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The millimeter -wave bolometric interferometer (MBI) for observing the cosmic microwave background polarization

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The Millimeter-wave Bolometric Interferometer
(MBI) for Observing the Cosmic Microwave
Background Polarization
by
Jaiseung Kim
Brown Univ. Sc.M.
Yonsei Univ. B.S.
Thesis
Submitted in partial fulfillment of the
requirements for the Degree of Doctor of Philosophy
in the Department of Physics at Brown University
Providence, Rhode Island
May 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 .
UMI Number: 3227866
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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 .
© Copyright 2005 by Jaiseung Kim
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 .
This dissertation by Jaiseung Kim is accepted in its present form by
the Department of Physics as satisfying the dissertation requirement
for the degree of Doctor of Philosophy.
Date
l o / u ) OS'
Gregory S. Tucker, Director
Recommended to the Graduate Council
Date j O j l l / O $
Ian P. Dell’antonio, Reader
Date
hard J. Gaitskell, Reader
Approved by the Graduate Council
Date
1 0
Sheila Bonde
Dean of the Graduate School
111
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 .
V ita
Brown University, Ph.D., Physics (Advisor: Gregory S. Tucker), August 2005
M ilitary/Community Service, October 2000 - April 2002
Brown University, M.S., Physics, Dec. 1999
Yonsei University, B.S., Electrical Engineering (Minor : Physics), February 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 .
Preface and A cknow ledgm ents
I appreciate the members of the Millimeter-wave Bolometric Interferometer (MBI)
project for their great works and collaboration, and believe the MBI will harvest a
scientifically significant result. For the simulation and analysis, we acknowledge the
use of Mathematical Libraries by ‘Numerical Recipe in C’ and Astronomical libraries
(HEALPix C library and Flexible Image Transport System (FITS) C library). We also
acknowledge and appreciate the use of the Legacy Archive for Microwave Background
D ata Analysis (LAMBDA). Support for LAMBDA is provided by the NASA Office
of Space Science.
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 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 ontents
V ita
iv
Preface and A cknow ledgm ents
v
List o f Tables
xi
List o f Figures
xii
1
2
T he
C osm ic M icrowave B ackground
1
1.1 Anisotropic Fluctuation of the CMB I n te n s ity ........................................
1
1.2 Polarization A nisotropies..............................................................................
5
1.3 D iscussion........................................................................................................
13
T he
M easurem ent by th e M illim eter-w avelength B olom etric Inter­
ferom eter (M B I)
15
2.1 Principle of an In te rfe ro m e te r.....................................................................
15
2.1.1
Adding In te rfe ro m e te r....................................................................
16
2.1.2
Correlating In te rfero m ete r.............................................................
16
vi
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3
4
5
2.2
Interferometer Observation of the CMB I n t e n s i t y ................................
17
2.3
Polarization Interferometer Observation of the C M B ............................
19
2.4
The Visibilities of the M B I.........................................................................
21
2.5
Noise-Equivalent-Power (NEP) of the MBI O b s e rv a tio n s ...................
25
2.6
D iscussion ......................................................................................................
27
A perture Syn thesis
29
3.1
The Relation between the Sky Brightness and V isib ilities...................
30
3.2
Optimal uv C overag e...................................................................................
31
3.3
The M B I..........................................................................................................
34
3.4
Simulation of Aperture Synthesis for the M B I ......................................
41
M axim um Likelihood E stim ation o f th e C M B E /B Pow er Spectrum
from th e sim ulated M B I observation
47
4.1
Spherical Sky and Flat Sky Approximation in the Small Angle Limit
48
4.2
Visibilities as the Linear Sum of Spherical Harmonic Coefficients . . .
49
4.3
Power Spectra E s tim a tio n .........................................................................
49
4.4
Computing on/off Diagonal Window Function N u m erically................
53
4.5
The Simulated MBI O bservation................................................................
54
4.6
Pointing E r r o r ................................................................................................
63
Foregrounds and th e M B I
70
5.1
70
The Physical Characteristics of Galactic F o re g ro u n d s.........................
vii
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5.2
The Simulated Q /U Map of the Polarizer F o reg ro u n d .........................
5.3
The Effect of Galactic Foreground Contamination on the CMB Power
Spectrum Estimation in the M B I .............................................................
6 A ll-sky Im aging o f th e C M B T em perature A nisotropy
7
73
76
83
6.1
Visibility Sourced by the CMB I n t e n s i ty ................................................
83
6.2
Determination of Individual Spherical Harmonic Coefficient...............
85
6.2.1
Simple Inversion in the Absence of N o is e ....................................
87
6.2.2
Maximum Likelihood Method in the Presence of Noise
....
87
6.2.3
Foreground C ontam ination.............................................................
89
6.3
Simulated O b se rv a tio n ................................................................................
89
6.4
Computation of bitim ...................................................................................
99
6.5
Computational L o a d ...................................................................................
100
6.6
Computing bim* in the Antenna Coordinate
.........................................
101
6.7
D iscussion ......................................................................................................
103
A ll-sky Im aging o f th e C M B P olarization A nisotropy
104
7.1
Visibility as the Linear Sum of Spherical Harmonic Coefficients . . . .
104
7.2
Determination of Individual Spherical Harmonic Coefficient...............
105
7.2.1
Foreground C ontam ination .............................................................
107
7.3
Simulated O b se rv a tio n ................................................................................
108
7.4
Computation of b±2,im ................................................................................
122
viii
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8
7.5
Computing in the Antenna C o o rd in a te ...................................................
123
7.6
D iscussion......................................................................................................
125
Future W orks
127
A A p p en d ix
A .l
129
Numerically Computing Spin 2 SphericalH arm onics..............................
129
A.2 Simulating v isibilities...................................................................................
131
A.2.1
W ith flat sky ap p ro x im atio n.........................................................
131
A.2.2
W ithout flat sky ap p ro x im a tio n ...................................................
133
A.3 Aperture S y n th e s is ......................................................................................
136
A.3.1
Search for the feed horn configuration by simulated annealing
136
A.3.2
Aperture Synthesis S im ulation......................................................
139
A.4 The Power Spectra Estimation by the Maximum Likelihood Method .
142
A.4.1
Quadratic Optimal E s t im a t o r ......................................................
142
A.4.2
Visibility covariance m atrix of the M B I ......................................
143
A.4.3
Fisher m atrix of the MBI
.............................................................
144
................................................................................
145
A.5 Antenna Temperature
A.6 Microwave frequency bands
......................................................................
145
A.7 Foregrounds Sim ulation................................................................................
145
A.7.1
Simulation of Polarization Angle for foregrounds......................
145
A.7.2
Simulation of Q and U map for fo re g ro u n d s .............................
146
ix
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A.8 All-sky Imaging of the CMB
T e m p e ra tu re .........................................
147
A.8.1 Resolving a .......................................................................................
147
A.8.2 Antenna pointings
147
..........................................................................
A.9 All-sky Imaging of the CMB
p o la riz atio n .............................................
148
A.9.1 Computing b for all sky-imaging of the CMB tem perature . .
148
................
150
A. 10 Computing b for all sky-imaging of the CMB polarization
x
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List o f Tables
1.1
Cosmological parameters of the A C D M cosmological model from WMAP+SDSS
s tu d y .................................................................................................................
2.1
The physical parameters of the MBI
3
.......................................................
25
2.2 The noise budget of the M B I.......................................................................
27
3.1 The baselines of the closepackhorn a r r a y ..................................................
39
3.2 The average error in synthesized m a p s ......................................................
46
A .l Microwave frequency bands
.......................................................................
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146
List of Figures
1.1 Power spectrum of tem perature a n iso tro p y ..............................................
4
1.2 Foreground cleaned tem perature anisotropy observed by the WMAP
satellite
..........................................................................................................
5
1.3 Power spectrum of E mode polarization
.................................................
9
1.4 Power spectrum of B mode polarization
.................................................
10
..............................................................................................
11
1.6 simulated Q m a p ...........................................................................................
12
1.7 simulated U m a p ...........................................................................................
13
2.1 the MBI horn c o n fig u ra tio n ........................................................................
23
2.2 the beam pattern of a MBI feedhorn
.......................................................
24
3.1 uv coverage of the MBI 1 ..............................................................................
35
3.2 The feed horn configuration of the MBI 2 .................................................
36
3.3 The feed horn configuration of 64 feed horns in the MBI 2 .................
37
3.4 uv coverage of the MBI 2 ..............................................................................
38
1.5 T E correlation
xii
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3.5
the closepack horn a r r a y ............................................................................
40
3.6 uv coverage of the closepack horn a r r a y .....................................................
40
3.7 30° x 30° input sky patch of CMB anisotropy T e m p e ra tu re .................
41
3.8 30° x 30° input sky patch of CMB Q p o la riz a tio n .................................
42
3.9 30° x 30° input sky patch of CMB U p o la riz a tio n .................................
42
3.10 uv coverage of the reference map
............................................................
43
3.11 Reference A T ................................................................................................
44
3.12 From noiseless visibilities............................................................................
44
3.13 From noisy visibilities...................................................................................
44
3.14 Reference Q ...................................................................................................
45
3.15 From noiseless visibilities............................................................................
45
3.16 From noisy visibilities...................................................................................
45
3.17 Reference U ...................................................................................................
45
3.18 From noiseless visibilities............................................................................
45
3.19 From noisy visibilities...................................................................................
45
4.1
Window function of the MBI 1 ....................................................................
52
4.2
Window function of the MBI 1 ....................................................................
56
4.3
E mode Power spectrum estimation from the simulated MBI 1 obser­
vation
4.4
.............................................................................................................
57
The upper bound on B mode Power spectrum by the simulated MBI
1 o b s e r v a tio n ................................................................................................
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58
4.5 Window function of the MBI 2 ....................................................................
59
4.6 E mode power spectrum estimation from the simulated MBI 2 observation 60
4.7 Upper bound on B mode power spectrum by the simulated MBI 2
....................................................................................................
60
4.8 window function of the closepack horn a r r a y ...........................................
61
observation
4.9 E mode Power spectrum estimation from the simulated closepack MBI
observation
....................................................................................................
62
4.10 B mode Power spectrum upper bound and estimation by the simulated
closepack MBI o b serv atio n ..........................................................................
63
4.11 the
RMS error of the visibility due to the
pointing error of the faceplate 67
4.12 the
RMS error of the visibility due to the
horn alignmenterror . . . . 68
5.1 free-free emission at K b a n d ........................................................................
71
5.2 dust at W b a n d ..............................................................................................
72
5.3 synchrotron radiation at K b a n d .................................................................
72
5.4 simulated Q map of polarized foregrounds at W band (brightness tem­
perature) ..........................................................................................................
74
5.5 simulated U map of polarized foregrounds at W band (brightness tem­
perature) ..........................................................................................................
5.6
E mode power spectrum of the CMB and
foregrounds.........
76
5.7
the
field location of the MBI o b s e rv a tio n ............................
77
xiv
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74
5.8 E mode power spectrum estimation from the simulated MBI observa­
tions of the field (D ec.=—49.34°, R .A .=19.38°).......................................
78
5.9 E mode power spectrum estimation from the simulated MBI observa­
tions of the North Celestial Pole (N C P )....................................................
79
5.10 E mode power spectrum estimation from the simulated MBI observa­
tions of the South Celestial Pole ( S C P ) ....................................................
80
5.11 E mode power spectrum estimation from the simulated MBI observa­
tions of the field (Dec.= —28.76°, R .A .= 2 6 5 .7 1 °)....................................
81
6.1 window function (normalized to its p e a k ) .................................................
90
6.2 antenna pointing assumed in the s im u la tio n ...........................................
91
6.3 simulated T m a p ...........................................................................................
91
6.4 foreground at W b a n d .................................................................................
92
6.5 Power spectrum estimation from noiseless visibilities..............................
93
6.6 Reconstructed tem perature anisotropy map from noiseless visibilities
93
6.7 the CMB tem perature anisotropy reference map (44 < / < 69) . . . .
94
6.8 Power spectrum estimation from noisy visibilities with one day inte­
gration time for each antenna p o i n t i n g ...................................................
95
6.9 Reconstructed tem perature anisotropy map from noisy visibilities with
one day integration time for each antenna pointing
.............................
95
6.10 Power spectrum estimation from noiseless visibilities in the presence of
fo re g ro u n d s ....................................................................................................
xv
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96
6.11 Reconstructed tem perature anisotropy map from noiseless visibilities
in the presence of fo re g ro u n d s..........................................................
97
6.12 Temperature anisotropy map (44 < I < 69), including foregrounds . .
97
6.13 Power spectrum estimation from noisy visibilities with 1 day integra­
tion time in the presence of fo re g ro u n d
98
6.14 Reconstructed tem perature anisotropy map from noisy visibilities with
7.1
1 day integration time in the presence of fo re g ro u n d s .................
98
antenna pointings assumed in the sim u la tio n ...............................
108
7.2 E mode window function (normalized to its p e a k ) .................................
109
7.3 B mode window function (normalized to its p e a k ) .................................
109
7.4
simulated Q m a p .........................................................................................
Ill
7.5
simulated U m a p .........................................................................................
Ill
7.6
simulated
Q map of polarized foregroundsat W b a n d .................
112
7.7
simulated
U map of polarized foregroundsat W b a n d .................
112
7.8 E mode Power spectrum estimation (noiseless v isib ilities)....................
113
7.9 B mode Power spectrum estimation (noiseless v isib ilities)....................
113
7.10 Reconstructed Q map from
noiseless
visibilities
....................
114
7.11 Reconstructed U map from
noiseless
visibilities.......................
114
7.12 the CMB Q map (44 < I < 69)
115
7.13 the CMB
115
U map (44 < / < 6 9 )........................................................
xvi
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7.14 E mode Power spectrum estimation from noisy visibilities with seven
days integration t i m e ....................................................................................
116
7.15 B mode Power spectrum estimation from noisy visibilities with seven
day integration time
........................................................................
116
7.16 Reconstructed Q map from noisy visibilities with seven day integration
t i m e .................................................................................................................
117
7.17 Reconstructed U map from noisy visibilities with seven day integration
t i m e .................................................................................................................
117
7.18 E mode power spectrum estimation map (no noise case in the presence
of foregrounds)
.............................................................................................
118
7.19 B mode power spectrum estimation map (no noise case in the presence
of fo re g ro u n d s ).............................................................................................
118
7.20 Reconstructed Q map from noiseless visibilities in the presence of fore­
grounds
..........................................................................................................
119
7.21 Reconstructed U map from noiseless visibilities in the presence of fore­
grounds
..........................................................................................................
119
7.22 Q map (44 < I < 69),
including fo re g ro u n d s .......................................
120
7.23 U map (44 < I < 69),
including foregrounds
120
.......................................
7.24 E mode power spectrum estimation map (noisy visibilities with seven
days integration time in the presence of fo reg ro u n d )..............................
xvii
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121
7.25 B mode power spectrum estimation map (noisy visibilities with seven
days integration time in the presence of fo reg ro u n d )............................
121
7.26 Reconstructed Q map from noisy visibilities with seven day integration
time in the presence of fo re g ro u n d s ..........................................................
122
7.27 Reconstructed U map from noisy visibilities with seven day integration
time in the presence of fo re g ro u n d s..........................................................
xviii
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122
Abstract of “The Millimeter-wave Bolometric Interferometer (MBI) for Observing the
Cosmic Microwave Background Polarization” by Jaiseung Kim, Ph.D., Brown Uni­
versity, May 2006.
This thesis describes the Millimeter-wave Bolometric Interferometer (MBI) to mea­
sure the Cosmic Microwave Background Polarization (CMBP) anisotropy at angular
scales 0.5° — 1° and a center frequency of 90 GHz. The measurement of the CMBP
anisotropy on these angular scales will put more stringent constraints on cosmo­
logical models and parameters. The prototype instrument employs four corrugated
feedhorns and cooled bolometers. Using a Butler beam combiner, beams from four
feedhorns are correlated, yielding interferometric measurements of the CMBP. From
these interferometric measurements, we can reconstruct the image of polarization by
aperture synthesis and estimate the power spectrum of the CMBP by maximum like­
lihood method. We describe aperture synthesis and maximum likelihood method.
We present the result of the image reconstruction and the power spectrum estimation
from simulated MBI observations. W ith the planned sensitivity of the MBI, the MBI
will be able to estimate the E mode power spectra of the CMBP in the multipole
range (150 < I < 300) and put upper bounds on the B mode power spectra in the
relevant multipoles. In the end, we describe all-sky imaging method from interfero­
metric measurements developed for the Einstein Probe Interferometer for Cosmology
(EPIC), which is the satellite version of the MBI.
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C hapter 1
T he C osm ic M icrowave
Background
The Cosmic Microwave Background (CMB) photons are primordial light which have
cosmological origin.* The CMB is quite close to uniform 2.725 K[l] blackbody radia­
tion, but there are very tiny intensity and polarization fluctuations dependent on the
incoming direction of the CMB photon. The anisotropy of the CMB is caused by the
energy density inhomogeneity of the last scattering surface C
1.1
A nisotropic F lu ctu ation o f th e C M B In ten sity
The intensity of the CMB is given by
f) T ^(u rF \
I Cm b ( n , v ) ^ B ( i s , T 0) + d B ^
dT
A T (n , i/),
T)
T=X0
‘These photons already existed before any astrophysical structures form. Photons from astrophysical origin are generated mainly by the formation of galaxies or stars.
^The last scattering surface is where the CMB photons has the last scattering with free electrons.
It is away from us by z ~ 1000.
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 .
where n indicates the direction in the sky, u is the frequency, B(u, To) is the Planck
function and To = 2.725 ± 0.002 K[l] is the monopole tem perature of the CMB. The
frequency spectrum of the CMB anisotropy is
dB[u, T)
dT
t
=t 0
\2
„. „ {I
xT"2
24.8
J y s r - ^ K ) - 1,
\s in h x /2 /
)
where x = h v / k T 0 ~ i//(56.8GHz) [2], All-sky tem perature anisotropy of the CMB
was mapped by the WMAP satellite [3]. The Mollweide projection of the foreground
cleaned map from the WMAP CMB tem perature map [4] is shown in Fig. 1.2.
W ith the completeness of spherical harmonics, any function f(9, (ft) of spin-zero
evaluated over the surface of a sphere can be expanded in a series of spherical har­
monics such th a t f(6,(ft) = Y2i,maimYim(9,4>) [5]- So, for all-sky analysis, the CMB
anisotropy tem perature can expanded in spherical harmonics [6]:
oo
A T (n) =
I
EE
1=1 m = —l
Inflationary cosmology, which is the most successful model for explaining the origin
of universe, postulates th at these fluctuations are seeded in the very early universe
by quantum perturbation with random phase, followed by a period of inflationary
expansion [6, 7]. In inflationary cosmology, the primordial anisotropies of the CMB
are Gaussian and can be completely characterized through their power spectrum [7].
W ith the Gaussianity of the CMB, ar j/m in Eq. 1.1 has the following stochastical
properties:
( a r, Zma 2q' m' ) =
CfT
,
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3
where the angle bracket denotes an average over the ensemble universe and C f T is
the tem perature power spectrum on the angular scale ~ 180°//. This angular power
spectrum of the tem perature anisotropies contains information on the cosmological
parameters and the structure th at existed at decoupling. In standard cosmological
models, the dominant processes are acoustic oscillation of the primordial plasma and
photon-diffusive damping, which give rise to a harmonic series of peaks in the CMB
anisotropy spectrum modulated by an exponential cutoff on small angular scales [6,8,
9]. The power spectrum of tem perature anisotropy with AC D M cosmological model
and cosmological parameters from WMAP+SDSS study [10] is shown in Fig. 1.1. The
values of cosmological parameters from W MAP+SDSS study [10] are summarized in
Table 1.1, where h is the Hubble constant and
is the ratio of dark m atter density
to the critical density and flb is the ratio of baryon density to the critical density.
parameters
KzDd
h2n b
Dark energy density Da
Scalar spectral index n s
Tensor spectral index n t
Optical Depth r
value
0.1222
0.0232
0.699
0.977
0
0.124
Table 1.1: Cosmological parameters of the AC D M cosmological model from
WMAP+SDSS study
As the CMB photon travels, its path is affected by the intervening mass through
gravitation, which results in the weak-lensing effect. The weak-lensing effect modifies
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 .
4
5000
original
weak-lensed
4500
4000
3500
=t
tn
O
k
3000
S. 2500
+_
~
2000
1500
1000
500
500
1000
1500
Figure 1.1: Power spectrum of tem perature anisotropy
the original anisotropy of the CMB on the last scattering surface *. The tem pera­
ture anisotropy power spectrum of the weak lensed CMB is shown also in Fig. 1.1.
The CMB photons traverse the large-scale structure of the universe and interact with
them, which generates the secondary anisotropies [8, 11]. The secondary anisotropies
*The CMB photon are most likely to be last-scattered at z ~ 1000.
R e p r o d u c e d w ith 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 r ep ro d u ctio n p ro h ib ited w ith o u t p e r m issio n .
5
Figure 1.2: Foreground cleaned tem perature anisotropy observed by the WMAP satel­
lite
are directly related to the structure formation, evolution and clustering in the uni­
verse and hence offer the prospect of studying the formation of large-scale structure
at recent times [8, 11]. While the primordial anisotropies, which are Gaussian, can
be completely characterized through their power spectrum, secondary anisotropies,
which are generated through non-linear effects, are not Gaussian and are not com­
pletely characterized through their power spectrum [8, 11]. So imaging the CMB is
very im portant for the sake of the study of secondary anisotropies.
1.2
Polarization A nisotropies
The Cosmic Microwave Background (CMB) is expected to be linearly polarized by
Thompson scattering on the last scattering surface and after re-ionization [6, 12, 8, 9].
Recently, the Degree Angular Scale Interferometer (DASI) collaboration has reported
the detection of the CMB polarization [13, 14, 15] and correlation between temper­
ature and polarization has been reported by the Wilkinson Microwave 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 .
Probe (WMAP) satellite team [16]. The CMB polarization can be decomposed into
gradient-like E mode and curl-like B mode [17, 18, 19, 20]. B mode polarization
is not induced by scalar density perturbation but only tensor perturbation while E
mode polarization gets induced by both [19, 20, 21]. Characteristic signatures on
the CMB polarization imprinted by re-ionization and primordial gravity wave (tensor
perturbation) can be used to constrain the epoch of re-ionization and the inflation
model parameters [19].
In the general description of electromagnetic wave polarization, there are four
Stokes parameters /, Q, U, V [22, 23]. In an all-sky study of the CMB polarization,
the Stokes parameters are measured with reference to (eg, ep) which are the unit
vectors of the spherical coordinate system [17, 18]. When Eg is the electric field along
9 and Ep is the electric field along <j>,
I =
{E]) + { El ) ,
Q =
<E * ) - { E l ),
U =
(2Eg Ep cos 5),
V
(2Eg Ep sin 5),
=
where 5 is the phase difference between Eg and Ep, and (...) indicates the time
average. Since in the early universe Thomson scattering does not generate circular
polarization but linear polarization, the phase difference between Eg and Ep is zero
[22] and circular polarization state V of the CMB polarization on the last scattering
surface is set to zero [6]. So Q and U completely specify the CMB 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 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 .
while I specify the CMB intensity. While I is invariant under rotation in the plane
perpendicular to the direction n, Q and U transform under rotation of an angle b
[18]:
Q;(n)
=
Q (n) cos 2ip + U (n) sin 2^,
(1.2)
U’{n)
=
—Q(n) sin 2-0 + t/(n )
(1.3)
008
2^.
Prom these, the following quantity can be constructed [18, 17]:
(Q ±iU y(n) = e ^ ( Q ± i U ) ( n ) .
(1.4)
For all sky analysis, Q(n) and U(h) are expanded in terms of spin ± 2 spherical
harmonics [17, 18, 20]:
Q(h) + iU(n)
= y \ 2tim 2 Vfm(n),
(1.5)
/,ra
Q (n)-iU (n)
= ^
a-V m - 2^m (n),
(1.6)
l,m
where
2Ylm{d,4>) =
- 2> W M )
^ ‘2^ [ F Urn{e) + F%lm(d)}eimt
- F%lm{0)\eim^.
=
(1.7)
(1.8)
Fi im and p 2 ,/m can be computed in terms of Legendre polynomials [24], The CMB
polarization can be decomposed into gradient-like E mode and curl-like B mode
[17, 18, 19, 20]. The E / B decomposition mode can be formed from spin ± 2 spherical
harmonics a±2 j m as follows [17, 18, 19, 20, 21]:
® E,lm
=
( a 2,lm "t" a - 2 , ( m ) / 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 r ig h t o w n e r . F urth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
8
&B ,lm
^(^2 ,/m
$ —2 , / m ) / 2 .
These two combinations behave differently under parity transformation [17, 18, 19,
20, 21]. So the correlation between B mode and E mode, and the correlation between
B mode and tem perature anisotropy vanish [21]. In the inflationary cosmology, E / B
mode of the CMB are Gaussian and can be completely characterized through their
power spectrum [6, 7]. W ith the Gaussianity of the CMB polarization,
— Cf
= Ci
(flT ,lm ® E ,l'm ' )
fill'fimm'i
C^
where the angle bracket denotes an average over the ensemble universe. W ith the
parity nature of the E / B mode, the cross correlation C EB and C TB vanish. In terms
of E / B decomposition mode, all-sky Stokes parameter Q and U can be expanded as
follows [17, 18, 19, 20, 21]:
l,m
s
;
£,m
where
x 2M(e,<f>) = ^ ± ^ F 2M(e)eimt
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 .
