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The cosmic microwave background: Gaussianity and polarization

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TH E COSMIC MICROWAVE BACKGROUND: GAUSSIANITY AND POLARIZATION
BY
DAVID LEONARD LARSON
B.Sc., University of M aryland at College Park, 2001
M.Sc., University of Illinois at U rbana-Cham paign, 2003
DISSERTATION
Subm itted in partial fulfillment of the requirem ents
for the degree of D octor of Philosophy in Physics
in the G raduate College of the
University of Illinois at U rbana-Cham paign, 2006
U rbana, Illinois
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UMI Number: 3242910
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© 2006 by D avid L eonard Larson. All rights reserved.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
CER T I F I C A TE OF C O M M I T T E E A P P R O V A L
U niversity o f Illinois at Urbana-Cham paign
G raduate C ollege
August 21, 2006
We h ereby recom m end that the thesis by:
DAVID LEONARD LARSON
Entitled:
THE COSMIC MICROWAVE BACKGROUND: GAUSSIANITY AND
POLARIZATION
Be accepted in p a rtia l fulfillm ent o f the requirements f o r the degree of:
Doctor of Philosophy
S ig n a tu res:
D irector 'of.Research -
Benjamin v A n d c lt
Head o f D epa/fm ent - Dale J. Vai'f’fhymngen
Committee on Final Examination*
____________________
C hairperson-
J ff e yyl'izQ________
Stuart Shapiro
Committee Member -
Committee Member -
Scott Willenbrock
_____
"
JTon Thaler
Committee Member - Benjamin Wandelt
Committee M em ber -
* Required for doctoral degree but not for master's degree
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 b stra ct
The Cosmic Microwave Background (CMB) is a window to th e earliest parts of the universe
th a t we can still directly see. As such, it provides excellent d a ta for testing the currently
accepted model of cosmology: an expanding Friedm ann-Robertson-W alker spacetime th a t
is statistically described by about six param eters. This dissertation presents three tools for
probing th a t standard model of cosmology and tests them on real CMB data.
The first tool checks the peaks and valleys in the CMB to make sure they statistically
m atch those expected from a Gaussian random field, as predicted by our standard model.
To do this, we analyze the one- and two-point correlation functions and compare those to
simulated Gaussian skies w ith the same power spectrum and noise. We find some discrepan­
cies in the W M AP data, and we can interpret these as either a detection of non-Gaussianity,
or some foreground or other overlooked detail in our model of the experiment. The second
tool provides a statistically sound and relatively rapid technique for estim ating the polarized
power spectrum of the CMB. This is the Gibbs sampler, applied to power spectra; it samples
the power spectra according to either the likelihood, or Bayesian posterior probability, as
desired. These samples accurately reflect the error bars on th e power spectra, and a correct
understanding of the error bars is essential to assuring th a t our stan d ard model properly
predicts the power spectra. We dem onstrate this tool on th e COBE data, polarized simu­
lations, and the polarized 3-year W M AP data. The third tool we present is a m ethod of
visualizing the CMB, which is useful for displaying polarized fields in an easily understood
fashion. While this does not directly test the standard model, it does provide a way to
more clearly understand our d ata and look for possible unwanted artifacts, such as polarized
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foregrounds. As a way to detect contam inants in the CMB d ata, it will be very useful when
trying to test the standard model of cosmology.
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For M o m and Dad
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A ck n o w led g m en ts
I would first like to thank my advisor, Ben W andelt, for his help and support through my
graduate studies. W ithout his occasional pep talks, constant willingness to answer questions,
and contagious enthusiasm for cosmology, I would not be where I am today.
My collaborators at the Jet Propulsion Laboratory and elsewhere have been industrious,
knowledgeable, and helpful: Hans Kristian Eriksen, Kris Gorski, Greg Huey, Jeff Jewell, and
Ian O ’Dwyer.
I would also like to thank a few of my fellow students (some graduated), who were very
helpful and did not mind when I asked them questions about cosmology, m athem atics or
computers: Charm aine Armitage, Rahul Biswas, Chad Fendt, Greg Huey, Tom McElmurry,
D am ian Menscher, Ian O ’Dwyer, and Amit Yadav.
Some of the results in this dissertation have been derived using the HEALPix : [1] package.
I also acknowledge use of the Legacy Archive for Microwave Background D ata Analysis
(LAMBDA). Support for LAMBDA is provided by the NASA Office of Space Science.
D. Spergel and O. Dore helped with the work presented in chapter 4 by reading m anuscripts
of my papers and providing useful feedback.
My work in chapter 4 was partially supported by National Com putational Science Al­
liance under MCA04N015 and utilized the Xeon Linux Cluster, tungsten. The work was
also partially supported by the Center for Advanced Study at Beckman.
I would like to thank G raca Rocha, Charles Lawrence, and the members of the Planck
C T P working group for comments and stim ulating conversations, particularly during the
1http: / /healpix.jpl.nasa.gov
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work on chapter 5. The work in this chapter was obtained in collaboration w ith Hans K ristian
Eriksen, Ben W andelt, Kris Gorski, Greg Huey, Jeff Jewell, and Ian O ’Dwyer. I acknowledge
support through NSF grant AST-0507676 and NASA JP L subcontract 1236748 for this work.
This work was supported in p a rt by the University of Illinois at U rbana-Cham paign and th e
N CSA /UIUC Faculty Fellowship program.
Finally, I would like to th an k John H art for teaching th e com puter graphics course in
which I learned about line integral convolution, discussed in chapter
6
.
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T able o f C o n ten ts
L ist o f T a b l e s ............................................................................................................................
xi
L ist o f F i g u r e s ............................................................................................................................
xiii
C h ap ter 1
In tr o d u ctio n
....................................................................................................
1
C h ap ter 2 C M B T h e o r y ....................................................................................................
2.1 The Expanding U n iv e rs e ..............................................................................................
2.2 I n f la tio n ............................................................................................................................
2.3 Linear P e r tu r b a ti o n s ....................................................................................................
2.4 Computing the CMB Power Spectra .............................
2.4.1 Scalar P erturbations of the M e tric ...............................................................
2.4.2 Tensor P erturbations of the M e t r i c ............................................................
7
9
10
14
C h a p ter 3 M a th e m a tica l P r o p e r tie s o f Spin 2 F ield s
..............................
3.1 Second Derivatives Are Spin-2 O b j e c t s ...................................................................
3.1.1 Analytic C a s e ....................................................................................................
3.1.2 General C a s e ....................................................................................................
3.2 S is a Covariant Complex Derivative on the Sphere .............................................
3.2.1 Geometry of the Sphere .................................................................................
3.2.2 A Projection .....................................................................................................
3.2.3 Deriving S ...........................................................................................................
3.2.4 D e f in itio n ...........................................................................................................
3.3 The E and B Modes ....................................................................................................
3.3.1 Connecting w ith HEALPix Conventions ..................................................
3.3.2 Dimension C o u n t i n g .......................................................................................
3.3.3 Tem perature-Polarization C o rre la tio n .........................................................
18
19
19
21
23
25
26
29
30
33
36
36
37
C h ap ter 4 T ests o f N o n - G a u s s ia n it y ............................................................................
4.1 Statistical A n a ly s is ........................................................................................................
4.2 One-Point M e t h o d ........................................................................................................
4.2.1 Monte Carlo S im u la t i o n .................................................................................
4.2.2 Analysis and Hypothesis T e s t .......................................................................
4.3 First One-Point Results ..............................................................................................
4.4 M ulti-Resolution One-Point, A n a l y s is ......................................................................
38
40
40
41
43
44
49
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5
5
6
4.5
4.6
4.4.1 W M AP M easurem en t.........................................................................................
4.4.2 Simulation P r o c e s s ............................................................................................
4.4.3 D ata R e d u c tio n ...................................................................................................
4.4.4 O ne-Point M ulti-Resolution R e s u lts ..............................................................
4.4.5 Varying the A m plitude of the N o is e ..............................................................
Smoothing One- and Two-Point A n a ly s is ................................................................
4.5.1
One-Point Statistics .......................................................................................
4.5.2
Two-Point S t a t i s t i c s .......................................................................................
4.5.3
Two-Point Results ..........................................................................................
C onclusion.........................................................................................................................
49
49
50
52
52
57
60
64
68
69
72
C h ap ter 5 G ib b s S am p lin g ............................................................................................
5.1 In tro d u c tio n ......................................................................................................................
73
5.2 Model and N otation ....................................................................................................
76
5.3 Signal S a m p lin g ..............................................................................................................
78
5.3.1 Overview ...........................................................................................................
78
79
5.3.2 Preconditioned Conjugate G radient T e c h n iq u e ..........................................
5.3.3 Realistic T re a tm e n t..........................................................................................
81
5.4 Power Spectrum S a m p lin g .........................................................................................
82
5.4.1 D e riv a tio n ...........................................................................................................
82
5.4.2 A lgorithm for Sampling From th e W ishart D is trib u tio n .........................
86
5.4.3 Low I is s u e s ........................................................................................................
86
5.4.4 B i n n i n g ...............................................................................................................
87
5.5 Param eter Estim ation and the Blackwell-Rao E s tim a to r....................................
88
5.6 C om putational C o n sid e ra tio n s...................................................................................
91
5.7 Application to the COBE d a t a ...................................................................................
93
5.8 Polarized Results on Simulated D a t a ..........................................................................101
5.8.1 Low-resolution B-mode experim ent (C M B P o l).............................................. 101
5.8.2 High-resolution T + E experiment ( P l a n c k ) ................................................... 105
5.9 Bayesian Analysis of W M A P Polarization D a t a ......................................................107
5.10 C onclusion............................................................................................................................ 118
C h ap ter 6 L ine In tegral C o n v o lu tio n ....................................................................... 121
6.1 The Algorithm for Line Integral C o n v o lu tio n .............................................................122
6.2 Adjusting the Param eters of the A lg o rith m ................................................................125
C h ap ter 7
C on clu sion
......................................................................................................
129
A p p en d ix A P r o p e r tie s o f 9 and 5 .............................................................................. 131
A .l A M athem atical Tangent into F o rm a lis m ................................................................... 133
A.2 Complex C o n ju g atio n........................................................................................................ 134
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A p p e n d ix B P e d a g o g ic a l I n tr o d u c ti o n t o S ta tis tic a l H y p o th e s is T e s tin g
137
B .l Statistical T e s tin g ................................................................................................................137
....................................................................................................... 137
B.1.1The Problem
B .l.2The Solution
....................................................................................................... 137
B .l.3
The Random Variable and Its D is trib u tio n ....................................................138
B .l.4 The H y p o th e s is .....................................................................................................139
B .l.5 The Test and Types of E r r o r ............................................................................. 139
B.1.6 T he Case of a One-Sided T e s t .......................................................................... 142
B.2 Connecting to Frequentist Confidence In te rv a ls......................................................... 143
B.3 The Next S te p ...................................................................................................................... 146
B.4 Users Guide for f a c t s ..................................................................................................... 147
A p p e n d ix C N p-lx o v e r 47 r ................................................................................................... 149
C .l Signal S a m p lin g ...................................................................................................................149
C.2 How the Problem was Solved B efo re.............................................................................. 153
C.3 P o la riz a tio n ......................................................................................................................... 154
A p p e n d ix D P r e c o n d itio n in g C o n ju g a te G r a d ie n t D e s c e n t ............................ 157
D .l Spherical Harmonic T r a n s f o r m s .....................................................................................158
D.1.1 F irst D efinitions..................................................................................................... 158
D .l.2 U nitarity C h e c k ..................................................................................................... 160
D.1.3 Double Checking the In v e rs e .............................................................................. 161
D.1.4 The Basis Conversion E q u a tio n s ....................................................................... 162
D.2 Redo Noise A p p ro x im atio n ...............................................................................................163
D.2.1 Redefining N .........................................................................................................163
D.2.2 Recom puting Spherical Harmonic Transforms .............................................. 164
D.2.3 Com puting N in Harmonic S p a c e .................................................................... 165
D.3 A few remaining details on s p i n .....................................................................................169
D.3.1 Spin 2 to EB and back a g a in .............................................................................. 169
D.3.2 A way around the spin-4 p r o b le m .................................................................... 169
R e f e r e n c e s .....................................................................................................................................
171
C u rric u lu m V i t a e .......................................................................................................................
180
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L ist o f T ables
4.1
4.2
The identifier column provides th e power spectrum , band, and mask used
and w hether the statistics are for m inim a or maxima. Power spectrum is the
best fit (bf) W M AP power spectrum . Columns labeled 0 through 4 give an
unbiased estim ate of the W M AP statistic ’s position among th e sorted Monte
Carlo sample statistics. The statistic in column 0 is number of hot spots. The
other columns correspond to: 1, mean; 2, variance; 3, skewness; and 4, kurtosis
of extrem a tem perature values. (For minima, mean and skewness statistics
are negated before estim ating probability of W M AP statistic being lower.)
Column 5 gives the num ber of M onte Carlo samples calculated. Probabilities
th a t indicate th at, “th e true value of p is at least 95% likely to be w ithin 0.025
of either 0 or 1,” are m arked w ith an asterisk. None are m arked in this table
because 99 samples is not sufficient to make this claim. The table shows th a t
the d ata fall low in th e mean tem perature distribution for almost every set of
simulations.........................................................................................................................
This is a continuation of table 4.1. The identifier column provides the power
spectrum, band, and mask used and whether the statistics are for minima or
maxima. Power spectra are best fit (bf), power law (pi), running index (ri),
or measured unbinned W M AP (w). Columns labeled 0 through 4 give an
unbiased estim ate of the W M AP statistic’s position among the sorted Monte
Carlo sample statistics. The statistic in column 0 is number of hot spots. The
other columns correspond to: 1, mean; 2, variance; 3, skewness; and 4, kurtosis
of extrem a tem perature values. (For minima, mean and skewness statistics
are negated before estim ating probability of W M AP statistic being lower.)
Column 5 gives the num ber of Monte Carlo samples calculated. Probabilities
th a t indicate th a t, “the true value of p is at least 95% likely to be w ithin
0.025 of either 0 or 1,” are marked w ith an asterisk. The table shows th a t
the d ata fall low in the mean tem perature distribution for almost every set of
simulations.........................................................................................................................
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46
47
Estim ates p of the probability p th a t a sim ulation statistic will be lower th an
the W M AP statistic. Column headings signify the statistic and whether the
noise was white or correlated in the simulations. For this table, we used
110 correlated noise simulations and 800 white noise simulations. Rows labels
signify the value of Wide and w hether the statistics are for m axim a or minima.
Values of p th a t are significant for our 95% level test have asterisks. Only
the white noise has enough simulations to enable a 95% detection. The first
6 columns of d ata are presented graphically in figure 4.3......................................
4.4 Number of standard deviations by which the am plitude of the correlated noise
was shifted. Hinshaw et al. [2] cite the error in th e noise am plitude to be
0.06%, so —3a corresponds to multiplying th e am plitude by exactly 0.9982,
and 3cr corresponds to m ultiplying by exactly 1.0018, etc. This multiplication
is carried out after the proper scaling of th e correlated noise, discussed in
section 4.4.2. The d a ta for the first two statistics is presented graphically in
figure 4.4.............................................................................................................................
4.5 These are our estim ates p of the position of the W M AP two-point statistic
among the simulated statistics. These results are for 50 arcm inute FWHM
smoothing, where 1 0 0 0 iterations went to create the covariance m atrix and
position of the W M A P statistic is found among the rem aining 3000. The
different rows show results for different masks, as well as th e minima-minima,
maxima-maxima, and minima-m axima statistics. The columns show the re­
sults for the nearest 7.2 degrees (first 40 bins) of our two-point statistics, as
well as for the full-sky 180 degree two-point statistics. Columns also show
the. different results for the spot-spot and tem perature-tem perature statistics.
Note the 0 value in the upper right corner................................................................
4.6 This is the same d a ta as figure 4.5, except for 180 arcm inute FW HM sm ooth­
ing........................................................................................................................................
4.3
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L ist o f F ig u res
3.1 The complex derivative of a ro ta te d scalar field. This figure is made with
the perspective of an active rotation. The lower colored patch represents
the complex values of f ( z ) in some patch, and the upper patch represents
the complex values for g(z) which have been rotated from the lower patch.
The rotation Rg takes point p to point q. Because the complex numbers are
rotating and the partial derivatives are not, th e derivative (dx —idy) / 2 changes.
3.2 The complex derivative of a ro tated scalar field. This figure is made w ith the
perspective of a passive rotation. We show two coordinate systems, prim ed
and unprim ed. Because the p artial derivatives for the two coordinate systems
are in different directions, the derivatives (dx —idy)/2 and (dx>— idy>)/2 differ.
3.3 A region of the “N orthern” unit hemisphere, where z > 0. It has been pro­
jected onto the plane. The x and y axes are geodesics, as are all lines intersect­
ing the origin. Angles at the origin are preserved by this mapping. Lengths
are also preserved at the origin, b u t only to first order. This projection was
made by rotating the desired (arbitrary) point 9, <f>= 0.7,1.5 up to the N orth
pole (z = 1) such th a t
= ex . T hen it was projected onto the x-y plane by
ignoring the z component, which was close to 1 anyway. Blue lines are lines
of constant 9 and 0; these were ro tated up to the N orth pole and projected in
the same fashion as the point in question. T he increments between blue lines
are 0.03 radians in b o th the 6 and <p directions.......................................................
3.4 This chapter’s sign conventions for looking at the sphere from the outside.
Note th a t they differ from the HEALPix conventions............................................
3.5 This chapter’s sign conventions for looking at the sphere from the inside. This
figure is redundant w ith figure 3.4 bu t is shown to emphasize the two pos­
sible viewpoints. Note th a t these sign conventions differ from the HEALPix
conventions.........................................................................................................................
3.6 A few spin s objects. They all live on the plane, or perhaps on a tangent
plane to the sphere. They are sym metric under a rotation by angle 2 t y / s , if
0, and a spin 0 object is sym metric under any ro tatio n .................................
3.7 The phase convention used to find the orientation of the spin s object from
the phase of a complex number. In the case of the sphere, the orientation is
with respect to the x and y directions, which are in tu rn aligned with the —<p
and —9 directions as per our phase convention........................................................
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24
27
31
31
32
32
4.1
4.2
•
4.3
4.4
4.5
4.6
4.7
Cumulative distribution functions (CDFs) of m ean tem perature value (in
units of millikelvins) of the local extrem a, found in sets of 5000 Monte Carlo
simulations for four G alactic masks: GS, GN, ES, and EN. Best fit power
spectrum and W -band d a ta are used. Means of th e m inim a are negated for
comparison. M axima CDF is dotted while m inim a CDF is solid. Note their
visual similarity. Statistics m easured on W M A P d a ta are shown as two verti­
cal lines, dotted for m axim a and solid for minima. Numbers on right are the
same probabilities as in Table 4.1; for each pair, probability for the m axim a
is higher on the page........................................................................................................
These are the 5 masks used for the one-point correlations. Left column is
mollweide projection of the sky; right column is HEALPix base tile 6 , where
the upper left corner is northernm ost. Tile 6 is directly opposite th e galactic
center; i t ’s solid angle is exactly 1/12 of the full sphere’s. Paranoid mask is
black, extended paranoid mask extends it in grey...................................................
Cumulative distribution functions of the hot and cold spot statistics from
various simulations. The grey (black) dashed line is the CDF for th e hot
(cold) spots from the 800 white noise simulations. The grey (black) solid line
is the hot (cold) spot CDF from the 110 correlated noise simulations. The
vertical grey (black) line gives the location of the W M A P statistic for the hot
(cold) spots. Estim ates p of the probability of a sim ulation’s statistic falling
below the W M AP statistic are printed on the graph and underlined w ith the
appropriate line. All simulations use the best fit power spectrum , the kpO
mask (degrading is detailed in the paper), and d a ta from the W band. Units
for the m ean statistics are rriK and units for the variance are m K 2. . . . . .
This figure shows the cumulative distribution functions for the statistics gen­
erated by shifting the correlated noise am plitude by ncr, where n ranges from
-3 to 3. 110 correlated noise m aps are used. The grey statistics (and dotted
lines) are for maxima, the black (and solid lines) for minima. From top to
bottom , the numbers are values of p for n = 3 to n = —3. The vertical lines
are the W M AP values. Note: Only the Mean statistic (plot on th e right)
shows the CDFs spread ap art from each other; the other 4 statistics have
the CDFs on top of each other, as in the left plot. Also, the detection of
non-Gaussianity in the mean is weaker when you consider a —lcr shift in the
correlated noise am plitude. The CDFs would become sm oother w ith more
than 1 1 0 sim ulations........................................................................................................
Power S p e c tr a ..................................................................................................................
These are the 5 masks used for th e two-point correlations. Left column is
mollweide projection of the sky; right column is HEALPix base tile 6 , where
the upper left corner is northernm ost. Tile 6 is directly opposite the galac­
tic center. The mask is black, the adjusted mask for 50 arcm inute FW HM
smoothing includes the mask and the thin grey region extending the mask.
These are images of base tile 6 in the HEALPix scheme at various steps in
the simulation process. N orth is in the upper left. Range of color scale varies
between some images.......................................................................................................
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4.8
4.9
These are cumulative distribution functions of selected one-point statistics
for 50 and 180 arcm inute sm oothing levels. Hot spot (maxima) statistics are
in red and are the upper num ber in each pair. Cold spot statistics are in
black. The vertical lines indicate th e position of the W M AP statistic. Mean
statistics are given in m K , variance is in m K 2, and skewness is dimensionless.
The mean and skewness values of the m inim a have been m ultiplied by —1
before creating CDFs, for easier com parison w ith the m axim a............................
Upper left: tem perature-tem perature correlation, minima, full sky. Upper
right: spot-spot correlation, maxima, full sky. The vertical axis is the excess
fractional probability density, for finding a pair of points at a given angular
separation. In the correlation functions, the black line is the W M AP data,
the white lines are the median sim ulation values, and the grey band is a 2 cr
error band calculated from the simulations. Lower: Cumulative distribution
functions for x 2 statistics. They correspond to the upper plots.............................
Computing time averaged over 30 iterations of the Gibbs sampler required for
solving 5.8 and 5.9 as a function of th e num ber of pixels in th e map. These
timings are for a single A thlonX P 1800+ CPU. Solid line: actual timings.
Dashed lines show n * for x E {3, 5/2, 2, 3/2} from the top to the bottom on
the right side of the figure..............................................................................................
5.2 The COBE-DM R power spectrum. The vertical bands display the m arginal­
ized densities at each £. Horizontal bars m ark off bins of constant probability.
These bins are assigned their color in Ci space and then projected into the
diagram. The bin w ith the highest probability density is shown in black.
The dark and light shaded regions are the 1 -cr and 2-cr highest posterior den­
sity regions, respectively. The Ci are measured in units m K 2 in this and all
subsequent figures.............................................................................................................
5.3 Marginalized posterior densities for each individual Cg from the COBE-DM R
data. At each t the fluctuations in the Ci at all other I were integrated out.
The axis ranges are the same for all panels...............................................................
5.4 Correlation m atrix of Ci estim ates from the COBE-DM R data. The diagonal
components have been set to zero to enhance the contrast of the off-diagonal
components. The surface is shaded according to height. We see th a t correla­
tions between the power spectrum estim ates vary between 8 % correlation at
{£,£') = (6,10) and 15% anti-correlation at {£,£')= (8,12). See Figure 5.5. .
5.5 2-D marginalized posterior densities. Each plot shows the full joint posterior
of the data, integrated over all dimensions except for the two shown. From
bottom left anti-clockwise: P ( C 2, C3), P ( C 2, C4), P ( C 8, C 12), and P ( C Gl C\o).
The latter two were chosen because these Ci pairs were maximally anti­
correlated and correlated, respectively........................................................................
63
66
5.1
xv
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93
96
97
98
99
5.6
Reconstructed signal m aps in G alactic coordinates. A: The signal component
of the COBE-DM R d a ta marginalized over th e power spectrum: (x)\P(x\m).
This is a generalized W iener filter which does not require knowing the signal
covariance a priori. B: The solution y of 5.9 at one Gibbs iteration. C: The
sample pure signal sky s = x + y at the same iteration (band-lim ited at
Cnax — 50). D: The W M A P internal linear com bination m ap smoothed to an
FWHM of 5 degrees. The corresponding features in parts A and D are clearly
visible. Note th a t in this m ap low galactic latitudes are not masked, which
leads to some artifacts th a t are not visible in th e masked COBE-DM R data. 100
5.7 Reconstructed E- and B-mode power spectra from the low-resolution analysis.
Input spectra are shown as dashed and d o tted lines, respectively, while the
reconstructed posterior distributions are indicated by solid curves (posterior
maximum) and gray regions (one and two sigma confidence regions). The
corresponding noise spectrum is given by a th in dashed line. The GelmanRubin convergence statistic as a function of m ultipole is shown in the bottom
panel.........................................................................................................................................103
5.8 The E x B cross-spectrum from the low-resolution analysis.......................................105
5.9 Reconstructed power spectra from the high-resolution Planck 100 GHz sim­
ulation. The true spectra are shown as dashed lines, and th e reconstructed
posterior distributions are given by a m aximum posterior value (solid lines)
and a 6 8 % confidence region. The Gelm an-Rubin convergence statistics are
shown in the bottom panels...............................................................................................107
5.10 Gibbs sampled signal maps. The three columns show, from left to right,
tem perature, Stokes Q and Stokes U param eters. The three rows show, from
top to bottom , the complete Gibbs samples, th e mean held (Wiener filtered)
maps, and the fluctuation maps. The mean field m ap provides the information
content of the data, and the fluctuation m ap provides a random complement
such th a t the sum of the two is a full-sky, noiseless sky consistent both w ith
the the current power spectrum and the d a ta .............................................................. 108
5.11 Close-up of the galactic center shown in Figure 5.10, emphasizing how the
algorithm separates E and B modes. Each of the sampled maps (the sum
of the fluctuation and mean field m ap) are full sky maps, so decomposing
the polarization into E and B modes is straightforw ard. The images show
tem perature as color and polarization overlayed as a fingerprint p attern of
stripes. The stripes are aligned w ith the direction of polarization. They
are darkest where the polarization is strongest, and they disappear where the
polarization goes to zero. See chapter 6 for more details on this representation
of polarization. The maximum am plitude of the polarization is given in y,K
and centered under each image. The maps have been sm oothed to 1 degree.
This is an orthogonal projection of the sky, about 60 degrees wide, centered
on the Galactic center. The W M A P KpO galactic mask is visible in the
fluctuation and mean term s........................................................................................... 109
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5.12 This is the same as figure 5.11, except it was made w ith IDL and uses line
segments to represent polarization instead of th e line integral convolution
stripes used in figure 5.11................................................................................................... 110
5.13 This figure shows the scalar E and B modes th a t are displayed in figure 5.11.
They represent how much “gradient” and “curl” is in each polarization field.
These are truly scalar fields because they are spin lowered versions of the spin 2
polarization fields. For this figure, th e images were created by overwriting the
ajm spherical harmonic coefficients w ith either the afm or the afm coefficients,
and then plotting the new “tem p eratu re” field.............................................................I l l
5.14 The posterior probability densities of C f E up to I = 32, made from a his­
togram of sampled C f E values. T he black histogram has 10,000 samples from
the QV analysis; red has 25,000 samples from Q only; blue has 25,000 samples
from V only............................................................................................................................ 113
5.15 The posterior densities of C f B up to I = 32, made from a histogram of
sampled C f B values. The black histogram has 10,000 samples from the QV
analysis; red has 25,000 samples from Q only; blue has 25,000 samples from
V only...................................................................................................................................... 114
5.16 The posterior densities of C f B up to t = 32, made from a histogram of
sampled C f B values. The black histogram has 10,000 samples from the QV
analysis; red has 25,000 samples from Q only; blue has 25,000 samples from
V only...................................................................................................................................... 115
5.17 The mean of 10,000 signal samples from the QV analysis, split ap art into E and
B modes. The stripes are aligned w ith the direction of polarization and their
darkness indicates the strength of polarization. The maximum polarization
for the sky is 2.3 j i K , for the E modes alone is 1.5 fxK, and for the B modes
alone is 1.6 fiK. The galactic mask is barely visible as a fading in polarization
on the center of the p lo t..................................................................................................... 116
5.18 The diagonal of the W M AP noise m atrix for the QV analysis. There are
correlations, but the diagonal is still a good approxim ation of the noise m atrix.
The square root of the diagonal element is shown, and units are /iK. From
top to bottom , we show the expected standard deviation in Q, then in U, then
these two maps added in quadrature. The galactic mask is clearly visible in
all three...................................................................................................................................117
6.1
This is how line integral convolution looks w ith white noise as a background
texture. This is the same polarized map as in figure 6.3, containing only E
modes correlated w ith the tem perature. Upper left: the raw texture. Upper
right: texture adjusted to reflect m agnitude of polarization. Middle left:
tem perature map. Middle right: tem perature overlayed with texture adjusted
by magnitude. Lower left: tem perature overlayed with raw texture. Lower
right: raw texture, color represents m agnitude of polarization............................... 126
xvii
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6.2
This figure shows line integral convolution w ith w hite noise, as does figure 6.1.
However, this m ap contains only B modes. Note th e difference in texture. Up­
per left: the raw texture. Upper right: texture adjusted to reflect m agnitude
of polarization. M iddle left: tem perature map. Middle right: tem perature
overlayed with tex tu re adjusted by m agnitude. Lower left: tem perature over­
layed w ith raw texture. Lower right: raw texture, color represents magnitude
of polarization...............................................................................................................127
6.3 This is the zebra striping technique applied to a polarized map. This m ap con­
tains only E modes, and they are completely correlated w ith the tem perature.
This m ap has also been smoothed w ith a 2 degree beam. U pper left: the raw
(thresholded) texture. Upper right: texture adjusted to reflect magnitude of
polarization. Middle left: tem perature map. Middle right: tem perature over­
layed w ith texture adjusted by m agnitude. Lower left: tem perature overlayed
with raw texture. Lower right: raw texture, color represents m agnitude of
polarization................................................................................................................... 128
B .l
This figure plots 7 , the probability of a false acceptance of th e hypothesis
(solid), and (3, the probability of a false rejection (dashed), as a function of p,
for a = 0.003,
= 13, n = 19000. In order to keep /3 below a , we find th a t
7 becomes quite large for some values of p. In order to avoid false rejections
of the hypothesis, we must allow false acceptances sometimes. The region in
which 7 is large becomes smaller as n increases................................................. 143
B.2 Double sided confidence intervals, a = 0.05, for all values of ifrom 0 to n = 50. 145
xviii
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C h ap ter 1
In tro d u ctio n
Ever since Georges Lem aitre proposed th e idea of an expanding universe in 1927 [3] and
Hubble found some evidence to support it in 1929 [4], we have had the Big Bang as a
possible model for our universe. Some tim e after this, Gamow [5], and Alpher and Herman
[6 ] did work which indicated th a t a present-day background of very cold radiation was a
consequence of the hot early universe. A hot universe in which all m atter was a plasm a
would eventually cool to where the m a tte r deionized into a gas. The m atter would become
transparent, and light would begin to travel in straight lines. This light, which was initially
blackbody radiation at high tem perature, would cool as the universe expanded, finally ending
up in the microwave band, today. This microwave radiation was eventually discovered in 1965
by Penzias and Wilson [7], and has become known as the Cosmic Microwave Background
(CMB).
In 1992, the Cosmic Background Explorer (COBE) mission discovered anisotropy in
CMB th a t was left over from initial density perturbations in the early universe [8 ]. Since
th a t discovery of anisotropy, a wide variety of other CMB experiments have also measured
this anisotropy. The logical successor to the COBE mission is the Wilkinson Microwave
Anisotropy Probe (WMAP), which has m ade megapixel maps of the full sky [9], far surpass­
ing C O BE’s thousands of pixels.
The CMB has opened up a window to the early universe th a t has shaped our models
of cosmology.
At present, we find our universe is well fit by an expanding Friedmann-
Robertson-W alker spacetime th a t is statistically described by about six param eters [10].
1
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These param eters are
^ 6 ; H q, t , fis, and A s.
Here, Qm and
(1.1)
are the fractions (today) of the universe’s critical density consisting of
m atter and baryons, respectively.
the universe is assumed to be flat.
