
2.3.1
Problems w ith the simple early-universe m o d e l ......................
14
2.3.2
Multipole e x p a n s io n .................................................................................................. 18
T h eory o f C M B P o larization
21
3.1
Quasi-monochromatic EM w a v e s ...................................................................................
23
3.2
Spin H arm o n ics............................................
24
3.3
Application of Spin-harmonics to P o la riz a tio n .................................................................26
3.4
Thomson S c a tt e r i n g .........................................................................................
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
28
ix
3.5
4
5
CMB Polarization and Cosmology
.........................................................
C urrent sta tu s o f C M B o b servation s
30
36
4.1
D e t e c t o r s ................................................................................................................................. 36
4.2
The Wilkinson Microwave Anisotropy P r o b e ................................................................... 37
4.3
The Degree Angular Scale Interferometer
.......................................................................38
In terferom etry
41
5.1
Overview
5.2
The M utual Coherence F u n c t i o n ....................................................................................... 41
5.3
The Coherence Function of Extended S o u rc e s ................................................................ 42
5.4
Visibility as a function on Intensity pattern on the s k y ................................................ 44
5.5
Interlude: A small discussion on in te rfe ro m e try .....................................
48
5.6
Visibility, the power spectrum and the beam
51
5.6.1
................................................................................................................................. 41
........................................
Window function for one baseline in an in te rfe ro m e te r......................................55
5.6.2 Effect of finite frequency bandw idth on w idth of window fu n c tio n ....................55
5.7
Visibility in the polarized c a s e ..............................................................................................56
5.8
W hy Use an Interferom eter?......................................................................
59
5.8.1 Angular R eso lu tio n ...........................................................................
59
5.8.2 No Rapid Chopping and S c a n n in g ......................................................................... 60
5.8.3 Clean O p t i c s .................................................................................................................60
5.8.4 Direct M easurement of Stokes P a r a m e te r s ............................................................ 61
5.9
Systematic Effects
..................................................................................................................62
5.10 The Adding Interferometer
6
................................................................................................. 64
T h e F izeau C om biner: A C o n cep t S tu d y
69
6.1
69
In tro d u c tio n .......................................................................................................
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
6.2
6.3
Spectral information from an interferometer using a Fizeau a p p ro a c h ...................... 73
6.2.1
Motivation
6.2.2
P re lim in a rie s ........................................................................................ ; ....................74
6.2.3
EfFect of non-zero detector s i z e ................................................................................76
6.2.4
Feasibility of using techniques in §6.2 for M B I ......................................................76
The Fizeau combiner as an i m a g e r ....................................................................................77
6.3.1
7
................................................................................................................... 73
Remarks about the Fizeau s y s t e m ......................................................................... 79
T h e M B I In stru m en t
83
7.1
A n t e n n a e ................................................................................................................................. 87
7.2
Fizeau Beam c o m b in e r...........................................................................................................88
7.3
Detectors, electronics and d a ta acq u isitio n .......................................................................88
7.4
C ryogenics................................................................................................................................. 89
7.5
Telescope and m o u n t.............................................................................................................. 89
7.6
Measurements 1: Analysis of d a ta from the Faraday-Effect Phase M odulator
7.6.1
. . 90
Estim ation - no l o s s e s ................................................................................................ 91
7.6.2 Estim ation w ith lo s s e s ................................................................................................ 91
7.6.3 Correcting for Ferrite l o s s ..........................................................................................94
7.6.4 O ver/under-estim ation of Ferrite lo s s ...................................................................... 94
7.7
Measurements 2: A ntenna Beam P a t t e r n s ....................................................................... 95
7.7.1 Loss in an overmoded circular waveguide
7.7.2 Introduction
8
.............................................................96
.................................................................................................................96
Sim u lation s o f th e C M B sk y and th e M B I In stru m en t
107
8.1
Simulation of the CMB sky p a tc h ......................................................................................107
8.2
Simulation of th e MBI I n s t r u m e n t ...................................................................................110
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
xi
9
8.2.1
Interferometry
...........................................................................................................110
8.2.2
Integration over the field-of-view (FOV)/ sky p a t c h .........................................116
8.2.3
Interference p attern in focal p l a n e ....................................................................... 118
8.2.4
Effect of finite b a n d w id th ...........................................................................
8.2.5
Implementation of formalism to the in s tr u m e n t..................................................121
8.2.6
Recovery of Cg from instrum ent s im u la tio n ................
C M B D a ta A n a ly sis
9.1
9.3
123
130
M a p m a k in g ............................................................................................................................130
9.1.1
9.2
119
The general m apmaking p r o b le m .......................................................................... 130
Power Spectrum Estimation: Bayesian A p p r o a c h ........................................................ 134
9.2.1
Detailed Bayesian F o rm a lis m .................................................................................135
9.2.2
The problem with the Bayesian a p p r o a c h .......................................................... 135
Interlude: T he Gibbs S a m p le r........................................................................................... 136
9.3.1
.................................................................................................... 136
The problem
9.3.2 Bayes’ Theorem
........................................................................................................136
9.3.3 Sampling T e c h n iq u e ................................................................................................. 138
9.3.4 Application to e x p e rim e n t........................................................................................138
9.3.5 Results
9.4
........................................................................................................................ 140
Cf extraction using Gibbs’ S a m p lin g ...............................................................
140
9.4.1 M e t h o d ........................................................................................................................140
9.4.2 Formalism
9.5
.....................................................................................................140
Application to simulated d a t a ............................................................................................ 146
9.5.1 Gelman-Rubin Test
..................................................................................................146
10 C on clusions
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
158
x ii
A D r.
P lan ck , or: H ow I L earned to S top W orrying and Love S ta t M ech .
161
A .l
The general p r o b le m ............................................................................................................ 161
A.2
Average E n e rg y .............................................................................................
A.3
Number of phase states available, or phase f a c t o r ........................................................ 162
A.4
Planck D istrib u tio n ................................................................................................................163
A.5
Distribution for particle num ber
161
......................................................................................165
B S- and T -m a trix form u lation
166
B .l
Two port devices and the S - m a tr ix .................................................................................. 166
B.2
The need for a T - m a tr ix ...................................................................................................... 167
B.3
Conversion between S- and T - m a tr ix ..................................
168
C R ela tio n sh ip b e tw e e n I and 0
170
D Inflaton field eq u a tio n o f m o tio n and slow -roll con d ition s
172
D .l
The equation of m o t i o n ...................................................................................................... 172
D.2
Slow-roll co n d itio n s...................................................................
E E -B d eco m p o sitio n
174
176
E .l
Stokes’ p a r a m e te r s ................................................................................................................176
E.2
Relationship between E-B and Q - U ...................................................................................177
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
x iii
List o f Tables
5.1
Comparison of various optical designs for the E IP ............................................................ 61
5.2
A Comparison of Systematic E f f e c t s .......................................................................... . 63
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
x iv
List o f Figures
2.1
Evolution of perturbations...................................................................................................... 14
2.2
Acoustic oscillations in the CM B.......................................................................................... 15
2.3
B-mode power spectrum compared with tem perature and EE power spectra[5]. . 17
2.4
W M AP 3 year power spectrum .........................................................................................
3.1
B-mode, E-mode and foreground levels................................................................................31
3.2
Scalar and Tensor modes w ith corresponding E and B components. .
3.3
The contribution of tensor modes to the tem perature power spectrum ........................ 32
3.4
W M AP 1st year power spectrum: cosmic variance at low £s.......................................... 33
4.1
A schematic of a bolom eter.....................................................................................................36
4.2
A schematic of how a bolometer is used.............................................................................. 37
4.3
W M AP param eters................................................................................................................... 38
5.1
A general interferometric setup..............................................................................................43
5.2
One baseline................................................................................................................................44
5.3
Schematic of an interferometer - one baseline.................................................................... 49
5.4
The u-v plane representation............................................................................................. 50
5.5
The u-v plane w ith several pixels.......................................................................................... 51
5.6
FOV and pixel in the image plane........................................................................................ 52
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
18
....................32
XV
5.7
The same FOV and pixel as in the previous figure...........................................................52
5.8
Adding interferometer. .
5.9
Block diagram of a planned CMB polarization experim ent............................................66
6.1
A simple multi-slit diffraction/interference experim ent........................................
6.2
A simple traditional interferom eter.......................................................................................70
6.3
A simple 1-d Fizeau system ....................................................................................................71
6.4
2-slit diffraction p a tte rn ...........................................................................................................72
6.5
8-slit diffraction p a tte rn ...........................................................................................................72
6.6
16-slit diffraction p a tte rn .........................................................................................................72
6.7
16-slit diffraction p a tte rn .........................................................................................................72
6.8
u-v spread.....................................................................................................................
6.9
Beam-width limit on pixel size reduction............................................................................80
7.1
A schematic of the m ain parts of the MBI instrum ent..................................................... 83
7.2
A schematic of the m ain parts of the MBI instrum ent..................................................... 84
7.3
A detailed schematic/view of how the Fizeau combiner system .................................... 85
7.4
CMB foreground spectra from the W M AP team [2].............................
7.5
The antenna arrangem ent....................................................................................................... 87
7.6
(a) Simulation of fringe patterns: 1 baseline.(b) Fringes :6 baselines............................. 88
7.7
A spider-web JP L bolom eter....................................................................................
89
7.8
The MBI m ount....................................................................................................
90
7.9
The Vector Network Analyzer(VNA) at the van der Weide lab a t UW-Madison. . 92
.................................................................................................... 66
69
73
86
7.10 Rotation angle and how it is related to S 21
93
7.11 Rotation angle vs. current, corrected for Ferrite loss...................................................... 100
7.12 The WR-10 to 0.2” transition (gold) connected w ith an adapter.
.
..................... 101
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
xvi
7.13 Schematics of the planned antenna beam te s t.............................................
101
7.14 Raw d a ta from the tube test for pipes of different lengths...................
102
7.15 The same d a ta as in fig.(7.14), but w ith resonances smoothed o u t..............................103
7.16 G raph of loss per 10 feet derived from smoothed d a ta ....................................................104
7.17 Resonances in the d a ta in a small frequency range..........................................................105
8.1
The power spectrum used to generate the simulated maps shown below................... 109
8.2
The tem perature map obtained from the power spectrum above.
8.3
Q m ap obtained from the power spectrum above............................................................. I l l
8.4
Tem perature m ap obtained from the power spectrum above.........................................I l l
8.5
Tem perature m ap th at a 6-baseline ideal interferometer is expected to output. . . 112
8.6
This is a basic check of the m ap in fig.(8.2).......................................... ...................... . 113
8.7
Schematic of the Quasioptical beam combination set-up inside the cryostat . . . 122
8.8
Schematic of the Quasioptical beam combination set-up inside the cryostat . . . 123
8.9
The power spectrum used for the sim ulation.....................................................................126
.
......................110
8.10 Tem perature map from the power spectrum shown in fig.(8.9) above.........................127
8.11 Recovered power spectrum from the Fizeau system sim ulation.................................... 128
9.1
Results from Gibbs’ sampling for the experiment mentioned above............................ 141
9.2
Simulated “flat-sky” CMB m ap..................
9.3
Power spectra: used for simulation and recovered............................................................149
9.4
Estim ates of Cp recovered from Gibbs’ sampling; beam effects not included. . . . 150
9.5
Map recovered from Gibbs’ sampling, no beam ................................................................ 151
9.6
Estim ates of Cp recovered from Gibbs’ sampling; beam effects included - 1....... 152
9.7
Estim ates of Cg recovered from Gibbs’ sampling; beam effects included - II. . . .
153
9.8
M ap recovered from Gibbs’ sampling, beam included - I......................
154
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
148
x v ii
9.9
Map recovered from Gibbs’ sampling, beam included - II............................................ 155
9.10 Histograms of recovered values of Cgs............................................................................... 156
B .l
Scematic of the 2-port d e v i c e ........................................................................................... 167
C .l
Stereographic p ro je c tio n ............................................
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
171
1
C hapter 1
Overview
We present here a brief overview of this thesis, followed by an overview of the author’s specific
contributions to the MBI project as described in this thesis.
1.1
T h e s is o v e r v ie w
In C hapter 2, we introduce cosmology and build up on first principles to get to the
Friedmann equation. An overview of the physics behind anisotropies in the CMB is given,
followed by a discussion of the serious problems in the model of the early universe, and inflation
is presented as a possible and logical solution to all of these problems. Observable signatures
of inflation on CMB polarization are mentioned.
In C hapter 3, we discuss a way to analyze CMB polarization using spin-harmonics, a
technique reviewed by W andelt et al [1], It is shown heuristically th a t the existence of B-modes
implies the existence of gravitational waves in the early universe, which had their origins in the
inflationary era.
In Chapter 4, we discuss the current state of CMB polarization experiments and briefly
discuss problems w ith imaging experiments. This leads into a discussion of interferom etry and
its merits in Chapter 5. C hapter 6 discusses a novel idea for beam combination th a t yields
spectral information in fourier space, unlike traditional interferometric systems.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
C hapter 7 is an overview of the MBI instrum ent. Chapter 8 discusses sky and instrum ent
simulations. D ata analysis techniques are discussed in Chapter 9.
1.2
Contributions
MBI is a collaboration between several institutions, UW-Madison, and Brown and Cardiff
Universities being the largest contributors in term s of manpower and resources. M BI’s m ount
has been designed and built a t UW-Madison by Peter Hyland. Tests of M BI’s tracking ability
are ongoing and have so far proven successful. The cryostat was built by Lucio Piccirillo and
tested extensively at Cardiff by Carolina Calderon. Corrugated antennae were tested at Brown
and UW-Madison by Andrei Korotkov and Melissa Lucero. The Fizeau beam combiner was
conceived and designed by Peter Timbie, Gregory Tucker, Lucio Piccirillo and Andrei Korotkov
and has been tested extensively at Brown by Andrei Korotkov. Spider-web bolometers have
been provided by JP L and have been tested at Brown by AK. Faraday effect phase m odulators
have been provided by Brian Keating at UCSD. These devices have undergone tests at UWMadison, done mostly by Am anda Gault, w ith support from the author and Peter Hyland.
Evan Bierman at UCSD has provided expert knowledge necessary to carry out these tests.
MBI started out w ith a Butler beam combiner. Analysis of simulated data from the
B utler version of MBI and extraction of bandpowers have been discussed in exquisite detail
in C. Calderon’s thesis [2]. CC has also studied non-linear methods to recover images from
incomplete u-v coverage. Jaiseung Kim provided the antenna placement th a t maximizes u-v
coverage.
The author’s main contributions MBI is a unique instrum ent: it is able to function
simultaneously as an imager and an interferometer. The realization th a t the MBI is capable of
this is a novel idea th a t has been introduced in this thesis. Also, simulation and analysis tech­
niques developed in this thesis greatly enhance the capability of this instrum ent and will allow
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
exquisite control of system atic effects and in future versions of MBI-4, the ability to characterize
foregrounds - the m ost im portant step towards B-mode detection and characterization. W ith
these as the broad aims of this thesis, the specific contributions of the author are as follows:
1. C hapter 6: The Fizeau combiner system is developed and its application to interferometry
to recover spectral information in the fourier plane, as well as the possibility of operating
an instrum ent w ith the Fizeau system as an imager and an interferometer simultaneously
are discussed in detail.
2. C hapter 7:
(a) A measurement of loss in an overmoded circular waveguide system, w ith a view to
testing antenna beam patterns for MBI.
(b) Characterization of a ferrite-based phase m odulator in the W -band (with Am anda
Gault).
(c) Plans to carry out tests of antenna beam patterns once tests described in 2a above
are complete.
3. Chapter 8:
(a) Simulation of a CMB sky patch - this was done with a lot of help from Carolina
Calderon.
(b) Simulation of the MBI instrum ent (specifically the Fizeau system) and crude power
spectrum recovery.
4. Chapter 9: Gibbs’ sampling is a robust, computationally efficient d a ta analysis technique
and is the only efficient m ethod th a t allows global inference of covariance. This has been
applied to imaging before [3, 4], and we adapt the technique to use it with interferometric
data. This work has been done w ith Benjamin W andelt.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
5. Chapter 10: M easurements of losses in m icrostrip lines with a view to replacing guided
wave systems by a compact beam combination scheme (with R. Pathak). This measure­
ment is mentioned only in passing, since this technique is being developed for future
versions of MBI and space-based interferometeric experiments.
M BI’s novelty lies not ju st in the fact th a t it is a new instrum ent w ith a novel combination
of interferometry and bolometry, but th a t its specific design allows it to achieve the capability
of characterizing both the CMB signal and foreground. This thesis explores how this is made
possible through design, instrum entation, simulations and d a ta analysis.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
5
B ibliography
[1] Y.-T. Lin and B. D. Wandelt, “A beginner's guide to the theory of CMB tem perature and
polarization power spectra in the line-of-sight formalism,” Astroparticle Physics, vol. 25,
pp. 151-166, Mar. 2006.
[2] C. Calderon, “SIMULATION OF TH E PERFORM ANCE OF THE MILLIMETRE-WAVE
BOLOM ETRIC IN TER FER O M ETER (MBI) FO R COSMIC MICROWAVE BACK­
GROUND OBSERVATIONS. Ph.D . Thesis, Cardiff.,” Ph.D. Thesis, 2006.
[3] B. D. W andelt, D. L. Larson, and A. Lakshminarayanan, “Global, exact cosmic microwave
background d a ta analysis using Gibbs sampling,” Phys. Rev. D, vol. 70, no. 8, pp. 083511—h,
Oct. 2004.
[4] B. D. W andelt, “MAGIC: Exact Bayesian Covariance Estim ation and Signal Reconstruction
for Gaussian Random Fields,” A rXiv Astrophysics e-prints, Jan. 2004.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
6
C hapter 2
Introduction
w z rw r
'^ m
- Rig Ved, Mandala I (Translation: The primeval atom gave rise to everything we know
in the universe. However, where did it come from, and if its source is unknown, does there even
exist anyone we can offer prayers to?)
It is only relatively recently (19th century onwards) th a t scientists have made predictions
about and observations of the early Universe and have come up with a successful paradigm th a t
explains the observations and reconcile them w ith physical theories.
Once upon a redshift (c. 1965), two scientists at Bell Labs decided to test their shiny new
antenna by pointing it to different parts of the sky. They ended up with a residual noise with
an equivalent tem perature of ~ 3K and a huge confusion on their hands. The puzzle about the
source of this seemingly uniform source was solved only when physicists at the nearby Princeton
University shared w ith them their ideas about the origins of the universe. Thus started the field
of CMB cosmology, one which has proved to be even more fundam ental to our understanding
of the universe over time.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
The rest of this chapter will briefly introduce two of the three “pillars” of Cosmology (we
do not discuss prim oridal nucleosynthesis here - see, e.g.[l]), as well as the background in Gen­
eral Relativity and discuss the physics and im portance of CMB tem perature and polarization
anisotropies and how they can acts as windows to the very early universe.
2 .1
H u b b le ’s L aw a n d F R W L C o s m o lo g y
In the 1920s[2], Hubble pointed his telescope to a few galaxies and discovered the fact
th a t each one of them was moving away from us, with a velocity proportional to the distance
between us and the galaxy we’re looking at. Since there is no reason to expect th a t the Milky
Way is at the centre of the Universe, it is reasonable to extend this result and say th a t every
galaxy is receding from every other w ith the same property of recession. This has been checked
w ith observations as well. It turns out th a t the formalism for expressing Hubble’s law is simple,
and the idea along w ith all its results remains the same in the General Theory of Relativity
(GR henceforth) as well as Newtonian mechanics. Clearly, Newtonian mechanics isnot up to
the task of dealing w ith the expanding Universe, for several reasons.
Let us denote by r the physical distance between two galaxies, and by v their relative
velocity. Then, Hubble law says th a t v o c r . We can then write the equation
v = Hr
(2.1)
where H is called the Hubble param eter (technically, it should be a “constant” , but we have
tacitly ignored curvature and every other issue associated with GR; H can be thought to en­
compass all these GR effects). We would do well to remember th a t this expansion is not ju st a
widening in distance between galaxies, it is a “stretching” of space (space-time, strictly speak­
ing, but the beauty of the presently-accepted Eriedmann-Robertson-W alker-Lemaitre (FRWL)
universe model is th a t one can view “spatial slices” or spatial hypersurfaces at different times; it
is possible th a t the Universe is not FRWL - there are other solutions to the Einstein equations
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
th a t are not homogeneous spatially or temporally, but while th at is an active area of research,
everyone in the astrophysics comuunity agrees th a t FRWL is by far the most likely model th at
the Universe obeys). In th a t case, we can (as a m atter of fact, we ought to, as we will see later)
reformulate the picture in the following way. We encode the expansion of the Universe in a
single variable which is a function of time, and define w hat is called a “comoving” frame of
reference in which the distance between, say, any two given galaxies is a constant, i.e. we are
“viewing” this distance from a pre-defined epoch. There is nothing th a t prevents us from this
pre-defined epoch to “now” - indeed, this is often a convenient choice as we will see. The vari­
able th a t encodes the expansion of the Universe is called the “scale-factor” , which we represent
here with a (t ). We can then write any given physical distance as
r = a (t) x
where x is the comoving distance between the two given points under consideration.
velocity is v =
(2-2)
The
meaning th a t
v= s<“<<>x>=xl
<2-3>
where the last equality holds because x (i.e. the comoving distance between between any two
given objects) is fixed by definition. We can then write the Hubble law as
v = x ^ = Hax. = H r
dt
^ H
= - ~
a dt
(2-4)
v ’
(2.5)
This, then, is the most general definition of the Hubble param eter. By calling it a param eter,
we have gotten away w ith proving this relation for any theory of gravity we might choose to
consider - Newtonian or Einsteinian.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Next, we look at a special case of the FRWL metric, namely, the Minkowski m etric in
spherical polar co-ordinates:
ds2 = c2dt2 —a (t)2 [dr2 4- r 2dfl2]
(2-6)
It is more convenient to set c — 1 so th a t
ds2 — dt2 —a (f)2 [dr2 + r 2dS72]
(2.7)
where clearly dQ2 = dO2 + sin2 Odeb1. This represents flat space-time only. Let us generalize
this to a space-time w ith positive curvature, in analogy with a 2-sphere (the object we know
and love as a “sphere” ). This is a 3-d surface, so in analogy with the “normal” or 2-d sphere
whose equation is
x2 + y 2 + z2 = r2
(2.8)
(where r is the radius of the sphere), we have
x 2 + y 2 + z 2 + w 2 = b2
(2.9)
Here, x, y and z are ordinary spatial dimensions, and w can be thought of as a fiducial variable,
whose physical interpretation is th a t it is a 3-sphere embedded in 4-d space. If we accept this
without much ado, we can go about expressing w completely in terms of r, b etc. in the following
way.
We first rewrite the above equation as
r 2 + w2 = b2 =>• w 2 = b2 —r2
(2.10)
Differentiating this equation, we get
rdr
, o r 2dr2
2rdr + 2wdw = 0 =4 dw — :------ =4 dw — —
w
ur
r 2dr2
~
fr —r z
.
(2-11)
Now, the m etric has to be modified to
ds2 — dt2 —a (t )2 [dr2 + dw2 + r 2dQ2]
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(2-12)
10
Let us evaluate a p art of the metric:
dr + dw = dr
The
£2
1+
b2 — r 2 + r 21
b2 — r 2
= dr2
b2
1
dr2
—r 2jb2
(2.13)
in the denominator is reminiscent of curvature, and so we call it exactly th a t and rewrite
it as k. Combining everything together, we then have
ds = dt — a (t )
1
dr2
—k r 2
(2.14)
where k is curvature. Notice th a t when k — 0, the FRWL metric reduces to Minkowski, as we
would expect it to.
This m ethod can be applied w ithout loss of generality to negative curvature as well, and
the only difference is th a t the fiducial variable will satisfy this equation
r 2 — w 2 = bl 2
(2.15)
so th a t we will end up with this metric
(2.16)
1
dr 2
+ r 2dfl2
+ kr2
(2.17)
1
dr2
—k r 2
ds1 — dt 2 —a (t)2
We can generalize and write
ds 2 = dt2 —a (t ) 2
r 2d n 2
where it is understood th a t k can take positive and negative values. We can write the metric
another way by substituting \fk r = sin V k x and working out th at
dr
dx2 =
1
cos V kxdx
(2.18)
dr2 =
cos V k x d x 2
(2.19)
=> dr2 =
( l —/.;r2) riy 2
(2 .20)
dr2
—k r 2
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
( 2 .21 )
11
Substituting for this and for r, we get th a t the metric is
d x <+
ds 2 = dt2 — a (t y
( 2 . 22 )
k
This excercise is useful because we can immediately extract the Angular Diameter Distance
from the new form of the metric - it is the square root of the factor th a t multiplies dfl2:
=
( 2 '2 3 )
In the case of flat space-time, k —> 0 such th a t
—> x — r which is w hat we expect.
Having studied the geometrical aspects of the metric, let us now tu rn our attention
to the dynamics of the Universe. The equations th at are derived below are again very useful,
especially in their m ost general form, and their beauty lies in the fact th a t though the derivation
has nothing to do w ith GR, these are the exact same result wewould get if weworked with
the Einstein equations instead. The G R approach will be outlined briefly after the
following
derivation. Let us sta rt from the first law of thermodynamics:
dU + p dV = 0
(2.24)
where, naturally, U = pa3, where p is the density (total energy density, but this can be simplified
for those epochs when the total energy density is dom inated by just one component) and a the
scale factor, which is a function of time. Substituting for U , we get
a3dp 4- 3a2pda + Sa2pda =
=> 3a2da (p + p) =
=k 3 (p + p) — =
a
\p
+
) a
0
(2.25)
—a3dp
(2.26)
dp
(2.27)
*
p
(2.28)
Now £ is w hat is referred to as the equation of state. It is usually denoted by w in the literature,
so we will follow the convention:
3 (w + 1) — = —
a
p
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(2.29)
12
=*■ T
T ^a = 3 (1 + w)
a In
(2-30)
This, then, is the most general expression relating p and a. Notice th a t we have not yet made
any assumption about w - it may very well be a function of a, and this equation will still hold.
If we assume a constant equation of state ui (as is true for baryonic m atter and radiation), we
get a simpler relation:
p ~ a - 3(1+u,)
(2.31)
There is another dynamical equation we can derive with our simplistic approach, but this
one requires a leap of faith on one count. S tart out with the classical statem ent for conservation
of energy
1
o GMm
- m v ------------- — constant
2
r
(2.32)
where m is the mass of a “particle” and M is the mass of the Universe in the shape of a sphere
of uniform density p and M = (4/3) n r 3p. Changing the above equation to represent quantities
per unit mass, we get
4 7 xG
r 3p
= constant
O V
(2.33)
-
1 2
—v
Z
Use Hubble’s law: v = H r to get
8nG
constant
H 2 = ~— p +
5—
(2.34)
This is called the Friedm an Equation. It is one of the most fantastic coincidences of Cosmology
th a t
a line of argum ent as weak as the preceding one can yield the same result asGR.
can derive this from Einstein’s Equations, w ith the only differenceth a t the second
We
term on the
right will be —^ where k is space-time curvature, as before, so th a t the final equation is
rr2
8 7 rG
k
H = -5 -P - I2
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(2-35)
13
2.2
Cosmodynamic calculations
Having introduced the basic concepts in cosmology, let us work through a few small
calculation th a t will be relevant in §2.3. It is conventional to write eq.(2.35) as
H2 = ^ P c rit
(2.36)
where we have incorporated curvature and the net energy density of the universe in the quantity
Pcrit * W hen studying cosmology, we are not always interested in the value of p for different
components - just w hat fraction of the energy density they make up. To this end, we define a
set of param eters denoted by Q such th a t for a component X,
(2.37)
P c rit
is the fraction of energy density in component X at a given time.
2.2.1
P relim in aries
Here is my notation: m , 7 , A, k denote m atter, radiation, vacuum and curvature respec­
tively.
= £lm,NOW = ^mO
(2.38)
etc., and
flm (t) is the same param eter a t t im e ’t ’.
Let us ju st write down the expressions for H (t ) and H q:
H =
&7tG
— [Pm “k P'y ~k PA “k Pk] ~
87 tG
^
[PmO*^
”k P-yO®
“k PA “k PkO® ]
Per (^)
(2.39)
and
87 tG
H q
= —
g
[PmO
+
P 7 O + PA +
Preo] ~
PcrO
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(2.40)
14
Now divide the two:
H2
Per (f)
i ?0
PcrO
Hmo 3 -|- Ha T fi7u 4
HKa
(2.41)
U rn 4“ H a 4“ H 7 4" H K ( — 1 )
And so:
Per (£) = (^m® 3 4- Ha + fi7a 4 + HKa 2) pcro
(2.42)
And also, for any general component I, fi/ (t) is:
Pi
q
z
Per (f)
_______________ P/oQ______ ___________
PcrO ( n ma ~ 3 + HA + H7a ~ 4 + HKa ~ 2)
^ 43)
and so finally:
^
2 .2.2
(Hma - 3 + HA 4- fi7a - 4 + n Ka~2)
H orizon size a t recom b in ation
Since light travels at a finite speed c, in a tim e t, only those spots th a t are within a
distance ct of each other are in causal contact. Therefore, if the age of the universe is t, then
parts as big as ct are causally connected. This is called the horizon size.
The universe is radiation-dom inated from the Big-Bang almost all the way upto recombi­
nation. M atter-radiation equality occurs just before recombination, so in principle, both m atter
and radiation term s m ust be kept while calculating the horizon size.
Let us write down the expression for the Hubble parameter:
H 2 =
4
[H7 (t ) + Hm (f)] P e r
(t ) =
[H 7 o T 4 +
4
fima~ 3] PcrO =
PcrO
PcrO
[H7 4- Hma] a -4
(2.45)
Replacing H* by 1 0 0 ^ , we get:
H = 0 f i 7 + Qma}a~2h ( 10°
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(2-46)
15
Now we get to w hat we started out to calculate, the horizon size at recombination:
[ a=10~3 dt
m=cL
f 10~3 da
,
.
{2A7)
« =c/o
Replacing the value of H from above, we get:
f l0~3
I
PR ~
da
100 J0
= -M pc =
+ n ma)h
P
3000M pc f 10^
V /
(f
da
..........................(2.48)
ha, , \
,
Jo
(ffe+ a)
The final result is:
6000
VR = ----------- r
(S lm / l 2) 5
/ [TL
rnZ\
\ h c r + 10
\ v fl™
- i r
(2 -49)
\t t m j
P u tting in S4m=0.3, Q7 = 4.8 x 10-5 and /i=0.72, we get r]R= 326 Mpc. This is the comoving
horizon size at recombination. Considering the age of the universe to be ~14G light years, we
get th at the angle th a t the horizon subtends on the sky should be
^ re c o m b in a tio n h o rizo n
326
— 14000 ^
^
(2.50)
This means th a t only 4.3° patches should have similar tem peratures on the sky! However, this
is not true - the CMB sky is very nearly uniform. This problem is discussed further in §2.3.1.
In the foregoing calculation, we have assumed th a t information is able to travel at the
speed of light. However, in reality, information travels at the speed of sound in the plasma,
which happens to be ~ ^=, so th a t the above estim ate revises to ~2°.
2.2.3
A g e o f th e U n iverse
From eq. 4, we have
Hl
=
Hq
Per (f)
PcrO
= Hl
4-
+ fhy 4"
(— 1)
(2.51)
and so
i
H = H0 [nma~3 + n A + fl7a “ 4 + 9.KaT2j 5
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(2.52)
Remember the definition of the Hubble param eter:
(2.53)
a dt
from where
dt =
da
da
(2.54)
aHo [flma 3 + $2a + fl7a 4 + ClKa 2]2
so th at
t =
/=-/
da
aHo [Qma 3 +
(2.55)
+ fl^a 4 + flKa 2]2
is a general expression for the age of the universe, without quintessence. Now, a = yrz so th a t
da = —, ,2 so th a t
(i+*)
t = -
2 .3
/
dz
i
(1 + z) H q
(1 + z)3 + Qa ~t~
(1 T z)^ +
(2.56)
(1 + z)^
The CM B
The CMB is another “pillar” of cosmology, and by far the most informative one. Before
we delve into w hat cosmological param eters can be constrained w ith the CMB, let us look
briefly at the CMB itself.