(1.10)
9
40
original
weak-lensed
^
25
A
in
LLI _
o
CK
NJ
+
500
1000
1500
Figure 1.3: Power spectrum of E mode polarization
The E and B mode power spectra of the CMB predicted by CMBFAST [25] are shown
in Figs. 1.3 and 1.4, together with weak-lensed E and B mode power spectra. In the
computation of the CMB E and B power spectra, we assume a ACDM cosmological
model whose parameters are from the W MAP+SDSS study [10] with the non-zero
tensor-to-scalar ratio (r = 0.161) This non-zero tensor-to-scalar ratio was obtained
with r = 7(1 —n s) and n s = 0.977, where this scalar spectral index n s is also from
the WMAP+SDSS study. The cosmological parameters assumed are summarized in
Table 1.1. Since weak lensing is generated by the structures formed at recent times,
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 .
10
0.08
original
0.07
weak-lensed
0.06
^
0.05
if °-04
CM
i
0.03
0.02
0.01
500
1000
1500
Figure 1.4: Power spectrum of B mode polarization
the effect of weak lensing is manifest at small angular scales. The boost of E and B
mode power spectra at low multipole (< 10) in Figs. 1.3 and 1.4 are attributed to
re-ionization [7].
Unlike the anisotropy tem perature map of the CMB, an all-sky map of the CMB
polarization is not available. So we will generate the simulated Q and U maps of the
CMB polarization and use them for the input sky maps in the simulation of the MBI.
To simulate all-sky Q and U maps of the CMB, aB im sets are drawn from a
Gaussian distribution with a variance which is given by CMBFAST [25]. Since aT irn
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
100
original
weak-lensed
*
=i
L
D
I—
o
-
CM
+
-50
-1 0 0
-150
500
1000
1500
Figure 1.5: T E correlation
and a,E,im are correlated with C f E, the generation of ajym and a,E,im sets require the
following procedure. First, a 2 x 2 m atrix M is constructed such th at the diagonal
components M u and M 22 are C f T and C f E, and the off-diagonal components M 12
and M 21 are C [ E. Then this 2 x 2 m atrix M is diagonalized by a rotation m atrix and
two independent Gaussian random numbers are generated with the variance whose
values are the diagonal values of the diagonalized matrix. These two independent
random numbers are rotated back by the inverse of the rotation m atrix used for
diagonalization.
Then, these two numbers give the ar,im,
a E ,im
with the correct
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 .
12
cross-correlation C j E [18]. W ith the set of aEj m and a/y/m, the simulated CMB sky
polarization maps can be generated by [18]:
Q (A ) + iU { h )
^
](
0'E ,lm
i'O'BJm) 2
(
1
.
1
1
)
l,m
To simulate an all-sky CMB polarization map §, spin 2 spherical harmonics need to be
computed as seen in Eq. 1.11. The mathematical form of spin 2 sphericalharmonics
and computer code, which computes spin 2 spherical harmonics, can be found in
Appendix A.I.
The Figs. 1.6 and 1.7 are the Mollweide projection of the simulated CMB Q and
U sky maps containing multipoles 2 < I < 1500.
Figure 1.6: simulated Q map
§All-sky CMB polarization map can be simulated by the widely used computer code such as
HEALPix [26, 27], but since we need to compute the spin 2 spherical harmonics for other
purposes such as window function computation discussed in the later chapters, we have developed
code to compute spin 2 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 .
13
Figure 1.7: simulated U map
1.3
D iscussion
The anisotropy of the CMB, imprinted at z ~ 1000 offers a view on the early Universe.
Since the prediction of the CMB anisotropy in given cosmological models for given
fundamental cosmological parameters can be made, the measurements of the CMB
anisotropy can constrain cosmological models and cosmological parameters. While the
CMB tem perature anisotropy has been measured up to multipoles I ~ 900 with good
accuracy by the WMAP [3], the CMB polarization anisotropy is still in the detection
stage [15]. Primordial CMB polarization anisotropy is even smaller than tem perature
anisotropy and has to be seen through various foregrounds in the presence of noise
just as the tem perature anisotropy observation
The measurement of primordial
CMB polarization anisotropy is even harder than the measurement of the tem perature
^Noise includes instrumental noise , atmospheric fluctuation and the Poisson noise of the CMB
photons themselves.
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 urth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
14
anisotropy measurement which has been unsuccessful for several decades.
We will discuss the Millimeter-wave Bolometric Interferometer (MBI), an instru­
ment being built to measure the anisotropy of the CMB polarization. Once we discuss
the instrument, we will present the data analysis method and the result of analysis on
simulated MBI observations. In the simulated observations, we will use the simulated
Q and U map of CMB polarization presented in this chapter.
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 urth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
C hapter 2
T he M easurem ent by the
M illim eter-w avelength B olom etric
Interferom eter (M BI)
The Millimeter-wavelength Bolometric Interferometer (MBI) is a ground-based interferometric instrument aiming to measure the Cosmic Microwave Background (CMB)
polarization. In this chapter, we will discuss what will be measured by the MBI,
which are called visibilities.
2.1
P rinciple o f an Interferom eter
The Degree Angular Scale Interferometer (DASI) [13, 14, 15], the Cosmic Microwave
Interferometer (CBI) [28] and the Millimeter-wavelength Bolometric Interferometer
(MBI) have used interferometers for the CMB tem perature or polarization observa­
tion.
Consider two identical apertures where one of the two apertures is positioned at
ri and the other at r2. The electric field signals from aperture A and from aperture
B can be expressed as E\ — E sin(;«A —k • rx) and E 2 = E s m ( w t — k • r2). Two
15
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 .
16
apertures are separated by r i —r 2 . This separation is called the ‘baseline’ [29, 30].
2.1.1
A d d in g Interferom eter
In an adding interferometer, the sum of two signals is squared and time-averaged [29]:
( ( £ 1 + £ 2)2>limi = / ( l + coS(k .B ) ),
where I is intensity, which is \E\2. By phase modulation* , only the interference term,
/c o s(k ■B ), is detected. To construct a complex quantity, I sin(k • B) is measured
as well by adding 90° phase delay to signal 1 [31, 32],
2.1.2
C orrelating Interferom eter
In a correlating interferometer, two signals are multiplied and time-averaged [30]. The
signal from aperture 1 and from aperture 2 are
51 oc E sin(wt —k • iq)
5 2 oc E sin(«;f —k • r2).
The time-averaged output at the multiplier is 1
(25x52}ttmeo c /c o s (k .B ).
*The relative phase of one signal with respect to the other is time-modulated with specific fre­
quency and picked up by locking on the modulation frequency. Often the optical path difference
is modulated.
^The DC signal is easily filtered 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 r ig h t o w n e r . F urth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
17
2.2
Interferom eter O bservation o f th e C M B In ten ­
sity
An interferometer measures the time-averaged correlation of two electric field from a
pair of receivers, which is called the visibility [29, 30]. W ith two apertures pointing
in the same direction of the sky, response of the interferometer to the sky is the
convolution of the interference-modulated intensity with the beam function.
For the observation of a small patch of sky, a flat sky approximation in the small
angle limit can be used. W ith a flat sky approximation [2, 29, 30], the visibility
associated with the intensity I is
V
=
J d" J
dx2A (x )/(x , i/)ei27ru x,
where A(x) specifies the two dimensional beam power pattern and u is the baseline
vector divided by wavelength of the signal and v is the frequency. Since the intensity
of the CMB is
AT(x,
I cmb{n, v) ~ B (u , T0) + —
is),
T=T0
the visibility associated with the CMB intensity is
V = J
d u B ( y ,T 0)
J
d x 2A(x)ei2,ru'x +
J
disf(is)
J
d x 2A (x)A T (x)ei2jru'x,
where B(is,T0) is the Planck function of the monopole tem perature of the CMB and
f{v ) is agC;r )
t and A T(x) is the two dimensional anisotropy pattern of the
t
=t 0
*/(«/) is explicitly given by Eq. 1.1.
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 urth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
18
CMB temperature.
In the derivation above, we used the flat sky approximation which is a good
approximation when the the field of view is limited to a small patch of sky. But, if
the field of view is large or the angular separations between the fields of observations
are large, the sky curvature should be properly taken into account. So two dimensional
linear space is replaced by two dimensional surface of a sphere of unit radius. W ith
the sky curvature taken into account, the visibility associated with the CMB intensity
is
i2'7TU‘n.
J du J
d fM (n —n 0)ei2?ru'A [B(u, T0) + f(u) AT (h )] ,
where n indicates the direction on the sky and A(n — n 0) is the primary beam
power pattern with its beam center pointing in the direction of no and A T(n) is the
anisotropy pattern of the CMB tem perature on the surface of a unit radius sphere.
All-sky CMB tem perature anisotropy is expanded in spherical harmonics [6]:
OO
A T (n) =
I
EE
1=1 m = —l
W ith the spherical harmonic expansion for the CMB tem perature anisotropy, the
visibility associated with the CMB anisotropy tem perature can be expressed:
v
.i27ru*n
1=1 m = —l
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19
2.3
Polarization Interferom eter O bservation o f th e
CM B
W ith linear polarizers, each baseline measures visibilities associated with < 2E x>Eyi >
where E indicates electric held, and x' and y' indicate directions of two orthogonal
linear polarization states.
For the observation of a small patch of sky, a hat sky approximation in small angle
limit can be used. W ith a hat sky approximation [2, 29, 30], the visibilities associated
with (2E xiEy/) are
V
J
J
=
=
dv f ( y )
J
dnA(x.)(2Ex,Ey,)i2™*
d v f ( v ) / d ^ ( x ) l m [ e - ^ ) ( g ( x ) + ^ ( x ) ) ]e- u'x,
where ip is the rotation of the polarizer frame with respect to a hxed reference frame
§. W ith the convolution theorem [5],
V
=
J
d v f ( v ) l m [ e - i2^A(u ) * (Q(u) + i U(u))],
where the star * indicates convolution and the tilde ~indicates Fourier transform.
In the hat sky approximation in small angle limit, the Fourier transform of Q /U
can be decomposed in terms of E and B modes as follows [17, 18, 21]:
Q(u) + iU(u)
=
e*2^u[E(u) + iB(u)],
defer to Eq. 1.2 and 1.3.
R e p r o d u c e d with p e r m issio n o f th e co p y 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 .
(2.2)
20
where Q and U are the Fourier transforms of the Q and U polarization fields and <pu
is the direction angle of u [18, 33, 34],
W ith Eq. 2.2, the visibilities associated with < 2E x>Ey/ > in terms of E and B
decomposition mode are
V = J d u f ( u ) l m [ e i2^ - ^ A ( u ) * ( l ( u ) + i B ( u ) ) ] .
W ith the sky curvature taken into account, Q and U are defined with respect
to the angular coordinate (9,<p) of a global coordinate (i.e. Galactic coordinate or
Equatorial coordinate). As mentined in chapter 1, Q and U at the given direction
(9,<p) are defined with the basis vectors (eg, e^) on spherical sky [18]. Projected on
the aperture plane, the basis vectors (eg, e^) have the following relation with respect
to the basis vectors (ex/, ey/) of the polarizer frame:
ex/
=
ey/ =
cos (ip - <f>)e0 + sin(ip —sin (ip —$)eg + cos (ip — $)e^,
where ip is the azimuthal rotation of the polarizer frame with respect to the antenna
coordinate and $ is the azimuthal angle of the given direction (9,<p) in the antenna
coordinate. When z axis of the antenna coordinate points in the direction of arbitrary
angular coordinate (9a,<Pa), $ is as follows:
* = t a W ? G ^ 'I = U n -'(
eg • e<jA /
\c o s 6 cos Qa cos(<p — 4>a ) —sin 0 sin 9a J
W ithout flat sky approximation, the visibility associated with < 2E xiEy>> is
y =
dv f ( v ) / dflA(n) < 2ExEy > e i27ru-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 .
21
=
J
duf{u)
J
dnA ( n)l m [ e- i2^ - ^ )\ Q ( h ) + iU(n))]ei2nu-il.
(2.3)
Since all-sky Q and U are described by
l,m
l,m
the visibility associated with < 2 E xtEy>> is
V = Y , aE’lm[ dvf ^ J d ^ ( n ) ei2™'A
l,m
x [sin 2 {ip - $ ( n ) ) X 1M (n) + i cos 2 (V» - $ (n ))X 2l/m(n)]
+ ^’ 2 , a B,im
l,m
f
duf{v)
J
dfM (h)e*27ru'n
x [—cos2(V> - $ (n ))X i,,m(n) - H s in 2 (r/> - $ (n ))X 2 ,<m(n)].
(2.4)
In chapters 3 - 7, we will carry out data analysis on simulated observations. We
have generated the simulated visibilities by the C language code we developed. In the
routines, the Gaussian random number generator of ‘Numerical Recipe in C’ [35] was
used to simulate noise. The visibilities with flat sky approximation were simulated by
the code in Appendix A.2.1. The simulated visibilities without flat sky approximation
were computed by the code in Appendix A.2.2.
2.4
T he V isibilities o f th e M BI
While conventional radio interferometric systems measure electric fields and correlate
two electric fields using an analog or a digital multiplier, the MBI measures the power
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
of the sum of two electric fields with a bolometric system^. The MBI measures the
CMB anisotropy at center frequency 90 GHz with 10 GHz bandwidth. The MBI em­
ploys four 8.1° Full W idth at Half Maximum(FWHM) corrugated feed horns. These
four feed horns are connected to four input channels of a Butler beam combiner [36].
Each channel is phase-modulated by a ferrite waveguide and alternating magnetic
field with a specific frequency [37, 38]. These four input are transformed into four
output channels through a Butler beam combiner by El out = TijE^injmt (in m atrix
notation, Eout = TE) where
Tij = ^exp[i<f>ij].
(2.5)
Egut is the beam at the output of the Butler beam combiner and faj is rcij/2 [36].
The outputs of the Butler beam combiners heat Neutron Doped Transmutation(NTD)
germanium bolometers heat-sunk by a Helium 7 fridge H.
Thermal change in bolometers are directly related to the radiation absorbed by
the bolometers. The feedhorn array configuration of the MBI is shown in Fig. 2.1.
Each feedhorn is linked to a rectangular waveguide which plays the role of a linear
polarizer. Two feedhorns marked with ‘x ’ are linked to the waveguide which pass
only a linear polarization along x axis while two feedhorns marked with ‘y ’ are linked
^Bolometers measure the power of radiation. The MBI belongs to the category of an adding
interferometer while conventional radio interferometer belongs to a correlating interferometer.
IHelium 7 fridge is a 2 stage (Helium 3 and Helium 4) sorption fridge [39].
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 .
23
18 sdeps_
Top view o f
MBI c r y o s t a d
Figure 2.1: the MBI horn configuration
to the waveguide passing a linear polarization along y axis. The correlation of the
co-polar states are (EXEX) = (T + Q)/2 and (EyEy) = (T — Q ) / 2. Since the co-polar
states contain the tem perature anisotropy term which overshadows much weaker CMB
polarization signal, we are concerned mostly with the correlation of the cross-polar
states such as (E xE y) = U/ 2. Since we have two feed horns sensitive to x polarization
and two feed horns sensitive to y polarization, we have 2 x 2 = 4 baselines associated
with the correlation of the cross polar states in the MBI.
As indicated in Fig. 2.1, the horn array of the MBI is rotated by 180° through 18
steps. For each rotated step, visibilities are measured. Visibilities are measured for
each rotation of the horn array. In this configuration, MBI has baselines of length
(9.02[cm], 10.72[cm], 12.26[cm], 13.74[cm]). The beam pattern of a MBI feedhorn
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
measured is shown in Fig. 2.2 [40]. To a good approximation, the MBI corrugated
10 12 U 16 18
Figure 2.2: the beam pattern of a MBI feedhorn
feed horns has the beam pattern as follows:
A(n) = exp [—1.387
d
ir
Though it is a good approximation, the actual beam pattern has sidelobes and asym­
metry. So in such cases, tabulated data of beam pattern from measurements instead
of the fitting function can be used in the data analysis. The physical parameters of
the MBI are summarized in Table 2.1.
The visibilities which will be measured by the MBI have the following form:
/»95GHz
IdB;
^oise ~l” /
785GHz
/*
du
/
q
\ 2
dflexp[-1.387( —— ) ]exp[i
J
\°A /
2-KvBi • n,
c
x { -sin 2 (V ' - 4>(n))/(^)Q (n) + cos2(ip - <f>(n))/(z/)Z7(h)},
(2.6)
where B* indicates the baseline vector formed in the MBI horn array **, Koise is
**The relative displacement vector between a pair of a feedhorn marked with ‘x ’ and a feedhorn
marked ‘y’ in Fig. 2.1 constitutes a single baseline vector.
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
Table 2.1: The physical parameters of the MBI
Physical Param eter
Full W idth at Half Maximum(FWHM)
8 .1 °
Expected center frequency
90 [GHz]
Expected bandwidth
10%
Number of feedhorns
4
Diameter of a feedhorn aperture
5.335 [cm]
Expected thermal conductivity G0
340 pW /K
Expected heat sink tem perature To
335mK
Expected detector responsivity S
-2 .4 x 108 V /W
Expected sky tem perature
20 K
the noise of the MBI observation and f{ y ) is the frequency spectrum for the CMB
anisotropy.
2.5
N oise-E quivalent-Pow er (N E P ) o f th e M B I Ob­
servations
The biggest noise sources in bolometric measurement of the CMB anisotropy consist
of the CMB photon noise itself, atmospheric fluctuation, Johnson noise, phonon noise
and amplifier noise [41, 42], Total Noise-Equivalent-Power (NEP) [41] is
N EP 2 = NEP 7 2 + N EP g 2 + N E P j 2 + N EP ^ 2 + NEPother2,
(2.7)
where NEP 7 : photon noise, N EP <3 : phonon noise, N E P j : Johnson noise, NEP^ :
the amplifier noise and NEPother : other sources of noise such as 1 / / noise tb The
^M ost electronic devices have increased noise at low frequencies [41]. There is no general under­
standing of it but empirical law:
K I a df
~ 1 E~ '
where / is the frequency, K is a normalization constant, I is the current and a « 2 and b « 1
[41]. The signals in the MBI are phase-modulated by ferrite-waveguide and alternating magnetic
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
NEP from photon noise without bunching term is [42, 43]
where h is the Planck constant, v is the frequency of photon, c is the speed of light, A is
the aperture area, fi is the solid angle, k is Boltzmann constant, T is the tem perature
of the photon noise source and
77
is the absorptivity of the bolometer. By the antenna
theorem, the throughoutput is Afl = A2 for a single mode diffraction limited beam
[42]. W ith AQ = A2, the NEP of the photon noise for the observation of bandwith
Ah' is
where u is the center frequency. The NEP from phonon noise is [41, 42]
N EP g 2 = i k T ^ G / r f -
(2.9)
where G is the thermal conductivity and Tb0 i0 is the tem perature of bolometer. The
NEP from Johnson noise [41, 42] is
NEP,,2 = 4fcrb<jo.R/|S|2,
(2.10)
where Tb0 i0 is the tem perature of bolometers and S is the responsivity of the detector,
and R is the resistance of bolometers. Since the MBI employs the bolometers of the
field [37, 38]. So the phase-modulation frequency, where 1/f noise is insignificant in comparison
with the signal, should be 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 .
27
same kind as those of the Arcminute Cosmology Bolometer Array Receiver (ACBAR)
[43], we can assume the detector responsivity of the ACBAR [43] so th a t S ~ —2.4 x
108 V /W .
In the MBI, the expected therm al conductivity, Go, is 340pW /K and
the expected tem perature of the heat sink for the bolometer, T0, is 335mK. The
resistance of bolometers is expected to be ~ 30M11 NEP7, N EPG and N E P j which
are estimated for the MBI with Eqs. 2.8, 2.9 and 2.10 are shown in Table. 2.2.
Table 2.2: The noise budget
the source of noise
magnitude
photon NEP7
phonon N EPG
Johnson noise N E P j
of the MBI
(10 17 W H z 1/72)
2.429
4.59
6.44
The expected Noise Equivalent Power(NEP) of the MBI is ~ 8.27 x 10-17 W Hz-1/2
and is comparable with the NEP of the QUEST on the DASI (QUaD) [44] and the
ACBAR[43]. Stochastic variance of the UnoiSe in Eq. 2.6 is the square of the total
MBI NEP such th at (UnoiseUnoiSe*) = (8.27 x 1 0 '17 W H z"1/2)2.
2.6
D iscussion
We discussed the visibilities of the MBI. In the simulation of Chapter 4, we will
numerically compute Eq. 2.6 to generate the simulated visibilities of the MBI. The
observational configuration such as (180° rotation of the feedhorn array through 18
rotation steps for a single field) and the observational configuration, discussed in t his
chapter, will be used in the simulation. In the simulation of the MBI, the Woise will
be simulated by adding random number drawn from Gaussian distribution whose
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
variance is the square of the total NEP of the MBI.
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 3
A perture Synthesis
Though the power spectrum of the CMB anisotropy can be estimated directly from
visibilities*, reconstructing an image from visibilities is crucial to distinguish point
sources and foregrounds from the CMB anisotropy. After imaging, we can mask
the point sources and foregrounds in the image and estimate the power spectrum
from the properly masked image. Since interferometer observations directly probe
Fourier components of the sky pattern, the sky pattern can be reconstructed by
applying inverse Fourier transformation to interferometer observations. This method
of imaging is called ‘Aperture Synthesis’ and it is widely used in astronomical radio
interferometry [22, 29, 30].
The relative displacement vector(baseline) between two apertures of the interfer­
ometer corresponds to the Fourier component probed. In the feedhorn array con­
sisting of multiple feedhorns where each pair of feedhorns forms a baseline, the horn
‘This will be discussed rigorously in Chapter 4.
29
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 urth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
30
array configuration which yields the ideal Fourier space sampling can be found only
numerically except for a few special cases. We have found the optimal horn array
configuration by Markov Chain Monte Carlo method. This horn array configuration
is reflected in the design of the MBI.
In this chapter, we will describe the aperture synthesis technique and how we op­
timized the feedhorn array of the MBI for aperture synthesis and present the aperture
synthesized image from the simulated MBI observation.
3.1
The R elation b etw een th e Sky B rightness and
V isibilities
For a narrow bandwidth observation of small patch of sky, the visibility associated
with the measurement of sky brightness can be approximated by
V & A vf(v)
JJ
dxdy A { x ,y )I { x ,y ) e i2'K(-ux+vv\
(3.1)
where A(x, y) specifies the two-dimensional beam power pattern, I(x, y) is the two
dimensional pattern of the sky brightness and u is the center frequency of the system,
f ( v ) is the frequency spectrum of the sky brightness. The baseline vector components
divided by wavelength of the signal are u = B x/ A, v = By/X. Eq. 3.1 shows th at an
interferometer directly measures the Fourier transform of the beam pattern modulated
sky brightness. When the interferometer has u = B x/X and v = B y/X, the Fourier
component at kx = u and kx = v is measured by the interferometer.
Hereafter,
uv sampling and Fourier space sampling will be used interchangeably. W ith enough
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 .
31
visibilities probing different Fourier components, the two dimensional pattern of the
sky brightness is obtained by inverse Fourier transform [28, 29, 30, 45] as follows:
I ( x , y ) = (Azq/(z/))_1T (x ,y )_1^ ^
JJ
dudv V(u, V)e~2m{xu+yv).
In the real world, visibilities with baselines of only finite number can be measured.
So the inverse Fourier transform is replaced by a discrete Fourier transform [5, 35]:
I(x,y) =
(3.2)
w
tr
This technique of interferometry is called ‘aperture synthesis’. W ith aperture synthe­
sis, interferometers are able to image the sky with an angular resolution of the single
filled-dish telescope whose diameter has the same dimension with the baseline length
of the interferometers [29, 30].
3.2
O ptim al uv Coverage
By NyquistShannon sampling theorem, signal of frequency range —f c < / < f c can
be properly sampled with the sampling interval A < l / ( 2 / c) [5, 35]. Since we have
a signal of range - x max < x < x max and - y max < y < Umax, sampling interval in uv
space should satisfy A u < 1/(2 x max), A v < 1/(2 y max) where x max and y max indicates
the maximum range of two dimension of the sky patch observed. Since x = sin 9 cos </>,
y — sin#sin<f>, x rnax and y max are sin(FOV/2), where 6, (j) are the angular coordinate
of spherical coordinates and FOV are the fields of view. Therefore, the optimal uv
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
coverage is such th at a single sample is made in each uv cell whose sizes are
1
1
2sin(FO V /2) X 2sin(FO V /2)'
In a CMB experiment, since large integration time is required due to low signal-tonoise ratio, very limited number of uv space samples can be made. Thus it is crucial
to find the instrument configuration which achieves the optimal uv coverage.
In most interferometric observations of the CMB, the feedhorn array consisting
of multiple feedhorns is employed and the beams from two feedhorns are correlated
or added [31]. By rotating the feedhorn array, the number of orientation of baseline
vectors increases, which improves uv coverage.
The horn array ( N > 2 horns) configuration which achieves the optimal uv cov­
erage cannot be found in general analytically. One way to find the solution for the
problems which have no analytical solution is the brute grid search, but the time
needed for the grid-search increase exponentially with the size of input. Another way
to find the solution for such problems is the Markov Chain Monte Carlo (MCMC)
[35, 46]. This method relies mainly on random guess and does not require time which
increases exponentially with the size of input while it finds the solution as good as
the brute grid search.
We have searched for the horn array configuration by ‘simulated annealing’ [35,
47, 48], which is the MCMC. For N feedhorns, there are 2N parameters which are
the two-dimensional coordinates of the N feedhorns. Let / be a merit function to be
maximized [35, 47, 48]. The merit function / should be chosen to quantify the degree
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33
of optimization.