In this simplest model still consistent with the data,
To m aintain flatness, the fraction of the universe’s
m ass/energy density which is not m atter is a cosmological constant. H q is the current rate
of expansion of the universe, usually given in kilometers per second per megaparsec.
governs the rate at which th e universe expands, and in conjunction w ith H 0l fixes the age
and past history of the universe’s expansion, r is the optical depth, which determ ines how
much scattering of the CMB occurred between when th e universe first deionized and today.
This is very im portant for the polarization of the CMB, because the polarization only occurs
through scattering. The quantities n s and A s describe the initial power spectrum of density
perturbations in the universe, which we believe was laid down by inflation. The m agnitude
of the fluctuations is given by A Sl and their power law by n s. These perturbations collapse
by gravitational instability, and oscillate in the very early universe because of radiation
pressure. Because the radiation only interacts with baryons, O5 governs the am plitude of
this oscillation. Knowledge of these six param eters allows us to predict th e power spectrum
of the CMB well enough to fit current d a ta [10]. Some of the details of how to obtain the
power spectra from these param eters are discussed in chapter
2
.
It is this model of the universe which we want to test, and this dissertation presents
three different tools th a t use the CMB to probe this present understanding of cosmology. A
chapter is devoted to each of these tools, and each chapter introduces its subject in more
detail.
In order to more thoroughly understand the cosmological model we propose to test,
a review of some CMB physics is presented in chapter 2.
There we briefly explain the
expanding universe, and m etric perturbations in it, and then we continue w ith a more
2
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detailed explanation of how th e polarized power spectra of the CMB are calculated from
these metric fluctuations.
C hapter 2 motivates an understanding of the m athem atics of spin 2 fields on the sphere,
because these are required to for understanding the polarized power spectra.
C hapter 3
therefore discusses these m athem atics in some detail. It explains how a spin 2 field can be
created by “raising” a scalar field w ith the S 2 operator, and how th e E and B harmonics,
which are commonly used when discussing polarization, are linear com binations of spin
2
spherical harmonics.
The first tool we present, in chapter 4, is a test of w hether the CMB is truly a Gaussian
random field, as predicted by inflation in our model of the universe.
This prediction is
discussed in section 2.2. We check to make sure the local extrem a in the CMB have the
same statistical properties as expected from our model. We check th e one-point and twopoint statistics at a variety of resolutions, and examine in detail the statistical claims th a t
can be made from these tests.
Our second tool, in chapter 5, is the Gibbs sampler applied to polarized power spectrum
estimation. It is the power spectrum of the CMB which constrains th e param eters of our
model and contains all of the statistical inform ation for Gaussian random fields, so it is
the power spectrum we which we must determine as accurately as th e d a ta will allow. The
Gibbs sampler provides a fast, statistically sound m ethod for obtaining error bars on the
power spectrum, by sampling power spectra from the likelihood. These samples describe the
uncertainties precisely. Inflation predicts a low level of tensor modes in the CMB which leave
their signature as B modes in the polarization. A detection of these prim ordial B modes
requires pushing the limits of w hat experiments can do, bu t it would be a highly interesting
result. Gibbs sampling may make it possible.
The third tool is the simplest of all—it is a m ethod of visualizing polarized maps of
the sky. This technique, called Line Integral Convolution (LIC), is described in chapter
6
.
Originally proposed for displaying vector fields [1 1 ], we extend it for use on the sphere with
3
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polarization fields. Being able to easily see th e structure of polarization fields is im portant
for a couple of reasons. First, it makes our research more accessible to the public. It is easy
to explain th a t the stripes are in the direction of the polarization, and they are darkest where
the polarization is strongest. For this reason, LIC was used on a poster for the January 2006
American Astronomical Society meeting to dem onstrate th e polarization th a t the PLANCK
space mission will be able to measure. Regarding research, LIC is useful because the po­
larization signal from the CMB is very weak and very easily contam inated by foregrounds.
W hen the polarization field is displayed, dram atic oddities caused by foregrounds will be
easily located by eye. For example, when looking tow ard the Milky Way, the polarization
field is oriented perpendicular to the plane of the galaxy, an artifact which is clearly visible
in these plots. W hile this visualization technique does not directly probe our standard model
of cosmology, it provides a sanity check th a t the d a ta we are using look reasonable before
running detailed analyses on them.
4
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C hapter 2
C M B T h eo ry
In this chapter, we discuss how the CMB is formed.
We start w ith the expansion of a
homogeneous, isotropic universe. After taking a short tangent to discuss inflation, why we
believe it happened, and its prediction of random Gaussian initial fluctuations, we take a
quick tour of linear perturbation theory. All this leads up to the m ain point of this chapter,
which is to define and show how to calculate the CMB power spectra from a knowledge of
the metric perturbations (both tensor and scalar). The power spectrum of the CMB is the
prim ary testable connection we have to th e param eters in our cosmological model.
2.1
T h e E xp an d in g U n iv erse
We write down the Robertson-W alker m etric in spherical coordinates for a homogeneous,
isotropic universe [1 2 ]:
c2ds 2 = c2dt2 — R 2(t)
r2 ° ^ 2
+
f2
sin 2 ^ dcj)2^
(2.1)
where k represents the curvature of the universe, and R(t) is a scale factor. For k > 0, the
universe is closed, for k <
0
the universe is open and for k =
0
, we have a flat universe—ours,
as far as we know. Typically, we rescale R ( t ):
a(t) = M
Kq
where R,0 is the scale factor today, so a = 1 today [13].
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 )
We will combine the Robertson-W alker m etric w ith the Einstein equation below, includ­
ing a cosmological co n stan t.[13]
G'a' + A
(2.3)
Here, c is the speed of light, G is the gravitational constant. This yields the FriedmannRobertson-W alker equation
R2
8irG
kc2
i p - ^ r p = - ip
.
.
{2A)
which governs the expansion of the universe [12, 13]. Here, p is the mass density of everything
(dark m atter, baryons, radiation) in the universe.
2.2
In flation
Inflation is a period of rapid expansion in the early universe. This section discusses some
of the reasons why it is believed to have happened and mentions its prediction of Gaussian
m etric fluctuations.
Several good argum ents for inflation are given in [13]; I discuss three of them here.
The first is the horizon argument. If the early universe was radiation dom inated, then
the distance a particle can travel since th e big bang is finite. This means th a t at th e time
of last scattering, when the CMB was released, there were causally disconnected regions. In
fact, these regions have a diam eter of about 1 degree on the sky. Because the CMB is highly
uniform in tem perature, one must question the assum ption of radiation dom ination at the
earliest times in the universe.
A nother argum ent concerns the flatness of the universe. From CMB, supernova, and
large-scale structure measurements, we know th a t the universe is very close to being flat.
This is a problem, because a flat universe is unstable— as it grows, the density of the universe
tends to move away from the critical density. Since the universe is currently very close to
6
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critical density, it m ust have been much, much closer in the past. A t 10 “ 43 seconds, the
universe must have been about 1060 tim es closer to critical density th a n it is now. This fine
tuning to critical density m ust be explained.
Third, the structure of the universe m ust be explained. The universe has structure in
it today, which presumably collapsed from initial density fluctuations. One m ust ask w hat
caused these density fluctuations originally.
These problems all indicate th a t a rapid expansion may have occurred in the early uni­
verse. The rapid expansion would allow our entire universe to be causally connected, thereby
explaining why the CMB is th e same tem p eratu re everywhere. This expansion would also
set the density of the universe extremely close to critical density, explaining the flatness
problem.
Finally, in the m ost popular inflation models (slow-roll inflation), the inflaton
field th a t drives inflation will remain in th e ground state. P erturbations are seeded by the
zero-point fluctuations of th e field modes of the inflaton field th a t drives inflation. In the
standard vacuum of a scalar field in de Sitter space (the Bunch-Davies vacuum) th e field
fluctuations have Gaussian statistics, like the n = 0 state of a quantum harmonic oscillator.
This prediction of Gaussianity in the CMB is a very specific prediction. C hapter 4 of
this dissertation is devoted to testing this assertion carefully.
2.3
Linear P ertu rb a tio n s
To give a flavor of how m etric perturbations evolve in the universe, we briefly describe
the evolution of scalar and tensor perturbations.
First we discuss scalar perturbations,
which are density fluctuations in the m atter, following [13]. Then we briefly explain tensor
perturbations, as in [14].
For perturbations in the prim ordial density,
Po
7
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one can w rite down the linearized relativistic equations for conservation of m omentum and
m atter, in comoving coordinates:
u + 2 - u = -----------a
a
po
8
(2 .6)
(2.7)
- V u
=
where u is th e fluid 4-velocity and g is the peculiar acceleration V JJ f/a caused by the
gravitational potential perturbations 5 $ obeying V 2h<f> =
47 rG<5 p.
Note th a t g is not the
m etric in this equation. We find th a t th e different Fourier modes decouple in linear theory,
and the equation of evolution for 5 is:
5+
c2s k 2
2—8 = 8 ( 4nGpo
a
\
( 2 .8 )
where c2 = d p/ dp is the speed of sound in the fluid. This is a wave equation, w ith b o th
growing and decaying modes th a t depend on the overall expansion of the universe. There is a
proper length, known as the Jeans length, which distinguishes between the long wavelength
modes th a t can grow exponentially and the short wavelength standing waves. This is given
by:
I
A j = c,
7TTr
Gp
(2.9)
After this brief overview of scalar perturbations, we move on to tensor modes.
For tensor perturbations, one can use the m etric [14]
/
-1
0
0
a2 J-
0
0
hx
0
\
( 2 . 10 )
g =
V
0
K
0
0
a2 — h +
0
0
a2
/
where a is the scale factor, to param eterize the perturbation w ith h + and h x . This pertur-
8
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bation corresponds to a gravity wave moving in the z direction. From linear perturbation
theory on Einstein’s equations, these modes behave according to
ha + 2—ha + k 2ha =
a
0
(2 -1 1 )
in Fourier space, where k is the wave number, and where a is either + or x [14]. Note th a t
this is a dam ped harmonic oscillator equation, w ith th e dam ping term proportional to the
expansion of the universe. An expanding universe dam ps out these tensor perturbations, so
it would be quite difficult to detect them directly today. However, we may be able to find
their signature in the CMB. It is the effect of scalar and tensor m etric perturbations on the
CMB th a t we address next.
2.4
C om p u tin g th e C M B P ow er S p ectra
We have ju st seen how m etric perturbations evolve. In this section, we discuss in much
greater detail how those perturbations determ ine the power spectrum of the CMB.
For both scalar and tensor perturbations, we will consider a p ertu rb atio n at one wave­
length, with the wave vector aligned w ith the z axis. We will com pute the power spectrum
in a universe containing only th a t one mode. Then, because we assume statistical isotropy
of the perturbations, we show how to compute the power spectrum of the CMB from the
power spectrum of the m etric perturbations.
The derivations here follow the work of [15], which was checked in detail by [16]. We
refer to both of these for further details and references. For an introduction to some of the
m athem atics behind spin 2 fields, see chapter 3.
9
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2.4.1
S calar P e r tu r b a tio n s o f th e M e tr ic
We have aligned the wave vector k of a scalar p ertu rb atio n w ith th e z axis. Let fj, be the
cosine of the angle between k and some point n on th e unit sphere.
fj, — n • k / k = cos 6
(2 . 1 2 )
This section will use the synchronous gauge [17]:
ds2 — a2(r) [—d r 2 + (<5y + hij ) dx tdx:j)]
(2-13)
Two new functions, rj(k, r ) and h ( k , r ) , are defined by the following equation in [17]:
hij(x, r) =
d ke
k i k j h ( k , r ) + ( kjkj - - 5 i3 ) 6rj(k,r
(2.14)
Because we already know the evolution of the perturbations, we know rj and h.
We now consider the light propagating through th e universe. The tem perature anisotropy
at conformal tim e r , location x, and angle fi is A T(r, x , /i).
Because of th e azim uthal
symmetry of a universe w ith only one mode of a scalar perturbation, /i is sufficient to
describe the direction in space. The Fourier transform of this is A r (r, k, fj). Likewise, we
have another quantity to determine polarization, A p (r, k , //). Because of the sym m etry of
the scalar mode, the polarization will be either aligned “N orth-South” or “East-W est” on a
hypothetical the sphere around an observer, bu t not “N ortheast-Southw est” or “NorthwestSoutheast” . (We take the N orth pole to be in the z direction.) This means there will only be
a Q Stokes param eter, and no U Stokes param eter, according to the HEALPix convention
[18]. This information will later be used to determ ine th a t scalar perturbations in th e m etric
do not produce B modes in the polarized CMB. (B modes are discussed in section 3.3.)
10
R e p r o d u c e d with p e r m issio n o f th e co p y 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 .
We can form th e L egendre expansion of th e azim uthally sym m etric A x and A p\
Ar (n,k) =
^ ( 2 £ + l ) H ) ^ A T,(k)P,(/i)
(2.15)
£=0
OO
AP (n,k) =
J ] ( 2 * + l)H)*Ap*(k)P*(/i)
(2.16)
£=0
where
is the Legendre polynomial of order £ [15].
We write down the Boltzm ann equation for light from [15]. The superscript (S) indicates
th a t we use scalar perturbations. This equation describes how light free-streams through
space and Thom son scatters off of charged particles.
~ \ h ~ \{h
O
n
0
+ 6i))P2(/i)
-\-k —A p ■
*+ ApQ + i\iv 6 + - P 2 (h )n
(2.17)
K - A p( ] + J[i - P2(//)]n
(2.18)
A g + A P(5)2
(2.19)
(5)
APO
The dot indicates differentiation with respect to conformal tim e r. Also, k = a ( r ) n e(r )x e(r)(j 7 ’,
where a (r) is the expansion factor, n e(r) gives the electron density, x e(r) is the fraction of
electrons not bound to an atom , and aT is the electron-photon (Thomson scattering) cross
section, k is the differential optical depth for electron-photon scattering. The to tal optical
depth is
CTO
= / k(r)dr
( 2 .2 0 )
= k ( r ) exp[—k ( t ) | .
(2.21)
k (t )
and the visibility function is
ff(r)
This function peaks at the epoch of last-scattering, when the CMB was released.
Zaldarriaga and Seljak integrate equations 2.17 through 2.19 along the line of sight [19]
11
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 .
to obtain:
A ^ ( t 0, k, n )
=
(2.22)
f ° d T e i(*T- ro)k» S {TS\ k , T )
Jo
A ® ( t 0, k , r f
=
T A . t)
=
n
=
£
g ( Ar.o +
2a
dTe^-^g(T)U (k,T)
+ A + __ ^
(2.23)
+ e k(t) + a)
+ j(7 + S ) + D
k
4k2)
4k 2
(2.24)
{s)
A*>®
g +' A gJ
T2 +
^ A (S)
P2
PO
(2.25)
where a = (h + 6rj)/2k2.
Note th a t only E modes are produced by this perturbation. E modes are discussed further
in section 3.3 and [15]. This is a consequence of only having a Q component of polarization.
A polarized sky must have some U Stokes param eter somewhere on th e sky to have B modes.
(If the wave vector k were not aligned w ith the
2
axis, then the resulting polarization held
would have nonzero U Stokes param eter. It would, however, still be an E mode, because if
it is an E mode in one orientation, it is an E mode in all orientations.)
Let £(k) be a random variable th a t describes the m agnitude of the initial tem perature
fluctuations. The tem perature in some direction is then:
T {5 )(h) =
J
d3k^(k)A^‘?) (r0, /c,//)
(2.26)
and the power spectrum P { k ) is given by
( r ( k 1)^(k2)) = P(fc)5(k1 - k 2)
At this point, knowing A ^
(2.27)
and Ap^ now is equivalent to knowing the power spectra.
To see this, we decompose the tem perature and polarization anisotropies into spherical
12
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 .
harm onics:
CO
I
A ^ ( t , x , n) =
flr,fa^m(n)
(2.28)
a 2/m 2 Yrm(n)
(2.29)
^=0 m = —i
oo
ApS)( r ,x ,n ) = Q = ^
f= 2
£
m = —£
where the harm onic coefficients a^m and a 2/ m m ust also be functions of r and x, and where
Yira is a usual spherical harmonic, and 2Ybn is a spin 2 harmonic, described in chapter 3.
The E and B coefficients can be obtained from a 2 ym by
0>E,im = “ (a 2 ,fm +
0-B,£m =
i(a2,em ~
a*2,J£-m)
(2.30)
( — l)" 1^ , £ - m ) / ^
(2-31)
It is the expectation value (over realizations of £) of products of these coefficients th a t
determines the power spectra:
C J T = <|aT/m|2)
C fE =
{\aEMl |2)
(2.32)
C f * = (ln B,fm| 2)
C jE =
(aT.,ma"Efm)
( 2. 33)
C f B = (aT,tmO*BIm)
C fB =
EJm^B ,im)
(2.34)
From this information, Zaldarriaga and Seljak compute these power spectra [15] and I
record the answer. Because there are no B modes, all the
coefficients are exactly zero,
13
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 .
and th e corresponding power sp ectra are om itted.
(2.35)
C\ S)EE =
(4tt ) 2
C\S)TE =
(47r) 2
J
J
k 2dkP{k) A g ( f c ) j 2
(2.36)
k 2d k P { k ) A $ ( k ) A $ ( k )
(2.37)
r^o
A Te(k)
/
=
do
(2.38)
d T S {TS\ k , T ) j e(x)
(2.39)
(2.40)
where x = kr0 — hr, and je(x) is the order £ spherical Bessel function.
2.4 .2
T en sor P e r tu r b a tio n s o f th e M e tr ic
The derivation for the power spectra of th e tensor modes follows a very similar path. Because
there are two orthogonal modes aligned w ith the z axis, we include both, bu t change bases
to
and
£2
instead of £+ and £ x.
e1 =
£2
=
2
(2.41)
(£+ + * £ x)A / 2
(2.42)
(e+ - ^ x ) / v
(C-fkOetk,)) = ( ^ ( k O ^ k , ) } =
(2.43)
- k 2)
(2.44)
« ‘*(k,)e2(k2)) = 0
Note th a t we have lost the azim uthal sym metry which was present with the scalar per(T)
turbations. To regain some symmetry, we define two new variables to replace A r
14
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 .
(T)
and A p .
~ (T)
These new variables, A ^
~ (T)
and A p , which have no (j) dependence, are defined by:
A ? V ,n ,k )
=
[ ( l - M2)e2^ 1(k) + ( l - /r2) e - 2^ 2(k)]
(2.45)
x A p{ (r,/x, k)
(A g ’) + i A < f )) ( r , n >k)
=
[(l-^ e^ H k J + a + ^ V ^ lk ) ]
(2.46)
x A p{ ( r , n , k )
(A g’) - * A S f )) ( r , n Jk)
=
[(l + /i)2e2^ 1(k) + ( l - ^ ) V 2^ 2(k)]
(2.47)
x A P(r,fjL,k)
W riting the Boltzm ann equation for CMB photons in these new coordinates yields:
A {P + i k f x A P
A P + ikfi A
(T)
=
—h — k a P - *
(2.48)
—K K P + T
(2.49)
J- l~ A TO
(T) +I 1r7 /-±T2
a (t) 4-' A
a T(t)4 - 3k A PO
(t) +' -n Al^ P(T)
- A
a -P4
(t)
7 ft
2
vn
10
7
70
70
(2 501
Again, we perform a line of sight integral. We obtain:
A^r)(r0,n , k )
[ ( l - MV
^
1 (k)
+ ( l - / i 2 ) e - 2^
2 (k)]
f T°
x / dreixflS P { k , T )
Jo
{ A {p
( A {p
+«Aj/T))(r0,n, k )
[ ( l - / r ) 2e2^ 1(k) + ( l + / i ) V 2^ 2(k)]
—i A P ) ( r 0, n, k)
x [ ° dTeix^ S P { k , r )
Jo
[(1 + / ^ V ^ k ) + (1 - /r)2e “ 2i<^ 2(k)]
S P ( k , t)
f T°
x / dreix^ S p \ k , r )
Jo
- h e - K+ g T
S {p ( k , r )
-0 *
(2.51)
(2.52)
(2.53)
(2.54)
(2-55)
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 .
This provides a solution in term s of the Stokes param eters Q and U. However, we would
like to find a solution in term s of E and B modes. This is done by acting twice with the
spin lowering operator, 9, on ( A ^ + «A[P)( t0, n, k). 9 acts on the n argument, since it is a
differential operator in 6 and <fi. One obtains a complex scalar field (which is also a function
of r 0 and k) on the sphere. The real p art of this field corresponds to the E modes and the
imaginary p a rt of the field corresponds to B modes. At least, th a t is how it would work if
we did not have complex factors because we are in Fourier space on k. In practice, we use
[15]:
E
=
(2.56)
3 ' {Q + iU) + & { Q - i U )
—
2
%
B
(2.57)
9 (Q + iU) - 9 2 (Q -iC 7 )
2 1
One finds [15]:
A ^V oA k)
(1 - /X2) [ e ^ e ^ k ) + e~ 2^ £ 2 (k)]
A ^ )(r0,n,k )
( 1 - V ) [e2 ^ £ 1 (k) + e ^ £ 2 (k)]
rro
/ dre^S^(k,r)
Jo
f
d° r S ( x ) e VXilS {p ( k , r )
(2.58)
(2.59)
Jo
A |V o , n , k )
(1 - p 2) [ e ^ e ^ k ) - e ^ £ 2 (k)]
S(x)
—12
B{x)
8x
+ x 2[l — <92] — 8xdx
[ ° d T B ( x ) e ^ S {p ( k , r ) (2.60)
Jo
(2.61)
+ 2x2dx
(2.62)
Since we now know A^P, A ^ , and A ^ , we have enough knowledge to com pute the
power spectra, which have the same definitions as in the scalar mode calculation.
16
R e p r o d u c e d with p e r m issio n o f th e co p y 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 .
The
nonzero power sp ectra from th e ten so r m odes are [15]:
C,
(T)TT
c.
(■T ) E E
_
c.
(T)BB
c,
(T ) T E
k 2d k P h(k) A $ ( k )
(2.63)
(4tr ) 2 f k 2d k P h{k) [ a ^ / c )
(2.64)
(4tt ) 2 I k 2d k P h(k) A f e (k)
(2.65)
(47r ) 2 / k 2d k P h( k ) A ^{ ( k ) A ^{ ( k )
( 2 . 66 )
{Air)2 j
_
This concludes the derivation of th e power spectra from the m etric perturbations (and a
few other bits of knowledge, such as th e ionization fraction as a function of tim e). These are
some of the core calculations th a t link the cosmological param eters in the standard model
to measurements of the power spectrum .
A proper understanding of the polarized power spectra is clearly im portant for under­
standing the CMB. This motivates th e next chapter, which discusses some of the m athem at­
ical details of spin
2
(polarization) fields on the sphere.
17
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 3
M a th em a tica l P r o p e r tie s o f Spin 2
F ield s
This chapter is a pedagogical introduction to some of the properties of the raising and
lowering operators (3 = edth and 3 = edth bar) for the spin-weighted spherical harmonics.
Since tem perature is a spin 0 (scalar) held, and polarization is a spin 2 held (invariant under
180 degree rotations), this chapter reviews some m athem atical background to make dealing
with these spin-weighted helds easier. The spin-weighted spherical harmonics were originally
invented by Newman and Penrose [20] and Goldberg et al. [21].
The m ain point of this chapter is to show how 3 can be considered a covariant complex
derivative on the sphere. Because of this, it naturally acts as a spin-raising operator. Thus if
we need to describe a spin
2
held, such as polarization, we can operate on the usual spherical
harmonics twice w ith 3, to create new spin 2 spherical harmonics. These spin 2 harmonics
now act as a basis for a polarization held, and they behave properly under rotations. An
alternative discussion of many ideas in this chapter can be found in appendix A of Okamoto
and H u’s 2003 Physical Review D article, “CMB Lensing Reconstruction on the Full Sky” ,
volume 67, number 083002.
For more formulas describing 3 and 3, see appendix A. For alternative m athem atical
detail on the relationships between spin 2 harmonics and the E and B modes of the CMB,
see appendix D. Many of the formulas toward the end of this chapter and in appendix A
come from [15], which is a useful reference.
Please note th a t the sign conventions for the Q and U Stokes param eters in this chapter
are not the same as those used in the HEALPix code.
18
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
3.1
S econ d D eriv a tiv es A re Sp in -2 O b jects
3.1.1
A n a ly tic C a se
In order to consider how a derivative behaves under rotations, we need to define w hat we
mean by a derivative and a rotation.
We will begin w ith the usual complex derivative. Given a function / : C —> C, we define
the derivative to be
f ( z ) = l + '» ) - / ( * )
h—>0
tl
where h and
2
(3.!)
are b o th complex num bers. For th e purposes of this section, we will only
work with analytic functions, where the limit exists.
We will want to consider rotations where the complex num bers ro tate as scalars. We
define a functional Re th a t maps a function / : C —> C to another function g : C —> C.
R$[f] = g
g(ze%e) = f ( z )
if and only if
for all
2
(3-2)
Think of g as the function / , ro tated counter-clockwise by an angle 9.
Now we investigate how the derivative behaves under rotations. Let R$[f] = g- Then, in
19
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 .
excruciating detail, we have:
^
i( \
=
=
v
lim
g{z + h ) - g ( z )
-----------
lim
/wO
(3. 3)
(3.4)
h
lim f ( z e ~10 + he~l9) ~ f ( z e ~l9)
=
^
/ ( « - « + ft)- /( » « - « )
heiB^ o
he10
=
lim n ‘ e - * + h ) - f ( z e - » )
ft—
+o
he10
=
e- » ^
+
ft-U)
=
(3>5)
- / ( « - )
(3.8)
h
e~ief ' { z e - id)
(3.9)
and
p '( ^ )
=
(3-10)
If <9Z is the differentiation operator, then we can rewrite equation 3.9 as
dzR ef = e~l0R edzf
(3.11)
R - e d zR e f = e~l0dzf
(3.12)
and equation 3.10 as
meaning th a t the operators dz and Re do not commute. In other words, ifR e f — g, then in
general, R e f f gr, and the derivative does not ro tate like a scalar; it rotates like“spin
1”
object.
Repeated application of the derivative shows th a t
g " W e) = e - 2i0f " ( z )
20
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(3.13)
or
R - e d z2 R ef = e~m d l f
(3.14)
R . e d zs R ef = e - ed zs f
(3.15)
and in general
for any positive integer s.
3 .1 .2
G en era l C a se
W hat happens if the complex function does not have a derivative in the above analytic sense,
but still is nice enough
Most
to have directional derivatives in the real
of the functions th a t will concern us have this property.
and imaginary directions?
To solve this problem we
define another form of differentiation. We use the following notation.
x + iy
z=
z
=
(3.16)
x — iy
(3-17)
Then
X
=
)(2
+ 2)
I 3 ’1 8 )
y
=
k z - z )
(3.19)
ll
and formally,
dz
dz d x
dz dy
2
We therefore make this (and a similar equation for z) a definition:
dz =
^{dx-idy)
(3.21)
d-z =
\ { d x + idy)
(3.22)
21
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 .
One can verify that
dz (x + iy) =
1
dz (x - i y ) =
0
(3.23)
dz(x + iy) =
0
d-z {x - i y ) =
1
(3.24)
so we are at least being self-consistent.
In fact, dz is equivalent to the previous definition of a complex derivative for analytic
functions. For example, one can show th a t for any product zpz q where p and q are integers,
dz (x + iy)p(x — i y)q = p ( x + iy)p 1(x —i y)q
(3.25)
dz(x + iy)p(x — i y)q = q{x + iy)p{x —iy)q~l
(3.26)
Now all we have to do is show th a t the derivative dz still raises the spin of the field.
Consider how it behaves under rotations of the coordinate system.
cos 6 — sin 9
X
y'
sin 9
cos 9
y
X
cos 9
sin 9
xf
— sin 9 cos 9
y’
x'
—
y
(3.27)
(3.28)
dx>
=
(cos 9) dx — (sin 9) dy
(3.29)
dy,
=
(sin 9) dx + (cos 6) dy
(3.30)
dx
=
(cos 9) dxf + (sin 9) dy.
(3.31)
dy
=
—(sin#) dxt + (cos9) dy>
(3.32)
22
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
^ ( d x, -idy>)
=
dX' + i d y >) =
e l8~{dx - i d y)
(3.33)
el9^ ( d x + idy)
(3.34)
The im portant thing to get out of this is th a t
dz, = ^
- idy,) = R - e dz R e.
(3.35)
The figures 3.1 and 3.2 graphically represent this equation for b o th active and passive inter­
pretations of the functional Re. We find th at
R - e dz Re = e~i0dz .
(3.36)
R- e d-z Re = el9d-z.
(3.37)
By a similar argument,
So our new derivative dz = \ { d x — idy) behaves like our old derivative under rotations,
except th a t this one can operate on a much larger space of complex functions. We also have
another derivative dz = ^(dx + idy) which behaves oppositely under rotations.
3.2
9 is a C ovariant C om p lex D eriv a tiv e on th e
Sphere
In order to derive the formulas for complex derivatives on the sphere, we start by recalling
the geometry of the sphere, then we write down the orthographic projection, and finally, we
use that projection to derive 5 and 5.
23
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 .
0
2
4
6
Figure 3.1: The complex derivative of a rotated scalar field. This figure is made w ith the
perspective of an active rotation. The lower colored patch represents the complex values of
f ( z ) in some patch, and the upper patch represents the complex values for g(z) which have
been ro tated from the lower patch. The rotation R q takes point p to point q. Because the
complex numbers are rotating and th e partial derivatives are not, the derivative
changes.
( d x
—
i d
y
) / 2
-1
1
0
1
3
2
4
5
6
Figure 3.2: The complex derivative of a rotated scalar field. This figure is made w ith the
perspective of a passive rotation. We show two coordinate systems, prim ed and unprimed.
Because the partial derivatives for the two coordinate systems are in different directions, the
derivatives
2 and
differ.
( d x
—
i d y )
/
( d x > —
i d y > ) / 2
24
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
3.2.1
G e o m e tr y o f th e S p h er e
Recall the definition of a connection coefficient:
(3.38)
This means it is the “a component of the change in ep relative to parallel tran sp o rt along
e7.” from page 209 of [22]. We param eterize the sphere by th e usual coordinates 6 and <f.
This calculation will be done with orthonorm al basis vectors, not coordinate basis vectors.
Recall how to calculate the connection coefficients:
f
2
(Wtta "f" 9h"/,i3
9f3"f,n T
T Wt/3
Abm)
(3.39)
where [eaj ep\ = ca/f fe1. Since we are using orthonorm al basis vectors, our m etric is the
identity, and so we can raise and lower indices w ith impunity. Also, all derivatives of the
metric vanish, so our connection coefficients only require us to calculate the com m utation
coefficients caff .
There are
8
connection coefficients,
6
of which are zero. The other two are:
r*w = c o t s
r%* = - c o t 0
(3.40)
This basically means th a t if one parallel transports (in the ep direction) two vectors fg and
fp which were originally identical to eg and e^>, then they begin to ro tate w ith respect to the
local ee and
at a rate of cot( 0 ) radians (clockwise as seen from outside the sphere) per
radian travelled. Perhaps it would be easier to think of the local eg and
as rotating in the
opposite direction, since the vectors fg and fp are being parallel transported. See figure 3.3.
25
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
3 .2 .2
A P r o je c tio n
It will be useful to consider an orthographic projection of the sphere onto a plane. In the
following, we write down a na ive orthographic projection, determ ine it has a wrong sign for
our purposes, and then correct it.
T h e First P r o je c ti o n
Suppose we are interested in points near (9*, 0*) on the sphere, where 9* and 0* are the usual
angular coordinates, ju st w ith asterisks. We apply two rotations: one about the z axis to
rotate the point to the x-z plane, where
0
=
0
, and one about the y axis to ro tate th e point
to the north pole (x = 0 , y = 0, z = 1). This will align the transform ed
w ith e x, and it
will align the transform ed e e w ith —e y . The explicit version of the ro tatio n is as follows:
X
/
y'
■y'
Ay
cos 9* 0 —sin 9*
=
sin 9* 0
cos <p
sin 0 *
0
X
— sin i
cos 0 *
0
y
0
1
Z
cos (
0
(3.41)
We originally have
sin# co s 0
X
y
=
sin 9 sin 0
(3.42)
cos 9
z
Now we can project the point (x', y', z!) onto the x-y plane by ignoring the z component,
which will be close to 1. We can w rite the x' and y' coordinates in term s of 6 and 0.
x! = —cos(0) sin(0*) sin(#) + cos(0*) sin(#) sin(0)
(3.43)
y' = cos(#) sin(#*) —cos(0*) cos(#*) cos(0) sin(0) —cos(#*) sin(0*) sin( 6l) sin(0)
(3.44)
26
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 .