Hubble’s law imples th a t as we go back in time, the size of the universe decreases monotonically. This means th a t the wavelength of photons decreases and the tem perature of the
universe increases. This implies th a t there must have been an epoch earlier th an which the
universe would have been ionized. This epoch is called “recombination” or “last scattering
surface” and we shall use these term s interchangably. Before recombination, the universe can
be thought of as a “primordial soup” of protons, electrons, neutrons (i.e. baryonic m atter) and
photons. Baryonic m atter experiences two opposing forces: the attractive force of gravity and
repulsive force of radiation pressure. These two opposing forces set up acoustic oscillations in
th e “primordial soup” . B ut these end at recombination, and the photons th a t travel freely after
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
17
recombination constitute the CMB. We need to remember, though, th a t the universe is expand­
ing even as these acoustic oscillations perm eate the universe. Keeping this in mind, and looking
a t comoving distances instead of physical ones, let us examine the acoustic oscillations in a little
more detail.
Ignoring the origin of the oscillations for the moment, we immediately see from
N
I
0
r
m
;o .5
1
First
Trough
z
e
d
0
First Peak
A
in
1 0 .5
I
S u p e r-H o riz o n
t
ti
d _ !
Figure 2.1: Evolution of perturbations. Shown here are three oscillation sizes which are impor­
tan t for extracting informatin from the CMB.
figs. (2.1) and (2.2) th a t every length scale ends up with a different amplitude. If the wavelength
of a “mode” (i.e. a length scale) is sufficiently large, small changes in the wavelength do not
produce an appreciable effect (this is the reason th at the power spectrum is nearly constant
for low is - see fig.(2.4)). As the wavelength decreases, however, the am plitude of the mode at
recombination increases until it reaches a maximum, and then decreases with decreasing wave­
length. The am plitude cannot possibly be measured today, but the power level can, and so this
is the quantity th at CMB cosmology aims to measure. The re a s o n th a t we can measure this
quantity (i.e. the power in fluctuations in m atter) is th a t the photons th a t we detect today as
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
18
Length Scale in comoving co-ordinates
Figure 2.2: Acoustic oscillations in the CMB. W hat we are able to measure today is proportional
to the square of the am plitude at recombination, via the CMB power spectrum.
the CMB were coupled to m atter before recombination. This is why fluctuations in the CMB
tem perature directly indicate fluctuations in the m atter before recombination. W hat makes the
study of the CMB fundam ental to our understanding of the universe is th a t it is these small
fluctuations in m atter th a t grow to become all the structure we see in the universe today. T h e
s tu d y o f flu ctu a tio n s in th e C M B is th e s tu d y o f th e origins o f all stru ctu re in th e
universe.
2.3.1
P ro b lem s w ith th e sim p le early-u n iverse m od el
We have explained the origin of the CMB, b ut there are problems w ith this model:
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
19
1. W hat is the origin of these oscillations? In particular, if there is no fixed phase relation
between the oscillations at different scales, the resulting spectrum turns out to be flat!
But this is not w hat we observe; w hat causes the initial phases of these oscillations to be
related to each other?
2. We k n o w th at these oscillations must have been small - but why?
3. The universe is very nearly spatially flat - w hat causes this particular value of curvature
to be chosen? But the W O R S T problem is:
4. W hy is the entire CMB sky nearly a t one tem perature when parts of it could not have
been in causal contact (as calculated in §2.2.2)?
It is possible to explain p art 4 above if the universe started out small, b u t was expanded out
by a large amount in a short period of time. This would cause parts th a t were in causal contact
before this expansion to be more than a comoving horizon away from each other.
This simple idea was p u t forth by A lath G uth in 1981 [3] as an elegant solution to all
the four problems mentioned above, and is called “Inflation” . Before we discuss how inflation
solves the problems mentioned above, let us look at its dynamics.
One of the simplest possible rapid expansions is exponential expansion, which can happen
in the following way. Look a t the definition of the Hubble param eter:
H = Ida ^
a dt
Exponential expansion = >
a ~ e«>nstantxt^
Thus, e x p o n e n tia l e x p a n s io n = >
f f d t = d h ia
(2.57)
can p e easily achieved if I I is a constant.
c o n s ta n t H . B ut w hat component of the universe can
satisfy this condition? Let us look at the Friedmann equation:
(2.58)
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
20
This means th a t the energy density of the component dom inating the total energy density would
have to be constant. However, from eq.(2.31), we get th a t p can be constant w ith tim e if and
only if w = —1, which implies a negative pressure. While the standard model of particle physics
does not provide us w ith a particle w ith this property, [4] shows a possible way to get w = —1:
a scalar field th a t is “slowly rolling” down a potential, such th a t the potential energy dominates
the kinetic energy a t first, b u t this slowly reverses. Certain criteria need to be satisfied in order
for this to happen, and these are discussed in Appendix E.
Let us now return to the three problems mentioned above and see how inflation can solve
them:
1. Quantum field theory tells us th a t there must be fluctuations at the level of
1(T30 in
classical vacuum. If these fluctuations in energy density can be expanded out by factors
of ~ 1025, we get classical fluctuations ~ 10~5, which can act as seeds for the acoustic
oscillations which lead to the formation of the CMB and large scale structure in the
universe. Furthermore, the spectrum of these fluctuations is flat.
2. Inflation expands EVERY scale by the same factor. Combined w ith the flatness of the
initial quantum fluctuations, this leads to all the acoustic oscillations starting out in the
same phase.
3. The universe can easily have a non-zero curvature pre-inflation. However, it is always
possible to find a small enough region of space which is spatially flat.
Inflation can
expand out this small section to the entire observable universe.
4. As stated before, inflation can get rid of the horizon problem with the correct am ount of
expansion.
Inflation doesn’t just solve the problems in early universe cosmology. It produces gravi­
tational waves as well - these are the tensor perturbations in Einstein’s equations of GR in the
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
21
100.00
10.00
¥
r'S
£
N
o
r*
+
0,10
0.01
10
100
1000
Multipole moment (/)
Figure 2.3: B-mode power spectrum compared w ith tem perature and EE power spectra[5].
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
22
early universe[4]. Scattering produces polarization before the LSS because photons have a small
quadrupole m om ent1. The gravitational wave passing through space-time while polarization
is being produced causes a certain “curl” pattern to be produced [6]. Thus, polarization over
the CMB sky can be split into two parts - one with a “gradient” pattern and the other with
a “curl” pattern. These are called “E-modes” and “B-modes” respectively. In the absence
of any interactions between LSS and now, the presence of B-modes indicates the presence of
gravitational waves in the early universe. Thus, th e d e tec tio n o f B -m o d es in C M B p o­
larization an isotrop y is th e m ost direct in d ication o f in flation and the B-mode signal
is proportional to the inflaton potential [4]. Slow-roll inflation (a model developed by Alath
G uth, Andrei Linde and Andreas Albrecht) and param eters associated w ith it are discussed in
Appendix D.
2.3.2
M u ltip o le exp an sion
Anisotropies in the CMB can be expanded over the full sky in term s of spherical harmonic
functions:
at
aemYem(0,cj>)
(2.59)
This is fine, b u t how do we extract useful information about cosmology from here? And how
do we relate this to measurements?
If early universe physics described in this section is correct, then the CMB is gaussian2,
so th at a two-point correlation function contains all the information in the CMB anisotropy
field. Thus,
C (9) =
1 'the
reason for this is discussed in detail in
(ATi {9, </>) A T2 (9, <£))
chapter
2
2In reality, there is some non-gaussianity, but little of it originates in the early universe
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(2.60)
23
6000
5000
<1 4000
£
-SL
\ 3000
o~
+ 2000
1000
0
200
400
600
800
1000
Mulfipola m o m e n t I
Figure 2.4: W M AP 3 year power spectrum.
contains all the information in the CMB. It turns out th a t the fourier transform of C (6) is
CiSu>Smm, =
(2.61)
where Ci is known as the pow er sp ectru m of the CMB. It tells us the am ount of power in
anisotropies at a given lengthscale specified by £, where for large enough I G,20), I = | . For
a detailed discussion of this relationship, see Appendix C. In fig. (2.1), the am plitude of the
oscillation a t the LSS is determined by the wavelength of the particular oscillation. Each Ci is
the square of the am plitude for a particular value of the wavelength, which is a function of I
and therefore an angle on the sky. T h is is th e reason th a t a pow er sp ectru m is a m ore
usefu l to o l for stu d y in g th e early un iverse th a n an im age - it probes individual angular
scales on the sky and therefore individual length scales in the early universe. We shall discuss
later in §5.6 how the power spectrum is related to the output of an interferometer. The power
spectrum from 3-year W M AP data is shown in fig. (2.4) [5].
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
24
Bibliography
[1] R. H. Cyburt, B. D. Fields, and K. A. Olive, “Prim ordial nucleosynthesis w ith CMB inputs:
probing the early universe and light element astrophysics,” Astroparticle Physics, vol. 17,
pp. 87-100, Apr. 2002.
[2] E. Hubble, “A Relation between Distance and Radial Velocity among Extra-G alactic Neb­
ulae,” Proceedings of the National Academy of Science, vol. 15, pp. 168-173, Mar. 1929.
[3] A. H. G uth, “Inflationary universe: A possible solution to the horizon and flatness prob­
lems,” Phys. Rev. D, vol. 23, pp. 347-356, Jan. 1981.
[4] S. Dodelson, Modem cosmology, M odern cosmology / Scott Dodelson. Am sterdam (Nether­
lands): Academic Press. ISBN 0-12-219141-2, 2003, X III + 440 p., 2003.
[5] L. Page, G. Hinshaw, E. Kom atsu, M. R. Nolta, D. N. Spergel, C. L. Bennett, C. Barnes,
R. Bean, O. Dore, J. Dunkley, M. Halpern, R. S. Hill, N. Jarosik, A. Kogut, M. Limon, S. S.
Meyer, N. Odegard, H. V. Peiris, G. S. Tucker, L. Verde, J. L. Weiland, E. Wollack, and
E. L. Wright, “Three-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations:
Polarization Analysis,” A p J Suppl., vol. 170, pp. 335-376, June 2007.
[6] W. Hu and M. W hite, “A CMB polarization prim er,” New Astronomy, vol. 2, pp. 323-344,
Oct. 1997.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
25
C hapter 3
T heory o f CM B Polarization
Most of the discussion in this chapter can be found in [1] and [2].
A monochromatic plane electromagnetic wave is characterized in the following way. The
x and y components of both E and H fields obey the wave equation.
If the direction of
propagation is z, then the electric fields are given by
Ex
=
E x0ei{-kz~wt+5x)
Ey =
E y0ei{kz- wt+5v)
(3.1)
where 5X and 5y are phases associated with the two components.
Despite the appearance of 4 variables, there really are only 3 independent ones in the
above equations: E xo, E xq and 5 = 5y — Sx . We therefore need 3 quantities to completely
characterize a monochromatic wave. The extension to quasi-monochromatic waves is discussed
in §3.1 - it will emerge there th a t we need 4, and not 3 param eters to completely characterize
a general wave.
Even though E xo, E yo and 5 = Sy — 5X completely characterize a monochromatic wave,
this param etrization/characterization is not satisfactory, because none of these quantities can
be directly measured by an instrum ent. Instrum ents can measure lE^ol2, |I?,;o|2 or their linear
combinations. For instance, it is possible to use waveguides and detectors to separate out and
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
26
measure |.Exo|2 and |E yo|2 for this wave. We therefore need 3 param eters in term s of |E xo|2 and
\Eyo\2 which contain all information about E xo, E yo and 5 — 5y —6X.
Simultaneously, we need to describe the state of polarization of the wave. These two prob­
lems are tightly coupled, and can be solved simultaneously as follows. One obvious param eter
is the total intensity of the wave, I = |Ex0|2 + |i?yoi2, or equivalently, I = Ix + I y , which is easily
measured by total-power detectors. Next, thinking only in terms of linear polarization, we can
define polarization as a “difference in intensity along two independent axes” . The preceding
sentence
\Exo\2 and
is strictly speaking, wrong, since I is a scalar. B ut itdoesmake sense
to compare
\Eyo j2to checkif there is more power on one axis than the other. B ut this
“extra
power along one axis” is precisely the definition of polarization! We can therefore define one
polarization param eter in the following way
Q = |£ x0|2 - l - M 2
(3.2)
We need to check w hat happens to Q under a rotation, since it is not guaranteed to be a
rotation-invariant quantity. We do this as follows.
Under a rotation by an angle, say 6, co-ordinates transform as
x
=
y' =
x cos 6 + y sin 9
—x s \n 9 + y c o s9
(3-3)
Electric fields will therefore transform the same way:
E'x
=
E x cos 0 + E y sin 9
Ey — —E x sin9 + E y cos9
(3.4)
In the rotated co-ordinate system, Stokes’ Q is
Q' = \E'x ? - \ E ' y \*
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(3.5)
27
and
\E'X\2 =
(Ex cos 9 + E y s in 9) (.E* cos 9 + E* sin 9)
\E'y \2 =
(—E x sin 9 + E y cos 9) ( - E * s in 9 + E* cos 9)
\E'X\2 =
\EX\2 cos2 9 + \Ey |2 sin2 9 + cos 9 sin 9 {EXE*}
\E'y \2 =
\EX|2 sin2 9 + \Ey \2 cos2 6 — cos 6 sin 0 (EXE*)
(3.6)
so that
(3.7)
B ut the quantity in the last bracket is ju st 23t(E*Ey). Subtract the two expressions to get
Q' =
(\EX\2- \Ey \2) (cos2 6 — sin2 d) + 2 sin 6 cos6 (23? (E*Ey))
(3.8)
Using the trigonometric identities, and the definition of Q: Q = \EX\2 —\Ey |2, we get
Q' = Q cos 29 + 23? (E*Ey) sin 29
(3.9)
W hen we compare this to the transform ation of co-ordinates above, we find th a t this equation
suggests th a t wedefine a quantity 27Z (E * E y) - we call this Stokes’ U. We
can check th a t U
transforms as
U' = —Q sin 29 + U cos 29
(3.10)
Q' = Q cos 29 + U sin 29
(3.11)
so th at
It is also possible to define a 4th param eter V :
V = 2T (E*Ey)
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(3.12)
28
We state the definitions of the 4 quantities:
I
=
Q
= . \EX\2 - \Ey \2
U
=
2R(E*xE y)
V
=
21(E*xE y)
I^ P + I^ I2
(3.13)
Before we proceed, we note th at these definitions in the x y co-ordinate system work well only
in the “flat-sky approxim ation” . For a general treatm ent of observations of radiation from the
sky, we would need to switch to the 6 — <f> co-ordinate system, where the definitions are as
follows
I
=
\Ee\2 + W 4
Q
=
\Ee\2 - |Erf
u = 23l(E*9E^)
V
=
21(E*eE (t>)
(3.14)
In w hat follows, we will work with E x and E y - the generalization to the 0 — <p co-ordinate
system is straightforward.
We state w ithout proof th a t V is a measure of circular polarization, and is hence = zero
for the CMB. Also,
I 2 = Q2 + U2 + V 2
(3.15)
An equivalent but more rigorous and interesting m ethod of defining Stokes’ Param eters - the
Poincare Sphere - is described in §3.2 of[l].
3 .1
Q u a s i-m o n o c h r o m a tic E M w a v e s
Regardless of the degree of polarization, the observable intensity of a wave is given by its
time-averaged Poynting Flux (PF henceforth). For the monochromatic case, the expression for
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
29
P F is straightforward:
I ( P ) = ExE* + EyE *y
(3.16)
For a non-monochromatic EM wave, th e electric and magnetic fields can be expressed
most generally as an integral over frequency:
J /*oo
' ■a { v ) S ^ ~ 2^ d v
o
(3.17)
(Mathematically, this can be thought of as an infinite sum over a finite frequency range.)
Then, the P F is given by
I (P) = (E (P, t) E* ( P , t )) =
( \EX\2 + I E / )
(3.18)
The rest of the Stokes’ Param eters can be defined exactly the same way. The 4 param eters are
I
=
{\EX\2) + {\Ey\2)
Q
=
(\EX\2) - (\Ey \2)
u
=
2fft(E*Ey)
V
=
2 1 ( K E y)
(3.19)
I 2 > Q2 + U2 + V 2
(3.20)
We find from equations 3.19 th a t
We can thus define the degree of polarization as
r = y / v + v, + v ,
(321)
Notice th a t there are 4 param eters needed to describe a quasi-monochromatic wave and we have
defined exactly 4 Stokes’param eters. These Stokes’ param eters can be measured by a variety
of instrum ents, and the 4param eters needed to characterize the wave can then be derived from
them , if needed.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
3 .2
S p in H a r m o n ic s
Equations (3.10) and (3.11) can now be w ritten using a compact notation. Using
e%e + e-io
cos 0 = ----- -----ei9 _ e ~ i0
sin 0 = -----—-----
(3.22)
we get
( Q ' ± i U ' ) = e :*i20( Q ± i U )
(3.23)
which is shorthand for: under a rotation by an angle 0, this is how the quantity (Q ± iU)
transforms.
However, this is the d e fin itio n of a spin-2 system! This implies, among other things,
th a t Q and U cannot be described by spherical harmonics, because they are not invariant under
rotation.
It turns out th a t there exists a class of functions th a t describe quantities with non­
zero spin -
these are called spin-weighted harmonics or spin-harmonics and they are related to
spherical harmonics. Weshall discuss them in brief here. For a more detailed
and complete
treatm ent, see[3].
The basic idea is this - there exist “spin-s” harmonic functions, sYim ( 0 , </>), which form a
complete, orthonormal basis on the sphere V|s| < I:
J dnsY?m (0 , 4>) sYlm {0 , 4>) = 6u>6mm>
E E ( ^ ^ ^ ^ ( 0,^ ,)) = <5( ^ - ^ ) M c o s 0 - cos0O
l m
(3-24)
For these spin-harmonic functions s]r)m (0, </>), there exist “spin-raising” and “spin-lowering”
operators, denoted here by jj and \>respectively, which, as the names suggest,“raise” or “lower”
the spin of a system. For instance, let a function f s = f s (0,<f>) have spin s and therefore
transform under a rotation ip as
■fs = e ^ f s
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(3.25)
31
Then,
(# /.)' =
(#/,)
(3.26)
(W = e'
(!»/«
(3.27)
and
Explicitly, the spin raising and lowering functions are [2, 4]:
- s i n 8 6 do
=
b
—sin
sin
sinf
de - ^ - „ B 4 sin
sin 0
(3.28)
(3.29)
These two operators can be used to raise (lower) the spin of the functions ~sYim ($> 4>) (s^im (&>4>))
to exactly zero. In other words, these spin-weighted functions (spin-harmonics) can then be
expressed as
sYlm
—
gYlm =
’G - * ) ! - ’
f Y lm
(l + s)l
(i + s)i
( - i ) 8 b“ 8y ir,
(3.30)
(3.31)
x i- m
These are spin-s harmonics. Spin— s harmonics can be expressed in a similar way:
—sYlm
—sYlm
-(Z-s)f 1
L(I + s)!j
(/ + *)!
L(/-S)!J
( - i ) s b8y ^
SYlr,
(3.32)
(3.33)
We end by stating some useful properties of spin-harmonics th at will come in handy later:
ttsYlm
— [(I — s )(l + S + l) ] |+ i Yim
(3.34)
bsYlm
=
(3.35)
- [ ( l + s ) ( l - s + l)]]_1Ylm
We are now ready to apply this formalism to polarization param eters over the sky.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
32
3 .3
A p p lic a tio n o f S p in -h a r m o n ic s t o P o la r iz a tio n
Let a position on the sky be defined by the co-ordinates (9, <f>). Let the unit vector along
the line-of-sight be n. The unit vectors on the tangent plane at any point (6, <i>) are given by
(e 0 ,e^). From equations (3.23)
(Q' ± iU ') = e ^ m {Q ± iU )
(3.36)
We can now expand Q ± i U in spin - 2 spherical harmonics:
(Q + iU) (n)
Y2 p-2,im 2 Yim (n)
=
(3.37)
lm
H J) (n)
(Q
=
^ ] a —2,lm —v Y im (h)
(3.38)
lm
Tem perature is characterized by spherical harmonics, which are spin-0, i.e. invariant under
rotation:
T ( h ) = ^ 2 almYlm (h)
(3.39)
lm
Since we wish to work with spin-0 quantities, we first lower the spin of Q + iU thus:
b2 (Q + iU)
^ ' 0*2,1
2Ylm
lm
E b
(l + s)\’
(—l ) 2 b 2Yim ) fromeq(3.31)
lm
E 1(1-s)'.
(3.40)
a2lmYln
lm
Similarly,
(Q —iU)
— ^ 2 a _ 2 ,imbb _ 2 Yin
lm
=
E i
( i + *)'■'
Ylm ) from eq(3.30)
lm
E
a —2 lm Y lr
lm
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(3.41)
33
Now, since our aim is to work w ith spin-0 quantities constructed from (Q ± iU), we can
in principle work w ith b2 (Q + iU) and (I2 (Q — iJJ). However, this is n ot a convenient choice for
the following reason. Q has parity even and U has parity odd, i.e. under a rotation n —> —n,
we get Q —* Q, U —» —U .
We would, therefore, like to work w ith two spin-0 quantities w ith well-defined parities,
i.e. one with parity even and the other with parity odd. However, the two quantities Q ± i U do
not have this property, and so we cannot expect the parities of b2 (Q -1- iU )and
work
jj2 (Q — iU) to
out to be even/odd. We need to construct two other quantities, say E and B from these
two thus:
E
=
a\>2 (Q + iU) + b f (Q - iU) (even)
(3.42)
B
=
cb2 (Q + iU) + d f (Q - iU) (odd)
(3.43)
where we need to determine the 4 quantities a, b, c, d.
We can write
E
=
(ob2 + b f ) Q + i (ab2 - b f ) U
(3.44)
B
=
(cb2 + d f ) Q + i (cb2 - d f ) U
(3.45)
Thus, we need even parities for ab2 + b'Z2 and cb2 — d f ’ as well as odd parities for ab2 — 6 }f2
and cb2 + d\j2. B ut under a parity transform ation b2 —> (—l / f 2 and Jp —> (—l / b 2. Thus, we
will have all the required parities as required iff a = b and c = —d. In particular, we choose
a = b=
and c = —d = 4- for reasons of normalization [2]. The expressions for E and B are
E =
-^[\> 2 (Q + iU) + f ( Q - i U ) ] ( e v e n )
(3.46)
B =
^ [b2 (Q + iU) ~ f ( Q ~ iU)] (odd)
(3.47)
These are the so-called “E and B-modes” in CMB polarization. The reason for the choice of the
letters E and B is primarily their respective parities: E-modes have parity even, like electric
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
fields, and B-modes have parity odd, like magnetic fields. This relationship between E and
B-modes and Stokes’ param eters is derived in a different way in Appendix E.
E and B-modes can now be expanded in term s of spherical harmonics:
E
'y ''j Q 'E lm Y lm
lm
( 3 .4 8 )
13
^ ^ Q'BlmYlm
lm
( 3 .4 9 )
where
(1 T 2 )! Cl2lm
0-E lm
((1I —
- 2 )!
(1
O-Blm
Q—2lm
2
2 )! Ct2/m
(1- 2)!
Q—2lm
2
( 3 .5 0 )
( 3 .5 1 )
We can now define the power spectra th a t provide a statistical description of CMB tem ­
perature and polarization anisotropies:
ka X t m a X l m )
( 3 .5 2 )
m
where X = T , E , B and (• • •) = ensemble average.
Here is another reason for working w ith E and B-modes instead of b2 (Q + iU ) and
U2 (Q — iU ): since E-modes are parity even and B-modes parity odd, the cross-correlations
B E , B T vanish. This means th a t we have to deal with fewer power spectra. Had we chosen
b2 (Q + iU) and f ( Q - i U ) , we would have had to analyze atleast two more power spectra,
w ithout gaining any additional physical insight.
3 .4
T h o m s o n S c a tte r in g
Scattering of a photon from an electron, when there is no change in photon energy, is
called Thomson scattering. Since electrons (and protons) are free before last scattering, this is
the dominant process th a t causes “communication” between photons and m atter.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
35
Thomson scattering cannot “produce” polarization if the incident radiation is completely
uniform. However, if there are anisotropies in the incident radiation (in particular, quadrupole
anisotropy, as we will later see) then the scattered radiation c a n have polarization. This is
the case w ith the CMB. In particular, a (temporally) thin slice of the last scattering surface
(LSS henceforth) causes polarization anisotropies to appear because of Thomson scattering
of radiation th a t has a quadrupole moment. Both tem perature and polarization anisotropies
depend on evolution before the LSS, albeit differently - Thomson scattering causes polarization
right before recom bination/LSS, b u t it also destroys polarization information before the LSS
([5] chapter 4).
To delve into the details of how Thomson scattering leads to E and B-modes, consider an
electron at the origin close to the LSS (or ju st before recombination). An incoming plane wave,
which consists of oscillating electric and magnetic fields will accelerate the electron which then
radiates EM waves. This can be viewed as scattering of radiation by an electron, and we will
refer to it as such.
Let us define co-ordinate systems first. Let x' — y' refer to the co-ordinate system of
the incoming (incident) radiation, which has wavevector k ,. Let x — y refer to the co-ordinate
system of the scattered radiation, which has wavevector k s. Scattering is represented in the
figure in [2 ],
If the electric field vector of the incoming linearly polarized wave is in the k, —k s plane
(we call this the “scattering plane” ) , the differential cross-section of Thomson scattering is [6 ]
da
dU
3 <7y
POL
- 2
k, • k.
(3.53)
%7T
where a r is the Thom son cross-section. If the elctric field is perpendicular to the scattering
plane,
da
d£l
POL
=
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
<3-54)
36
where the solid angle dfl = d(cos 6)d(f> is defined in the usual spherical coordinates.
Now, consider unpolarized radiation, which is = many linearly polarized waves a t all
angles to each other. We can thus regard an incoming E-field as consisting of one E-field
polarized parallel to the scattering (i.e. k,; —k s) plane and the other polarized perpendicular
to it. The net differential cross-section is ju st the sum of the two cross-sections:
da
dQ
3 ax
1 + k,; • k.
(3.55)
1®7r
UN POL
Thus, for right-angle scattering (i.e. 6 = | ) , scattered radiation is completely linearly polarized
perpendicular to the scattering plane. Eqs. (3.53) and (3.54) tell us w hat happens to 7 i and
7|| respectively. Expressions for these two, i.e. I± and 7|| will immediately give us two Stokes’
param eters - 7 = 7 i + 7|| and Q — 7|| — 7x (where the definition of Q is arbitrary up to an
overall -ve sign). The other two scatter as follows
U
=
U'7 ( k s • k j)
(3.56)
V
=
V'' ( k s • k i)
(3.57)
B ut the CMB has V = 0, and our choice of co-ordinate systems and geometry for this one
particular angle ensure th a t U = 0. Thus, the four Stokes’ param eters are
7(2)
=
^ ( 1 + c o sV )^
16-tt
(3-58)
Q ( z)
=
^ g s i n 2 e'I'e, ^
(3.59)
U (z)
=
0
(3.60)
V( z )
=
0
(3.61)
However, this geometry is defined only for <f>' — 0. For any general angle <j>' ^ 0, we will have
Q (z, <j>'} =
Q (z) cos 2(f) + U (z) sin 2<f> = Q (z) cos 2<f>'
(3.62)
U (z, cf>') =
Q (z) sin 2(f> + U (z) cos 2(f) = Q (z) sin 2(f)
(3.63)
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
37
We can now integrate over the solid angle to get
J(z)
=
3<
tt
j d,Q' (1 + cos2 O') I()l (y
167T
(3.64)
Q( z)
=
J (]Q' sin2 Q' cos 2(f) I q, ^,
(3.65)
U ( z)
=
J
(3.66)
3(7J1
167r
dQ! sin2 6' sin 2
We can now expand the incoming intensity by spherical harmonics
(3-67)
lm
and remembering th a t sin2 0 = 1~c°s26> and cos2 0 = b iiv*20, and th a t / dSl cos riOsin q4>Yim
picks ou t anq, we get th at
Q ± iU oc 0 ^
2
(3.68)
This is the result we had quoted earlier: polarization anisotropies in the CMB are caused on ly
because of the quadrupole moment in the radiation ju st before the LSS.
3 .5
C M B P o la r iz a tio n a n d C o s m o lo g y
We have shown in the preceding sections th at
1. Scattering produces polarization - both Q and U modes
2. Both E and B modes are thus produced in polarization due to scattering
While (2) is true in general, Hu and W hite [8] have shown th a t there are only on ly two
ways to produce B-modes: by having either tensor or vector perturbations before recombina­
tion (also, see fig.(3.2)). Both vector and tensor modes decay after recombination, b u t vector
modes decay faster such th a t none survive to the present time. Thus, ten so r m o d es are th e
o n ly reason for B -m o d es to sh ow up and a m easurem en t o f B -m o d es in d ica tes th e
p resen ce o f ten so r m od es in th e early universe. These tensor modes are equivalent to (or
lead to) gravitational waves, which could only have been produced during inflation, according
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
38
Polarized CMB and Foreground Spectra
Angular Scale
10 -
ma
1
o.2 !
1.00
5T
■J3;
,
B
3.10
J '" ” ’'
|
"
f '" t 'f
0.01 hr
/
£ K>**rer«aojf'i5
t
10
100
Multipole moment, /
Figure 3.1: B-mode level compared with the levels of E-modes, foregrounds and the lensing
contribution to B-modes[7]
to our present understanding. Schematically, the relation between spin-2 spherical harmonic
coefficients for B-modes and the energy scale of inflation quantified by the inflaton potential
(since it is the potential th a t drives inflation - see chapter 6 in [5]) is given by
± 2 a<m
*
J j( .( r ) r 2drk2dkV,j,T^aG (a)
(3.69)
where
je
= Bessel function of order I
r
= Comoving distance
k
= Wavenumber of a mode
T^ a
= Transfer function : quantifies the change at a mode transition
G (a)
= Growth function : describes behaviour of mode at late times
V®
= The Inflaton potential
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(3.70)
39
(a) Polarization Pattern
(b) Multipole Power
Figure 3.2: Scalar and Tensor modes w ith corresponding E and B components.
s
<au
\
o
1500
Al
<9
530
8
800
1209
1406
Figure 3.3: The contribution of tensor modes to the tem perature power spectrum (in green).
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
40
Angular S c a le
05'
0 ,2 “
TT Cross Power
Spectrum
-—
A - COM AS Cara
}
WMAP
*
ACBAR
| €»
4000
3000
1000
0
10
40
100
400
MiJBpote moment (ft
Figure 3.4: W M AP 1st year power spectrum , showing cosmic variance at low is. Notice th at
the cosmic variance shown here is significantly larger th an the tensor mode contribution in
fig. (3.3)
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
41
The actual relation is more involved and is given by, e.g. eq.(70) in [2]. The param eter r in
fig.(3.1) is the ratio of the average power in tensor modes and the average power in the scalar
modes of perturbation in the early universe before recombination. C urrent estimates of the
highest value of r are ~0.3. This corresponds to U/>
1015GeV (i.e. the GUT scale), well out
of reach of the capabilities of current particle accelerators by more th an a decade in order-ofmagnitude! T h is is t h e re a s o n w e n e e d m o re s e n sitiv e cosm o lo g ical p r o b e s o f th e
e a rly u n iv e rse .
Tensor modes in th e early universe contribute to tem perature and polarization power
spectra. However, they decay away exponentially w ith time, and the smaller the scale (i.e. the
higher the value of f), the faster they decay away[5]. Thus, they have a small effect on the low-f
p a rt of the tem perature power spectrum as shown in fig.(3.3). However, this is the p art of the
power spectrum dom inated by cosmic variance (the fact th a t we have only one sky to look at
implies th a t the sampling error is high at low fs) as shown in fig.(3.4), which is large enough
th a t the effect of the tensor modes cannot possibly be distinguished from th a t of scalar modes.
Thus, B -m o d e s a r e t h e m o s t d ire c t in d ic a to r s o f co sm o lo g ical in fla tio n . The
expected level of B-mode signal is shown in fig.(3.1). However, all th a t is stated about the
connection between B-modes and cosmological inflation above holds true when there are no
foregrounds. There are two ways foregrounds can produce a spurious B-mode signal:
1. Emission: All processes th at produce polarization, e.g. synchrotron can produce polarized
foregrounds in the presence of inhomogeneous magnetic fields.