Let T be “a user-controlled tem perature” [48] and let a be some constant smaller than
1. The procedure of the simulated annealing search for the horn array configuration
is as follows.
1) Assign random values to 2N parameters, which are the coordinates of the feed
horns.
2) Generate the uv coverage for the current configuration i and compute merit func­
tion fi.
3) Assign a new random value to a single parameter randomly chosen out of the 2N
parameters.
4) Generate the uv coverage for the current configuration i + 1, compute the merit
function f i+\.
5) Accept the new configuration i + 1 if f.t f i > fi and multiply T by a.
6)
If fi+i < fi, accept the new configuration i + 1 with the probability oc exp (—^’f / ' ).
7) Repeat from 3) until a new configuration is not found after some number of itera­
tions.
The purpose of accepting a worse configuration in 6 ) with the probability oc exp (—A ri?)
is to avoid getting stuck in a local maxima. T and a are chosen empirically so th at
the search converges in reasonable amount of time. W ith a bigger T and a, it takes
longer to converge while with too small value of T and a, there is higher chance of
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34
not reaching the global maximum, stuck in a local maxima h In the search for the
MBI feedhorn array configuration, we chose the merit function such th at
/
=
Y 2 ( Ui ~ Ui+i ~
+ (Vi ~ Vi+i ~ ^
2
+ (Ui ~ Ui+2 ~
+
_
Vi+2 ~
2
i
+ {«. - tli+ 3 - A u ) 2 +
(V i
— Vi+ 3 ~ A v ) 2 + (U i
-
U i+ 4
- Au)2 +
(V i
- fi + 4 - A v f ,
(3.3)
where Au = l/(2sin(F O V /2 )), A v = l/(2sin(F O V /2)) and the index i spans all uv
points and i + 1,
+ 4 indicates the four uv points in the vicinity of uv point i.
This merit function favors the horn array configuration whose uv samplings are close
to the crystal lattice structure of spacing l/(2sin(F O V /2)). The search for the MBI
horn configuration was made by the C routine we developed and the C routine can
be found in Appendix A.3.lb
3.3
T he M BI
The MBI described in Sec. 2.4 has four feedhorns and two of them are sensitive to
linear polarization along the x-axis while the others are sensitive to linear polarization
along the y-axis. Hereafter, we call this configuration ‘the MBI 1\ The MBI 1 has
four baselines associated with (E xE y} and the feedhorn array of the MBI 1 is rotated
^The convergence was tested by repeating the search with a different starting guess. The different
starting guess was made by feeding different seed numbers to a random number generator.
*Header file, variable and function declaration in the actual routine are omitted.
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35
by 180° through 18 steps. V ( —u, —v ) is the complex conjugate of V(u, v). which can
be seen easily in Eq. 3.1. If we measure the visibility V(u,v), we can also obtain
the measurement of V ( —u, —v) with the conjugate relation. So the number of uv
samples in the MBI 1 is 4 x 18 x 2 = 144. W ith simulated annealing and Eq. 3.3 for
the merit function, we have found the horn configuration, which is shown in Fig. 2.1.
This feedhorn array configuration and the rotation yields the uv coverage in Fig. 3.1,
where each small blue circle corresponds to a single uv sampling and each gridded
cell has dimensions A u x Av.
5 0 1--------------- '------------------------'----------- '----------------'----------<-------------- 1--------------- '-------------- '----------------1-------------40
-
30
-
20
-
1°>
o-
.
-4 0
-
.
‘
•
_ • • •
' ’ •
• ’ • ;
•
-2 0 -
-
• • • . ‘
10-
-3 0
•
.
; • ’ •
'.
• . ••.
. • • . •
_50l----- ,-------- ,----,----- ,--- ,-----,----- ,-----,----- ,-----5 0 -4 0
-3 0 -2 0 - 1 0
0
10
20
30
40
50
u
Figure 3.1: uv coverage of the MBI 1
After the successful run of the MBI 1, the MBI is planned to be upgraded to have
64 feed horns and
8
beam combiners. Hereafter, we call this configuration ‘the MBI
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36
2’. Each beam combiner is attached to
feedhorns. Among
8
8
feedhorns and combines beams from the
feedhorns attached to each beam combiner, 4 feedhorns pass x
polarization and the other 4 feedhorns pass y polarization. Since we are concerned
with the cross polar states (xy), we have 4 x 4 = 16 baselines from each beam combiner.
The cluster of
8
feedhorns attached to each beam combiner has the same horn array
configuration except th at each cluster is rotated around the center of each cluster and
the centers of each cluster are positioned distinctively in the whole horn array consist­
ing of 64 feedhorns. Fig. 3.2 shows the two clusters of
8
feedhorns where the second
50
X
40
+
+
30
+
20
+
X
10
I>.
X
0
X
+
-10
+
X
-20
+
-30
X
+
-40
-50
-50
0
50
x[cm]
Figure 3.2: The feed horn configuration of the MBI 2
cluster is rotated and displaced, compared with the first cluster. The marks ’+ ’ and
’x’ indicates the distinct polarization state each feed horn is sensitive to. The purpose
of rotating the cluster of
8
feedhorns is multiplying the orientations of the baseline
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37
vectors formed. The faceplate of 64 feedhorns are rotated to 180° through 4 rotation
steps, which multiplies the orientations of the baseline vectors by a factor of 4. So the
total number of uv samplings are 16 x 8 x 4 x 2 = 1024. The factor
beam combiners and the factor 2 is because V ( —u, —v ) = V(u. v)*.
8
is because of 8
Fig. 3.3 shows
50
40
30
20
10
I>.
0
-10
-20
-30
-40
-50
-50
0
50
x[cm]
Figure 3.3: The feed horn configuration of 64 feed horns in the MBI 2
the feed horn configuration of 64 feedhorns in the MBI 2. The same color of the marks
indicates th at the feed horns are attached to the same beam combiner. Since we as­
sumed 45° rotation for each rotation steps for the faceplate, the rotation of each cluster
around its center should be smaller than 45° to avoid rotational redundancy. So each
cluster of 8 feedhorns was rotated by i 45/8 [°] where i = 0 , 1 . . . , 7 index the clusters.
The clusters of 8 feedhorns are displaced so th a t any pair of 64 feedhorn have distance
more than 5.335 [cm] which is the diameter of the feed horn. If any pair has smaller
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38
100
50
>
0
-50
-100
Figure 3.4: uv coverage of the MBI 2
distance, the two feed horns physically overlap each other, which make the horn array
configuration unrealizable. The displacements were found by Monte Carlo search with
preference for the smaller total area occupied by 64 feedhorns. The uv coverage of the
MBI 2 resulting from the horn configuration in Fig. 3.3 is shown in Fig. 3.4. Color
indicates a distinct beam combiner. The MBI 2 have 16 distinct baseline lengths
(0.0661cm, 0.104cm, 0.128cm, 0.162cm, 0.187cm, 0.207cm, 0.222cm, 0.230cm,
0.249cm, 0.268cm, 0.278cm, 0.285 cm, 0.312 cm, 0.323cm, 0.328cm, 0.342cm). The
alternative for the MBI 2 is a single closepack horn array consisting of 64 feedhorns.
Hereafter, we call it ’the closepack MBI’. It is also investigated with the same weight
with the MBI 2. The closepack feedhorn array is shown in Fig. 3.5. The red circle
indicates the horn is connected to the receiver sensitive to x linear polarization state
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39
Table 3.1: The baselines of the closepack horn array
baseline length [cm] number of the baselines
8 .8
55
1 0 .2
75
13.4
93
15.2
70
17.6
62
18.3
92
20.3
25
2 2 .1
56
23.3
69
25.4
34
26.4
31
26.9
51
28.3
47
30.4
8
30.9
34
31.7
24
33.3
22
35.2
11
35.6
25
36.6
15
38.4
6
39.7
2
40.3
4
40.6
3
and the blue circle indicates y linear polarization state. Thirty-two feedhorns are
sensitive to the x polarization state, and the other 32 are sensitive to the y polariza­
tion state. The beam combiner combines beams from all 64 feedhorns. Since we are
interested in the cross polar states (xy), there are 32 x 32 = 1024 pairs. Since we have
enough baselines of diverse orientations in a closepack feedhorn array configuration,
we do not need to rotate the faceplate of the horn array. So the total number of
uv samplings are 1024 x 2 = 2048. The uv coverage of the closepack horn array is
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40
shown in Fig. 3.6.
Table. 3.1 shows baseline lengths and number of baselines for
0.3
0.2
0.1
I>.
o
-
0.1
-
0.2
-0.3
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
x[m]
Figure 3.5: the closepack horn array
150
O OOOO O
O O O O O O O
O
O
O O O O O O O
100
o o o o o o o o o o o
o o o o o o o o o o
o o o o o o o o o o o o o
o o o o o o o o o o o o o
o o o o o o o o o o o o o o o
o o o o o o o o o o o o o o o
o
50
>
0
-5 0
-100
O
O O O O O
0 0 0 0 0
o
o o o o o o o o o o o o o o o
o o o o o o o o o o o o o o o
o o o o o o o o o o o o o
o o o o o o o o o o o o o
o o o o o o o o o o
o
o o o o o o o o o o o
O O O O O O O
o
O
o
-1 5 0
- 1 5 0 -1 0 0 -5 0
O O O O O O O
o o o o
0
u
o
50
100
150
Figure 3.6: uv coverage of the closepack horn array
the closepacked array.
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41
3.4
Sim ulation o f A perture Synthesis for th e M BI
In the flat sky approximation, depending on which Stokes parameters are measured,
visibilities have the following forms:
VT *
Vnoise + A v f ( v ) J J dxdy A ( x , y ) A T ( x , y ) e i2^ ux+vy\
VQ «
Vnmse + A „ m J
Vu ~
J
Vnoise + Avf(v) J J
(3.4)
dxdy A ( x , y )Q (x ,y )e i2^ ux+vy \
(3.5)
dxdy A ( x ,y ) U ( x ,y ) e t2niux+vy\
(3.6)
where Vn0ise is the noise, A T ( x , y) is the tem perature anisotropy of the CMB, Q(x, y)
and U (or, y ) are the Q and
U polarization of the CMB. The visibilities are computed by
Eq. 3.4, 3.5 and 3.6 with the instrumental configuration of the MBI 1. The visibility
Figure 3.7: 30° x 30° input sky patch of CMB anisotropy Temperature
noise Vnoise is simulated by generating a random complex number whose real and
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42
imaginary parts are drawn from a Gaussian distribution^.
The simulated 30° x 30°
Figure 3.8: 30° x 30° input sky patch of CMB Q polarization
Figure 3.9: 30° x 30° input sky patch of CMB U polarization
sky patch for CMB tem perature anisotropy AT, Q polarization and U polarization
^The variance of the Gaussian distribution is chosen to be the expected NEP of the observation.
We assumed 8.27 x 10-17 [W/v/Hz], which is the NEP of the MBI 1.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
43
of the CMB are shown in Figs. 3.7, 3.8 and 3.9 respectively. These two-dimensional
maps are used in Eqs. 3.4, 3.5 and 3.6 to generate the simulated visibilities. Two
dimensional maps of the CMB anisotropy tem perature AT, Q polarization and U
polarization can be generated by aperture synthesis from the visibilities as follows:
A T ( x , y ) = (A ^ / ( ^ ) ) - 1 A (o;,y)- 1 ^ ^ J ] c T( n ,,^ ) e - 2^
hj
+^ ) ,
(3.7)
+^ ) ,
(3.8)
' Y ^ V u {ui,vj )e 2i7r( ^ + ^ ) .
hj
(3 .9 )
Q(x, y) = (Azz/(i/) ) - 1 A (x , 2/) - 1 ^ ^ ^ E
(2 t t )2
U(x,y) = ( A v f ( v ) ) l A{x,y)
: An An
1
0
(nJ, ^ ) e - 2i^
The horn array of the MBI 1 shown in Fig. 2.1 is rotated by 180° through 18 rotation
steps. For each rotation step, visibilities are measured with 21 days integration time.
So the total observational time 21 x 18 = 378 days. The simulated visibilities of the
>
-
U
Figure 3.10: uv coverage of the reference map
MBI 1 are generated by Eq. 3.4, 3.5 and 3.6. From simulated visibilities of the MBI
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44
1, two dimensional map of A T , Q and U are generated by aperture synthesis with Eq.
3.7, 3.8, and 3.9. To simulate the aperture synthesis of the MBI, we have developed
the C routine and used it for the simulation. This C routine can found in Appendix
A.3.2.
Figure 3.11: Reference A T Figure 3.12: From noiseless Figure 3.13: From noisy visvisibilities
ibilities
The fluctuations on scales bigger than 2 n \ / B max = 0.024 and on scales smaller
than 2 ir \/B min = 0.037 were filtered out from the input sky patches and the filtered
maps are shown in Fig. 3.11, 3.14 and 3.17 (The circle has 10° diameter). In other
words, the input sky patches were Fourier transformed and these Fourier components
were masked with Fig. 3.10 where the blue dots correspond to 1 and zero otherwise.
The masked Fourier components were inverse Fourier transformed. These are the
reference maps to compare the synthesized map with to quantify the fidelity of the
aperture synthesis for the MBI 1. Fig. 3.12 is the synthesized map of A T from the
MBI 1 observation without noise and it has average error 8.75 pK, compared with
the reference A T in Fig. 3.11. Fig. 3.13 is the aperture synthesized map of A T from
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45
the MBI 1 observation with noise and it has average error 8.765 pK, compared with
the reference A T in Fig. 3.11. Fig. 3.15 is the aperture synthesized map of Q from
Figure 3.14: Reference Q Figure 3.15: From noiseless Figure 3.16: From noisy visvisibilities
ibilities
the MBI 1 observation without noise and it has average error 0.113 [iK, compared
with the reference Q in Fig. 3.14. Fig. 3.16 is the aperture synthesized map of Q
from the MBI 1 observation with noise and it has average error 0.189 n K, compared
with the reference Q in Fig. 3.14. Fig. 3.18 is the aperture synthesized map of U
Figure 3.17: Reference U Figure 3.18: From noiseless Figure 3.19: From noisy visvisibilities
ibilities
from the MBI 1 observation without noise and it has average error 0.1 /jK , compared
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
46
with the reference U in Fig. 3.17. Fig. 3.19 is the aperture synthesized map of U
from the MBI 1 observation with noise and it has average error 0.187
/jK,
compared
with the reference U in Fig. 3.17. The reconstruction errors in aperture synthesize
____________ Table 3.2: The average error in synthesized maps____________
associated Stokes parameter
AT
U
Q
reconstruction error [fiK] (from noiseless visibilities) 8.75 0.113
0 .1
reconstruction error [//K] (from noisy visibilities)
8.765 0.189 0.187
maps are summarized in table 3.2. In the case of the aperture synthesis from noiseless
visibilities, the errors in the reconstructed maps with respect to the reference maps
are attributed to the imperfection of uv coverage while in the case of noisy visibilities
the errors are attributed to visibility noise and the imperfection of uv coverage. Table
3.2 shows th at imperfection of uv coverage is the source of reconstruction error as
big as the noise of the MBI 1 (with 21 days integration time). Though the feedhorn
array of the MBI 1 is optimized for the aperture synthesis by simulated annealing,
the limited number of feedhorns still makes the uv coverage of the MBI 1 insufficient
in comparison to idealistic uv coverage.
By the aperture synthesis which is described and demonstrated by the simulation
in this chapter, we can reconstruct the sky image from visibilities.
R eproduced with permission of the copyright owner. Further reproduction prohibited w ithout permission.
C hapter 4
M axim um Likelihood E stim ation
of th e C M B E /B Power Spectrum
from th e sim ulated M BI
observation
The CMB angular power spectra for a wide range of cosmological models are now well
predicted and can be computed quickly for a given cosmological model by a widely
used computer code such as CMBFAST [25]. By estimating the power spectra of
the CMB anisotropy, we can constrain the cosmological models and the cosmological
parameters [6 , 8 , 9, 11, 12].
In the CMB interferometric observation [13, 49, 50], it has become standard to
estimate the power spectra [51] by the maximum likelihood method [52, 53, 54], The
likelihood function is defined as the probability th at the theory is right, given the
experimental data [6 ]. By maximizing the likelihood function, we can estimate the
most probable parameters of a theory [6 ]. In the power spectrum estimation from the
interferometric observation, the experiment data are visibilities and the parameters of
47
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
48
the theory are the power spectra of the CMB anisotropy. In this chapter, we present
the maximum likelihood estimation of the CMB power spectra from the simulated
MBI observation.
4.1
Spherical Sky and Flat Sky A pproxim ation in
th e Sm all A ngle Limit
The power spectra from a flat sky in the small angle limit are a good approxima­
tion to the exact power spectra from a spherical sky, but they shows discrepancies
at large angular scales. Especially power spectra of E mode polarization by tensor
perturbations shows notable discrepancies even at multipole I ~
100
[18], and the
discrepancy increases at lower multipoles for both of E and B mode. In the analy­
sis of interferometer observations, using an exact spherical harmonic description of
tem perature/polarization does not result in much computational load. On the large
angular scales where the B mode has its peak, using the exact spherical harmonic
description is desirable. Although there are several previous works on the analysis
of the CMB tem perature/polarization from interferometer observations, they were
done with the flat sky approximation in the small angle limit [2, 34, 45]. Instead of
sacrificing the generality by using flat sky approximation, we take into account the
sky curvature by carrying out all the analysis on a spherical sky.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
49
4.2
V isibilities as th e Linear Sum o f Spherical Har­
m onic Coefficients
Since visibilities are given by Eq. 2.1 and 2.4, we can express the CMB visibility as
the linear sum of noise,
v
a,T,im i a E ,im
= Vnoise +
and
(p T ylm O'Tylm “1“
^E ^lm “1“
&B,lrri) •
l,m
W ith Eq. 2.1, the visibilities associated with (E% + Ey) have
b T ,im
=
J
b E ,lm
=
0,
bB,lm
dflY;m(h)
J
d v f( v )A ( fi —hi)e*27ru'n,
0-
W ith Eq. 2.4, visibilities associated with (2E xE y) have
b r ,im
=
bE,im =
0,
j
dflfsin 2(-ip - $( n)) Xhim(ri) + i cos 2(V> - $ (n ))X 2,im(n)]
x J dvf(u)A(h)ei2™-A,
bB,im =
4.3
J
dft[i sin 2 (V> - $ (n ))X 2,im(n) - cos 2(xp - $( n)) X1M(ri)]
x
J
dvf{v) -A(n) ei2nu ii.
Power Spectra E stim ation
In inflationary cosmological models,
a r ,im
,
a E ,i m
and
a E ,im
are stochastically drawn
from a Gaussian distribution. The visibilities, which are the linear sum of noise, ax^im,
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
50
a E ,im
and
a,E ,im ,
are also stochastically Gaussian. The probability function th a t the
interferometer measures A (the sum of a visibility signal and noise) is
£ =
jv—
r exp[—
(27r) 2"(det C ) 2
2
where N is the number of data and A is the data vector and C is the visibility covari­
ance matrix. The probability function £ is the probability th at the interferometric
experiment would get such data given a theory, and this is often called the likelihood
function. By exploring the parameter space of a theory to maximize the likelihood,
we can determine the param eter values of a theory within some confidence interval.
One simple and powerful way of finding the parameters that maximize the likelihood
is the optimal quadratic estimator [2, 53]. By the optimal quadratic estimator, the
parameter A„ which maximizes the likelihood can be found iteratively, starting with
an initial guess A°:
A C - ^ C ^ A - TrfC” 1! ^ ]
’
+ ( F - % , -------- ^ -------2 --------------
A. =
(4 1 )
where Xa are the parameters, and the Fisher matrix, F is given by
” '5
a2(In£)
1
d x aa \ g
2
8C
,a c
W
,
9X S
[2, 6 ]. Evaluated at the maximum of the likelihood, the diagonal part of the inverse
Fisher m atrix yields the marginalized 1 —a error on the parameter estimation [53, 55].
W ith
( a T ,lm a T ,lm )
=
i
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
51
(a E,lmQ-*E,lm)
=
( a T ,lm a E ,lm )
=
C [E,
( a B,lm O > B,lm )
=
C BB,
the visibility covariance m atrix is given by
i
where
WIS
= E
W
W
(4.2)
m = —l
W lEi f
= Y
bEi,lmb*Ej,lrn^
(4 ‘3)
m = —l
=
wm
Y . ( b T i M b *EUm
m = —l
= E
W
+
b E i,lm b T j , l m ) i
W
(4 -4)
(4-5)
m = —Z
and TVjj- is the noise covariance m atrix and index i, j runs in measured visibility data
set. The diagonal parts of the window function W [ T, W EE, W f E, W BB shows the
relative sensitivities to signals on different multipoles. Fig. 4.1 shows the diagonal
part of the window function for the MBI 1 observation. An interferometer is sensitive
over the multipole range I «
2-k u
± A t/2, where u is a baseline divided by wavelength
and At is the FWHM of the diagonal part of the window function*. In the estimation
of power spectra by maximum likelihood method, it is usual to estimate the band
*For a circular Gaussian beam, A I = 4-\/21n2/0p\vHM [28].
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
52
9.02 [cm]
10.72 [cm]
12.26 [cm]
13.74 [cm]
0.9
0.8
0.7
&0.6
2
0.4
0.3
0.2
0
50
100
150
200
250
multi pole
300
350
400
Figure 4.1: Window function of the MBI 1
power t. In the flat bandpower assumption *, visibility covariance m atrix is
Cv
=
Ei K
+ W i T Xe e + W ™ \ TE + W g f \ BB) ,
(4.6)
where At t , AE e , At e and Ab b are the band powers of the CMB power spectra. W ith
Eq. 4.1 and Eq. 4.6, the most likelily values of band powers can be found iteratively
tMost experiments have not been sensitive to individual C i, but rather to the average of C i over
a range of multipoles, i.e. in a given band [6]. The average of of C i over a range of multipoles
is called a band power [6, 51].
*Ci is assumed to be constant over the given multipole range.
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 .
53
as follows:
A°
Ay
A#
_
A%
A C -' £ , w y c - ' A
+ V
L
A°c
Ac
As j
1
- I V [C -' £ , W P I
A C - 1 £ , W y c - ' A - T r [C -' £ , W p ]
A C ’ 1 £ , W y c - ' A - T r[C 1 £ , W f E]
^ A C - 1 E , W f BC - ' A - T t [ C - ' E i W f B]
v X°B )
The Fisher m atrix F is [2, 6 ]
(F U =
w ? c_1 E
1
w T ^ i,
1
where W f = { W f T, W f E, W f E, W f" } and W f = { W fT, W f E, W f E, W f B}.
4.4
C om puting o n /o ff D iagonal W indow Function
N um erically
Computing on and off diagonal parts of the window function with Eqs. 4.2, 4.3, 4.4
and 4.5 with spherical sky formalism requires huge numerical computation, since it
needs an integration over all-sky and computation of spherical harmonics. In com­
puting bT,im, &E,im an<3 5b,;toj integration over azimuthal angle becomes equivalent to
discrete Fourier Transformation [56], which can be computed quickly by Fast Fourier
Transformation [35]. An interferometer is sensitive to a finite range of multipoles
[28]. Legendre polynomial of multipole I corresponds to roughly the angular scale of
0 ~ 180°//. Choosing the optimal number of integration cells and using Fast Fourier
Transform render the numerical computation of Eqs. 4.2, 4.4, 4.3 and 4.5 feasible
in a reasonable amount of time even for interferometer which is sensitive to high
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 .
54
multipoles. Another way is computing window functions (off-diagonal part as well as
diagonal part) in the coordinate of the z axis coinciding with the antenna pointing.
For azimuthally symmetric primary beam and coplanar baseline §, integration over
azimuthal angle is given by ordinary Bessel function, which reduces the dimensions
of numerical integration from two to one. W ith unitary transformation, the antenna
pointing can be rotated to general direction from z axis. Numerically, it turns out
that this method has a numerical precision problem for multipoles higher than 60
even when using C language variable type long double (96 bits). So in the simula­
tion we did not choose to compute window functions with this method, but with the
method mentioned first.
4.5
The Sim ulated M B I O bservation
The simulated visibilities of the MBI observation are generated by numerically com­
puting
/>95GHz
Vbj =
Koise+ /
./8 5 G H z
p
/
n
\ 2
O t t i j T K- ■ f i
d«//(i/) / dftexp[—1.387 ( —— ) ] exp[i
J
\o .l J
x {—sin 2(t/> —$(n))Q(n) + cos 2(^ —$(n))C7 (n)},
where
---- ]
C
(4.7)
indicates the baseline vectors formed from the horn array and l/n0 isc is the
instrumental noise of the MBI and f{ v ) is the frequency spectrum for the CMB,
which is given by Eq. 1.1. The Q /U maps in Fig. 1.6 and 1.7 are used for the values
§The baselines sit on the aperture 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 r ig h t o w n e r . F urth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
55
of Q(n) and U(n) in Eq. 4.7. The visibility noise, Kioisc, are randomly drawn from
Gaussian distribution of the variance which is the square of the noise equivalent power
(NEP) of the MBI (8.27 x 10- 1 7 W H z” 1^2). The estimation of E and B mode power
spectrum can be made from the MBI observation iteratively, using the quadratic
optimal estimator:
Ae
AC'
1
E z W f EC _1A - T r[C -x E z W f -E]
\
(4.8)
BC~l
A C " 1 Ez W f ■
A - Tr[C _ 1 E z W f B]
\ Xb )
X °B /
\
/
We developed the C routine to realize the iterative quadratic optimal estim ator above.
The C routine can be found in appendix A.4.1.