- 0 . 05
- 0.15
0 .1
0 . 05
0
0 . 05
0 .1
0 .15
Figure 3.3: A region of the “N orthern” unit hemisphere, where z > 0. It has been projected
onto the plane. The x and y axes are geodesics, as are all lines intersecting the origin.
Angles at the origin are preserved by this mapping. Lengths are also preserved at the origin,
but only to first order. This projection was made by rotating th e desired (arbitrary) point
6,(j) = 0.7,1.5 up to the N orth pole (z = 1) such th a t
= ex . Then it was projected onto
the x-y plane by ignoring the z component, which was close to 1 anyway. Blue lines are
lines of constant 9 and (ft; these were ro tated up to the N orth pole and projected in the same
fashion as the point in question. The increments between blue lines are 0.03 radians in b o th
the 9 and (j) directions.
This allows us to m ap a point on the sphere with angular coordinates (9, (f>) to a point on the
plane w ith coordinates (x', y' ). The point (9*, 4>*) defines the projection, and it gets m apped
to the origin of the plane. This projection is least distorted when the points (9, <p) are close
to (9*, 4>*). See figure 3.3 for the mapping of several lines of latitude and longitude to the
plane.
For ease of typing, we now ignore the primes on x and y. Under this m apping from (9, (f>)
to ( x , y ) , the following holds at the point ( x , y ) = (0 , 0 ):
dx = do
d„ =
sin 9
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 .
(3.45)
and so our current form alism w ould find
dx
-
i d y
=
d
B
-
i - r
(3.46)
— ^ d t) ,
sin U
This has the wrong relative sign between dg and 8$, if we want to m atch dz w ith 5.
C or recti ng the P r o j e c t i o n
So we switch our viewpoint to th a t of someone inside the sphere instead of someone outside.
(ex , e y) is a right-handed 1 coordinate system as seen from outside the sphere. As viewed
from the origin ( x , y , z ) = (0 , 0 , 0 ), which is inside the sphere, it is a left-handed coordinate
system. We can swap coordinates x and y to tu rn it back into a right-handed coordinate
system.
For aesthetic purposes, we will also negate both directions.
This gives us the
following projection:
x
=
—cos(9) sin( 6 **) + cos(</>*)
y
=
cos{(p) sin(</>*) sin( 6*) —cos(</>*) sin(0) sin(0)
0 0 8 (6**)
cos(</>) sin(0) -I- cos( 6**) sin(<^>*) sin(0) sin((^.47)
(3.48)
Now we obtain (at x, y = 0, 0):
dy = - d e
y
dx = —
s md
dx — idy = i (dg + i — -d<t,)
V
sin 8 J
(3-49)
(3.50)
This has the correct relative sign between dg and d^, and we can begin.
Before we do, here are a few last comments about the projection. The lines of longitude
and latitude intersect at right angles at the origin (x, y) = (0, 0) of the plane. They do not
1W hen th e re are only tw o vectors, we tak e “rig h t-handed” to m ean th a t th e second vector is ro ta te d 90
degrees counter-clockw ise from th e first. T his explicitly depends on w hether they are viewed from th e front
or th e back, and so requires an implied direction from which they are viewed.
28
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 .
intersect at right angles away from th e y axis, b u t th a t is a second order effect. T h at is, the
angle between lines is 7r / 2 + 0 ( x 2). T he coefficient of x 2 may depend on y. Also, all lines
intersecting the origin of the plane are geodesics when m apped back to the sphere. This
m apping preserves angles at the origin of the plane.
3 .2 .3
D er iv in g 9
If the complex field behaves like a scalar field, invariant under rotations, then our complex
derivative is
dz = \ (dx ~ idy) = l- ( d e +
-
(3.51)
Suppose th a t our field is not a scalar, and it instead changes phase under rotations. Specif­
ically, let it be a spin s field. Hence if the physical object the field represents is rotated
counter-clockwise (positive sense about an inward pointing radial vector) by angle a , then
the phase of number is multiplied by e~lsa. If instead we ro tate the complex plane (passive
transform ation) by angle a, the num ber in the new coordinate system should be eisa times
the old number.
A general map will have some rotation in it, and in particular, our m ap does. Let the
m apped complex function be f ( x , y) if it is m apped as a scalar, bu t let it be f ( x , y)e~isa(x’v')
if it is m apped properly. Here, a(x, y) provides th e rotation of the map, and s is the spin of
the field. Then
)
( 8 ,
-
t d y )
/ e - ‘“ =
< T ‘ “
i
(4 -
i d y )
f
+
f
1-
( 8 X
-
i d y )
e ~ “ °
(3.52)
The rotation is zero in the dy direction, and zero at the origin, where the derivative will be
evaluated. Therefore
^ (dx - idy) f e ~ tsa = ~ (dx - idy) f - f U s ( d xa)e~lsa = ^ ( d x - idy) f - f 7-s cot 9 (3.53)
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 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 so
2idzf e tsa = i ( d x - %8y) / + s f cot 0 = - ( dg + v—^
) f + s f cot 0
(3.54)
We claim th a t this is the operator <3 acting on a spin s field. It is proportional to a complex
derivative in the tangent plane.
Note the proportionality constant. T he factor of %could be removed by choosing different
coordinates. For aesthetic purposes, we d on’t do th a t, so as to keep the im aginary direction
pointing north. In any case, the phase of the proportionality constant can be removed by
rotation, and doesn’t mean much. The factor of 2 cannot be removed except by scaling the
tangent plane. Perhaps it could be removed by using a sphere of radius 2 or 1/2. In any case,
the proportionality constant does not affect th e behavior of th e <3 operator as a spin-raising
operator.
3 .2 .4
D e fin itio n
The operators <3 and 5 act on complex functions z(9, 0) on the sphere.
%sf{0,<t>)
=
- s i n s(0 )
3 s /(M )
=
- s i n ~s(6)
8
86
8
89
i
8
sin 3{9)f(9, i
sin( 0 ) d(p
(3.55)
i d
sin s(9)f(9,,
sin( 0 ) d(p
(3.56)
They can be re-w ritten as
3S/ ( M )
3 J(M )
8
89
8
89
8
f ( 9 , 0) T s cot(6)f(9, (
sin(0) 8(f>
i d
f(9, 0) - s co t(9)f(9,~
sin( 6l) 30
30
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(3.57)
(3.58)
Looking down
(looking at the sphere from outside)
N
-u
-Q
W
y
s
y
cp
V
e
z=x+ iy
Figure 3.4: This chapter’s sign conventions for looking at the sphere from the outside. Note
th a t they differ from the HEALPix conventions.
Looking up at the sky
(looking at the sphere from inside)
N
-Q
A
-u
W
¥
^
e
X
:+ i y
Figure 3.5: This chapter’s sign conventions for looking at the sphere from the inside. This
figure is redundant with figure 3.4 bu t is shown to emphasize the two possible viewpoints.
Note th a t these sign conventions differ from th e HEALPix conventions.
31
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 .
spin s objects:
s=0
s =1
s=2
s=4
s=5
o
s=3
V
Figure 3.6: A few spin s objects. They all live on the plane, or perhaps on a tangent plane
to the sphere. They are symmetric under a ro tatio n by angle 2?r/s, if s ^ 0, and a spin 0
object is symmetric under any rotation.
z=x+iy
-i
y
t
1
---------------------- X
i
s=2
s=3
-l
-l
Figure 3.7: The phase convention used to find the orientation of the spin s object from the
phase of a complex number. In the case of the sphere, the orientation is w ith respect to
the x and y directions, which are in tu rn aligned w ith the —(/> and —6 directions as per our
phase convention.
32
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
3.3
T h e E an d B M o d es
One can now obtain th e spin s spherical harmonics by applying the raising operator s tim es
to the usual spherical harmonics. The details (and constants of proportionality) are laid out
in appendix A.
The E and B modes are raised versions of pure real and pure imaginary spherical h ar­
monics. They describe how much “gradient” and “curl” there is in the field.
The usual decomposition of the polarization field into spin 2 spherical harmonics is:
Q + iU -
a.2 ,lm 2 ypn
(3.59)
im
and the alternative decomposition into spin
-2
Q — iU =
harm onics is:
0 - 2 , Im - 2 Y i m
(3.60)
im
Zaldarriaga and Seljak [15] define two coefficients which are combinations of the a 2/ m and
0 - 2 ,i m-
a 2,im + a - 2 , i m
oE,tm
=
------------- 2 ------
a,B,em
=
---- 1-------w:-----Zl
02.i m
U—2 ,£m
/0
\
(3.61)
/ 0 ^r,\
(3.62)
This can also be w ritten as follows, w ith only the a 2 ym coefficients:
a-E/m =
0-2,em + ( - 1 )m« 2 y - m
----------------- 2-----------
OBim =
---------------- tt.----------Z%
0 2 , em ~
( “ 1)
a 2 , i —m
,
,
(3.63)
a A -^
(3.64)
Observe also th a t
0-2,im. =
Ofi/m
33
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(3.65)
One m ight naively th in k th a t we could now decom pose th e p o larizatio n field as follows:
Q + iU = E
(3.66)
Q-E/m. E^im + ^B/m BYem
£m
for some (yet to be defined) E and B spherical harmonics. Unfortunately, this is not m athe­
matically possible. The most visible problem w ith this decomposition is th a t m ultiplication
by i rotates a polarization field by 45 degrees and therefore turns an E mode into a B mode.
We would have to restrict our coefficients to being purely real, which is not possible because
they have already been defined.
Instead, we must be content w ith the more detailed decomposition:
Q + i U = Re(aE/m)
£m
E Y [m
+ lrn.(aE/ m) sY/m +
BYem +
R
^m ( a B / m
s Y lm (3.67)
)
To determ ine the details of this expansion, we note the scalar E and B fields defined by
Zaldarriaga and Seljak [15]. The expansion in spherical harmonics can be derived from the
above definitions of aB/ m and (iBim-
E = —
2
{£ +
3 ‘ (Q + iU) + S 2( Q - z U ) J
h
B = - d2(Q + iU) - & { Q - i U ) \ = Y ,
2
2 )\
1/2
a-E/mYim
(£-2)y
(£ + 2)!1 1/2
£m L (* -2 )!J
2
~
a B/mY£m
(3.68)
(3.69)
~
Note th a t since (9 (Q + iU))* = 3 2(Q — iU), we see th a t b o th the E and B scalar fields are
pure real2. This is more clear if we w rite this as:
E
=
—Re 3 ‘ {Q + iU)
(3.70)
B
=
—Im B2{Q + i U)
(3.71)
2See appendix A on p roperties of S an d 3.
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 .
where Im(;z) is a real num ber representing the im aginary p a rt of z. z = Re(z) + i Im(z).
Because E and B are b o th pure real, we m ust have
aE,tm =
(-1
(3.72)
O'B/m =
(3.73)
which can also be seen from the original definitions of ciE.im and as/mThe polarization field can be recovered from E and B as follows:
2 (1
-
2
)!
-E -iB
(£ + 2)1
where
(3.74)
is really an operator defined in spherical harm onic space. This equation allows
one to construct all of the E modes by simply raising the spin of a real scalar field on the
sphere. To get the B modes, raise the spin of a pure im aginary field. This idea comes from
[20], equation 3.32.
One can now see (probably in several different ways) th a t
Q + iU = ^ ^ ( ~ a E ,£ m
—
iaB,em) 2 Y i m
(3.75)
Using equations 3.72 and 3.73 as well, one can construct the polarization modes created
from the real and im aginary p arts of the E and B coefficients.
£
OO
Q + iU = E
E
Ro((lE.£m) E^irn ~f~ ^-^(^E.tm) E^im 4” Be(fl.fjym) B^trn T
B^irn
£=Q m = 0
(3.76)
We are now in a position to find E ^lmi etc. Simply set all coefficients to zero except the
real or imaginary p a rt of one of them (and one other coefficient is required to be nonzero by
35
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 .
equations 3.72 and 3.73) an d d eterm in e w h at th e p o la rizatio n field is.
EY[m =
- ( 2^ m + ( - l ) m 2^
eyl
=
“ *( 2 ^ - ( - i r 2^ - m)
(3.77b)
BYL
=
- i ( 2^ m + ( - l ) m 2^ - m )
(3.77c)
bY
=
+ hYfin - { - l ) m 2Yt- m)
(3.77d)
L
Remember th a t Im (^) is pure real. Note th a t e Y^0 = b YIq -
3 .3 .1
(3.77a)
m)
0
.
C o n n e c tin g w ith H E A L P ix C o n v e n tio n s
The sign convention for the current im plem entation of HEALPix is exactly opposite the one
chosen here. This am ounts to redefining the aE,em and a s / m coefficients with the opposite
sign, which has the effect of m ultiplying the harmonics in equations 3.77 by —1.
This
m aintains the t;E-ness” and “B-ness” of the modes, so it is an arb itrary choice of sign.
Equation 3.76 still holds. We will ignore the HEALPix sign convention for th e rest of this
chapter. Note th a t all of the formulas in appendix A still hold, though.
3 .3 .2
D im e n sio n C o u n tin g
A complex field on the sphere requires
2^
+
1
complex coefficients to represent it at the
angular scale given by t. This is equivalent to 4£ + 2 real coefficients. For a real field on the
sphere, we need half of the information, which comes to 21 +
1
real coefficients.
The polarization can be represented by two scalar fields on the sphere, or one complex
field on the sphere. W hen w ritten out as a sum of EY[m, e Y ^ , BY[m■ and bT /„( terms,
ranging over 0 < m < t, we find 47 + 4 real coefficients. We subtract two because e Y^0 and
b Y£q
are exactly zero, and this gives exactly the 4£ + 2 necessary degrees of freedom. The
degrees of freedom are also conveniently evenly split between E and B modes.
36
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
3 .3 .3
T e m p e r a tu r e -P o la r iz a tio n C o rr ela tio n
We are interested in the case where th e E coefficients exactly correspond to the tem perature
coefficients. T hat is, we have a (pure real) tem p eratu re map
(3.78)
and only electric modes of polarization
(3.79)
where the coefficients are related by:
a-T/m
OC
(3.80)
—a,E,em
where the proportionality constant is a positive function of only I. Thus we see th a t T oc —E .
By plotting the tem perature field underneath the polarization lines, one can see th a t
they represent the polarization field one would expect from the tem perature. The d operator,
when acting twice on the tem perature field, raises it to a spin
2
field th a t exactly m atches the
polarization field one would expect from th a t tem perature distribution, up to a norm alization
function of £. (This statem ent may need to be modified under different sign conventions,
but it is true for this discussion.)
37
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 h ap ter 4
T ests o f N o n -G a u ssia n ity
The Wilkinson Microwave Anisotropy Probe (W M AP) d a ta provide the most detailed d a ta
on the full sky cosmic microwave background (CMB) to date.
This inform ation about
the initial density fluctuations in th e universe allows us to test the cosmological standard
model at unprecedented levels of detail [9]. A question of fundam ental im portance to our
understanding of the origins of these prim ordial seed perturbations is w hether the CMB
radiation is really an isotropic and Gaussian random field, as generic inflationary theories
predict [25, 26, 27],
In this chapter we check the prediction of Gaussianity, following the work in [23] and
[24], We look at the pattern of hot and cold spots (local extrem a) seen in the CMB sky. We
use several methods to determ ine if th a t p attern is statistically similar to the p attern s of hot
and cold spots th a t we simulate for Gaussian isotropic random fields on the sky. First, we
find an anomaly of the one-point statistics of hot and cold spot excursions. Then, we extend
this analysis using correlated noise and multi-resolution studies, and we add a detailed study
of two-point statistics: the spot-spot and tem perature-w eighted correlation functions of hot
and cold spots.
Most frequentist searches for non-Gaussianity follow a set recipe: first compare a statistic
computed on observed d ata to a set of Monte Carlo simulations. An assessment of goodnessof-fit then leads to a significance level at which Gaussianity is rejected. This assessment is
often rather informal and prone to false detection. One of the main goals of our paper is the
presentation of a robust hypothesis test th a t is applicable to all tests of Gaussianity in this
xT his chapter contains m aterial w hich has been previously published in [23, 24].
38
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 .
category.
A number of groups have applied this recipe to various statistics of the CMB anisotropy.
For example, Vielva et al. detect non-G aussianity in the three- and four-point wavelet mo­
m ents [28], as do Liu and Zhang [29], who claim it may be a residual foreground. McEwen
et al. investigate wavelets th a t aren’t azim uthally symmetric, and find non-G aussianity us­
ing the skewness and kurtosis of their wavelet coefficients [30]. Chiang et al. detect nonG aussianity in phase correlations between spherical harmonic coefficients [31, 32, 33], and
Park finds it in the genus Minkowski functional [34], Eriksen et al. find anisotropy in the
n-point functions of the CMB in different patches of the sky [35]. O thers discuss possi­
ble m ethods of detecting non-Gaussianity. Aliaga et al. look at studying non-G aussianity
through spherical wavelets and “sm ooth tests of goodness-of-fit,” [36].
Cabella et al. re­
view three m ethods of studying non-Gaussianity: through Minkowski functionals, spherical
wavelets, and the spherical harmonics [37]. They propose a way to combine these m ethods.
More recently, Cabella et al. constrain one generic type of non-Gaussianity using spherical
wavelets and local curvature of the CMB tem perature field [38]. K om atsu et al. discuss a
fast way to test the bispectrum for prim ordial non-G aussianity in the CMB [39], and do not
detect it [40]. Using a generic model for non-Gaussianity, Babich shows th a t the bispectrum
is the best way to constrain it, and therefore claims th a t the bispectrum test used by the
W M A P team [39] is optimal [41]. Finally, G aztanaga et al. find the CMB to be consistent
w ith Gaussianity when considering the two and three-point functions [42, 43].
This chapter is laid out as follows. First, we discuss the initial m ethod of simulation, our
m ethod of testing our hypothesis, and then our first results. After th a t, we discuss a more
detailed analysis of the 1-point statistics. Then, we go on to discuss our two-point statistic
results. For more details on our statistical tests, see appendix B.
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 .
4.1
S ta tistica l A n a ly sis
The problem we tackle in this chapter is how to find non-G aussianity in the CMB when
we do not have a specific model for the non-Gaussianity. We have a m easured CMB sky
(W M AP data) and the ability to make m any Gaussian simulations of CMB skies, and we
w ant to determine if the measured CMB sky looks like it came from th e distribution of skies
we simulate.
Because we do not have a model for non-Gaussianity, we do not have a second distribution
of skies to compare to the G aussian distribution. This limits the testing we can do to merely
determ ining how large a statistical fluctuation our currently measured CMB sky is.
We approach the problem numerically as follows: we find some way to reduce the an
entire CMB sky to a single number, a single statistic com puted on the hot and cold spots.
We compare the statistic for the measured CMB sky to the distribution of statistics for the
sim ulated CMB skies. If the measured statistic falls significantly higher or lower th an all
of the others, then we have a large statistical fluctuation, which we quantify. It is then
up to the reader to determ ine if this should be interpreted as merely an unlikely statistical
fluctuation, an indication of non-Gaussianity, a residual foreground, or a m ism atch between
our simulations and the actual observations. To eliminate th a t last possibility, we describe
our simulations in detail.
The specific simulation m ethods and statistics we use are detailed in the corresponding
sections. A detailed discussion of w hat constitutes a statistically significant detection is
given later in the chapter. In addition we make available a freely distributed Mathematica
notebook and C code.
4.2
O n e-P oin t M eth o d
We test the W M AP data of the CMB sky by comparing the one-point statistics of its extrem a
to those same statistics on several sets of Monte Carlo-sim ulated Gaussian skies. Our null
40
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 .
hypothesis is th a t the statistics of the W M A P d a ta are draw n from the same probability
density function (PD F) as the statistics of the M onte Carlo skies. If some W M AP one-point
statistic falls lower or higher th a n most of th e M onte Carlo statistics, this indicates th a t our
hypothesis may be false.
We examine several inputs to our Monte Carlo sim ulation to see how those change the
M onte Carlo distribution of one-point statistics around th e W M A P one-point statistics. We
sta rt very generally, looking at different frequency bands and G alactic masks, and then
narrow our search. Initially, we look at sim ulations including the three frequency bands (Q,
V, and W) and the three published Galactic masks for the W M AP data. Then we check
to see if changing to a different published theoretical power spectrum affects our results.
Finally, we look for anisotropy between the statistics of the ecliptic and Galactic north and
south hemispheres.
4 .2 .1
M o n te C arlo S im u la tio n
A general outline of our Monte Carlo sim ulation process follows. Each set of skies is labeled
by its theoretical power spectrum , frequency band (Q, V, or W ), and Galactic mask. The
frequency band determines b o th the (azim uthally averaged) beam shape function and the
noise properties on the sky. The simulated CMB skies are created as follows:
1. A Gaussian CMB sky is created w ith SYNFAST, using a power spectrum and a beam
function. The H EALPix2 pixelization of the sphere is used, w ith N sicie = 512.
2. Random Gaussian noise is added to th e sky (at each pixel) according to the pub­
lished noise characteristics of the band being simulated. The W M AP radiom eters are
characterized as having white Gaussian noise [44],
3. The monopole and dipole moments of th e sky (outside of the chosen Galactic mask)
are removed.
2See h ttp ://w w w .eso .o rg /scien ce/h ealp ix /
41
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 .
We make no attem pt to simulate any foregrounds, including th e galaxy; our analysis ignores
d a ta inside a G alactic mask and uses the cleaned m aps published on LAMBDA (Legacy
Archive for Microwave Background D ata Analysis) [45].
For each Monte Carlo set, one of four power spectra is used. These are the power spec­
tra published by the W M AP team on LAMBDA [45]. We prim arily use the best-fit (bf)
theoretical power spectrum to a cold dark m a tte r universe w ith a running spectral index us­
ing the WMAP, Cosmic Background Imager (CBI), A rcm inute Cosmology Bolometer A rray
Receiver (ACBAR), Two-Degree Field, and Lycr data. In addition, we check the unbinned
power spectrum (w) directly measured by W M AP, the power law (pi) theoretical power spec­
trum fit to WMAP, CBI and ACBAR, and a running index (ri) theoretical power spectrum
fit to WMAP, CBI and ACBAR. See [46], [9], and [45] for more information.
The Galactic masks used are the KpO, Kp2, and K p l2 masks published by th e W M A P
team [47]. To check for differences between the north and south ecliptic hemispheres, we
define additional masks th a t extend the KpO Galactic mask to block either the n o rth or
south hemisphere as well. For example, th e ecliptic south (ES) mask blocks the northern
ecliptic sky as well as the galaxy. As a control, we also extend the KpO mask for G alactic
north and south hemispheres (GN and GS) to bring the to tal number of masks up to seven:
KpO, Kp2, K pl2, GS, GN, ES, EN. We use th e same masking and dipole removal procedure
for the W M AP d ata as for the Monte Carlo skies.
The W M AP d a ta th a t we use are the cleaned, published maps. They are published by
channel, so we calculate an unweighted average over (for example) all four W -band channels
to get a m ap for the W band. The noise variance is calculated accordingly. We com pute an
unweighted average of the maps so th a t we can combine the W M AP beam functions through
a simple average.
42
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 .2 .2
A n a ly sis and H y p o th e s is T est
Our analysis of b oth the M onte Carlo and W M A P skies involves th e following: We find
the local m axima and m inim a of the H EA LPix grid using H O TSPO T. Then we discard
the extrem a blocked by the Galactic mask.
We calculate the statistics (number, mean,
variance, skewness, and kurtosis) on the tem peratures of the m axim a and minima, and
then statistically analyze the significance of the position of th e W M A P statistic among the
M onte Carlo statistics. Because we consider only the one-point statistics, we consider only
the tem perature values, not their locations.
We calculate our five statistics for the m axim a and m inim a separately. The two statistics
which are typically negative for the minima, the m ean and skewness, are multiplied by —1
in our results, to make comparison w ith th e m axim a statistics more clear.
For the rest of this section, we explain our analysis of the statistics in detail. To simplify
the discussion, we consider the analysis of only one statistic on either m axim a or minima,
as we analyze the results for each statistic separately.
O ur Monte Carlo simulations are binomial trials, where the statistic calculated on a
simulation can lie either above or below the W M AP statistic.
It lies below the W M A P
statistic with probability p, and for some set of n trials, i of the trials will have statistics
below the W M AP statistic. G ivenp, the probability of i is P(i\p) = [n\/i\{n —i ) ] pl (l —p)n~lThe value p = i / n is both an unbiased and maximum likelihood estim ator of p.
We are interested in w hether p is near 0 or 1, since th a t indicates th a t our hypothesis—
th a t the W M AP statistic came from the same PD F as the Monte Carlo statistics—may be
false. Because we do not have an alternative distribution for the W M A P statistic th a t we
can test against th e Monte Carlo distribution, we do not test our hypothesis as phrased. We
only look at a hypothesis H 0 th a t claims p is in some interval, p € (a /2 , 1 —a /2 ), where we
have arbitrarily chosen a = 0.05.
We devise a statistical test, of this hypothesis.
Given our experimental result i, we
43
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 .
construct the 1 — a = 95% symmetric confidence interval for p, as described in Kendall &
S tu art (1973). If this confidence interval lies entirely w ithin the interval [0, a /2 ] or entirely
w ithin [1 — a / 2 , 1] then we reject our hypothesis, H 0. We reject H 0 for no values of i when
n = 99, for 0 < i < 15 or 985 < i < 1000 when n = 1000, and for 0 < i < 103 or
4897 < i < 5000 when n = 5000.
This interval is a “95%” confidence interval in the following frequentist (non-Bayesian)
sense. Suppose we repeat the experiment (with the same num ber of M onte Carlo runs, and
the same W M AP data) many times and get many values of i. We recalculate the confidence
intervals each time, for each particular value of i. Ninety-five percent of the confidence
intervals we calculate will contain the true value of p.
Our test is biased in favor of H 0. Let Hi be the alternative hypothesis p E [0, a /2 ] U [1 —
a / 2 , 1]. Then, for some values of p where Hi is true (for example, p = 0.02, n = 1000), our
test will choose H 0 more often than Hi, given th a t i is a random variable with probability
P(i\p). If desired, we can make the test unbiased by changing our value of a in the hypotheses
H q and Hi, b u t keeping the test (range of i for which H 0 is accepted) the same.
For n = 1000, we have an unbiased test if a = 0.0313, and for n = 5000, we have an
unbiased test if a = 0.0415. Note th a t these values are less th an a = 0.05. For any value
of p, these tests are at least as likely to choose the correct hypothesis as the incorrect one.
This is a 50% confidence, as opposed to our previous 95% confidence. This interpretation
of the test does not change our results; it merely provides the different perspective th a t our
test may be considered an unbiased 96.9% test, for n = 1000; or an unbiased 95.9% test, for
n = 5000.
4.3
F irst O n e-P oin t R esu lts
We display our results in Figure 4.1 and Table 4.1 and 4.2. The figure shows where the
means of the W M AP maxima (and minima) he in the Monte Carlo cumulative distribution
44
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Mean
0.044
CO
o
0 .0 0 9
0.002
z
o
0.012
CO
I_lJ
0 .0 0 7
LxJ
0 .2 6 6
0.267
0 .2 6 8
0 .2 6 9
0 .2 7 0
Figure 4.1: Cumulative distribution functions (CDFs) of mean tem perature value (in units
of millikelvins) of the local extrema, found in sets of 5000 Monte Carlo simulations for four
Galactic masks: GS, GN, ES, and EN. Best fit power spectrum and W -band d a ta are used.
Means of the minima are negated for comparison. M axima CDF is dotted while minima
CDF is solid. Note their visual similarity. Statistics measured on W M AP d ata are shown as
two vertical lines, dotted for maxima and solid for minima. Numbers on right are the same
probabilities as in Table 4.1; for each pair, probability for the m axima is higher on the page.
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 .
Identifier
bf, Q, KpO, max
bf, Q, KpO, min
bf, Q, Kp2, max
bf, Q, Kp2, min
bf, Q, K pl2, max
bf, Q, K pl2, min
bf, V, KpO, max
bf, V, KpO, min
bf, V, Kp2, max
bf, V, Kp2, min
bf, V, K pl2, max
bf, V, K pl2, min
bf, W, KpO, max
bf, W, KpO, min
bf, W, Kp2, max
bf, W, Kp2, min
bf, W, K pl2, max
bf, W, K pl2, min
0
0.374
0.010
0.333
0.010
0.020
0.000
0.091
0.030
0.061
0.040
0.000
0.020
0.475
0.414
0.495
0.293
0.434
0.364
1
0.000
0.000
0.000
0.000
0.000
0.000
0.616
0.182
0.384
0.202
0.384
0.232
0.000
0.000
0.000
0.000
0.000
0.010
2
0.030
0.232
0.152
0.293
0.576
0.798
0.182
0.303
0.061
0.263
0.212
0.444
0.141
0.343
0.131
0.323
0.242
0.505
3
0.566
0.919
0.545
0.919
0.929
0.919
0.495
0.990
0.364
1.000
0.343
0.980
0.364
0.879
0.182
0.808
0.313
0.707
4
0.889
0.707
0.899
0.768
1.000
1.000
0.848
0.960
0.727
0.960
0.889
1.000
0.475
0.646
0.283
0.616
0.253
0.737
5
99
99
99
99
99
99
99
99
99
Table 4.1: .The identifier column provides the power spectrum, band, and mask used and
w hether the statistics are for minima or maxima. Power spectrum is th e best fit (bf) W M AP
power spectrum. Columns labeled 0 through 4 give an unbiased estim ate of the W M AP
statistic ’s position among the sorted M onte Carlo sample statistics. The statistic in column
0 is number of hot spots. The other columns correspond to: 1, mean; 2, variance; 3, skewness;
and 4, kurtosis of extrem a tem perature values. (For minima, mean and skewness statistics
are negated before estim ating probability of W M AP statistic being lower.) Column 5 gives
the number of Monte Carlo samples calculated. Probabilities th a t indicate th a t, “the true
value of p is at least 95% likely to be w ithin 0.025 of either 0 or 1,” are marked w ith an
asterisk. None are m arked in this table because 99 samples is not sufficient to make this
claim. The table shows th a t the d a ta fall low in the mean tem perature distribution for
almost every set of simulations.
46
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Identifier
pi, W, KpO, m ax
pi, W, KpO, min
ri, W, KpO, m ax
ri, W, KpO, min
w, W, GS, m ax
w, W, GS, min
w, W, GN, m ax
w, W, GN, min
w, W, ES, m ax
w, W, ES, min
w, W, EN, m ax
w, W, EN, m in
bf, W, GS, m ax
bf, W, GS, min
bf, W, GN, m ax
bf, W, GN, min
bf, W, ES, m ax
bf, W, ES, min
bf, W, EN, max
bf, W, EN, min
0
0.427
0.377
0.388
0.396
0.633
0.685
0.436
0.243
0.607
0.176
0.470
0.702
0.603
0.641
0.371
0.209
0.560
0.151
0.436
0.668
1
0.002*
0.000*
0.000*
0.000*
0.023
0.007*
0.000*
0.006*
0.010*
0.003*
0.011*
0.005*
0.044
0.009*
0.002*
0.012*
0.011*
0.007*
0.012*
0.011*
2
0.129
0.224
0.216
0.406
0.362
0.981
0.168
0.060
0.869
0.923
0.019
0.067
0.240
0.852
0.091
0.035
0.472
0.586
0.002*
0.011*
3
0.461
0.880
0.285
0.821
0.045
0.247
0.504
0.832
0.103
0.244
0.416
0.958
0.152
0.405
0.648
0.883
0.198
0.3.76
0.529
0.960
4
0.410
0.704
0.310
0.618
0.304
0.213
0.159
0.610
0.429
0.152
0.119
0.861
0.434
0.284
0.284
0.697
0.487
0.253
0.188
0.869
5
1000
1000
1000
1000
1000
1000
5000
5000
5000
5000
Table 4.2: This is a continuation of table 4.1. The identifier column provides the power
spectrum , band, and mask used and w hether the statistics are for m inim a or maxima. Power
spectra are best fit (bf), power law (pi), running index (ri), or measured unbinned W M AP
(w). Columns labeled 0 through 4 give an unbiased estim ate of the W M A P statistic’s position
among the sorted Monte Carlo sample statistics. The statistic in column 0 is num ber of hot
spots. The other columns correspond to: 1, mean; 2, variance; 3, skewness; and 4, kurtosis
of extrem a tem perature values. (For minima, mean and skewness statistics are negated
before estim ating probability of W M AP statistic being lower.) Column 5 gives the num ber
of Monte Carlo samples calculated. Probabilities th a t indicate th a t, “the true value of p is
at least 95% likely to be w ithin 0.025 of either 0 or 1,” are m arked w ith an asterisk. The
table shows th a t the d a ta fall low in the mean tem perature distribution for almost every set
of simulations.