2. Conversion: Gravitational lensing of the CMB by galaxies and galaxy clusters produces
distortions because lensing depends on the 2-D surface density, which is necessarily nonuniform for clusters. This produces a “torsion” effect ([5] chapter 11) which converts a
portion of E-modes to B-modes. Since B-modes are an order of m agnitude smaller, even
a small percentage of conversion leads to a large spurious B-mode effect.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
42
These systematics will challenge the next generation of CMB polarization experiments.
In the next two chapters, we discuss results from recent experiments and the reason we
prefer interferometry over imaging.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
43
B ibliography
[1] K. Rohlfs and T. L. Wilson, Tools of Radio Astronomy, Tools of Radio Astronomy, XVI,
423 pp. 127 figs., 20 tabs.. Springer-Verlag Berlin Heidelberg New York. Also Astronomy
and Astrophysics Library, 1996.
[2] Y.-T. Lin and B. D. W andelt, “A beginner's guide to the theory of CMB tem perature and
polarization power spectra in the line-of-sight formalism,” Astroparticle Physics, vol. 25,
pp. 151-166, Mar. 2006.
[3] M. Zaldarriaga, Fluctuations in the cosmic microwave background, Ph.D . thesis, MAS­
SACHUSETTS IN STITU TE OF TECHNOLOGY, 1998.
[4] N. Goldberg,
J. Math. Phys., vol. 8, pp. 2155+, 1966.
[5] S. Dodelson, M odem cosmology, M odern cosmology / Scott Dodelson. Am sterdam (Nether­
lands): Academic Press. ISBN 0-12-219141-2, 2003, X III + 440 p., 2003.
[6] G. B. Rybicki and A. P. Lightman, Radiative processes in astrophysics, New York, WileyInterscience, 1979. 393 p., 1979.
[7] J. Bock, S. Church, M. Devlin, G. Hinshaw, A. Lange, A. Lee, L. Page, B. Partridge,
J. Ruhl, M. Tegmark, P. Timbie, R. Weiss, B. W instein, and M. Zaldarriaga, “Task Force
on Cosmic Microwave Background Research,” ArXiv Astrophysics e-prints, Apr. 2006.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
44
[8] W. Hu and M. W hite, “A CMB polarization prim er,” New Astronomy, vol. 2, pp. 323-344,
Oct. 1997.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
45
Chapter 4
Current statu s o f CM B observations
Detection of CMB anisotropy has always been a challenge because of its low am plitude ~ 10/iK
out of a background of 2.7K. In fact, it took over two decades to discover anisotropies in the
CMB [1] from the tim e the CMB tem perature was first measured in 1965 by Penzias and Wilson.
T he reason is th a t CMB anisotropies are smaller than the CMB by a factor of ~ 10s , i.e. at the
level of ~10/iK. COBE (the COsmic Background Explorer) was the first experiment to measure
anisotropy in the CMB[1], It was also the first experiment th at proved conclusively th a t the
spectrum of the CMB is Planckian. Since COBE, a lot of CMB experiments (e.g. W M AP)
have constrained the CMB tem perature power spectrum to exquisite precision. We discuss the
two most successful of these post-COBE experiments - W M AP and DASI.
4.1
Detectors
Detectors used in CMB cosmology can be divided into two broad categories:
1. C oherent R eceivers - These detect the am plitude and phase of the incoming signal.
This is why they are used in interferometric CMB probes.
Amplifiers th a t use High
Electron Mobility Transistors (HEMTs) have been the coherent receivers of choice for
CMB experiments. However, their sensitivity is low above 100 GHz.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
46
Incoming
radiation
Thermometer
W eak Link
Figure 4.1: A schematic of a bolometer, showing how it works.
Feedhorn or Planar
Antenna \
Waveguide or
Microstrip
Figure 4.2: A schematic of how a bolometer is used.
2. Incoh eren t d e te c to r s - These are total power detectors and are unable to detect phase.
Bolometers are an example of incoherent detectors. These consist of an absorber, a ther­
mometer, a cold reservoir and a therm al link from the absorber to the reservoir. The
radiation incident on the absorber warms it up and changes its tem perature, which is
measured by the therm om eter. This heat is then drained into the cold reservoir and the
cycle is repeated. Bolometers can work at any tem perature; however, they are most sensi­
tive when cryo-cooled. Since bolometers cannot detect any phase or spectral information,
the instrum ent th a t they are p art of has to incorporate some m ethod th a t enables phase
detection. A novel technique th a t discusses one such arrangement is discussed in chap­
ter 6. Fig.(4.1) is a cartoon of a bolometer and fig.(4.2) shows how a typical bolometer
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
47
operates.
4.2
The W ilkinson Microwave Anisotropy Probe
Named in honor of its pioneer, Prof. David T . Wilkinson, the Wilkinson Microwave
Anisotropy Probe (WMAP) is a satellite th a t orbits the sun at the second Lagrange point.
W M AP uses differential radiom eters, meaning th a t it differences the input from two horns
th a t point 140° away from each other. It takes six m onths to image the entire sky. W M A P’s
radiometers use a series of Orthom ode Transducers, Hybrid T ’s, HEM T amplifiers and phase
shifters. A pair of horns each has its polarization components separated, processed, amplified
and the signals recorded by a pair of detectors th a t are a combination of a single polarization
from both of the horns. Differencing the two detector signals then produces a result th a t is
proportional to the difference in polarization between the two horns. From these measurements
the W M AP team then reconstructs the am plitude and orientation of the CM B’s polarized signal
a t each point on the sky [2]. W M AP was optimized for CMB tem perature measurements, which
it has done with unprecedented precision. A table of cosmological param eters constrained by
W MAP is given below.
Parameter ACDM + Tensor ACDM + Running +Tensors
flj/r2
sv * 2
h
ns
dn,/dhk
r
7
0.0233 ±0.0010
0.1195™
0.787 ±0.052
0-984! “ Jj
set to 0
< 0.65 (95% CL)
0.090 ±0.031
0.702 ±0.062
0.0219 ±0.0012
0.128 ±0.011
0.731 ±0.055
1.16 ±0.10
-0.085 ±0.043
< 1.1 (95% CL)
0.108!®;“
0.712 ±0.056
Figure 4.3: W M AP param eters.
While W M AP is a phenomenal success in term s of the tem perature power spectrum
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
48
it has recovered, its sensitivity is not enough to enable it to detect B-modes.
The upper
limit it has placed on B-modes is shown in fig.(2.3). Clearly, we need more sensitivity by a
factor of about 10 to get down to the expected B-mode levels shown. For th is reason , a
com b in ation o f in terferom etry and b o lo m etry is preferred over th e tech n iq u es used
b y th e in stru m en ts m en tion ed in th is chapter. W e b eliev e th a t th is com b in ation
w ill provide us w ith th e s e n sitiv ity required to d e te c t th e w eak B -m o d e sign al.
4 .3
T h e D e g r e e A n g u la r S c a le I n te r fe r o m e te r
The Degree Angular Scale Interferometer (DASI) is a 13 element co-planar interferom eter
array. It operates with HEMTs in the 26-36 GHz range w ith the frequencies broken into ten,
one GHz wide bands [3]. DASI uses right and left circular polarizers to separate polarizations as
opposed to linear polarizers in WMAP. This turns out to be desirable for control of system atic
effects. DASI focused on 140 < I < 900. DASI found a 6.3 a significant detection of EE power
spectrum and a 2.9 a significant detection of the TE cross correlation power spectrum[4]. D ata
from DASI enabled the detection of the second peak in the tem perature power spectrum , but
shows no evidence of B-modes.
Most CMB experiments have used imaging techniques to estimate the power spectrum of
the CMB. We discuss power spectrum estimation from CMB imaging in some detail in chapter
9, but the essential steps are as follows:
1. Image the CMB using a certain scan-strategy and beam
2. Use a computationally optim al technique to extract the signal from the equation
d = A ■s + n
(4-1)
where d is imaging data, A is a m atrix th at describes the beam and the scan strategy and
is called the “pointing m atrix” , s the signal and n is the noise. This is where the image
is “pixelized” . Care has to be taken not to pixelize the data beyond the beam resolution.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
49
3. Define likelihood. Using an optim al com putational technique, estim ate the values of Ces
th a t maximize the likelihood.
The trouble w ith imaging lies in points 2 and 3 above. In 2, we could decide to pixelize too
coarsely. This would certainly increase the signal-to-noise ratio, leading to lower errors in the
power spectrum , but the Cg estim ates cannot be made for high fs. W hat is needed for future
CMB experiments is a system th a t can sample the power spectrum more directly and with
b etter control of systematics. The connection between an image and the power spectrum is
indorect; the power spectrum is the fourier transform of the two-point correlation function in
image space.
However, we show in chapter 5 th a t the power spectrum can also be expressed as a twopoint correlation function of the visibility (the output from one baseline of an interferometer).
This means th a t the interferom eter samples f-space directly - in fact, it turns out th a t every
unique baseline length corresponds to a unique f-band where the width of the band depends
on the bandw idth of the instrum ent. Thus, there is n e v e r any confusion about the location of
the values of t where the power spectrum is sampled - these are fixed in interferometry.
These and other characteristics of interferom etry make its use preferable for CMB cos­
mology. The advantages of interferom etry are discussed in detail in chapter 5. Additionally, in
chapter 6, we explore a new technique to utilize the information in a particular kind of interferometric system to enhance resolution in Cspace, leading to better estimates of Cgs, as well
as better images.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
50
Bibliography
[1] C. L. Bennett, A. J. Banday, K. M. Gorski, G. Hinshaw, P. Jackson, P. Keegstra, A. Kogut,
G. F. Smoot, D. T. Wilkinson, and E. L. W right, “Four-Year COBE DM R Cosmic Mi­
crowave Background Observations: Maps and Basic Results,” A p J Lett., vol. 464, pp. L1+,
June 1996.
[2] G. Hinshaw, M. R. Nolta, C. L. B ennett, R. Bean, O. Dore, M. R. Greason, M. Halpern, R. S.
Hill, N. Jarosik, A. Kogut, E. Kom atsu, M. Limon, N. Odegard, S. S. Meyer, L. Page, H. V.
Peiris, D. N. Spergel, G. S. Tucker, L. Verde, J. L. Weiland, E. Wollack, and E. L. Wright,
“Three-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Tem perature
Analysis,” A p J Suppl., vol. 170, pp. 288-334, June 2007.
[3] N. W. Halverson, J. E. Carlstrom, M. Dragovan, W. L. Holzapfel, and J. Kovac, “DASI:
Degree Angular Scale Interferometer for imaging anisotropy in the cosmic microwave back­
ground,” in Proc. SPIE Vol. 3351, p. Jt 16~J,23, Advanced Technology M M W , Radio, and
Terahertz Telescopes, Thomas G. Phillips; Ed., T. G. Phillips, Ed., July 1998, vol. 3357 of
Presented at the Society of Photo-Optical Instrumentation Engineers (SPIE) Conference,
pp. 416-423.
[4] E. M. Leitch, J. M. Kovac, N. W. Halverson, J. E. Carlstrom , C. Pryke, and M. W. E. Smith,
“Degree Angular Scale Interferometer 3 Year Cosmic Microwave Background Polarization
Results,” ApJ, vol. 624, pp. 10-20, May 2005.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
51
C hapter 5
Interferom etry
5 .1
O v e r v ie w
The observing wavelength of the Millimeter-wave Bolometric Interferometer (M BI)(~3m m ,
W -band) places it in the category of a radio telescope. However, MBI is not an imaging tele­
scope, but an interferometer. Even though it can be used as an imaging instrum ent (as shown
in the following chapter), we will discuss it here only as an interferometer.
Classically, interferometers were preferred over dish antennae for the following reason.
The angular resolution of a dish is given by 6 ~ jj. However, radio-waves are long-wavelength
and so for radio astronomy, we require huge single dishes for any reasonable angular resolution.
Interferometers are fundamentally different in th a t they produce diffraction patterns of the fieldof-view (‘FO V’ henceforth), and not images. Interferometers can achieve high angular resolution
by combining signals from widely separated small dishes. To a good first approxim ation, we can
tre a t them the way we treat diffraction slits. It will be shown later th a t there are fundam ental
differences between a simple 1-slit diffraction of a point source and the diffraction of an extended
source through an interferometer.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
52
5 .2
T h e M u t u a l C o h e r e n c e F u n c tio n
While interferom etry has many advantages (as described in §5.8), we need to be able to
relate the output of an interferom eter to the image on the sky. It turns out th a t this connection
can be m ade via the study of coherence properties of the source. As will become clear later in
this chapter and the next, an interferometer makes use of both intensity and phase information,
so th a t this is not surprising. The following discussion on can be found in greater detail in several
texts, e.g. [1].
The sim p lest wave-held th a t can be imagined is the plane monochromatic wave. For
this wave, if we know the field a t a point A, we can find the field at any other point B - all
we need is the phase-difference between A and B. This is a c o m p letely coheren t wave-held.
The other extreme is th a t of a random polychromatic wave - for this wave, the held at any two
points is completely uncorrelated. In general, though, all real wave-helds lie between these two
extremes, i.e. they are p a rtia lly coherent.
For a general wave, therefore, we require some measure of coherence. This measure m ust
be a time average, and we will want to compare the held at two different points, say P\ and P 2 .
Let the (Electric) helds at the two points be E ( p , t,\) and E (P2, h)- The M utual Coherence
Function is dehned as
r (P1, P2, r) = Limr-^oo ^ J ^ E ( ^ , t ) E* (P2, f + r) = (E ( P , t) E* (P2,t + r ))
(5.1)
where we have used () to indicate a time-average and recognized the fact th a t the difference in
the helds at the two different points due to a single point source is ju st a time-delay. Note th at
the intensity is a special case of this definition:
I (P) = T (P , P, 0) = (E (P , t) E* (P, f))
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(5.2)
53
5 .3
T h e C o h e r e n c e F u n c tio n o f E x t e n d e d S o u r c e s
Single point-sources have limited use in astrophysics - we need to extend the definition of
the M utual Coherence Function to extended sources, especially if we want to study the CMB,
a diffuse source over the whole sky. We can do this - we ju st need to remember th a t any two
points on an extended source, which is necessarily very distant, are completely independent. In
short, the source is spatially incoherent.
Consider two waves originating at two different points on an extended source and therefore
w ith two different wavevectors k a and kb, incident on the observing plane. The resulting field
at any point in the observing plane is given by E = E a + E b. The M utual Coherence function
for two points on the observing plane then is
T ( P 1, P 2, t )
=
( E ( P l , h ) E * ( P 2, t 2))
=
([Ea (P l5P ) + E b (Pa, p )] [E*a (P2, t2) + E*h (P2,f 2)])
=
(Ea (P i,h)E*a (P2, 12)> + (E b ( P i, h ) E l (P 2, f2))
+
(Ea (P i, p ) E l (P2, t 2)) + (Eb (P i, p ) E l (P2, t 2))
^
s
—0
(5-3)
The last two term s are zero because of the assured spatial incoherence of the source. Thus, the
M utual Coherence Function for two points a and b on the source becomes
r ( P i, P2, r )
= (Ea ( P i, 11 ) E*a (P2, t2)> + (Eb ( P i,fi) E l (P 2, f2))
(5.4)
We want to define the M .C.F. for the whole source; b u t now we can imagine the extended
source being made of a huge num ber of point sources, and sum over all of them thus:
N
r ( P 1, P 2, t ) = Y ,
E i ( P i, t) E * (P2, t + t )
(5.5)
i= l
It is more practical to define it as an integral over the FOV:
F ( P i ,P 2, t ) =
J E (Pi, t) E* (P2,t + T)dfl
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(5.6)
In other words, for an interferometer, we need to sample the field from a distant source at two
different points on the observation plane (which we can do with antennae) and then multiply
these together. This can be achieved by the following simple setup:
Detector
Figure 5.1: A general interferometric setup
In the combiner marked C, the two electric fields E x and E? are ju st added. T he detector then
squares this sum to get
{El + Eh) (E l + E*2) = |Eh |2 + \Eh|2 + E xE*2 + E 1E 2
^
-> v
"v" 1 >
Total Power
Visibility
(5.7)
where only the last two term s indicate interference. This is the basic idea in interferometry,
and we will keep using this schematic to treat interferometry in the following chapters.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
55
Also, we have not yet assigned a name to the last two term s on the RHS in eq(5.7); it is
called the “Visibility” . W hat follows is a m athem atical derivation of the Visibility expressed
as a functional transform of the intensity p attern on the sky.
5 .4
V is ib ilit y as a f u n c tio n o n I n t e n s it y p a t te r n o n t h e sk y
Consider two horn antennas / radio telescopes separated by a distance B . These define
one baseline. Let them be oriented as shown to receive a signal from an extended object in the
sky.
Then, as shown in figure 2, consider a single point on the extended object or source, P.
Let the distance from P to each of the telescopes be d\ and (p.
The reason we consider a single point on the source is th a t rays originating in different
parts of the source are not m utually coherent, i.e. their relative phases are random . So it doesnt
make sense, for instance, to calculate the net electric field at one telescope due to rays from the
object as a whole. We do need to make an image of the whole object, however, and for this
reason we scan across it with our baseline. Mathematically, this is equivalent to calculating the
net field and then integrating over the source.
Let (x , y ) be the co-ordinate system on the source. Let I (x,y ) be the intensity as a
function of position on the source, and let E (x, y) be the electric field due to the source on
both the antennas. D is the distance between the telescopes and the source.
There is a tim e delay of (d2~dl) between the signals received by the two telescopes (see
fig.(5.2)). T he electric fields at the two telescopes can be w ritten as:
E 1 = E ( x , y ) e iuJ(t- ^ )
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(5.8)
56
Figure 5.2: One baseline
E 2 = E (x, y) e
(5.9)
Consider now the product of these two electric fields. In brief, it is only by multiplying
the two signals th a t we can get interference, as discussed in §5.3.
E l E l ^ E 2 { x , y ) e iuit
^
* ^ +) = /
(x,y) e
(5.10)
Looking at figure 3, the two distances d\ and d2 can be w ritten as
d\ = ( x - "-B ) + y 2 + D 2 = D 2
i\= ( x + J s )
+ y2 + D 2 = D 2
1 + J L +
^
+ D2 + I
1 +
- —
D2
2
IB
D
I-
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(5.11)
(5.12)
57
Clearly,
1; in other words, the distance between us and the source is much greater
th an the length of the baseline. We assume th a t f),
<S 1, or th a t the size of the source is
much smaller than the distance between us and the source.
Then,
di ~ D
1 y2 ' 1 ( x —
1+ 2 D2 + 2
D
(5.13)
d,2 — D
1 m l i 'L + I ( X + \ B '
2 D2 2 I
D
(5.14)
so th a t
d2 - d l = D . \ . ^ . 2 . 2 . \ . B x = ^
(5.15)
=£■ E 1 E 2 ~ I {x,y) el c n — I (x,y) el27r
(5.16)
All this is fine, b ut we would like to work in term s of angles, so we change variables to a =
P=
so th at
ExE^ ~ I (a, (3) ei2w%a
(5.17)
However, the basis (a, /?) is relative to the source, and not the observation plane or the sky. We
therefore slip the formalism into something more comfortable and convenient - the equitorial
co-ordinates, thus:
a = cos Ox + sin Oy
(5.18)
P — sin Ox + cos Oy'
(5.19)
?*
z>»'27r#-(cos0a:,-t-sin<V)
EiE*z
~ IT (IV
x',y ') e1
(5.20)
u = -— cos 9
(5.21)
Then,
W rite
A
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
58
v = — sm&
(5.22)
A
These are w hat are called the u, v co-ordinates. Now we integrate over x' and y' to get:
JJ
E xE ld x 'd y ' = K
J J I (x'./y1)
dx'dy'
(5.23)
B ut the right side of the equation is ju st the fourier transform of I (x ' , y '). The left hand side
is w hat we call visibility.
In this discussion, we ignored the effect of the diffraction patterns of the telescopes /
antennas themselves. We can introduce it in equations 1 and 2 above:
E x = y / A { x , y ) E (x, y) eK t - ^ )
-® 2
= y / A ( x , y ) E (x, y)
c)
(5.24)
(5.25)
and then follow it through, to get:
J J E xE ld x 'd y ' = K j J A (x ',y ') I (x', y ) e ^ ^ ’+ ^ d x ' d y 1
(5.26)
We do not n e e d to make the small-sky patch approximation to get a useful result, though.
It ju st so happens th a t in this approxim ation, the o utput of the interferometer, the “Visibility” ,
or as we defined it earlier, the “M utual Coherence Function” happens to be the fourier transform
of the intensity pattern on the sky. If the approxim ation is relaxed, th e visibility becomes a
general m athem atical transform of the intensity pattern, and not necessarily a fourier transform .
The m a in idea is th a t one baseline, i.e. one p a ir of antennas gives us a sin g le p o in t in
the fourier transform of the intensity pattern on the sky, convolved, of course, w ith the beam.
To get more distinct points, we need more baselines, each with a different length or orientation.
The most general form of eq(5.26) is then
JJ
E xE ld x 'd y ' = K
j J
A ( x \ y') I (xr, y ) e ^ ^ d x ' d y '
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(5.27)
59
(x' and y' are really 0 and <fi on the sky) where u is the vector tiu + v v and the unit vectors
u and v span w hat is called the “u-v” plane, which is the fourier-transform equivalent of the
0 —<f>plane on the sky. W hat this means is th a t visibility, which is a function of u and v, i.e.
V = f (u,v) is the fourier transform of the image (or intensity pattern) on the sky (for small
patches) i.e.
V ( u ,v )= ^ ( I (0 ,< /> ))
(5.28)
which is exactly w hat eq(5.27) says above.
Then, to find an expression for |u|, consider eqs(5.21) and (5.22):
|u| = ^
A
(5.29)
, th a t is, |u| ex the baseline length. For the same baseline, though, a different orientation will
give us the same |u| but different values of u and v. W hat this means is th a t if we were to track
a single patch on the sky and rotate the instrum ent w .r.t. the patch, we will be observing at
all those points in the uv plane th a t lie on a circle w ith the radius |u| = y .
Now, tem perature on the sky can be expanded out as
T (0> 4>) = X ] ^ 2
l m
(Q, 4>)
(5-30)
The power spectrum is defined as
Cf
(5.31)
Eqs.(5.30) and (5.31) are m eant for the full-sky case. However, the quantity th a t the interfer­
ometer measures, i.e. the visibility, is the flat-sky equivalent of a,frn. Therefore, in the flat-sky
case, the power spectrum is ju st the two-point correlation function of the visibility.
Recall th a t the power spectrum is the fourier transform of the two-point correlation
function [Chapter 1]. It can also be w ritten
as the two-point correlation of the fourier transform
of the intensity pattern on the sky (later in this chapter). B ut the visibility
is the F T of the sky
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
60
image! Therefore, with an interferometer, all we need to do is find the two-point correlations
between observations from different baselines!
Furtherm ore, we are looking for the two-point correlation function of the d e v ia tio n from
the mean tem perature. Now every baseline, which has already defined an angular scale on the
sky, gives us this correlation between several pairs of points separated at a certain angle. If we
find the variance of these values, th a t w ill be the power spectrum we are after.
T h e p o w e r s p e c tr u m is j u s t t h e v a ria n c e o f t h e v isib ility and visibility is the
output of the interferometer. Naturally, in real instrum ents one has to extract the visibility
from the detectors.
The foregoing discussion is summarized in the statem ent I n te r f e r o m e te r s d ire c tly
m e a s u re fo u rie r m o d e s o n t h e sky.
We describe a few characteristics of the u-v plane in the next section and discuss polarized
interferom etry in the following section.
5 .5
I n te r lu d e : A s m a ll d is c u s s io n o n in te r fe r o m e tr y
As was shown in §5.4, visibility (i.e.
transform of the image on the sky.
the output of an interferometer) is the fourier
It is useful then to compare w hat an imager and an
interferometer “see” - both in the image plane and the fourier plane (which we shall refer to as
the “u-v plane” henceforth). Fig.(5.4) shows this comparison. From §5.4, recall th a t a single
visibility from one baseline of an interferometer is o n e point in the u-v plane. The length of
the baseline determines the spatial frequency of the fringe, which is the same as a length in
the u-v plane1. T he exact form of this relationship is as follows. For a baseline B , the angular
resolution is 6 =
Then, the value of I th a t this corresponds to is I —
1Just as frequencies appear as lengths in the fourier plane
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
i.e. I ~ B,
61
Figure 5.3: Schematic of interferomentric observation - one baseline. The two antennas are at
G and D.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
62
long
very lo ng
baseline
baseline
small
baseline
single
dish
Figure 5.4: The u-v plane coverage of an imager and an interferometer. Figure courtesy Dr.
Carolina Calderon[2].
or, longer baselines correspond to higher t-modes or higher angular frequencies. By comparison,
an imager is an interferom eter with B = 0. This is illustrated in fig. (5.4).
It is also pertinent to mention th a t each baseline produces its own fringe pattern. The
fourier transform of a real quantity is necessarily complex (property of FTs) and therefore the
u-v plane image is always complex. This means th a t the fringes have a real and an imaginary
part. The precise combination depends on the phase of the fringe, which is determ ined by the
relative orientation of the baseline and the sky. Thus, rotating the instrum ent through 360°
w.r.t. the sky allows each baseline to cover a circular ring in the u-v plane and shifts the phase
of the corresponding fringe continuously through 360°. B ut what effect do these fringes have
on the image? Effectively, every baseline chooses a Fourier mode from the image on the sky. In
other words, each baseline modulates the image with a fringe p attern whose spatial frequency
depends on the length of the baseline and whose phase depends on instrum ent orientation.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
63
Figure 5.5: The u-v plane with several pixels. Pixels marked “1” and “2” have the same distance
from the origin, but differ only in their angular position (this corresponds to the phase of the
fringe). Pixels marked “3” and “4” differ in their distance from the origin and angular position.
Let us look at several different pixels in the u-v plane. In fig.(5.5), vectors marked “3” and
“4” clearly have different lengths. These different lengths imply different lengths of baselines
and hence different spatial frequencies of the fringe pattern on the focal plane. However, “1”
and “2” are of equal length, and differ only in their angular position. This angle in the u-v
plane corresponds to a phase in the image plane, fn other words, this angle represents the
phase of the fringe p a tte rn produced by a particular baseline. The only way th a t a baseline can
produce fringe patterns th a t differ in phase is by rotating it w.r.t. the image on the sky. Thus,
angle in the u-v plane is the same as the phase of the corresponding fringe or the orientation
of the instrum ent w .r.t. the sky.
Figs.(5.6) and (5.7) illustrate th a t the image plane and the u-v plane are “inverses” of
each other in a sense - the smaller dimension in the image plane (a pixel) becomes the larger
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
64
V
u
Figure 5.7: The same FOV and pixel as in
e
the previous figure. The size of the inter­
ferometer’s FOV determines its resolution
Figure 5.6: FOV and pixel in the image
in u-v space. Notice th a t the two objects
plane. In this figure and the one alongside,
have swapped their dimensions. If N pix­
red represents a pixel in image space and
els fit in the FOV in the image plane, then
green the FOV in image space.
the u-v plane is also divided into N pixels
whose size is inversely proportional to the
FOV.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
65
dimension in the u-v plane and vice-versa.
5 .6
V is ib ility , t h e p o w e r s p e c tr u m a n d t h e b e a m
In this section, we prove the claim in §5.4 th a t the power spectrum is the variance of
visibility. The effect of the beam has to be taken into account, and it turns out th a t a new
quantity, called the W in d o w F u n c tio n needs to be defined.
For an imager, the output signal is given by
Si = j
dnQ (n)
(n)
(5.32)
This is equivalent to the expression:
JJ
E ^ d x 'd y ' = K
j J A ( x ' , y ' ) l { x ' , y ' ) ^ 2'K{uxl+vy')dx'dy'
(5.33)
in from §5.1 earlier in this chapter; which
= ^V i= S i=
j
d a E rE% = K
J
dnAi (n) I (n) ei27run
(5.34)
R e m e m b e r, h o w e v e r, t h a t th is is O N E p a ir o f a n te n n a s , a n d so O N E b a s e lin e , u
is th e r e fo r e fixed; it s h o u ld b e la b e lle d ux So,
Si = Vi — K
j
dnAt (n) I (n) ei27ru‘ n
(5.35)
Now,
dB
AX’
I c m b (n, v ) ~ B (v, T0) + — ItoT o- ^ t
(5.36)
from Jaiseung Kim ’s thesis [3]. We will consider only the perturbed part, and therefore
Qr>
K = ^ \ t 0T o
b ut let us NOT write this down and use K instead.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(5.37)
66
Let us rewrite (rework) Cs,ij — (V iV ?^ now. Remember the full-sky decomposition of
tem perature anisotropies:
00
^
^
rp (n ) = ^ ' 'y ' O'lmXlm (h) = y ^
1=1 m = —l
(n )
(5.38)
Im
so then
(V iV * )
irirv l —
rr
I dn I dn'Ai (n) A* (n ') E E Ylm (n) Y?m, (n ') (alma*Vm,)
J
J
lm I’m1
(5 39)
B ut remember, cqm’s are like fourier coefficients, and th at Power Spectrum = square of fourier
coefficients. More precisely,
(almal'm') =
(5.40)
( v i Vf )
J dn [ dn'Aj (n) 4 ( * 0 E E CtYlm (n) Y{m (n ')
l m
K 2T 2
({U1)
B ut from properties of spherical harmonics,
9 / -1-1
Ylm (n) YCm (n ') = - ± - I \ (n • n ')
= Jdn
K 2T 2
j
dn'Ai (n) A* (n ') £
(5.42)
(n • n 0
(5.43)
Again, we rem ind ourselves th at Via Visibility, so th a t |V |2 should give us Q x another quantity.
(V , v ■ ;)
^2 (^4“ ') Cl j dnJ
K 2T 2
dn>Ai (n ) A J
(n0 p i (n• n ') ei27r(“i n“ ^
n')
(5.44)
For various reasons, we always prefer to plot ^ j p - C i instead of Cq. Let us therefore m anipulate
this equation to get those factors; i.e. multiply and divide by I (I + 1)
( V iV - )
K 2T 2
=
E ( y^ ) C'‘^ r !r i ) / <in/
i
-
<‘" ' M n ) A ' j ( n ' ) P l (n .n ') e ™ t< ‘‘- < ‘‘ " ,'>
--
(5.45)
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
67
+
(5.46)
= Ci instead of C'i - the reason is buried in the m ath in W hite et
where we have chosen
al [4], and this is the reason we end up w ith 2i(i+i) ~ an<^ we have defined
/ dn
J
dn'Ai (n) A* (n ') Px (n • n ')
(5.47)
as the W IN D O W F U N C T IO N . It really is ju st the fraction of Cx th a t the antenna ’’lets in”
at every 1. This expression is completely general, i.e. w ithout any approximations:
(v*vi)
W
^
i
= ^ ( 2! + i ) CiH,«-‘ 2 i ( i T I )
(5'48)
We could also have defined the Window function another way:
1
= 0w , ,
21 ( [ + 1)
(5.49)
where W^y is now the ‘n et’ Window function, and we have
(w A
= £ ( 2 t + l)C,W yy
(5.50)
Aside: this implies th a t the height of the ‘n e t’ window function decreases w ith increasing
I. Physically, w hat this means is th a t by increasing the baseline, the amount of light we let
into the telescope system decreases compared to the amount th a t would have been let in had
the telescope been a filled aperture w ith the length of baseline as the diam eter. Succinctly, the
‘filling-factor’ (the ratio of the total area of antennas in one baseline to the area of a dish with
the baseline as a diam eter) decreses w ith increasing I (equivalent to increasing the baseline).
The foregoing argum ent implies th a t W^y ~ l~2, which is indeed the case, as can be seen
in the above expression.
In the following, we will derive an expression for Wijj,, and will finally write down the
expression for
at the end of this section.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
68
We can now apply the FLAT-SKY APPROXIM ATION (small 0, large 1):
1. x and x ' are vectors in directions n n ' respectively. Approximating them to 2-D vectors
on the sky
=>•
Pi (n • n ') ~ Pi (cos|x —x '|)
(5.51)
J dn J dn' ~J d 2x J d2x'
(5.52)
and
=> W ijj
This is the
=j
J
d2x
d2x'Ai (x) A* (x') Pi (cos|x - x '|)
modified version of
( 5 .5 3 )
eq. 11.40 in Dodelson.