Visibility covariance m atrix of the MBI is
Cij =<
(w%f\E+ wg? \ B) ,
>= N5tj + £
I
where N is the noise variance of the MBI, which is (8.27 x K) - 17 W H z"1^2)2. The
visibility covariance m atrix of the MBI can be computed by the C routine in appendix
A.4.2.
Fisher m atrix F is
iTYE,wpc-1 £,w pc-'] iTrE,wfi:c- 1 s;,wfBc-'] ^
F = f
v I ^ E z W f SC - XE z
wf ^ C - 1] |Tr[E/ wf 5 C - XE z W f BC - X] /
We computed the Fisher m atrix of the MBI by the C routine in appendix A.4.3. The
window function
and W fff of the MBI are
wuf
= E
m = —l
wuf
= E
m = —l
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 .
56
where
bE i,im
J
=
dH[sin 2(i/j - $ (n ))X 1)/m(n) + i cos 2 ( - 0 - $ (n ))X 2 ,/m(n)]
/*95GHz
/
duf{y) exp [—1.387 f
x /
J
=
9 _ ; /-d
) ] exp[i
V 8 -1
J 85G H z
b B i,im
/i \ 2
'n]
C
/
dO[« sin 2(i(j - <h(n))X2iirra(n) - cos 2(i> - $ (n ))X i>/m(n)]
;>95GHz
/
n \ 2
9-7r7/'R
x /
d » M exp[—1.387 (
) ] exp[i
J 85GHz
V 8 -1 /
ri
'
].
c
We have simulated the MBI 1 observation. The diagonal part of the window function
for the MBI 1 is shown in Fig. 4.2. Since the MBI 1 has four distinct baseline
9.02 [cm]
10.72 [cm]
12.26 [cm]
13.74 [cm]
0.9
0.7
S’
(0 0.6
I
0.5
0.4
0.3
0.2
0
50
100
150
200
250
multipole
300
350
400
Figure 4.2: Window function of the MBI 1
lengths, the MBI 1 is sensitive to 4 regions of multipole space, which are seen in
Fig. 4.2. So the MBI 1 observation can make estimation of four band powers. Since
there are 18 baseline orientations for each baseline length, 18 visibilities are used to
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 .
57
estimate each corresponding band power
Each visibility are measured with 21 days
integration time. We assumed a single field observation, whose equatorial coordinate
is (29.21°,205.4°). We assumed total observation time to be 21 x 18 = 378 days.
Since 21 days integration time was assumed, visibility noise Woise were drawn from
the Gaussian distribution whose variance is (6.21 x 10~ 20 J)2.
After generating simulated visibilities with Eq. 4.7, we have estimated the four
band powers with Eq. 4.8. The estimation of the E mode power spectrum
C e ,i
is shown in Fig. 4.3. The height of the vertical bars from the center of the cross
corresponds to 1 —a error on the E mode band power estimation. The horizontal bar
shows the width of the band power. The MBI 1 does not have enough sensitivity for
4.5
3.5
LU
LU
“
CN
T
2.5
0.5
140
160
180
200
220
I
240
260
280
300
Figure 4.3: E mode Power spectrum estimation from the simulated MBI 1 observation
^Each baseline length probes a band power in distinct multipole region.
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 .
58
the measurement of B mode power spectrum but it can impose upper bounds. The
upper bounds on the B mode power spectrum is shown in Fig. 4.4.
C
Q
m
o"
K
C
N
+
10"4
140
160
180
200
220
I
240
260
280
300
Figure 4.4: The upper bound on B mode Power spectrum by the simulated MBI 1
observation
We have also simulated the MBI 2 observation. The MBI 2 have the 16 distinct
baseline lengths. The diagonal part of the corresponding window function of the MBI
2 is shown in Fig. 4.5. In the horn array of the MBI 2, there are
8
distinct orientations
for each baseline length. Since the horn array of the MBI 2 is planned to be rotated
by 180° through 4 steps, there are
each baseline length.
8
x 4 = 32 baselines of different orientations for
Since the MBI 2 have the 16 distinct baseline lengths, the
MBI 2 is sensitive to 16 region of multipole space, which is seen in Fig. 4.5. So
the MBI 2 observation can make estimation of 16 band powers. Since there are 32
baseline orientations for each baseline length, 32 visibilities are used to estimate each
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 .
59
1.1
1
0.9
0.8
0.7
ra 0.6
1 0.5
0.4
0.3
0.2
0.1
0
100
200
300
400
500
600
700
800
multipole
Figure 4.5: Window function of the MBI 2
corresponding band power. For each rotation step of the horn array, 16 x
8
= 128
visibilities are measured with 91 days integration time. Since 91 days integration time
was assumed, visibility noise Woise were drawn from the Gaussian distribution whose
variance is (2.985 x 10~ 20 J)2. The total observation time is 91 x 4 = 364 days. After
generating simulated visibilities with Eq. 4.7, we have estimated the 16 band powers
with Eq. 4.8. The estimations of the E mode power spectrum from the simulated
MBI 2 observation are shown in Fig. 4.6.
The MBI 2 does not have enough sensitivity for the measurement of B mode power
spectrum but it can impose upper bounds. The upper bounds on the B mode power
spectrum which will be imposed by the MBI 2 are shown in Fig. 4.7.
The closepack MBI have 64 feed horns and beam combiners combines beam from
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 .
60
50
45
40
35
30
25
20
15
10
5
0
■5
100
200
300
400
I
500
600
700
800
Figure 4.6: E mode power spectrum estimation from the simulated MBI 2 observation
CD
CD _
o
K
CM
T+
10~4
100
200
300
400
I
500
600
700
800
Figure 4.7: Upper bound on B mode power spectrum by the simulated MBI 2 obser­
vation
32 feed horns sensitive to x polarization and 32 feed horns sensitive to y polarization.
The closepack horn array is not rotated since there are many baselines of distinct
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 .
61
orientations. Its horn array configuration and uv coverage were discussed in chapter
3, and are shown in Fig. 3.5 and 3.6. In the closepack horn array, the baseline
lengths and the number of the baselines for the baseline length are shown in table
3.1. The diagonal part of the corresponding window function of the closepack MBI
is shown in Fig. 4.8. Since the closepack MBI have the 24 distinct baseline lengths,
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0 .2
-
0
100
200
300
400
500
multipole
600
700
800
900
Figure 4.8: window function of the closepack horn array
the MBI 2 is sensitive to 24 region of multipole space, which is seen in Fig. 4.8. The
closepack MBI observation can make estimation of 24 band powers. As shown in table
3.1, the number of baselines varies, depending on the baseline length. So the varying
number of visibilities were used for the estimation of each band power. Visibilities are
measured with 365 days integration time. So visibility noise Vnoise were drawn from 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 .
62
Gaussian distribution whose variance is (1.49 x 10- 2 0 J)2. The total observation time
is 365 days. After generating simulated visibilities with Eq. 4.7, we have estimated 24
band powers with Eq. 4.8. The estimations of the E mode power spectrum from the
simulated closepack MBI observation are shown in Fig. 4.9. As shown in the number
of baselines, specific baseline length has too few baselines. Bandpower corresponding
to such baseline length do not have enough sensitivity. So only upper bound can
be imposed on such bandpowers, which are indicated by a horizontal bar without a
vertical bar.
The estimations of the B mode power spectrum from the simulated
10
,4
10
CM
*
LU
111
_
o
K
,2
10'
CM
10
10
o
■2
0
500
1000
1500
Figure 4.9: E mode Power spectrum estimation from the simulated closepack MBI
observation
closepack MBI observation are shown in Fig. 4.6. There will be upper bounds on 21
band power of B mode and estimation of 3 band power of B mode by the closepack
MBI observation.
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
CM
CD
m
O
CM
+
-1 0
500
1000
1500
Figure 4.10: B mode Power spectrum upper bound and estimation by the simulated
closepack MBI observation
4.6
P oin ting Error
There are inevitably pointing errors due to the mispointing of the horn array and
misalignment of feed horns. Imagine we believe th at two feedhorns forming an inter­
ferometer are pointing in the direction of z axis and one of the feed horn is actually
pointing in the direction with the angular deviation 591 from the z axis and the other
with 562 - Since the MBI measures the visibility associated with the linear cross polar
states, the visibility of the MBI is
V = b E ,lm U E,lm +
a B,lrm
where
bsM
=
[ dfijsin 2 ( ip - $ (n ))X 1)Jm(n) + i cos 2 (tp - $ (n ))X 2>/m(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 .
64
bB,im =
X
J
J
dO[i sin 2 (^ - $ ( n ) ) X 2,im(n) - cos 2 (ip - $ (n ))X i>;m(n)]
x
J
d v f( v )A ( 6 u e2)ei2™ &
d u f (u)A(9i, 92)ei2nu'il.
9\ is the angular distance between the beam center of the feed horn
1
and n, and
92 is the angular distance between the beam center of the feed horn 2 and n. The
deviation of the visibility due to the mispointing and misalignment is
dV
69
01=0
dV
d9,2
62=0
The sources of these deviations can divided into the pointing error of the feedhorn
array and the misalignment of individual feedhorns. The variance of 89\ and 602 are
< S9\ 89\ > = Ng1 + N q
< S92 592 >— Ng2 + Ng
< 89\ 892 > = Ng,
where Ngx is the variance of the alignment error of the feed horn 1, Ng2 is the variance
of the alignmenterror of the feed horn 2 and Ng is the variance of the pointing error
of the horn array.To keeptrack of the same sky (mainly for the compensation of the
earth rotation), the pointing structure keep adjusting the pointings. So adjustment of
pointings is equivalent to new trials of pointing, therefore the variance of the pointing
error of the horn array, Ng is oc 1 /T where T is the integration time of the visibility
measurement. But, the alignment of horns are fixed and the misalignment of two
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 .
65
horns are almost invariant over timel'. So unlike the pointing error, Ng1 and N s,2 are
independent of the integration time. The visibility deviation 5V is
nr
xo
SV
- 59
1
“EM+.
dbB,im
aBM\) +. 592\ ^ rdbE,im aem + dbB,im aB,im)-x
l,m
l,m
The covariances between 59, aB,im and aBj m are
< a E ,lm a E ,lm ^
=
^ E ,l
< a B ,l m a *B ,lm >
=
C B tl
< a E ,lm a *B,lm >
—
0
< 59\ a*E lm >
=
0
<
592 a *E jm
>
=
0
<
59\ a*B im
>
=
0
<
592 a*B lm
>
=
0,
where CB,i, CBj are the CMB E and B mode power spectrum. We can assume 591
and 592 follow Gaussian distribution. Then, with the covariance between 59, aEj m
and aB>im, the variance of 5V is **
< 5V5V* >=
IIThough there is a slight variation due to thermal contraction and expansion.
**If A,, A j , A* and A; are Gaussian, the expectation value of four As is [6]
< AjAjAfcA; > = CijCki + CikCji + CuCji,
where C is the covariance of two 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 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 .
66
d b E .Im
J^U ^+ N eXC E,
l,m
+
89 1
\
C b ,i
d b B .Im
2
2
i
AT
!
8 b E ,lm
8 b E
+ J y e ^ E A —Ea
89i
im
8 b E
+
862
d b s im
+ C,B.l
+{N q2 + N$)(C e ,i ^ E,lm
89o
(4.9)
89i
im
)
89o
8 b E%l m \
891
.
sy
/ 8 b B ,lm
+
892
8 b B ,lm
89x
bb892
d b B ,lm
8 b B ,lm
N I
+ W15-------891
892
(4.10)
In the MBI observation,
b E ,im
J
=
J
=
and
b B j/m
have the following forms:
dfi[sin 2 (ip - $ (n ))X liJm(n) + i cos 2 (</> - $ (n ))X 2,im(n)]
J
b B ,im
b E ,im
d v f i y ) exp[— (^~
2 + 922\-,
r.27r^Bj • h .
] exp[i4cr2
dft[i sin 2 (V> - $ ( n ) ) X 2>im(n ) - cos 2 (V> - 4 > (h ))X ii;m(h)]
J
2 + 922
d v f (v) exp[—
4 cr2
\n
r.27rz/Bj • rL
exp [t-------------J
where a is 0.4245 FWHM and the FWHM of the MBI is 8.1°. Then
8 b E ,lm
8b E M
89\
I
X
89o
@
1=0y$2=0
dfi[sin 2(ip - § ( h ) ) X 1M (h) + i cos 2(ip - $ (n ))X 2,/m(n)]
J
d v f ( v ) ( —^ ) ex.p[-
92
2 -KuBi -r i
expf,— ^----- ]
(4 .1 1 )
d b B ,Im
d b B Im
891 0\ =#, $2=0
I
61=6, $2=0
892
9i= 6, $2=0
dfi[*sin2(V' - 4> (h))X 2i;m(h ) - cos 2 (?/> - 4 > (n ))X i)im(n)j
n
\ t
6
\
r
1
r .2 7 r i/B j • n
' ex p [- — ]exp[z
x I d v f { y ) { - J—-M
2cr2 '
2 <r2
1
-----
(4.12)
We have estimated the RMS error of the visibility due to pointing/alignm ent error
in the MBI 1 observation by numerically computing Eq. 4.11 and Eq. 4.12, 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 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 .
67
using them in Eq. 4.10. Since visibilities with distinct baseline lengths are sensitive
to the different multipole space, the visibility RMS error due to pointing/alignment
error has different values for distinct baseline lengths (equivalently saying distinct
bands) and the MBI
1
have 4 bands (equivalently four baseline lengths). So the RMS
error should be estimated respectively for each baseline length. The RMS error of
the visibility due to the pointing error of the horn array is shown in Fig. 4.11 for
various assumed magnitude of the pointing error. In Fig. 4.11, the RMS errors of
the visibility due to the pointing error of the horn array for four baseline length in
the MBI are shown respectively together with the NEP of the MBI. Comparing them
.-16
.-17
-18
-19
“
10
-2 0
-21
-22
d e te cto r noise of th e MBI
.-23
RMS error of th e facep late pointing [degree s 1/2]
Figure 4.11: the RMS error of the visibility due to the pointing error of the faceplate
with the NEP of the MBI detector noise (8.37 x 10- 1 7 [W s1/2]), it is seen th at the
pointing error of the horn array is much less significant than the detector noise. Since
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 .
68
both of the detector noise and the visibility RMS error due to the mispointing of the
horn array are oc 1/ T , specific integration time is not required for the comparison.
The alignment of individual feedhorns is fixed and the misalignment of individual
feedhorns is almost invariant over time. So the RMS error due to horn misalignment
is independent of the integration time. The RMS error of the visibility due to the misx 10
de te cto r noise of th e MBI
0.8
0.6
0.4
0.2
RMS error of th e horn alignm ent [degree]
Figure 4.12: the RMS error of the visibility due to the horn alignment error
alignment of individual horns are shown in Fig. 4.12 for various assumed magnitude
of the alignment error. For the 21 days integration time, the NEP of the detector
noise in the MBI is 6.214 x 10~ 20 [W]. In Fig. 4.12, the RMS errors of the visibility for
four baseline length in the MBI due to the alignment error of individual feed horns are
shown respectively together with the NEP of the MBI (21 days integration time). It
can be seen in Fig. 4.12 th at the RMS error of the visibility due to the misalignment
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 .
69
of horns is comparable to the detector noise of
21
days integration time when the
RMS error of individual horn misalignment is 5°. We can see th a t in the MBI 1, as
far as the misalignment is order of arc minutes, the horn misalignment is much less
significant than the detector noise, therefore the horn misalignment can be ignored
as well as the pointing error of the horn array.
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 5
Foregrounds and th e M BI
The CMB tem perature and polarization anisotropy have to be observed in the pres­
ence of various foregrounds. The contamination by foregrounds is one of the biggest
systematic effect in the CMB anisotropy observation. By understanding the nature
of foregrounds, we can establish the observation strategy to minimize the foreground
contamination and separate them from the CMB signal using multi-frequency chan­
nel data. In this chapter, we discuss the nature of diffuse foregrounds and present
simulated Q and U all-sky maps of polarized forgrounds. In this chapter, we investi­
gate foreground effects on power spectrum estimation by the MBI 1 observation using
simulated Q and U maps of polarized forgrounds.
5.1
T he Physical C haracteristics o f G alactic Fore­
grounds
Foreground signals differ from th at of the CMB anisotropy in their spectral and spatial
distribution. According to the foreground definition of (Tegmark et al. 2000) [57],
a foreground is “an effect whose dependence on cosmological parameters we cannot
70
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
compute accurately from first principles at the present time” . By this definition,
gravitational lensing of the CMB [58, 59], the last ISW effect are not foregrounds,
while patch re-ionization and the thermal SZ effect are foreground [57].
Figure 5.1: free-free emission at K band
Free-free emission arises from electron-ion scattering and has antenna tempera­
ture* Ta
rise of
T
at high frequency {v > 10GHz) with f3 = —2.15, and a low-frequency
oc
a
oc
v
2
due to the optically thick self-absorption [60]. The frequency de­
pendence of the free-free emission is best known of all diffuse Galactic foregrounds
[57]. The free-free emission does not generate polarization in general, though it can
become polarized by Thomson scattering within Hn regions [57, 61]. Since free-free
emission doesn’t dominate the sky at any radio frequency, radio astronomy provides
no free-free emission map of the sky [60]. But Hcc (hydrogen n = 3 —> 2 ) emission
can serve as an approximate template for the free-free emission except in the region
‘Antenna tem perature T a is defined as T a = c 2/ 2 k v 2I , where c is the speed of light, v is the
frequency, fc is Boltzmann constant and I is radiation intensity [23]. For more information, refer
to Appendix A.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 .
72
of high optical depth (r > 1 ) [60].
Figure 5.2: dust at W band
The thermal dust emission arises from the black body radiation of dust, and the
dust emission is dominant at W band F
The full sky template for dust emission is provided by (Schlegel 1998) [62] and
extrapolated in frequency by (Finkbeiner 1999) [63]. Synchrotron emission arises from
Figure 5.3: synchrotron radiation at K band
tW band refers to 75 GHz ~ 110 GHz, and K band refers to the frequency range 18 GHz ~ 26
GHz. For the complete table of microwave frequency bands, refer to Appendix A.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 .
73
the acceleration of cosmic ray electrons in magnetic fields [60]. Since a relativistic
electron number density distribution varies across the sky, as does the magnetic fields,
the resulting synchrotron emission can be characterized by a wide range of spectral
behaviors [57, 60].
The Wilkinson Microwave Anisotropy Probe (WMAP) team has separated fore­
grounds of different sources by Maximum-Entropy-Method (MEM) [29, 35] from the
WMAP observations [60]. For synchrotron emission, Haslam map [64] scaled to K
band was used for the prior spatial distribution, and for free-free, H a map by [65],
and for dust, extrapolated dust map by [63]. In Figs. 5.1, 5.2, 5.3, all-sky maps of
free-free emission at K band, dust at W band, synchrotron at K band constructed by
the WMAP team out of the WMAP observations are shown [60].
5.2
T he Sim ulated Q /U M ap o f th e Polarizer Fore­
ground
Unlike the intensity map of foreground, there is very little known about the distribu­
tion of polarized foregrounds. We are going to rely on simulated polarized foregrounds
to simulate the MBI observation and investigate the effect of the foreground contam­
ination on the power spectrum estimation of the CMB.
The Stokes parameter Q and U of polarized diffuse foregrounds can be modeled
as
Q =
/co s(
27)
/
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 .
(5.1)
74
Figure 5.4: simulated Q map of polarized foregrounds at W band (brightness tem ­
perature)
Figure 5.5: simulated U map of polarized foregrounds at W band (brightness tem ­
perature)
U =
/s in (2 7 ) / ,
where / is the fractional polarization and
7
(5.2)
is the polarization angle and I is the
intensity of foregrounds. We assumed the mean fractional polarization (/) to be
5% for dust, no polarization for free-free emission and 30% for synchrotron [6 6 ] and
simulated the polarization angle and fractional polarization across the sky by the
method of [67].
By [67], we first populate each pixel with two component [x,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 .
75
vector drawn respectively from a uniform distribution and normalize each vector
to unit magnitude, and then smooth the resulting maps with 10° Gaussian fullwidth at half-maximum (FWHM) to simulate the coherent structure of foregrounds.
We have realized these by the C routine in Appendix A.7.1.
It generates ran­
dom two component vectors from a uniform distribution, and save x and y compo­
nents in ‘foreground_x.fits’ and ‘foreground_y.fits’ respectively. W ith HEALPix util­
ity ‘smoothing’, we have smoothed ‘foreground j x . fits’ and ‘foreground_y.fits’ with
10°
Gaussian FWHM each. The smoothed maps are saved as ‘foregroundjx_smooth.fits’
and ‘foreground_x_smooth.fits’. After smoothing, we generate a new normalization
n = \J x' 2 + y'2 and assign
7
= ta n ~x{y'fx') for each pixel and use the smoothed
normalization n to define a spatially varying fractional polarization such that
f(l,b) = a n(l,b),
where the scale factor a is chosen so th a t the mean fractional polarization ( /) agrees
with the mean fractional polarization assumed. All-sky intensity map of diffuse fore­
grounds in W band from [60] are used for intensity 1(1, b) in Eq. 5.1 and 5.2. Simu­
lated Q /U maps of polarized foreground in W band by the method described above
are shown in Fig. 5.4 and 5.5. We have generated the simulated Q /U maps of polar­
ized foreground by the C routine in Appendix A.7.2. Although we do not expect these
simulated Q and U maps of polarized foregrounds to be true in details, it provides
a simple way to model polarized foregrounds while retaining the basic features of
polarized foregrounds [67]. The E mode power spectrum of these simulated polarized
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
foregrounds is shown together with th a t of the CMB polarization anisotropy in Fig.
5.6.
10
— CMB theory
9
8
•
CMB sky
x
foreground
260
280
X X
7
X
6
x X ^
Xv
X
Xx
xx
X )
xxX>xAXx
v\
5
4
3
2
1
0
140
160
180
200
220
240
300
I
Figure 5.6: E mode power spectrum of the CMB and foregrounds
5.3
T he Effect o f G alactic Foreground C ontam ina­
tion on th e C M B Pow er Spectrum E stim ation
in th e M BI
For all-sky intensity maps of dust and synchrotron at W band constructed by WMAP
team [60] shown in Fig. 5.2 and 5.3, we have assumed uniform fractional polariza­
tion to be 5% for dust and 30% for synchrotron [6 6 ] over sky. Since the polarized
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
foreground map is not available, we can only indentify the spot of least polarized
foregrounds based on the intensity map of foregrounds and mean polarization of fore­
grounds. From this, we expect the least polarized foreground at the equatorial coor­
dinate (Dec. = —49.34°, R.A. = 19.38°) and the most at the equatorial coordinate
(Dec. = -28.76°, R.A. = 265.71°).
Figure 5.7: the field location of the MBI observation
These two fields are indicated over the foreground intensity map in Fig. 5.7.
The underlying map is the mollweide-projection of the foreground intensity map in
equatorial coordinate weighted by the uniform fractional polarization (5 % for dust
and 30% for synchrotron) over sky. The white circle corresponds to the location of the
least foreground-contaminated field and the dark circle corresponds to th at of the most
foreground-contaminated field. The radius of the circle corresponds to the beamsize of
the MBI observation. Since the guess was made with insufficient knowledge (ignoring
the fluctuation in the fractional polarization), the least foreground contaminated fields
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
M B I : sin g le pointing, 1 8 b a s e lin e o rie n ta tio n , 21 d a y s in teg ratio n
5 .5
4.5-
3.5
in _
0.5
140
160
180
200
220
240
260
280
300
I
Figure 5.8: E mode power spectrum estimation from the simulated MBI observations
of the field (Dec.= —49.34°, R.A.=19.38°)
is not necessarily the least foreground-contaminated field in real or in the simulated
Q /U maps of foregrounds*. The simulated visibilities of the MBI observation can be
generated by numerically computing
/*95GHz
^
=
K o ise+ /
J 85GH z
p
/
a
\
2
O ' t t v TK
T1
dz/ / dQexp[—1.387( —— ) ] exp[i——-———]
J
V 8 -1 /
c
x { - s i n 2 (ip - $ (n ))[/(i/)Q (n ) + B fg{v)Q{g(h)}
*We have taken into account the fluctuation in the fractional polarization in simulating the
foreground Q /U maps. The simulation of polarized foregrounds was discussed in chapter 5.2.
R e p r o d u c e d with p e r m issio n o f th e co p y 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
M B I : s in g le pointing, 1 8 b a s e lin e o rie n ta tio n s , 21 d a y s in teg ratio n
5 .5
4.5
CM
3.5
LU_
5
2.5
0.5
140
160
180
200
220
240
260
280
300
I
Figure 5.9: E mode power spectrum estimation from the simulated MBI observations
of the North Celestial Pole (NCP)
+ cos2(^ - $(n))[/(i/)C7(n) + B fg(u)Qig(n)}}.
(5.3)
The simulated Q /U maps of polarized foreground in Fig. 5.4 and 5.5 are used for the
values of Qfg(n) and f/fg(n) in Eq. 5.3.
After generating the simulated visibilities for the specific fields, we estimated the E
mode power spectrum with Eq. 4.8 iteratively. We had simulated the 10 independent
MBI 1 observations by using different seed numbers for random number generator in
visibility noise generation. Fig. 5.8 shows 10 estimations of E mode power spectrum
for 4 band powers from 10 independent observations of the field (Dec. = —49.34°,
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 .