47
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functions for th a t statistic. T he table contains our estim ates p of p. W hen the result rejects
the hypothesis H 0, this is noted with a footnote.
We find th a t the mean tem perature of the W M A P extrem a, and in some cases the
variance, differs significantly from th e sim ulations, b u t number of extrem a, skewness, and
kurtosis are modeled fairly well by the simulations. There is ecliptic north-south asym m etry
in the variance of the extrem a.
For the mean, all of our results reject our hypothesis H 0 for the power law and running
index power spectra, and all four of our results for the ecliptic n o rth and south hemispheres
reject H 0 when n = 1000 and n = 5000. Since the statistics for th e m inim a are negated, this
means th a t the W M AP m axim a are too cold and the W M A P m inim a are too hot. This is a
very significant result, regardless of w hether our tests are considered 95% tests biased away
from detection, or a 96.9%, and a 95.9% test.
Even more significant are two 99% (a = 0.01) level detections. In Table 4.1, they are
rows bf, W, GN, max, and mean, and bf, W, EN, max, and variance. For this level of
detection, we only accept values of i where 0 < i < 12 or 4988 < i < 5000.
No 99%
confidence detection was possible with only 1000 iterations.
Using our initial 99 simulations, we find qualitatively low mean excursions in the Q and
W bands, b u t not the V band. We chose th e W band to examine further because it had the
best signal to noise ratio, and it had the least chance of foregrounds outside the KpO mask
th a t we use in our final analysis.
It has been suggested [35] th a t there is statistical anisotropy between the ecliptic north
and south hemispheres.
We see this in th e variance of the extrem a tem peratures.
The
ecliptic north hemisphere has a statistically low variance (in one case at the 99% level) while
the ecliptic south is normal. To compare, we find the Galactic north to be slightly low while
the south is again normal.
48
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 .4
M u lti-R e so lu tio n O n e-P o in t A n a ly sis
4 .4 .1
W M A P M e a su r e m e n t
We construct a single tem perature m ap of th e CMB to represent the W M A P te am ’s m ea­
surements. M otivated by our multi-frequency study in LW04, we take this m ap to be an
unweighted average of the four channels of the W -band foreground cleaned tem perature
m aps (available on the LAMBDA web site3. We use an unweighted, average of the tem pera­
tu re maps so th a t we can take the (azim uthally symmetric) beam to simply be the average
of the beams for each of the channels. This is the m ap whose properties we concentrate on
simulating.
4 .4 .2
S im u la tio n P r o c e ss
We take legitimate shortcuts in our G aussian sky simulation process. The brute-force way to
simulate the W M AP team ’s m easurem ent of a Gaussian CMB sky is to sim ulate th e CMB
sky, add foregrounds, simulate the tim e-ordered-data stream from th e W M A P satellite, and
then run it through the full m ap-m aking and foreground removal pipeline. We take a simpler
approach; we simulate the result of th a t process by a Gaussian CMB sky and then add either
white or correlated noise to th a t sky. This is acceptable because the full W M AP pipeline
does a good job of reconstructing some characteristics of the tru e CMB sky, and we are
careful to make sure our statistics depend only on those characteristics.
One possible criticism of LW04 is th a t we used uncorrelated noise in our CMB simula­
tions. There are good reasons for using uncorrelated noise: the noise is stated to be white
noise, and the primary correlations are at angular scales of about 141 degrees and are on the
order of 0.3% [2]. For this paper, however, we compare the results of white and correlated
noise.
The W M AP team has provided 110 publicly available [45] correlated noise simulations
3h ttp : / /la m b d a .gsfc.nasa.gov/
49
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th a t can be used to calculate the correlated noise on our averaged W -band map. We take
each of the simulated noise maps provided by the W M A P team and average and rescale
them. Since there were 4 channels in the W band, we do an unweighted average of the noise
maps. Then we rescale the noise, pixel-by-pixel, so th a t the num ber of effective observations
(amount of noise in th a t pixel) exactly m atches the am ount quoted in the m easured W band.
This is necessary because the number of observations in the sim ulated d a ta provided by the
W M AP team does not exactly m atch the number of observations in the measured d a ta [48].
This requires about a 2.5% decrease in th e noise. Since the correction is approxim ately the
same in each pixel, this rescaling should m aintain the correlations in the noise.
Our white noise simulations model the noise as an unweighted average of Gaussian noise
in each of the four W -band channels.
4 .4 .3
D a ta R e d u c tio n
Care must be taken in our reduction of the d ata to a single statistic.
Specifically, the
reduction process should ignore d ata contam inated by galactic foregrounds, and should be
insensitive to monopole and dipole moments. In this section we describe our d a ta reduction
process: lowering the resolution of the HEALPix4 m ap, ignoring d a ta contam inated by
Galactic foregrounds, removing the monopole and dipole moments, finding the local maxima
and minima in tem perature, and calculating statistics on these local extrema.
Lowering the resolution allows us to investigate the dependence of our results on different
angular scales. Specifically, we can see how the w hite noise and correlated noise models
behave at different resolutions. In this case, lowering the resolution simply means doing an
unweighted average of all the smaller pixels inside the larger degraded pixel.
We use several masks to be sure we have removed all the effects of the Galactic fore­
grounds. See Figure 4.2. We start with the kpO mask, which we must degrade to a lower
resolution. Every degraded (larger) pixel which contains any of the original kpO mask is
4h ttp : / / www.eso.org/scien ce/h ealp ix /
50
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also masked; we call this degraded m ask the “paranoid” mask. We extend this m ask by one
pixel in all directions to get a “paranoid extended” mask, which will be useful later. We do
not apply our analysis to different hemispheres for our one-point statistics, because at low
resolution the degraded masks would block too much of the sky.
We remove the monopole and dipole m oments from the map outside the paranoid mask,
and then find the local extrem a. Since these extrem a are defined by their neighboring pixel
values, extrem a next to the paranoid mask are dependent on pixels we wish to mask. To be
completely independent of masked pixels, we ignore all extrem a inside the paranoid extended
mask.
Finally, we calculate statistics on the local extrem a as in LW04. We use a one-point
analysis involving statistics th a t ignore their angular distribution. For the hot spots, we
use the number of hot spots, as well as the m ean tem perature, and variance, skewness
and kurtosis of the tem peratures. We calculate the same statistics for the cold spots. For
completeness, we give the formulas here, where angle brackets represent an average over all
hot (or cold) spot tem peratures t:
mean
=
(t)
variance
=
((t — (t))2)
skewness
=
at - m
variance3^2
kurtosis
=
- <*»)>
variance
(4.1)
The process of reducing the d ata to a single statistic is summarized by the following list.
We use identical methods to reduce both the W M A P d ata and the simulations.
1. Degrade the map resolution the desired resolution.
2. Remove the dipole from the map outside of the paranoid mask.
3. Find the local extrema.
51
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4. Ignore local extrem a inside the paranoid extended mask.
5. Calculate statistics of the rem aining local extrem a as usual.
4 .4 .4
O n e-P o in t M u lti-R e s o lu tio n R e su lts
From the cumulative distribution functions (CDFs) in figure 4.3, one can see how the differ­
ence between correlated noise and w hite noise changes w ith resolution. The resolution has
a dram atic effect on the num ber of extrem a and on their mean value. At lower resolutions,
the CDFs for the white noise and correlated noise look very similar, as one would expect.
It is true th a t switching from white noise to correlated noise reduces our original detection
of the hot and cold spots not being hot and cold enough. However, the switch to correlated
noise reduces the number of hot and cold spots in th e simulations, so now we have too many
extrem a. W hether or not there are too m any extrem a also seems to be heavily dependent
on the resolution at which we find th e extrem a.
At all scales, we do find the variance of the extrem a to be slightly low.
Because we have only 110 correlated noise samples, we cannot claim any 95% fluctuations
for this noise model. We can claim occasional 95% fluctuations for the 800 white noise
samples we have, but in light of the dram atic differences between the CDFs, unlikely statistics
from the white noise simulations are more likely to be indications of an incorrect noise model
th an of non-Gaussianity. Nonetheless, we do calculate where 95% fluctuations occur and
m ark them in figure 4.3.
4 .4 .5
V ary in g th e A m p litu d e o f th e N o ise
We also check the effect of varying the am plitude of the noise. The W M A P team quotes an
uncertainty of the noise am plitude of 0.06% (at one standard deviation) [2], W hen including
this in our analysis of the m ean of the extrem a, we find th a t the WA1AP mean statistics are
52
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N sid e
=
32
Figure 4.2: These are the 5 masks used for the one-point correlations. Left column is
mollweide projection of the sky; right column is HEALPix base tile 6, where the upper left
corner is northernm ost. Tile 6 is directly opposite th e galactic center; it’s solid angle is
exactly 1/12 of the full sphere’s. Paranoid mask is black, extended paranoid mask extends
it in grey.
53
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Variance, nside = 32
Mean, nside = 32
Num ber of Extremo, nside = 32
9.-142
0:274 :
0 ,3 4 0 0 .2 2 7
252
281
309
338
0 .0 5 9
Number of Extrema, nside = 64
2269
2342
0 .0 6 7
0 ,7 0 2
0,078
0.309
0J36
0 .7 7 3
0 .0 8 2
0 .0 7 5
n-3
0 .0 8 3
Variance, nside
Mean, nside = 64
64
0,027
9,582
0,018
P:Q54.
9.-334
9 :036.
0.055
0,475
0.000
0 .1 0 9
0 .1 5 5
2414
2487
Num ber of E xtrem a, nside = 128
0 .0 8 7 4
0 .0 9 0 0
0 .0 0 9
0 .0 0 4 1
0 .0 9 2 5
0 .0 0 5 0
0 .0 0 5 9
V ariance, nside = 128
Mean, nside = 128
0 ,0 4 4
acne.
0 .0 2 7
0 .0 4 5
0 .1 1 2 8
Num ber of E xtrem a, nside = 2 5 6
0 .0 0 6 0
0 .1 1 4 6
Variance, nside = 2 5 6
Mean, nside = 2 5 6
0.044
9:961.
0 .3 7 4
0 .0 6 5
a .191
0.027
0 .0 5 5
0 .5 3 6
5 .4 8 6 x 1 0 4
5 .5 1 8 x 1 0 4
0 .0 0 7 6
0 .0 0 6 8
5 .5 5 0 x 1 0 4
5 .5 8 2 x 1 0 4 0 .1 4 9 5
Number of Extrema, nside = 5 1 2
0 .1 5 0 6
0 .1 5 1 7
0 .1 5 2 9
0 .0 0 9 1
Mean, nside = 5 1 2
0 .0 1 0 1
0 .0 1 1 2
0 .0 1 2 2
Variance, nside = 5 1 2
0,006
0.-000
0.082
0 .1 9 1
0 .0 2 7
2 .5 6 8 X 1 0 5
2 .5 7 6 X 1 0 5
2 .5 8 4 X 1 0 5
2 .5 9 2 x 1 0 5 0 .2 6 7 6
0 .2 6 8 1
0 .2 6 8 6
0 .2 6 9 1
0 .0 1 9 2
0 .0 2 1 7
0 .0 2 4 2
0 .0 2 6 8
Figure 4.3: Cumulative distribution functions of the hot and cold spot statistics from various
simulations. The grey (black) dashed line is the CDF for the hot (cold) spots from the 800
white noise simulations. The grey (black) solid line is the hot (cold) spot CDF from the
110 correlated noise simulations. The vertical grey (black) line gives the location of the
W M AP statistic for the hot (cold) spots. Estim ates p of the probability of a sim ulation’s
statistic falling below the W M AP statistic are printed on the graph and underlined w ith the
appropriate line. All simulations use the best fit power spectrum, the kpO mask (degrading
is detailed in the paper), and data from the W band. Units for the m ean statistics are m K
and units for the variance are r n K 2.
x
54
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, max
32, min
64, max
64, min
128, max
128, min
256, max
256, min
512, max
512, min
Number
white
corr.
0.274
0.291
0.340
0.227
0.027
0.055
0.054
0.109
0.995* 0.982
0.983
0.936
0.044
0.191
0.374
0.536
0.973
0.080
0.741
1.000
Mean
corr.
white
0.407
0.309
0.702
0.773
0.582
0.473
0.334
0.155
0.675
0.318
0.109
0.045
0.750
0.155
0.264
0.829
0.006* 0.082
0.000* 0.027
Variance
white corr.
0.142 0.136
0.078 0.082
0.018 0.000
0.036 0.009
0.018 0.018
0.044 0.027
0.061 0.027
0.065 0.055
0.136 0.082
0.301 0.191
Skewness
white corr.
0.391 0.427
0.928 0.927
0.390 0.482
0.887 0.827
0.141 0.100
0.794 0.800
0.610 0.591
0.822 0.745
0.426 0.527
0.853 0.845
Kurtosis
corr.
white
0.114
0.118
0.136
0.161
0.964
0.940
0.441
0.491
0.990* 0.973
0.950
0.936
0.900
0.900
0.683
0.709
0.385
0.445
0.682
0.627
Table 4.3: Estim ates p of the probability p th a t a sim ulation statistic will be lower th an the
W M AP statistic. Column headings signify the statistic and w hether the noise was white
or correlated in the simulations. For this table, we used 110 correlated noise simulations
and 800 white noise simulations. Rows labels signify the value of N side and w hether the
statistics are for m axim a or minima. Values of p th a t are significant for our 95% level test
have asterisks. Only the w hite noise has enough simulations to enable a 95% detection. The
first 6 columns of d a ta are presented graphically in figure 4.3.
still qualitatively low, b u t our results are very sensitive to this value. The numerical results
are given in figure 4.4 and plots of the CDF functions are shown in figure 4.4.
Again, w ith only 110 samples, our statistics are not strong enough to claim a 95% fluc­
tuation. In some cases, it is clear th a t more samples will not help.
For example, three
simulations out of 110 have more cold spots th an th e W M AP data, at N side = 512. The
number of cold spots in the W M AP d a ta will not likely be a three sigma fluctuation; it will
probably be just under two sigma. For the hot spots, however, all of the simulations have
fewer maxima than the W M AP data. In this case, more correlated noise simulations would
be useful to determine the significance of this result.
Use of the correlated noise has two effects. It makes the fluctuation in the mean statistic
much less significant (if we also allow a one sigma shift in the noise am plitude). Secondly, it
indicates th a t there are too many extrem a in the W M AP measured CMB, a result not seen
in the white noise. This second effect is not affected by our shifts in the am plitude of the
noise.
55
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 .
statistic
Number of Local E xtrem a max
Number of Local E xtrem a min
M ean max
Mean min
Variance max
Variance min
Skewness max
Skewness min
K urtosis max
K urtosis min
—3a
0.991
1.000
0.782
0.345
0.145
0.291
0.455
0.873
0.373
0.718
—2cr
1.000
1.000
0.509
0.155
0.136
0.273
0.509
0.909
0.355
0.682
—l a
0.982
1.000
0.255
0.064
0.127
0.191
0.464
0.864
0.382
0.727
0<7
0.991
1.000
0.109
0.009
0.091
0.227
0.473
0.927
0.464
0.709
la
0.991
1.000
0.027
0.000
0.055
0.245
0.373
0.900
0.409
0.673
2a
0.982
1.000
0.000
0.000
0.127
0.209
0.455
0.827
0.464
0.618
3a
0.982
1.000
0.000
0.000
0.091
0 .2 0 0
0.409
0.882
0.355
0.691
Table 4.4: Number of standard deviations by which th e am plitude of the correlated noise
was shifted. Hinshaw et al. [2] cite the error in th e noise am plitude to be 0.06%, so —3a
corresponds to multiplying the am plitude by exactly 0.9982, and 3a corresponds to mul­
tiplying by exactly 1.0018, etc. This m ultiplication is carried out after the proper scaling
of the correlated noise, discussed in section 4.4.2. The d a ta for the first two statistics is
presented graphically in figure 4.4.
Mean
N u m b e r of Local E xtrem a
Figure 4.4: This figure shows the cumulative distribution functions for the statistics gen­
erated by shifting the correlated noise am plitude by n a , where n ranges from -3 to 3. 110
correlated noise maps are used. The grey statistics (and do tted lines) are for maxima, the
black (and solid lines) for minima. From top to bottom , th e numbers are values of p for
n = 3 to n = —3. The vertical lines are the W M AP values. Note: Only the Mean statistic
(plot on the right) shows the CDFs spread apart from each other: the other 4 statistics have
the CDFs on top of each other, as in the left plot. Also, the detection of non-G aussianity in
the mean is weaker when you consider a —l a shift in the correlated noise am plitude. The
CDFs would become sm oother with more than 110 simulations.
56
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 .
4.5
S m o o th in g O ne- and T w o -P o in t A n a ly sis
The process of simulation and d a ta reduction is very similar to the previous one, except
th a t we sm ooth instead of changing resolution. The sm oothing increases the signal to noise
ratio and therefore reduces our sensitivity to the noise model. Also, we are free to choose
the smoothing scale w ithout being tied to the discrete pixel sizes of the HEALPix scheme.
We again describe the simulation and d a ta reduction process.
Our simulation process has only two steps: creating a CMB sky and adding white noise.
Again, this is an acceptable approxim ation of the W M A P d a ta if our final statistics only
depend on the accurate characteristics of this approximation. To assure this in our d ata
reduction, we again ignore the monopole and dipole moments, as well as the region contam ­
inated by galactic foregrounds.
To mask the sky and check for large-scale anisotropies in the statistics, we extend th e kpO
mask to different hemispheres, as in LW04. This yields four other G alactic masks: G alactic
N orth and South masks (GN, GS) and Ecliptic N orth and South masks (EN, ES). See figure
4.6. W hen masking a CMB map, we set the tem perature fluctuations inside the m ask to
zero to assure th a t later smoothing does not allow contam ination in the masked region to
leak out. To remove dependence on the monopole and dipole, we also remove these moments
outside of the galactic mask we use.
To remove dependence on the small scale structure of the noise, we sm ooth the sky, with
either a 50 arcm inute or 3 degree Full W idth at Half Maximum (FWHM) beam. To choose
the sm oothing scale, we arbitrarily decided to suppress the power by a factor of 10 at the
multipole £ value where the signal to noise has dropped to a ratio of 1. The signal to noise is
1 at about £ = 350, so we choose a Gaussian smoothing FW HM scale of 50 arcminutes. As
seen in figure 4.5, this suppresses the CMB power spectrum by a factor very close to 10 at
£ = 350. We also check our results with 3 degree smoothing scale, where the noise is entirely
subdom inant.
57
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 .
After the smoothing, we find the m axim a and minima. However, the smoothed tem per­
ature map and therefore these extrem a will be affected by the zeroed pixels inside the mask,
so we want to ignore extrem a th a t are significantly affected by the presence of the mask.
To do this, we create an adjusted mask. We sm ooth th e original mask (kpO, GS, GN, ES,
or EN) with the same FW HM Gaussian beam as we will use to sm ooth the CMB, and we
mask all areas with values less th an 0.9. Recall the convention th a t unmasked pixels have a
value of 1 and masked pixels have a value of 0. W hen we ignore extrem a inside this adjusted
mask, we ignore most of the extrem a which have been significantly affected by being close
to a region of zeroed pixels. O ur value of 0.9 is less conservative th a n Eriksen et al. [49] who
use 0.99. This value affects how strictly we want to ignore mask effects. It does not affect
the significance of our results, because the simulations and th e W M A P d ata are treated
identically.
The process for sim ulating and and reducing th e m aps is as follows:
1. Simulate a map w ith N side = 512, £max = 700 (or £max — 300, for 3 degree FW HM
smoothing), and the W M A P m easured power spectrum .
2. Add in white noise, according to the effective num ber of observations on each pixel.
3. Set the tem perature fluctuation to zero inside a galactic mask.
4. Remove the monopole and dipoles outside of th a t same mask.
5. Smooth with a 50 (or 180) arcm inute FW HM Gaussian beam.
6. Find the local extrema.
7. Discard extrem a inside the adjusted version of the mask in step 2.
8. Calculate statistics on the extrem a for further analysis.
58
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Power s p e c t r a and su p p re s s io n of power by sm oothing
h-
m u ltip o le m o m e n t
I
Figure 4.5: The solid line is the m easured tem perature-tem perature power spectrum from
the first year W M AP data. The dots are the average (over 110 simulations) of the W
band correlated noise power spectrum . Note the ringing, which is due to th e pixelization
interacting w ith the scanning strategy. The dashed line is the unitless suppression of power
(due to a 50 arcm inute FW HM Gaussian sm oothing), which has been rescaled to fit the
height of the graph. Note th a t at I = 350, where signal to noise is about 1, smoothing
suppresses power by a factor of about 10.
59
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 few images of the various stages of this sim ulation process are shown in figure 4.7. We
compare these simulations to the W M A P cleaned tem perature m ap d ata, which goes through
the same process, starting at step 3.
4 .5 .1
O n e-P o in t S ta tis tic s
We performed 4000 simulations of Gaussian CMB skies. A 99% detection would require fewer
th a n 10 of the statistics to be below (or above) th e W M AP statistic. A 95% detection only
requires fewer than 84 of the statistics to be below the W M AP statistic. See th e appendix
for details.
Some results for the one-point statistics are shown in figure 4.8.
There are no 99%
detections, but there are several 95% detections. For the mean statistic, the hot spots do
not seem particularly unusual, b u t th e cold spots are too warm in the G alactic N orth at 50
arcm inute smoothing. We also have 95% detections in the Ecliptic N orth w ith the hotspots
not having enough variance at 50 arcm in sm oothing and the cold spots not having enough
variance at 3 degree smoothing. W ith respect to skewness, the cold spots have too much
negative skewness at bo th smoothings. It is possible th a t this is caused by a single very cold
cold spot, perhaps th a t described by [50]. The hot spots have too little skewness at the 3
degree smoothing.
We calculate 100 one-point statistics, and 7 of them give 95% detections, so one could
argue th a t these detections are not highly significant.
Nevertheless, it is interesting to
see th a t the detections support previous results, such as the lack of power in the Ecliptic
North[35].
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 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 .
kpO
g alac tic so u th
galactic n o rth
ecliptic so u th
ecliptic n o rth
Figure 4.6: These are the 5 masks used for the two-point correlations. Left column is
mollweide projection of the sky; right column is HEALPix base tile 6, where the upper left
corner is northernm ost. Tile 6 is directly opposite the galactic center. The mask is black,
the adjusted mask for 50 arcminute FWHM smoothing includes the mask and the thin grey
region extending the mask.
61
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Figure 4.7: These are images of base tile 6 in the HEALPix scheme at various steps in the
sim ulation process. North is in the upper left. Range of color scale varies between some
images.
62
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 .
Mean, 180 arcmin
Mean, 50 arcmin
o
a.
o
a.
cn
o
0, 3.62 .
0-017
0 . 7.68
z
o
w
)
UJ
oz
0-045
0.477
z
UJ
0.064
0.066
0.070
0.068
0 . 336 .
z
0, 0.76..
0-123
0-447
UJ
0-069
0.062
i/i
LiJ
0.072
0*030
0.035
0 , 3.25
i/i
o
0, 8.78
Q.887
0, 0.18
o
a.
0-419
oz
i/i
o
0, 8.54
0, 01.4
z
0-073
0.0045
0.0050
2 . 12 .
0.Q4Q
in
UJ
0-897
0.0040
.0 ,
z
o
Q.031
0.0035
0.055
0.050
Variance, 180 arcmin
Variance, 50 arcmin
o
a.
0.0030
0.045
0.040
UJ
0.0055
0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040
Skewness, 180 arcmin
Skewness, 50 arcmin
oa.
a
i/i
o
oz
0, 0.84
0, 5.86 .
0.4
-
0.2
0.0
0.2
Q.994
0, 7.8 4 .
z
0-458
-
0 ,. 0.1.8 .
i/i
UJ
0 QQ7
0-315
UJ
0.4
-
1.0
-
0.5
0.0
0.5
1.0
Figure 4.8: These are cumulative distribution functions of selected one-point statistics for
50 and 180 arcm inute smoothing levels. Hot spot (maxima) statistics are in red and are
the upper number in each pair. Cold spot statistics are in black. The vertical lines indicate
the position of the W M AP statistic. Mean statistics are given in m K } variance is in m K 2,
and skewness is dimensionless. The mean and skewness values of th e m inim a have been
multiplied by —1 before creating CDFs, for easier comparison w ith the maxima,
63
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
4 .5 .2
T w o -P o in t S ta tis tic s
C a lc u la tin g T w o -p o in t F u n ction s
For the two-point functions, we perform two general analyses on the extrema. The first is
related to the m ethod used by Heavens and Sheth [51] where they look at the point-point
correlation function of the locations of the m axim a above a certain threshold (and minima
below a threshold). We arbitrarily pick a threshold of 2a, where a is the standard deviation
of all the tem peratures in the map outside the (non-adjusted) mask.
The other analysis is where no threshold is applied to the extrem a and the two-point
statistics of the tem perature field at the locations of the extrem a are calculated.
It has
been proposed th a t this two-point function is very close to th e two-point function on the full
sphere [52],
In both analyses, the statistics we calculate are m otivated by the concept of a correlation
function. We simply find the average number of pairs at a given angular separation, or
the average product of spot tem peratures at some separation. We calculate three statistics
of this form: between m axim a and maxima, between m inim a and minima, and between
m axim a and minima.
Consider the spot-spot statistics between m axim a and minima, for example. We select
all pairs w ith one hot and one cold spot and find the angles between the spots.
These
angles we bin into 1000 equally spaced bins of angular separation between 0 and it radians.
To remove dependence on the number of spots, we normalize the histogram we just m ade
by dividing by the to tal number of counts.
This makes the bins sum to 1.
Eriksen et
al. [53] have already studied the Minkowski functionals on the CMB, which are related to
the number of spots above a threshold, so we did not feel th e need to study it further by
including inform ation about, the number of spots in our d a ta set. The normalized histogram
contains the correlation information, bu t it is slightly dependent on the pixelization and
highly dependent on the geometry of the mask we used.
64
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 .
Suppose we wanted to calculate th e tru e underlying correlation function, independent of
the geometry of the mask. We do not need to do this in our statistical analysis (and in fact
we do not), since we have exactly th e same geometric masking effects included in both the
W M AP d a ta and simulations. Nonetheless, if we w anted to determ ine the mask-independent
correlation function, we would need to know the effects of the mask. For this purpose, we
bin the angles between pairs in a random distribution of “m axim a” and “m inim a” for each
CMB simulation. The number of random ly placed extrem a is determ ined by the number
of extrem a found in th a t simulation.
The underlying correlation function is the excess
probability of finding pairs at a given angle. To obtain this for each angular bin, we divide
the normalized number of pairs from th e CMB sim ulation by the average normalized num ber
of random pairs in th a t bin, and sub tract 1. This gives us a mask independent correlation
function for each iteration, which we can use to visually examine our results.
C alculating the tem perature-tem perature tw o-point statistics is very similar to calculat­
ing the spot-spot statistics. Instead of counting th e number of angles th a t fall into a given
bin, we find the average product of tem peratures for pairs of spots in th a t angular bin. This
can also be turned into a correlation function, if only for visual examination.
R e d u c in g T w o -p o in t F u n ction s to a S in g le N u m b er
Now we m ust reduce a 1000 dimensional discretized two-point statistic £(6) into a single
statistic, a single real number. We tre a t £ as a 1000 dimensional vector. First, we reduce th e
dimension by ignoring some of the data. We do this in two ways: by re-binning the vector
into 40 bins, and by ignoring all b u t the first 40 of the 1000 bins. After this, we calculate
a x 2 value for the lower dimensional £ vector, based on the covariance C of those vectors.
We define y 2 =
Since we m ust be sure to tre a t the W M A P two-point vector in th e
same way as the simulation vectors, and we do not want to include it in the calculation of
the covariance matrix, we must use some sim ulation vectors to define the covariance m atrix
and calculate our x 2 statistics with the rest. We calculate the covariance m atrix C w ith the
65
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 .
cold-cold,
temp-temp,
kpO mask
hot-hot, point-point, kpO mask
5
0 . 007
0. 0065
1
0 . 006
im
0 . 0055
£
0.005
5
0.0045
0
0 . 004
0 . 0035
0
n
7T
rad ian s
i/n = 0.0007
71
i/n = 0.6452
1
.
8
.
6
6
.4
.4
2
.2
0
7T
rad ian s
1
.
3
.
20
40
X
60
80
100
statistic
120
8
0
50
X
100
150
statistic
200
Figure 4.9: Upper left: tem perature-tem perature correlation, minima, full sky. Upper right:
spot-spot correlation, maxima, full sky. The vertical axis is the excess fractional probability
density, for finding a pair of points at a given angular separation. In th e correlation functions,
the black line is the W M AP data, the white lines are the median simulation values, and the
grey band is a 2er error band calculated from the simulations. Lower: Cumulative distribution
functions for y 2 statistics. They correspond to the upper plots.
first 1000 vectors £ and then find where the W M AP d a ta lies in the distribution of y 2 values
of the rest of the £ vectors. We also visually check th a t the distribution of y 2 values from
the vectors used to make the covariance is not excessively different from th a t of the rest of
the vectors. This verifies th a t we are using enough vectors to define C.
66
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 .
kpO
kpO
kpO
GS
GS
GS
GN
GN
GN
ES
ES
ES
EN
EN
EN
min
max
cross
min
max
cross
min
max
cross
min
max
cross
min
max
cross
7.2 degree
p t-pt
tem p-tem p
0.14133
0.69100
0.09400
0.01000
0.41833
0.18700
0.03567
0.85800
0.14633
0.10367
0.90200
0.59900
0.87400
0.28033
0.30600
0.21100
0.10233
0.06967
0.42667
0.77000
0.16633
0.11500
0.36467
0.38833
0.21833
0.20033
0.03100
0.07367
0.23700
0.18700
180 degree
p t-p t
tem p-tem p
0.05033
0.00000
0.38933
0.11567
0.01633
0.02067
0.87500
0.16300
0.11533
0.43467
0.89000
0.80567
0.07767
0.31033
0.49200
0.72667
0.22933
0.83233
0.73033
0.51933
0.33967
0.65267
0.54033
0.51900
0.19933
0.20533
0.23033
0.14067
0.04800
0.12700
Table 4.5: These are our estim ates p of the position of the W M A P two-point statistic among
the simulated statistics. These results are for 50 arcm inute FW HM smoothing, where 1000
iterations went to create the covariance m atrix and position of the W M AP statistic is found
among the remaining 3000. The different rows show results for different masks, as well as
the minima-minima, maxima-maxima, and m inim a-m axim a statistics. The columns show
the results for the nearest 7.2 degrees (first 40 bins) of our two-point statistics, as well as
for the full-sky 180 degree two-point statistics. Columns also show the different results for
the spot-spot and tem perature-tem perature statistics. Note the 0 value in the upper right
corner.
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kpO
kpO
kpO
GS
GS
GS
GN
GN
GN
ES
ES
ES
EN
EN
EN
min
max
cross
min
max
cross
min
max
cross
min
max
cross
min
max
cross
7.2 degree
tem p-tem p
pt-pt
0.75833
0.69500
0.26333
0.16533
0.95433
0.69967
0.40567
0.80267
0.98967
0.88833
0.00067
0.32900
0.34700
0.11933
0.59533
0.80900
0.65267
0.73033
0.81800
0.10300
0.62600
0.54967
0.82933
0.82333
0.67800
0.20933
0.75033
0.73200
0.04600
0.69700
180 degree
tem p-tem p
pt-pt
0.83800
0.18100
0.62600
0.27000
0.83500
0.14833
0.42267
0.78467
0.93700
0.52133
0.66933
0.75400
0.03267
0.08233
0.17967
0.60667
0.26500
0.80100
0.51967
0.84533
0.63733
0.84300
0.45500
0.71167
0.41333
0.17833
0.36067
0.33567
0.12133
0.17533
Table 4.6: This is the same d a ta as figure 4.5, except for 180 arcm inute FW HM smoothing.