2.
-I PZ7T
Pt (cos|x - x 'l) -> Jo (I|x - x '|) = —
d(f>e-illx~x'lcos^>
2tt J o
(5 .5 4 )
where the last equality is the definition of the Bessel function of order I.
Now, l|x — x '| cos <j) can be written as 1 • (x —x'), because 4> is really ju st a param eter
we are integrating over. We are therefore free to provide our own physical interpretation of it.
T he one convenient for us is: <j} is an angle in 1-space, and is defined as c6 = ta n -1
and then
l= y flf+ q .
So then
=►W ijti = ^
J
d2x j d2x A{
(x)A* (x') ^
^ i ^ - u r P)
(5 5 5 )
j * * J ^ e- i((‘- 1i) x- ( 1j-O x')
(5.56)
Again, remember, 2-ku = I, so th a t 2-kui = U and 2-nUj = lj
W m = n1 j d2x j d2x'Ai (x) A* (x ')
%3,i 2ir
r2w
W ijjL =
± - j J d 4> J
.
.
d2x'A* (x') e+i,xV aJ x'J
d2x A l (x) e_ i l - x e
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(5.57)
69
The quantity in the left square-bracket is the fourier transform of A* (x) x a phase factor and
the one inside the right square bracket its complex conjugate. Recall th a t T ((f(x ) e ~ zk°x) =
f (k — ko). If we denote the T (Bi (x))) as Aj (I), we end up with
=* W ijtl = ^
5.6.1
£ * d<f>A* (1 - lj) At (It - 1)
(5.58)
W in d ow fu n ctio n for one b aselin e in an in terferom eter
Suppose we ju st want to calculate W u j for gaussian beams. In th a t case,
Ai (x ) = A* (x) =
^
(5.59)
and
A( l ) = e =^ ~
A \ A ( l - h) I2 =
(5.60)
^ W U>1
(5.61)
where the last equality holds because there is no angular dependance in a gaussian distribution.
5.6.2
E ffect o f fin ite freq u en cy b a n d w id th on w id th o f w indow fu n ction
Our beam-combiner-detector system works in the following way. For two antennas which
output electric fields E \ and £)>, it first adds them and then squares the sum, so th a t w hat we
record in the detector is (£ ) + E 2 ) (E j + E^)- This is all very well, b u t when there is a finite
bandw idth, the detector sums this up over all frequencies, and we get J dv {E\ + E 2 ) (£)* + E*2)
instead.
Now recall from §5.1th a t E \ E 2 is proportional to the visibility.Therefore,
we need to
integrate over all frequencies to get the visibility and the expression for a single visibility now
becomes
Vi = K
J dv J
dnA (n) I (n) e*27rUi'n
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(5.62)
70
So, we can follow through the entire last section with two integrals over the previous expression
thus:
(ViV*)
K 2T’2 ~
r
J
r
r
J dUJ
r
J ^ V>
^*i (n O E E Ylm (n) ^T'rra' (n )
e
^
lm I'm,1
(5.63)
Now, from the 1-D relation for angular resolution
A0 = A
(5.64)
- where in this case, B is the baseline - we can deduce the following:
27r
B
B
c 1
l ~ —- — 27r— = 2ir— v => dv = — —dl
A6
A
c
2vr B
(5.65)
K
’
Substituting this in the above expression for the window function, we get:
Wij,i =
1 ( (2^)2C d* J dlJ dl'A*i(1“ 1j)Al (li “ 1}
B {B j \ ( 2 tt)
(5-66)
This is the general expression for two different baselines. B ut w hat are the limits of integration
over I and I'? Define the center frequency to be vq. Let the band be defined by the lower and
upper frequencies vq —A
m
and uq +
A
m
respectively. The lower and upper limits of integration
for the baseline labelled i are then
B
li i = 27r — (mo —A v )
(5.67)
li2 — 2 7 r ( m o + Am)
(5.68)
•
and
respectively, and similarly also for the baseline labelled j .
5 .7
V is ib ilit y in t h e p o la r iz e d c a se
We discuss now how the output of a polarized interferometer is related to the Stokes’
param eters.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
71
Consider two horn antennas / radio telescopes separated by a distance B . These define
one baseline. Let them be oriented to receive a signal from an extended object in the sky.
Then, the electric fields at the two antennas are the same except for a phase factor th at
depends on their separation: B sin
Both E \ and E 2 can be w ritten in term s of x- and y-
polarized states thus:
E i = E xx + E y y
(5.69)
E 2 = (E x± + E y$r) e~i2-?Ba
(5.70)
The reason we do this is th a t we wish to express all measurable quantities in term s of the Q,U
param eters, which are very easily expressed in terms of E \ and E 2.
In general, waveguides can be coupled to some combination of linear polarizations, so:
E i = a iE x± + a2E yy
(5-71)
E 2 = ( h E x± + b2E vy ) e - T Ba
(5.72)
If a2 = b2 = 0, then one linear polarization is chosen; if ~ = ± i then a circular polarization is
chosen.
The Stokes’ param eters are defined as follows:
T
=
(\E x \2 + \Ey \2)
(5.73)
Q
-
(\E X\2 - \Ey\2)
(5.74)
u = (25ft (E*Ey)}
(5.75)
(21(E *E y))
(5.76)
V
=
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
72
Then, E x and E y can be expressed in term s of the Stokes’ parameters:
\EX?
= \ { T + Q) .
(5.77)
\Ey ?
= \(T-Q )
(5.78)
E*xE y = l- { U + iV )
(5.79)
E xE*y = l- ( U - i V )
(5.80)
Then, the o utput of the multiplying interferometer is:
{E \E 2) = ^ e ~ ^ Ba [alh (T + Q) + a2b\ (T - Q) + a\b2 (U - iV ) + a*2h (U + *V)]
(5.81)
Simplifying,
(E *E 2} = —e * A
[(aj^i + a2b2) T + (a*5j —a2b2) Q + (a\b2 + a2bi) U + i ( a ^ i —a i52) V-]
(5.82)
We can now do an integration over x ' and y', as shown in §5.1, to end up with:
JJ
{E {E 2) dx'dy'
=
K
Jj
A (x 1, y') [ ( a f r + a\b2) T + ( a f r - a*2b2) Q
+ (a\b2 + a*2bi) U + i (a*2bx - ojfc) V]
or
V = K A * [(ajfri +
a2b2) T + (a\b i - a*2b2) Q + (a\b2 + a*2bi) U + i (a*2bx- a\b2) v ]
where A (x r, y') is the antenna pattern, tildes denote a fourier transform and
(5.83)
asterisk denotes
convolution.
We need to assign one kind of polarization, i.e. either linear or circular, in order to figure
out the four different visibilities. Let us consider two horns; one th at outputs left circular
polarization and the other th a t outputs right. Define left and right polarization states thus:
R = —
a\
=
-i
(5.84)
L = —
ai
=
i
(5.85)
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
73
for E i and
bo
R= fb\
bo
L = ^bi
=
~i
(5.86)
=
i
(5.87)
for EoSubstituting these values in the above equations leads to:
Vr l = K A * (Q + iU)
(5.88)
Similarly, we get the other visibilities:
VLR
= K A * { Q ^ iU )
(5.89)
Vr r
= K A * ( T + V)
(5.90)
VLL
= K A * (T~—V )
(5.91)
Eqs 19-22 are visibilities for the circular polarization case. For linear polarization,
X =
Y =
a2
ai
ai
ao
=
0
(5.92)
=
0
(5.93)
=
0
(5.94)
=
0
(5.95)
for E i and
bo
bi
bi
Y =
bo
X =
for Eo and so
Tv)
(5.96)
~iV)
(5.97)
VxY
=
VYX
— KA
V xx
=
K A * ( T + Q)
(5.98)
VYY
=
K A * ( T ^ Q)
(5.99)
K A *(£/ +
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
74
5.8
W hy U se an Interferometer?
The preceding section describes the output of an interferometer and how it relates to
the power spectrum . B ut why build an interferometer instead of a more traditional imaging
system for studying CMB polarization? There are a number of reasons th a t have m otivated the
construction of the interferometers listed in Table 5.2. The main reason is to control system atic
effects, which in some cases are more manageable than in imaging systems. There are additional
factors, especially aperture size, th a t favor interferometric approaches over imaging for spacebased systems. For equivalent angular resolution, an interferometer can be substantially simpler
and less costly than a single aperture.
5.8.1 A n gu lar R eso lu tio n
For a monolithic dish of diameter, D, equal to the length of a two-element interferom eter
baseline, B , the interferometer has angular resolution (fringe spacing) roughly twice as good
as th at of the monolithic dish. The reason for this difference in angular resolution is th a t
the filled dish is dom inated by spacings th a t are much smaller than the aperture diam eter.
The full w idth to the first zero for a uniformly illuminated aperture of diam eter D is 2.4X/D.
T he full w idth to the first zero for a two-element interferometer, when the baseline B is much
larger than the individual aperture diameter, is X/ B. It is helpful to consider the difference
between the systems in /-space as well. For an interferometer the window function peaks at
/ = 2-k B /X . For an imaging system with a Gaussian beam the window function is IF) = e
12 2
a .
The beam width a — 0.42 FW HM and FW HM = (1.02 + 0.0135Te)A/H where Te is the edge
tap er of the antenna in dB [5]. For an edge tap er of 40 dB (typical for CMB instrum ents),
FWHM = 1.51 A/D, a = 0.66A/D and the window function falls to 10% of its peak value at
I = 2.29D/X, which is less th an half of the peak /-value for an interferometer baseline of the
same size.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
75
This angular resolution factor is im portant because the size of the aperture is a cost-driver
for the E IP mission. Angular resolution is im portant for CMB polarization measurements in
two ways. First, imperfections in the shape and pointing of beams couple the CMB tem perature
anisotropy into false polarization signals. These problems can be reduced significantly if the
CMB is sm ooth on the scale of the beam size, which happens for beams smaller than ~ l(y [6].
Second, removing contam ination of the tensor B-mode signal by B-modes from weak lensing
requires maps of the lensing at higher angular resolution th an the scale at which the tensor
B-modes peak [7].
5.8.2
N o R ap id C h op p in g and Scanning
Imaging systems w ith either coherent or incoherent detectors typically use some form of
“chopping,” either by nutating a secondary m irror or by steering the entire prim ary at a rate
faster than the 1 / / noise in the atmosphere and detectors. Similar approaches are used with
arrays of detectors. Since an interferometer does not require this rapid chopping, the time
constants of the bolometers used can be relatively long. W hen using an imaging system to
form a two-dimensional (2D) m ap w ith minimal striping or other artifacts, the scan method
m ust move the beam (or beams) on the sky at a rapid rate. Interferometers provide direct
2D imaging and do not require such scanning strategies. In the interferometer, only correlated
signals are detected, so it has reduced sensitivity to changes in the total power signal absorbed
by the detectors [4].
5.8.3
C lean O ptics
The simplicity of an interferometric optical system eliminates numerous system atic prob­
lems th a t plague any imaging optical system. Instead of a single reflector antenna, the in­
terferometers we have studied use arrays of corrugated horn antennas. These antennas have
extremely low sidelobes and have easily calculable, symmetric beam patterns. Furthermore,
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
76
there are no reflections from optical surfaces to induce spurious instrum ental polarization, an
unavoidable problem for any system w ith imaging optics [8, 9]. In principle, one could con­
struct an imaging instrum ent w ithout reflective optics; an array of horn antennas, each coupled
directly to a polarimeter, could view the sky directly. Each horn aperture would be sized to
provide the required angular resolution. However, such a system uses the aperture plane ineffi­
ciently. A single horn antenna in such an imaging system will have angular resolution ~ 2X/ D,
where D is the horn diameter. An N - element interferometric horn array th a t achieves the
same angular resolution will have a maximum baseline length of B = D (and require the same
aperture size), but will collect N modes of radiation from the sky and hence be more sensitive.
Another advantage over an imaging system is the absence of aberrations from off-axis
pixels: all feed elements are equivalent for the interferometer. In contrast to an imaging system,
the field-of-view (FOV) of an interferometer is determined by the prim ary beam w idth of the
array elements, not by beam distortion and cross-polarization at the edge of the focal plane.
One can choose to increase the sensitivity of the instrum ent by collecting more modes (optical
throughput) of radiation from the sky.
In the interferometer this can be done by adding
additional antennas; the only lim itation is the size of the aperture plane rather th an optical
aberrations in the focal plane. The largest usable FOV for an off-axis Gregorian reflector is
approximately 7° [10]. See Table 5.1 for a comparison of imaging and interferometric optical
systems.
5.8.4
D irect M easu rem en t o f S tokes P aram eters
Interferometry solves many of the problems related to mismatched beam s and pointing
errors raised by Hu et al. (2003) [6]. This advantage arises because interferometers measure
the Stokes param eters directly, w ithout differencing the signal from separate detectors.
Imaging instrum ents for CMB polarization measure the power in each linear polarization
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
77
Table 5.1: Comparison of various optical designs for the EIP. To achieve the same angular
resolution each instrum ent allows different amounts of throughput (number of modes) and
requires different aperture diameters, D . For the Gregorian the edge taper on the prim ary
m irror illumination is assumed to be —40dB, the diam eter of the FOV is given in degrees and
the number of modes is approximately [FOV/(angular resolution)]2, assuming all the modes
reaching the focal plane are coupled to detectors. For the imaging horn array, the horn diameter
= D. For the interferometric horn array, D = B , the diam eter of a close-packed array of horns,
each of diam eter d, and the number of modes is given by the number of horns ~ (D /d )2. In
the last three columns, for all cases, the angular resolution = 1° and A — 3 mm.
Instrum ent
Angular resolution
FO V
n
Aperture D
(cm)
Modes
(FWHM)
Gregorian telescope
1.51A/D
~ 7
26
49
Imaging horn array
2X/ D
2X/ D
34
1
Interfer. horn array
A/2 D
2X/d
8.6
16
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
78
on separate bolometers and then form the difference of the two signals to determine the linear
polarization.
This approach requires careful m atching of the bolometers. Moreover, if the
signals being differenced come from two different antennas, then the beam patterns and pointing
of the two antennas m ust coincide precisely. Any mismatch converts power from the total
intensity into a spurious polarization signal [6]. In an interferometer, differences in antenna
patterns for the different horns do not couple intensity to polarization in this way (See §5.9).
An interferometer measures the Stokes param eters by correlating the components of the
electric field captured by each antenna w ith the components from all of the other antennas. If
the output of each antenna is split into E x and E y by an orthomode transducer (OMT), on the
baseline formed by two antennas, 1 and 2, the interferom eter’s correlators measure (E \XE 2 X),
{E \yE 2y), (E \xE 2 y), and {E\yE 2 X)- The first two are used to determine I and the latter two
measure U . R otating the instrum ent allows a measurement of Q. Stokes V can be recovered in
a similar m anner bu t is expected to be zero for the CMB. Alternatively, the antenna outputs can
be separated into left- and right-circular polarization components by a combination of an OMT
and a polarizer. Correlating these signals also allows recovery of all four Stokes param eters.
DASI uses a switchable polarizer to accomplish this [11].
Separation of E and B Modes. A significant challenge in CMB polarization measurements
is separation of the very weak B modes from the much stronger E modes. Unless a full-sky
map (an impossibility because of Galactic cuts) is made w ith infinite angular resolution the
two modes “leak” into each other [12, 13].
It has been shown [14, 15], however, th a t an
interferometer can separate the E and B modes more cleanly than can an imaging experiment,
although detailed calculations of this advantage in realistic simulations remain to be done.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
79
5 .9
S y s t e m a t ic E ffe c ts
Hu et al. (2003) [6] have reviewed system atic effects relevant to CMB polarization mea­
surements, mainly in the context of imaging instrum ents. Bunn (2006) [16] performs similar
calculations for interferometers. Table 5.2 outlines a variety of system atic errors and how they
can be managed in imaging and infererometric instrum ents. The relative im portance of these
effects is quite different in interferometric systems: some sources of system atic error in imaging
systems are dram atically reduced in interferometers. As an example we consider the effects of
pointing errors and mismatched antenna patterns.
In a traditional imaging system, the Stokes param eters Q and U are determined by
subtracting the intensities of two different polarizations. For example, Q might be measured by
splitting the incoming radiation into x and y polarizations, determining the intensities I x and
ly of the two polarizations, and subtracting. In such an experiment, any mismatch in the beam
patterns used to determine Ix and I y (including differential pointing errors as well as different
beam shapes) will cause leakage from total power (T ) into polarization (Q, U).
In an interferometer, the signals are combined before squaring to get intensities. In such a
system, mismatched beams do not lead to leakage from tem perature into polarization. Suppose
th a t the signal entering each horn of an interferometer is split into horizontal and vertical
polarizations. Working in the flat-sky approximation, let E « ( r ) and E iy (r) stand for the x and
y components of the electric field of the radiation entering the <th horn from position r on the
sky. The signals coming out of each horn are averages of the incoming electric fields weighted
by some antenna patterns Gi(x ,y) (r ).
In an interferometer, these signals are multiplied together to obtain a visibility.
To
measure the Stokes param eter U, for example, we would multiply the x signal from horn i with
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Table 5.2: A Comparison of Systematic Effects
Systematic Effect
Imaging System Solution
Interferometer Solution
Cross-polar beam response
Instrum ent rotation
& correction in analysis
Instrum ent rotation
& non-reflective optics
Beam ellipticity
Instrum ent rotation
& small beamwidth
No T to E and B leakage
from beams; inst. ro t’n
Polarized sidelobes
Correction in analysis
Correction in analysis
Instrum ental polarization
Rotation of instrum ent
& correction in analysis
Clean, non-reflective optics
Polarization angle
Construction
& characterization
No T to E and B leakage
from beams; construction
& characterization
Relative pointing
Rotation of instrum ent
& dual polarization pixels
No T to E and B leakage
from beams; inst. ro t’n
Relative calibration
Measure calibration using
tem perature anisotropies
Detector comparison
not req’d for m apping or
measuring Q and U
Relative calibration drift
Control scan-synchronous
All signals on all detectors
drift to 10~9 level
Optics tem perature drifts
Cool optics to ~ 3 K
& stabilize to < ffK
No reflective optics
1 / / noise in detectors
Scanning strategy
& phase m odulation/
Instant, measurement of
power spectrum
lock-in
without scanning
M ultiple frequency bands
Multiple frequency bands
Astrophysical foregrounds
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
81
the y signal from horn j to obtain the visibility
V ij = j d 2r\ d2r 2 Gix(r{)Gj y(r 2 )(E ix{r\)E*jy (r2 )).
The angle brackets denote a tim e average. The electric fields due to radiation coming from two
different points on the sky are uncorrelated, and the product of x and y components of the
electric field gives the Stokes U param eter:
(Eix(n )E * y (r2)) = £ /(n )e w
^ 5 ( n - r2),
so the visibility is
V? =
Note th a t the visibility
J
d2r G ix(r)G jy(r )U (r)e2rtiu'r.
does not contain any contribution from the total intensity
(Stokes I), even if the two antenna patterns are different. This means th a t differential pointing
errors and different beam shapes for different antennas do not cause leakage from T into E and
B . Antenna pattern differences do cause distortion of the observed polarization field, so errors
in modeling beam shapes and pointing may cause mixing between E and B .
Coupling between intensity and polarization will arise if the beams have cross-polar con­
tributions. In th a t case, the visibility VU, which is supposed to be sensitive to ju st polariza­
tion, will contain contributions proportional to (EXE*) and (EyE*), to which Stokes I does
contribute.
The same considerations apply if the incoming radiation is split into circular rather
than linear polarization states. The visibility V ^ L, obtained by interfering the right-circularlypolarized signal entering horn i w ith the left-circularly-polarized signal entering horn j , contains
only contributions from Q and U if the beams are co-polar, even if the two horns have different
beams. Again, cross-polarity induces leakage from intensity into polarization.
In short, in an interferometer, beam mismatches are less of a worry th an cross-polar
contributions. The reverse is true for an imaging system.
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82
5 .1 0
T h e A d d in g I n te r fe r o m e te r
In a simple 2-element radio interferometer, signals from two telescopes aimed at the same
point in the sky are correlated so th a t the sky tem perature is sampled w ith an interference
pattern with a single spatial frequency. The output of the multiplying interferometer is the vis­
ibility (defined in the last section). W ith more antennas these same correlations are performed
along each baseline. To recover the full phase information, complex correlators are used to
measure simultaneously both the in-phase and quadrature-phase components of the visibility.
In interferometers th at use incoherent detectors, such as an optical interferometer, E PIC
and MBI, the electric field wavefronts from two antennas are added and then squared in a
detector — an “adding” interferometer as opposed to a “multiplying” interferometer [17]. (See
Figure 5.8.) The result is a constant term proportional to the intensity plus an interference term.
The constant term is an offset th a t is removed by phase-m odulating one of the signals. Phasesensitive detection at the modulation frequency recovers both the in-phase and quadraturephase interference term s and reduces susceptibility to low-frequency drifts ( 1 / / noise) in the
bolometer and readout electronics. The adding interferometer recovers the same visibility as a
multiplying interferometer.
In an interferometer with an array of N > 2 antennas, the signals axe combined in such
a way th a t interference fringes are measured for all possible baselines ( N ( N — l) / 2 antenna
pairs). This combination can occur in two different ways: pairwise combination or Fizeau (or
Butler) combination [18]. Pairwise combination involves splitting the power from each of the N
antennas in the array N — 1 ways, adding the signals in a pairwise fashion, and then squaring
the signals and separating out the interference term as described above. In optical systems
this approach is analogous to Michelson interferometry. In Butler combination the signals from
each of the antennas are split and then combined in such a way th a t linear combinations of all
the antenna signals are formed at each of the outputs of the combiner (Figure 5.9). To allow
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
all the Stokes param eters to be determined simultaneously, orthomode transducers (OMTs)
are inserted after corrugated horn antennas. In this case, the Butler combiner delivers the
signals from 2N antenna outputs to 2N detectors. Each detector squares these amplitudes,
creating interference signals from all baselines simultaneously on each detector. Effectively, the
signals from all baselines are multiplexed onto each of the N detectors. Only 2N detectors are
required, rather th an the 2 N (2 N — l) /2 detectors required for pairwise combination. Butler
combiners are commonly used for phased array antennas w ith coherent systems using either
waveguide or coaxial techniques. The optical analog is Fizeau combination, which is typically
used for incoherent systems at optical wavelengths. We have developed both Butler and Fizeau
approaches and have decided to concentrate on the Fizeau m ethod because of its relative sim­
plicity and low-loss. However, in a coherent system, with amplifiers, the Butler approach is still
an attractive option for forming a large-N interferometer.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
84
1L.
low
Figure 5.8: Adding interferometer. At antenna A 2 the electric field is
E q.
and at A \ it is EqE^,
where <j>= k B sin a and k = 2tt/X. B is the length of the baseline, and a is the angle of the
source with respect to the symm etry axis of the baseline, as shown. (For simplicity consider
only one wavelength, A, and ignore tim e dependent factors.) In a multiplying interferometer
the in-phase output of the correlator is proportional to
Eq
cos <j>. For the adding interferometer,
the output is proportional to Eq + Eq cos(cf>+ A M o d u l a t i o n of A 4>(t) allows the recovery
of the interference term , Eq cos <p. which is proportional to the visibility of the baseline.
f
i
l
l
Figure 5.9: Block diagram of a planned CMB polarization experiment. Light enters the instru­
m ent from the left. Each phase switch is m odulated in a sequence th a t allows recovery of the
interference term s (visibilities) by phase-sensitive detection at the detectors. T he signals are
mixed in the beam combiner and detected on cold bolometers at the right. The beam combiner
can be implemented either using guided waves (Butler combiner, as shown here) or quasioptically (Fizeau combiner, see below). The triangles represent corrugated conical horn antennas,
which connect through transitions to rectangular waveguide. Orthom ode transducers (OMTs)
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
85
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Degree Angular Scale Interferometer,” Nature, vol. 420, pp. 763-771, Dec. 2002.
[12] A. Lewis, A. Challinor, and N. Turok, “Analysis of CMB polarization on an incomplete
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[13] E. F. Bunn, “Separating E from B,” New Astronom y Review, vol. 47, pp. 987-994, Dec.
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[14] C.-G. Park, K.-W . Ng, C. Park, G.-C. Liu, and K. Umetsu, “Observational Strategies
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[17] K. Rohlfs and T, L. Wilson, Tools of Radio Astronomy, Springer, 2004.
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Chapter
6
The Fizeau Combiner: A C oncept Study
6 .1
I n t r o d u c tio n
The advantages of interferometry have been stated/discussed in the preceding chapter.
However, extraction of visibilities from an interferometer is not a unique process - there are
many different techniques th a t can be employed to do this. A general introduction to “adding
interferometry” was provided in §5.8, and one of the adding techniques was discussed (the
Butler beam combination technique). In general, we wish to obtain the highest signal-to-noise
ratio for every baseline, and based on this and other design-related criteria, it is possible to
choose an extraction technique th a t suits us best.
In this chapter, we introduce one such technique, which we refer to as “Fizeau beam com­
bination” and the beam combiner as a “Fizeau system” . This is a wavefront-division interferom­
etry technique which is analogous to the simplest interferometer in 1-D: the Young’s double(or
multiple)-slit interference/diffraction set-up, an example of which is shown in fig.(6.1). While
this is very well-known, we stress here the fact th a t there is a path difference (and therefore a
phase difference) introduced in sid e the instrum ent, i.e. different rays entering the instrum ent
suffer a phase difference a f te r they pass through the slits/antennae (these two term s will be
used interchangably in the remainder of this chapter). Compare this to a traditional interfer-
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
89
Light from a m onochromatic source
No path difference
here
Path differences
due to aperture
Positions (8)
Screen / Focal plane
Figure 6.1: A simple multi-slit diffraction/interference experiment. Phase differences occur
after light has passed through the slit, inside the instrum ent.
ometer (also shown in 1-d, though the extension to 2-d is straightforward) as shown in fig. (6.2),
where rays entering the instrum ent have already undergone a phase difference before they enter
the antennae.
A “Fizeau system” is one th a t combines b o th the aforementioned instrum ents, quite
literally. A simple Fizeau system is shown in fig. (6.3). Notice th a t rays entering the instrum ent
suffer phase differences both before and after they pass through the slit. It is th is fact th a t
makes Fizeau combination a powerful tool. Let us explore w hat this combination achieves. We
sta rt by noting th a t the “external” phase difference as shown in fig.(6.2) is the reason th a t
visibility is a fourier transform of the image on the sky. As mentioned in §5.4, this implies th a t
the output (visibility) is essentially the intensity m odulated by a fringe, where the fringe is a
function of baseline length, and therefore selects one “mode” from the image. In the Fizeau
system, we have an additional set of phase differences. W ithout loss of generality, we may
assign a -ve sign to the phase introduced inside the instrum ent. Now, if we sum over both the
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
90
External path differcnc£(<j>)
Correlator
Figure 6.2: A simple traditional interferometer. Rays suffer phase differences b efore they enter
the slits.
phases, we get a fourier transform followed by an inverse fourier transform - but this is the
image itself! Thus, F izeau com b in ation en ab les im agin g in an in terferom eter. This is
discussed later in this chapter in §6.3.
Just as a traditional interferometer multiplies the image with the fringes produced by its
baselines (i.e. the fringes due to the “outer” phase differences), so the Fizeau system multiplies
v isib ilities w ith in tern al fringes, where each fringe is a function of baseline length, and is
produced due to “internal” phase differences. This is not an added complexity - visibility is a
complex quantity, but detectors measure only real and positive quantities. By m odulating the
visibility by two different known phases, we can recover both the real and imaginary parts of
the visibility. B ut we can do more - if there is a large number of detectors on the focal plane, we
can m odulate each visibility by many known phases and extract much more information than
is posible in a traditional interferometer. Let us explore how. Irresp ectiv e of which technique
we choose to employ, CMB interferom etry will alw ays require th at we use as wide a bandw idth
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
91
External path difference^#
Internal path difference (8)
Focal plane / detector array
Figure 6.3: A simple 1-d Fizeau system. Notice th a t there are two sets of phase differences.
as possible, since the CMB polarization signal is very low (~ pK). Let v be the center frequency
and A v the bandw idth. Then, a baseline of length B will measure CMB polarization at
7tB
itv B
A
(6 .1)
where the w idth in Aspace is
A£ =
7tA v B
(6 .2)
Thus, the larger the bandw idth, the larger the radial w idth of the pixel in the u-v plane. This
is shown in fig. (6.8) Herein lies one of the m ain problems w ith CMB interferometry: a large
bandw idth ensures high enough signal-to-noise, but increases the size of a pixel in the u-v
plane. The additional information th a t is available to us as mentioned above can be utilized
to sub-divide the band in the u-v plane. Thus, th e F izeau sy ste m en ab les e x tr a c tio n o f
sp ectra l in form ation v ia geom etry, w ith o u t a d d ition al co m p o n en ts like filters. We
discuss this aspect in detail in §6.2
To summarize, in this chapter, we study the aforementioned Fizeau combiner approach
to interferom etry and find th a t it is more useful th an traditional interferom etry in two ways:
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
92
•too
-50
0
50
100
0
50
100
Figure 6.4: 2-slit diffraction pattern. The
Figure 6.5: 8-slit diffraction pattern. The
large envelope is caused by the single-slit
p attern is more “focused” , leading to betdiffraction and the fine features by the in­
ter image recovery.
terference between 2 slits.
1. Possible to get spectral information within a single frequency band
2. Possible to use the instrum ent as both an imager and an interferometer
Before we begin to discuss the two aspects of the Fizeau system in detail, we note th a t the
simple 1-d multi-slit system acts as an imager as well. Figs.(6.4) and (6.5) show the diffraction
pattern due to 2 and 8 slits respectively, illustrating the fact th at a larger num ber of slits leads
to better image recovery. This can be explained in term s of interferometry as follows. Each
baseline detects a mode in the image. The greater the number of baselines, the greater the
number of modes th a t can be recovered and hence the better the recovered image. Figs.(6.6)
and (6.7) illustrate th at even in a simple multi-slit system, the image formed on the focal plane
traces the actual image faithfully.
While the idea of using a Fizeau system is a novel one in CMB cosmology, and while the
Fizeau system employed in the MBI was developed by th e MBI collaboration (and the following
ideas by the author), this is by no means the first tim e this technique has been employed
[1],[2],[3]. B ut our attem pt to extract spectral information using the Fizeau system and use it
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
93
1 6 s lits
16 slits
1.0
0.8
0.6
0.4
0.2
-100
Figure 6.6:
-5 0
L _.
0
A
.
50
100
0.0
........................... 1 .
-100
-5 0
16-slit diffraction pattern, Figure 6.7:
source 10° away from center.
. .
0
50
100
16-slit diffraction pattern,
source 20° away from center.
to run the instrum ent as an imager and an interferometer are certainly new developments, to
the best of our knowledge.
6 .2
6.2.1
S p e c tr a l in fo r m a tio n fr o m a n in te r fe r o m e te r u s in g a F iz e a u a p p r o a c h
M o tiv a tio n
As mentioned earlier in the chapter, cosmological signals are very weak; therefore, a
wide bandw idth helps increase the power input from the cosmological source.
However, a
wide bandw idth also means poor u-v coverage as shown in fig.(5.4) and fig.(6.8). This can be
explained as follows. Consider a two-slit experiment w ith a monochromatic point source. This
experiment yields fringes th at can be computed given the exact param eters of the experiment.
Now, if the same point source emits two different wavelengths th at differ by a small percentage,
the fringes are slightly shifted. If a lot of such wavelengths are used, each only slightly different
th an the one preceding it, then a “fringe-band” is produced instead of clear, sharp fringes. We
call this effect a “fringe wash-out” . The wider the bandw idth, the greater the wash-out.
Now, a single sharp (i.e. monochromatic) fringe corresponds to a single point on the u-v
plane. If the effect of the bandw idth is to add many such fringes, w hat it means is th a t we are
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
94
V
Spread caused by observing time
One pixel
Size of FOV:
This determines
resolution
Spread caused by bandwidth
U
Figure 6.8: The u-v plane coverage of one baseline of an interferometer for a single pointing in
a single baseline orientation angle. The two causes of spread in a single pixel in the u-v plane
are shown. Also shown is the size of the FOV, which is the fundam ental limit to u-v resolution.
measuring the average visibility over a certain finite area on the u-v plane, where the radial
stretch is due to a finite bandw idth and the angular stretch represents the integration tim e for
the interferometer.