80
M B I : s in g le pointing, 1 8 b a s e lin e o rie n ta tio n s , 21 d a y s in teg ratio n
5 .5
4.5
CM
3.5
LU
LU _
cn
2.5
0.5
140
160
180
200
220
240
260
280
300
I
Figure 5.10: E mode power spectrum estimation from the simulated MBI observations
of the South Celestial Pole (SCP)
R.A. = 19.38°). Each dot is the estimate made by 10 independent observations. The
vertical bar corresponds to 1-a error in the absence of foregrounds. The horizontal bar
corresponds to the width of each band power (The multipole range the interferometer
is sensitive to.). The observation is likely to be made at Madison, Wisconsin. W ith
relative simplicity of tracking, the NCP is the most likely field to be observed in the
observation at Madison, Wisconsin. Since the latitude of Madison, Wisconsin is 43°,
the feedhorn array will be tilted by 47° from the vertical to point to the NCP. To
compensate for the earth rotation, the feedhorn array is planned to be rotated by 15°
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 .
81
M B I : s in g le pointing, 1 8 b a s e lin e o rie n ta tio n , 21 d a y s in teg ratio n
CM
HI
111 _
O
CM
1 0 '1
140
160
180
200
220
240
260
280
300
Figure 5.11: E mode power spectrum estimation from the simulated MBI observations
of the field (D ec.=—28.76°, R.A.=265.71°)
azimuthally per hour §. Fig. 5.9 shows estimations of E mode power spectrum from 10
independent observations of the North Celestial Pole (NCP). Then the observation
is likely to be made at the South Pole.
One of candidate fields observed at the
South Pole is the South Celestial Pole (SCP). Fig. 5.10 shows estimations of E mode
power spectrum from 10 independent observations of the South Celestial Pole (SCP).
Fig. 5.11 shows 10 independent estimations of E mode power spectrum of the field
§Since feedhorn arrays are housed inside the MBI cryostat, the pointing structure of the MBI is
designed so th at the cryostat can be rotated from the ground mount.
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 .
82
(Dec.= —49.3468°, R.A.=19.3869°). As seen in Fig. 5.11, the estimation of the band
power is biased to several orders higher.
As shown in the simulation, the effect of the foreground contamination on the
CMB E mode power spectra estimation can be reduced to tolerable level by observing
proper fields where the foreground contamination is expected to be small,
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 6
A ll-sky Im aging o f th e CM B
Tem perature A nisotropy
In this chapter, we present novel method of imaging all-sky CMB tem perature anisotropy
from interferometer observations. We also present the results of this method using
the simulated observations. This method is developed for the Einstein Probe Inter­
ferometer for Cosmology (EPIC), which is the satellite version of the MBI.
6.1
V isib ility Sourced by th e CM B In ten sity
As shown in Eq. 2.1, the visibility sourced by the intensity of the CMB is
V
= J dv J dQA(n - h 0) e -i2™'h [B(u, T0) + /(^ )A T (h )]
=
J dfl J duB(u, To)A(n —h 0)ei2wu'h
OQ I
p
p
+ ^ 2 ^ 2 aT,im / dQ y;m(h) / d u f ( u ) A ( n - n 0 )el27ru'n.
1=1 m=—l
Since spherical harmonics are the complete and orthonormal sets of functions, the
complex conjugate of f d u f( u ) A ( h —h 0 )e*27ru'n can be expanded in terms of spherical
83
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
harmonics such th at
( J d v f ( v ) A ( n - h )e*27run^
0
'
blmYim(n),
=
(6.1)
l,m
where
blm =
J
dQ Ylm*(n) ( y d v f ( v ) A ( n - n 0
.
(6 .2 )
W ith Eq. 6.1 and Eq. 6.1, the visibility associated with the CMB intensity is
V
J
=
dfl
J
OO
+E
dvB(v, T0 )^4(n —h 0 )ei27ru'“
I
E
OO
I'
E
p
aTMrti>m>/ dfi Ylm(n)Y*m,(n).
£
l—l m = —l V=1 m '——V
W ith the orthogonality of spherical harmonics J dQYim(9, cj))Yl*m,(8, 0 ) = biiArnm/, the
visibility associated with the CMB intensity can be expressed as
OO
I
^ = EI rn=—l
E
where
aoo —
boo*
blm
The coefficients
1
J
=J
=
dQ
J
dv B(v, To)v4(n —n 0 )ei27ru'fi
dflV /m(h)
J
dv f ( v ) A ( n — n 0 )ei27ru'A
(I > 1 ).
depend on the characteristics of the experiment and contain all
of the information of the experiment. Since Vcmb = Y^tm blm aim, the diagonal part of
window function [6 8 ] is
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 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 .
85
6.2
D eterm ination o f Individual Spherical H arm onic
Coefficient
Visibilities are
I
OO
Vi = J 2 Y l alm h lm \
I
(6.3)
m = —l
where
him* =
Index i
J
dfLYlm(n)
J
dv f {y)A(p. — nj)el27rUi'n
(I > 2 ).
indicates the different observational setup of the same
as an antenna pointing and a
baseline orientation. Individual
(6.4)
interferometer such
sphericalharmonic
coefficient are linearly weighted in the way unique to the spherical harmonics number
I, m and the distinct antenna pointing and baseline orientation. We split visibilities
into real and imaginary part such th at
I
OO
EE
Re[a/m]Re[6 Mm] - lm[aim]Im[bitim]
1=2 m = —l
I
oo
EE
lm[Vi ,cm&]
1=2 m = —l
ai-m = a%m should be satisfied to make the sky antenna a real function[18, 56]. W ith
a l - m — a lm'>
Re[V^cmk]
oo
^ ^ R^[n/o]R'^[^i,io]
1=2
I
oo
+EE RJe[ci/m]R,e[ )/m H62
Im[u/m]Im[6 ^;m
1=2 m= 1
oo
lm[VitCrnb\ =
Re[azo]Im[6HM
i,
.
.
.
1=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 .
86
OO
I
+EE
1=2 m=1
Since an interferometer is sensitive to the finite range of multipole as seen in the
window functions such as Fig. 6.1, there are finite number of spherical harmonic
coefficients which makes contribution to the visibilities. In the multipole range lo <
I < h, there are [(/i + l ) 2 —Iq + ( —lo +
1 )]/2
for the real part of aim and [(/i + l ) 2 —
II — (li — lo + l)]/2 for the imaginary part of a;m. We can convert the index l,m to
unidimensional index by assigning index 1(1 + 1) + m + 1 —Z2 for the real part of a;m
and 1(1 + 1 ) —m +
1
—Iq for the imaginary and assign index i to the real part of the
visibility and n + i to the imaginary part of the visibility where n is the number of
the complex visibilities such th at
Vi
=
(6 -5 )
hi3 a-j
i
Vn+i
^
' ^n+i j
>
(® -6)
where
Re[a/TO]
: j = 1(1 + 1 ) + m +
lm[aim\
: j = 1(1 + 1) - m + 1 - 1%
1
- %
aj —
Re[bi,im +
bij — <
_m]
Re[6 i^o]
-Im[bijm ~ biti _m]
: j = 1(1 + 1 ) + m +
1
—Z2
: j = 1(1 + 1) + 1 — Iq
: j = 1(1 + 1 ) - m +
1
- 1%
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 .
87
Ini[&i,*m
bn+ij
^
R®[^i,irn
l —m]
. j — 1(1 + l) +
Tfl +
Im[6 j^o]
: j — 1(1 + 1 ) +
1
: j — 1(1 “h i )
771+1
+
bi
biti —TO]
1
Iq
—I,
Iq
In m atrix notation, it can be written: V = b a.
6.2.1
Sim ple Inversion in th e A b sen ce of N o ise
In the extreme case of no noise (though unrealistic), the spherical harmonic coef­
ficients aT,im can be determined with the visibilities whose number is equal to the
number of spherical harmonic coefficients by Eq. 6.7.
a = b^V ,
(6.7)
where V is the vector of the measured visibilities.
6.2.2
M axim u m L ikelihood M eth o d in th e P resen ce o f N o ise
In the presence of noise which is Gaussian, the likelihood function is
£
=
( ^ f ( k i j i expI^
( A ^ v )T N ' 1 (A - v )1 ’
where n is the number of data, A is data (the sum of visibility signal and noise), N is
the noise covariance m atrix and V is the vector whose elements are given by Eq. 6.5
and 6 .6 . By exploring the parameter space to maximize likelihood, we can determine
the parameter values within certain error limits. The likelihood is maximized when
the log of the likelihood is maximized. We differentiate the log of the likelihood
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 .
88
function and and set it to zero
<91n£
= b i:i( N
daj
In matrix notation Eq.
6 .8
)jj/(A j/ - b e y a y ) = 0
(6.8)
is
b r N _1( A - b a ) = 0 .
Then the likelihood function is maximized at
a = (bTN ~ 1b)~1bTN ~ 1A.
(6.9)
Since b is not necessarily a square matrix, Eq. 6.9 is not necessarily reduced to Eq.
6.7. In case the noise covariance N is diagonal, we can get the same result with Eq.
6.9 by minimizing y 2 with respect to a for
X2 — (A —b a)T(A —b a).
Computationally, a can be obtained faster by solving the linear equation A x = y
where A — bTN _1b, x = a and y = bTN _1A than computing Eq. 6.9 through
m atrix inversion. For the diagonal and uniform noise covariance m atrix N , which is
true in the interferometer [2, 34], N in the A and y cancel each other. We determined
a by the C routine in appendix A.8.1.
The minimum possible variance on the parameter estimation can be computed
from the Fisher m atrix [55]. The 1-a error on estimation of a, is
A«i = VO^N-ib)::1
‘Einstein summation notation and the fact th at noise covariance matrix is symmetric were 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 .
89
6.2.3
Foreground C on tam in ation
W ith the contamination from foregrounds,
Bi{vo)
.
a ~ a^mt, +
y
.
i
. a.j,
f(yo)
where Bi is the frequency spectrum of ith foregrounds and vq is the peak frequency of
the system response and the sum is over all species of physically distinct foregrounds
and aj is spherical harmonic coefficient of ith foreground. Using foreground templates,
foreground contributions can be reduced down to residual level [60].
6.3
Sim ulated O bservation
We have generated simulated visibilities by numerically computing the following:
/
(
Q,v^k
dv J d 0 4 ( n - n 0) e ~ ^ uil [ f(v )A T (n ) +
(6 .10)
where Uioise is the noise, f( v ) =
and TL is the antenna tem perature
°
of the foreground^.
T=T0
The noise is simulated by generating random complex num­
bers whose real and imaginary parts are drawn from Gaussian distribution with
the variance which is the square of the noise equivalent power (NEP) of the MBI
(8.37 x 10“ 17 W H z“ 1/2). The observational frequency was assumed to be 85—95GHz
with 10 GHz bandwidth. A 20° FWHM Gaussian primary beam was assumed and
^The foreground observed by the WMAP satelite are represented in the unit of the brightness
tem perature [60].
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 .
90
baselines of length 3 [cm] with various orientations were assumed. The corresponding
window function for the assumed interferometer is shown in Fig. 6.1. In Fig. 6.1, the
numerically computed window function is shown. As seen in Fig. 6.1, the interferom-
0.8
0.6
IH
—
§f
0.4
0.2
multipole I
Figure 6.1: window function (normalized to its peak)
eter is sensitive to the multipoles 44 < I < 69 with peak at I ~ 56. So the number of
spherical harmonics to be determined is (69+ 1)2—442 = 2964. The antenna pointings
are selected to point to the center of discrete 768 partitions of equal area *, which
is shown in Fig. 6.2. The angular coordinate of the center of each HEALPix pixel
is obtained by the C routine in appendix A.8 . 2 which employs C HEALPix [26, 27]
*The partitions of equal area are made by HEALpix [26, 27]
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
Figure 6.2: antenna pointing assumed in the simulation
routine. In Fig. 6.2, each dot corresponds to an individual antenna pointing and
the circle has the 20° diameter. The visibilities of 16 baseline orientations were as­
sumed to be measured simultaneously with one day integration time for each antenna
pointing. So total number of visibility measurements is 768 x 16 = 12288, which is
about 4 times more than the number of the spherical harmonics to be determined.
Greater constraints (the number of visibilities) than the number of unknown(spherical
harmonics) is desirable in the presence of the noise.
In Eq. 6.10, the anisotropy
Figure 6.3: simulated T map
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 .
92
Figure 6.4: foreground at W band
tem perature map shown in Fig. 6.3 was used for the values of A T(n) and the antenna
tem perature map of foreground shown in Fig. 6.4 was used for the values of T/a(n).
Since it is advised th a t the internal linear combination (ILC) CMB tem perature map
shown in 1.2 should not be used for the CMB study other than for visual presenta­
tion because of the complicated noise properties of the map [60], we have used the
simulated CMB tem perature anisotropy map generated by Healpix [26, 27], which is
shown in Fig. 6.3.
W ith Eq. 6.9 §, we have estimated spherical harmonics coefficient cqm (44 < I <
69). W ith the estimated values of cqTO, we have generated all-sky map and computed
the power spectrum C f 1 by
CF =
m
(6-n )
First, we started with the noiseless visibilities in the absence of foreground and
§b is computed by the C routine shown in appendix A.9.1.
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
, n -i°
x 10
2.5
2
1.5
1
— theory
0.5
0
45
50
55
60
multipole I
o
our universe
x
observation
65
70
Figure 6.5: Power spectrum estimation from noiseless visibilities
Figure 6 .6 : Reconstructed tem perature anisotropy map from noiseless visibilities
increased the degree of reality one by one. We have estimated a;TO (44 < I < 69)
from the simulated visibilities which contain no noise and no foreground. The power
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 .
94
spectrum estimation from these values of a/m is shown in Fig. 6.5. ’Theory’ are the
values computed by CMBFAST and ’our universe’ are the C [ T of one realization of
ensemble universe which is assumed to be our universe, ’observation’ are the estimated
values by Eq. 6.11 from simulation. All-sky CMB anisotropy tem perature (44 < I <
69) map reconstructed is shown in Fig.
6 .6
. Fig. 6.7 is the reference map, which
is generated by keeping fluctuation only on the multipole (44 < I < 69) from the
input CMB map in Fig. 6.3. Compared with the reference map in Fig. 6.7, the
reconstructed map in Fig.
6 .6
has the standard deviation 0.182/iK.
Figure 6.7: the CMB tem perature anisotropy reference map (44 < I < 69)
For the next stage, we have estimated cqm (44 < I < 69) from noisy visibilities
in the absence of foreground and with one day integration time for each antenna
pointing assumed. The power spectrum estimation from these values of a/,m is shown
in Fig.
6 .8
. All-sky CMB anisotropy tem perature (44 < I < 69) map reconstructed is
shown in Fig. 6.9. Compared with the reference map in Fig. 6.7, the reconstructed
map in Fig. 6.9 has the standard deviation 1.1883//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 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 .
95
An -10
x 10
2.5
2
1.5
1
— theory
0.5
0
45
50
55
60
multipole I
o
our universe
x
observation
65
70
Figure 6 .8 : Power spectrum estimation from noisy visibilities with one day integration
time for each antenna pointing
Figure 6.9: Reconstructed tem perature anisotropy map from noisy visibilities with
one day integration time for each antenna pointing
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 .
96
We have estimated
(44 < I < 69) from noiseless visibilities in the presence
of foreground. The power spectrum estimation from these values of a;TO is shown in
Fig. 6.10. The power spectrum which are indicated by ‘our universe’ in Fig. 6.10 is
-9
x 10
3
2.5
11——
2
o
ti
51
\—
+_ 1.5
1
0.5
0
45
55
60
multipole I
50
o
our universe
x
observation
65
70
Figure 6.10: Power spectrum estimation from noiseless visibilities in the presence of
foregrounds
computed by
2v2kB/(?
~ 21 +
where
1
5 1 \aim + ~ 1 W
( 6 . 12 )
is the spherical harmonic coefficients of foreground map and 2u
is the ratio of foreground frequency spectrum and the CMB anisotropy frequency
spectrum. All-sky anisotropy tem perature (44 < I < 69) map reconstructed is shown
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 .
97
Figure 6.11: Reconstructed tem perature anisotropy map from noiseless visibilities in
the presence of foregrounds
in Fig. 6.11. Fig. 6.12 is the reference map, which is generated by keeping fluctuation
on the multipole (44 < I < 69) from the sum of the input CMB map in Fig. 6.3 and
the input foreground map in Fig. 6.4. Compared with this reference map in Fig.
Figure 6.12: Temperature anisotropy map (44 < I < 69), including foregrounds
6.12, the reconstructed map in Fig. 6.11 has the standard deviation, 0.59//K.
We have estimated cpm (44 < I < 69) from noisy visibilities of one day integration
time (for each antenna pointing) in the presence of foreground. 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 .
98
estimation from these values of aim is shown in Fig.
6.10.
All-sky anisotropy
x 10"
3
©
2.5
2
1.5
1
0.5
0
45
50
55
60
o
our universe
x
observation
65
70
Figure 6.13: Power spectrum estimation from noisy visibilities with 1 day integration
time in the presence of foreground
Figure 6.14: Reconstructed tem perature anisotropy map from noisy visibilities with
1 day integration time in the presence of 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 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
tem perature (44 < I < 69) map reconstructed from the o/m is shown in Fig. 6.14.
Compared with the reference map in Fig. 6.12, the reconstructed map in Fig. 6.14
has the standard deviation, 1.989/iK.
6.4
C om putation o f bijm
The determination of o/m by Eq. 6.9 requires the computation of b*lrn first, which can
be computed from
boo* = Y0 J du B ^ To) / d n A ^ ~ “ o)e<2,rU'fi
blm* =
J
dvf(v)
j
d fM (n —h 0 )Y/m(h)el27ru'n
(I > 1 ).
(6' 13)
(6.14)
We can compute Eq. 6.13 and 6.14 in the antenna coordinate where the z axis coin­
cides with the antenna pointing and xy plane coincides with the aperture plane. Since
the CMB is referenced to the fixed global coordinate which is Galactic coordinate in
this chapter, the spherical harmonic expansion of the CMB anisotropy tem perature
should be rotated to the antenna coordinates of each antenna pointing with rotation
m atrix D m
l ,m(lZ) where 1Z is the rotation which brings the Galactic coordinate to the
antenna coordinate. More rigorous mathematical discussion will be made in chapter
6 .6
and the results are given by Eq. 6.16 and 6.17. We found th a t computing b*[m in
the antenna coordinate suffers from serious numerical precision problem especially for
high multipole, which are due to machine floating-point rounding error occurring in
the multiplication of the rotation m atrix D lm,m(1Z). Besides the numerical precision
problem, the huge time required for computing the rotation m atrix D lm,m(7Z) makes
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 .
100
Eq. 6.17 lose most of merit gained by the availability of analytic integration over
azimuthal angle. But since the monopole of the CMB tem perature is independent
of the coordinate, Eq. 6.13 can be conveniently computed in the antenna coordinate
without the rotation m atrix D m
l /m(TZ). So in the simulation, we computed Eq. 6.13
in the antenna coordinate and Eq. 6.14 directly in the Galactic coordinate instead of
the antenna coordinate.
6.5
C om putational Load
When a single interferometer is sensitive to the multipole range lo < I < h, there are
(li + l ) 2 —/q unknowns, as mentioned earlier in section 6.2. To determine spherical
harmonic coefficient cqTO, the following equation should be solved for a:
(bTN _ 1 b) a = b TN - 1 A.
(6.15)
a is the vector of length m, b is a n x m m atrix and N is a n x n m atrix where n
is the number of the visibilities and m is the number of unknowns, which is (h +
l ) 2 —Iq. Unlikes the noise of a single dish experiment, the noise covariance m atrix of
interferometer observation is diagonal to a good approximation [34, 2]. For a diagonal
noise covariance N , b TN -1A can be computed in 0 ( nm). Since b r N -1b is m x m
matrix, inversing the m atrix requires 0 ( m3) steps [2, 35].
The same formalism we developed and presented can be applied to non-interferometric
single dish experiment to determine spherical harmonic coefficient a/m since a single
dish experiment is identical with the interferometer of zero length baseline. W ith
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 .
101
zero length baseline, it is sensitive over the multipole range ^ 0 < I ;$ A l/2 where A I
is the FWHM of the window function. Unlike the interferometer of non-zero length
baseline, a lower bound on the multipole range is fixed to be zero. So it is not quite
feasible in terms of computing capacity to determine spherical harmonics from single
dish experiment by Eq. 6.9 unless it probes low multipoles with a large FWHM.
6.6
C om puting bim * in th e A ntenna C oordinate
Let’s choose Galactic coordinate as the global coordinate for the CMB and assume
the azimuthally symmetric primary beam and coplanar baseline ^. For I = 0,
b00* =
^ y ’^ JB(i/,T 0 ) y 'd U A ( h - h o ) e i27ru-fi
p 2 tt
/
d u B ( u ,T 0) J
— 2-k
J
pi
d</>V2™cos(^
di/B(i',To)Jo(2nu)
J
u) j
d(cos 0')A{8')
d(cos 6')A(0'),
(6.16)
where 8', f t are the angular coordinates of the antenna coordinate. Since monopole
is independent of the coordinate, b0o is independent of the antenna pointing as seen
in Eq. 6.16. For I > 1,
*= J
blm
dvf{y)
J
d fM (n —ho)F/m(h)e*2,ru'n
^W ith beam switching, monopole is rejected.
IIThe baselines stays on the aperture plane. In other words, the baseline vector has no component
in parallel with the antenna pointing.
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 .
102
/
pi
dv f ( v )
J
p27T
d(cos 8')A{6')
J
d<//Yim(d, <f>)
i 2 n u cos 4>f
where 6 and 0 are the angular coordinate of Galactic coordinate. Galactic coordinate
coincide with the antenna coordinate after rotation of Euler angles (£, | —6 , 0) where
I and
6
are the galactic longitude and latitude of the antenna pointing. Rotation
m atrix of Euler angles (£, | —6 , 0) for spherical harmonics [69, 5] is
n'
m'm
1 1 *-™+™'
’
2
■“
x
r.i
26 —7T \\
cos — -—
\ / ( / + m)\{l — m)\{l + m')\(l — m')\
(I + m — k)lkl(l —k — m')\(k — m + m')\
2l—2 k+ m —m! /
m.
/ 26
—7TX\ 2k—m+m!
sm
where we take the sum over k whenever none of the arguments of factorials in the de­
nominator is negative. Then, spherical harmonics transform under the Euler Rotation
(£, | —6 ,
0
) as follows:
Ylm(8, 0) = £
Dlm,m(£, | -
6 ,0 )YM
{ff, 00
m'
Since spherical harmonics have the following form [35]
y
(O' Jj\
/
2 / + 1(1
771 ) ! pm! /
4ir (l + my P , («*9)e
,
b*m can be expressed as
=
j
dv f W) Y , Dt-n.(<.
xJ
=
E
m'
xJ
\
- b, 0) ^
1 [] ~
jT
d(cos 9') A ^ P ^ ' ( c o s 8’)
V
+
e' ” ’<fc+l) /
d,/
d(cos 0)A(9)P[m' (cos 0O>
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 .
(6-17)
103
where Jm>is the m 'th order ordinary Bessel function and </>u is the orientation of the
baseline in the antenna coordinate.
6.7
D iscussion
All-sky CMB tem perature anisotropy is described by spin 0 spherical harmonics [6 ].
Visibilities weighs spherical harmonic coefficient a;m in a peculiar way which depends
on the observational configuration such as antenna pointing and baseline orienta­
tion. Since an interferometer is sensitive over the finite range of multipoles, we can
determine individual a;m from multiple visibilities. Since all-sky map of the CMB
tem perature anisotropy on the corresponding angular scales can be reconstructed
straightforwardly from the determined set of cqm, determination of individual a/m
is equivalent to all-sky imaging of polarization on the corresponding angular scales.
Since determination of individual spherical harmonics coefficient cqm requires visibil­
ities as many as the number of cqm in the multipole range, the method described in
this chapter suits the satelite observation more than ground-based observation since
it is very hard for ground-based interferometers to measure enough visibilities in rea­
sonable amount of time due to much longer integration time required in ground-based
observations.
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
A ll-sky Im aging o f th e CM B
Polarization A nisotropy
We present the novel method of imaging all-sky CMB polarization from interferometric observations. This method is developed for the Einstein Probe Interferometer
for Cosmology (EPIC), which is the satellite version of the MBI. In the end of the
chapter, the result from simulated observations will be presented.
7.1
V isib ility as th e Linear Sum o f Spherical Har­
m onic C oefficients
W ith Eq. 1.5, 1.6 and 2.3, the visibility associated with cross linear polar states is
V
=
J
duf {v)
J
dfL4(n - h 0 )ei27ru'fi
Introducing
him
=
y
J duf {v) J d n A ( n - n o)e8(27ru-fl- 2^
b - 2 ,im =
Y
J
dvfly)
J
(ft))2 ^ m(h),
(7.1)
d ^ h - h o ) ^ ^ ' ^ - 2^ . ^ ^ ) ,
(7.2)
104
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 .
105
visibilities sourced by cross linear polar states of the CMB can be expressed as the
linearly weighted sum of <22 ,Zm as follows:
V
All the instrumental and configuration information are contained in b±2 j rn-
7.2
D eterm ination o f Individual Spherical H arm onic
Coefficient
When the interferometer is sensitive to the multipole range lo < I < h, there are
(Zi+ 1 )2—Iq spin
2
spherical harmonics to which the interferometer is sensitive to. Since
individual spin
2
spherical harmonic coefficient (22 ,
(Zq < I < h) is linearly weighted
in the way unique to its spherical harmonics number Z, rn. the antenna pointing and
the baseline orientation, spin 2 spherical harmonic coefficient set <2 2 ,im (Zo < Z < Zi)
can be determined from visibilities in various observational configurations.
Consider multiple visibilities associated with the cross linear polar states of the
CMB such that
where index i indicates a different observational configuration such as an antenna
pointing and a baseline 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 .