4 .5 .3
T w o -P o in t R e su lts
. O ur results here are for 4000 simulations. The first thousand go to define th e covariance
m atrix, so the W M AP statistic is compared to the statistics for the remaining 3000. A 95%
detection requires 0 < p < 0.02 or 0.98 < p < 1, and a 99% detection requires 0 < p < 0.002
or 0.998 < p < 1, where p = i / n is the same as in our previous paper and is also defined in
the appendix.
The most interesting result is in the tem perature-tem perature correlation function for
the kpO masked full-sky correlation function. For 3000 iterations, the W M A P statistic fell
lower than all of them. To b etter determ ine the significance of this result, we ran a set
of 20,000 simulations for this particular mask. Again, the first 1000 went to determ ine the
covariance m atrix. Of the remaining 19,000 simulations, 13 of their statistics fell lower th an
the W M AP statistic. This is extremely close to a 3<r detection, it comes to be 2.989(7.
One interesting point about this result is th a t our \ 2 statistic is too low. This means th a t
instead of fitting the distribution of two-point functions too poorly, it instead fits them too
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well. In fact, it fits the covariance m atrix describing the two-point functions b etter th an the
two-point functions th a t went into m aking th a t covariance m atrix. This is a very unusual
result.
It is also curious th a t we only see this for the kpO mask and not for the masks in which
we cut out half of the sky. This suggests th a t our effect is different from those th a t led to
recent claims of anisotropy in the CMB.
To check to see if this result is an effect of foregrounds, we repeat our simulations for 3000
iterations in the V-band, ju st for the kpO mask. We take 1000 simulations for the covariance
m atrix, and find the position of the W M AP statistic among the remaining 2000. We find
th a t the min-min tem p-tem p W M A P statistic on the full sky then falls just above 5% of
the simulation statistics. However, we find th a t if instead of the min-min statistics we look
at the min-max (cross), then we find th a t it now sits lower th a n all 2000 of the simulation
statistics.
4.6
C on clu sion
We initially find the W M A P d a ta to have maxima th a t are significantly colder and minim a
th a t are significantly warmer th an predicted by Monte Carlo simulation. For almost all
simulations, we have 95% confidence th a t the mean of the W M A P hot spots or cold spots is
in a 5% tail of the Monte Carlo distribution. In one case, we are 99% confident the m axim a
statistic is in a 1% tail. Since we find the same lack of extreme tem perature when we use
the directly measured W M A P power spectrum, we are not simply restating th a t the W M AP
power spectrum has a lack of power at large angular scales. The effect is independent of the
G alactic mask or power spectrum used. However, it does tu rn out to be dependent on the
noise model.
We also find some anisotropy between the ecliptic north and south hemispheres. The
W M AP d ata in northern hemisphere have a low variance statistic (95% confident th a t the
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variance statistic is in a 5% tail). In one case, we are 99% confident the variance of the
m axim a is in a 1% tail. There is less asym m etry between the north and south Galactic
hemispheres.
O ur results may not be a detection of prim ordial non-Gaussianity.
They could still
be an effect of the W M A P instrum ent or d a ta pipeline not modeled in our simulations or
an as yet undiscovered foreground. O ur result is still highly significant. We have detected
something, whether it is prim ordial non-G aussianity or some other effect in the data. Having
anomalous mean tem perature values for the m axim a and minima in b o th the north and south
ecliptic hemispheres is unlikely to occur if the W M A P d ata were consistent w ith theoretical
expectations.
We will present a complete treatm en t of the one- and two-point extrem a
statistics for the W M AP d a ta set in a future publication.
We use and advocate a robust statistical test th a t reduces the probability of a false
detection of non-G aussianity to a level com m ensurate with the significance of the detection.
A Mathematica notebook and small user-friendly C program are available for determ ining
the significance in this robust hypothesis test where a m easurement is compared to MonteCarlo simulations. The m ethod and C code are described in appendix B. The code was
previously available at the web page h ttp ://c o sm o s.a stro .u iu c .e d u /~ d la rso n l/fa c ts/, and it
can be made available upon request.
While the W M AP hot spots are too cold com pared to the white noise simulations, this is
no longer as dram atically true when we use the correlated noise simulations. The detection
drops to below 2a. Instead, we find th a t there are now too m any hot and cold spots in the
W M AP data. We cannot give this qualitative statem ent a significance at or above 2a because
we only have 110 correlated noise simulations. We also find th a t the variance continues to
be qualitatively low.
While the nature of our result is sensitive to changes in th e noise model, the significance
is not. Allowing the am plitude of the white noise to vary by two stan d ard deviations puts
the CDF squarely around the W M A P measurement. This means our result did not go away;
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we may have error either in the am plitude of the noise or from statistical fluctuations in the
CMB, but we still need to move something by two stan d ard deviations to make them match.
W hen we switch to the properly correlated noise samples, regardless of their amplitude
(within three standard deviations) we definitely get too many cold spots, and probably too
many hot spots. This was unexpected, since the w hite noise did not show an unusual number
of hot or cold spots.
Our main result comes from our investigation of the two-point statistics. We find an
anomaly in the full-sky m inim a-m inim a tem perature-tem perature two-point function using
the kpO mask. This is a very large fluctuation, unlikely at the 3 sigma level. We observe
this anomaly only on the full sky. This suggests th a t our effect is distinct from those th a t
led to recent claims of anisotropy in th e CMB.
In addition to this 3-sigma result, we also have several 2-sigma results. For example, we
see low variance of the hot spots at 50 arcm inute smoothing, the high skewness of the cold
spots at both 50 and 180 arcm inute smoothings, and a low “x 2” statistic for th e point-point
function between m axim a and m inim a for the 180 arcm inute sm oothing (a 2.7 sigma result).
We have dem onstrated th a t there is a statistically highly significant difference between
the W M AP d ata and our Gaussian M onte-Carlo simulations. This can be interpreted in one
of three ways: it is just a large fluctuation, it is caused by non-Gaussianity, or it caused by
some other unknown foreground or systematic effect th a t we do not consider in our model
of the W M AP data.
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C hapter 5
G ibb s S am p lin g
A typical CMB experiment has a detector th a t scans across th e sky and measures the tem ­
perature of the CMB in each direction th a t it points. The scientists who run the experiment
are then faced with the problem of turning this stream of tim e-ordered d a ta into knowledge
about the early universe, such as its curvature, or m atter or baryon density.
In the past this has been done by converting th e tim e-ordered d ata into a map of the
sky, while propagating the error in the time dom ain to error on each pixel.
Then the
power spectrum of the CMB was estim ated from this map. Finally, a Markov chain is run
which tries random values of cosmological param eters, predicts the CMB power spectrum
from those values, and maps out w hat range of param eters gives reasonable m atches to the
current power spectrum. In each step of the process, the error bars are an approxim ation
of those in a previous step, so while error is propagated p retty well, it is not propagated
perfectly.
A Bayesian approach has the potential to propagate the error perfectly from the timeordered data domain through to the cosmological param eters.
This is discussed in [54],
However, this goal has not yet been implemented, because it is a large task. Instead, we
have concentrated on making one of these steps Bayesian: finding the power spectrum from
a m ap of the sky. Even this step has turned up some some questionable propagation of
errors in previous work. My work in this project has been to help implement a version of
the Gibbs sampler and to extend it from tem perature-only to polarization.
This chapter gives an overview of w hat Gibbs sampling is, then it discusses each sampling
1T his chapter contains m aterial which has been previously published in [54, 55],
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step in detail, mentions a m ethod for using the signal samples to get a continuous distribution
for the power spectrum , and concludes w ith some results.
The m athem atics are w ritten out for polarization, where th e integer p is 3: for the T, E,
and B modes of the CMB. W hen using tem perature only, set p = 1, and use this specialized
case of the formulas. (This modification will come into play in th e power spectrum sampling
step.)
5.1
In tro d u ctio n
The observation and analysis of cosmic microwave background (CMB) anisotropies have a t­
tracted a great deal of attention in recent years due to their unique relevance for cosmological
theory (see [56] for a recent review). A slew of observational results have been published
[57, 58, 59, 60, 61, 62, 63]. These were obtained from maps of the microwave sky at ever
increasing sensitivity and resolution. Since the recent release of th e first year W M AP d ata,
an all-sky microwave survey has been available down to angular scales of 12 minutes of arc
[9]. By the end of the decade the Planck satellite is expected to generate 1 Terabyte of high
resolution, high sensitivity all-sky data.
The basic assum ption is th a t the CMB anisotropy signal and the instrum ental noise are
Gaussian and th a t the signal statistics are isotropic on the sky. Contact between theory
and observation is then best made by extracting the angular power spectrum Ct from the
data[64, 65, 66]. M ethods for efficiently estim ating the power spectrum have been investi­
gated since the com putational unfeasiblity of using the brute-force approach was realized
[67, 68, 69]. This effort has yielded two classes of methods: exact methods, applicable only
to two narrowly defined classes of observational strategies [70, 71], and approximate bu t
more broadly applicable m ethods [72, 73, 74]2.
We will describe here a solution to the problem of inference from microwave background
2An exception to th is classification is a hybrid m ethod which has been proposed very recently and which
com bines a m axim um likelihood approach on large scales w ith an approxim ate approach on sm all scales [75],
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d a ta which combines the advantages of exact m ethods with the practicality of the approx­
im ate m ethods. The com putational cost of our m ethod scales like the best approximate
m ethod for the same experiment, albeit w ith a larger pre-factor. Power spectrum estim ates
and any desired characterization of the (m ultivariate) statistical uncertainty in the esti­
m ates can be com puted free from any approxim ations in the estim ator which could lead to
sub-optim ality or biases.
The solution we propose is to sample the power spectrum (as well as other desired
quantities, such as the underlying CMB signal, foregrounds or the noise properties of the
instrum ent) directly from the joint likelihood (or posterior) density given the data. We can
efficiently sample from this multi-million dimensional density using the Gibbs sampler. This
approach obviates the need to evaluate th e likelihood or its derivatives in order to analyze
CMB data.
O ur approach shares certain algorithmic features with the approach independently dis­
covered in [76] which describes a maximum likelihood estim ator of the power spectrum using
Bayesian Monte Carlo methods. However, our goal from the outset was to design a m ethod
th a t allows a full exploration of the m ultivariate probability density of the power spectrum
and the param eter estimates, given the data.
Our m ethod seamlessly integrates w ith param eter estim ation w ithout recourse to semianalytic Gaussian, offset log-normal [77], x 2 [78] or hybrid [79] approxim ation schemes. If
desired, theoretical priors can be applied in th e analysis by restricting the space of power
spectra to those which arise from a physical model of the CMB anisotropy.
By design, the sample of power spectra and reconstructed sky maps will reflect th e
statistical uncertainty given the d a ta through the full non-Gaussian statistical dependence
structure of the Ce estimates. This inform ation can be propagated losslessly to the cosmo­
logical param eter estimates.
One aspect of our method which is of general interest in astrophysics beyond CMB
analysis is th a t it generalizes the results on globally optimal interpolation, filtering and
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reconstruction of noisy and censored d a ta sets in [80] to self-consistently include inference of
the signal covariance structure. This defines a generalized W iener filter th a t does not need
a priori specification of the signal covariance. A byproduct of our m ethod is a prescription
for “unbiasing” the W iener filter which clearly reveals the tight relation between W iener
filtering and power spectrum estim ation.
Our m ethods differ from traditional m ethods of CMB analysis in a fundam ental aspect.
Traditional m ethods consider the analysis task as a set of steps, each of which arrives at
interm ediate outputs which are then fed as inputs to the next step in the pipeline. Our
approach is a truly global analysis, in the sense th a t the statistics of all the science products
are com puted jointly, respecting and exploiting the full statistical dependence structure
between the various components.
In summary, our m ethod is a M onte Carlo technique which samples power spectra and
other science products from their exact, m ultivariate a posteriori probability density, and
which does so w ithout explicitly evaluating it.
The result is a detailed characterization
of the statistics of the CMB signal on the sky, reconstructed foregrounds, the CMB power
spectrum , and the cosmological param eters inferred from it w ith a cost which is proportional
to the cost of a least squares m ap-making algorithm for the same set of observations.
This chapter is starts w ith a discussion of our model of a generic CMB experim ent and
introduces our notation and the Gibbs sampler. Then we discuss the two prim ary sampling
steps in sections 5.3 and 5.4. We go on to investigate the Blackwell-Rao estim ator and how
to use it for param eter estim ation from the Gibbs samples in section 5.5. After a discussion
of com putational considerations in section 5.6, we go on to discuss our analysis of the COBE
CMB experiment in section 5.7, and two different polarized CMB analyses in section 5.8.
We conclude with an analysis of the W M AP d a ta with a full noise covariance m atrix in
section 5.9.
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5.2
M o d el and N o ta tio n
We begin by defining our model of CMB observations and introduce our notation.
We
imagine th a t the actual CMB sky s is observed w ith some optical system and according to
some observing strategy encoded in a pointing m atrix A, which maps the signal on the sky
into a collection of n 0 tim e-ordered observations of the sky. This results in the “raw” d a ta
d, represented by a vector w ith n 0 elements (an n 0-vector). Our model of this process is
encoded in the model equation
d = A (s + f) + n tod,
(5.1)
where n tod is a realization of Gaussian instrum ental noise added to the d a ta and f = JT f*
is the sum of a collection of foregrounds (assumed spatially varying and constant in tim e).
We represent maps on the sky w ith APjX resolution elements (pixels) as jVpix-vectors. Note
th a t while we do not explicitly consider m ulti-channel d ata, the model is easily generalized
to th a t case by adding a frequency index to d, A, n tod and f.
The “m ap” vector m is the least squares estim ate of th e signal s + f. from d. Because
we assume Gaussian noise with zero mean, this is also the maximum likelihood estim ate (or
maximum a posteriori estim ate assuming a flat prior). It can be found as the solution of the
norm al equation
= ArN - >
Here
N tod
the m atrix N todiis the covariance
(5.2)
m atrix of the noise in the tim e ordered d a ta space
= (ntodn todT).Then m = s + f + n where n describes the residual
noise on th e m ap
estim ate w ith covariance m atrix N = (nnT) = (A TN ^ A ) ”1.
The cosmological model specifies the signal covariance m atrix S. For isotropic theories
S is diagonal in the spherical harmonic basis, w ith the special form Seme>m' = C^5w5nimt
where
is a scalar when dealing with only tem perature, and a 3 x 3 m atrix when dealing
with polarization. This is discussed further in section 5.4.
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In keeping w ith the m ajority of the literatu re in th e field, we restrict our discussion to
theories which predict a Gaussian CMB signal s.
For a cleaner exposition of the m ethod, we will ignore th e foregrounds f. We are trying
to explore the a posteriori density
P ( S |m ,N )
(5.3)
since S contains the inform ation about th e power spectrum , and its probability density
contains all the inform ation about its error bars.
Traditionally, the approach to exploring th e a posteriori density has been to define an
estim ator, such as the least squares quadratic (LSQ) estim ator [65] or the maximum likeli­
hood (ML) estim ator [66]. Then some m easure of uncertainty in the values of this estim ator
was defined, for instance by approxim ating the shape of P (S |m ) around the maximum by a
m ultivariate Gaussian and evaluating elements of th e curvature m atrix at the extremum.
Evaluating the LSQ or ML estim ators is a very costly operation, taking 0 ( N pix) opera­
tions3. In general, evaluating the curvature m atrix is even more costly because it has 0 ( N p-lx)
elements each of which requires 0 ( N ^ ix) operations, making the overall operation count of
order 0 ( N pix). In addition a Gaussian approxim ation fails at low £ where the small num ber
of degrees of freedom makes the posterior significantly non-Gaussian, and also at high £ in
the regime of small signal-to-noise ( S / N < 1).
Instead, we propose to sample the signal covariance S from the posterior directly. There
is no known way to directly sample from equation 5.3, but if a way can be found to sample
s and S from the joint distribution P ( S ,s |m N ) then the S taken by themselves are exact
samples from the marginalized distribution.
At first, sampling from the joint distribution seems even less feasible.
But powerful
theorem s can be proved [82] th a t show th a t if it is possible to sample from the conditional
distributions P ( s |S ,m , N ) and P (S |s , m , N ) oc P (S |s ) then one can sample from the joint
3See however [81] for fast num erical techniques th a t were applied successfully to weak lensing d a ta .
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distribution in an iterative fashion.
Begin with some startin g guess S°. Then iterate the following equations
sm <- P ( s |m ,S i ,N )
(5.4)
Si+1 <- P ( S |s m , m, N ) = P ( S |s i+1)
(5.5)
and after some “burn-in” the ( S \ sl) converge to being samples from th e joint distribution
P ( S ,s |m N ).
This technique of sampling from th e joint distribution is called the Gibbs
sampler.
Note th a t if a continuous distribution for P (S |m , N ) is desired, as opposed to a set of
individual samples, one may take advantage of the known analytical form of the distribution
P (S|s) by applying the Blackwell-Rao estim ator. This procedure was discussed by [54] and
[83] for the tem perature only case. The required modifications for polarization are w ritten
out in Section 5.5.
5.3
Signal S am p lin g
5.3.1
O v erv iew
The signal sampling step of the Gibbs sampler requires us to sample a value of s from
equation 5.4. Since the signal and noise are constrained by m = s + n, sampling the signal
also requires us to sample a noise realization. The log probability density for this distribution
is
—2 log P (s|m , S, N ) = (m —s)TN _1(m —s) + sTS _1s + const.
(5.6)
This equation is clearly quadratic in s, so the density for s is a m ultivariate Gaussian. By
either completing the square or taking two derivatives with respect to elements of s, one
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finds
- 2 log P ( s |m , S, N ) = (s - [S-1 + N - 1] - 1N " 1m ) T
x [S_1 + 1NG1]
(5.7)
x (s - [S” 1 + N ^ ^ N ^ m ) + const,
This shows th a t s
is Gaussian distributed, w ith a m ean of [S-1 + N _1]_1N _1m and
a
covariance of [S-1 + N -1]-1 .
The sky signal s is sampled by solving for th e m ean m ap x and a fluctuation map y w ith
m ean zero and the proper covariance. Then s = x + y. The following equations satisfy this
requirement.
[ l + S 1/2N _1S 1/2] S “ 1/2 x
=
5 1/2N _1m ,
(5.8)
[l + S 1/2N _1S 1/2] S~1/2y
=
£ + S 1/2N ~ 1/2y,
(5.9)
where £ and y are random maps containing Gaussian unit variates (zero mean and unit
variance) in each pixel for each of the / , Q, and U components. One additional requirem ent
is th a t S 1/ 2 and S -1/2 must be symmetric square roots for these equations to be valid. On
the other hand, N -1/2 only has to satisfy N _1/2(N _1//2)T = N -1 , and may be chosen to be
the Cholesky decomposition.
There remains one subtlety with the above equations because a spherical harmonic tra n s­
form is not unitary, but rather only proportional to a u nitary m atrix. This is discussed in
appendix C.
5 .3 .2
P r e c o n d itio n e d C o n ju g a te G ra d ien t T ech n iq u e
Equations 5.8 and 5.9 can be w ritten differently, bu t here they have the advantage of being
in the form D x = b, where D is a symmetric m atrix. Because of the symmetry, the equation
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can be solved by conjugate gradient descent, as discussed, for example, in [84]. The benefits
of this m ethod of solution are discussed in section 5.6.
The efficiency of the CG technique depends critically on the condition number of the
m atrix under consideration. For our case, this is simply the relative signal-to-noise ratio of
any mode in the system (so th a t for scan strategies which observe a p art of the sky w ith
high signal-to-noise and another region of the sky w ith vanishing signal-to-noise will take a
long tim e to converge w ithout preconditioning). As an example, for a fixed pre-conditioner
in the commander code, it takes about 60 iterations to solve for the first-year W M AP data,
about 120 iterations to solve for th e three-year W M AP data, and about 300 iterations to
solve for the Planck 100 GHz data.
This is a particularly serious issue for CMB polarization measurements.
While these
signatures by themselves have a very low signal-to-noise ratio, and therefore should be easy to
determ ine on their own, the corresponding tem perature signal-to-noise ratio is tremendous.
Consequently, if a m ain goal is to estim ate the T E cross-spectrum, by far m ost of the CPU
tim e is spent on tem perature m ap convergence. On the other hand, if all interest lies in E and
B modes, the tem perature d a ta may be disregarded completely (or alternatively conditioned
on by sampling from P{afm, afm |d, ajm)), and convergence is then achieved very rapidly even
for CMBPol type missions.
It is possible to reduce the com putational expense of a CG search significantly by pre­
conditioning. One approach th a t has proved successful so far is to pre-com pute a subset of
the coefficient m atrix in equations 5.8 and 5.9, and multiply b o th sides of the equations by
the inverted sub-m atrix. Thus, by inverting th e most problem atic p arts of the m atrix this
way by hand, the effective condition number is greatly reduced, and significant speed-up
may be achieved.
Currently, our pre-conditioner is independently constructed for th e tem perature and
polarization states. (See appendix D for full details). For the polarization components, it is
a diagonal m atrix in EE and BB independently, while for the T T correlations, it consists of
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a low-£ m atrix th a t includes all coefficients up to some £majX, and then the diagonal elements
at higher Us [84], For W M A P-type applications, we typically used £max = 50, which requires
52 MB of memory and about 1 m inute of CPU tim e for inversion. For upcoming Planck d a ta
it will clearly be desirable to use a significantly larger pre-conditioner, and more realistic
numbers are £max ~ 150 or 200. This will require extensive parallelization, and has not
yet been implemented in our codes. We therefore still use a serial pre-conditioner up to
f’max = 70 in this paper, and pay the extra cost in CG iterations.
5 .3 .3
R e a listic T r ea tm e n t
In section 5.3.1, the formulas for signal sampling were greatly simplified, for clarity when
describing the algorithm. A more realistic treatm en t will involve multiple channels, sym­
m etric beams, the pixel window function, and a cutoff at some value of £. For reference we
write out the log likelihood for s and the sampling equations th a t one derives with these
additions.
Let the index i run over channels. Let Bj be the beam sm oothing function, and W be
the HEALPix pixel window sm oothing function. If all channels are a t the same resolution,
then there is only one pixel window function; otherwise W will need an i index as well. Let
P be a projection operator th a t removes all modes with £ above some cutoff. Note th a t P,
W , and Bj all commute, and P commutes w ith S. As before, m , are th e maps and s is the
signal. For generality, we also include a foreground com ponent f,:, which is not otherwise
discussed in this paper.
- 21 ogP (s|m i,f 7;,Si, N i,Bj, W ) =
sTS _1s + ^ 2 (mi - B .W s - B lW fl)T P N ~ 1P ( m l - B 8W s - BiWf-)
i
-(-constant
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(5.10)
From the above equation, it is clear th a t W can be absorbed into Bj, so we do this and drop
W from the equations. The equations for sampling s = x + y become:
S~1/2P x = P 5 1/2
P + P S 1/2 ^ ( B iP N “ 1P B j) S 1^ P
B iP N ~ 1P ( m i - Bifi)
(5.11)
i
i
P + P S 1/2 ^ ^ ( B j P N _1P B j) S 1/2P
S “ 1/2P y = P £ + P S 1/2 Y , B i P N r 1/2Xi,
(5-12)
where now we have several m aps Xi °f G aussian unit variates. In this equation, we assume
square roots of S, namely S~G2 and S 1/ 2, are chosen to be symmetric. This must be true
for the equations to hold.
5.4
Pow er S p ectru m S am p lin g
This section describes how to sample the power spectra in equation 5.5.
After a brief
discussion on factoring over £, we take th a t equation and work through the m ath until we
have an algorithm for sampling from it, given in section 5.4.2. The m athem atics are presented
here for polarized power spectra; when working w ith only tem perature, the equations reduce
to the simpler version presented in [54].
5.4.1
D er iv a tio n
It will be very convenient to factor our equations over i. Let us see how this works. We
order the spherical harmonic coefficients a f m as follows:
T
E
B
a o,o> a o,o> a om
T
a l,-l>
B
E
al.-li
F
B
T
E
B
T
E
a l , - l > a l.Cb ° 1. 0> a l,0> a l . l > a l , l J
B
T
E
B
T
B
al,l’
E
(5.13)
B
T
E
B
° 2 , - 2 > a 2 ,-2 > a 2 , - 2 ) a 2 , - l ’ a 2 , - l ) a 2 , - H a 2,0> °2,0> a 2,0’ a 2,l> a 2 , l) a 2 , l ’
T
° '2 , 2 )
E
B
a 2,2> a 2,2)
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The letters T, E, and B stand for the tem perature, electric/gradient, and m agnetic/curl
modes of the polarized CMB, respectively. For the sake of th e p attern , we have ignored the
fact th a t a f m and a f m do not exist for £ < 2, and we have simply set these coefficients to
/
oo
zero. Under this ordering of the afm values, the S m atrix is in block diagonal form:
0
0
0
0
0
Ci
0
0
0
0
0
Ci
0
0
0
0
0
Cl
0
0
0
0
0
C2
\
(5.14)
V
J
where there are 2£ + 1 identical C t m atrices for each value of C and where each
is the
3 x 3 , symmetric, positive definite, covariance matrix:
/
C,
r< T T
/~t T E
rVTE
^<TB
/^<EE
^EB
W
c7B c,
EB
°e
\
(5.15)
c,
BB
This discussion will make use of the W ishart distribution [85]. Because of this, it will be
necessary to use S -1 and
instead of the more custom ary S and C^. Conveniently, the
block diagonal nature of S still allows factoring over £.
The distribution in equation 5.5 can be rew ritten with S"1 and simplified.
P ( S ~ 1|s, m, N ) - P ( S - 1 |s)
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(5.16)
By B ayes’s theorem , we know th a t
P ( S - '|s ) =
p
W s - ^ p i s -')
' 1 p 's ) '-------------------------------------------(5.17)
cc JP (s |S _ I)JP (S “ J)
(5.18)
where the constant of proportionality. P (s). is independent of S h
Now we factor over £ and rewrite as follows:
00
1
/ I
P ( s |S '') P ( S - ‘) - I I
00
\
exp ( “ 2 £
f
1
=
e x p
h
\
sh C ^ ls " “ ) P <C 4 )
(5-19>
\
tia,cjl)
p(c‘")
< 5 -2 0 )
OC
n f f o r l C j - 'J P I C , - 1)
e=o
(5.21)
where tr is the trace operator, and for convenience we have defined erg to be the real valued,
symmetric, m atrix
t
=
^ ^
(5 .2 2 )
m = —£
Each
Sgm
Sgm
is a vector of the ajm, afm, and afm coefficients for the signal sky.The
is a three dimensional, complex valued, column vector. The m atrix
og
quantity
is real because
sgm satisfies
SO., = ( - l ) ’"So_„.
(5.23)
Because our equations factor nicely, it is sufficient to sample each C j l individually when
sampling S -1 . Our problem is reduced to sampling C j 1 from
P { C j l \ag)
(x1C^r 1 12^+1 e x p
^ - I t r ^ C , " 1) P(C~l )
where tr is the trace operator.
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( 5 .2 4 )
However, we m ust be careful about the prior P { C j l ). This is subtle, because a flat prior
on Ce is not the same as a flat prior on C ^ 1. Since the power spectrum is an easier object
to think about th an its inverse, we will set our prior on C ? and then change variables to
convert to a prior P ( C ^ 1). Specifically, we choose our prior to be
P ( C ,) =
(5.25)
1^1
Note th a t q — 0 is a flat prior and q = 1 is the Jeffreys prior. W hen we change variables
between symmetric matrices from Cb to C ^ 1, we find the Jacobian to be
H
< 5 ' 2 6 )
from equation 1.3.13 of [85], yielding
P { C J 1) = |C 7 T “p“ 1
(5.27)
We now consider the W ishart distribution. From definition 3.2.1 of [85], “A p x p random
positive definite m atrix [D] is said to have a W ishart distribution with param eters p, n, and
£ . . . , if its p.d.f. is given by”
2?npFp ( ~ n ) | S | tn \
2 ) 1 1 J
|D |5(n_P_1) exp ( —- tr S _1D )
11
V2
(5.28)
where n > p, and where both £ and D are positive definite.
Our distribution is the W ishart distribution with
n
= 2£ — p + 2q
(5.29)
£
= a '1
(5.30)
D
= C -1
(5.31)
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where factors like Tp ( |n ) , being independent of D (or CjT1), are p art of th e proportionality
constant— they make sure the probability density is normalized. The only thing remaining
is to determ ine how to sample from the W ishart distribution.
5 .4 .2
A lg o r ith m for S a m p lin g F rom th e W ish a rt D is tr ib u tio n
One samples from the W ishart distribution as described in theorem 3.2.2 of [85]: Let the
columns of a m atrix U be drawn from a m ultidimensional Gaussian w ith covariance S = o f 1.
Let U have n = 21 — p + 2q columns. Then U U T is a value of D =
desired W ishart distribution
sampled from the
We use (U U T)_1 as our sample of Ce, noting th a t
changing variables after sampling does not require a Jacobian.
This concludes construction of our algorithm for sampling polarized powerspectra. Next
we will discuss a few issues th a t arise, and the benefit of E-B decoupling.
5 .4 .3
L ow
£
issu es
There is a caveat for 1 = 2. The W ishart distribution, is only defined if n > p; if not, the
sampled m atrix is singular. This is a problem for £ = 2 and a flat prior, since we would only
sample one vector to form a 3 x 3 m atrix. Thus, the algorithm breaks down for this particular
case. Fortunately, this is not a m ajor problem in practice. Three straightforw ard solutions
are: 1) sampling the 2 x 2 T E block and the B block of the m atrix separately, assuming no
TB or EB correlations; 2) using a Jeffreys prior (q = 1); or 3) binning the quadrupole and
octopole together. Note th a t all other multipoles may be sampled individually by the above
algorithm w ithout modifications.
In the limiting case, when n = p, we have found the spectra to have extremely large
error bars. If this is not desired, then one can again sample a smaller m atrix, or bin the
quadrupole and octopole together.
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5 .4 .4
B in n in g
As discussed by [86], it is highly desirable for the Gibbs sam pler to be able to bin several
power spectrum multipoles together.
The m ain advantage of this is improved sampling
efficiency: As currently implemented, th e step size taken between two consecutive Gibbs
samples is given by cosmic variance alone. T he full posterior, however, is given by both
cosmic variance and noise. Therefore, in th e low signal-to-noise regime, one m ust take a
larger num ber of steps to obtain two independent samples. The easiest way of improving on
this is simply to bin many multipoles together, and thereby increase th e signal-to-noise ratio
of the power spectrum coefficient. In practice, we choose bins such th a t the signal-to-noise
ratio is always larger th an some limit, say, 3.
Since the CMB power spectrum is roughly proportional to l/ £( £-\ -1), it is convenient to
define uniform bins in Ct£{£ + 1). We therefore redefine ay for a bin b = {tAm, • • •, Gax} as
i
at =
^
+ 1)S^ SL -
(5-32)
rn = —l
Note th a t there are now
M = '$2 2 £ + l =
£eb
(A n ax + l ) 2 ~ 4 i n
(5-33)
independent spherical harm onic modes contributing to this power spectrum coefficient.
Thus, the W ishart distribution has n = M — p — I + 2q degrees of freedom rath er th an
n = 2£ — p + 2q. W ith this modification, the basic sampling algorithm remains unchanged,
but since we have sampled C b = £{£+l)Qb and not Ce, the actual power spectrum coefficients
are given by Cf = C b/£(£ + 1) for each £ in bin b.
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5.5
P a ra m eter E stim a tio n and th e B lack w ell-R ao
E stim a to r
Currently power spectrum estim ation algorithm s rely on approxim ate representations of the
posterior density P ( C b |d ) 4, for example in term s of m ultivariate Gaussian, shifted log-normal
or hybrid representations. These approxim ations have to be fitted to sets of M onte Carlo
simulations [79]. Since they take simple analytical forms they can only be expected to be
accurate near the peak of the posterior density.
In order to faithfully propagate all the
information in the Cb estim ates through to th e param eter estim ation step, efficient ways
m ust be found to accurately represent and comm unicate P ( C b |d ) .
The Bayesian estim ation technique described in this paper provides a n atural answer
to this problem. The m ethod generates a set of samples from P ( C b |d ) which can simply
be published electronically. Meaningful summaries of th e properties of P (C b ( d ) can all be
calculated arbitrarily exactly, given a sufficient number of samples.