6 .2.2
P r e lim in a rie s
The output measured at the bolometers in MBI contains the following phase information
integrated over the entire bandw idth (75 — 110GHz )
1. Phase introduced because of the path difference between any two rays th a t arrive from
the same p art of the sky on the the two outward-facing antennae th a t make up a baseline
2. Phase introduced because of the p ath difference between any two rays th a t arrive from
two different antennae on to the same point in the focal plane
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95
The phase in point 1 is due to the fact th a t MBI is an interferometer, and so the visibility th at
we measure must, by definition, include this phase. However, the phase in 2 above introduced
by the beam combiner needs to be eliminated to recover visibility from each bolometer. I f
there were a way to calculate the net phase introduced by the beam combiner over the whole
bandw idth, then all we would need to do is to divide the output at each point in the focal plane
by this net phase, and we would get visibility directly.
However, calculating this n e t phase is not easy, since integration over the bandw idth
turns our calculation into an unrecognizable beast. So we choose instead to work with fringe
p attern s1. In order to do so, we need to realize th a t w hat we observe at every detector is the
visibility on the sky times the phase factor, summed over a p art of the fringe pattern.
But the fringe p a tte rn is different for every single frequency in the bandw idth. Visibility
is also different for every different frequency. This can be reasoned as follows. Every single
frequency defines a single value of £ for a single baseline as follows. The angular resolution of
a single baseline of length D is
A
27tD
^
(6.3)
c
Thus, a range of values of v will produce a range of As, or a band in
space. A finite-bandwidth
interferometer thus measures w hat is called a “bandpower” instead of a single value of the power
spectrum at one value of £. B ut the power spectrum is just the variance of the visibilities for
a circle (ring) in the u — v plane, as proved in the previous chapter. And so we get different
visibilities for the same baseline and orientation but for different frequencies [fig. 4.8?].
1This is exactly the sam e as saying that the visibilities in each sub-band are modulated by a fringe which
depends on baseline length. To extract these visibilities, we need to separate out the fringes.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
96
We can therefore think of the effect of the instrum ent on the visibilities in the following
way. Let us divide the entire bandw idth of the instrum ent into N sub-bands and let v \, v2...vn
be the centre-frequencies of these N sub-bands. Then for one baseline, one orientation, and
one detector position, these will correspond to visibilities Vi, 14 --V v and to phase differences
(where j represents the detector). If we represent the output at the j t h detector
as O then we get
N
O j = J 2 Vae ^ a
(6.4)
ct—0
Given just one detector, it is impossible to extract every Va for every sub-band, even though we
know precisely w hat the (/>ja ’s are. However, if we have N detectors, then we can easily write
the following system of equations
Q1
=
02 =
On
=
V je ^ 11 + V2e ^ 12 + • - • + V v e * ^
Vie ^ 21 + V2ei f e + • •• + VNei<t>2N
Vie i(t>N1 + V2e ^ JV2 + - • • + VNeic^NN
(6.5)
This is a system of N equations with N unknowns - Vj, V2 • • • Vv >and so we can get the values
for each one ofthese “sub-band visibilities” . The beam combiner thus
achieves far more than
ju st separating the real and imaginary parts of visibilities. (In fact, there are 2N equations,
since visibilities are complex quantities, b ut this has been overlooked to simplify the equations
for the discussion).
6 .2 .3
E ffect o f non -zero d e te c to r size
In the discussion above, we assumed th a t the collection area of the detectors is negligible,
and we completely
ignored the effect of the fringe pattern. Let us account for these effects
in the following way. Let A be the effective collecting area of each detector. Let /
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(x,va) be
97
the value of the fringe pattern (i.e. ju st a fraction) at a point on the focal plane
x and in a
frequency sub-band marked by a. Then, equations (6.5) become
Oj. = J v 1ei M x ) f ( x , v 1)d2x + - - - + J v Nei*1NWf ( -x, vN)d2x
0 2
=
J
Fie ^ 2l(x)/ (x, 1/1 ) d2x + • • • +
J
VN e ^ 2N^ f (x, vN )
d2x
On = j v 1eirt>N1Wf(x,is1)d2x+--- + J v Nei<l>NNWf(x,vN)d2x
(6.6)
where it is understood th a t integration is done over the area of the detector.
This still leaves us with a problem - th at of deconvolving the F ’s from the integrals.
However, if the area of the detector is small compared to the width of fringes, then we can
assume th a t the phase differences remain roughly constant over the collecting area of one
detector, so th a t we may write
where F
Ox
=
02
=
(x,v x) + • • • + Fjve^liv(x)F (x,v N )
A [vie ^ 2l(x)F (x,vx) + • • • + Vweifev(x)F (x, v N )
On
=
A [F ie ^ Jvl(x)F ( x , ^ ) + --- + Five ^ iv(x)F ( x ,i/ Jv)]
A [v ie ^ n(x)F
(6.7)
(x,v a ) represents an “average” value of the fringe pattern, perhaps the value at the
centre of the detector.
Equations (6.7) again have N variables and can be solved to get N visibilities over the
bandw idth.
6 .2 .4
1.
F easib ility o f u sin g tech n iq u es in §6.2 for M B I
Bandw idth Issues
For MBI, we are using ~ 20 detectors, meaning th a t we can extract visibilities for 20
“sub-bands” . Bandw idth for MBI is 35 GHz, so th a t the width of every sub-band is 1.75
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
98
GHz, such th a t we get, for every sub-band
^
=
0.002
(6.8)
which is really small. Thus, the sm all-bandwidth approximation holds for equations (6.7).
2. Detector Area Issues
We need to compare the area of every detector to the width of the fringes, in order to
estimate whether the area approxim ation holds in equations (6.7). We first estim ate the
width of fringes on the focal plane thus (distance to focal plane = L ~ lm ):
A
03
A w = —L ~
x lm ~ 3cm
(6-9)
The diam eter of a detector area is ~ 1” = 2.54 cm, so th at one detector covers about
one entire fringe. This reduces the spectral resolution th a t can be obtained using this
technique with MBI-4. Future versions of MBI will have much smaller collecting areas,
and will thus provide better resolution.
6 .3
T h e F iz e a u c o m b in e r a s a n im a g e r
In addition to acting as an interferometer, the Fizeau system can also be used directly
as an imager and additionally be used to extract spectral information in the u-v plane not
normally possible w ith conventional interferometer systems. But how is this possible? After
all, an interferom eter chooses certain fourier modes specified by the lengths of its baselines.
However, in the Fizeau system, every fourier mode from every baseline falls on every detector
in the focal plane. In addition, the phase differences introduced within the instrum ent correct
for the phase differences introduced outside the system. This way, every detector detects ALL
possible modes in the right phase - b u t this is exactly a p art of the image!
Thus, the Fizeau system acts naturally as an imager. As a m atter of fact, one has to
make a greater effort to operate it in the interferometer mode, for precisely the reason mentioned
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
99
above - th a t the output from every baseline occurs at every detector. So, in order to be able to
distinguish between visibilities from different baselines at every detector, we need another level
of modulation. We use a phase m odulator based on the Faraday Effect, and discuss it in the
following chapter. A more detailed discussion and m athem atical description of the operation
of this phase m odulator combined with the Fizeau system will be provided in a subsequent
publication.
In w hat follows, we denote the output at the bolometers as O and a fourier transform as
3.
Let e be the phase difference introduced outside the instrum ent and 5 the phase difference
inside the instrum ent. If E \ and E 2 are the electric fields a t the two antennae th a t make a
baseline, then the output from one baseline at any detector is:
(02
r4>2
0 = \ 2 r
E^ E 2ei<-5+£'>+ E iE ^ e-^ 5^ sin 9d(f>d9
J0=$i
J(b—(bi
= 0i J<P=<t>i
(6.10)
The units of this “interm ediate” visibility are W m ^ H z - 1 , since we have integrated over the
solid angle.
Equivalently,
O =
f2
J0=01
[** 2M ( E l E 2ei(-5+e)) sin 9d<f>de
v
'
pd2
f02
r4>2
O tt
,
v
2 M ( E t E 2ei(-5+end<t> de
0=01
J 00=0\
=0t J
Jd>=<bi
(6.11)
(6.12)
'
'
in the flat-sky case. We can now integrate over the focal-plane area in the following way: if the
area on the focal plane being integrated over is
Ap,
c
r<t>2
O « -Jf
f
f
02
f
h 25R (' E l E
A f J J Je= 0i J0=0,
then
02
2S
5(x’v)+d0^ ])) dtpdOdxdy
'
=01 J 0=01
Let us consider ju1st one term in the expression
(02
O « 4 A
f
/f f2
J J J0=o
(...):
r4>2
[ * 2 (e*1E 2 (M 0A'>) dcbd9e-lMx'v)dxdy
=0i k
-01j JJ 00=01
(6.13)
/
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(6.14)
where we have changed the sign on S w ithout loss of generality, since phase differences inside
the cryostat are independent of phases due to the skyward horn antennae.
Now, E 1 E 2 oc I s where I s is a linear combination of Stokes’ param eters [3]. In addition,
we need to take into account the antenna beam:
(6.15)
E 1E%<x.B(0,<l>)Is
We can thus write
0 = - ^ ~ f f B(x,y)
Af J J
B (9, <j>) I s ei€(9’^d0d(f>e~i5(x’v^dxdy
(6.16)
Now, if e (0 ,(p) is linear in 6 and (b (this is true in the flat-sky case), then
(6.17)
° = J ^ J / B ( x , y ) $ ( B I s ) e - i5^ d x d y
Now, if the distance from the inward-facing antennae to the focal plane 3> the collecting area
for each bolometer, 5 (x , y) is linear in (x, y) and
O=
(6.18)
(B $ (B is ))
IF th is is correct, the beam needs to be deconvolved from the above expression in order to
obtain the image from MBI.
Now, eq(6.17) can be split up over the focal plane:
j J B $ (B IS)
j J^B $ (B Is)
£ -iS(x,y)
(6.19)
where 1 • • • N are labels for bolometers on the focal plane.
Each of the bolometer outputs then represents a pixel in image space. However, the total
num ber of pixels depends on the resolution of the instrum ent, and not the num ber of bolometers
on the focal plane. Therefore, if the num ber of bolometers on the focal plane are g reater than
the number of pixels in the image, we need to “repixelize” the image obtained.
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In general, this is how the beam is convolved w ith the image on the sky for the Fizeau
beam combiner:
O =
= ~
(6.20)
[(S-'B) * (Bis)]
(6.21)
There are two assum ptions inherent in the foregoing discussion:
1. The focal plane is large enough to receive m ost of the power from the inward-facing
antennae
2. There are no “blank” areas on the focal plane for which the incident power is not absorbed
by a bolometer
This approach
can be extended to include a finite bandwidth. Also,
this with what isknown
it ispossible to do
as a Butler beam combiner as well [4, 5]. In th a t case,
5 is fixed for
every pair of antennae; therefore, we can do one of two things:
1. Construct a Butler combiner th a t produces several different values of <5 for the same pair
of antennae
2. Devise a phase-switching scheme for the phase m odulator which allows us to recover
visibilities for a certain tim e and an image for some time while observing the sky
In MBI-4, the longest baseline is ~ 15cm which translates to an angular resolution of
~1-1.5°. Since the FOV is ~8°, this implies ~36 pixels for an image. However, there are only
19 bolometers on the focal plane, and so we can have only a 19-pixel resolution in the image.
6.3.1
R em arks a b o u t th e F izeau sy ste m
1. The Fizeau system acts naturally as an imager.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
102
2. By introducing phase m odulators discussed in the following chapter, we can measure
visibilities for all baselines in a Fizeau system.
3. The Fizeau system makes it possible to recover spectral information w ithout the need for
filters.
While it is possible to divide the bandw idth into many different sub-bandw idths, it isn’t
possible to do this indefinitely. The beam for a single antenna determines the FOV of the
instrum ent and limits the resolution in the u-v plane, as shown in fig.(6.9).
>,
Beam of single antenna
Fine repixelization due to Fizeau combiner
Figure 6.9: The u-v coverage of a single baseline has been divided into m any pixels; however,
the beam of a single antenna is larger than a single pixel, so th a t this division is not physical.
There exists a way to reduce the effective u-v beamsize below th a t determined by the
FOV: “super-resolution coverage” . Further discussion on this is left to a future publication.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
103
It is also possible to operate the interferometer s im u lta n e o u s ly as an imager. The
additional m odulation mentioned above opens up a range of possibilities, including the simul­
taneous measurement of visibilities and images. This shall also be explored further in a future
publication.
In conclusion, the Fizeau system introduced here is a powerful tool for CMB cosmology:
it allows the recovery of more information than is possible with traditional interferometers or
imagers and does not need significantly more resources to build. A discussion of the simulation
of a simple Fizeau system is given in §8.2.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
104
Bibliography
[1] D. Loreggia, D. Gardiol, M. Gai, M. G. Lattanzi, and D. Busonero, “Fizeau interferom­
etry from space: a challenging frontier in global astrometry,” in New Frontiers in Stellar
Interferometry, Proceedings o f SP IE Volume 5491. Edited by Wesley A. Traub. Bellingham,
WA: The International Society fo r Optical Engineering, 2004-, p .255, W. A. Traub, Ed.,
Oct. 2004, vol. 5491 of Presented at the Society of Photo-Optical Instrum entation Engineers
(SPIE) Conference, pp. 255—h
[2] M. R. Swain, C. K. Walker, M. Dragovan, P. J. Dumont, P. R. Lawson, E. Serabyn, and
H. W. Yorke,
“A Fizeau Spatial-Spectral Imaging Submillimeter Interferometer for the
Large Binocular Telescope,” in Bulletin o f the American Astronomical Society, Dec. 2002,
vol. 34 of Bulletin of the Am erican Astronomical Society, pp. 1302—h
[3] M. L. Cobb, “A Comparison of Michelson and Fizeau Beam Combiners for Optical Inter­
ferometry,” in Bulletin of the American Astronomical Society, Dec. 2000, vol. 32 of Bulletin
of the American Astronomical Society, pp. 1429—h
[4] J. Zmuidzinas, “Cramer-rao sensitivity limits for astronomical instruments:im plications for
interferometer design,” J. Opt. Soc. Am . A, vol. 20, no. 2, pp. 218, 2003.
[5] C. Calderon, “SIMULATION OF TH E PERFORM ANCE OF TH E MILLIMETRE-WAVE
BOLOM ETRIC IN TER FER O M ETER (MBI) FO R COSMIC MICROWAVE BACK­
GROUND OBSERVATIONS. Ph.D . Thesis, Cardiff.,” Ph.D. Thesis, 2006.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
C hapter 7
T he M B I Instrum ent
Liquid nitrogen tank
Liquid helium tankSecondary minor
3He refrigerator
Primary mirror
Bolometer unit
Figure 7.1: A schematic of the main parts of the MBI instrum ent.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
106
The Millimeter-wave Bolometric Interferometer (MBI) is a ground-based instrum ent de­
signed to measure b oth intensity and polarization of astronomical sources. The first version of
MBI has 4 antennae and is called MBI-4. MBI-4 does not have adequate sensitivity to detect
CMB polarization. R ather, the current instrum ent is a technology demonstor. MBI measures
visibilities using a kind of incoherent detector called “bolometer” (fig. (7.7)). These are more
sensitive th an coherent receivers (e.g. amplifier systems like HEMT) at A <3mm , the region
where the CMB spectrum peaks1. Ultimately, instrum ents with 100s of apertures at m ulti­
ple wavelengths are envisioned. One such proposed instrum ent is the Einstein Polarization
Interferometer for Cosmology (EPIC) [1].
The MBI consists of 4 outward-facing horn antennae and each antenna selects a single
linear polarization. The configuration of MBI-4 optics and cryostat is shown in Fig.(7.3). A
photograph of the MBI-4 optics is also shown in Fig. (7.3). The cryostat is attached to an
altitude-azim uth mount. This mount has a third axis to rotate the instrum ent about its optical
axis.
The feed horn configuration is chosen to provide uniform uv coverage. The instrum ent
is sensitive to CMB tem perature and polarization fluctuations in a medium multipole range
( £ '"-' 200 ) .
The phase of each of the four inputs is sequentially modulated between -90° and +90°
using ferrite-based m odulators [3] implemented in circular waveguide. The m odulation rate is
~1-10 Hz and the loss is < 1 dB. The phase shifters dissipate negligible power, ~ 1 m W each.
Differential loss between the two phase states will produce an offset after demodulation of the
detector signal, so the differential loss betweent the two phase states must be small. Details of
1The choice of the frequency band in which the instrument operates is very important. Fortunately, there
exists a window in which the foreground emission is a minimum - the CMB spectrum happens to be a maximum
there as well, as shown in fig.(7.4).
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107
Outward-facing an ten n ae
P h a s e modulators ------Inward-facing antennae
S econ d ary mirror----------I
Primary mirror
Bolometer u n it
►
Figure 7.2: A schematic of the main parts of the MBI instrum ent.
the phase m odulators are discussed in §7.6. Light is interfered on an array of 16 bolometers
at the focal plane of the prim ary mirror. MBI-4 uses spider-web bolometers, provided by JPL,
with NTD germanium therm istors. The bolometers are coupled to the incoming radiation with
conical horns; the horns form a hexagonally packed array. The bolometers and horns are cooled
to ~ 330 mK w ith a 3He refrigerator.
MBI-4 will be dem onstrated a t the Pine Bluff Observatory (PBO) near Madison, Wis­
consin. Key tests include measuring the interferometric beam patterns, observing bright object
such as the moon, and during the winter, when atmospheric conditions are good, carrying out
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108
Figure 7.3: A detailed schem atic/view of how the Fizeau combiner system fits inside the MBI
instrum ent.
long integrations on test fields.
We follow this w ith a brief discussion of MBI operation and then examine parts of the
MBI in more detail in the rest of this chapter.
Fig. (7.2) shows the Fizeau combiner system discussed in the previous chapter. It is easy
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
109
Ka
^ 100
10
1
20
40
60
80
100
200
Frequency (GHz)
Figure 7.4: CMB foreground spectra from the W M AP team [2]. The frequency range of MBI
is indicated by the last yellow column on the right marked “W ” for the W -band, which is very
close to the minimum of the combined foreground spectrum . This is the frequency band in
which the MBI operates.
to trace the p ath of a ray from the sky into the instrum ent all the way to the detector. A
ray entering an outward-facing horn produces an electric field in the antenna; this electric field
is the same as th a t on the sky weighted by the beam pattern of the outward-facing antenna.
The E-field then gets m odulated as it passes through the phase m odulator and weighted by the
beam p attern of the inward-facing antenna, before being reflected by the prim ary and secondary
mirrors onto the detector array/unit.
7 .1
A n te n n a e
The observation of the sky directly with feed horns rather th an telescope has several
advantages. The optical design is simple and clean. A large number of feed horns, not limited
in number by a telescope design, can be used to increase sensitivity. The cost of this approach
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110
Antennae inside cryostat
Antenna assembly - full view
Figure 7.5: The antenna arrangem ent (right) and how it looks from atop the cryostat, covered
by filters.
is the loss of angular resolution unless extremely large feed horns are used. MBI uses this
approach, but adds interferom etry between feed horns to recover some of the angular resolution
lost by dispensing w ith a telescope. In MBI-4 we have used electroformed corrugated conical
feed horns with aperture 5.3 cm for the input elements. These feed horns have a symmetric
beam p attern with measured beam FW HM of ~7°. MBI-4 only collects a single polarization
for each feed selected by the rectangular WR-10 waveguide attached to the horn output. In a
future MBI instrum ent a waveguide ortho-mode transducer will be used. The relative placement
of the feed horn is chosen in order to provide uniform u v coverage for polarization sensitive
channels w ith 10° step rotation of the instrum ent around its optical axis (Fig. 4). This set of
baselines makes the instrum ent sensitive to CMB polarization fluctuations over the multipole
range t = 150 — 270. Tem perature channels will be used for calibration by comparison with
tem perature maps of WMAP.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(a)
(b)
Figure 7.6: (a) Simulation of fringe patterns formed in the focal plane of the Fizeau beam
combiner from a single baseline, (b) Superposition of fringes from 6 baselines (as expected in
MBI). Fringes are separated by phase m odulation sequence.
7.2
Fizeau Beam combiner
The signals from each of the input units2 (IUs) are interfered using a so-called Fizeau
beam combiner. The Fizeau combiner acts as an image-plane correlator or interferometer, as
described in the previous chapter. In our instrum ent, the Fizeau combiner is essentially a
Cassegrain telescope. All signals from the IUs illuminate the prim ary mirror, and the light is
correlated or interfered on the array of 16 bolometers at the focal plane behind the prim ary
m irror. For MBI-4 an alternative version of the beam combiner based on a 4 x 4 waveguide
Butler m atrix has also been developed and will be tested. Simulations of fringes from a Fizeau
system set-up are shown in figs. (7.6(a)) and (7.6(b)).
7.3
D etectors, electronics and data acquisition
MBI-4 uses 16 traditional spider-web bolometers, provided by JPL , w ith NTD germanium
therm istors. The bolometers are placed in an optical cavity (see fig.(7.7)) and coupled to the
2An input unit consists of an outward-facing antenna, a phase modulator and an inward-facing antenna
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
112
Figure 7.7: A spider-web JP L bolometer, with NTD germanium therm istor.
incoming radiation via 30° flare sm ooth wall conical horns w ith 2.54 cm diam eter. The horns
form a hexagonally packed array w ith spacing 2.8 cm in the image-plane of the beam combiner.
The whole unit is suspended from the supporting frame by Kevlar threads and connected to
the cold plate of the 3He refrigerator. The optical efficiency for this configuration is expected
to be ~50%.
The MBI-4 bolometers are read out w ith a standard AC-biased differential circuit. The
readout circuit demodulates the detector signals to provide stability to low frequencies (j30
mHz).
The bolometer bias and readout electronics are based on those of BLAST38.
The
preamplifiers consist of Siliconix U401 differential JF E T s w ith 57 n V / y / H z noise at z/^100 Hz
and 120 (j,W power dissipation per pair. They are suspended on a lithographed silicon nitride
membrane, using fabrication techniques similar to those used to make the bolometers and self­
heat to the optim al operating tem perature of 120 K. The total power of the JF E T s for 16
channels is only 4 m W which allows them to be placed close to the detectors. The d a ta are
read by two FPG A boards NI-7833R.
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113
7.4
Cryogenics
A schematic of the MBI-4 instrum ent is shown in Fig. (7.1) and a photograph of the
receiver is shown in Fig.(7.3). The cryostat holds 17 liters of liquid nitrogen and 25.7 liters
of liquid helium. In its operational configuration the liquid helium lasts for 50 hours. The
detectors are cooled by a self-contained 3He refrigerator m anufactured by Simon Chase. The
3He condenser is cooled by a self-contained charcoal-pumped 4He pot. The base tem perature
of the 3He refrigerator in its operational configuration is 330 mK and lasts at least 90 hours.
Cycling the refrigerator takes about one hour. The refrigerator is designed so th a t an additional
3He stage can be attached to the first 3He stage, which would provide lower tem peratures ( 200
mK).
7.5
Telescope and mount
Figure 7.8: The MBI mount.
The MBI pointing platform , shown in Fig. (7.8), consists of a fully-steer able altitudeazim uth m ount. In addition, the entire cryostat can be rotated around the optical (9) axis.
Tracking of the sky occurs under computer control using feedback from 17-bit absolute optical
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114
encoders on each of the three axes altitude, azim uth and theta. Absolute pointing is estab­
lished using a bore-sited optical telescope. This altitude azimuth mounting scheme was used
successfully on the COMPASS experiment.
7.6
M easurements 1: Analysis of data from the Faraday-Effect Phase M od­
ulator
In order to separate the interference (visibility) signals from the total power signal (see
chapter on Interferometry) detected by each bolometer, the phase of the signal from each
antenna m ust be m odulated. The phase is sequentially m odulated between -90° and +90°, and
a “lock-in” amplification is done in software to recover the signal. For MBI-4 we use ferritebased phase modulators; these waveguide devices have been fabricated by the Observational
Cosmology team a t TJCSD and are a modification of the Faraday rotators used in BiCEP [4],
The m odulation rate is ~10-100 Hz. The loss in the phase shifter is <1 dB. The magnetic
field in the ferrite is controlled by the small superconducting coil. The phase shifters dissipate
negligible power, ~1 mW each. Also, the differential loss between the two phase states must
be small. Differential loss will produce an “offset” signal after demodulation of the detector
signal.
This section discusses the tests performed on the UCSD-made Ferrite Phase M odulators
(FRMs henceforth). This work was carried out in the Electrical Engg. lab. of Prof. Dan van
der Weide w ith A. Gault [5]. It was necessary to measure not just the in p u t/o u tp u t ratio, but
also the relative phase of the outgoing phase m odulated signal. For this reason, we had to use
a device th a t can not only generate an input signal for the FRM and measure the o u tp u t/in p u t
ratio (S21 henceforth (Appendix B) b u t also measure the relative phase of the outgoing signal.
This device is called a Vector Network Analyzer (VNA henceforth) because it can measure the
v e c to r (i.e. both m agnitude and phase) of outgoing signal values. Fig.(7.6) shows an early test
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
115
of one of the FRM s in MBI. The FRM is inside the cryostst.
7.6.1
E s tim a tio n - n o losses
Let us suppose th a t we have a p e rf e c t measuring device; in particular, a perfect VNA.
In this case, the o n ly reason where loss can occur is because the phase shift is not 90°. This
is illustrated in fig. (7.10). Looking at fig. (7.10), we see th a t the fraction of the signal (in term s
of electric fields) th a t gets through is sin# where 9 is the am ount of phase shift/rotation angle.
However, the VNA measures the power ratio, so th a t the ratio of input and output powers is
OutputPow er
9„
T
—-------- = sm 9
InputPower
.
(7.1)
However, this ratio is expressed in dB ’s by the device where
S 2 1 (dB) = 101og10
= 101°gio (sin2 (?) = 201og10 (sin<?)
(7.2)
where
S 21 (dB) = 101og10 (S2i (ratio))
(7.3)
Then, the rotation angle can be extracted from S 2\ by the formula
9 = sin
in-1 (lQ T » )
7.6.2
(7.4)
E s tim a tio n w ith losses
If, however, there are losses in other parts of the set-up, e.g. waveguides, then S 2\ is no
longer a measure of the angle. We need to subtract this loss (the loss being represented by a
negative num ber in dBs) from S 2 1 >and then th a t quantity will be the true measure of rotation
angle.
The losses th a t we expect are as follows
1.
Adapter losses
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116
Figure 7.9: The Vector Network Analyzer(VNA) at the van der Weide lab at UW-Madison.
The FRM is inside the gold cryostat.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
117
O utput w /g o rientation
Faraday rotation angle
( \th e ta )
In p u t w /g o rientation
Figure 7.10: Rotation angle and how it is related to S 21
2. Waveguide losses
3. Ferrite losses
We will discuss Ferrite losses in §7.6.4 For now, we limit ourselves to correcting for adapter and
waveguide losses, which we represent by ‘adloss’ and ‘wgloss’ respectively. We stress again th a t
both these quantities - ‘adloss’ and ‘wgloss’ are n e g a tiv e numbers in dB which we subtract
from S 21 ■ After correcting for these losses, the rotation angle is given by
1
/
•S'ot —w g lo ss—a d lo ss \
9 = sin " 1 (10
----- 20
J
(7.5)
or
sin# = 10
S21 —w g lo ss—ad lo ss
20-----
( 7 .6 )
Now, the VNA d o e s n o t give us numbers in dB. Instead, for each S-param eter, it gives us the
5R
(real) and X (imaginary) parts of the ra tio s . Thus, w hat the VNA gives us is.ft (S^iratio)
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
118
and X (S 21 ratio)• We can then easily extract S 21 ratio thus
ratio = \ J (5ft (S‘2lr a tio ) ) "t" (X ( 5 2 1 r a t i o ) )
(^*^)
Obviously, we can convert this, as well as adloss and wgloss into dBs and estim ate #. However,
we do not really need to convert to dBs, because eq(7.6) can be w ritten as
1 0 'S’2idB
sin# — | 0 WgiOSSrfB-LQa<n0 ssdB
(^-8 )
However, each one of the factors on the right is really a ratio, so th a t
sin# = — ---- l^21rat’°--------wglossratioadlossratio
i.e. we ju st need to divide
7.6.3
£21
(7.9)
by the modulus of the loss ratios th at we get from the VNA.
Correcting for Ferrite loss
In principle, we need to correct for ferrite loss in exactly the same way, i.e.
a
•
^ l r a t i o
sm # — — :----------- ----------- - ---------
.
A \
(7.10)
W gl0SSra tio a d l0 S S r atio n0S S r atio
where ‘flossrati0’ is the ferrite loss as a ratio. However, it cannot be measured in any obvious
way unlike adloss and wgloss, which are measured by an adapter calibration and a separate
baseline test respectively. Instead, we need to make an estimate indirectly as follows
1. From the ferrite-uncorrected angle vs. current graph, find a current for which the rotation
angle is zero
2. Find the S n for this current, as a ratio
3. This S ii is the result of the wave traversing the entire length of waveguide once, plus
traversing through the ferrite twice (after correcting for adapter loss).
4. Subtract wgloss obtained from the baseline test from this S n
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
119
After the aforementioned operations are done, this S n is a good estim ate of twice the ferrite
loss (in dB) = square of the ferrite loss ratio (as a ratio). N ow , we are ready to obtain the
corrected 9:
s ■m a0 = — -----------
<^21ratio
1s
(7.11)
—
w S lo s s ratioa d lo s s ratioflossratio (c o r r e c te d )
This is w hat was done, and the result is in fig.(7.11).
7.6.4
O v er /u n d e r -estim a tio n o f Ferrite loss
If we pick out the current at which the phase shift angle is zero, and if S n and
S 22 are
the same, then loss estim ation is exact. However, this is rarely ever the case. In reality,
the
current chosen alw ays has some non-zero phase-shift associated w ith it. If so, we have actually
u n d erestim a ted the loss in the ferrite. Then there is the question of whether the hysteresis
loop is symmetric.
To summarize, the estim ation error could be a combination of the following factors:
1. Asymmetry in the reflection at the zero-phase angle point
2. The supposed “zero-phase angle” point not being at exactly zero angle
3. The asymm etry of the hysteresis loop because of other reasons
The following possibilities exist about point 2 above:
1. If 9 is the phase angle at the current we have chosen, S n is off by a factor of sin 9, so th at
we need a corrective multiplicative factor of 1_^ing
2. The factor in point (1) above is actually sin2 9 instead of sin 61, so the corrective factor is
1
1—sin26
_
1
cos26
3. The factor is
—^-3
COS V
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
120
This will be discussed in greater detail in [5]. Concerning points 1 and 3 above, an iterative
approach is a good solution in the absence of a detailed knowledge of the m odulator. In this
iterative scheme, we shift the hysteresis loop in every iteration until it is approxim ately centered
and then recalculate all param eters. We can continue to repeat these steps until the required
accuracy is reached.
7 .7
M e a s u r e m e n ts 2: A n t e n n a B e a m P a t te r n s
Since MBI is ground-based, it has to detect sub-pK signals in the vicinity of warm sources
(the E arth, the sky). This level of accuracy has never been studied before in any ground-based
telescope system. We need an excellent understanding of the beam pattern of MBI for the
following reasons.
1. Hu et al.
[6] have shown th a t even if the errors in the main beam are very low, the
corresponding error in m easuring polarization is huge, because the tem perature signal,
which is 2-3 orders of m agnitude higher than polarization for the CMB, “leaks” into the
polarization signal, and even a small leakage causes huge errors in observed polarization,
and changes one form of polarization into another (true especially of sidelobes, however
low). This makes it extremely difficult to extract useful cosmological information from
the data.
2. A ntenna sidelobes can couple signals from warm objects (the sky, the E arth).
These are problems th a t will challenge the next generation CMB polarization probes. For this
reason, antenna beam patterns need to be measured to exquisite precision in order to eliminate
mixing the CMB tem perature signal into polarization (to ~1 p art in 108).