106
We split the visibilities into real and
imaginary part such th at
R e[W ]
=
Y ( R e [a 2,;m]Re[&2,*m - V_2,i-m] ~
Im[W]
=
Y
(
-
I m h,Zm]Im[K>)Zm + b '_2j,_ J ) ,
V_2tl_m) + Im[a 2 iim] R e [ ^ TO + b i2>l_ J ) .
l,m
We can index a2j/m with unidimensional index by assigning index 2 ( / 2 + l + rn —Z2) +
1
for the real part and 2(Z2 + Z4- m — Zq + 1) for the imaginary:
Reform]
: j = 2(I2 + Z+ m - 1$) + 1
aj —
Im[a2,;m] : j = 2(Z2 + Z+ m, - 1%+ 1)
Index i isassigned to the real part of the visibility and n + i to the imaginary part of
the visibility where n is the number of the visibilities such that
■ij aj
j
vn+i = Y h>
'n + ij
a j-
When the visibilities are associated with cross linear polar states, Vi, Vn+*, b,j and
bn + ij
are as follows:
V
=
Re[W]
(7.3)
K +i =
Im[W]
(7.4)
M b h m - b-2,l-m]
: j = 2(l2 + I + m - Iq) + 1
bij —
^m
[b2,lm + b - 2 yl - m ]
■ j = 2(Z2 + l + m - l % + 1 )
Im [b^lm
j = 2 (Z2 + l + m - l l ) + 1
R e [ &2 ,lm + b - 2 ,l-
j = 2(Z2 + Z+ m —Z2 + 1)
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 .
107
In m atrix notation, it can be written V = b a where V is the vector of length 2n, b
is a n x [2(Zi + l ) 2 —2Z2] m atrix and a is the vector of legnth 2(Zi + l ) 2 —2Z§.
Through the same derivation with Sec. 6.2.2, we can show th at the most proable
value of a in the presence of Gaussian noise is
a = (bTN -1b )-1bTN -1A,
(7.5)
where A is data (the sum of visibility signal and noise) and N is the noise covariance
matrix. The 1-cr error on estimation of a, is
Aai =
The E /B mode decomposition [17, 18] are
O'EJm
a B ,lm
(®2,lm T U—2
i( fl2 ,lm
=
® -2 ,im )/2 -
Since a_2,/m = o*2l _m. we can get the coefficients of E /B mode polarization from a 2 ym
as follows:
7.2.1
® E,lm
=
( ® 2 , l m t ffi2 , I - m ) / ^
(7 6 )
& B,lm
=
®2,/—m ) / ^ '
(7 7 )
Foreground C on tam in ation
W ith the contamination from foregrounds,
.
a ~ acmb t
)
i
Biiyo)
~F7 r- a j,
/W o )
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 .
108
where £?; is the frequency spectrum of ith foregrounds,
is the peak frequency of
the system response, the sum is over all species of physically distinct foregrounds,
a* is spin 2 spherical harmonic coefficient of ith foreground. In the observation with
the multi-frequency channels, the foreground contribution can be separated from the
CMB down to residual level, which is limited by frequency spectral incoherence and
incomplete knowledge of polarized foregrounds [57, 70].
7.3
Sim ulated O bservation
Figure 7.1: antenna pointings assumed in the simulation
We have generated the simulated visibilities by numerically computing the follow­
ing:
v
= Vnoise +
j du J
dfM (n - n 0, e - ^ u .n {_ sin2(v, _ $ ( fi))[/ (I/)Q (n) +
+ cos 2 (^ —<h(n))[/(z/)f/(n) +
Vnoise is noise, f ( v ) =
8B(i/,T)
dT
„2 Q /s(n )
2u2kf
, and Qfg and t/fg are the antenna tem perature
T=T0
of foreground Q and U polarization.
Noise in interferometric measurements is
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 .
(7.8)
109
L
LU
ii
0.6
0.4
0.2
100
multipole I
Figure 7.2: E mode window function (normalized to its peak)
CO
0.6
0.4
0.2
100
multipole I
Figure 7.3: B mode window function (normalized to its peak)
uncorrelated [34, 45]. So the noise is simulated by adding a random complex number
to each complex visibility whose real and imaginary parts are drawn from Gaussian
distribution whose variance is the square of the noise equivalent power (NEP) of
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 .
110
the MBI (8.27 x 10- 1 7 W H z” 1/2).
The observational frequency was assumed to
be 85—95GHz with 10 GHz bandwidth, 20° FWHM Gaussian primary beam and
baselines of length 3 [cm] were assumed. The corresponding window functions for
E /B mode polarization in the assumed interferometer observation are shown in Fig.
7.2 and 7.3. These are computed with Eq. 4.3 and 4.5. As seen in Fig. 7.2 and 7.3,
the interferometer is sensitive to the multipole range 44 < I < 69 with peak at I ~ 56.
So the number of spherical harmonics to be determined are is 2[(69 + l ) 2 — 442] =
5928. The antenna pointings are selected to point to the center of discrete 12 x 162
elements which whole sky are partitioned into with equal area by HEALPix [26, 27].
In Fig.7.1, each dot corresponds to each antenna pointing and the circle has the 20°
diameter.The visibilities of 20 baseline orientations were assumed to be measured
simultaneously with seven day integration time for each antenna pointing. So total
number of visibility measurements is
10
times more than the number of the spherical
harmonics to be determined, making the number of constraints
10
times more than
the number of unknowns, which is desirable in the presence of noise.
In Eq. 7.8,
the Q and U polarization map shown in Fig. 7.4 and 7.5 were used for the values of
<5(n) and U (n ) of the CMB. The antenna tem perature maps of foreground Q and U
polarization shown in Fig. 7.6 and 7.7 were used for the values of Qf g{n) and Ufg(n)
in Eq. 7.8.
W ith Eq.
7.5, we have estimated spherical harmonics coefficient a 2 ,im (44 <
I < 69). W ith the estimated values of a 2 j m, we have generated all-sky map which
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 ll
Figure 7.4: simulated Q map
Figure 7.5: simulated U map
contains the fluctuation in the multipole range. The a 2 ,im can be decomposed into
dEJm and
with Eq. 7.6 and 7.7. The power spectrum C EE and C 8B of E /B
mode polarization are estimated by
= 21 + 1 53 \aE,lm\2^
m
c,BB = ^
T E l “ s.‘"'l2m
The Qand Umaps in the multipole range can be reconstructed
the estimated values of
sets.
(7‘9)
<7-10)
with Eq.1.5 from
We started with the noiseless visibilities 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 .
112
Figure 7.6: simulated Q map of polarized foregrounds at W band
Figure 7.7: simulated U map of polarized foregrounds at W band
absence of foreground, increasing the degree of reality. First, we have estimated a 2,/m
(44 < I < 69) from the simulated visibilities with no noise and no foreground. The
power spectrum estimation of the E /B mode polarization in the relevant multipole
range are shown in Fig. 7.8 and 7.9. ‘Theory’ denote the ensemble universe and were
computed by CMBFAST [25] in a given cosmological models, ‘our universe’ denote
one realization of ensemble universe and ‘observation’ are the estimated values by Eq.
7.9 and 7.10. All-sky CMB Q and U maps (44 < I < 69) reconstructed are shown in
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 .
113
x 10"14
— theory
3.5
o
our universe
x
observation
LU
111
o
C\l
2.5
+
0.5
multipole I
Figure 7.8: E mode Power spectrum estimation (noiseless visibilities)
2.5
m
O
J£N
CD
+
— theory
0.5
o
our universe
x
observation
multipole I
Figure 7.9: B mode Power spectrum estimation (noiseless visibilities)
Fig. 7.10 and 7.11. Fig. 7.12 and 7.13 are all-sky CMB Q and U maps which contains
the CMB Q and U polarization in the multipole range (44 < I < 69). Compared 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 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 .
114
Figure 7.10: Reconstructed Q map from noiseless visibilities
Figure 7.11: Reconstructed U map from noiseless visibilities
the reference Q map in Fig. 7.12, the reconstructed map of Q polarization in Fig.
7.10 has average error 8.831 x 10_4/iK and compared with the reference U map in
Fig. 7.13, the reconstructed map of U polarization in Fig. 7.11 has average error
7.821 x 10~VK.
Increasing the degree of reality, we have estimated a 2 j m (44 < I < 69) from noisy
visibilities of seven days integration time in the absence of foreground.
The power
spectrum estimation of E /B mode polarization are shown in Fig. 7.14 and 7.15.
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
Figure 7.12: the CMB Q map (44 < I < 69)
Figure 7.13: the CMB U map (44 < I < 69)
All-sky CMB Q and U maps (44 < I < 69) reconstructed from noisy visibilities are
shown in Fig. 7.16 and 7.17.
Compared with the reference Q map in Fig. 7.12, the
reconstructed map of Q polarization map in Fig. 7.16 has average error 0.1188/jK
and compared with the reference U map in Fig. 7.13, the reconstructed map of U
polarization map in Fig. 7.17 has average error 0.1181/jK.
We have estimated a^,im (44 < I < 69) from noiseless visibilities in the presence of
foreground. The power spectrum estimations of E /B mode polarization are shown in
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 .
116
8
x 10, - 1 4
— theory
7
o
our universe
x
observation
6
5
4
3
2
1
0
45
50
55
60
65
70
multipole I
Figure 7.14: E mode Power spectrum estimation from noisy visibilities with seven
days integration time
3.5
x 10
theory
3
o
our universe
x
observation
2.5
m
m
_ 2„
O
e
oj
7
1.5
1
0.5
o o-e0
45
-^-450
§ u 5 o u $ 5 n u Ou n o n n^
55
60
65
70
multipole I
Figure 7.15: B mode Power spectrum estimation from noisy visibilities with seven
day integration 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 .
117
Figure 7.16: Reconstructed Q map from noisy visibilities with seven day integration
time
Figure 7.17: Reconstructed U map from noisy visibilities with seven day integration
time
Fig. 7.18 and 7.19. The power spectra which are indicated by ‘our universe’ in Fig.
6 .1 0
is computed by
r EE
_
1
~
r BB -
^i
~
1
21
+
|
2 V*kB /(?
fiy )
1
2is2kB/c 2
I,
1
21 + 1
j
|2
af9,Ei™\ >
’lm
f{y)
/7 1 1 v
I '- i i j
2
f9,Blm' ’
where afgtEim and £J/s,b/to are the E and B mode decomposed spherical harmonic coef­
ficients of foreground map, and 2
u
is the ratio of foreground frequency 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 .
1-
O'
o
our universe -
x
observation
'----------- '----------- '----------- '----------- '-----------
45
50
55
60
multipole I
65
70
Figure 7.18: E mode power spectrum estimation map (no noise case in the presence
of foregrounds)
1.2
x 10"
1
aa _
O
s.
0.8 - ° o
o o
X X
o
o
our universe
x
observation
X
O O
Oo
°-6
±
o
0.4
0.2
45
50
55
multipole I
60
65
70
Figure 7.19: B mode power spectrum estimation map (no noise case in the presence
of 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 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
and the CMB anisotropy frequency spectrum. All-sky Q and U map (44 < I < 69)
reconstructed are shown in Fig. 7.20 and 7.21.
Fig. 7.22 and 7.23 are all-sky
Figure 7.20: Reconstructed Q map from noiseless visibilities in the presence of fore­
grounds
Figure 7.21: Reconstructed U map from noiseless visibilities in the presence of fore­
grounds
CMB Q and U maps which contain the polarization of the CMB and foreground in
the multipole range (44 < I < 69).
Compared with the reference Q map in Fig.
7.22, the reconstructed map of Q polarization in Fig. 7.20 has average error 4.26//K
and compared with the reference U map in Fig. 7.23, the reconstructed map of U
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 urth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
120
Figure 7.22: Q map (44 < I < 69), including foregrounds
Figure 7.23: U map (44 < I < 69), including foregrounds
polarization in Fig. 7.20 has average error 5.22/zK.
We have estimated «2 j m (44 < I < 69) from noisy visibilities of seven day integra­
tion time in the presence of foreground. The power spectrum estimation of E /B mode
polarization are shown in Fig. 7.24 and 7.25.
All-sky Q and U map (44 < I < 69)
reconstructed are shown in Fig. 7.26 and 7.27. Compared with the reference Q map
in Fig. 7.22, the reconstructed Q map in Fig. 7.26 has average error 4.267/dK and
compared with the reference U map in Fig. 7.23, the reconstructed U map in Fig.
R e p r o d u c e d w ith p e r m is s io n o f th e c o 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 .
121
x 10
o
our universe
x
observation
uLUj — 5
o
CNj 4
0
o
0
O
o
oo
° ° o o ° o o
45
50
55
multipole I
60
65
70
Figure 7.24: E mode power spectrum estimation map (noisy visibilities with seven
days integration time in the presence of foreground)
1.2
x 10
0.8 - ° o
o
5
o o
o
o
o
0.6f-
Oo
o
0 .4 -
0.2
-
45
50
55
multipole I
60
o
our universe
x
observation
65
70
Figure 7.25: B mode power spectrum estimation map (noisy visibilities with seven
days integration time in the presence of foreground)
R e p r o d u c e d w ith p e r m is s io 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 .
Figure 7.26: Reconstructed Q map from noisy visibilities with seven day integration
time in the presence of foregrounds
Figure 7.27: Reconstructed U map from noisy visibilities with seven day integration
time in the presence of foregrounds
7.27 has average error 5.226pK.
7.4
C om putation o f b± 2 j m
The determination of cqm by Eq. 7.5 requires the computation of b±2 ,im first, which
can be computed from
b2M
=
y
J
duf{u)
f
df2 R ( h - h o ) e i^ u'fi- 2^ 2 $(fi» 2 y/m(h),
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 .
(7.13)
123
6 - 2 ,im
=
y J duf ( u) J d ^ A (n - n 0 )ei(27ru'ft+2V,“ 2 $(ft))_ 2 l / m(n).
(7.14)
Though we can compute Eq. 7.13 and 7.14 in the antenna coordinate, which are
given by Eq. 7.15 and 7.16, computing b±2 tim in the antenna coordinate suffers from
the same problems discussed in chapter 6.4. So in the simulation, we computed Eq.
7.13 and 7.14 directly in the Galactic coordinate instead of the antenna coordinate.
7.5
C om puting in th e A n ten n a C oordinate
Let’s choose Galactic coordinate as the global coordinate for the CMB, assume the
azimuthally symmetric primary beam. O' and
0
' are the angular coordinates of the
antenna coordinate. Then, Eq. 7.13 and Eq. 7.14 are
,
02 ,Zm
=
"2 " J d v f(y ) J dfL4(n - n Q)e^™ -h- ^ +2^ \ Y lrn{n)
—l
~2 ~
/
p i
du f( y ) J
b-2M = y J dv^ v) J
=
y Jdvf(v)J
p 2n
d(cos e')A{6') J
d ^ 2Ylm(6, <
j>y(2*u<x»W-M-24>+W)i
d ^ (n -n 0)ei(2ffU-fi+2,/’-2$(fl))_211m(n)
d(cos 0')A{Q')
d<j7 - 2Ylm{d,
,
4>u is the orientation of the baseline in the antenna coordinate, and angular coordi­
nate of galactic coordinate (6, </>) coincides with the antenna coordinate (O', (j)') after
rotation of Euler angles (£, | — b, 0) where I and b are the galactic longitude and
latitude of z axis of the antenna coordinate. Non-zero spin spherical harmonics trans­
form under coordinate rotation 7Z £ SO(3) in the same way with zero spin spherical
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 .
124
harmonics as follows [71, 72]:
±2Y,m (0, *) = J 2 D 'm, J K ) ±2Y,m,(ff, 4>'),
m!
where Dim,m(77) is the rotation m atrix [69, 5] and angular coordinate (9,6) corresponds
to {O',6 ) in the rotated coordinate. Spin ± 2 spherical harmonics defined in Galactic
coordinate is related to the spin ± 2 spherical harmonics in the antenna coordinate as
follows:
±2Ylm(0, 6) = Y .
\ ~&
,0
) ±
2
, 6)
m'
Rotation m atrix Dlm,m{7Z) has the following form:
n*
m
(fl-hrt
’2
=
i6-m+m>
’ ;
k
V O + m ) ! ( i - m ) ! ( Z + m ,) ! ( Z - m Q !
{l + m —k)\k\(l —k —m')\{k —m + m')\
d
\ 2l—2 k+ m —m ' /
Z O — 7T \
X COS
;—
/.
sin
cv
1 0 — 7T
\ 2k —m + m f
where we take the sum over k whenever none of the arguments of factorials in the
denominator is negative. Spin
- 2 Y l m { 0 , 6 )
spherical harmonics have the following form [17]
±2
=
) J ^ ~ [ F i , i m(0) + F2M (0)\eimt
=
^
^
[
F
i , i m
{
0
)
-
F
2 M
{ 9 ) } e i m 6
where F ijm and F2j,m can be computed by Eq. A . 2 and A.3 in terms of Legendre
polynomials [24], fr±2 ,/m are
^2,1m
Jdvf(v)T0
J
d(cos 0')A{9')
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 .
125
T
X
I
Dlm ’m (Z , 2 -
^
=
6’ °)
0 ' ) e i (2™ « » ( * , - ‘M - 2* + 2* ')
m'
Dlm,m(£, ^ - 6 , 0 )
y e - i2^ v ^ ( 2 i + l ) E
ml
X y ,'d(cOS0/)^ (^ /) [ ^ m ' ( ^ ) + ^ TO'(^)]
J
d v f ( v ) J m,+2(27ru),
(7.15)
5 -2 ,im
=
J
Y
d u f ( u)
J
d (C0S 6')A{0')
f 2«
X I dcf>'J2 Dto'to(^ 2 “ 6’ °) - 2 ^ ( 0 ', 0 ')e * (2 ™ W'- ^ ) +2^-2^)
®
to'
=
^
V
^
+ l)
ei(TO'-2)(4>u+7r/2) D ^ E
_ 6>Q)
m!
x
j
d{c^ff)A {& )[F Um!{&) - F2M,{&)\
J
d v f ( v ) J m,_2(27ru),
(7.16)
where J n is the n th order ordinary Bessel function.
7.6
D iscussion
All-sky CMBpolarization is
described by spin ± 2 sphericalharmonics
and visibilities weighs a±2,/m in a peculiar
[17, 18, 24]
way whichdepends on the observational
configuration such as an antenna pointing and a baseline orientation. Since an inter­
ferometer is sensitive over the finite range of multipoles, we can determine individual
a± 2,im from multiple visibilities. As shown in Sec. 6.5, determination of a requires
0 (m 3) where m is the number of unknowns, which is 2(l\ + l ) 2 — 212. All-sky Q
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 .
and U map of the CMB polarization on the corresponding angular scales can be re­
constructed straightforwardly from the determined set of a± 2 ,im■ Determination of
individual a±2 j m is equivalent to all-sky imaging of polarization in the corresponding
angular scales. Since determination of individual spin
±2
spherical harmonics re­
quires visibilities as many as the number of a±2 ,im in the multipole range, the method
described in this chapter suits satelite observations more than ground-based observa­
tions.
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 8
Future Works
In real experiments, what are measured are time-ordered raw data. They should
be converted into visibilities. The procedure to convert raw data into an individual
visibilities depends on the details of the experiment. In the MBI, visibility signals of
various baseline lengths are multiplexed to detectors. By a planned phase-modulation
scheme for the MBI, specific linear combinations of signals from channels of distinct
phase modulation yields an individual visibility. The conversion of raw data to visi­
bilities should be taken into account in the simulations for future works.
In the simulation of the noise for the MBI, the other dominant sources of noise
such as amplifier noise and atmospheric fluctuations were not taken into account and
absorptivity of bolometers were simply assumed to be unity.
In the simulation of polarized foregrounds, polarization angle and fractional polar­
ization were randomly drawn from uniform distribution and they were smoothed by
10° FWHM Gaussian beam. Thermal dusts which are dominant foreground sources
at W band are unlikely to have coherent structures at these scales. While simulation
127
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
of polarized foreground relied on a Monte-Carlo simulation with some imitation of
coherent structures for polarization angle and fractional polarization, individual char­
acteristics of distinct diffuse foregrounds should be taken into account for a realistic
simulation of polarized foregrounds.
In the all-sky imaging of the CMB tem perature and polarization anisotropy, dis­
crete dot-like antenna pointings were assumed, but in real observation by satelite, an­
tenna pointings are in constant motion, sweeping over sky. Dot-like discrete antenna
pointings should be replaced by antenna pointings sweeping a finite area. W hat are
measured by the satelite interferometer observations are the averages of visibilities of
antenna-pointings over the finite area. The derivation described in the chapter
6
and
7 can be still used by replacing delta function with a continuous time-varying function
for antenna pointings. All-sky imaging of the the CMB tem perature and polarization
anisotropy contains huge contribution from foregrounds. Separating foregrounds by
using multi-frequency channels with foreground templates or frequency spectrum of
foregrounds, should be added to the simulation for future works.
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 ppendix A
A ppendix
A .l
N um erically C om puting Spin 2 Spherical Har­
m onics
Spin 2 spherical harmonics have the following form:
2> W M ) =
+ FVmW™**
(A-1)
where F ijm and F2j m can be computed in terms of Legendre polynomials [24]:
/„s
F u " '(" ) =
„
(I ~ 2 )!(Z —m)! r/,
\J (l + 2 )!(l + m)! +
sm 0
F2 M (9) =
Ncos 0
,
..
C°
+ h ( ( - l ) W ’"(coS«)],
2
2^ [| + 2 )|(j + ^
j
(A.2)
+ m )p i-i(cos0) - (^ “ !) cos0P,m(cos0)].
(A.3)
In Eq. A .l, e”7^ = cos(mb) + isin(m ^) can easily be computed with the built-in
trigonometric function in C language. To compute Legendre polynomials, we used
the following recurrence relations which are numerically stable [35]:
(l-m)Pr(x)
=
x { 2 l - l ) P ^ { x ) - { l + m - l ) P ^ 1{x)
129
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
P%{x)
=
( - l ) m(2 m - l ) ! ! ( l - x 2 )(m / 2 ).
The following C routine we developed computes P[n(cos0) and
F%im(P)]- For (lmin < I < lmax, x = cos(9), the values of y ^
1
^
^ ^ [ F \,im{9) +
^
["{cos9) are
stored in Nlm_Plm and the values of J ^ ^ [ F \ j , m(ff) + F2tim(9)\ are stored in Nlm_P21m.
void get_Nlm_Plm(double *Nlm_Plm,double *Nlm_P21m,long Lmin .long Lm ax.fioat x)
{
float l,m;
float Nl,y2;
in t m m .m lm l.m lm ;
in t lm,llm,12m,L,M;
in t Plm_n,P21m_n;
d o u b le *NlmPlm,*NlmP21m;
//th e number of Legendre polynomials in the range of Lmin and Lmax
P lm _n=((l_m ax+l)*(L m ax+2)-L m in*(L m in+l))/2;
//th e number of (Fllm+F21m) in the range of Lmin and Lmax
P21m_n= (L m ax+1) * (Lm ax+1)—Lmin*l_min;
Nlm Plm =(double*)m alloc((Lm ax+1 )*(Lmax+1 )*sizeof(double)); / /Legendre polynomials
Nlm P21m =(double*)m alloc(2*(Lm ax+l)*(Lm ax+l)*sizeof(double)); / / (Fllm+F21m)
NlmPlm[0]=0.282; / / the value of Legredre polynomial (1=0,m=0)
//com puting Legedre polynomials by the recurrence relations
for (m =l;m <=l_m ax;++m )
{
m m =(int) (m * (m + l)/2 + m );m lm l= (in t) ((m—l)* m /2 + m —1);
NlmPlm[mm]=—sqrt((2*m +l)/(2*m ))*sqrt(l—pow(x,2))*NlmPlm[mlml];
}
for (m =0;m <l_m ax;++m )
(m lm = (in t) ((m +l)*(m +2)/2+m );
m m =(int) (m*(m+l)/2+m);NlmPlm[mlm]=x*sqrt(2*m+3)*NlmPlm[mm];}
for (m =0;m <=(l_m ax—2);++m )
{
for (l=m +2;l<=l_m ax;++l)
{
lm = (int) (l* (l+ l)/2 + m );llm = (in t) ((l-l)*l/2+ m );12m = (int) ((l-2 )* (l-l)/2 + m );
NlmPlm[lm]=(x*NlmPlm[llm]—sqrt((l+m —1)*(1—m —1)/((2*1—3)*(2*1—1)))
*NlmPlm[12m])*sqrt((4*l*l—1)/(1*1—m*m));
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
}
}
y2=l—x*x;
/ /computing (Fllm+F21m)
for (l=2;l<=l_m ax;++l)
{
N l=2/sqrt((l+2)*(l+l)*l*(l—1));
lm = (int) (l* (l+ l)/2 );llm = (in t) ((l-l)* l/2 );
NlmP21m[(int) (l*(l+l))]=N l*(-(l/y2+ l*(l-l)/2)*N hnP lm [lm ]
+sqrt(l*l*((2*l+l)/(2*l—l)))*x/y2*NlmPlm[llm]);
for (m = l;m < l;+ + m )
{
lm = (int) (l* (l+ l)/2 + m );llm = (in t) ((1—l)*l/2+m );
NlmP21m[(int) (l*(l+ l)+ m )]= N l*(-((l-m *m )/y2+ l*(l-l)/2.0+ m /y2*(l-l)*x)*N lm P lm [lm ]
+sqrt((l*l—m *m )*((2.0*l+l)/(2.0*l—l)))*(x+m)/y2*NlmPlm[llm]);
NlmP21m[(int) (l* (l+ l)-m )]= N l* (-((l-m * m )/y 2 + l* (l-l)/2 .0 -m /y 2 * (l-l)* x )* N lm P lm [lm ]
+sqrt((l*l—m *m )*((2.0*l+l)/(2.0*l—l)))*(x—m)/y2*NlmPlm[llm]);
}
lm =(int) (l*(l+ l)/2+ l);
NlmP21m[(int) (l* (l+ l)+ l)]= N l*(-((l-l*l)/y2+ l*(l-l)/2.0+ l/y2*(l-l)*x)*N lm P lm [lm ]);
lm =(int) (l*(l+ l)/2+ l);
NlmP21m[(int) (l* (l+ l)-l)]= N l* (-((l-l* l)/y 2 + l* (l-l)/2 .0 -l/y 2 * (l-l)* x )* N lm P lm [lm ]);
}
/ / copy the computed values to the address of the variable with which the function is called with
memcpy(Nlm_Phnp&NhnPhn[Lm in*(Lm in+l)/2],Plm _n*sizeof (double));
memcpy(Nlm_P21m1&NhnP2hn[Lmin*Lmin],P21m_n*sizeof (double));
free(NlmPlm);free(NlmP21m);
}
A .2
A .2.1
Sim ulating visib ilities
W ith flat sky ap proxim ation
idum =—1; gasdev(&idum);
//initializing the Gaussian random number generator of ’Numerical Recipe in C’ .