The disadvantage of using this sample set for param eter estim ation is th a t it does notlend itself easily to com puting a numerical probability density for a theoretical Cb power
spectrum com puted from a set of cosmological param eters 6.
However, a fortunate circumstance solves the problem of finding an arbitrarily exact
numerical representation of P ( C b |d ) . At each iteration of the Gibbs sampler the Cb are
drawn from P (C b |s ) which is in fact P (C b |o y ) We can therefore write
P ( C b |d )
=
/ d s P (Q ,s |d )=
/ d s P ( C b |s ) P ( s |d )
■ ^ g ib b s
daeP{Ci\ae)P{crl \d)
(5.34)
gibbs
The sum (where the index runs over fVgjbbs Gibbs samples) becomes an arbitrarily exact
4We w rite P (C b |d ) as sh o rth an d for th e m ultivariate p o sterio r density, a function of {Cb
b • • • >^max}-
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:
i
approxim ation to the integral as the num ber of samples increases. It is called th e BlackwellRao estim ator for the density and can be shown to be superior to binned representations.
This sum yields a numerical representation of the posterior density of th e power spectrum
given the signal samples. All the inform ation about P (C b |d ) is contained in th e a\, which
generate a d a ta set of size 0 ( £ maxN gfo\yS).
The polarized Blackwell-Rao estim ator can be w ritten out explicitly, now w ith th e prod­
uct over i which was previously only implied:
(5.35)
Gi bbs
(5.36)
2t+l+2q
where j is the index over Gibbs samples.
An intuitive understanding of the Blackwell-Rao estim ator may be found in term s of the
usual Gibbs sampling algorithm. W ithin the theory of Gibbs sampling (or more generally
Markov Chain Monte Carlo), it is perfectly valid to sample one param eter more often than
others, so long as the sampling scheme is independent of the current “sta te ” of th e Markov
chain. In particular, one may choose to sample the Cb values a thousand times for each
tim e one samples s, and thereby obtain more power spectrum samples (although not sky
signal samples) w ith negligible cost. The result is a sm ooth power spectrum histogram . The
Blackwell-Rao estim ator takes this idea to the extreme, and replaces the power spectrum
sampling step by the corresponding analytical distribution.
The result is a highly accu­
rate and sm ooth description of P (S |m , N ) th a t is very useful for the previously m entioned
estim ation of cosmological param eters [54, 83, 86].
It is noteworthy that, in the limit of perfect data, using 5.36 returns the exact posterior
density after only one iteration of the Gibbs sampling algorithm.
In addition to being a faithful representation of P (C b |d ) it is also a com putationally
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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 .
efficient representation. Evaluating the Gaussian or th e shifted log-normal approxim ations
to P ( C ^ d ) takes 0 { ^ ax) operations, while our approach requires only 0 ( £ maxN g^ s) oper­
ations. Note also th a t any moments of P (C b |d ) can be calculated through
^ g ib b s
( C f 'c P ’ ... C ? : r " ) \ „ c M
* ^
7 v gi bbs
£
{crcp" ... C?,"y")|P(C(kj).
(5.37)
j =1
This is a far more efficient representation th an would be afforded by a Monte Carlo sample
of a pseudo-Cb estim ator since each of th e term s on the right hand side can be computed
analytically.
A nother feature of this framework is th a t is possible to include cosmological param eter
estim ation in the joint analysis of the d ata. If we assume a class of theoretical models, we can
solve the estim ation problem of power spectrum and cosmological param eters concurrently.
The assumption of such a class of models which am ounts to choosing a prior for the power
spectra which excludes spectra th a t could not possibly be the result from a solution of the
Boltzmann equation for any combination of the param eters about which we wish to make
inferences.
W ith such an assumed class of models the relationship between Cb and the cosmological
param eters 9 is a non-stochastic one, Cb = Cb(#), and P(C b|#) is a delta function. We can
integrate out this delta function in the posterior and th en obtain the conditional density for
sampling the cosmological param eters given the data. This procedure results in removal of
the C t sampling step (equation 5.5) in the Gibbs sampler, and its replacement with:
di+1 < - P(Ce(Q)|sz+1).
(5.38)
Here P (C b(0)|s!+1) differs from 5.24 by only a change of variables, and C i(8) is defined
through cosmological theory. Instead of sampling from the £max power spectrum coefficients
given the
we sample from 9 assuming th a t we just m easured the cp on a perfect signal
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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 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 .
sky (the last draw). In practice, th a t can be achieved by running a Markov Chain using the
M etropolis Hastings algorithm until one independent 6 sample is produced.
If we believe strongly in the theoretical framework, using this prior information is desir­
able: it reduces the num ber of param eters in the problem and therefore improves th e signal
and hence also the foreground reconstruction from th e data. The set of Cb for th e draws
of 6 represents stochastically w hat is known about th e theoretical power spectrum . This
m ethod defines an optim al non-linear filter which returns the best power spectrum and a
characterization of the error while including physical constraints on th e analysis (for example
the smoothness of the Cb which is related to th e n atu ral frequency of oscillations modes in
the prim ordial plasma).
However, just as we are interested in making m aps from the d a ta w ithout inputting
inform ation about the foregrounds and the statistical properties (e.g. isotropy) of th e CMB,
we are also interested in the model independent power spectrum constraints.
5.6
C o m p u ta tio n a l C o n sid era tio n s
The com putationally most demanding p a rt of implementing this m ethod is solving 5.8 and
5.9 at each iteration of the Gibbs sampler. Each of these is a linear system of equations of
the form M v = w, where M = (1 + S ^ N -1 S s). It should be noted th a t these systems are of
the same size as the map-making equation, 5.2. Maps also have to be made for approxim ate
estim ators. Therefore we expect the com putational complexity of the Gibbs sampler to be
no worse th a n the com putational complexity of an approxim ate method.
For large iVpix (Npix > 105) , on the largest supercom puters available at the time of
writing, direct solution of either of these equations becomes infeasible, because neither of
them are sparse. This means the operation count scales as N^-JX and because th e memory
requirem ents for storing the coefficient m atrices scales as iVpix. Therefore large systems of
this type are usually solved using iterative techniques, such as the conjugate gradient (CG)
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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 .
technique [87]. The memory savings can be very large: the com ponents of M do not have
to be stored as long as m atrix vector products M v can be com puted somehow. In term s of
CPU time, iterative techniques outperform direct techniques if either M v can be com puted
in less th an A^ix operations or the num ber of iterations required to converge to a solution of
sufficient accuracy is much less th a n N p\x .
We chose to write 5.8 and 5.9 in a form which satisfies all of theses requirements. The
memory required is of order N p,x since we never need to store the components of the coeffi­
cient m atrix.
The action of any power of S on a vector can be com puted in much less th a n AUix
operations using spherical harm onic transform s (or FFTs in the flat sky approxim ation).
The action of ISU1 = A TN to^A on a vector is generally easier to com pute th an the action
of N on a vector. As long as noise correlations can be modeled in a simple way in the
tim e-dom ain (e.g. as piecewise stationary) the time required for applying N -1 to a vector
is similar to th a t required for a forward sim ulation of the data.
The number of CG iterations until convergence can be reduced far below th e theoretical
maximum N pix if M is nearly proportional to the unit m atrix. This goal can be approached
by finding an approximate inverse for M , a preconditioner.
If N ”1 were diagonal in the spherical harmonic basis, M would be, too. Therefore, as
long as this is approximately true on scales where S
N , a good preconditioner for this
system would be the inverse of the diagonal part of M in the spherical harmonic basis.
These are easy to compute if we approxim ate the diagonal components of N -1 by counting
the number of T O D samples in each pixel and weighting by the current noise tem perature
of the detector. Due to the way 5.8 and 5.9 have been w ritten, the structure of IND1 in the
noise dom inated regime does not m atter, since if S <C N , M ~ 1.
3
This preconditioner can be com puted in G (A p2ix) operations. Figure 5.1 shows the results
of a tim ing study for simulated d a ta sets of varying size with W MAP-like scanning strategy
and uncorrelated noise. The preconditioner performs well. The num ber of iterations does
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5
4
3
CO
X)
2
c
o
o
CD
00
0
10
10
2
1 CT
10
4
c5
Figure 5.1: Com puting tim e averaged over 30 iterations of the G ibbs sampler required for
solving 5.8 and 5.9 as a function of the num ber of pixels in the m ap. These timings are
for a single A thlonX P 1800+ CPU. Solid line: actual timings. Dashed lines show rip for
x G {3, 5/2, 2, 3/2} from the top to the b o tto m on th e right side of th e figure.
not increase with problem size over three orders of m agnitude in _/Vpix and the computing
time is is dom inated by the spherical harm onic transform s.
5.7
A p p lica tio n to th e C O B E d a ta
In order to test our m ethod we applied to the well-studied COBE-DM R tem perature-only
data. The exact maximum likelihood estim ator [88, 89, 66] and th e least square quadratic
estim ator [65] have been com puted for this d a ta set. However, even for this small d ata set,
the marginalized probability densities of each individual C/, or the joint posterior density of
pairs of C/ have not been computed because doing so would require numerical integration
over ~ 20 dimensions. We will show these densities here for the first time.
The COBE-DM R d ata [90] is published in the quadcube d ata structure, at a resolution
which has 6144 pixels on the sphere. We use a noise-weighted average of the 53GHz and
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90GHz maps. Because much of our code was already w ritten for a H EA LPix d a ta structure,
we put the COBE d ata into a H EALPix pixelization at resolution n side = 64 w ith 49152
pixels. HEALPix pixels whose centers lie w ithin th e same quadcube pixel get th e same d ata
(tem perature) value.
Because the noise is completely correlated between sets of HEALPix pixels in th e same
quadcube pixel, the noise covariance m atrix N is block diagonal, where each element of the
block is cr2, the published (noise) variance of th a t quadcube pixel. This m eans th a t N is
not strictly invertible, so we have to use a pseudo-inverse for N ~ 1. Our pseudo-inverse is
also block diagonal, w ith const ant-valued blocks, and correctly inverts the action of A” on a
vector th a t is constant valued on the same blocks as N .
We project out the mean and dipoles from the uncut region of the COBE-DM R map, and
model the d a ta within the custom galactic cut as G aussian random white noise w ith large
variance. This corresponds to claiming complete ignorance of the foregrounds at low galactic
latitudes (within the custom cut) and assuming th a t no residual foregrounds are present at
high latitudes (outside the cut region). This is th e simplest possible way of treatin g the
monopole, dipole, and galactic foregrounds.
O ur noise m atrix has values published by the CO BE team , bu t w ith the cr2 noise variance
increased to 1000 m K 2 in the galactic cut region, a numerically large value th a t exceeds any
other variance in the problem.
For the first iteration of the Gibbs sampler we choose
Ct =
Co = Cj = 10 -30 m K 2
lfC 4
m K 2.
(5.39)
We chose these values because they very roughly approxim ate the true Ce values to reduce
burn-in time. The first two are numerically small, because we consider the monopole and
dipole to be non-cosmological. D uring the Ce estim ation step of the Gibbs sampler, the C0
and Ci values are not changed. This corresponds to enforcing the prior th a t the cosmological
94
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signal does not contain such components.
The Gibbs sampler is run through 10,000 iterations (sets of Ct values).
approximately 24 hours on an A thlon X P1800+ w orkstation.
This takes
We ignore th e first 1000
iterations to ensure th a t the Gibbs sampler has converged to th e true distribution. This
is very conservative—in fact by com puting correlations of our Ct draws along the chain we
infer th a t about every 20t/l sample is uncorrelated.
We plot the power spectrum in figure 5.2. For each I value, we display vertically a binned
representation of the marginalized posterior densities P(Ct\m). The bins all hold an equal
number of points. The bins th a t are thinnest (points are densest in Ct space) are colored
more darkly. The top 68% are dark gray; from 68% to 95% are lighter gray, and the rest are
white. The highest density bin is shown in black.
To explore the marginalized posterior Ct distribution in more detail we plot their his­
tograms in Figure 5.3. It is notew orthy th a t not a single one of these is even nearly Gaussian.
W ithin the context of the discussion of th e lack of large scale power in the CMB, it is worth
pointing out th a t all inferences about C2 from COBE-DM R can be based on the P ( C 2\d)
shown here.
The correlation structure of the estim ates contains inform ation about how well we were
able to account for the effects of the galactic cut. It is clear from figure 5.4 th a t the residual
correlations are at most of order 10% even at very small I.
However, since the posterior densities are non-Gaussian, the two-point correlations do
not contain all the information. We therefore show the marginalized posteriors for four pairs
of Cts in figure 5.5. Again, all four of these densities are strongly non-Gaussian.
Lastly, we show the reconstructed signals. Figure 5.6A shows the expectation of the
signal component (the solution of 5.8 at each iteration of the Gibbs sampler) of the COBEDMR d ata with respect to the posterior density marginalized over the power spectrum:
{x)\p(x\m)- This is a generalized W iener filter (GW F) which does not require knowing the
signal covariance a priori. The sm oothing of the map autom atically adapts locally depending
95
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 .
0 .175
0.15
0.125
S
C'
0.1
+
'=0
^ 0. 0 7 5
O
0.05
0 .0 2 5
0
5
10
15
20
I
Figure 5.2: The COBE-DM R power spectrum . The vertical bands display the marginalized
densities at each £. Horizontal bars m ark off bins of constant probability. These bins are
assigned their color in Cg space and then projected into the diagram. The bin w ith the
highest probability density is shown in black. The dark and light shaded regions are the 1-a
and 2-a highest posterior density regions, respectively. The Cg are measured in units m K 2
in this and all subsequent figures.
96
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\I"-.,
<=17 :
1= 13 :
<= 1 8 ;
1=9
<= 14
<=19 J
I l l ] ..................
<• • 12
oo
II
.
1*11111......
CM
II
1=7
1
l-s
>s
'(/)
c
(D
X)
>N \
\
1=4
- 4—'
?
cn
...... . ......
<= 11
h
<=15.
1 M' 1
<= 6
.
...... 1......1 1r 1r
O
CN
IIo
<=
<= 10
.
/V
o
o
_Q
Q_
.
.
.
300
200 L
Tv
100
0
0. 00
0.01
<=21
*=
0.02
0.03
.
1 6
C« l(L+ 1)
Figure 5.3: Marginalized posterior densities for each individual Ce from the COBE-DM R
data. At each £ the fluctuations in the Ce at all other £ were integrated out. The axis ranges
are the same for all panels.
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Figure 5.4: Correlation m atrix of Ce estim ates from th e COBE-DM R data. The diagonal
components have been set to zero to enhance the contrast of th e off-diagonal components.
The surface is shaded according to height. We see th a t correlations between the power
spectrum estim ates vary between 8% correlation at (£,£') = (6.10) and 15% anti-correlation
at (£,£') = (8,12). See Figure 5.5.
on how much detail the d a ta support. The more strongly sm oothed central horizontal band
was obscured by the galaxy. Still the G W F reconstructs large scale modes in the galactic
cut.
The power spectrum of figure 5.6A would be biased low, since th e W iener filter removes
everything th a t could be noise. At each iteration of th e Gibbs sam pler the solution to 5.9
(shown in figure 5.6B) adds in a fluctuating term th a t replaces filtered noise w ith synthetic
signal. It is noticeable th a t this fill-in signal is larger in the regions of the m ap where the
Galaxy obscures the CMB. The resulting draw s from 5.4 is shown in figure 5.6C. Every s
draw is one possible pure signal sky th a t could have given rise to the data. Since we know
th a t the COBE d a ta has no statistical power above an £max of about 20, we imposed a
bandlim it of £max — 50.
For comparison w ith the inferences we draw from th e COBE-DM R data, we show in figure
5.6D the internal linear combination map from the W M AP satellite [9] smoothed down to
five degrees FW HM, an interm ediate scale between th e slightly larger average smoothing
98
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0. 015
o
0 .0 0 8
0.00 6
0. 010
o
o
0 .0 0 4
0.005
0.00 2
0.000
0.000
0 .0 0 0 0
CN
0 . 0 1 02
C6 6(6 + 1;
0 .0 2 0 4
0 .0 0 0 0
0.00 15
0 .0 0 1 5
0.00 10
Tf 0 . 0 0 1 0
Cxi
o
0 .0 0 8 8
CR 8 (8 + 1 '
0 .0 1 7 6
o
0 .0 0 0 5
0 .0 0 0 5
m ,
0 .0 0 0 0
0 .0 0 0 0
m s
■
X
0 .0 0 0 0
0 .0 0 0 0
0 .0 1 4 8
C3 3 ( 3 + 1 )
0 .0 297
m
m
:
0 .0 1 4 3
C4 4 (4 + 1 )
:
0 .0 285
Figure 5.5: 2-D marginalized posterior densities. Each plot shows the full joint posterior of
the data, integrated over all dimensions except for the two shown. From bottom left anti­
clockwise: P (C 2,C 3), P (C ,2 ; C4)j P ( C s , C i 2), and P ( C e, C \ 0). The latter two were chosen
because these Ct pairs were maximally anti-correlated and correlated, respectively.
99
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Figure 5.6: Reconstructed signal maps in Galactic coordinates. A: The signal component
of the COBE-DM R d a ta marginalized over the power spectrum: (x}\p(x\m). This is a gener­
alized W iener filter which does not require knowing the signal covariance a priori. B: The
solution y of 5.9 at one Gibbs iteration. C: The sample pure signal sky s = x + y at the
same iteration (band-lim ited at £max = 50). D: The W M AP internal linear com bination
m ap sm oothed to an FW HM of 5 degrees. The corresponding features in parts A and D are
clearly visible. Note th a t in this m ap low galactic latitudes are not masked, which leads to
some artifacts th a t are not visible in the masked COBE-DM R data,
100
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of panel A and the somewhat smaller sm oothing of panel B and C due to the bandlim it of
tmax — 50. Nearly every hot and cold spot th a t is identified by th e G W F can be found in the
high signal-to-noise W M AP data. Figure 5.6C fills in signal very plausibly up to the imposed
bandlim it. Even more striking is the sim ilarity of our figure 5.6A to th e com bination of Q and
V band W M AP d a ta shown in figure 8 of [9], which is intended to mimic the COBE-DM R
53GHz map.
5.8
P olarized R e su lts on S im u la ted D a ta
We now apply the Gibbs methodology to sim ulated d ata and W M A P data. Two different
cases of simulated d a ta are considered to highlight different features. In the first, we consider
a low-resolution but high signal-to-noise experiment, aimed at detecting prim ordial B modes.
The main goal of this exercise is to dem onstrate the fact th a t th e so-called E /B coupling
problem th a t plagues approximate m ethods is not an issue for exact m ethods. Second, we
consider a high-resolution simulation based on the Planck 100 GHz channel to dem onstrate
th a t Gibbs sampling is feasible even for very large CMB d ata sets. This is good preparation
for the W M AP analysis, which will follow in section 5.9.
5 .8.1
L o w -reso lu tio n B -m o d e e x p er im en t (C M B P o l)
Our first case corresponds to a possible future mission targeting the prim ordial B-modes
th a t arise during the inflationary period.
Such modes are expected to have a very low
amplitude, and also to be limited to large angular scales. Some case studies for a B-mode
mission therefore emphasize extreme sensitivity rath er than angular resolution, and we adopt
similar characteristics for this exercise.
As discussed in Section 5.3, the convergence ratio for the Conjugate G radient search
depends critically on the signal-to-noise ratio of the data. Therefore, in order to achieve
acceptable performance when analyzing tem perature observations w ith the sensitivity re-
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quired for detecting B-modes, a much b e tte r pre-conditioner th a n w hat we have currently
implemented is required. We therefore only consider the E- and B-mode spectra here, and
not the tem perature spectrum .
The simulated d a ta set consists of a sum of a CMB component and a white noise compo­
nent. The CMB realization was drawn in harm onic space from a Gaussian distribution w ith
a ACDM spectrum 5 having a tensor contribution of r ~ 0.03. M ultipoles up to £max — 512
were included. This realization was then convolved w ith a 1° FW HM Gaussian beam and
N side = 256 pixel window, and projected onto a H EALPix6 grid. Next, uniform noise of 1 y K
RMS was added to each pixel. Finally, the W M AP3 polarization mask [91] was applied, re­
moving 26.5% of the sky from the analysis.
We adopted a binning scheme logarithmic in £, such th a t 6j = [2*, 21+l — 1]. Note th a t
this is not directly connected to the signal-to-noise ratio of th e d a ta themselves, and this will
have consequences for the convergence properties of the highT B-mode bins. However, our
main focus in this paper is the m ethod itself, and this scheme is chosen to illustrate the effect
of b oth high and low signal-to-noise binning, not to obtain an optim al power spectrum .
The simulation was then analyzed w ith the Gibbs sampler described earlier, producing
1000 samples in each of five independent Markov chains. The CPU cost for producing one
sample was 10 minutes, or a wall clock tim e of 2.5 minutes when parallelized over four
processors. The to tal running time was thus 42 hours using 20 processors.
We first consider the reconstructed auto-spectra, which are shown in the top panel of
Figure 5.7. The input (unbinned) spectra are given by dashed and dotted for E- and Bmodes, respectively, and the reconstructed (binned) posterior maximum spectra are shown
by solid black curves. One and two sigma confidence regions are marked by gray regions.
Finally, the beam deconvolved noise spectrum is indicated by a th in dashed line.
In the bottom panel, we show the Gelman-Rubin convergence statistic [92] (as conserva5dow nloaded from th e W M A P3 p aram eter tab le a t LAMBDA.
6h ttp ://h e a lp ix .jp l.n asa.g o v /
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(N
E m ode spectra
N o ise
B m ode spectra
0.9
10
30
100
300
M ultipole m om ent, I
Figure 5.7: Reconstructed E- and B-mode power spectra from the low-resolution analysis.
Input spectra are shown as dashed and dotted lines, respectively, while the reconstructed
posterior distributions are indicated by solid curves (posterior maximum) and gray regions
(one and two sigma confidence regions). The corresponding noise spectrum is given by a thin
dashed line. The G elman-Rubin convergence statistic as a function of multipole is shown in
the bottom panel.
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tively com puted from the <r^’s, not Ct s) is shown for the corresponding spectra. A general
recom m endation is th a t this statistic should be less th an 1.1 or 1.2 in order to claim con­
vergence, although the particular value is dependent on the initialization procedure. In this
particular case we thus find excellent convergence everywhere for th e E-mode spectrum , and
up to £
60 for the B-mode spectrum .
As discussed in Section 5.4.4, this behavior can be intuitively understood in term s of
signal-to-noise ratio: Since the step size between two signal samples is given by cosmic
variance alone, while the full posterior distribution is given by b o th cosmic variance and
noise, it takes a large num ber of G ibbs steps to diffuse efficiently in the very low signal-tonoise regime. Further, the noise spectrum is about three orders of m agnitude larger th an
the B-mode spectrum at £ > 100, and the Gibbs sampler is therefore completely unable to
probe the full distribution w ith a reasonable num ber of samples.
To resolve this issue, we binned th e power spectrum . However, the binning scheme was
not tuned to obtain constant signal-to-noise in each bin, but was rath er arbitrary. The result
is clearly seen in the Gelm an-Rubin statistic: For £ < 60 the signal-to-noise p er bin is high,
and convergence is excellent. At £ > 60, it is low, and the convergence is very poor. The
way to resolve this would have been to choose larger bins at higher As.
In Figure 5.8, we show the E x B cross-spectrum. As expected, this is nicely centered on
zero.
We end this section by commenting on the applicability of this formalism to a possible
future CMBPol type mission. As is well known, the m ain problems for such a mission will not
prim arily be statistical issues of the type discussed above, b u t rath er systematics in various
forms. Two im portant examples are correlated noise and asymm etric beams. However, if it is
possible to pre-compute the complete A TN -1A m atrix for the d a ta set under consideration,
then these two im portant effects may be fully accounted for using the m ethods described
here. And for a low-resolution CMBPol mission this may perhaps be possible: For an upper
multipole limit of, say, £max = 300 there is a total of 2x90 000 = 180 000 polarized spherical
104
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ExB spectrum
<N
100
M ultipole mom ent, /
Figure 5.8: The E x B cross-spectrum from the low-resolution analysis.
modes to account for. In other words, one has to store and invert a 180 000 x 180 000 m atrix
in order to do this. A lthough this is a considerable com putational problem today, it is not
unrealistic in the tim e frame of these experiments. Thus, if it is possible to compute this
m atrix in the first place, an exact and complete analysis will be feasible through the m ethods
described in this paper.
5.8.2
H ig h -reso lu tio n T + E e x p e rim e n t (P lan ck )
In order to dem onstrate the feasibility of this m ethod for analyzing even the largest planned
data set, we now consider a simulation with properties similar to those of th e Planck 100
GHz instrum ent. Specifically, the grid resolution is chosen to be N side = 1024 (corresponding
to a 3.4' pixel size), the maximum multipole moment is Aide = 1500, the beam size is 14', and
the noise level is 38.2 f i K RMS per pixel for tem perature and 61 /J.K per Q /U pixel. These
noise levels correspond to the requirem ent specifications for the 100 GHz channel, which are
lower than the goal by a factor of two. No B-modes were included in this case, and the sky
cut was chosen to be the W M AP Kp2 mask.
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Such large d ata sets are certainly a challenge for the Gibbs sampling algorithm , and
the com putational requirem ents are considerable. Specifically, th e CPU tim e for generating
one sample (requiring ~ 250 CG iterations) is about 16 CPU hours when using the low-f
preconditioner described by [84] up to £ = 70. B etter preconditioners will of course improve
this significantly.
However, it is im portant to notice th a t even though this is an expensive operation, it
is by no means prohibitive. To obtain a reasonably converged posterior distribution, one
requires on the order of ~ 103 independent samples, and this would then require ~ 104 CPU
hours. Of course, this number m ust be m ultiplied w ith a significant factor for an actual
production analysis (e.g, number of frequency bands or d ata combinations), b u t considering
the trem endous efforts spent on obtaining the Planck d ata in the first place, this am ount of
CPU tim e is a most reasonable cost for analyzing them .
For the high-resolution analysis presented in this paper, we produced a to ta l of 800 sam ­
ples, divided over eight independent chains. The results from these com putations are sum­
marized in Figure 5.9, showing bo th the reconstructed power spectra and the corresponding
convergence statistics. This provides is a direct dem onstration th a t power spectrum estim a­
tion through Gibbs sampling is feasible even for Planck-sized polarized d a ta sets.
S ep a ra tio n o f E and B m od es
In the context of the high-resolution E-mode dom inated simulation, we make a brief comment
on the so-called “E-B coupling problem ” th a t plagues most approxim ate m ethods, such as
the pseudo-Q m ethods (see, for example, [93]). Briefly put, the problem lies in the fact th a t
the spherical harmonics are not orthogonal on a cut sky, and this results in leakage from the
(much larger) E-modes into the B-modes.
However, for exact methods such as exact likelihood analyses or Gibbs sampling, this
is a non-existing issue. This may be understood intuitively in terms of the signal sampling
process illustrated in Figures 5.10, 5.11, 5.12, and 5.13: O btaining a complete sky sample
106
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10,2'
2
- TT spectrum
TE spectrum
10.0
0
i
u
o
_ EE spectrum
10
10'
2
10'-2
■2
0
a 1.2
1,2
1 1,1
1, 1
£ 1,4
ts 1,2
a l
°
1
10
Multipole moment, /
M ultipole moment, /
100
1000
M ultipole moment, I
Figure 5.9: Reconstructed power spectra from the high-resolution Planck 100 GHz simula­
tion. The true spectra are shown as dashed lines, and the reconstructed posterior distribu­
tions are given by a maximum posterior value (solid lines) and a 68% confidence region. The
Gelman-Rubin convergence statistics are shown in the bottom panels.
for the Gibbs sampler is a two step process. First, one filters out as much inform ation as
possible from the observed d ata using a W iener filter. Second, one replaces th e lost power
due to noise and partial sky coverage by a random fluctuation term . The sum of the two
is a full-sky, noiseless sample th a t is consistent w ith the data, w ith emphasis on “full-sky” .
Because it is a full-sky sample, no E-B coupling arises.
5.9
B ayesian A n a ly sis o f W M A P P o la riza tio n D a ta
We have applied the Gibbs sampling algorithm to a low-resolution, polarization-only case
where the full covariance m atrix is available. On LAMBDA, th e QU polarization maps and
full covariance m atrix are available at Aside = 16 for the Q and V bands separately, and
for Q and V averaged together. We modified the MAGIC code to be able to handle full
covariance matrices, and ran it on this data.
The full covariance matrices are constructed from a knowledge of the scanning of the
detectors across the sky and the 1/ / noise in the detectors. This has a tendency to cause
some small correlation between nearby pixels, or pixels at certain special distances from each
other (such as the angle between the two detectors in the differential W M AP experiment).
107
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Figure 5.10: Gibbs sampled signal maps. The three columns show, from left to right,
tem perature, Stokes Q and Stokes U param eters. The three rows show, from to p to bottom ,
the complete Gibbs samples, the mean field (W iener filtered) maps, and the fluctuation
maps. The mean field m ap provides the inform ation content of the data, and the fluctuation
map provides a random complement such th a t th e sum of the two is a full-sky, noiseless sky
consistent both with the the current power spectrum and the data.
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Fluctuation
Mean
Gibbs Sample
polarization amplitude: 2.2 uK
Figure 5.11: Close-up of the galactic center shown in Figure 5.10, emphasizing how the
algorithm separates E and B modes. Each of the sampled m aps (the sum of the fluctuation
and mean held map) are full sky maps, so decomposing the polarization into E and B modes
is straightforward. The images show tem perature as color and polarization overlayed as a
fingerprint pattern of stripes. The stripes are aligned with the direction of polarization. They
are darkest where the polarization is strongest, and they disappear where th e polarization
goes to zero. See chapter 6 for more details on this representation of polarization. The
maximum amplitude of the polarization is given in fiK and centered under each image. The
maps have been smoothed to 1 degree. This is an orthogonal projection of the sky, about
60 degrees wide, centered on the Galactic center. The W M AP KpO galactic mask is visible
in the fluctuation and mean terms.
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Fluctuation
M ean
Gibbs Sample
polarization amplitude: 2.2 uK
0.39 uK
0.04 uK
0.38 uK
Figure 5.12: This is the same as figure 5.11, except it was m ade w ith IDL and uses line
segments to represent polarization instead of th e line integral convolution stripes used in
figure 5.11.
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Figure 5.13: This figure shows the scalar E and B modes th a t are displayed in figure 5.11.
They represent how much “gradient” and “curl” is in each polarization field. These are truly
scalar fields because they are spin lowered versions of the spin 2 polarization fields. For this
figure, the images were created by overwriting the ajm spherical harmonic coefficients w ith
either the afm or the afm coefficients, and then plotting the new “tem perature” field.
Ill
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The param eters for the runs were as follows. The C% values were all sampled separately,
except for 1 = 2 and £ = 3, which were binned together. The Jeffreys prior was not used.
For the Q and V analyses, the beam s for the Q l and V I channels were used. For the QV
analysis,
110
beam was used. The Q and V beam s drop off to 98.7% and 99.2% of their full
strength, respectively, at i = 32. This means th a t failure to include the beam for the QV
analysis will mean the sampled values of C% (black histogram ) are too low by about 2% at
£ = 32, and by less th a n 2% at lower £. As one can see from figures 5.14, 5.15 and 5.16,
this does not dram atically affect the histograms. The Q and V chains were run for 28,170
and 26,738 Gibbs iterations, respectively, and the QV chain for 12,719 iterations. For the
histograms, we used the final 25,000 iterations of the Q and V chains, and the final 10,000
iterations of the QV chain.
The mean of 10,000 signal sky samples from the QV chain are shown in figure 5.17,
split apart into E and B modes. Because of the level of noise, the B modes are ju st as
strong as the E modes. An approxim ation of the noise given by the diagonal of the noise
m atrix is shown in figure 5.18. The noise is clearly much higher th an the sampled E and
B modes, making a detection at any given I very difficult. The galactic polarization mask
which shows up clearly in figure 5.18 is also ju st barely visible has a faded region in figure
5.17. This indicates the lack of information in the masked region—the Gibbs sampler m ust
interpolate into the masked region based on the d a ta outside th e mask. Because the direction
of polarization is less certain inside the mask, it averages down over many iterations.