Requirements of beam -m apping measurement:
The top of the MBI instrum ent is about 3 m above the ground and the antennae receive
signals in a band of a wavelengths centered on 3mm. All cosmological sources are at a distance
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
121
of many billions of light-years from earth, so th at the source for these antennae is always in the
far-held. Therefore, all beam patterns need to be measured w ith the test source in the far-held.
For a 3mm antenna with a 15cm diam eter, the far-held is ~14m away. However, placing a test
source on the ground a t this distance is not an option because we cannont tilt MBI any more
than 45-degrees from the zenith for the following reasons:
1. Our lim itation in tilt is caused by the dewar - the refrigerator th at cools the detectors
stops working well when the dewax is tipped more than 45 degrees.
2. Signals from the ground sta rt to interfere with th a t from the standard source. Since the
ground signal is signihcantly stronger than th at from the standard source, this will result
in appreciable distortion of beam measurements, even with an AC-modulated source. For
high-precision beam-mapping, we thus need to place the source about 14m above the
ground.
The signal thus needs to be conducted 14m without appreciable loss. Standard sweepers output
~0.1m W of power, and the sensitivity of MBI is 10 ~u W \/s . At ~15m , we expect an attenuation
of at most 60dB (factor of 106). If the source power is O.lmW, we expect 10“ 10 W a t the MBI.
We can therefore tolerate a maximum conduction loss of 40dB (factor of 10000) in the apparatus
th a t conducts the signal 14 m. Placing a source on the tower poses problems, since power or
frequency cannot be adjusted easily. Therefore, all the frequency and loss properties of the
conducting m aterial need to be characterized before we begin to measure the beam.
To summarize, requirements for the measurement axe:
1. 14 mappaxatus to conduct signal w ith maximum loss of 40dB
2. A towex to hold the appaxatus steady
We discuss a technique to minimize conduction loss below.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
122
7.7.1
Loss in an overm od ed circular w aveguide
7.7.2
In trod u ction
Microwave signals often need to be carried over large distances, e.g. in precision astro-
physical applications (interferometry for instance) In our case, to make precision m easurements
of beam patterns, we need to transport R F power to a tower ~20 m high. We describe a simple
technique to propagate a signal in the W -band over ~20m or more w ithout appreciable loss.
The technique is easy to implement and does not require elaborate fabrication. It involves
propagating the signal through a small section of standard WR-10 waveguide and then transi­
tioning to a wider circular waveguide (i.e. overmoding) for ~20m . We then transition back to
WR-10 and detect the loss through the entire section.
This overmoding technique depends on low-loss transitions. In order to be low-loss, these
transitions had to be smooth and gradual. We used a 2” transition from WR-10 to a 0.3” inner
diameter circular waveguide. The reason for this choice was th a t copper tubes of this w idth are
readily available commercially.
7.7.2.1
T h eory - Loss in a w aveguide at room tem p era tu re
We now calculate the loss in a waveguide th a t occurs due to the resistive element of the
waveguide material. We do this calculation for two waveguide systems:
1. Rectangular W -band waveguide made of silver
2. Circular 0.3” Waveguide made of copper
A naive first assesment would assume a lower loss in (1) above, because silver is a better
conductor than copper. We show below th a t resistive losses depend on the dimensions of the
waveguide as well as m aterial conductivity.
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123
For a section of waveguide of length z, the ratio of the amount of power in the TEjo
mode, which carries almost all the transm itted power, to the input power P q is given by
^
= e~2a*
Po
(7.12)
where a is the “attenuation constant” and is measured in N p/m ([7] p p .188). Let us calculate
the attenuation constants for the two cases, followed by an estimate of the loss through a
waveguide length of 60’ in each case. This is the maximum length for which we measured the
loss through a circular copper waveguide.
R e c ta n g u la r W -b a n d w a v e g u id e
For a rectangular waveguide, the attenuation constant is given by ([7] pp.188):
a =
B rn \
1
Z 0 ) abfiioko
(2 bk2cl0 + ak20)
(7.13)
where
Rm
=
Zq =
a,b
ko
kcio
=
Real p art of surface impedance of waveguide
Impedance of free space
Dimensions of waveguide
— Wavenumber in free space
=
Wavenumber corresponding to cutoff frequency
Ac, f c — Cutoff wavelength and frequency respectively
,/3 =
P iq =
Propagation factor
Propagation factor for the TEjo mode
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(7-14)
124
For the W -band, the param eters are as follows:
/
=
fc =
a
Centre frequency = 93GHz
60GHz
=
0.10" = 0.254cm
b =
0.05" = 0.127cm
k0 =
Pio
=
kcW =
Z0 =
— f = 1947.79m -1
c
— V P - / I = 1488.20m-1
c
— f c = 1256.64m-1
c
3770
(7.15)
W ith these values, the a tte n u a t i o n c o n s ta n t, a. is calculated to be
a = 0.303Np/m
(7.16)
The ratio of transm itted power for a 60’-long waveguide section is given by
= 1.54 x 10-5 = - 4 8 dB
(7.17)
C ir c u la r W a v e g u id e : 0 .3 ”
The attenuation constant for a circular waveguide is given by ([7] pp. 196):
a
(7.18)
where the only changes from eq(7.13) are:
Diameter of waveguide = 0.3" = 0.00762m
a
Rm
=
0.0795 for copper
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(7.19)
125
W ith these values, the a tte n u a tio n co n sta n t, a is found to be
a = 0.0121Np/m
(7.20)
for a 0.3” circular copper waveguide. The ratio of transm itted power for a 60’-long waveguide
section is given by
£*“ > = g —2 xO.0121 X 18.288 = q g 4 2
= _l_Q2dB
(7.21)
Po
7.7.2.2
M easu rem en ts, d a ta and con clu sions
The two transitions were attached together to measure the loss through them .
This
can then be subtracted from the d a ta to get an estim ate of loss through ju st the 0.3” tube
section. Raw
d a ta from experiments is shown in fig.(7.14). It is clear th a t the loss increases
monotonically w ith the length of the copper tube at all frequencies.
Fig.(7.16) shows the average loss per unit length calculated from the sm oothed data. The
net loss is about 1 dB per 10 feet of tube length; however, this estim ate holds for frequencies
below ~105 GHz.
Fig.(7.17) shows the signal in a small frequency range (90.0-90.4 GHz). Notice th a t the
frequency interval between resonances decreases w ith increasing tube length. Calculations [8]
show th a t these frequency intervals correspond exactly to what is expected for the corresponding
tube lengths.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
126
1 r
)K
i
i
i
i
i
i
O
O
r
*
*
o
o
CM
*
IN DEGREES
VS. C U R R E N T
~r
*
LlJ
yn
o CC
*
Cd
ZD
CD
*
SHIFT
*
CD
O
PHASE
CM
)K*
J
1
I
I
I
I
_J
I
I
I
I
1
o
I
I
O
UO
I
LO
I
I
L_
o
o
M"
I
S33d03Q Nl 330NV
Figure 7.11: R otation angle vs. current, corrected for Ferrite loss, as described in the text.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
127
Figure 7.12: The WR-10 to 0.2” transition (gold) connected with an adapter which then con­
nects to the circular copper tube.
Scalar Network Analyzer
Signal Generator
60' Copper tube
WR-10 to 0.2" preciaon transition
Frequency multiplier
Figure 7.13: Schematics of the planned antenna beam test.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
128
Raw Data
b aseline
10ft
20ft
30ft
40ft
SOft
60ft
-3 5 -----------------------------------------------------------------------------------------------------------------------------------------------
Freq(GHz)
Figure 7.14: Raw d a ta from the tube test for pipes of different lengths. The oscillations are
caused by standing waves in the pipes. Notice th a t the signal from different lengths decreases
monotonically w ith increasing length.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
129
Smoothed Data
Power (d B )
— baseline smooth
10ft smooth
— 20ft smooth
— 30ft smooth
— 40ft smooth
— 50ft smooth
— 60ft smooth
Freq (GHz)
Figure 7.15: The same data as in fig.(7.14), but with resonances smoothed out.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
130
Power Loss/1 Oft (dB)
Power Loss per 10 ft of Tube
Freq. (GHz)
Figure 7.16: G raph of loss per 10 feet derived from smoothed data.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
131
Resonace Graph
90.05
90.1
90.15
90.2
90.25
90.3
90.35
— ion
-18
— 50ft
basehne
Freq (GHz)
Figure 7.17: Resonances in the d a ta in a small frequency range. These are consistent with
standing waves in the tube lengths used.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
132
Bibliography
[1] P. T. Timbie, G. S. Tucker, P. A. R. Ade, S. Ali, E. Bierman, E. F. Bunn, C. Calderon,
A. C. G ault, P. 0 . Hyland, B. G. Keating, J. Kim, A. Korotkov, S. S. Malu, P. Mauskopf,
J. A. M urphy, C. O ’Sullivan, L. Piccirillo, and B. D. W andelt, “The Einstein polarization
interferometer for cosmology (EPIC) and the millimeter-wave bolometric interferometer
(MBI),” New Astronomy Review, vol. 50, pp. 999-1008, Dec. 2006.
[2] C. L. B ennett, R. S. Hill, G. Hinshaw, M. R. Nolta, N. Odegard, L. Page, D. N. Spergel,
J. L. Weiland, E. L. Wright, M. Halpern, N. Jarosik, A. Kogut, M. Limon, S. S. Meyer,
G. S. Tucker, and E. Wollack, “First-Year Wilkinson Microwave Anisotropy Probe (WMAP)
Observations: Foreground Emission,” A p J Suppl, vol. 148, pp. 97-117, Sept. 2003.
[3] J. Bock, S. Church, M. Devlin, G. Hinshaw, A. Lange, A. Lee, L. Page, B. Partridge,
J. Ruhl, M. Tegmark, P. Timbie, R. Weiss, B. W instein, and M. Zaldarriaga, “Task Force
on Cosmic Microwave Background Research,” ArXiv Astrophysics e-prints, Apr. 2006.
[4] K. W. Yoon, P. A. Ade, D. Barkats, J. O. Battle, E. M. Bierman, J. J. Bock, H. C. Chiang,
C. D. Dowell, L. Duband, G. S. Griffin, E. F. Hivon, W. L. Holzapfel, V. V. Hristov, B. G.
Keating, J. M. Kovac, C. Kuo, A. E. Lange, E. M. Leitch, P. V. Mason, H. T. Nguyen,
N. Ponthieu, and Y. D. Takahashi, “R eport on B IC E P’s First Season Observing the Cosmic
Microwave Background from South Pole,” in Bulletin of the American Astronomical Society,
Dec. 2006, vol. 38 of Bulletin of the American Astronomical Society, pp. 963—K
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
1 33
[5] A. C. Gault and S. S. Malu,
“A measurement of the Faraday-effect Phase m odulator
performance,” In preparation, 2007.
[6] W. Hu, M. M. Hedman, and M. Zaldarriaga, “Benchmark param eters for CMB polarization
experiments,” Phys. Rev. D, vol. 67, no. 4, pp. 043004—
Feb. 2003.
[7] R. E. Collin, Foundations for Microwave Engineering, W iley-IEEE Press. ISBN-13 9780780360310, 2000, X III + 944 p. 2nd ed., 2000.
[8] L. Levac, S. S. Malu, and P. T. Timbie, “Loss in an overmoded circular waveguide over
medium distances,” In preparation, 2007.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
134
C hapter 8
Sim ulations of th e CM B sky and th e M B I
Instrum ent
In chapter 7, we described the MBI instrum ent in detail. Before the MBI can be p u t to use in
CMB observations, though, we need to know:
1. its response to a simulated CMB sky, in order to perform checks on its various parts
2. its response as a function of t, which depends on its antenna beams patterns.
To achieve this, we need the following calculations/simulations:
1. simulation of the CMB sky over a patch as large as the MBI beam
2. a calculation of the Window functions of MBI for C x i where X = T. E, B
3. simulation of the MBI instrum ent
In addition, we need to describe the analysis of data from the FRM.
We describe these three calculations/sim ulations below.
8 .1
S im u la tio n o f t h e C M B s k y p a tc h
As described in §5.6, the power spectrum is a statistical description of CMB anisotropies.
T h a t is, it does not contain information about the amount of power in every single anisotropy
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
135
over the sky as a function of position. Instead, it tells us how much power there is in the
anisotropy a t a given angular scale. This can be pictured as follows. Imagine a point on the
sky, say 0 = 0, i.e. the NCP. Now consider all the points (ideally, infinite, but practically, a
large number) at some 0 ^ 0 . If we compare the tem perature at 0 = 0 w ith the tem peratures
at points 0,4> = 0 —> 2ir (i.e. find the angular two-point correlation function or the power
spectrum ), and take the standard deviation, we end up with the value of the power spectrum
at £ = jj. Thus, information in a CMB m ap is “compressed” , so to speak, to form the power
spectrum.
So if we are given a set of Cosmological param eters, and therefore a power spectrum , which
can be calculated via software packages like CMBFAST[1], we need to “add a dimension” to
it in order to get a sim ulated map. But it isn’t possible to just generate random numbers to
get the tem perature of the points at 0, <j> = 0 —> 2tt, w ithout knowing anything else about the
CMB. There is one property of the CMB th a t we have not recalled yet - its gaussianity. If we
include this property, we need to do the following in order to generate a simulated m ap of a
patch of the CMB sky:
1. Generate N (depends on the desired resolution of the simulated map) gaussian random
numbers with unit variance for every angle and therefore every I.
2. M ultiply the vector containing the random numbers with the value of \[CTt (the standard
deviation)
3. Repeat the above steps for all values of 0 - this forms a map in fourier space
4. Take the inverse F T of this m ap to get a m ap of CMB anisotropies in real space
Since only a small patch of the sky is observed, the curvature in this patch may be ne­
glected, and this is also the reason we can use fourier transform s instead of spherical harmonics.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Under this assumption, this is how fourier decomposition works[2]:
(a* (u) a (u')) — S (u ) 5 (u —u')
(8.1)
where
( 8 .2 )
and u = 1/ (2n).
The fourier definitions are as follows:
e - 2 * i u x d 2x
(8.3)
a (u) e-2*iv*d2u
(8.4)
a (u) =
/ a (x)
a (x) =
J
for forward and
for reverse FT.
To generate a small CMB map, we first s ta rt w ith a power spectrum - the one used is
shown in fig.(8.1).
We then derive the fourier transform of the m ap ajnp in the following way:
rr {C F )
(8.5)
and then take the inverse fourier transform to get the real map, shown in fig.(8.2).
Q and U maps are also shown below in figures (8.3) and (8.4) respectively.
We can also generate the m ap we should expect to see with an ideal (no noise) interfer­
ometer, given 6 baselines (like the MBI) - this is shown in fig.(8.5).
We can also perform a very basic check on the maps generated as follows. As described
in [Knox], the error bars expected on the power spectrum , given an ideal instrum ent observing
a fraction of the sky
f
s k y
are
(21 + 1) fs K Y
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
( 8 .6 )
137
POWER SPECTRA USED TO SIMULATE MAPS.
6x10
0
500
1000
1500
Figure 8.1: The power spectrum used to generate the simulated maps shown below. This was
obtained by choosing a set of cosmological param eters in CMBFAST[1].
To perform this check, we recover the power spectrum from the simulated map, w ith the given
mapsize, and compare with the formula above. In fig. (8.6)
8 .2
S im u la tio n o f t h e M B I I n s tr u m e n t
Aim: “observe” a simulated CMB sky patch w ith MBI-4 and recover bandpowers for
different baselines, given a nearly ideal instrum ent, i.e. no noise or system atic effects.
8.2.1
In terferom etry
In §8.1, we discuss how a polarization interferometer works and the relation between
observable quantities (Stokes’ T, Q, U and V) and sky signal. In section 2, we calculate w hat
we see at one frequency by integrating over the field-of-view. In section 3, we integrate over the
bandw idth th a t the antenna / waveguide system and presumably the detectors (in the case of
the MBI, the bolometers) are sensitive to. For MBI, we needn’t worry - spider-web bolometers
being used are not sensitive to any particular bandwidth. Sections 1 through 3 are general and
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
138
6
4
2
0
-2
-4
-6
-6
-4
-2
0
2
4
6
Figure 8.2: The tem perature m ap obtained from the power spectrum above and the m ethod
described in this chapter. The size of the m ap is in degrees, indicated on the two axes. Tem­
peratures are in K.
can be applied to any interferometric observations of CMB polarization.
Section 4 describes the beam combiner system being used in MBI, and calculates the
phase difference between two rays from two different antennas, i.e. it calculates the fringe
p attern produced at the focal plane by one baseline.
Notation: 9 and <j>represent a direction on the sky, and i/j is an angle inside the cryostat. e
and S are phase differences of a “pixel” on the sky and a position in the focal plane respectively.
These will be useful later, x denotes an orientation of the instrum ent.
We follow here the discussion in §5.7, and write the output electric fields of the two horns
as
E i = E xx. + E yy
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(8-7)
1 39
6
4
2
0
-2
-4
-6
-6
-4
-2
0
2
4
6
Figure 8.3: Q m ap obtained from the power spectrum above and the m ethod described in this
chapter.
E 2 = (Exx. + E vy) eie
(8.8)
In general, waveguides can be coupled to some combination of linear polarizations, so:
E i = axE x-k + a2E yy
(8.9)
E 2 = (b1E xii + b2E yy ) e ie
(8.10)
If a2 = b 2 = 0, then linear polarization is chosen; if
then circular polarization is chosen.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Figure 8.4: The tem perature m ap obtained from the power spectrum above and the m ethod
described in this chapter.
The Stokes’ param eters are defined as follows:
T
=
(\E x \2 + \Ey \2)
(8.11)
Q
=
(\EX\2 - \Ey \2)
(8.12)
U
=
m ( E * xE y))
(8.13)
V
=
(21 (E*Ey))
(8.14)
in term s of the Stokes’ parameters:
|£ * |2 =
\ ( T + Q)
(8.15)
\Ey?
\(T -Q )
(8.16)
=
= \ ( U + iV )
(8.17)
= \(U -iV )
(8.18)
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
jH=H
A N G L E IN D E G R E E S
Figure 8.5: The tem perature m ap th a t a 6-baseline ideal interferometer is expected to output,
given the sky map shown in fig.(8.2).
In general, the multiplying interferometer works in the following way. The two electric
fields are first added using a beam combiner (in the present MBI configuration, this is the
Fizeau scheme) and then detected on the focal plane. W ith no other phase differences, e.g. at
exactly the middle of the focal plane the output at the detector will be {E\ + E 2) (E * + E^).
However, there will be an additional phase factor due to the difference in p ath length between
the two paths to the focal plane from the two antennas, as shown in figure 1. There is a relative
phase of e due to the position of the two antennas looking towards the sky and 5 between the
rays from antenna two and antenna one inside the cryostat, and therefore between E \ and E 2.
Now recall th a t the p art of the detected signal th a t has been phase m odulated is oc EiE%
and its conjugate. Let us work out the detected quantity explicitly:
(Ex + £ 2ei(<5+£)) ( E l + £ £ e -<(5+£)) = |Fd|2 + \E2\2 + E x E ^ e ^ 5^
+ E l E 2ei{~5+^
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(8.19)
142
1000
1500
Figure 8.6: This is a basic check of the m ap in fig.(8.2). The curves on the top and bottom
indicate the 1-cr error bars expected from eq.(8.6), and the marked points make up the recovered
power spectrum. Note th a t the vertical scale is different from the power spectrum in fig.(8.1).
The first two term s are easily evaluated:
l ^ l 2 = E ^ l = \a i \2\Ex \2 + \a2\2\Ey \
( 8 .20 )
\E2\2 = E 2E l = \ h \ 2\Ex \2 + \b2\2\Ey \2
(8.21)
We can substitute for \EX\2 etc. from the above equations to get
\Ei\2 =
l [ T ( \ a 2\ + \ a 2\) + Q ( \a 2\ - \ a 2\)}
(8.22)
m 2 =
\ [ T { \ b i \ + \ b 2\ ) + Q ( \ b 2\ - \ b 2\)]
(8.23)
If we want to study interference, we wish to look a t only the last two term s, which will
have been phase-modulated. However, they are ju st complex conjugates of each other.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
So we
143
need evaluate only one, and the other will follow. W LOG, we consider the last term:
( E t E h j W } = ^ e i(5+e) [a\b i {T + Q) + a2b*2 (T - Q ) + a ^ 2 (U - i V ) + a*2bi (U + iV)}
(8.24)
Simplifying,
( E l E 2e i{5+€)Sj = i e i(5 + e ) x
[(a{bi + a\b2) T +
- a*2b2) Q + ( a ^ 2 + a*2h ) U + i (a^h - a\b2) V]
(8.25)
Similarly,
x
[(a i bt + a2b2) T + (a ib\ - a2b2) Q + (a ib*2 + a2b\) U - i {a2b\ - a ib*2) V]
(8.26)
We need to rem ind ourselves th a t the four quantities T, Q, U and V already have the effect of
the prim ary antenna beam included. Ju st so we are clear, let us replace T etc. by T where
T = A ((f), 6) T etc. thus:
(jE JE a e " ) = \ e i{S~e) [Kfcj + a*2b2) T + ( a ^ j - a*2b2) Q + (a\b2 + a*2h ) U + i (a*2h - a\b2) V]
(8.27)
and
=
l e - i{S~ £) x
[(aibl + a2b*2) T + (afy? - a2b*2) Q + (a ib*2 + a2b\) U - i (a2bl - a ib*2) V]
(8.28)
We need to assign one kind of polarization, i.e. either linear or circular, in order to figure
out the sum of these two quantities. Let us consider the case of linear polarization first, where
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
1 44
X = (“?=?) and y = (£=?) for E i and x = (£=?) and Y = (£=°) for E 2, so th a t
(E .E ie -* ) ^
=
l- e- ^ ) { r + Q )
(8.29)
(E rE le ^ )^
=
l- e~ ^
{T - Q)
(8.30)
(e . E * ^ ) ^
=
| e - ^ - £)(W + tV)
(8.31)
( E xE*2e - i5) Y x
=
l- e- ^ { U - i V )
(8.32)
And similarly, the complex conjugate term gives us
( E l E 2ei5) x x
=
l j V ~ ‘) ( T + Q )
(8.33)
( E \ E 2ei5) YY
=
I e* ( « - 0 ( r - Q )
(8.34)
( E l E 2ei5\
=
\ e ^ ~ ^ { U + iV)
(8.35)
/ £ 1* £ 2eW')
=
\ e ^ s- ^ { U - i V )
(8.36)
\
/ XY
\
8.2.1.1
/ YX
2
2
A p p lica tio n to sim u lation s o f tim e-ord ered d a ta (T O D )
In an interferometer, the only diference between the electric fields at the different antennas
is a phase factor th a t depends on the p ath difference between the photons th a t arrive at those
antennas. Therefore, we need to find the electric field at only one antenna; the field at the
others will follow easily.
Following equations 9 and 10, we get for the two components of the electric field:
m
2 =
\ ( T +£)
\E y\2 =
(8-37)
(8-38)
It seems a little strange a t first th a t the electric fields be maps instead of ju st a num ber, but
we have to remember th a t they do not get added/averaged over until they reach the detectors.
A t the detector, they are summed over w ith all the appropriate phase factors.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
145
8.2.2
I n te g r a tio n o v e r t h e field-of-view (F O V ) / s k y p a tc h
This is not a single step, since there are several things involved:
1. Q a n d U a s fu n c tio n s o f o r ie n ta tio n y: As the instrum ent is rotated, the response
from every baseline changes - Q and U are functions of orientation angle y - this relation
is described in the appendix.
2. R e la tiv e p h a s e o f e a c h p o in t o n t h e sk y : Every antenna’s position is a point, and
the distance of every point in the sky patch / field-of-view to th e antenna is different;
consequently, each point on the FOV has a unique phase associated w ith it, which needs
to be calculated. In other words, we need to calculate the exact functional form of e,
which is a function of <p. 0 and x
Let us perform each operation, one by one. Before performing integration, though, we need to
get the units of each quantity right. The units of the incoming power are W m _2Hz-1 Sr_1.
8.2.2.1
I n te g r a tio n
We can simply integrate over an “area” on the sky. An “area” on the sky is given by
fQ2 r<t>2
A = I
/
sin 0d<j)d0
J&=$i J4>—<f)i
(8.39)
where 0\ and (l>\ can have any value depending on which p art of the sky we are looking at, and
02 —9i and <j>2 —4>\ are determined by the FOV - these will be calculated in sub-section 4 (to be
added later). We want to integrate the last two term s in (8.19); let us call the “interm ediate”
visibility V - this is clearly a function of the orientation of the instrum ent, y. So
(8.40)
0 = 0 i J <)>=4>i
The units of this “interm ediate” visibility are W m 2Hz x, since we have integarted over the
solid angle.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
146
8.2 .2 .2
T h e r ela tiv e p h ase difference
Look at fig.(8.8). This
e
figure shows the orientation of a baseline w.r.t. the co-ordinate
system. As the instrum ent (and therefore the baseline) rotates, the two points labelled ‘A ’ and
‘B ’, i.e. the two antennas th a t form the baseline, also rotate. Their position a t an orientation
angle x is given by
x\
= xi cos x + Vi sin x
(8-41)
y[
= - x i sin x + yi cos x
(8.42)
x'2
= X2 cos x +
(8.43)
y2
= —x 2 sin x +
2/2
sin x
2/2
cos x
(8.44)
The distance of each antenna from the origin remains constant, though, so th a t the two quan­
tities
bi = y/A + vl
=
b2
\Jx\ + y\
(8-45)
(8.46)
rem ain constant. For w hat we are about to do, it is useful to define the unit vectors along the
direction of the antennas thus:
=
^ ( x ' 1,y[, 0)
(8.47)
b2 =
^ {x'2, y 2,0)
(8.48)
br
02
W rite out all the quantities explicitly:
bi
=
i ( x i c o s x + y i s i n x ,- a : is i n x + 2/icosx,0)
0\
(8.49)
b2 =
— (x2 c o s x + y 2 s m x , - x 2 s m x + y2cosx,0)
(8.50)
02
Let us also write out the unit vector at a point on the sky:
r = (cos (j) sin 0, sin <f>sin 6, cos 6)
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(8.51)
147
Now,
look at figure(to be drawn) - the two antennas clearly form
a triangle w ith the point
on the sky under consideration. Since we know the unit vectors to all the three points on the
triangle, we can find the three angles a, a\ and a2:
cos ai
=
b i •r
(8.52)
cos a 2 =
b 2 •?
(8.53)
( f - b x ) • ( f - b 2)
i
„
| f - b i | | r - b 2|
(8.54)
cos a
=
The path difference can then be calculated with the help of the sine identity for triangles
(suggested by Peter H.):
B
so
si
- — = -r* — = - 7 ^ —
sm a s m a j
s in a 2
,
8.55)
so th a t the p ath difference is
B
s2 —si = — (sin ax —s in a 2)
sm a
(8.56)
_ 2-k B (sin ai - s in a 2)
e — — ------------ :—-------A
sm a
.
.
(8.57J
and the phase difference is
Since each one of the angles a, ai and a2 is a function of <f>, 9 and x, e = e ((?), 9, x).
We are now ready to write the integral over the FOV. For simplicity, let us not write the
normalization or the integration limits for now:
V (x ) =
8.2.2.3
JJ
E*1E 2ei(5+e^ ' 9’x)) + E ^ e - ^ + ^ ' ^ s m O d ^ d e
(8.58)
L im its o f in teg ra tio n
The most convenient thing to do is to assume th a t the centre of the FOV has 9 — 0, and
then, 92 - 9i =
(f>2 - 4>1 = 2-tt
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
148
8 .2 .3
Interferen ce p a ttern in focal plane
First,
let us recall th at we have an antenna radiating out to the focalplane.
We m ust
account for the prim ary beam of the antenna again, i.e. we must multiply our result w ith the
antenna beam.
Now, notice th a t it really does not m atter which configuration we wish to work out; they
will have the same factor of
e i(S~e) + e - i ( S - c ) = 2 cog ^
(g 5 9 )
A ntenna beam can be w ritten generically as
A(<f>) = e
(8.60)
2CT2
where
a =
FWHM
(8.61)
y/2 In 2
However, when we calculate the pattern due to a baseline at a single point in the focal plane,
we must remember th a t the
signal at th a t point is coming from two different antennas, and the
beam p attern of each antenna at th a t point is, in general, different. We label the antennas by i
and j and each one of them has a beam th a t is a gaussian. We will evaluate each angle (pi and
(!>j separately in section 4. For now, we bring together all the factors for finding an expression
for the interference pattern as below.
<6?
4>^
4>?+4>2Atj = e ~ ^ e ~ ~ ^ = e 2 ^ "
(8.62)
The net interference pattern then is
I (x,y) oc 2A {(pi, <f>j) cos (<5 —e) = e
2^2
cos(<5 —e)
(8.63)
If we wish to calculate the p attern for a single pointing, we could include the receiving antenna
beam as well:
I (x , y) oc 2A (cpi, <pj) A (6) cos (<5 —e) = e
n + fi
^2e
i?2
2^2
cos (§ _
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(8.64)
149
8.2 .4
E ffect o f fin ite b an d w id th
In the frequency range v —> v + d v , the proportion of intensity th a t an instrum ent receives
is P (v) dv where P (is) is the Planck Brightness Function. The total amount of energy th a t a
detector receives in a bandw idth is then
[V2
/ ( i q , i / 2)oc / f ( v ) P ( v ) d v
J V\
(8.65)
where / (is) is the interference p attern above. The final expression isthen
Looking at the document ‘distribution.pdf’, all we need to do isweigh the interferenceterm in eq(8.58) function w ith the distribution function
=
(8‘66)
and divide by the integral
fCz2
L
ZS
(8-67>
' Z1
where z\ =
and z2 =
. Therefore, changing variables to z =
in the above expression
for I (i/i,t/2), we get
f^V ix )
)dz
H w 2;x) = - - - 7 ^24 : . —
(8.68)
Jzi ez - l dZ
or, equivalently, in a short form,
f f 2 V (x) P (z) dz
I (vi,V 2 ',x) =
1 rz2 p , > .
JZ1 p (z ) d z
(8.69)
In its full glory, the expression is
f Z2 f f E *E 2ei(s+£tt'e’X» + E i E Z e - ^ + ^ ’XV sin QdcjsdO (
dz
H » u » r , x ) = - -------------------------------------------------------------------------------------J z i e z—
1
We will evaluate an expression for 6 as a function of z in section 4.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(8.70)
150
8.2.4.1
D ep e n d en ce o f F W H M on freq u en cy
The above expression for the fringe p a tte rn is not final. We have to take into account the
FWHM of the beam and the fact th a t F W H M ~
where D is some aperture w idth associated
with the antennae. In other words, F W H M (v) ~ i . All our beam pattern measurements have
been at 90 GHz (more generally, the central frequency, call it uc)\ therefore,
FW HM (i/)
FW HM (v = vc)
=» FW HM (!/) = — • FW HM (v = vc) =
(8.71)
• -F W H M (v = vc)
(8.72)
The same expression will then hold for a, since it is related to F W H M by a constant factor:
(8.73)
Now,
= D (say), and call a (v = vc) = ag
D
=S> a (y) = —ctq => <r
(8.74)
The net interference pattern then is then
I (x, y) oc 2A (fa, <j>j) cos (8 — t) = e
202^2o
cos (<5 —e)
(8.75)
and the final expression becomes
(8.76)
8.2.5
Im p lem en ta tio n o f form alism t o th e in stru m en t
The foregoing formula is difficult to implement in the case of an actual instrum ent, because
it contains two phase angles instead of positions of the antennas or the detectors in the focal
plane. Let us consider the internal antennas first, and let us define a convenient co-ordinate
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
151
system in the following way. Choose a point on the focalplane and call th a t the origin. Let
the optical distance from the antennas to the focal plane be d. Let the antennas be labeled by
the numbers i and j (we need two labels since this is an interferometer and the basic unit is
one baseline, i.e. two antennas). Then the position of one of the antennas is specified by the
vector
rm = (xi,yi,d)
(8.77)
(the notation rtb may seem a little intriguing; after all, w hat it means is “position of the ith
bolometer” ; however, we will need to define another vector
r;0which represents a point on the
focal plane with the same (x, y) co-ordinates as the bolometer - we will need this to calculate
angles). Positions on the focal plane are specified by
r = (x ,y ,0 )
(8.78)
The path length from one antenna to any point in the focal plane is then given by
|rjb - r | = ((x - X i f + { y - y i f + d2^
(8.79)
For the secong antenna in a baseline, similar equation can be w ritten down:
lrjb - r| = ((x - X j f + ( y - y j f + d2^ f )
(8.80)
Then, the p ath difference between the two antennas in one baseline is:
kib - r | - |rjb - r | = ((x - x;)2 + (y - y{)2 + d2^ / } - ((x - x j f + (y - y j f + d2^ / } = r ^
(8.81)
The “phase angle” associated with the p ath difference i\j is
5 -
— nj
( 8 .82 )
This is the definition of cj>th a t needs to be substituted in the last equation in the previous
sub-section.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
152
nb = (a*, yi)
r jb = ( Xj , Vj )
ANTENNAE
Path difference
FOCAL PLANE
0
r
f + V x 1 + y2
Figure 8.7: Schematic of the Quasioptical beam combination set-up inside the cryostat
8.2.5.1
C alcu lation o f A n gles
Using the above geometrical setup, we can also evaluate the angles introduced in section
3,
cf>i
and
<f)j.