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
for (i=0;i<n_y;++i)
{
y = —y_max+i*delta_y; //coordinate y
for (j=0;j<n_x;++j)
{
x = —x_max+j*delta_x; //coordinate x
beam=exp(—(x*x+y*y)/(2*pow(sigma,2))); //b e a m pattern of the primary feedhorn
Re_V_T[k]+=band_width*cos(2*PI*(u[k]*x+v[k]*y))*delta_x*delta_y
*(anisotropy_spectrum*beam*T_cmb[i*n_x+j]+foreground_spectrum*beam*T_fg[i*n_x+j]);
/ /R eal part of visibilities associated with intensity
Im_V_T[k]+=band_width*sin(2*PI*(u[k]*x+v[k]*y))*delta_x*delta_y
*(anisotropy_spectrum*beam*T_cmb[i*n_x+j]+foreground_spectrum*beam*T_fg[i*n_x+j]);
//Im aginary part of visibilities associated with intensity
Re_V_Q[k]+=band_width*cos(2*PI*(u[k]*x+v[k]*y))*delta_x*delta_y
*(anisotropy_spectrum*beam*Q_cmb[i*n_x+j]+foreground_spectrum*beam*Q_fg[i*n_x+j]);
//R e a l part of visibilities associated with Q polarization
Im_V_Q[k]+=band_width*sin(2*PI*(u[k]*x+v[k]*y))*delta_x*delta_y
*(anisotropy_spectrum*beam*Q_cmb[i*n_x+j]+foreground_spectrum*beam*Q_fg[i*n_x+j]);
//Im aginary part of visibilities associated with Q polarization
Re_V_U[k]+=band_width*cos(2*PI*(u[k]*x+v[k]*y))*delta_x*delta_y
*(anisotropy_spectrum*beam*U_cmb[i*n_x+j]+foreground_spectrum*beam*U_fg[i*n_x+j]);
/ /R eal part of visibilities associated with U polarization
Im_V_U[k]+=band_width*sin(2*PI*(u[k]*x+v[k]*y))*delta_x*delta_y
*(anisotropy_spectrum*beam*Q_cmb[i*n_x+j]+foreground_spectrum*beam*U_fg[i*n_x+j]);
/ /Im aginary part of visibilities associated with U polarization
}
}
/ /adding noise
Re_V_T[k]=Re_V_T[k]+NEP/sqrt(days)*gasdev(&idum)*sqrt(.5);
Im_V_T[k]=Im_V_T[k]+NEP/sqrt(days)*gasdev(&idum)*sqrt(.5);
Re_V_Q[k]=Re_V_Q[k]+NEP/sqrt(days)*gasdev(«^idum)*sqrt(.5);
Im_V_Q[k] =Im_V_Q [k] +NEP/sqrt(days)*gasdev(&idum)*sqrt(.5);
Re_V_U[k]=Re_V_U[k]+NEP/sqrt(days)*gasdev(&idum)*sqrt(.5);
Im_V_U[k]=Im_V_U[k]+NEP/sqrt(days)*gasdev(&idum)*sqrt(.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 .
133
n_x and n_y are the number of pixels along x and y axis. Index i is the index
for the x coordinate, j is the index for y coordinate and index k is the index for the
baselines.
A .2.2
W ith o u t flat sky ap proxim ation
idum =—1; gasdev(&idum);
//initializing the Gaussian random number generator of 'N um erical Recipe in C'.
//integrating over the descending angle
for (i=0;i<theta_steps_m ap;++i)
{
k_z=cos_theta[i];
solid_angle=sin_theta[i]*delta_theta*delta_phi; //com puting solid angle
//integrating over the azimuthal angle
for (j=0 ;j <phi_steps_map;+ + j)
{
k_x=sin_theta[i]*cos_phi[j];k_y=sin_theta[i]*sin_phi[j];
/ / A_i is the index of antenna pointings
for (A_i=0;A_i<A_n;++A_i)
{
/ / Computing beampattern
cos_beam_theta=cos_antenna_theta[A_i]*cos_theta[i]
+sin_antenna_theta[A_i]*sin_theta[i]*(cos_antenna_phi[A_i]*cos_phi[j]
+sin_antenna_phi [A_i] *sin_phi [j]);
if (cos_beam_theta<cos_beam_theta_cutoff) c o n tin u e ;if (cos_beam_theta<0) co n tin u e;
if (cos_beam_theta<l)
{beam_theta=acos(cos_beam_theta);beam=exp(—pow(beam_theta,2)/(2*pow(sigma,2)));} e ls e beam = l;
theta_dot_theta_A=cos_theta[i]*cos_phi[j]*cos_antenna_theta[A_i]*cos_antenna_phi[A _i]
+cos_theta[i]*sin_phi[j]*cos_antenna_theta[A_i]*sin_antenna_phi[A_i]
+sin_theta[i]*sin_antenna_theta[A_i];
theta_dot_phi_A=—cos_theta[i]*cos_phi[j]*sin_antenna_phi[A_i]+cos_theta[i]*sin_phi[j]*cos_antenna_phi[A_i];
N_rotation=sqrt(theta_dot_theta_A*theta_dot_theta_A+theta_dot_phi_A*theta_dot_phi_A);
cos_rotation=theta_dot_theta_A/N_rotation;sin_rotation=theta_dot_phi_A/N_rotation;
cos_2_rotation=cos_rotation*cos_rotation—sin_rotation*sin_rotation;
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 .
134
sin_2_rotation=2*sin_rotation*cos_rotation;
/ /B_i is the index of baselines
for (B_i=0;B_i<B_n;++B_i)
{
k_dot_B=k_x*B_x[A_i*B_n+B_i]+k_y*B_y[A_i*B_n+B_i]+k_z*B-z[A_i*B_n+B_i];
spectrum_interference_r=0;spectrum_interference_i=0;
foreground_spectrum_interference_r=0;foreground_spectrum_interference_i=0;
cos_f_k_dot_B=cos(phaseO*k_dot_B);sin_f_k_dot_B=sin(phaseO*k_dot_B);
cos_df_k_dot_B=cos(delta_phase*k_dot_B);sin_df_k_dot_B=sin(delta_phase*k-dot_B);
//integrating over bandwidth
for (Li=0;f-i<frequency_n;++f_i)
{
spectrum_interference_r+=spectrum[f_i]*cos_f_k_dot_B;
spectrum_interference_i+=spectrum[f_i]*sin_f_k_dot_B;
foreground_spectrum_interference_r+=foreground_spectrum[f_i]*cos_f_k_dot_B;
foreground_spectrum_interference_i+=foreground_spectrum[f_i]*sin_f_k_dot_B;
cos_Lk_dot_B=cos_f_k_dot_B*cos_df_k_dot_B—sin_f_k_dot_B*sin_df_k_dot_B;
sin_Lk_dot_B=sin_f_k_dot_B*cos_df_k_dot_B+sin_df_k_dot_B*cos_f_k_dot_B;
}
/ / Visibilities associated with intensity
Re_V[A_i*B_n+B_i]+=solid_angle*df *beam*spectrum_interference_r*cmb_T_map[i*phi _steps _map+j];
Im_V[A_i*B_n+B_i]+=solid_angle*df*beam*spectrum_interference_i*cmb_T_map[i*phi_steps_map+j];
Re_V[A-i*B_n+B_i]+=solid_angle*df*beam*foreground_spectrumJnterference_r
*foreground_T_map[i*phi_steps_map+j];
Im_V[A_i*B_n+B_i]+=solid_angle*df*beam*foreground_spectrum_interference_i
*foreground_T_map[i*phi_steps_map+j];
/ / Visibilities associated with <E_x~2 — E_y~2>
Re_Vr[A_i*B_n+B_i]+=solid_angle*df*beam*spectrum_interference_r
*((cos_2_psi[A_i*B_n+B_i]*cos_2_rotation+sin_2_psi[AJ*B_n+B_i]*sin_2-rotation)
*cmb_Q_map[i*phi_steps_map+j]
+ ( —sin_2_psi[A_i*B_n+B_i]*cos_2_rotation+sin_2_rotation*cos_2_psi[A_i*B_n+B_i])
*cmb_U_map[i*phi_steps_map+j]);
Im_Vr[A_i*B_n+B_i]+=solid_angle*df*beam*spectrum_interference_i
*((cos_2_psi[A_i*B_n+B_i]*cos_2_rotation+sin_2_psi[A_i*B_n+B_i]*sin_2-rotation)
*cmb_Q_map[i*phi_steps_map+j]
+ ( —sin_2_psi[A_i*B_n+B_i]*cos_2_rotation+sin_2_rotation*cos_2_psi[A_i*B_n+B_i])
*cmb_U_map[i*phi_steps_map+j]);
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
Re_Vr[A_i*B_n+B_i]+=solid_angle*df*beam*foreground_spectrum_interference_r
*((cos_2_psi[A_i*B_n+B_i]*cos_2_rotation+sin_2_psi[A_i*B_n+B_i]*sin_2_rotation)
*foreground_Q_map[i*phi_steps_map+j]
+ ( —sin_2_psi[A_i*B_n+B_i]*cos_2_rotation+sin_2-rotation* cos_2_psi[A_i*B_n+B_i])
*foreground_U_map[i*phi-steps_map+j]);
Im_Vr[A_i*B_n+B_i]+=solid_angle*df*beam*foreground_spectrum_interference_i
*((cos_2_psi[A_i*B_n+B_i]*cos_2_rotation+sin_2_psi[A_i*B_n+B_i]*sin_2-rotation)
*foreground_Q_map[i*phi_steps_map+j]
+ ( —sin_2_psi[A_i*B_n+B_i]*cos_2_rotation+sin_2_rotation*cos_2_psi[A_i*B_n+B_i])
*foreground_U_map[i*phi_steps_map+j]);
/ / Visibilities associated with <2 E_x E_y>
Re_Vi[A_i*B_n+B_i]+=solid_angle*df*beam*spectrum_interference_r
*((—sin_2_psi[A_i*B_n+B_i]*cos_2_rotation+sin_2_rotation*cos_2_psi[A_i*B_n+B_i])
*cmb_Q_map[i*phi_steps_map+j]
+(cos_2_psi[A_i*B_n+B_i]*cos_2_rotation+sin_2_psi[A_i*B_n+B_i]*sin_2_rotation)
*cmb_U-map[i*phi_steps_map+j]);
Im_Vi[A_i*B_n+B_i]+=solid_angle*df*beam*spectrum_interference_i
*((—sin_2_psi[A_i*B_n+B_i]*cos_2_rotation+sin_2_rotation*cos_2_psi[A_i*B_n+B_i])
*cmb_Q_map[i*phi_steps_map+j]
+(cos_2_psi[A_i*B_n+B_i]*cos_2_rotation+sin_2_psi[A_i*B_n+B_i]*sin_2_rotation)
*cmb_U_map[i*phi_steps_map+j]);
Re_Vi[A_i*B_n+B_i]+=solid_angle*df*beam*foreground_spectrum_interference_r
*((—sin_2_psi[A_i*B_n+B_i]*cos_2_rotation+sin_2_rotation*cos_2_psi[A_i*B_n+B_i])
*foreground_Q_map[i*phi_steps_map+j]
+(cos_2_psi[A_i*B_n+B_i]*cos_2_rotation+sin_2_psi[A_i*B_n+B_i]*sin_2_rotation)
*foreground_U_map[i*phi_steps_map+j]);
Im_Vi[A_i*B_n+B_i]+=solid_angle*df*beam*foreground-spectrum_interference_i
*((—sin_2_psi[A_i*B_n+B_i]*cos_2_rotation+sin_2_rotation*cos_2_psi[A_i*B_n+B_i])
*foreground_Q_map[i*phi_steps_map+j]
+(cos_2_psi[A_i*B_n+B_i]*cos_2_rotation+sin_2_psi[AJ*B_n+B_i]*sin_2-rotation)
*foreground_U_map[i*phi_steps_map+j]);
}
}
}
}
//adding noise
for (A_i=0;A_i<A_n;++A_i)
{
for (B_i=0;B_i<B_n;++B_i)
{
Re_V[A_i*B_n+B_i]=Re_V[A_i*B_n+B_i]+NEP/sqrt(days)*gasdev(&idum)*sqrt,(.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 .
136
Im_V[A_i*B_n+B_i]=Im_V[A_i*B_n+B_i]+NEP/sqrt(days)*gasdev(&idum)*sqrt(.5);
Re_Vr[A_i*B_n+B_i]=Re_Vr[A_i*B_n+B_i]+NEP/sqrt(days)*gasdev(&idum)*sqrt(.5);
Im_Vr[A_i*B_n+B_i]=Im_Vr[A_i*B_n+B_i]+NEP/sqrt(days)*gasdev(&idum)*sqrt(.5);
Re_Vi[A_i*B_n+B_i]=Re_Vi[A_i*B_n+B_i]+NEP/sqrt(days)*gasdev(&idum)*sqrt(.5);
Im_Vi[AJ*B_n+B_i]=Im_Vi[A_i*B_n+B_i]+NEP/sqrt(days)*gasdev(&idum)*sqrt(.5);
}
}
theta_dot_theta_A is
beam is
A .3
A .3.1
A
e g ■e g A
and theta_dot_phi_A is
e g ■e^A.
(n —no), k_dot_B is B • n, phaseO is 27r / 85 GHz and df is d u .
A perture Synthesis
Search for th e feed horn configuration by sim u lated
annealing
# d e fin e horn_n 4
# d e fin e horn_n_p 2
# d e fin e baseline.n 4
//checking if the minimum baseline is smaller than the 2.5 times of the feedhorn radus.
flo a t check_minimum_B()
{
flo a t B _x,B_y,violation;
violation=0;
for (i=0;i<horn_n;++i)
{
for (j= 0;j< i;+ + j)
{
B_x=horn_trialO[i]—horn_trialO[j];B_y=horn_trialO[i+horn_n]-horn_trialO[j+horn_n];
i f ((B_x*B_x+B_y*B_y)<0.004032) violation—=2*theta_n;
// 0 .004032 is the square of 0.0635 [m], 0.0635 is 2.5 times of the radius of a single feedhorn used.
//T h e baseline should be longer than 0.0635 [m] to avoid the physical overlapping
//betw een a pair of feedhorns.
}
}
r e tu r n violation;
}
/ / rotated feedhorn array by the given rotation steps
v o id rotate_horn_array()
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 .
137
{
for (theta_i=0;theta_i<theta_n;++theta_i)
{
for (i=0;i<horn_n;++i)
{
horn_trial[i]=horn_trialO[i]*cos_theta[theta_i]—horn_trialO[i+horn_n]*sin_theta[theta_i];
horn_trial[i+horn_n]=horn_trialO[i]*sin_theta[theta_i]+horn_trialO[i+horn_n]*cos_theta[theta_i];
}
fo r (i=0;i<horn_n_p;++i)
{
for (j=0;j<horn_n_p;++j)
{
u[theta_i*baseline_n+i*horn_n_p+j]=(horn_trial[i]—horn_trial[j+horn_n_p])/lambda;
v[theta_i*baseline_n+i*horn_n_p+j]=(horn_trial[i+horn_n]—horn_trial[j+horn_n_p+horn_n])/lambda;
}
}
}
/ / V(—u, —v) is the complex conjugate of V(u,v).
fo r (i=0;i<theta_n*baseline_n;++i) {u[i+theta_n*baseline_n]=—u[i];v[i+theta_n*baseline_n]=—v[i];}
}
/ / compute the merit funtion f
flo a t get_mean_spacing()
{
violation=check_minimum_B ();
spacing-var=—violation*delta_u*delta_u* 1000000000;
/ /W hen the feedhorn configuration violates the constraints such as the minimum length requirement
//o f the baseline, very big number (see 1000000000) is substracted for penanlty
//s o that the configuration is never accepted.
rotate_horn_array ();
fo r (i=0;i<array_n*vis_n;++i)
{
for (j=0;j < array_n*vis_n;+ + j) {spacing|j+l]=(u[i]-u[j])*(u[i]-u[j])+(v[i]-v[j])*(v[i]-v[j]);}
sort(array_n*vis_n,spacing);
diff[0]=sqrt(spacing[2])—delta_u; diff[l]=sqrt(spacing[3])—delta_u;
diff[2]=sqrt(spacing[4])—delta_u; diff[3]=sqrt(spacing[5])—delta_u;
spacing-var+=diff [0] *diff [0] +diff [l]*diff[l]+diff [2] *diff [2] +diff [3] *diff[3]
spacing_var=spacing_var/(4.0*vis_n);
r e tu r n spacing-var;
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
}
in t m ain(int argc, ch a r * argv[])
{
lam bda=C/(f90Ghz); / /wavelength
/ / delta_u and delta_v are constrained from the FOV of the experiment
delta_u=.5/sin(.5*10.0/180*PI);delta_v=.5/sin(.5*10.0/180*PI);
vis_n=2*theta_n*baseline_n; / /th e total number of uv points.
i f (initial_n>0) {file=fopen("h.orn_position_p.dat","w ");
T=100; //tem perature set to 100.
idum =—1 ;printf ( " \n°/.ld\n" ,idum);
//procedue #1
for (i=0;i<horn_n;++i)
{
horn_trial0 [i]= —array _radius+array _radius*ran2 (&idum);
horn_trial0 [i+horn_n]= —array _radius+array_radius*ran2(&idum);
}
//procedure #2
E_trial=get_mean_spacing();
E=E_trial;memcpy(horn,horn_trialO,2*horn_n*sizeof (float));
w hile(true)
{
i=(int)floor(2*horn_n*ran2(&idum)); //procedue #3
horn_trialO[i]=—.13+.26*ran2(&idum); E_trial=get_mean_spacing(); //procedue #4
i f (E_trial>E) //procedure #5
{ T*=0.999;
E=E_trial;memcpy(horn,horn_trial0,2*horn_n*sizeof (flo a t));
printf("T l g E 7.f \n\n",T ,E );
fprintf(filep"7»f
",E);
if (T<0.1)
{fo r (i=0;i<horn_n;++i) fprintf(file,"70f 7,f ",horn[i],horn[i+horn_n]);fprintf(file,"\n");fllush(file);}
}
else //procedure #6
{
P=ran2(&idum);
i f (P < = ex p (—abs_value(E_trial—E )/T ))
{T*=0.999;E=E_trial;memcpy(horn,horn_trial0,2*horn_n*sizeof (flo a t));}
}
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139
}
/ / save the feedhorn array configuration found
for (i=0;i<horn_n;++i) fprintf(file,"%f %f ",horn[i],horn[i+horn_n]);fprintf(file,"\n");fflush(file);
fclose(file);
}
re tu r n 0;
}
A .3.2
A p ertu re S yn th esis Sim ulation
#define FWHM PI/180.0*8.1 / / the FWHM of the feedhorn
#define days 21 //integration time : 21 days
v o id generate_visibility(char *filename)
{
flo a t frequency;
frequency=90*le+9; //90GHz
//com pute the frequency spectrum of the CMB anisotropy
get_cmb_anistropy_spectrum(& anisotropy .spectrum, &frequency .l.false);
file=fopen("uv_m bi.dat","r"); //reading uv coverage of the MBI
for (i=0;i<vis_n;++i) fscanf(file,"’/,f y(f\n'',&u[i],&v[i]);fclose(file);
reset_array(Vr,vis_n);reset_array(Vi,vis_n); //initializing the visibilities
read_fits(filename,map,&status); //reading inputsky maps
for (k=0;k<vis_n;++k) / / k is the index fo r the baselines
*for
(i=0;i<n_y;++i) / / i is the index fo r y coordinate
{
y = —y_max+i*delta_y;
for (j=0 ;j<n_x;++j) / / j is the index fo r x coordinate
{
x = —x_max+j*delta_x;
beam =exp(—(x*x+y*y)/(2*pow(sigma,2))); //beam pattern for the primary beam
Vr[k]+=band_width*anisotropy_spectrum*beam*cos(2*PI*(u[k]*x+v[k]*y))
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140
*map[i*n_x+j]*delta_x*delta_y;
Vi[k] +=band_width*anisotropy_spectrum*beam*sin(2*PI*(u[k] *x+v[k] *y))
*map[i*n_x+j]*delta_x*delta_y;
}
}
}
}
v o id n oise(float *V r,float *Vi)
{
flo a t noise_R,noise_I;
idum =—1; gasdev(&idum);
for (k=0;k<vis_n/2;++k)
{
noise_R=sqrt(8.33818*(le+19)/days)*gasdev(&idum)*sqrt(.5);
noisc_I=sqrt(8.33818*(le+19)/days)*gasdcv(&idum)*sqrt(.5);
Vr[k]=Vr[k]+noise_R; Vi[k]=Vi[k]+noise_I; //ad d in g noise to real and imaginary visibilities
}
for (k=0;k<vis_n/2;++k)
{Vr[k+vis_n/2]=Vr[k];Vi[k+vis_n/2]=—Vi[k];} / / V ( —u ,—v) is the complex conjugate of V(u,v)
}
v o id aperture_synthesis(char *filename)
{
reset_array(synthesized_map,n_x*n_y); //initializing the synthesized map
for (i=0;i<n_y;++i) / / i is the index fo r y coordinate
{y= —y_max+i*delta_y;
for (j=0;j<n_x;++j) / / j is the index fo r x coordinate
{x=—x_max+j*delta_x;
beam =exp(—(x*x+y*y)/(2*pow(sigma,2))); //beam pattern for the feedhorn
for (k=0;k<vis_n;++k)
{synthesized_map[(i-n_y/4)*n_x/2+j-n_x/4]+=(cos(2*PI*(u[k]*x+v[k]*y))*Vr[k]
+sin(2*PI*(u[k]*x+v[k]*y))*Vi[k])*delta_u*delta_v;}
synthesized_map[i*n_x+j]=synthesized_map[i*n_x+j]/(band_width*anisotropy_spectrum*beam);
}
}
write_fits(filename,synthesized_map,n_x/2,n_y/2,&'status); //saving the synthesized map in fits file
}
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141
in t m ain(int argc, ch ar * argv[])
{
lambda=C/(f90Ghz); //wavelength of the signal
sigma=0.4245*FWHM;
band_width=10*le+9; //10GHz bandwidth
x_max=0.2588;y_max=0.2588;
n_x=512;n_y=512; //th e number of pixels
delta_x=2*x_max/n_x;delta_y=2*y_max/n_y;
delta_u=.5/sin(.5*10.0/180*PI);delta_v=.5/sin(.5*10.0/180*PI); //delta_u and delta_v are given by the FOV.
vis_n=144; //th e number of visibilities 4*18*2=144
map=(float*)malloc(n_x*n_y*sizeof(float)); //th e input sky map
synthesized_map=(float*)malloc(n_x*n_y*sizeof(float)); //th e synthesized map
Vr=(float*)malloc(vis_n*sizeof(float)); //th e real visibilities
Vi=(float*)malloc(vis_n*sizeof (float)); //th e imaginary visibilities
u=(float*)m alloc(vis_n*sizeof(float)); / / u values of the uv points
v=(float*)m alloc(vis_n*sizeof(float)); / / v values of the uv points
generate_visibility("T.fits");noise(V r,V i);aperture_synthesis("T_reconstructed.fits"); //T em perature map
generate_visibility("Q .fits");noise(V r,V i);aperture_synthesis("Q _reconstructed.fits"); / / Q polarization
generate_visibility("U .fits");noise(V r,V i);aperture_synthesis("U _reconstructed.fits"); / / U polarization
//freeing memory
free(map);free(synthesized_map);free(Vr);free(Vi);free(u);free(v);
re tu r n 0;
}
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 .
142
A .4
A .4.1
T he Power Spectra E stim ation by th e M axi­
m um Likelihood M ethod
Q uadratic O ptim al E stim ator
The following C routine, which we developed, realizes the iterative quadratic optimal
estimator above until the change from the iteration are less 0.1%. In the C routine,
Cl [0] and Cl [1] are Ae and XB, and W_ij array stores J2i
and E / W f 5 .
inverse_F and inverse_C _ij are F _1 and C -1 .
while(change>0.001)
{
reset_array(vl,V_complex_n); //V_complex_n : the number of visibilities
for (CLi=0;C l_i<C Ln;++C Li) //C L n : # of band powers, CLi; index of bandpowers
{
for (i=0;i<V_complex_n;++i)
{
for (j=0;j<V_complex_n;++j) vl[i]+=(inverse_C_ij[i*V_complex_n+j]*Vp]);
}
}
reset_array (v2,V_complex_n);
for (i=0;i<V_complex_n;++i)
{
for (j=0;j<V_complex_n;++j)
{
v2[i]+=(W_ij[Cl_i*V_complex_n*V_complex_n+i*V_complex_n+j]*vlp]);
}
}
reset_array (v l,V_complex_n);
for (i=0;i<V_complex_n;++i)
{
fo r (j—0;j<V_complex_n;++j)
{
v l [i]+=(inverse_C_ij [i*V_complex_n+j]*v2 p]);
}
}
delta_Cll[CLi]=0; for (i=0;i<V_complex_n;++i) {delta_Cll[CLi]+=V[i]*vl[i];}
reset_array(ml,V_complex_n*V_complex_n);
for (i=0;i<V_complex_n;++i)
R e p r o d u c e d w ith p e r m issio n o f th e c o 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 .