The likelihoods of the power spectra are available in figures 5.14, 5.15 and 5.16 for the
EE, BB, and EB power spectra, respectively. Note th a t the QV histogram puts a stronger
constraint on the power spectrum th an either the Q or the V histogram s individually, as one
should expect. Note also th a t the EB spectra are consistent w ith zero, as expected. While
one could almost claim some preference for nonzero EE power spectra, there are no strong
detections at individual I values.
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1=20
1= 22
= 24
= 27
'= 2 6
£= 29
0
2
4
6
C, «(M -1)/2n
8
0
[/xK2]
2
4
6
C, < (t+ 1 )/2 n
8
0
[/xK2]
2
4
6
8
0
C ,« (« + 1 )/2 tt [ m K2]
2
4
6
C, «(«+1 ) / 2 n
8
0
[/xK2]
2
4
6
8
C, J(« + 1 )/2 n [/xK!]
Figure 5.14: The posterior probability densities of C f E up to I = 32, made from a histogram
of sampled C f E values. The black histogram has 10,000 samples from the QV analysis; red
has 25,000 samples from Q only; blue has 25,000 samples from V only.
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 .
£=22
£= 2 4
0
2
4
6
C, t ( t+ 1 ) / 2 n
8
0
[/iK2]
2
4
6
C, 1(1+1 ) / 2 n
8
0
[/xK!]
£=26
2
4
6
C( «(M -1)/2n
8
0
[/^K2]
2
4
6
8
0
C ,« (M -1 )/2 n [>K2]
£=27
2
4
6
8
C, «(!+1 ) / 2 n [^K 2]
Figure 5.15: The posterior densities of C BB up to I — 32, made from a histogram of sampled
C f B values. The black histogram has 10,000 samples from the QV analysis; red has 25,000
samples from Q only; blue has 25,000 samples from V only.
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1= 1 2
ji£
1=20
= 19
n j= 2 4
= 27
JT-
= 28
-6 -4 -2 0 2 4 -6 -4 -2
C, «(«+1 ) /2 tt [MK!]Ce «(«+1 ) /2 tt
,= 2 9
0
2
OK1]
4 -6 -4 -2
C, 1(1+ 1) / 2 n
0
2
4 -6 -4 -2 0 2
4 -6
[>K3]
C, e(«+1) /2 tt OK3]
-4 -2 0 2
4
Cj 1( 1+ 1) / 2 n O K 2]
Figure 5.16: The posterior densities of C f B up to I = 32, m ade from a histogram of sampled
C f B values. The black histogram has 10,000 samples from the QV analysis; red has 25,000
samples from Q only; blue has 25,000 samples from V only.
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Figure 5.17: The mean of 10,000 signal samples from the QV analysis, split apart into E
and B modes. The stripes are aligned with the direction of polarization and their darkness
indicates the strength of polarization. The maximum polarization for th e sky is 2.3 /iiF, for
the E modes alone is 1.5 fiK, and for the B modes alone is 1.6 jiK. The galactic mask is
barely visible as a fading in polarization on the center of the plot.
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Figure 5.18: The diagonal of the W M AP noise m atrix for the QV analysis. There are
correlations, but the diagonal is still a good approxim ation of the noise m atrix. The square
root of the diagonal element is shown, and units are pAT. From top to bottom , we show
the expected standard deviation in Q, then in U, then these two maps added in quadrature.
The galactic mask is clearly visible in all three.
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5.10
C on clu sion
We have described a framework for global and lossless analysis of cosmic microwave back­
ground data. This framework is based on a Bayesian analysis of CMB d ata. It has several
advantages compared to traditional m ethods.
It is com putationally feasible.
It is opti­
mal and exact under the assumption of G aussian fields and the ability to encode our prior
knowledge about foreground com ponents in term s of m ultivariate Gaussian densities. It uses
controlled approximations (e.g. the num ber of samples of the Gibbs sampler controls the
accuracy of the result but this can be increased by spending more computing time). It allows
joint analysis of the CMB signal, foregrounds and noise properties of th e instrum ent, while
modeling and exploiting the statistical dependence between these different inferences.
Traditional methods of inference from CMB d a ta divide the d ata analysis into several
steps: map-making from TOD, component separation, power spectrum estim ation from the
CMB signal and cosmological param eter estim ation. Our m ethod allows treating all these
inferences jointly and self-consistently, if desired. The traditional results can be understood
as special cases of our m ethod for certain uninform ative prior choices. For example, pure
m ap-m aking could be viewed as applying this framework w ith P ( s | C ^ ) P ( f ) P ( C b ) = const.
In spite of this generality, the framework for analyzing CMB d a ta described here is
very modular: the structure of the Gibbs sampling scheme separates the different steps of
the inference process focusing on each com ponent in turn. The framework described here
therefore holds the promise of making more d a ta analysis steps part of a self-consistent
framework rather th an sequential stages in a d a ta pipeline.
Our m ethod turns out to give an unbiased W iener filter and generalizes the global filtering
and reconstruction methods in [80] to include power spectrum estimation, obviating th e need
for a priori knowledge of the signal covariance.
We require the use of iterative techniques to solve the most com putationally dem anding
step in this method.
We find th a t our simple-minded preconditioned gradient iteration
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works well over 3 orders of m agnitude in problem size. It remains to be studied w hether
other preconditioners may be even more effective (for example [81]).
We applied our formalism to the well-studied COBE-DM R d a ta set. We dem onstrate
th a t our m ethods enable new analyses for even such a small d a ta set. We quote posterior
densities for individual Cg as well as posterior densities for pairs of Cg as examples of results
th a t would be prohibitively expensive to obtain w ith traditional algorithms. O ur results
are consistent with the most sophisticated bru te force 0 ( N p ix) analyses available in the
literature.
The approach has been extended straightforw ardly to polarized m aps and d a ta th a t
spans different frequency bands. We have detailed both the necessary generalization steps
relative to the original tem perature only descriptions given by [76], [54] and [84], and we
have considered com putational aspects of polarized analysis.
The polarized version of the algorithm was dem onstrated w ith three specific examples.
First, considering a possible CMBPol type mission, we showed th a t the Gibbs sampler
cleanly separates E- and B-modes, and no special care is required. This is in sharp contrast
to approximate m ethods such as so-called the pseudo-Q for which great care must be taken
in order for the larger E-modes not to compromise the m inute B-modes.
Second, we analyzed a Planck-sized d a ta set, dem onstrating th a t the algorithm is useful
for analyzing the quantity of d a ta which will come from near-future CMB experiments.
Third, the low resolution, polarization-only, W M AP d ata was analyzed with full covari­
ance matrices.
The Gibbs sampling results presented here use symmetric beams and, except for the lowresolution W M AP analysis, noise which is uncorrelated between pixels. However, the Gibbs
sampling algorithm has potential to analyze considerably more complicated d ata sets th an
these. For Planck, the solution lies in exploiting the very regular scanning strategy, which
reduces the com putational burden of a time-ordered d ata analysis. For a future CMBPol
mission, the solution lies in the relatively large angular scales required. Since it is possible
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to invert the noise covariance m atrix for m ultipoles up to several hundreds, one may pre­
com pute the all-im portant A T~N~1A m atrix. A fter paying this high one-time cost, efficient
and exact analysis is feasible using the m ethods described in this chapter.
Finally, we re-emphasize th a t the Gibbs sampler provides a direct route to the exact
likelihood (and to the Bayesian posterior), and it is much more reliable th an approxim ate
methods. This issue has been dem onstrated explicitly through the analysis of the threeyear W M A P data, where an approxim ate likelihood between I = 13 and 30 caused a nonnegligible bias in the spectral index n s [86]. Using Gibbs sampling, such worries are greatly
reduced. Further, this paper dem onstrates th a t the m ethod is in fact capable of analyzing
the am ount of d a ta th a t will come from the Planck mission w ith reasonable com putational
resources. It therefore seems very likely th a t this m ethod will play a significant role in the
analysis of future Planck data.
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C h ap ter 6
Line In tegral C o n v o lu tio n
As we have seen, the polarization of the cosmic microwave background contains much useful
inform ation about the early universe. Many experim ents are planning to measure the po­
larization of the CMB in hopes of finding the holy grail of cosmology: B mode fluctuations
due to gravitational waves from inflation. Based on this level of interest, we should have a
good way to visualize this information. It would be nice to be able to plot polarized maps as
easily as one plots m aps of tem perature on the sky. Not only would this be useful for public
outreach, but the ability to make easily understood m aps of the sky would be extremely
helpful when looking for unusual artifacts or when debugging code.
One approach for plotting polarization is to p u t down m any little line segments (doubleheaded vectors) to represent the m agnitude and orientation of the polarization. This has
some disadvantages. Usually, it is on a regular grid, and the p attern of the grid can over­
whelm or obscure the patterns of polarization. Also, one only sees information about the
polarization at a discrete array of points. This is reasonable if the polarization is sm ooth on
the inter-vector length scales, and usually one sm ooths the polarization map to make sure
this is the case. However, it does m ean one has to interpolate by eye, and the grid p attern
can make this difficult.
This section discusses line integral convolution, a much b e tte r technique for displaying
polarized maps of the CMB. This m ethod is very impressive for its original purpose of
visualizing vector fields on a plane, [11], and, realizing its potential, we have extended it to
polarization fields
011
the sphere. This technique creates a texture th a t appears to be smeared
in the direction of the vector held. It has the advantage of using every pixel available, so it
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packs th e m axim um am o u n t of in form ation a b o u t th e p o larizatio n into th e p lot of th e sky.
In fact, line integral convolution (LIC) is even b e tte r for polarization, a “double headed”
vector field, than for a regular vector field. LIC removes all inform ation about the direction
of the vector. One loses inform ation when plotting a vector field in this fashion, b u t not
when plotting polarization.
T he texture also has the advantage of transform ing as a scalar under various projections,
so th a t there are no projection effects to remove, as there are when overplotting lots of little
line segments.
6.1
T h e A lg o rith m for L ine In teg ra l C o n volu tion
The technique for performing line integral convolution is described in [11]. Cabral and Leedom create a monochrome texture in which the pixels are highly correlated in the direction
of the vector field and uncorrelated in a direction perpendicular to the field. This gives
the appearance of a texture which is smeared in the direction of the field, and it is highly
effective for seeing pattern s in the polarization. See, for example, th e differences between
the E and B modes in figures 6.1 and 6.2.
Here, we rewrite the formulas given by C abral and Leedom for use on the sphere, and for
use w ith a double-headed vector field (polarization). I ignore the discreteness/pixelization
of all of the functions in use, because the C + + version of the HEALPix code has an inter­
polation function. This allows a pixelized function on th e sphere be treated as a continuous
one. Let
Q (n) + iU (n)
(6.1)
be the polarization field, where n denotes the location on th e sphere. Let 6(h) be a real
valued function th a t represents the background texture. For example, it can be consist of
white noise: random Gaussian variates on a HEALPix grid.
For the line integral convolution, it is necessary to create lines th a t lie in the direction
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of the polarization field. This is done in the fashion of a naive differential equation solver,
by determ ining the direction of polarization and then taking small steps on the sphere in
th a t direction. Let a be the angle of the polarization (electric field of the incoming CMB
radiation) w ith respect to “N orth” . Then
a = - a rc ta n (Q /[/)
(6.2)
where the 180 degree am biguity will be dealt w ith in th e next paragraph. If we wish to take
a small step on the sphere in this direction, we increm ent the usual angles 9 and 0 by
d 6 ( h ) = e cos <n(n)
sin <a(n)
(6.3)
.
.
where e parameterizes the size of the step. Then if one starts at a point ($„,</>„) on the
sphere, one finds successive points w ith indices i > n by
■ (0j,0j) = (0i-u fa-i) + { d O ^ ^ . d c p i h i ^ ) )
(6.5)-
and points with indices i < n by
{9i,(f)i) = (0i+i , 0 i+i) - ( d 9 ( h i+1),dcj)(hl+1))
(6.6)
In this fashion, one can extend the line in both directions from the startin g point, and obtain
a set of points {n0, rq,..., n 2n}.
A difficulty will become immediately apparent when implementing the above steps. At
every point, two choices exist for a: a and a + n. Since the polarization is effectively a
double-headed vector field, there is no obvious “forward” and “backw ard” direction to make
the choice easy. It is therefore necessary to choose a forward direction arbitrarily for the
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starting point, and keep track of th e forward angle ati as one computes successive values
n i+i from fq. The new forward angle a i + 1 is the one th a t is less th an 90 degrees away from
the old forward angle cp. The same trick is used when going backward, for indices i < n.
The polarization field m ust be sufficiently sm ooth and e must be sufficiently small for this
to work well, but in practice these constraints are easily met.
Next, we require a kernel for the convolution. This is a list of values /q where i runs from
0 to 2n. A simple flat kernel is /q = 1, which actually works quite well, b u t a slightly more
useful one is
(6.7)
which tapers nicely at the edges.
To compute the line integral convolution, one computes
2n
LIC(rf„) = V / . ! i i V
( 6.8 )
for every starting point n„. This is the convolution of the kernel w ith the background texture
along a line aligned with the polarization. W hat results is a real valued field LIC(n) on the
sphere, which we can display, such as with a greyscale.
From this prescription, one can see th a t nearby points (in the final texture) lying along
field lines will be highly correlated, because their line integrals will be nearly the same.
Nearby points lying on different field lines, however, will have uncorrelated values.
This algorithm was implemented in C + + . It is in the CVS (Concurrent Version Sys­
tem) repository for the HEALPix package [94], and will be available in a future release of
HEALPix. The algorithm is rath er slow right now, since a separate field line is com puted
for every pixel in the final image. The code could be sped up, perhaps by a factor of 10, by
following the work in [95]. In th a t paper, Stalling and Hege describe how to reuse field lines
for neighboring points.
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6.2
A d ju stin g th e P a ra m eters o f th e A lg o rith m
This line integral convolution has a number of free param eters which can be adjusted to
improve the image. The background texture doesn’t have to be white noise; it can be ju st
about anything. For CMB maps, it is useful to have a background texture w ith power at only
one value of i. This puts the texture all at one angular scale on the resulting figure. W hen
combined with a thresholding operation where values below zero go to black and values
above zero go to white, the resulting texture is a black and white p attern th a t resembles
zebra stripes, or a fingerprint. See figure 6.3.
A fter the line integral convolution has been computed, one is left w ith a monochrome
texture.
This can be overlayed on a map of the tem perature to show both polarization
and tem perature on the same graphic.
For this application, the zebra-stripe texture is
particularly useful, since the texture is easily visually distinguishable from the tem perature
information.
If desired, the transparency of the texture can be adjusted to reflect the m agnitude of
the polarization. This is also particularly effective w ith the zebra stripes, because the stripes
are otherwise uniformly black. It is easy to remember and interpret th a t where the stripes
are darkest, the polarization is strongest, and where the stripes fade out, the polarization
goes to zero.
The kernel shape and size are also adjustable. Non-uniform kernels (th at tap er off at the
ends) tend to give aesthetically b e tte r results th an uniform kernels. Also, the farther the
kernel extends along the held line (the larger 2ne is), the farther the correlation will extend
in the final image. It is useful to have this length be at least as long as the scale at which
the image was smoothed.
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Figure 6.1: This is how line integral convolution looks w ith white noise as a background
texture. This is the same polarized map as in figure 6.3, containing only E modes correlated
w ith the tem perature. Upper left: the raw texture. Upper right: texture adjusted to
reflect m agnitude of polarization. Middle left: tem perature map. Middle right: tem perature
overlayed with texture adjusted by m agnitude. Lower left: tem perature overlayed w ith raw
texture. Lower right: raw texture, color represents m agnitude of polarization.
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Figure 6.2: This figure shows line integral convolution w ith white noise, as does figure
6.1. However, this m ap contains only B modes. Note th e difference in texture. Upper
left: the raw texture. Upper right: texture adjusted to reflect m agnitude of polarization.
Middle left: tem perature map. Middle right: tem perature overlayed w ith texture adjusted
by magnitude. Lower left: tem perature overlayed with raw texture. Lower right: raw
texture, color represents m agnitude of polarization.
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Figure 6.3: This is the zebra striping technique applied to a polarized map. This m ap
contains only E modes, and they are completely correlated w ith the tem perature. This m ap
has also been smoothed with a 2 degree beam. Upper left: the raw (thresholded) texture.
Upper right: texture adjusted to reflect m agnitude of polarization. Middle left: tem perature
map. Middle right: tem perature overlayed w ith texture adjusted by magnitude. Lower
left: tem perature overlayed with raw texture. Lower right: raw texture, color represents
m agnitude of polarization.
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C h ap ter 7
C on clu sion
This dissertation has discussed three tools for testing the standard model of cosmology w ith
the CMB, and dem onstrated how to pu t them to work on real data.
The statistical tests of Gaussianity are very useful not only to test cosmology, bu t also
to look for defects in our understanding of the CMB experiment, or im proper subtraction of
foregrounds, or any other number of details th a t may have been missed in an initial model
of the data.
Dealing with foregrounds will be one of the largest problems when analyzing the polarized
CMB in the near future. If we believe we have removed all the foregrounds from some region
of the sky, and our statistical G aussianity tests all fail, we know something is wrong—w hether
it is our assumption of Gaussianity, or our foreground removal technique.
As another tool, the Gibbs sampling algorithm shows much promise for analysis of future
polarized CMB missions. The polarized CMB is a hot area in cosmology right now, w ith a
num ber of experiments preparing to make high-resolution or high-sensitivity studies of the
polarized CMB. The Planck space mission is scheduled to launch in 2008, and it plans to
make full-sky maps of the tem perature and polarization of the CMB w ith unprecedented
accuracy. The low-£ polarization has the potential to break the degeneracy between the
cosmological param eters A s, n s, and r , and the currently available W M AP d a ta is starting
to constrain these.
There is also the hope to detect prim ordial B modes w ith the upcoming Planck mission,
and others. A detection of prim ordial gravity waves through the CMB would be a highly
interesting test of inflation. Untangling the E and B modes—or rather, sampling full skies
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so these modes are never tangled in the first place—is one of the great benefits of the Gibbs
sampler.
For purposes of constraining cosmological param eters, or being able to say with certainty
whether we have detected B modes, it will be more necessary th a n ever to m ap out and
propagate the error bars on the CMB d a ta accurately. The Gibbs sampler is a reasonably
fast algorithm th a t is actually capable of sampling th e true likelihood (or Bayesian posterior,
if desired), and approxim ating th a t likelihood continuously and w ith high accuracy using
the Blackwell-Rao estim ator.
The Gibbs sampler also shows promise for incorporating other d a ta sets besides the CMB.
The approach can be extended to joint estim ation of different d a ta sets. There is nothing
th a t prevents the application of these ideas to random fields on manifolds other th an the
sphere, such as one-, two- or three-dimensional Euclidean space. We are investigating the
formalism for joint inference from CMB d ata about the power spectrum and map of the pure
CMB sky w ith the power spectrum and m ap of the projected gravitational potential.
Finally, the line integral convolution m ethod has the potential to completely change the
way we visualizing polarized maps of the sky. It doesn’t suffer from visual artifacts caused
by a grid of line segments at different angles. The sm ooth curves in the texture make it
much easier to see p atterns in the polarization. It is also highly adjustable, w ith a number
of free param eters, so plots can be made to suit almost any need, from quick-look analysis
on a desktop to presentation quality materials. This texture provides a new way to the the
cosmic microwave background.
It is an exciting time to be in cosmology. W ith the availability of new d a ta every few
years and an increasingly sophisticated toolbox w ith which to analyze it, it may not be long
before we have to modify the standard model of cosmology.
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A p p en d ix A
P ro p e r tie s o f <3 an d 5
The following formulas describe the spin-raising and spin-lowering operators 5 and 5.
(5sz (9 : <fi) =
—sins(0)
dsz(9,<p)
— sin~s(9)
=
d
89
8
89
i
8
sin s(9)z(91(p)
sin($) dcf)
i d
sin(0)
sins(9)z(9, 8)
(A .l)
(A.2)
3.5 sYim — a/(£ — s)(£ + S + 1)
S + 1 Y(_m
(A.3)
3.5 s^lm =
) s-lTem
(A.4)
—
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From those equations, we can derive th e following relations:
9 s+i 9s
sYim— 99 sYtm — —{£ — s){£ + s + l) sY£m
(A.5)
9s- i 9s
s ^ m= 99 sYem = - ( £ - s + 1)(£ +
(A.6 )
s) 8Y£m
s¥tm = 2s sYlm
s
a
a
h + p —1 • • • O s + 1 O s
y
s*£m
_
*
0
y
_
, (£ +
S Ylm
(A-7)
5 + p ) ’
s)!
{£ -\-
{£ ~ s)\
(£
S
p)!
s^ p
=
- , +i ... g,-i g , , v lm
= t? , y lm =
( A
( - i ) q /
(* + ?)!
oYem = ( - 1
„
^
( i {e "£+s ) f
-M m
,
/
V
' 9 )
(a.io)
(A.11)
, lV,(^ + s + P ) ! ■l { £ - s ) \ ( £ - s - p + q)\
ij
6 c r s Y£ m - {
{£_ s _ p)l^ {i + s)l{i + s + p _ q)l
s+p - g y t m
(A.12)
«Pff Y
f Y ^ £ - S + q^1 K £ + s )'-(t + s - q + p)\
,Yem - ( i) v + s _ qy \ j y _ s) l { ( _ 3 + q _ p),.+v-,y^
(A. 13)
5SP?SP
^
( 1\p(£
+
S +
^ ) !
~
S) !
W
g w , Y lm = ( - i ) ” ( £ _ s _ p ), (€ + - ^ ^
^
ys
{ iAp(^ “ S + P ) ' (^ + S)' v
= i - 1? (( + s _ p H e - s ) , -Y‘«‘
(W<5p - d pW ) oY£m = 0
(\
1 /I ^
(A -14>
( \ ^<\
<A -15>
(A .16)
where we are constrained to —£ < s < £ and —£ < m < £ whenever the spin s spherical
harmonic sY£m occurs. There are no problems if £
|s|, |m |,p , q.
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A .l
A M a th e m a tic a l T an gen t in to F orm alism
Note th a t we occasionally drop the subscript s from the d and d operators. This is because
we mentally assign a spin value s to every complex field on the sphere th a t we work with.
This tells how it transform s under rotations, and is prim arily in place for physical intuition.
We originally define complex fields on the sphere as maps
/ : S 2 -> C
and we
can take the set of all(suitably differentiable) such
(A. 17)
m aps to be F.Then we could
extend our definition of complex fields to elements of F x Z, where in addition to specifying
the field, we also specify an (integer) spin. W rite an arb itrary element as (/,
s)
G
F x Z.
Then define the new d and d in term s of the old ones. The new operators are functions
S :F x Z ^ F x Z
d : F x Z ^ F x Z
(A .18)
■
(A.19)
and are defined by
d ( /,s )
=
(da/ , s + l)
(A.20)
3 ( /,s )
=
( 3 J , s - 1)
(A.21)
Fields of nonzero spin can be formed by raising and lowering scalar fields of the form (/, 0),
with the operators d and d. Although this prescription is too tedious for everyday use, it
does allow the functions to explicitly carry around their spin weight s so th a t the operator
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does not require it. For example, one can now define 3 3 without ambiguity, since
33(/,a)
3 3 3 ( /,s )
(A.2 2 )
3 3 (3 s/ , s + l)
(A.23)
3 (9 s+iSs/ , s +
( S s+ 2 3 s + i 3 s / , s
A .2
(A.24)
2)
(A.25)
+ 3)
C om p lex C on ju g a tio n
Perhaps by abusing terminology, one can take the complex conjugates of the <3 and 5 oper­
ators:
5*
-
3_s
(A.26)
K
=
3 -,
(A.27)
Take a look at the definitions of these operators again to check th a t this makes sense. We
may find the formalism of section A .l to be useful. We define complex conjugation on F x Z
as
(/.«)* = ( / • ,- .? )
(A.28)
Let us try an example.
[S(/, »)]* = ( 3 ,/. S + 1 )* = ((3 J T , - s - I ) , (a ;/* .
<
3
-a -
1)
(A.29)
If we now define
3*
=
3
(A.30)
3*
=
3
(A.31)
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for th e operators on F x Z, we see th a t th e definition is self-consistent.
P ( /. s)]* = 5"(/, s)* = 3 ( / - . - s ) = (
3
- s - 1)
(A.32)
Now we come to equation 3.34 of [20]. Newman and Penrose claim th a t
(A.33)
and previously define
Ve = SsRe(£)
(A.34)
ds5sRe(£) = ds[dsRe(0]*
(A.35)
Hence they claim
The real part of £ is understood to be a spin 0 field, so this is just
I f d sRe(£) = 0s5sRe(£)
(A.36)
which we have already proven in equation A. 16. The corresponding equation for r/m is
S V = - 3 s*
(A.37)
skipping steps ...
dsSsIm(£) = - - Ssg sIm(£)
where our
(A.38)
signscancel because Im(£) changes sign under complex conjugation. If desired,
we can specialize to the case of E and B modes of polarization:
B \ Q e + tUE) = d2(QE - iUE)
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(A.39)
and
32( Q b + iUB) = - 3 2( Q b ~ iUB)
2
(A.40)
—2
These statem ents ju st mean th a t 9 ( Q e + ^U e ) is a pure real quantity and 9 ( Q B + i U s ) is a
pure imaginary quantity. These quantities are closely related to Re(£) and Im(£), b u t there
are different proportionality constants for different £ values in spherical harmonic space.
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A p p en d ix B
P ed a g o g ica l In tro d u ctio n to
S ta tistic a l H y p o th e sis T estin g
This appendix contains a detailed explanation of our statistical test used in chapter 4,
following [24], The statistical test is discussed in section B .l. In section B.2 we connect our
discussion to the derivation of frequentist confidence intervals in [23], and then add a brief
conceptual comment on tests of non-G aussianity in section B.3. We close in section B.4 with
a brief user’s guide for the f a c t s program, which helps com pute the frequentist statistical
tests presented here. The code has been made publicly available.
B .l
S ta tistica l T estin g
B .l.l
T h e P r o b le m
The statistical analysis we discuss in this paper can be reduced to the following problem.
We are given a few thousand (= n) random numbers {xj}, j = 1 .. . n which have come
from some random number generator (distribution), w ith some probability density function
(PD F) f ( x ) . We are also given a single number
£0
and asked to determine if we have any
reason to believe, statistically, th a t it may not have come from th a t same random number
generator.
B .l.2
T h e S o lu tio n
Because this is a statistical problem, we cannot do w hat we naturally want to do: prove
th a t Xq was or was not chosen from the same distribution as the {xj}. Instead we must
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settle for a weaker statem ent about how large a fluctuation x 0 is, if it were drawn from the
distribution of the {x j} . This appendix describes how to make statistical statem ents about
the size of this fluctuation.
We begin by assuming th a t the single num ber xo did come from the random number
generator th a t produced the { x j} . Given this, we attem p t to determ ine how large a statistical
fluctuation x 0 is. There are several ways to do this. We will use a completely standard
(frequentist) hypothesis test. O ur hypothesis concerns a param eter, p, which describes how
large the fluctuation is. We perform a test on a random variable, i, whose distribution is
affected by p, and, based on the results of th a t test, decide w hether or not to accept the
hypothesis.
This m ethod is useful if the pdf is sufficiently complex th a t it is not practical to evaluate
it analytically.
B .l.3
T h e R a n d o m V a ria b le an d Its D istr ib u tio n
The m ethod we use requires knowing only how m any of the random numbers Xj fell below
xq
.
Let there be n random num bers and let i of them fall below x0. Then i is our random
variable, chosen from the binomial distribution P(i\p,n), where
77I
p ( A p ’ n) = i (
^
p 'p "~'
< a i)
and where p gives the position of Xq in the PD F f ( x ):
/
x0
f { x ) dx
(B.2)
-OO
We can estim ate p w ith p = i / n , which is a maximum likelihood and unbiased estim ator.
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B .l.4
T h e H y p o th e s is
O ur hypothesis concerns the value of p. As previously stated, we would prefer to pick “x 0 was
chosen from the same distribution as the { x j } values” as our hypothesis. Since th e negation
of this hypothesis is extremely difficult to work with, we choose a simpler hypothesis, about
p:
H0 :
p E (a /2 ,1 - a / 2 )
(B.3)
where a is much less th an 1. Here we use a double sided hypothesis for our test; th e case
of a single-sided test is discussed in section B . 1 .6 . To a frequentist, th e hypothesis H 0 is
either true or false, so it either has a probability of 1 or 0. The statistical test described
in this appendix will then be useful to decide w hether to accept or reject the hypothesis.
The frequentist statem ent is th a t UH 0 is tru e in a fraction 1 — a of all possible Gaussian
Universes” .
It is certainly true th a t we could have chosen our hypothesis to be anything of the
form p G S where the set S C [0,1] has to tal length (or measure) 1 — a .
S = (a /2 ,
1
We choose
—a / 2 ) because of our natu ral inclination to think th a t values of p far out in the
tails of the distribution are unusual. Also, if we had reason to believe th a t xq were drawn
from a distribution whose m ean was many standard deviations away from the distribution
of the {xj}, our hypothesis would be a powerful test of w hether x 0 was drawn from the same
distribution as the {xj}.
B .l.5
T h e T est an d T y p e s o f E rror
Our test must be of the form where we accept Hq for certain values of i, i G / , and reject it
for all others, i G I . Here, I and I are disjoint and I U I = { 0 ,1 ,2 ,..., n}. W ith a statistical
test of this form, one is interested in the errors of type I (rejection of a true hypothesis) and
type II (acceptance of false hypothesis). I will call their probabilities (3 and
7
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, respectively.
test accepts H q,
test rejects H q,
i e I
i e 7
H 0 is true
1 -/3
P
H 0 is false
1
1
-
7
Explicit calculation of these probabilities requires us to specify a test. Our test is similar
in form to th a t of our hypothesis. We accept H 0 whenever
i G {?o + b o + 2 . . . , n —io — 1}
for some specified value of i0. Since calculation of f3 and
the test used, (3 and
7
7
(B-4)
requires knowledge of n, p, and
are functions of these three values.
(3 = P(i0, p , n )
7
=
7
(?0 ,P ,n )
It is desirable to adjust the test (the value of z0) so th a t (3 and
(B.5)
7
are as small as possible
for all values of p. Unfortunately, these cannot b o th be made small. The probability of
the test accepting H 0 must vary continuously as p varies continuously from the region from
where H 0 is true to where it is false. This means th a t f3(io,p, n) = 1 — 7 (^0 , P, n) at p = a / 2
and p = 1 — a/2 . We must decide which we want to be small. For this paper, we choose
to make [3 small. This all but eliminates the possibility of rejecting H 0 when it is actually
true. As a trade-off, we have the problem of accepting H 0 when it may be false.
We can construct an explicit expression for (3\
n
?;°
P(io,P,n) = ^ P { i \ P , n ) +
(B. 6 )
i = n —io
i= 0
We want to limit ,d(zo,P, n) for any value of p satisfying H q, which means we want to limit
0(i 0 , a/2 , n ). There are two approaches to this: to fix the hypothesis (a) and change the test
(io), or to fix the test and change the hypothesis. Suppose we are given the hypothesis and
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want to determine a test th a t keeps (3 below some bound. This requires one to progressively
increase io while checking th a t f3(i0, a / 2 , n ) rem ains below th e desired bound. Alternatively,
if we have done our experiment and want to determ ine the highest possible significance we
can assign to it, then we know io and want to find a. Specifically, we set io to our measured
value of i, and find the smallest value of a such th a t [3 is still below some desired bound. For
simplicity, we could set a = (3, since we want b o th values to be low for a highly significant
result, and numerically solve for a\
f3{i0, a / 2 , n ) = a
(B.7)
We can then claim th a t our test has rejected the hypothesis, and has a probability of a type
I error (rejecting a true hypothesis) of a = [3.
For com putational purposes, it is useful to note th a t one of the sums in equation B . 6 will
contribute very little to the value of (3. Let us consider the contribution of the second sum,
when it is the smaller of the two. Its maximum value (while still being smaller) occurs when
p = 1/2. We will also estim ate the integer io + 1 ~ n a /2 . The value of io + 1 will typically
be lower than this.
n
P(i\a/2,n)
<
(i0 + l ) P ( n - i0\a /2 ,n )
<
(i 0 + l)rfi 0+1 ( l / 2 )n
<
( n a / 2 ) n na/2( l / 2 ) n
<
e x p { ln (n a / 2 ) + {na/2) ln(n)
i = n —io
+ n ln (l/2 )}
(B. 8 )
We want this to remain small. To assure th a t the a n In n term does not become larger than
the n ln ( l/2 ) term as n increases, we must have a < k j ln(n) for some k. Now we can check
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th e contrib u tio n of th e second sum (eq u atio n B.6) to (5 for some reasonable bounds:
a < 1 /ln n
n > 100
io < a n / 2 .