Looking at figure 1, we can define two vectors th a t represent two points w ith the
same (x , y ) co-ordinates as the two bolometers respectively, but both of them on the focal plane.
Let us call these vectors r i0 and r j 0 ; their co-ordinates are (x i,y u 0) and (X j ,y j , 0) respectively.
Again, looking at figure 1, we can figure out the angles
4>i
and
(j)j
w ith the help of the two
vectors we ju st defined. Notice th a t the two vectors r i b — r io and r j b —r enclose the angle (pi
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
153
( 0 ,0 ,0 )
Figure 8.8: Schematic of the Quasioptical beam combination set-up inside the cryostat
between them . We can therefore easily figure out the cosine of the angle:
^
cos fa =
( r i b - r i 0 ) • ( r ib - r )
-------------—j---------r i b - r i0 • r j b - r
fo c o ^
(8.83)
But rjb —rjo = d so th at
cos-' (r‘» ~ r ; ° ) ' (r"’ - r)
d • |r ib - r |
(8.84)
Similarly,
(r jb - rj 0 ) • (r Jb - r )
3
-------jd—i
---------• |rjb
- r| i------
(8.85)
We can now substitute the values of S, <jjtl (f)j and e (which depends on the baseline and pointing).
Given a set of n antennas and their positions, we can cycle between all the baselines to get the
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
1 54
net interference pattern as a function of x and y, the co-ordinates on the focal plane.
8 .2.5.2
S ize and p lacem en t o f b olom eters
Let the placement of bolometers on the focal plane be characterized by
X f, : ,
and let their
lengths along th e x and y axes be a and b respectively. To get the signal from one bolometer,
we need to integrate the expression for the signal as a function of x and y over this range.
8.2.6
R ecovery o f Cg from in stru m en t sim ulation
8.2.6.1
S im u lation
Let us sta rt w ith the expression for the output at the Fizeau combiner’s focal plane:
I(v UWX) =
P 2 f f E ?E 2ei(s+e^ ’d’x» +
sin dd^dd ( - ^ ) dz
-------------------------------------------------------------------------"...Z .....
4 7Z=i*z
(8‘86)
Excluding normalization, and denoting Planck distribution effects as P(z),
I
x) = j * 2 j
J
E t E 2ei(s+cM ' x)) + E 1E Z e -iV+eM ' x'» sin0d(j>d9P(z)dz
(8.87)
In the actual simulation, the / ’s are replaced by X /s:
I {V\,V2\X) =
E E
z
E *E 2ei(-s+^ ’9’x)) + E 1E^e-'i(-s+t(-<t>’d'x^ A n P ( z ) A z
( 8 .88)
n
Let H A = collecting area of the horn antenna; F P A = area of the focal plane. Let (x, y) be
co-ordinates on the focal plane. Then,
HA
O {yx ,v 2\ x; x, y (N D); N B ) = | ^ S U M
(8.89)
where
SUM =
E E E
x,y
where (x,y )
z
E l E 2ei{s+e('<t>’e’x ^ B)) + E i E l e - i^ +t^ ' e’x ’NB))A Q .P { z )A z A x A y
(8.90)
P
specify the position of one detector, x 1S the orientation of the instrum ent and
N b and N o specify a baseline and a detector respectively; O is the output at a detector for a
particular baseline and orientation.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
155
To recover visibilities, we need to “undo” the effect of all the factors above. Let
Qo
= Solid angle on the sky observed by the instrum ent
Da
= Area of one detector
P (v) Az/
= Net Planck factor
^ (x ,y (N o ))
/ sky
= Net phase introduced inside Fizeau combiner
=
Fraction of sky covered by instrum ent
(8.91)
Then, the visibility V for a given baseline, detector and orientation is given by
w , . AT
v fc n k
¥ ^
b) =
x; x ,y { N D); N B ) / F P A \
n o D A e .» (WD)
(h a )
0 { y x ,V 2 \
n
(8-92>
To obtain an estim ate of the power spectrum , Cg, recall from §5.6 the relation between
visibility and power spectrum:
K 2T 2
=
CeJ dnJ
dn'Ai
Ai
(n 0
Pl (n
'
ei27r(“i'n- ^ ' n')
(8.93)
i
or, equivalently,
( ViVi )
v '
f2t+l\
(8-94)
An estim ate of the power spectrum is then obtained by
C( = ( i m
)
< v fe ;
N
a
N
b
) V ' {x
'■ N k Nb})
(8' 95)
where we have completely disregarded
1. The antenna beam and therefore the window function, which is assumed to be a deltafunction above.
2. Finite sky coverage.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
156
An estim ate of errors on these estimates for the power spectrum is also needed, to find
out whether the recovered values of Q are consistent w ith the values th a t the sim ulation used.
A very small fraction of the sky is used, so we expect the cosmic variance to be high. Let us
list the various errors on this estim ate and find out how they contribute to the net error:
Finite sky coverage
/ s k y (2^ + 1 )
Ci Sampling variance
Simulation sampling variance
: J —-—
V Apix
(8.96)
Errors due to finite sky coverage and finite num ber of instrum ent orientations are couple
to each other, whereas the num ber of pixels on the simulated sky is independent:
" S E T = /sKY(2< + l ) S * + j v S ;
(8-97)
The plotted error bars are given by
" " ET =
/ sk y(2 < + 1 )
A sample recovered power spectrum is shown in fig.(8.11).
( 8 '9 8 )
Notice th a t the normalization
is different from the power spectrum th a t the simulation started out with.
The recovered
spectrum , does, however, present the same features as the input power spectrum . The beam
of each antenna was assumed to be a “top-hat” , which leads to a window function w ith large
sidelobes w ith which bandpowers are convolved. This convolution has not been reversed in
the recovered spectrum . Also, it was not possible to make baselines th a t would correspond to
I >200 because of the size of the antennae.
This simulation shall be extended to dem onstrate the u-v plane spectral resolution ability
of the Fizeau combiner.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
157
8 .2 .6 .2
Sim u lation p aram eters
Frequency Range
93 - 94GHz
A ntenna 1 position(in cm)
( - 5 ,- 5 )
Antenna 2 position(in cm)
(1 0 ,-5 )
Antenna 3 position (in cm)
(4,6)
Antenna 4 position(in cm)
( - 8 ,3 )
Baseline Lengths (in cm)
Cooresponding values of i
15.0,14.2,12.5,8.5,19.7,12.4
147,139,123,84,193,121
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(8.99)
158
Power S p e c t r u m for simulation
8000
4000
3000
0
200
600
40 0
I
Figure 8.9: The power spectrum used for the simulation.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
800
1000
1 59
T—1
T—1
1
o
+
u
05
O
CO
o
+
H
CO
C\2
c\i
o
+
H
CO
CO
T—<
o
o
+
m
02
in
o
o
+
00
C\2
CO
1
1
o
+
H
CO
T—1
T—1
1
H
o
+
H
CO
q
oi
1
II I I I I I I I | I I I I I II I I | II I I I I I I I | I I I I I I I I I | I I I I I I I I I j M I I i m i i j
co
.................................II .............................I ....................................I I I I I I I I I II ................................... I I I I I M I I l l
CO
co
a
—i
o
—i
cq
co
1
Figure 8.10: Tem perature m ap from the power spectrum shown in fig.(8.9) above. Used as
input for the instrum ent simulation. Tem perature anisotropies are in /j,K.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
160
"i,T r r r » : T T T ,| " m " ,r r i T T T |
TTTT.il...r|.T.p,.r.rrT
T T T T J T T T T T T T T ”q T T T T T T T T T
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Figure 8.11: Recovered power spectrum from the Fizeau system simulation.
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161
Bibliography
[1] M. Zaldarriaga and U. Seljak, “CMBFAST for Spatially Closed Universes,” A p J Suppl.,
vol. 129, pp. 431-434, Aug. 2000.
[2] M. W hite, J. E. Carlstrom , M. Dragovan, and W. L. Holzapfel, “Interferometric Observation
of Cosmic Microwave Background Anisotropies,” ApJ, vol. 514, pp. 12-24, Mar. 1999.
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162
Chapter 9
CM B D ata A nalysis
For almost three decades after Penzias and W ilson’s discovery, the task of finding anisotropies
in the CMB remained a challenge.
COBE brought about a revolution when it reported a
detection on a 7° angular scale. A num ber of smaller instrum ents focused on smaller angular
scales followed, and in a short period of time, the size of the datasets exceeded capabilities of
the techniques used to analyze the data. Developement of analysis techniques has been the
prime focus of theorists in CMB ever since. In addition, experiments like DASI have upped the
stakes because they use interferometry. In a way, interferometry has some advantages, since it
enables sampling directly from fourier space, i.e. directly from the I-modes themselves, which is
precisely w hat we aim for in power spectrum estimation. However, interferometers come with
their own challenges.
In this chapter, we start with a concise discussion of linear m apmaking techniques in §9.1.
We then move to Bayesian Maximum-likelihood Analysis of interferometry d a ta to recover C^’s
and show th a t a full Bayesian approach is com putationally unfeasible.^In section 4, we explore
a novel approach to likelihood analysis th a t enables computationally efficient calculation within
a fully Bayesian framework, without having to make approximations. This is the first tim e this
technique (called “Gibbs sampling”) has been applied to interferometry. We then present results
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163
and show th a t Gibbs sampling as used here is indeed robust by applying the Gelman-Rubins
test for convergence.
9 .1
9.1.1
M a p m a k in g
T h e g e n e ra l m a p m a k in g p ro b le m
This section is a concise summ ary of the detailed discussion in [1],
In general, every instrum ent has its own unique scan strategy, and w hat we receive (in our
case, the visibility from some baseline) depends on the signal on the sky, and the convolution
with the beam, combined w ith the scan strategy. Instrum ental noise has to be added to th a t
later. All this is summarized in the nice equation
d = PA + n
(9.1)
where d is TO D (“time-ordered d a ta ”), A is the signal we are trying to recover, n is instrum ental
noise, and P has all the information about our scan strategy. Bear in mind in the discussion
th a t follows th a t d, A and n are vectors, i.e. column matrices (one could take them to be row
matrices ju st as well, since the final results will be exactly the same), and P is a m atrix with
the correct dimensions.
In order to make a m ap, then (or, for th a t m atter, do any analysis), we need to recover
A , the signal on the sy; in our case, the visibility from different baselines. We therefore need
to invert the above equation to recover A . While it isn’t clear to me whether there is a finite
number of non-linear m ethods, our first task should be to find linear solutions. All linear
solutions can be expressed as
A ' = Bd
(9.2)
W .L.O.G., where A ' is an estimate of A , and therefore, there is an estim ation error involved,
no m atter how good our technique. Before we jum p into calculating B , let us define a few basic
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164
quantities. Let S = ( A A T) denote the theory covariance m atrix, and N = ( n n r ) the noise
covariance m atrix. It seems reasonable to me (do let me know if you object, and why, since
I may have missed something) to suppose th a t noise and signal are uncorrelated, since their
sources have nothing to do w ith each other, i.e. ( A n r ) = ( n A T) = 0
We are now ready to address the m apm aking problem. W hat follows is a discussion of
my own analysis, and unfortunately, I haven’t been able to compare it to any reference to see
whether it has any element of sensibility. W hat I found is th a t broadly, there are three different
conditions on can impose, and each will lead to a different (and unique) definition of B . The
three conditions are:
1. Ease of calculation
2. Minimizing the estim ation error
3. Minimizing y 2
Let us look at these in detail.
9.1.1.1
T h e B ru te -fo rc e s im p lis tic m e th o d
In the equation
d = PA+n
(9.3)
the simplest thing to do to recover A is to multiply throughout by P ~ l to get
P “ 1d = A + P 1n
(9.4)
A ' = P ad
(9.5)
<y = P _1n
(9.6)
so th at
with an estimation error
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165
However, we C A N N O T possibly do the inverse operation on P , since it is not a square m atrix.
B ut we can do other operations, like taking the transpose. So we recall th a t P r P is the modulus
of P . This is a square m atrix, and we can take its inverse, which is, schematically P - i ( p
t}- i
(I say schematically, because the operation inverse is not perm itted on the individual matrices).
Clearly, if we multiply this by P 7 , we can recover P . T he net m atrix is then ( P r P ) 1 P 7 .
If we look at the last equation more carefully, we may perhaps be persuaded to feel a
little less silly for adopting such a simplistic approach, since it tells us th at the estim ation error
is independent of signal, and depends only on instrum ental noise. While this is nice, we may
not entirely be happy with it and want to minimize |<5|2.
9.1.1.2
M in im iz in g t h e e s tim a tio n e r r o r
In general, we want to estimate A by a “correction m atrix” ; call it B:
A' = Bd = BPA + Bn
(9.7)
so th at the estim ation error is
5 = A' -
A = B P A + B n - A = (B P - I) A + B n
(9.8)
Therefore,
\S\2 = (S5t ) = ([(B P - I) A + Bn] [A r ( P t B t - I) + n r B r ])
(9.9)
Multiplying out explicitly,
|<5|2 = ((B P - I) A A t ( P r B r - I) + B n n TB T + [B n A r ( P TB T - I) + (B P - I) A n r B r ] )
(9.10)
The two term s in the square brackets are oc either ( A n T) or its transpose, both of which are
zero. Therefore,
|6|2 = ((B P - I) A A t ( P TB r - I) + B n n r B r )
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(9.11)
166
Substituting S = ( A A t ) and N = ( n n r ), we get
\6\2 = (B P - I) S ( P r B r - I) + B N B t
(9.12)
Now, we need a B such th a t |<5|2 is minimized. Therefore, we need a solution to the equation
S|4|2
dB
= 0
(9.13)
Differentiating w .r.t. B , we get
P S (P t B t - I) + N B r = 0
(9.14)
(P S P r + N ) B r - P S = 0
(9.15)
(P S P T + N ) B t = P S
(9.16)
Collecting term s w ith B 7
or
On a first glance, it seems like solving this equation is impossible for any value of B T , because
if we define B 7 such th a t it cancels out either term inside the brackets, it is impossible to make
the LHS equal to P S.
However, wecan always cancel out the bracket in the LHS by defining a B 7 th a t
is oc its
inverse. If wealso require th at B 7 is simultaneously oc P S , then we have essentially solved the
equation. Formally, the solution is
B t = (P S P T + N ) - 1 P S
(9.17)
All we need to do now is to take the transpose of the RHS, and we will have our solution, i.e.
B = (P S )r ( ( P S P r + N ) T) _1
(9.18)
where we have used the fact th at (A B )T = B TA T. We can now expand out to get
B = (Sr P T) ( ( P S P T)T + N r ) _1
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(9.19)
167
B ut for S, ST = ( A A t )T = ( A t )T A t = A A T = S, and similarly, N T = N .
Now
(P S P T) r = (S P )T P r = (P T) r Sr P r = P S P T. So the final expression is
B = S P T [P S P r + N ] _1
(9.20)
We could have differentiated |t>|2 w .r.t. B r instead, and we would get precisely the same result.
9.1.1.3
Minimizing x2
Following the discussion in [2] §11.5 (equations 11.129 to 11.131),
X2 = (d - P A ) N _1 (d - P A )
(9.21)
dy2
^
= 0
(9.22)
A ' = (P r N _1P ) 1 P r N _1d
(9.23)
=> B = (P TN ~ 1P ) ~ 1 P t N _1
(9.24)
To minimize y 2, we set
which gives us
This map-making m ethod can be employed to extract a “fourier m ap” ; in other words, visi­
bilities, from an interferometer. However, the m ethod involves a num ber of m atrix inversions,
each of which costs ~ Nf> operations where N p is the number of pixels in the m ap (in fourier or
real space). W andelt et al ([3, 4]) have introduced fully Bayesian m ethods th a t allow a global
3
inference of covariance and allow m ap recovery at the same time, and costonly N p operations.
This m ethod (called
“Gibbs sampling”) will be discussed in §..The true advantages of Gibbs
sampling come to light when power spectra need to be evaluated.
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168
9 .2
P o w e r S p e c tr u m E s tim a tio n : B a y e s ia n A p p r o a c h
Let d = pixelized d a ta from an interferometer. We wish to explore the posterior density
P (Ct \d) oc P (d\Ce) P (C » <- Prior
(9.25)
P (d\Ce) oc exp ( d t C ^ d )
(9.26)
C ^ = S ( C £) + N
(9.27)
where
where
is the covariance m atrix.
Traditionally, least-squares [5] and maximum-likelihood [6] estim ators have been em­
ployed to explore the posterior. However, evaluating either is computationally very costly,
requiring O (N p ) operations.
9.2.1
D eta iled B ayesian Form alism
In the Bayesian approach, we wish to compute the posterior density
(9, 8)
If S and N are signal and noise covariance matrices respectively,
P ^
= ~J^ p \n \ exp ( ~
JV_1 ^ “ s 0
^9'29^
Also, since we know th a t the CMB is very nearly gaussian
P{s\Ci) = — ± ^ exp
(9.30)
so th at
- In P ( C t , s\d) = \ { d ~ s)+N ~ l (d - s ) +
+ i In |5 |
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(9.31)
169
The best estim ate of the signal s can then be found by
d ( - l n P ( C e,s\d))
dst
=
- N ~ 1 ( d - s ) + S - 1s = 0
=►
( S ' 1 + N ~ 1) s BE = N ~ 1d
(9.32)
M atrix inversions are com putationally costly, and in the final expression for
sBe
,
there are
three inversions, each costing O (N p ) operations. If we were to transform the above equation
into the form A x = B , we can employ efficient techniques to solve for
However, an efficient m ethod of extracting
sBe
s Be
-
leaves the problem of Cg extraction being
extremely time-consuming computationally.
9 .2.2
T h e p roblem w ith th e B a y esia n approach
Interferometry has been used to detect CMB tem perature and polarization anisotropy
(VSA, DASI, CBI references). Here are some of the advantages of interferometry:
1. Direct sampling of Fourier space
2. No leakage T —> Q,U, so better control of system atic effects
However, an exact Bayesian analysis analysis m ethod is ju st as unfeasible as for an imaging
experiment.
However, W andelt et al[3] have introduced a fully Bayesian approach - Gibbs
sampling - th a t allows a global inference of covariance. Among the m any advantages of using
the Gibbs sampler is th a t it is easily extended for foreground removal.
Let us first illustrate G ibbs’ sampling by applying it to a simple problem in the next
section.
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170
9 .3
I n t e r l u d e : T h e G ib b s S a m p le r
9.3.1
T h e p ro b le m
Here is the statem ent of the problem: The problem is to infer the variance of a fluctuating
signal when you only have noisy measurements of this signal: Given 10 d a ta values d,, say, and
given th at these values are independent samples of the sum of two Gaussian variates each s,
(the signal) and n, (the noise), and th a t
s
and n have zero mean, and the variance of n, b ut we
don’t know the variance of s, a\. W rite down Bayes’ theorem for this case, compute
• the conditional density for the vector
s
(with components
Sj)
given
and
• the conditional density for a 2 (up to normalization).
• then sample from the joint density of s and <jg using Gibbs sampling.
We spell out Bayes’ Theorem in §2 and the sampling technique in §3.
9.3.2
B a y e s’ T h e o re m
Since we are working w ith an interferometer, let us assume th a t all the formalism we
write down is for an interferometer. For instance, st stands for visibility from a baseline, free
from noise, and dj is the measured visibility etc.
Our aim is to find the posterior density, i.e. the probability of the theory given the data.
We can write, from Bayes’ theorem
P ^ ^ = P^ p } d l ^ '
(9'33)
which can be w ritten schematically as
^
Likelihood x Prior
Jom tPostcnor = ——------------ :-----Normalization
.
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(9.34)
171
Assume a flat prior and then, up to a normalization, we get
P t f \ d i) = P { d i \<%)
(9.35)
However, the w h o le p ro b le m is th a t a'j cannot directly be related to data, d{. In other words,
p (o 'sK ) = P (di\si) P
(9.36)
Now, we know th at
1. si and d, are related through noise, so th at
p
1
( (S i - d i f
/------- exP ( — ho r
I
TN
=
(9-37)
2. P (silc^) is Gaussian in Sj, so th at
exp( _ 4
)
(938)
Thus,
p ("M
=
7 ^ “ p( ' ii^
) exp( - ^ )
(9-39)
Since there are N observations, the right hand side is really a product of N factors
<»*»
B ut we know the form of this function, up to a normalization assuming th a t the m ean is zero:
r « « P ( .,k S =
exp
exp ( - A )
(9.41)
And so
N
P (al ld) = Norm/ •'•/
eXPf “
i= l
x
, x2\
“ UN
/
) /6XP\
N
o
S± _
^2 ) dSl '''dSN
i=1 ~a
s ,
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(9'42)
172
where Norm = normalization:
N
Norm =
1
(9.43)
where we have marginalized over s to get the posterior. This N-dimensional integral is hard
to evaluate for large values of N , i.e. Evaluating this probability requires huge am ounts of
com putational time. So we sample from the joint probability instead.
It seems like an even more difficult task, b ut it is made easier if we are willing to undergo
a paradigm shift in the way we visualize probabilities. Normally, we look a t probability as a
fu n c tio n of N variables. If we stick to this interpretation, we will h a v e to evaluate the function
a t some point. We can, however, choose to see this probability as a “density of points” , kind of
like the way we see the electron probability density inside an atom. If we can make this step,
then there exist sampling techniques in Statistics th a t make our job easier. One of them is the
G ibbs’ sampling technique, described in §9.3.3 below.
9.3 .3
S a m p lin g T e c h n iq u e
Let us state our problem in a general way first. We have two variables, x and y, and we
know the functional form of P (x\y) and P (y\x) , and we wish to find either P (x) or P (y) or
both. Normally, we would marginalize over x or y thus:
(9.44)
However, we Gibbs sample instead in the following way:
• Start with an initial guess of x, say
xq
• Sample y\ from P (y\xo) <— this can be done since we know the functional form of P (y\x)
• Sample x\ from P {x\y\ )
• Repeat the last two steps, keeping track of all the values of x and y sampled
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173
After a certain number of iterations, the density of values of x and y represents P (x
9.3.4
P i y ).
A p p lic a tio n t o e x p e rim e n t
In §3, let x = (7^, y = s and follow through. There is ju st one small issue: there are two
Gaussians multiplied to each other, and not ju st one:
= i v m ) ( ^ r ) p('i'|s‘)p(s‘l‘T') =exp("(£!^ T !) exp( ' A )
(9.45)
We re a lly need to reduce this to one Gaussian w ith a mean and a variance. The solution to
this problem is purely algebraic but simple. In general, when there are two Gaussians g\ and c/2
with different means m \ and m 2 and different variances <J\ and sigma 2 respectively, we want
to find o n e set of (mean,variance) for
gt92 oc exp
j exp ( - <X; ” 2) )
(9.46)
We need to remember th a t the logarithm of the Gaussian function is quadratic. Therefore, we
can differentiate the log of the Gaussian and equate it to zero to get the mean. For example,
w ith the above distribution g i ,
<9'47>
=
d\ng\
dx
so th at
=
0
=£• x — m \ .
(9.48)
(x — m i )
a
(9.49)
This is the general way to get the mean of a complicated
Gaussian.
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174
Also, we can easily differentiate twice to get rid of all dependence on the variable x, and
only a combination of mean and variance will remain. So, in our present example, differentiate
eq( 9.49) again to get
e 2 ln ®
dx2
-4
( d2 In g\ \
=>a j =
dx2
\
We apply this to the product
3132
(9.50)
1
/
(9.51)
above in eq( 9.46) and get
(x —m i)
2(jj
ln (5 1 5 2 )
(x — rri2 y
2a 2
(x —m 2 )
(x - m i)
d In (gig2)
dx
(9.52)
(9.53)
Now, equate this to zero, which means th a t we have to solve the following equation for x (the
n e t average):
( n~ — m
I'r —
r r > 2)
o \ __ n
(x
m i)i , (x
- m
O '
O
—”
CTn
1
(9.54)
Collect all the term s containing x on one side:
1
(9.55)
:T ( t + f )
This is the mean for
3152.
(9.56)
Now, differentiate eq( 9.53) to get
d2ln (gig-j)
(J _
J_
dx2
V<Tj
°2
(9.57)
where a 2 is the n e t variance. Solve to get
at2l u2
a;‘1
( i +i )
Now we have everything we need for implementing Gibbs’ sampling.
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(9.58)
175
9.3.5
R e s u lts
9 .4
Q e x t r a c t io n u s in g G ib b s ’ S a m p lin g
This section is a concise version of [7].
9.4.1
M e th o d
Let d = pixelized d a ta from an interferometer. We wish to explore the posterior density
P (Ce\d) oc P (d\Ce) P ( Q ) «- Prior
(9.59)
where
P (d\Ct)
exp
(9.60)
C ^ = S ( C e) + N
(9.61)
oc
where
is the covariance m atrix.
Traditionally, least-squares [5] and maximum-likelihood [6] estim ators have been em­
ployed to explore the posterior. However, evaluating either is computationally very costly,
requiring O {Np) operations.
Instead, we use the G ibbs’ sampler introduced to CMB data analysis by W andelt et al
[3, 4], and s a m p le from the joint distribution
P (Ct , s , d ) = P (d|s) P {s\Ci) P (Ce)
since there
(9.62)
is no known way to sample directly from P (C( \d) ineq.(9.59) above.
point of using G ibbs’sampling is th a t it can be proved [8] th a t if
The main
it ispossible to sample from
P (s\C t, d) and P (C(\s, d) oc P (CV|s) then we can sample iteratively from the joint distribution
[3].
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176
o
o
Gibbs s a m p l i n g
CN
O
of v a r i a n c e
Variance
samples
o
o
sample
value
recovered
from
in
o
Histogram
m
o
o
OO
o
o
o
o
CO
o
o
^
S 1!M
10
o
C\l
-o n
Figure 9.1: Results from G ibbs’ sampling for the experiment mentioned above.
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o
177
9 .4 .2
F o rm a lism
In the Bayesian approach, we wish to compute the posterior density
P ( C ( iS ,
(M 8 )
If S and N are signal and noise covariance matrices respectively,
P (d|s) = exp
- (d — s)f N ~ x (d —s)^
(9.64)
Also, since we know th a t the CMB is very nearly gaussian [ref]
p
= yjffexp (- £stsrls)
^9-65^
so th at
- In P (C/.sId) = ^ ( d - S)t l T 1 ( d - s ) + ^ S - 1s + ^ l n \ S \
(9.66)
The best estim ate of the signal s can then be found by
3 (-ln P (O ,4 0 )
<9st
- N - 1 ( d - s ) + S ~ 1s = 0
( 5 _1 + N - 1) s BE = N - l d
(9.67)
M atrix inversions are computationally costly, and in the final expression for s b e , there are
three inversions, each costing O (N p ) operations. If we were to transform the above equation
into the form A x — B, we can employ efficient techniques to solve for
sbe
-
We also need to remember th a t eq.(9.67) gives us a value for the average of the signal.
The real signal on the sky is of the form x +
where £ is the variation and is a gaussian
variable and C is the covariance. C is evaluated easily by noting th a t P is a multiple of two
gaussians, w ith covariances S and N ; therefore the covariance of P is
C ~ l = S ~ l + N ~ l = > C = (S - 1 + N - 1) - 1
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(9.68)
178
Eq.(9.67) can be recast in a more suitable format thus
(S '-1 + IV- 1 ) s be
=
=► S~* ( l + S ^ N - ' S ^ S ~ * x =
=►
where we have replaced
sbe
( l + S s A T 1,?*) S ^ x =
by x. We now need to find
N-'d
N ~ 'd
s'zN -'d
(9.69)
a similar equation for the fluctuating
Call this p art b such th a t s = x + b. b then needs to satisfy the following
p art of the signal.
properties
(b)
(“’)
b
0
(9.70)
C, since
(9.71)
1
(9.72)
==► C,-16 = C,_5f
(9.73)
( S - 1 + N - 1) b = S - ? £ i + N - * Z 2
(9-74)
We claim then th a t if
then b has the properties outlined in eq.(9.73). To prove the first property, take the average of
the LHS in eq.(9.74):
( S ' 1 + AT-1 ) (b) = S ~ i (6> +
<&) =
0
(9.75)
The second property is proved thus
(btf)
=
c ((s -k i + N ~ k2)(s-k\ + N -h i))c
= C
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(9.76)
179
where we have used the fact th a t
£1
and
£2
are independent gaussian variates w ith unit variance:
( $ 1$ )
=
(eka)=0
(9.77)
(ad )
=
(ad ) = i
(9-78)
The equation for b is then
(,s - 1 + N - 1) b = S ~ k i +
=► 5 " 5 ( l + S s N - ' S * ' ) S ~ h = S ~ k i +
=*■ ( l + S ^ N ^ S ^ S - h ^ t i + s i N - ^ i
(9.79)
To summarize, the two equations for simulating the signal are
( l + S * N ~ 1S * ' ) S - * x
=
S sN -'d
( l + 55 J V" 1
= f i + 5 5 j\T 5 f 2
(9.80)
(9.81)
These can be solved to obtain x and b for every iteration, and s = x + b.
9.4.2.1
B eam / W in d ow Fun ction
B ru te-force Im p lem en ta tio n The foregoing discussion ignores the beam of the instru­
ment,
which isassumed to be flat in fourier space. This assum ption is unreal and the beam
needs to be included
in signal and power spectrum estimation. Let us denote the signal co-
variance m atrix w ith the “flat” beam mentioned above as S d - S d is clearly diagonal, and its
elements are the different C /s . If S is the signal covariance m atrix, then, schematically,
S = BSd
where B is the beam m atrix of the instrum ent in fourier space.Recalling th a t
(9.82)
S = ss* and
th a t this is a fourier space representation of the signal, B isreally the window-function of the
instrum ent. Let us therefore denote the beam in fourier space as B p such th a t
S = B jF S DB F
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(9.83)
180
In other words, we have replced the signal s with Bps .
Let us work out relations for the best estimate of the average signal s b e or x and the
fluctuation b, retracing steps in §9.4.2. In particular, we sta rt w ith the modified version of
eq.(9.66) w ith s —> B p s and S o =
-\nP{C e,s\d) =
i ( d - B p s ) f TV” 1 ( d - Bps)
+
l ^ S j s + ^lnlSl
(9.84)
The best estim ate of the average signal can be found as before:
9 ( ~ l n a , (tQ ’ a |< i ) > = ~ B > - ' (d - B ? ‘ ) +
=>
= o
( S p 1 + B iF N - 1B F'jsBE = B FN - 1d
(9.85)
As before, we can recast eq.(9.85) into a more suitable form:
f a
+ B ^ N ^ B p J sb e = B p N -'d
=>
SD* ^1 + S l B FN - l B FS]yj S ~ ^ x = B FN ~ l d
=►
^1 + S l B p N - ' B p S ^ J S ^ x = S ^ B p N - ' d
(9.86)
where we have again replaced s b e by x.
To obtain an equation for the fluctuations, we note th a t the covariance of P is still C
where
C - 1 = S p 1 + iV-1 ==>■ C = ( S p 1 + N - ' y 1
(9.87)
Prom eq.(9.79), we get
f a
+ B ^ N - ' B p ) b = S ~ k i + N-*H2
=»
s y ( l + S lB p N -'B p S l) s y b = S~ ki +
=*
( l + S Z B FN - 1B Fs i ' ) s ~ h = t i + S!)B FN - k 2
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(9-88)
181
Results are shown in figs.(9.6,9.7,9.8,9.9). Both Ces and maps seem to have considerable
loss in power, implying perhaps th a t the beam has not been accounted for properly?