143
{
for (j=0 ;j < V-complex_n;+ + j )
{
for (n=0;n<V_complex_n;++n)
{
ml[i*V_complex_n+j]+=(inverse_C_ij[i*V_complex_n+n]
*W_ij [CLi* V_complex_n* V_complex_n+n* V_complex_n+j]);
}
}
}
Tr=0; fo r (i=0;i<V_complex_n;++i) {T r+ = m l [i*V_complex_n+i];}
delta_C12 [CLi] = T r;
}
for (Cl_i=0;CLi<CLn;++CLi)
{
delta_Cl[CLi]=0;
for (C L j=0;C L j< C L n;++C L j)
{
delta_Cl[CLi]+=inverse_F[CLi][CLj]*(delta_Cll[CLj]—delta_C12[Cl_j])/2;
}
Cl[Cl_i]=Cl[CLi] +delta_Cl[CLi];
}
change=abs_ value (delta_ C 1[0] / Cl [0]);
}
A .4.2
V isib ility covariance m atrix o f th e M B I
Cl_i is the index for band powers and in the MBI, it runs from EE to BB. V_complex_n
is the number of visibilities and C_ij stores the visibility covariance m atrix CVJ.
for (Cl_i=0;CLi<CLn;++Cl_i)
{
for (i=0;i<V_complex_n;++i)
{
fo r (j=0 ;j < V_complex_n;+ + j )
C_ij [i*V_complex_n+j]+=Cl[Cl_i]* W_ij [CLi* V_complex_n* V_complex_n+i* V_complex_n+j];
}
}
for (i=0;i<V_complex_n;++i) C_ij[i*V_complex_n+i]=pow(NEP,2)+C_ij[i*V_complex_n+i];
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
144
A .4 .3
F isher m atrix o f th e M B I
Array F stores the Fisher m atrix F.
//C l_i runs from EE to BB
for (C Li=0;C Li<C Ln;++C Li)
{
for (C l_j=0;CLj<C Ln;++C Lj)
{
reset_array(m l,V_complex_n*V_complex_n);
for (i=0;i<V_complex_n;++i)
{
for (j=0;j<V_complex_n;++j)
{
fo r (n=0;n<V_complex_n;++n)
ml[i*V_complex_n+j]+=(W_ij[Cl_j*V_complex_n*V_complex_n+i*V_complex_n+n]
*inverse_C_ij [n* V_complex_n+j]);
}
}
reset_array (m2, V_complex_n* V_complex_n);
for (i=0;i<V_complex_n;++i)
{
for (j=0 ;j < V_complex_n;+ + j )
{
fo r (n=0;n<V_complex_n;++n)
m2[i*V_complex_n+j]+=(inverse_C_ij[i*V_complex_n+n]*ml[n*V_complex_n+j]);
}
}
reset_array(ml,V_complex_n*V_complex_n);
for (i=0;i<V_complex_n;++i)
{
for (j=0;j<V_complex_n;++j)
{
fo r (n=0;n<V_complex_n;++n)
ml[i*V_complex_n+j]+=(W_ij[CLi*V_complex_n*V_complex_n+i*V_complex_n+n]
*m2 [n*V_complex_n+j]);
}
}
Tr=0; for (i=0;i<V_complex_n;++i) Tr+=ml[i*V_complex_n+i];
F[CLi][CLj]=Tr/2;
}
}
R e p r o d u c e d w ith p e r m issio n o f th e c o 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
A .5
A ntenna Tem perature
The spectral distribution of a black body radiation is given by the Planck law [23]
n m
= 21,1/3
1
c2 ehv!kT — 1
If hv <C kT . which is the Rayleigh-Jean limit, the spectral distribution of a blackbody
radiation is approximated to [23]
2;y2
1 3 rj(^ T ) = — kT,
c2
where c is the speed of light, v is the frequency, k is Boltzmann constant and I is
radiation intensity. In the millimeter and submillimeter range, one frequently defines
antenna tem perature as T& = c2/2 ku2I [23].
A .6
M icrowave frequency bands
Microwave Frequency Bands are defined in the table in table A .l [73]:
A .7
A .7.1
Foregrounds Sim ulation
Sim ulation o f P olarization A n g le for foregrounds
for (i=0;i<number_of_pixels;++i)
{
x=2*ranl (&ddum)—1.0 ;y=2*ranl (&idum)—1.0;
x_norm[i]=x/sqrt(x*x+y*y);y_norm[i]=y/sqrt(x*x+y*y); //norm alization to unit magnitude
}
R e p r o d u c e d w ith p e r m issio n o f th e c o 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
Table A .l: Microwave frequency bands
Microwave frequency bands designation Frequency range [GHz]
1- 2
L band
S band
2 —4
C band
4 —8
8-12
X band
12 - 18
Ku band
K band
18 - 26
Ka band
26 - 40
Q band
30 - 50
U band
40 ~ 60
V band
50 - 75
E band
60 - 90
W band
75 - 110
F band
90 - 140
D band
110 - 170
write_healpix_map(x_norm, nside," f o r eg r ound_x . f i t s " , " NESTED"," G");
write_healpix_map(y_norm,nside," f oreground_y. f i t s ", "NESTED"," G");
A .7.2
Sim ulation o f Q and U m ap for foregrounds
//reading dust intensity map constructed by the WMAP
dust=read_healpix_map("map_w_mem_dust_yrl_vl . f i t s " ,&nside,coordsys, ordering);
//reading free—free emission intensity map constructed by the WMAP
fr eefree=read-healpix_map("map_w_mem_f r e e f r e e _ y r l_ v l. fits",& nside, coordsys.ordering);
//reading synchrotron emission intensity map constructed by the WMAP
synch=read_healpix_map("map_w_mem_synch_yrl_vl . f i t s ",&nside, coordsys.ordering);
//m ean polarization of dust : 5%, mean polarization of synchrotron : 30%
for (i=0;i<12*nside*nside;++i) {foreground[j]=le—3*(.05*dust[i]+.3*synch[i]);}
x=read_healpix_map("foreground_x_smooth. f i t s ",&nside, coordsys.ordering);
y=read_healpix_map("f oreground_y_sm ooth.f i t s " ,&nside, coordsys.ordering);
a=0; for (i=0;i<number_of_pixels;++i) {a+=sqrt(pow(x[i],2)+pow(y[i],2));} a=a/number_of_pixels;
R e p r o d u c e d w ith p e r m is s io n o f th e co p y r ig h t o w n e r . F u rth er r ep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
147
for (i=0;i<number_of_pixels;++i)
{
gamma=atan2(y[i],x[i]); f=sqrt(x[i]*x[i]+y[i]*y[i])/a;
I[i]=f*foreground[i]; Q[i]=f*cos(2*gamma)*foreground[i]; U[i]=f*sin(2*gamma)*foreground[i];
}
A .8
A .8.1
A ll-sky Im aging o f th e CM B Tem perature
R eso lv in g a
Noise covariance m atrix is assumed to be uniform and diagonal and b is the matrix
of size V-Ji x blm_n and A is the vector of size V_n.
/ / Computing A
for (V_i=0;VJ<V_n;++V_i)
{
for (i=0;i<blm _n;++i)
{
for (j=0;j<blm _n;++j)
{
A[i*blm_n+j]+=blm[V_i*blm_n+i] *blm[V_i*blm_n+j]);
}
}
}
//C om puting y
for (i=0;i<blm _n;++i) for (V_i=0;V_i<V _n;++VJ) {y[i]+=blm[V_i*blm_n+i]*V[V_i];}
//T h e ludcmp and lubksb from ’Numerical Recipe in C’ solves Ax=y
ludcmp(A_m,n,indx1&d);lubksb(A_mln,indx,b-l);m em cpy(x,bpn*sizeof (float));
A .8.2
A n ten n a p oin tin gs
for (A_i=0;A_i<A_n;++A_i)
{
//th e HEALPix routine which computes the angular coordinate
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
//o f the A_i th pixel in HEALPix pixellization.
pix2ang_ring(nside,A_i,&theta1&phi);
//A ngular coordinates are saved in the array of antenna_theta and antenna_phi.
antenna_theta[A_i]=theta;antenna_phi[A_i]=phi;
}
A .9
A .9.1
A ll-sky Im aging o f th e CM B polarization
C om p u tin g b for all sk y-im agin g o f th e C M B tem p e r­
ature
//com puting baselines vectors in the galactic coordinate
for (A_i=0;A_i<A_n;++A_i)
{
cos_antenna_theta[A_i]=cos(antenna_theta[A_i]);sin_antenna_theta[A_i]=sin(antenna_theta[A_i]);
cos_antenna_phi[A_i] =cos(antenna_phi[A_i]); sin-antenna _phi[A _i] =sin(antenna_phi[A_i]);
for (B_i=0;B_i<B_n;++B_i)
{
B _phi=PI*BJ/(B _n)+PI/(2*B _n);
B_x_a=Baseline_length*cos(B_phi);B-y_a=Baseline_length*sin(B-phi);B_z_a=0;
antenna2generaLcoordinate(B_x_a,B-y-a1B_z_al&B_x[A_i*B_n+B_i],<&'B_y[A_i*B_n+B_i]l
&B_z[A_i*B_n+B_i],antenna_theta[A_i],antenna_phi[A_i]);
}
}
/ / A_i is the index fo r antenna pointings and B_i is the index for baseline orientations
for (A_i=0;A_i<A_n;++A_i)
{
reset_array(blm_Vr,B_n*bhn_n);reset_array(blm-Vi,B-n*blm_n);
for (B_i=0;B_i<B_n;++B_i)
{
//Integrating over theta
for (i=0;i<theta_steps_blm ;++i)
{solid_angle=sin_theta[i]*delta_theta*delta_phi; //solid angle
k-z=cos_theta[i];
get_Nlm_Plm(Nlm_Plm,Lmin_blm,l_max_blm,cos_theta[i]); //C om puting Legendre polynomial
reset_array(bm_T_Vr,2*phi_steps_blm+l);reset-array(bm_T_Vi,2*phi-Steps_blm+l);
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 .
149
//Integrating over azimuthal angle
for (j=0;j<phi_steps_blm;++j)
{
//C om puting the beam pattern of a feedhorn
cos_beam_theta=cos_antenna_theta[A_i]*cos_theta[i]
+sin_antenna_theta[A_i]*sin_theta[i]*(cos_antenna_phi[A_i]*eos_phi[j]
+sin_antenna_phi [A_i] *sin_phi [j]);
if (cos_beam_theta<cos_beam_theta_cutoff) continue;if (cos_beam_theta<0) continue;
if (cos_beam_theta<l)
{beam_theta=acos(cos_beam_theta);beam=exp(—pow(beam_theta,2)/(2*pow(sigma,2)));}
else beam =l;
k_x=sin_theta[i] *cos_phi|j] ;k_y=sin_theta[i] *sin_phi[j];
k_dot_B=k_x*B_x[A_i*B_n+BJ]+k_y*B_y[A_i*B_n+BJ]+k_z*B_z[A_i*B_n+BJ];
beampattern_spectrum_interference_r=0;beampattern_spectrum_interference_i=0;
cos_Lk_dot_B=cos(phaseO*k_dot_B);sin_f_k_dot_B=sin(phaseO*k_dot_B);
cos_df_k_dot_B=cos(delta_phase*k_dot_B);sin_df_k_dot_B=sin(delta_phase*k_dot_B);
//integrating over finite bandwidth (85GHz~95GHz)
for (Li=0;f_i<frequency_n;++f_i)
{
beampattern_spectrum_interference_r+=beam*spectrum[f_i]*cos_f_k_dot_B;
beampattern_spectrum_interference_i+=beam*spectrum[f_i]*sin_f_k_dot_B;
cos_fl_k_dot_B=cos_f_k_dot_B*cos_df_k_dot_B—sin_f_k_dot_B*sin_df_k_dot_B;
sin_fl_k_dot_B=sin_f_k_dot_B*cos_df_k_dot_B+sin_df_k_dot_B*cos_f_k_dot_B;
cos_f_k_dot _B=cos_f1_k_dot _B;sin_f_k_dot _B=sin_f1_k_dot _B;
}
bm_T_Vr[2*j+1] =beampattern_spectrum_interference_r;
bm_T_Vi[2*j+l]=beampattern_spectrum_interference_i;
}
fourl(bm_T_Vr,phi_steps_blm,l);fourl(bm_T_Vi,phi_steps_blm,l);
for (l=Lmin_blm;l<=l_max_blm;++l)
{
blm_ Vr [B_i*blm_n+1* (1+1)—Lmin_blm*Lmin_blm]+ =
Nlm _Plm [i*Plm_n+l*(l+l)/2—l_min_blm*(l_min_blm+l)/2]*bm_T_Vr[l]*solid_angle*df;
for (m = l;m < = l;+ + m )
{
blm_Vr[B_i*blm_n+l*(l+l)+m—l_min_blm*l_min_blm]+=
Nlm _Plm [i*Plm _n+l*(l+l)/2+m —l_min_blm*(l_min_blm+l)/2]*2*bm_T_Vr[2*m+l]*solid_angle*df;
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 .
150
blm_Vr[B_i*blm_n+l*(l+l)—m-l_mm_blm*Lmin_blm]—=
N lm _Plm [i*Plm _n+l*(l+l)/2+m —Lmin_blm*(l_min_blm+l)/2]*2*bm_T_Vr[2*rn+2]*solid_angle*df;
}
}
for (l=Lmin_blm;l<==l_max_blm;++l)
{
blm_ Vi[B _i*blm_n+l* (1+1)—Lmin_blm*l_min_blm]+ =
Nlm _Plm [i*Plm_n+l*(l+l)/2—Lmin_blm*(Lmin_blm+l)/2]*bm_T_Vi[l]*solid_angle*df;
for (m = l;m < = l;+ + m )
{
blm_Vi[B_i*blm_n+l*(l+l)+m—Lmin_blm*Lmin_blm]+=
Nlm _Plm [i*Plm _n+l*(l+l)/2+m —l_min_blrn*(l_min_blm+l)/2]*2*bm_T_Vi[2*in+l]*solid_angle*df;
blm_Vi[B_i*blm_n+l*(H-l)—m —l_min_blm*Lmm_blm]—=
Nlm _Plm [i*Plm _n+l*(l+l)/2+m —Lmin_blm*(Lmin_blm+l)/2]*2*bm_T_Vi[2*m+2]*solid_angle*df;
}
}
}
}
}
A. 10
C om puting b for all sky-im aging o f th e CM B
polarization
//com puting baselines vectors in the galactic coordinate
for (A_i=0;A_i<A_n;++A_i)
{
cos_antenna_theta[A_i]=cos(antenna_theta[A_i]) ;sin_antenna_theta[A_i] =sin(antenna_theta[A_i]);
cos_antenna_phi[A_i]=cos(antenna_phi[A_i]);sin_antenna_phi[A_i]=sin(antenna_phi[A_i]);
for (B_i=0;B_i<B_n;++B_i)
{
B_phi=PI*B_i/(B_n)+PI/(2*B_n);
B_x_a= Baseline_length*cos(B_phi);B_y_a=Baseline_length*sin(B_phi);B_z_a=0;
antenna2general_coordinate(B_x_a,B-y-a,B_z_a,&B_x[A_i*B_n+B _i],&B_y[A_i*B_n+B_i],
&B_z[A_i*B_n+B_i],antenna_theta[A_i],antenna_phi[A_i]);
psi=B _phi+psi_deg*PI /180.0;
cos_2_psi[A_i*B_n+B_i]=cos(2*psi);sin_2_psi[A_i*B_n+B_i]=sin(2*psi);
}
R e p r o d u c e d w ith p e r m is s io n o f th e c o 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
}
/ /A_i is the index for antenna pointings and B_i is the index for baseline orientations
for (A_i=0;A_i<A_n;++A_i)
{
reset_array(b21m_Vr,B_n*b21m_n);reset_array(b21m_Vi,B-n*b21m_n);
for (B_i=0;B_i<B_n;++B_i)
{
//Integrating over theta
for (i=0;i<theta_steps_blm ;++i)
{
get-Nlm_Plm(Nlm_P21m,Lmin_blm,Lmax_blm,cos_theta[i];//Computing (Film + F21m)
solid_angle=sin_theta[i]*delta_theta*delta_phi; //solid angle
k_z=cos_theta[i];
reset_array(bm_Q_Vr,2*phi_steps-blm+l);reset_array(bm_U_Vr,2*phi_steps_blm+l);
reset_array(bm_Q_Vi,2*phi_steps_blm+l);reset_array(bm_U_Vi,2*phi_steps_blm+l);
/ /Integrating over azimuthal angle
for (j=0;j<phi_steps_blm;++j)
{
,
//C om puting the beam pattern of a feedhorn
cos_beam_theta=cos_antenna_theta[A_i]*cos_theta[i]+sin_antenna_theta[A_i]*sin_theta[i]
*(cos_antenna_phi[A_i]*cos_phi[j]+sin_antenna_plii[A_i]*sin_phi[j]);
if (cos_beam_theta<cos_beam_theta_cutoff) continue;if (cos_beam_theta<0) continue;
if (cos_beam_theta<l)
{beam_theta=acos(cos_beam_theta);beam=exp(—pow(beam_theta,2)/(2*pow(sigma,2)));}
else beam =l;
theta_dot_theta_A=cos_theta[i]*cos_phi[j]*cos_antenna_theta[A_i]*cos_antenna_phi[A_i]
+cos_theta[i]*sin_phip]*cos_antenna_theta[A_i]*sin_antenna_phi[A_i]
+sin_theta[i] *sin_antenna_theta[A_i];
theta_dot_phi_A=-cos_theta[i]*cos_phi[j]*sin_antenna_phi[A_i]+cos_theta[i]*sin_phi[j]*cos_antenna_phi[A_i];
N_rotation=sqrt(theta_dot_theta_A*theta_dot_theta_A+theta-dot_phi_A*theta_dot_phi_A);
cos_rotation=theta_dot_theta_A/N_rotation;
sin_rotation=theta_dot_phi_A/N_rotation;
cos_2_rotation=cos_rotation*cos_rotation—sin_rotation*sin_rotation;
sin_2_rotation=2*sin_rotation*cos_rotation;
k_x=sin_theta[i] *cos_phi[j] ;k_y=sin_theta[i] *sin_phi[j];
k_dot_B=k_x*B_x[A_i*B_n+B_i]+k_y*B_y[A_i*B_n+B_i]+k_z*B_z[A_i*B_n+BJ];
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
152
beampattern_spectrum_interference_r=0;beampattern_spectrum_interference_i=0;
cos_f_k_dot_B=cos(phaseO*k_dot_B);sin_f_k_dot_B=sin(phaseO*k_dot_B);
cos_df_k_dot_B=cos(delta_phase*k_dot_B);sin_dLk_dot_B=sin(delta_phase*k_dot_B);
//integrating over finite bandwidth (85GHz~95GHz)
for (f_i=0;f_i<frequency_n;++f_i)
{
beampattern_spectrum_interference_r+=beam*spectrum[Li]*cosJLk_dot_B;
beampattern_spectrum_interference_i+=beam*spectrum[f_i]*sin_Lk_dot_B;
cos_Lk_dot_B=cos_f_k_dot_B*cos_dLk_dot_B—sinJLk_dot_B*sin_df_k_dot_B;
sin_f_k_dot_B=sinJLk_dot_B*cos_dLk_dot_B+sin_df_k_dot_B*cos_f_k_dot_B;
}
bm_Q_Vr[2*j+l]=beampattern_spectrum_interference_r*(—sin_2_psi[A_i*B_n+B_i]*cos_2_rotation
+sin_2_rotation*cos_2_psi[A_i*B_n+B_i]);
bm_Q_Vi[2*j+l]=beampattern_spectrum_interference_i*(—sin_2_psi[A_i*B_n+B_i]*cos_2 .rotation
+sin_2_rotation*cos_2_psi[A_i*B_n+B_i]);
bm_U_Vr[2*j+l]=beampattern_spectrum_interference_r*(cos_2_psi[AJ*B_n+B_i]*cos_2 .rotation
+sin_2_psi[A_i*B_n+B_i]*sin_2_rotation);
bm_U_Vi[2*j+l]=beampattem_spectrumJnterference_i*(cos_2_psi[A_i*B_n+B_i]*cos_2 .rotation
+sin_2_psi[A_i*B_n+B_i]*sin_2_rotation);
}
fourl(bm_Q_Vrlphi_steps_blm,l);fourl(bm_U_Vr,phi_steps_blm,l);
fourl(bm_Q_Vipphi_steps_blm1l);fourl(bm_U_Vilphi_steps_blm,l);
for (1=l_min_blm;1< =Lm ax_blm ;++1)
{
b21m_Vr[B_i*b21m_n+(l*(l+l)—Lmin_blm*Lmin_blm)*2]
+=Nlm_P21m[i*P21m_n+l*(l+l)—Lmin_blm*Lmin_blm]*bm_Q_Vr[l]*solid_angle*df;
b21m_Vr [B_i*b21m_n+(1* (1+1) - Lmin_blm*l_min_blm) *2+1]
+ —Nlm_P21m[i*P21m_n+l*(l+l)-Lmin_blm*Lmin_blm]*bm_U_Vr[l]*solid_angle*df;
for (m = l;m < = l;+ + m )
{
b21m_Vr[B_i*b21m_n+(l*(l+l)+m-l_min_blm*l_min_blm)*2]
+=Nlm_P21m[i*P21m_n+l*(l+l)+m—Lmin_blm*l_min_blm]
*(bm_Q_Vr[2*m+l]+bm_U_Vr[2*m+2])*solid_angle*df;
b21m_Vr[B_i*b21m_n+(l*(l+l)+m—Lmin_blm*Lmin_blm)*2+l]
+=Nlm_P21m[i*P21m_n+l*(l+l)+m—l_min_blm*l_min_blm]
*(bm_U_Vr[2*m+l]—bm_Q_Vr[2*m+2])*solid_angle*df;
b21m_Vr[B_i*b21m_n+(l*(l+l)—m —Lmin_blm*Lmin_blm)*2]
+=Nlm_P21m[i*P21m_n+l*(l+l)—m —l_min_blm*Lmin_blrn]
R e p r o d u c e d w ith p e r m is s io 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 .
153
*(bm_Q_Vr[2*phi_steps_blm—2*m+l]-fbm_U_Vr[2*phi_steps_blm—2*m+2])*solid_angle*df;
b21m_Vr[B_i*b21m_n+(l*(l+l)—m —Lmin_blm*Lmin_blm)*2+l]
+=Nlm_P21m[i*P21m_n+l*(l+l)—m —l_min_blm*Lmin_blm]
*(bm_U-Vr[2*phi_steps_blm—2*m +l]—bm_Q_Vr[2*phi_steps_blm—2*m+2])*solid_angle*df;
}
}
for (l=Lm in_blm ;l<=Lm ax_blm ;++l)
{
b21m_Vi [B_i*b21m_n+(1*(1+1)—Lmin_blm*l_min_blm) *2]
+=NlmJP21m[i*P21m_n+l*(l+l)—Lmin_blm*LminJblm]*bm_Q_Vi[l]*solid_angle*df;
b21m_Vi[B_i*b21m_n+(l*(l+l)—Lmin_blm*Lmin_blm)*2+l]
+=Nlm_P21m[i*P21m_n+l*(l+l)—Lmin_blm*l_min_blm]*bm_U-Vi[l]*solid_angle*df;
for (m = l;m < = l;+ + m )
{
b21m_Vi[B_i*b21m_n+(l*(l+l)+m—l_min_blm*Lmin_blm)*2]
+=Nlm_P21m[i*P21m_n+l*(l+l)+m—Lmin_blm*l_min_blm]
*(bm_Q_Vi[2*m+l]+bm_U_Vi[2*m+2])*solid_angle*df;
b21m_Vi[B_i*b21m_n+(l*(l+l)+m—Lmin_blm*Lmin_blm)*2+l]
+=Nlm_P21m[i*P21m_n+l*(l+l)+m—Lmin_blm*Lmin_blni]
*(bm_U_Vi[2*m+l]-bm_Q_Vi[2*m+2])*solid_angle*df;
b21m_Vi[B_i*b21m_n+(l*(l+l)—m —l_min_blm*Lmin_blm)*2]
+=Nlm_P21m[i*P21m_n+l*(l+l)—m —Lmin_blm*l_min_blm]
*(bm_Q_Vi[2*phi_steps_blm-2*m+l]+bm_U_Vi[2*phi_steps_blm-2*m+2])*solid_angle*df;
b21m_Vi[B_i*b21m_n+(l*(l+l)—m —Lmin_blm*Lmin_blin)*2+l]
+=Nlm_P21m[i*P21m_n+l*(l+l)—m —Lmin_blm*Lmin_blm]
*(bm_U_Vi[2*phi_steps_blm-2*m+l]-bm_Q_Vi[2*phi_steps_blm—2*m+2])*solid_angle*df;
}
}
}
}
}
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 .
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