C o m p u ta tio n a lly E fficient Im p lem en ta tio n T he foregoing discussion about includ­
ing the beam is easily implemented; however, the com putational cost is higher th an before. In
order to reduce com putation time, we employ the following trick. Let us denote the beam in
pixel (or real) space as Bp . It is more desirable to work w ith B p since
1. It is a sparse m atrix, diagonal in the ideal case when there is no “leakage” of signal from
one pixel into another
2. It is a measured quantity and so any non-ideal behaviour can be directly inferred from
measurement.
However, the quantities th a t an interferometer measures (visibilities) are in the fourier domain.
Therefore, a convenient representation of the beam in fourier space is
Bp ~ $ B p $ 1
(9.89)
where # represents a fourier transform . The idea is as follows: when the fourier-beam B f
multiplies another quantity, take the inverse fourier transform of the quantity, m ultiply with
the pixel-beam B p and then fourier transform the result.
This representation changes the quantities in §9.4.2 above. The signal covariance m atrix
becomes
5
=
B ]f S d B f
(9.90)
=
S^BpZSodBpZ-'
(9.91)
This can also be w ritten as
(9.92)
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182
and since the quantities in the two brackets are equal,
S 2 = B ]F S l
(9.93)
We can then replace S ^ and S in all the equations in §9.4.2 above to include the beam.
The advantage of using this representation of the instrum ent becomes apparent when we
look at the modified posterior density
- In P ( C e,s\d)
=
^ { d - B Fs)] N ~ l { d - B Fs)
+
^ s t5 Dls +
(9-94)
Notice th at in the second term , there is no factor th a t depends on the beam. This leads to the
possibility of combining two or more datasets from different instrum ents for a joint analysis via
Gibbs’ sampling. For two datasets, the posterior density becomes
- I n P ( C e,s\d)
=
] ^ { d - B Fls)] N ~ l { d - B Fls)
+
^ ( d - B F2s )] N - 1 ( d - B F2s)
+
^S^s+^lnlSol
(9.95)
where B Fland B Fl are the fourier-beams of the two instrum ents.
9 .5
A p p lic a tio n t o s im u la te d d a t a
We simulated a 7°x7° patch of the sky with CMB signal. The simulated m ap is shown
in fig.(9.2). Histograms of recovered values of Ces are shown in fig.(9.10).
9.5.1
G elm an -R u b in Test
In order to test the convergence of the Gibbs’ sampling setup, we perform the Gelman-
Rubin test [9, 10] in the following way.
1.
Let n be the number of samples in a G ibbs’ sampling algorithm. Let there be m param eters
we wish to estim ate - in our case, these are the num ber of bins.
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183
2. Run the G ibbs’ sampling algorithm with m any different initial values. These should be
sampled from a wide range. Effectively, run N “chains” of the G ibbs’ sampling algorithm
where N = num ber of different initial values.
3. Compare the “in-chain” and “between-chain” variances. These should be approximately
equal in order for the Gibbs’ sampling algorithm to converge.
T he last step is completed by calculating the following quantities
1. “W ithin-chain” variance
^
= ^
-
m
r v
i ) E
n
E
W
j=i «=i
- % ) 2
«»■«<»
2. “Between-chain” variance
m
B = ^ r i E ( si - lf
(9 -97)
K ( « ) = |l - - W + - B
\
n/
n
(9.98)
j =i
3. Estim ated variance
4. The Gelman-Rubin Statistic
^ s^
<
9'99)
where || indicates trace.
T he Gelman-Rubin statistic was evaluated to be ~0.999951 - sufficiently close to indicate con­
vergence.
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184
Figure 9.2: Simulated “flat-sky” CMB map.
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185
o
o
o
o
o
o
o
o
H
Oh
O
O
o
H
O
O
W
o
I
O
I
O
CO
I
o
03
o
I
o
I
o
I
o
CO
I
o
( ^ r l ) i i 2 / !3(l+ ?)?
Figure 9.3: The power spectrum used for m ap simulation and the spectrum recovered from the
simulated map.
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186
LO
CO
Figure 9.4: Estim ates of Q recovered from Gibbs’ sampling; beam effects not included.
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187
cv
I
CO
I
Figure 9.5: M ap recovered from G ibbs’ sampling, no beam.
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188
CO
o
multipole
moment
CO
o
CO
O
o
co
O
o
O
O
O
Td > / l 3 ( l + l ) l
Figure 9.6: Estim ates of Ci recovered from G ibbs’ sampling; beam effects included - I.
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189
o
m
cv
multipole moment '
CO
O
o
o
O
CD
o
O
O
O
Idt/I3(T + I)l
Figure 9.7: Estim ates of Ci recovered from G ibbs’ sampling; beam effects included - II.
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190
CO
CO
00
I
I
CO
I
Figure 9.8: M ap recovered from G ibbs’ sampling, beam included - I.
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191
05
C
O
o
o
o
o
M
CO
T
—
1
c\i
K
ii
K
CO
cq
cd
CO
^
C\2
1
1
o
i
ki
1
T
—1
05
1
O
1
U
J
o
I
w
2>
l
O
1
1
O
w
cv
C
O
C \2
U
J
O
1
w
■
«
—
1
C
O
c\i
l
CO
Figure 9.9: Map recovered from Gibbs’ sampling, beam included - II.
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192
0
•0
L
*L
0
>0
30
3®
1 0 '
3®
30
#&
Sf
.1 0
• S3
30
Hs-
m
0
10
" 30 .
• 36.
« j.
$6
10
' ft
. 36.
*0
$*
Figure 9.10: Histograms of recovered values of Ces: beam NOT included.
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193
Bibliography
[1] M. Tegmark,
“How to Make Maps from Cosmic Microwave Background D ata w ithout
Losing Inform ation,” A p J Lett., vol. 480, pp. L87+, May 1997.
[2] S. Dodelson, M odem cosmology, M odern cosmology / Scott Dodelson. Am sterdam (Nether­
lands): Academic Press. ISBN 0-12-219141-2, 2003, X III + 440 p., 2003.
[3] B. D. W andelt, D. L. Larson, and A. Lakshminarayanan, “Global, exact cosmic microwave
background d a ta analysis using Gibbs sampling,” Phys. Rev. D, vol. 70, no. 8, pp. 083511+ , Oct. 2004.
[4] B. D. W andelt, “MAGIC: Exact Bayesian Covariance Estim ation and Signal Reconstruc­
tion for Gaussian Random Fields,” A rX iv Astrophysics e-prints, Jan. 2004.
[5] J. R. Bond, A. H. Jaffe, and L. Knox,
“Estim ating the power spectrum of the cosmic
microwave background,” Phys. Rev. D, vol. 57, pp. 2117-2137, Feb. 1998.
[6] M. P. Hobson and K. Maisinger, “Maximum-likelihood estimation of the cosmic microwave
background power spectrum from interferometer observations,” Monthly Notices of the
RAS, vol. 334, pp. 569-588, Aug. 2002.
[7] B. W andelt and S.S. Malu, “Gibbs’ sampling for Interferom etry,” Work in progress, 2007.
[8] Tanner, Tools fo r Statistical Inference: Methods fo r the Exploration of Posterior Distribu­
tions and Likelihood Functions, Springer Verlag, Heidelberg, Germany., 1996.
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
[9] G. Huey, R. H. Cyburt, and B. D. W andelt, “Precision prim ordial 4He m easurement from
the CMB,” Phys. Rev. D, vol. 69, no. 10, pp. 103503—f-, May 2004.
[10] Gelman, Andrew and Rubin, Donald B., “Inference from iterative simulation using multiple
sequences,” Statistical Science, vol. 7, no. 4, pp. 457-472, nov 1992.
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
195
C hapter 10
Conclusions
In this thesis, we have introduced a novel interferometer, the Millimeter-wave Bolometric Inter­
ferometer, which combines the advantages of interferom etry with the sensitivity of bolometers.
Furthermore, MBI has a novel quasi-optical beam-combination arrangement th a t will allow it
to simultaneously function as an interferometer and an imager. In addition, an efficient sta­
tistical technique already employed by W andelt et al [1, 2] for imagers was adapted for an
interferometer. Several instrum ental aspects were also studied in chapter 5 and two different
m easurem ent/instrum ent optim ization techniques explored (§7.7.1,§7.6,§7.7). These techniques
will provide MBI w ith the sensitivity required to provide upper limits to B-mode levels. B ut
MBI-4 has ju st six baselines and a 7° beam - these param eters imply large pixels in u-v space.
Future versions of MBI and another planned space-based version [3] will have larger beams
(~15°), leading to smaller pixels in the u-v plane. A space-based version (called EPIC ) is also
intended to have many more detectors in the focal plane in its Fizeau system. EPIC is also
planned to have several “units” , each in a different bandwidth, each w ith several dozens of
antennae, providing both the f-space coverage and a means to characterize foregrounds.
This thesis has thus introduced a new kind of instrum ent with exquisite control over
system atic effects, brought about through instrum ent optimization. In particular, it can simul­
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196
taneously observe as an imager and an interferometer. Power spectra can thus be computed
with both approaches and compared.
But comparing power spectra is not all. MBI can make images of those parts of the
sky where foregrounds are known to dom inate the CMB signal. This same information can
simultaneously be obtained in the u-v plane, split into several sub-bands. This provides us w ith
the ability to perform a unique comparison of foregrounds and systematic effects in image plane
and the u-v plane. If we add the advantage of several bands to this instrum ent, we gain the
a b ility t o c h a ra c te riz e fo re g ro u n d s a s w ell a s d e te c t th e fa in t B -m o d e sig n al. This
is a unique ability, not yet achieved by an experiment in CMB cosmology. This eliminates the
need to cross-correlate data from several experiments to eliminate foregrounds, thereby allowing
greater control over instrum ent systematics. In this sense, the MBI is a c o m p le te instrum ent,
supported by the analysis and simulation techniques developed in this thesis.
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197
Bibliography
[1] B. D. W andelt, D. L. Larson, and A. Lakshminarayanan, “Global, exact cosmic microwave
background d a ta analysis using Gibbs sampling,” Phys. Rev. D, vol. 70, no. 8, pp. 083511—b,
Oct. 2004.
[2] B. D. W andelt, “MAGIC: Exact Bayesian Covariance Estim ation and Signal Reconstruction
for Gaussian Random Fields,” A rX iv Astrophysics e-prints, Jan. 2004.
[3] P. T. Timbie, G. S. Tucker, P. A. R. Ade, S. Ali, E. Bierman, E. F. Bunn, C. Calderon,
A. C. Gault, P. O. Hyland, B. G. Keating, J. Kim, A. Korotkov, S. S. Malu, P. Mauskopf,
J. A. M urphy, C. O ’Sullivan, L. Piccirillo, and B. D. Wandelt, “The Einstein polarization
interferometer for cosmology (EPIC) and the millimeter-wave bolometric interferometer
(MBI),” New Astronomy Review, vol. 50, pp. 999-1008, Dec. 2006.
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198
A ppendix A
Dr. Planck, or: H ow I Learned to Stop W orrying
and Love Stat M ech.
A .l
The general problem
We wish to figure out how particles are distributed among N states, n l , each w ith energy
E n - this is the general problem th a t Statistical Mechanics addresses. In these notes we will
restrict ourselves to photons.
In general, the distribution function U (E) is given by U (E) dE = num ber of available
states x average energy per unit state. The number of available states is usually referred to in
literature as the ‘phase factor’. We m ust then, deal with two separate calculations.
A .2
Average Energy
For a photon, E 1 — his. In general, the n th state will have energy E n = nhis. To find
the average energy per state, we sum over the entire distribution, weighing each state w ith a
factor given by the distribution function in this case the Boltzmann distribution function given
by e~PEn where 3 =
The average energy then is
E E ne - ^
E =
STe-PEn
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(A -1)
199
In Statistical Mechanics, the sum ^ e " ^ En is called the Partition function, and all physical
n
quantities like (average) Energy, Entropy etc. can easily be related to it. Below is a simple
example of such a relation th a t is useful to us. Call Z the P artition function, so th a t Z =
En Then differentiate Z w.r.t. [3 to get:
n
H
= -£ E „ e - »
(A .2 )
n
We immediately see th a t this is very close to the expression for E above; all we need to do is
e~,3En, which is exactly Z . So we get, adding a minus sign,
to divide this by
*
=
Z 8(3 ~
(A.3)
8(3
Now, for photons,
CO
Z =
e~0En =
(A-4)
ft—0
Recall now th at
CO
(A.5)
n= 0
Applying th at here, we get, with r = e Phv
z =
( A-6)
^ • I n Z = —In jl - e ~ phv\
d ln Z _ ,
dj3
A .3
~
+
1 _
1
c-0hu
kv_
e -p hv
e
n v
-
(A.7)
hu
(i _
e - P la , ) e [ ) h v
_
hv
e 0hu _
j
Number o f phase states available, or phase factor
The Uncertainty Principle decides the ‘minimum size’ of a ‘phase cell’ because
dxdp > h
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(A-9)
200
=> d3x d 3p > h3
(A. 10)
Therefore, given a certain phase-space volume d3xd3p, the num ber of ‘cells’ available in th at
phase space is
C=
d3xd3p
(A.11)
W riting the volume in real space as V , we get
Vdh, =
For photons, E = pc from Special Relativity =>■p =
^
/h2p2 jkdu
1
47r
C — 4w— x— ■-------• —x ■V — -xxv dv ■V
c2
c
/i3
c3
(A.13)
Now, we account for the fact th a t there are two unique polarization states possible for any given
direction of propagation. The above expression becomes
Q'TT
' C = — lA ii/.y
A .4
(A.14)
Planck Distribution
Planck distribution can then be w ritten as a multiple of the two factors in the two sections
above
U (i/) dp = ^
p2
• -J ' " _ ^ dpV
(A.15)
We can express the LHS in term s of the energy density instead of energy, so th a t
u (vp >
) dp =
rJ W—
dp
c3 p2 ■ef)hv
_ i
v(A.16)>
This is the expression we wanted.
W hat we really need in calculations for the MBI, though, are integrals of this functions
x other functions, like the
interference p attern on the bolometers.W hile doing an integral, it
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201
is always very convenient to separate out unitless quantities th at we integrate over, and factors
th a t depend on the physics of the situation. In this case we define a unitless quantity x = ^
and strive to present this distribution in term s of x thus:
Replacing ^
by x, and collecting all other factors together, we get
, .
8 7 rh ( k T \
u M du = - j - { T )
4
x3
^~=7
(
.
}
where u has units JH z- 1 m - 3 , i.e. energy per unit volume per unit frequency interval. B ut the
energy radiated out in all directions is the same. Moreover, we are interested in the spectral
intensity, i.e. power per unit area per unit solid angle per unit frequency integral. We denote
the spectral intensity by I , and this is how I is related to u:
(A.19)
such that
2h f k T \ 4
=
x3 ,
— i dx
(A20)
We can also write this as
l ( v , T ) = B v {T)dv
(A.21)
for a black body, where
Bv =
(ex - l ) - 1 [W m - 2 Sr_ 1 Hx_1]
(A.22)
so th at
0 7 , ,,3
I { v , T ) = - j - (ex - I ) - 1 du [Wm_ 2 Sr_1]
(A.23)
However, if the signal varies across the sky, then B u is a function of (0, <f>). Thus, if we write
T = T0 + 6T(9,4>)
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(A.24)
202
for the CMB, then
f)n
B v (T + ST) = B v (T0) + — ST = B u (T0) + A B
(A.25)
If we are studying the anisotropies in the CMB, we are interested only in the second term ,
which we can write as
dB dx d ( ± )
d x d ( ^ ) dT
2k v 2 x 2ex
c2 (ex — l ) 2
(A.26)
so th at
?k /
A B 09, </>) = ^
r 2px
\
W (*> *)
(A.27)
M ultiply and divide by yjr
kT0 (in order to end up w ith dx instead of du in the expression for
intensity, I):
2k2T0 (
x 2ex
ST (6 ,4>)
[W m -2Sr-.1H z - 1]
(A.28)
Now, because of the extra factor of -gjr at the end, we can write the intensity as
AI
0k2Tn /
^ = " W
t 2px
( (ex _
\
if) ST
^
dx[W m _2Sr_1]
(A -29)
This is the quantity th a t we wish to calculate in the instrum ent simulation, given a simulated
map of the sky.
A .5
D istribution for particle number
We can follow through §A.2 for number of particles too. Similar to eq(A .l), we can write
E ne
-P E n
e~PEn
” =
(A.30)
n
Recall th a t Ej, = n h v , so th at
hud/3
and we can write
_ _ E _ _ _________ = ___ 1 d l n Z
hu
hu Z d/3
hu d/3
1
e@hv — 1
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203
Using the phase space factor from §A.3, we get th a t the number of particles between v and
v + dis, given by n (v) du is
n(v)du= ^~
_
1du
(A.33)
The author is forever indebted to late Dr. Swaminathan for his expositions on Statistical
Mechanics.
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204
A ppendix B
S- and T -m atrix form ulation
B .l
Two port devices and the S-matrix
Most of the microwave devices we use in our lab are 2-port devices, and are usually used
in series, e.g. a w /g twist with a w /g straight piece. Any 2-port device has two possible inputs
and two outputs. We label the inputs w ith
a
and outputs w ith
For all practical purposes, we are interested
not
b.
in the values of the outputs
b,
but what
they are compared to the inputs. In other words, we wish to look at the generic ratios
output
input
(B .l)
for all four quantities.
The most simple-minded approach would be to define the four ratios ^ etc., b u t we can
write these out systematically as:
S i 2d 2
(B.2)
i>2 — S 2 l d \ + S 2 2 O .2
(B.3)
b \ — S lid } +
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205
b2
at
2 -p o ii d e v ic e
bi
a2
Figure B .l: Scematic of the 2-port device
The above equations can also be in vector form w ritten as:
b = S ■a
(B.4)
or, better still, in m atrix form, which is more useful for our purpose:
61
&2
S u
_ S21
ai
S12
S22
_
az
Note for the m athem atically inclined (a.k.a nerdy): each of the four quantities bi,
6 2 , 0 .1 ,
and a 2 is independent of all others, and so these are four linearly-independent quantities. This
purely m athematical fact deduced from common-sense will help us later.
This definition of the so-called S-matrix is good-enough for anyone involved in making
measurements, and the four S-parameters have the (by now obvious) meanings:
B.2
The need for a T-matrix
All this is fine for a single device, but w hat if there is a series of 2-port devices? Taking
the familiar example from our lab of many waveguide devices in series, we see immediately th a t
while we care about characterizing every single device, our eventual aim is not to slog away
tediously trying to figure out how the input from one device becomes the output of another,
b u t to figure out the effect of all the series devices at the same time.
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206
Note, however, th a t this is not really possible w ith the S-matrix, since the inputs and
also the outputs are on both sides of the device. Therefore, we need to change into a system
where the inputs are both on the left (right) and the outputs on the right (left). The simplest
thing to do then would be to have this formalism worked out such th a t the net effect of all
devices would be:
Neteffect = devicei x device 2 x devices x ... x device„
This is why we need the so-called T-m atrix. Here is how the formalism is defined:
instead of going from input to output (this is w hat the S-matrix does), we want to go
ai
bi
to
the m atrix
.. 1
*0CN
1
CL2
Now look at fig 1. The two quantities on the left are <i\ and b\ , and the two quantities
on the right are
<12
and
62-
So, very naively, we wish to go from
ai
b\
to
02
&2
we can w rite this as:
Right = T • Left
(B.6 )
or, a little more clearly, as:
B.3
C12
Tn
T 12
^2
T 21
T22
ai
(B.7)
.
61
.
Conversion between S- and T-m atrix
W hen we make measurements of a device, it makes sense to think in term s of S-parameters,
especially since those are w hat all network analyzers output. So, we need to figure out a way
to change from S-parameters to T-param eters and back. Lets try to figure out the former first:
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207
Essentially,there are four equations we need
to work with for the four T-parameters:
bi = S 1 1 O1 + S \ 2a2
(B-8)
b2 = S2iai + S 2 2 Q2
0^-9)
«2 = T \ \ cl\ + T 12&1
(B.10)
T22b\
(B -ll)
62 = r 21ai +
Equations 9 and 11 imply
Sna-i + 522^2 = Taiai 4- I 2 2 (S’n a i + S i2a2)
(^-12)
Similarly, equations 8 and 10 imply
0-2
— T u ^ i + 7i2 (S'hOi + S 1 2 O2 )
(B.13)
Equation 12 is
-S^ioi + S 220-2 — 72iai + T22S\\a\ + T 22 S 12 C12
or, grouping term s
w ith a\ and
Oil [521 — 721 —
0,2
(B-14)
separately:
7 2 25 h ]
+ 0 2 [ 5 22 — T
^ -^ ] = 0
(B.15)
Each ofthe two brackets must equal separately, since a\ and a 2 are independent, so th a t the
second bracket yields
Soo
S 22 - T 2 2 S 12 = 0 ^ T 22 = ~
012
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(B.16)
208
Now substitute this value of T22 into the equation we get from the first bracket
S 1 2 S 2 1 — ^ 22 ‘S'r 1
$21 —T 21 — T 2 2 S 11 — 0 =>• T21 — S 21 —T 2 2 S 11 =>■ T 21
* S 'l2
Now look at equation 13, which reads
0-2 = T n a \ + T \ 2 S n a \ + T i 2
As above, we collect term s with
5i 2a 2
(B.18)
and a 2
01
a2 [1 —T125 i 2] —
Tn
’ 11
=
+
0
(B.19)
’ 12
Again, using the linear independence of a\ and a2 we get from the first bracket
Ii2 =
1
(B.20)
S 12
and from the second bracket
Tn
+
~
=
0
= *
Tn
<->12
(B.21)
S l2
We can now write out our T -m atrix in term s of elements of the S-m atrix thus
Tn
. T21
t
—S l l
12
T 22
S l2
S iq S ^ I—S^^S m
Si2
1
S l2
S 22
S l2
(B.22)
_
It turns out th at we can m anipulate the same equations to express the S-matrix in term s
of the T-m atrix thus
S21
S22
T
_
h l
T12
J-
T 1 2 T 2 1 -T 2 2 T 1 1
Z22.
Ti2
T l2
|------
A
S 110
2 1
1^
SSA
n
(B.23)
.
A nother note for the vector-space inclined: w hat we have done essentially is changed
from the Input-O utput basis to the Right-Left basis, and found the corresponding change in
the transform ation m atrix.
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209
A p p en d ix C
R elationship b etw een t and 6
CMB anisotropy is usually “broken-down” in spherical harmonics
at
w
0
<m > = E E
£
m
with the power spectrum
Q = X > ™ !2
(c -2)
m
However, how are ‘9:(angular scale on the sky) and I related? The physical intuition is th a t the
sky is divided into i parts, sothere m ust be an inverserelationship between the two.
Looking
a t how ‘0’ is defined in spherical-polar co-ordinates, we notice th at 9: 0 —» n. We are then
essentially dividing this angle into LP parts, so th at
e= j
(C.3)
In reality, this relation is approxim ate and holds only for small-enough (~ 5 — 10°) angles.
W hat is the most general relationship between 9 and £?
To answer this question, we first have to make sense of scales on a non-flat geometry; since
for us, relative scales make sense when represented in a flat geometry. So let us represent our
sphere on a flat sheet of paper - the only way to do this is via the Stereo g ra p h ic p r o je c tio n
(figure C .l).
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210
We
place a sheet of paper so
th a t it forms a tangent and draw a line
from point ‘A ’
through the point ‘P ’(at an angle 9 to the origin ‘O ’) onto the sheet. ‘BC ’is then the
length
we wish to calculate. We can write
^
=> r
=> r
~
(C.4)
=
A
2 f ? c o t-
(C.5)
=
Q
2 cot -
(C.6)
with a unit circle.
r is oc £, and this is the relationship between them.
90-0 /2
0/2
e/2
2R
e/2
Figure C .l: Stereographic projection
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211
A pp endix D
Inflaton field equation o f m otion and slow -roll
conditions
This is my attem pt to heuristically ‘derive’ the infiaton-field equation of motion (which is fairly
straightforward) and the slow-roll conditions, which every set of notes or book/s define/quote in
their own way. Fed up of the lack of consensus in literature, I attem pt to follow the convention
th a t makes most sense to me. Usual health warnings apply: this is my attem pt a t understanding
these topics, so I make no claim about these notes being right.
D .l
The equation of motion
Let us start from the first law of thermodynamics:
dU + p d V — 0
where,
(D .l)
naturally, U= pa3,where p is the density (total energy density, b u t since we are in
the inflation-era, p isdom inated by the energy density of the inflaton field <■/>) and a the scale
factor, which is a function of time.
Substituting for U, we get
a3dp + 3a2dap + 3a2dap = 0
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(D.2)
212
=>■ 3a2 (p + p) = —a3dp
(D.3)
Now, for a field, we have the following expressions for pressure and energy density:
p = K E + P E = l ( ^ 2 + V(0)
(D.4)
p = K E - P E = l- { ^ j
(D.5)
-V(<f>)
where the second equation can be derived from the general expression for the energy-momentum
tensor. See, for instance, [1], Prom these two expressions, we get
(E \2
(D.6)
) da = —a3dp
(D-7)
p + e, == l a )
which we can now substitute in eq. 3 to get:
2
3a2 (
dp
da
3 / d<p\2
a \d t J
(D.8)
Now, if we differentiate the expression for p w.r.t. time, we get
B ut 7ft =
dp = d4(Pp
dV^dcp
dt
d<p dt
dt dt2
>so the above equation becomes, after substituting for ^
3 da { d<p\
a dt \ d t )
A fter cancelling
d(pcP(j)
d V dtp
dt dt2 ^ d<f> dt
from every term ,
_ 1 dad4
a dt dt
dV
dt2 + d<t>
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m m
1
'
213
• H , the Hubble param ter, so th a t
B ut 1 d a is
Scj)
nTTd<j>
dV
J +3Hi +^ =0
(cu2>
This, then, is th e equation of motion.
Let us now slip into a more comfortable notation: ^ = <j>and ^
= V '. The equation of
motion is
0 + 3 ff0 + 7 ' = O
D .2
(D.13)
Slow-roll conditions
In order to sustain inflation for long enough to solve the horizon problem etc., we need
the inflaton field to move slowly. The two conditions can be w ritten as follows:
1. Define slow:
, or, KE < < PE
2. Keep it slow:
\<j>\ «
\ZH4>\
(D.15)
The equation of state then changes to
3H<j> + F ' ~ 0
(D.16)
Aside: while sliding from one sordid equation to the other, remember th at
TTo
8-ttG
87tG
H 2 = - — - p ~ — V (<f>)
(D.17)
constant during inflation. The approxim ate equation of motion means th a t
(D.18)
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214
1 12
V*
2 ^ ~ 18H 2
^
(D.19)
Substituting for H 2 from above,
V '2
18^V
1
2
V '2
48ttG V
(D.20)
V
(D.21)
Applying the first slow roll condition, we get
y>2
«
487tG V
or,
1
, V' .
«
V487rG V
1
(D.22)
This leads us to define our first slow-roll param eter
e = —r
V’
|— | «
1
1
(D.23)
To work out the second slow-roll condition, differentiate equation 18 w .r.t time again to
get
Y
(D.24)
3H
so th a t the second slow-roll condition is
-V"(b
•
ITT «
V"
91J3 « 1
(D-25>
Substituting for H 2 again from eq 17
V"
«
24ttG V
1
(D.26)
This leads us to define our second slow-roll param eter
" S 2 U V <<1
<D '27)
The use of 7/ is unfortunate, since this is the same greek letter used in literature to denote
conformal time. However, this is the convention followed in literature, unfortunately.
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215
A p p en d ix E
E-B decom p osition
E .l
Stokes’ parameters
W hen dealing w ith tem perature anisotropies, it is conventional to ignore polarization.
However, CMB photons are polarized, and we need to think about how to characterize tem ­
perature and polarization at the same time. The reason we need to consider all of them at the
same time is because our instrum ents are capable of measuring only intensities, and not the
amplitudes of radiation falling on them.
The most “common-sense” way to characterize polarization is to figure out the difference
between the intensities along the two rectangular-coordinate axes. This is referred to as Stoke’s
Q and its definition is easily extended to the case of circular polarization. All this is very well,
b u t how many independent quantities do we need to characterize the radiation field?
Consider this: detectors are sensitive to intensities, which ~ amplitude-squared. We are
therefore dealing with two electric fields and their phases, i.e. four quantities. We therefore
need four quantities to completely characterize radiation. (Logic suspect).
But how do we represent these four quantities? In quantum mechanics, we represent
observables by herm itian matrices.
The four quantities, then, should be w ritten as a 2x2
herm itian m atrix of observable quantities, two of which are intensity and Stoke’s Q.
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216
In general, a 2x2 herm itian m atrix can be w ritten as
a
b + ic
b — ic
d
In our case, this m atrix happens to be
I +V
Q + iU
Q -iU
I +V
where the four quantities are defined as
I = {El ) + {El)
(E .l)
Q = <%)-<%)
(E.2)
U = (2ExE y cosS)
(E.3)
V = (2ExE y s m S )
(E.4)
where 5 is the phase difference between E x and E y and the unit vectors are (ex, ey). The
definitions are very similar in the (e 0 ,e^) basis:
I = (El) + (El)
(E.5)
Q = <£?) - ( E l )
(E. 6 )
U = (2EgE^cos5)
(E.7)
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217
V — (2EgE^sind)
W hen V = 0, as is the case with CMB (i.e.
(E.8)
the CMB is not circularly polarized),
polarization of the CMB is completely characterized by Q and U. Here are the transform ation
properties of Q and U:
y
\
/
V
E .2
cos 2x1)
sin 2xp
\ /
—sin 2xp cos 2xp
Q .
(E.9)
U
R e l a t i o n s h i p b e tw e e n E - B a n d Q -U
Looking at the expression Q + iU gives us the idea th a t they could be represented by
Qx + U y
(E.10)
However, we must remember the transform ation properties of Q and U stated above. These
imply th a t Q and U transform into each other after a rotation of 45°, and therefore in this
basis we cannot write the abve expression. However, if we define <j>= 2xp, and work in a basis
/ co-ordinate system where angles go from 0 — 180°.
We can now proceed with the m ath of E-B decomposition.
In very simple term s, w hat
we want is to split both Q and U into a gradient component and a curl component. B ut th a t
is easily done, for any vector field can be w ritten as a sum of the two. For an arbitrary vector
field A , we write
A = V -/ + V x B = 0 + C
( E .ll)
where / is a scalar field and B is a vector field, and Q and C are the gradient and curl components
of A respectively. Using vector calculus identities, we get
V •A = V •Q
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(E.12)
218
V x A = V xC
(E.13)
Substitute Q x + U y for A and E and B for Q and C respectively, we get
V •E = V •
V
x
B = V
x
(Qx+ Uy)
( Qx + Uy)
(E.14)
(E.15)
We cannot really do very much else in real space, so lets take the fourier transform of the first
equation, changing all derivatives to factors of I, and take the V out of the integral:
V • J d2x E e ilr = V • J d2x (Qx. + U y) eil r
(E.16)
Remember the definition of V:
V = | x + | y
(E.17)
and with r — x x + y y we get
/ d2X
+
’C,' (COS(X^j+Sin(^ )) = J d2X ^ X
+ | ^ y ) •e*Kcos(x0)+sin(?/</,))c^ + v ~
(E.18)
But
( — x + — y ] eiilcos(^ )+ sin(^)) = il (cos 0x + sin</>y) e«(cos(^)+sin(^))
\d x
dy J
,E _19j
=$■ji J. J d2x (cos <j>k + sin <f>y) - Ee*l r = / / J d2x (cos 0 x + sin</>y) • (Q x + U y) e*l r
(E.20)
J d2x E e lXr = J d2x (Q cosfi + U sm<f)) eJ, r
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(E.21)
219
Denoting fourier transform s like this: E , we get:
E = Q cos (f>+ U sin <f>
(E.22)
E = Q cos 2ip + U sin 2ip
(E.23)
B = —Qsin2'ip + U cos2ip
(E.24)
Now restore <f>= 2ip:
Similarly, for B , we get:
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