(B.9)
We find the value of the second sum in equation B . 6 to be well below IO- 4 0 whenever these
conditions are satisfied. Since the probabilities we test are all much higher th an 10“ 40, it
is safe to ignore the second sum in equation B . 6 when p <C 1/2. By symmetry, it is safe
to ignore the first sum when p
1/2. The f a c t s code makes use of this inform ation by
ignoring the second sum and mapping its input to a problem where only the first sum is
im portant.
B .1 .6
T h e C a se o f a O n e-S id ed T est
It may be useful for other applications to do a one-sided analysis, for example if one will
only consider a statistic x 0 to be unusual if it is lower th an m ost simulated statistics. Our
work can be repeated for th a t case:
Ho:
p e { a , 1]
(B.10)
We accept the hypothesis when i > ioio
P(io,p,n) = ^ 2 P ( i \ p , n )
(B .ll)
i=0
To specify a test, we require io to be small enough th a t the
P(io, a , n), is below some bound.
maximum value of /?, which is
Alternatively, to find the significance of previously obtained
results, we set i0 to be our measured value of i, set a = (3, and solve
f3(io,a,n) = a
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(B.12)
a = 0 .003,
i 0 =13,
n=19000
1
8
6
4
2
0
0
0
.
001
0 .002
0 .003
0 .004
P
Figure B .l: This figure plots 7 , the probability of a false acceptance of th e hypothesis (solid),
and (3, the probability of a false rejection (dashed), as a function of p, for a = 0.003, io = 13,
n = 19000. In order to keep (3 below a, we find th a t 7 becomes quite large for some values
of p. In order to avoid false rejections of th e hypothesis, we m ust allow false acceptances
sometimes. The region in which 7 is large becomes smaller as n increases.
to find the m aximum significance of our test. As in the double sided case, we can claim
th a t our test rejects the hypothesis Hq, and our test has a probability of a type I error
(rejecting a tru e hypothesis) of a = (3. The analysis for a single sided confidence interval
Hq : p
B .2
6
[0,1 — a ) can be m apped by sym m etry to the above problem.
C o n n ectin g to F req u en tist C on fid en ce Intervals
For an understanding of our statistical test th a t is sufficient to do calculations, the previous
section is enough, and the practical reader can stop here. For the enthusiastic reader, we
describe in this section how our test can also be derived using frequentist confidence intervals.
This connects the preceding discussion to our alternative derivation for the calculations in
LW04.
We use the same formalism in the previous section. There are n simulated statistics,
{xj}.
Exactly i of these fall below the test statistic x 0. The true probability of another
simulated statistic falling below the test statistic is p, which can be estim ated by p = i/n.
We construct an interval [p~ pp+] such th a t the true value of p will be inside the interval
at least a fraction 1 — (3 of the time. Note th a t this doesn’t mean we think the probability
of p € [p~ , p +] is at least
1
—[3 for the specific interval we construct, since p is not a random
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variable. For some given interval, [p~ , p +], either P(p E [p~,p+}) = 0 or P( p E [p~,p+]) = 1,
and we do not know which is correct. Instead it is helpful to think about many sets of
numbers {or,-} and the same xq. Each set will have n numbers, where i& of them fall below
x 0. We construct ^ = h / n as before. For each set we also construct the interval [p/,Pk\W hen we say the probability P (p E [p~,p+]) > 1 - / 3 , this is to be interpreted w ith p~ and
p + as the random variables, since they are b o th functions of the random variable
It is a
statem ent about w hether the interval falls around the true value of p and not about w hether
p falls in the interval.
The procedure for constructing this interval [p~ , p +] is detailed in chapter 20 of Kendall
& S tu art [96]. We state a few relevant results here.
We define a cumulative distribution function over the index i and a reverse cumulative
distribution function as follows:
i
cdi (i , p, n) = ^ 2 P ( j \ p , n )
(B .1 3 )
j=o
n
rcdf(*,p,n) = Y ^ P ( j \ p , n )
(B.14)
j=i
From here, we can pick a significance 1 —(3 for our test and then define p~ — p~(i, n, (3) and
p+ = p+(i,n,f3) by
Note
cdf (i, p+,n)
=
(3/2
(B.15)
rc d f(i,p ~ ,n )
=
(3/2
(B.16)
th a t here wehave constructed a double sided confidence interval.
single sided interval,for example p E [0 ,p +] with probability
If we wanted a
>1 — (3, then wewould have
p+ =p+(i^ n, (3) defined by
cdf (i,p+,n) =
P
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(B.17)
Probability confidence intervals
1
0.8
0.6
a
0 .4
0.2
0
Figure B.2: Double sided confidence intervals, a = 0.05, for all values of i from 0 to n = 50.
We will not prove this in m athem atical detail, bu t we will restate an argum ent given by [96]
for the double sided confidence intervals.
As [96] describes, figure B.2 can be constructed horizontally and read off vertically. We
construct it by fixing p and varying i (as we sum cdf() and rcdfQ) while we read off the
numbers p~ and p + at fixed i.
Consider a single value of p. Then look across at th e values of i th a t are m arked in blue
at th a t p value. By construction, they have probabilities th a t sum to > 1 — (3. Hence for
any p value, the probability of being in the blue band is > 1 — (3. B ut one is in the blue
band if and only if p E [p~,p+]- Hence the interval [p~,p+] will fall around the correct value
of p with probability > 1 — (3. I ’ll rephrase it once again: At any specific p value, values of
%th a t cause the interval [p~ , p +] to surround p will be chosen at least
1
—(3 of the time.
This frequentist confidence interval can now be used to construct a test of our hypothesis,
H q: p E ( a / 2 , 1 —a /2 ). We reject the hypothesis if and only if our interval [p~7p +] is entirely
outside of the interval ( a / 2 , 1 —a /2 ). In LW04, we required a double sided confidence interval
[p~jP+] to be outside of ( a / 2 , 1 —a /2 ). This was too conservative, since we only needed a
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single sided confidence interval to be outside of ( a / 2 , 1 — a /2 ). T h at is, for low values of p,
we reject H 0 if [0, p +] is completely below the interval ( a / 2 , 1 —a /2 ), which means p + < a /2 .
For high values of p, we reject H q if [p~, 1] is completely above the interval ( a / 2 , 1 — a /2 ) ,
which means
1
—a / 2 < p~.
This is equivalent to the test previously discussed, as long as the reasonable assum ptions
in equation B.9 hold so th a t one of the sums in equation B . 6 can be dropped.
B .3
T h e N e x t S tep
We can now quantify how large a statistical fluctuation the number xq is among the random
numbers {xj}. Given the number of statistics th a t fell below the test statistic, we can test
the hypothesis th a t x 0 is not in the tails (which have to tal probability a ) of the distribution.
If our test says th a t x 0 is indeed far out in the tails of the distribution, then we can decide
not to believe th a t Xq came from the same random number generator as the {xj} . This is
in fact the route taken in our paper.
A general caveat of any blind search for anomalies is th a t one cannot distinguish between
a large statistical fluctuation and a genuinely different parent distribution. This is often
casually referred to as the “non-dog” problem of non-G aussianity searches, since the class
of things which are not dogs is as large and as difficult to deal w ith as the class of CMB
models which are not Gaussian. The difficulty lies in defining something by negation.
To illustrate this idea, suppose we find a detection for some very small value of a .
Claiming th a t x 0 was not produced by the original random num ber generator (RNG) is
a dangerous thing to do w ithout specifying exactly w hat random num ber generator did
produce
X q.
Consider, for example, th a t the {ay} are Gaussian distributed w ith zero mean
and unit variance, and x 0 = 10. If the only alternative RNG is one which also has zero
mean, but has a variance of IO-100, then w hat we had originally claimed as evidence against
the original RNG would now be considered strong evidence for it.
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The ambiguity made apparent in this example holds true generally for any blind test of
non-Gaussianity and is not ju st a feature of our work. Consequently, we m ust either come
up w ith an alternative model to Gaussianity, or leave the interpretation to the reader.
B .4
U sers G u id e for f a c t s
To aid in the calculation of significance, we provide a publicly available code w ritten in c. It
is named f a c t s , which stands for a Frequentist’s Ally for the Calculation of Test Significance.
To compile the code on a typical Linux system, unzip and u n tar the file, enter the f a c t s
directory which was created, and type make. This will make th e executable f a c t s .
The
program is small enough th a t a makefile is not necessary, b u t I include it for convenience.
The syntax for the command can be retrieved by executing f a c t s w ithout any command
line arguments. The program takes from three to five argum ents, facts s n i [alpha [beta]]
• s: this value is either
1
or
2
for a single or double sided test.
•
n: this is the number of simulated statistics calculated.
•
i:
this is the number of simulated statistics th a t fell below the test statistic. If i is a
negative number, the code prints out information on w hat values of i will reject the
hypothesis.
• a lp h a : this is a number between 0 and 1 th a t param eterizes our hypothesis.
For a
single sided test, the hypothesis is p G (ct, 1]. For a double sided test, the hypothesis
is p £ (a / 2 , 1 — a/2). If not specified, the default value is a = 0.05.
• b e ta : this is a num ber between 0 and 1 th a t gives an upper limit on the probability
of a type I error (rejection of the hypothesis when the hypothesis is actually true). If
not specified, it takes the value of a.
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The code takes the values of s, n, a, and (3 and constructs a test. If i satisfies th a t test
then the hypothesis is accepted, otherwise it is rejected. The code prints out inform ation
w hether i satisfies the test, and what the minimum values of a and (3 are for which i will
satisfy the test.
A few examples follow. The next com mand determines w hether a two-sided test w ith a
0.02 maximum probability of a type I error will accept the hypothesis th a t p £ (0.005, 0.995)
when only 17 out of 10000 statistics fell below the test statistic.
f a c t s 2 10000 17 0.01 0 .0 2
The command below determines if 5 out of 1000 statistics falling below th e test statistic is
few enough to reject the hypothesis p £ (0.025,0.975) when the test has a 5% maximum
chance of a type I error.
f a c t s 2 1000 5
While this program is reasonably fast, it is not optimized for pathological requests. It is
slower for larger values of n, i, a , and (3.
148
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A p p e n d ix C
N
over 47r
n i v
This appendix discusses the factor of N p;x/4 tt th a t arises because the spherical harm onic
transform s are not unitary m atrices. They are only proportional to unitary m atrices, and
the constant of proportionality is ^ N p-lx/Air. This is relevant for our MAGIC code, discussed
in chapter 5.
C .l
Signal S am p lin g
We have a signal, denoted s, which is distributed (by cosmic variance) according to
p ( s | s ) = ^ f e r xp (
We have multiple channels.
channel is
-
^
)
(gi)
The m ap for each channel is m ,, and the noise on each
, distributed according to
(G2)
The beam smoothing (by an azim uthally symmetric beam) is represented by th e m atrix Bj
for each channel.
m i = B is + rij
m ; is the observed map of the sky.
149
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(C.3)
P ( s, { m ills , {N j}) = P ( s, K } |S , {N j}) = P (s|S ) n
(C.4)
21ogP(s, {nij}|S, {N j}) = sr S *s + y ^ ( m j —B js)r N- 1 (m i — B js) + const
(C.5)
where the constant of proportionality is independent of s, {nij}, or {nj}.
Now, let us pause and think about th e transform ation between pixel space and spherical
harmonic space. Everything can be done in either pixel space or spherical harmonic space,
b u t it will be com putationally convenient to switch between the two. Let us deal w ith s, Bj,
and S in harmonic space, and n ij,n j, and N j in pixel space.
Our inner product of two scalar functions / : S 2 —>K and g : S'2 —>■R. on th e sphere will
be
(C. 6 )
Let T be the transform ation from pixel space to harm onic space. Harmonic space is
orthonorm al, and pixel space is orthogonal, but pixel space is not normalized. T h at means
T is not an orthogonal transform ation. T h a t is,
T
1 7^
T r . If we normalize pixel space
then the orthogonal transform ation from normalized pixel space to harm onic space is U ,
where L E 1 = U T, so U is orthogonal. Additionally, we have
and
Nw
t
(C. 8 )
which will be useful momentarily.
Now let us rewrite the log likelihood w ith the transform ation matrices. Let C = P ( s, {m,} [S, {Nj}).
21og£ = sTS 1s + ^ ( m j —T
1 B js)TN
i ^ m ; —T
1 B is)
+ const
150
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.9)
Now expand th e p ro d u ct an d com plete th e square.
— 21og£ = sTS _1s + ^ m f N ^ r r i i — (T _1B is)TN l" 1m i
i
m f N ^ T ^ B i S + ( T _1B is )TN ^ 1T _1BjS + const
-
(C.10)
Since N j and B t are symmetric for all i,
- 2 log C = sTS -1s +
- 2sTB i (T ~ 1)TN r 1m i
i
+ sTB i(T - 1 )TN “ 1 T ~ 1BjS + const
21og£
= sT ■ S ” 1 + Y
(C .ll)
B i(T _ 1 )TN “ 1 T ^ 1 B i S+
i
Y \ —2sTB i (T _ 1 )TN “ 1 m i + m f N ^ n i j + const
(C.12)
and by completing the square we find
T
lo g £
a
S^ 1 + Y
s
B j(T ” 1 )TN “ 1 T _1Bj
i
x
S-
1
+ y ^ B i ( T - 1 )r N - 1 T - 1 B J
i
X
s
S ' 1 + J ] B i( T - 1)r N - 1T - 1B i
The proportionality constant in this equation may now depend on rrq, bu t not on s. This
151
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 .
likelihood is clearly G aussian, w ith m ean
-l
(0 .1 4 )
x =
and covariance
(C.15)
It only remains to sample from this distribution. We construct a random variable y
which has m ean
0
and the desired covariance by
-l
y=
S ~V2£
+ 5Z
Xi
(C.16)
where £ and the Xi are random vectors, where each component is an independent Gaussian
unit variate (has mean zero and unit variance). Note th a t £ is in spherical harm onic space
and the Xi are in pixel space. To construct
we create it in pixel space and then transform
w ith the U operator. We will later use T to transform , and this will give us a factor of
y 7N pix4n.
After a few m atrix m ultiplications to make things symmetric, we sample (x + y) from
the following equation:
^1 + S 1/2
B i(T _ 1 )TN b 1 T _ 1 B jS 1//2^ S -1/2(x + y)
= £ + S 1/ 2
B j(T _1)t Cn ~1/2X i + N - 'm , )
152
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.17)
w here 1 is th e identity m atrix .
+ A ^ S 1/2 ^
47T
1
B iT N r^ -^ iS
172 j
S _1/2(x + y)
^+ ^ s
An
C .2
1/ 2
Y
B *T ( N r V2Xi + N - Xm i)
(C.18)
H ow th e P ro b lem w as S olved B efore
The Ce values are stored in the code w ith the assum ption of normalized spherical harmonics.
Whenever the m atrix S is applied, though, the Cb values are rescaled so as to make the pixels
normalized. T hat is, values C'e are used instead of Cb, where
C'f =
*
4?r
' 1 , N pix S1/2
N pix \
An
4tt
iVpix
Ct
S' =
iVpix
S
(C.19)
V
;
BiT N r 1T"1BiS1/2j S~1/2(x + y)
477 h + ^ S
AW \
47r
^1 + S 1 /2
47t
1/ 2
J ] B , T ( N r 1/ 2 x i + N 7 1 m ,) )
(C.20)
B lT N r 1 T - 1 B iS 1/2 j S “ 1/2(x + y)
4
+ S V2
1Nvpix
B lT ( N r 1/2Xt + N ^ m , )
(C.21)
where factors of N v\x/An have now been absorbed into the S matrices. At present, £ is in
spherical harmonic space. If we sample £pix in pixel space and then convert to spherical
153
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 .
harmonic space, we get the desired factor of yj 4i r/ N pix.
^1
+ S 1 /2
B iT N ^ 1 T _ 1 B iS 1/2^ S “ 1/2(x + y)
= T £pbt + S 1/ 2 ^
C .3
B jT ( N - 1/2Xl + N r 1!!!,)
(C.22)
P o la riza tio n
We now extend our definition of an inner product on th e sphere to complex functions:
( f , 9) =
J
f*(°> 4>)g(9; 4>) sin 0 d6 dp
(C.23)
The { 2 Wm} spherical harmonics are a normalized, orthogonal set of basis functions for
complex fields on the sphere. The {Yem} spherical harmonics are another such set. Let
be a set of non-normalized, orthogonal basis functions on the sphere. All of these
basis functions are maps from location on the sphere to the complex plane, and %= y[—\.
.
ft = {
1
inside pixel i
(C.24)
0
(Pj,Pk)
(pv Yem)
otherwise
= S3,k^~
^pix
(C.25)
Atx
= Yem( j ) — -
(C.26)
*pix
(1a ,
2W n )
=
47T
2 Yim( j ) - —
’ pix
154
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(C.27)
We’ll need to represent a complex field X on th e sphere. O ur definitions:
£m
O'lm — y ^ 'f-imjQ'j
(C.29)
3
Working w ith these equations, we find:
(Ylm, X ) = ( h m, ^
^ ^ Ql'm' (Wmi W'm')
I'm !
^im
(C.30)
I 'm '
/~T7~
(£;,
= (&> £ a^ ' ) =
j'
^im = {^£mi X ’j
(hm ; ^
S
jP j)
3
a3 = ^ ( P v x ) = ~ ~ ( P j > Y l aimY^
dm
" X l ’rP j') =
h
(C -31)
P“
^ ^
3
=
tmi Pj}
^ Imj O'j
^
(C.32)
3
im
= I]
dm
(C.33)
Hence
1 , m, = ( W ; >
( T - ' ) (m. = ^ ( p „ Y lm) =
( T - ‘)» =
(C.34)
The above is also true if we replace the spin 0 spherical harmonic w ith a spin 2 spherical
harmonic. We find the same factor of N ^ /A -n th a t we found in the tem perature-only case.
Our pixel space is T, Q, U, and our spherical harmonic space is T, E , B . We can use
similar reasoning to the above for this case too. The pixel space vectors each have length
V * 7r/Npix, as above, and the spherical harmonics are normalized in each of the T, E, and B
spaces. So we have exactly the same equation as before, but now all the vectors are 3 times
155
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as long.
1 + S1/2 ^
v
B . T N ^ T ^ B i S ^ 2\ S ” 1/2(x + y)
i
J
= T
+ S 1/2 J ] B ,T ( N “ 1/2x, + N r 1!!!,)
156
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.35)
A p p en d ix D
P reco n d itio n in g C o n ju g a te G rad ien t
D escen t
The equation th a t m ust be solved in the signal sampling step of the Gibbs sampler is the
sum of equations 5.8 and 5.9,
[l + S 1/2N -1 S 1/2] S -1/2(x + y) = £ + S 1/2N " 1/2x + S 1/2N _1m
(D .l)
The symmetric, positive-definite m atrix on the left hand side is large and not sparse, so it
is much easier to solve this equation for S ~ 1,/2(x + y) by preconditioned conjugate gradient
descent th an by directly inverting it. This involves creating an approxim ation of the m atrix
1 + S 1/2N ” 1S 1/2, which we do in spherical harm onic space.
In this appendix, we provide the details for a polarized generalization of the precondi­
tioner of [84]. The m ajor p art of this work is to com pute the noise covariance m atrix N
in harmonic space. First, we discuss the spherical harm onic transform s. Then we redefine
N in a more useful pixel basis, namely [T, (Q + i U ) / y / 2, (Q — iU)/y/2] instead of [T, Q, U}.
Next, we determine w hat the spherical harm onic transform s are for this new basis. Finally,
we take block diagonal elements of N in pixel space, and from th a t com pute block diagonal
elements of N in harmonic space. These elements are used to approxim ate N in harmonic
space—this approxim ation is used for our preconditioner.
This appendix will use the sign conventions of [97] and the HEALPix primer [18]. Due
to some ambiguity as to whether the spherical harmonics should be denoted as matrices,
because they have multiple indices, or functions, because they are functions of 8 and 0, this
appendix will now depart from the previous convention of making all vectors and matrices
157
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bold.
D .l
Spherical H arm on ic Transform s
D .l.l
F irst D e fin itio n s
The following guide may help make sense of th e indices used throughout this appendix.
pixel numbers
I, H!
(D-2)
—» m ultipole moment
(D-3)
—>angular m om entum component
(D-4)
p, p
—»polarization: T, E, or B
(D-5)
q, qr
—>polarization: T, Q or U
(D.6)
rri, m!
n
N pix
(D.7)
We will use commas to separate array indices when it makes the expression less ambiguous.
158
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From equation 9 of th e H ealP ix prim er, we have
£
OO
(D-8)
£=0 m = —£
X l,£ r n
=
X2,im =
(2Y im
+ -2 Yirn) / 2 = “ 2Y I m +
( 2 Yem
—_2 Ypm) / 2 = - 2^m “
OO
(D.9)
(D.10)
YYp_r
£
EE
(m i)
£=0 m ——£
00
^
£
h
5 3 a E ,£ m { — ~ 2 Y l m ~
•
2Y £ _ m ) + a B , £ m { - - 2Y p m +
2 ^ -m )
0=0 m = —£
(D.12)
11 = y]] y t Q£^m(lX 2,£m) +
£=0 m = - l
00
u =
£
55 53 aE^m ^n YY£m —
(D.13)
Xi^m)
2^*_m)+ a B,£m,{—y 2Ypm —
( l)m
2E£*_m)
^=0 m = —£
(D.14)
So we find:
Ypm(9, (/>)
Yf£ m piq
if p = T
and
q —T
2Ylm ~ K “ 1)m 2 Ye*_m
if p = E
and q = Q
2 V«m +
if P = B
and q = Q
P =E
and q = U
if p = B
and q = U
K _ 1 )m 2y /-m
+ | 2 Ygm 2 Yim
- \
2^t-m
2Ye*_m
0
if
otherwise
where the spherical harmonics are evaluated at (9, </>), the location of pixel i.
159
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(D.15)
We see th a t Ytmi is a 3 x 3 m a trix for each value of £, m , an d i. Specifically:
^
o
Ypm
1 v
0
2 2U m
0
v
21 2-*£?n
— 1 (_i\m v*
2 1
1
2
*(
2
v ^/
i
_i_ i (
V
1 \m
(D.16)
v *
'2 2^m -r 2 V XJ 2-** _m
£,—r,
V*
2-*r
For clarity, we can label th e rows and columns.
(
y
1 tm i
T
_
\
D .1 .2
T
E
B
Ynm
0
0
Q
0
u
0
2 2b m
+|
2 b m
K
- i r
v*
Xim +
— i( - l) " * 2 v*
■‘ £ , —777
(D.17)
i(
1
2V
\m
7
y *
2
£ ,— m
X*-n,
2 2Y im
U n ita r ity C h eck
Now, the question might arise: is this unitary?
Y,m
y
y t
_
0
0
-\iY „ n+ \(-\T Y Y l
0
/
X
\
y
0
\
0
*
1 im
n
u
0
- i 2^ - | ( - l ) m 2 ^ ,-r
0
+ i 2Y;m - i ( - i r 2Yt,-r
y
YtmY;m
0
0
\ ( 2 Yim X l m + X t - m X l - r ^ )
0
)
+ y u m - | ( - i ) m2 r y m
- h Y L - \ ( - Y r * Y t,-m j
\
0
1 ( v
2
V*
\2-*im 2-*
£m
_
V
V*
)
2-*!rm
2^i-
(D.18)
^ (- 2^m 2^m + 2b,-m 2^*_m) \ { X l m X X l , - m X l - - ,
by M athem atica.
p27T
pTX
/
d<f> / sin 0 dfl y ,mi(0, 0) Y L (0 , 0) =
'0
do
1
160
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(D.19)
w here th e 6 , </> n o ta tio n is red u n d a n t w ith i. H ence
47T
E
N ■ (
JVpIX
y '"“y h , = 1
(D 20)
means the same thing. The integral is the sum over i, and is w hat we mean by a m atrix
m ultiplication, here. This shows th a t the change of basis m atrix Y is unitary, up to a factor
of A/pix/47r, which shows up when converting th e integral to a sum. This factor is dealt w ith
in detail in appendix C.
More formally, this is w hat we mean by unitarity:
Y Y ^ = —j
-'*pix
D .1 .3
^ ^ YgmpjiqYj'm'p',i,q ~ ^£mp,£'m'p' =
(b-21)
i,<7
D o u b le C h eck in g t h e In v erse
Now we try a different approach to getting the inverse relation. See equation 7 of the Healpix
Primer.
a 2,£rn
~
/ (2y ;j (q + zU)
Js2
(D.22)
0-2 ,im
=
( - 1 )ma*2A- m
(D.23)
(
(D.24)
®E ,£m
d'2,£m
2,On)/2
(D.25)
& E ,im
& B ,lm
2 Ye^
m) { Q - i U )
(D.26)
“
=
{—0-2 ,tm + tt_ 2.^m) / ( 2 *)
(D.27)
2* n0*2,im
(D.28)
0'B,£m
& BXm
1
=
=
[ W J i Q + iU )z Js2
21
>s2
* f 1 lm/T
1 2,i,—m
1 1 U Y £J(Q + iU ) ~
2 7 s2
(D.29)
IS2
161
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 .
/
0
0
& E ,£m
\
/
t
\
i
y ^B/m J
This looks very similar to
/
(D.31)
Js2
r
Z_^
Z_^
Y i m p i q M iq
P“
i= 0
q& {T ,Q ,U }
(D.32)
as desired.
Now we have w ritten down the formalism for converting between pixel space and spherical
harmonic space using spin 2 spherical harmonics.
D .1 .4
T h e B a sis C o n v ersio n E q u a tio n s
Now we summarize. Let Memp be a m ap in spherical harmonic space. T he coefficients Memp
are complex. Then
t
OO
M iq =
E
E E
£=0
YimpiqMgrnp
(D.33)
m = —£ p = T ,E ,B
is the m ap in pixel space. The values M iq are real.
To invert the process, we have:
5^
pix
5]
Y;mpiqMi
J=0 qe{T,Q,u}
162
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(D.34)
D .2
R ed o N o ise A p p ro x im a tio n
D .2 .1
R e d e fin in g N
We will find it very useful to redefine N for a different pixel basis. Instead of [T, Q, U], we
will use [T,{Q + iU )/y /2 , ( Q - i U ) / y / 2 \ .
Let us define N to be
(
N =
T
^
(D.35)
(Q + iU )/y /2
oQ + i U ) / V 2
y (Q - i U ) / V 2 )
y (Q -iU )/y /2 )
(
\
w
T
(T T)
<:T{Q - i U ) / V 2)
+iU)/V2)
(T (Q + iU )/y /2 )
{{Q + i U ) { Q - i U ) / 2). {{Q + iU ){ Q + i U ) / 2}
( T ( Q - i U ) / y / 2)
{ { Q - i U ) { Q - i U ) / 2)
/
2 (T T )
V 2 (TQ) + i V 2 (TU)
(Q2) +
\
(D.36)
((Q - i* 7 )(Q + *J7)/2)
\/2 (T Q ) - iV 2 (T U )
1
2
(T (Q
(t/2)
V v^(Tg) - iv/2(TC7) (Q2) - (t/2) - 2 z([/g)
(TQ) + i^/2(TU )
\
(D.37)
(g2) - (f/2) + 2 z([/g)
(Q2) + (C72)
/
which can be calculated from the previous inform ation. Note th a t N is herm itian, as one
would expect from a covariance m atrix of complex vectors. Also, A^_1 is also completely
determined, as long as det N is nonzero (which we will assume).
163
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D .2 .2
R e c o m p u tin g S p h er ica l H a r m o n ic T ran sform s
We retu rn to the following equation:
Yftm\
i/2 2^ m
Y^rnpiq
^
(
-
f
i
if p = T
and q = T
if p = E
and
q = Q+
and
q = Q~
f
P = £7
(D.38)
<
- ^
if p = B
2
+ U - l ) m 2Ye:_m if P = B
0
and q = Q +
q = Q-
and
otherwise
Here, Q + refers to the (Q + iU ) / \ / 2 basis element and Q - refers to the (Q
iU )/V 2
basis element.
The next equations derive from equations D.22 and following
/
&Tlm
\
(
&E,£m
0
I s2
0
&B,£m
/
\
^E,im
0
1 v*
2 2 £m
12 2v*On
0
0
\
T
(D.39)
+ iU
4 ( - 1)m 2T€,
\
-iU
/
T
On
IS2
\ /
1 v*
'^2 2 O™
2 V*
~^/2 2 £rn
\
(D.40)
{Q + iU )/V 2
(Q -zU )/V 2
164
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 .
J
£
OO
Q + iU
EE
2 2^ £ m ^ B ,£ m
(D-41)
y y —2^ £ m ° * E ,£ m + * 2Y £ m a B',£m
r
(D.42)
2 ^ fort®E ,£ m
1=0 m = —£
oo
Q -iU
£
^=0 m = —£
oo
£
£=0 m = —£
oo ^
V
^ £ ~ m a * E ,t- m
(D.43)
+ Z2 ^ * -ma
- ( - l ) m2Yl_maE,,m+ i( - l) m
(D.44)
£=0 m = ~ £
And so
T
\ /
dfrri
E
E
■T=0
(■Q + iU )/y/2
0
V2 2^ m
o-r/m
\
& E .£m
m = —£
{Q -iU )/V 2
0
/
y
& B ,£m
J
(D.45)
In short,
oo
£
E
E E *
£=o m=-^Pe{r,£;,B}
(D.46)
M itm p
y
(D.47)
Mi£mp
An
N,
M lq
£mp,iq-^J-£m p
[ Y£mpiqM iq
Js2
yyy ^m p M ^ q
(D.48)
P'x z=0 g
D .2 .3
C o m p u tin g N in H a rm o n ic S p a ce
We have the inverse noise covariance m atrix, diagonal over pixels:
Wig.W =
165
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 .
(D.49)
We w ant to find th e b ehavior of th is m a trix in spherical harm onic space.
^£m p,£'m 'p' =
Nr
X ^ ^ m p iq ^ iq p 'q ’^ 'm 'p 'i'q ’
-^Vpix
^ i,z,q,q'
.
(D.50)
,
We have assumed TV-1 to be block diagonal. Now we decompose th e elements of each
block in harmonic space. We will use a spin-0 harm onic for the diagonal elements of each
block, and other harm onics for the other components according to their spins.
N i T ,i T
— X>
(D.51)
£m
N iQ+,iQ+
=
E
(D.52)
£m
N iQ -,iQ -
=
E
q _q _
(D.53)
£m
^ iT ,iQ +
= XZ —
2Wm iP'i ) ^ £ m 1TQ+
£m
^ iT,iQ~
=
E
(D.54)
(D.55)
£m
^ iQ +,iQ~
=
E
£m
Note the spin —4 harm onic in the last equation.
Now we will w rite out N as several m atrix multiplications.
166
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(D.56)
All Yem values are evaluated at (9,</>), which is at the location of pixel j.
(
-i
N,im p ,im p '
t v -1
N£~mlT ,£ m E
N£ m T ,£ m B
N £m .E ,£m T
At-1
Im E /m
E
A/-1
£ m E ,£ m B
£ m B ,£ m E
l s £ m B ,£ m B
£ m T ,£ m T
Y AIm B /m T
AT” 1
(D.57)
A/--1
On
0
0
N jT ,jT
1 2v*
^ /2
i
V*
V22
IV
-1
jT,jQ+
N jT,jQ-
-1
\
N j Q +,jT
N jQ+JQ+ N jQ+,jQ~
N jQ - ,jT
N jQ~,jQ+
~x
N jQ~jQ~ J
Y£m
0
0
0
“ 71 2 ^ m
“ 73 2Y^m
x
^
0
(D.58)
2y£_ro + ^ ( _ i ) ™ 2y^_m ^
/ y(-m
o
o
x
'5 2
oYf''m" A \ « m " T T
2Y£"m"Ni„m,,Q+T
O'm"
\
Yrm
X
0
V 0
-1
£"m "Q ~ T
-2 X £ "m "
A \ n m " T Q+
0^£”m"Ni„rn„g+Q+
-4
0
Q-
/
^
(D.59)
- 72 2^ m
+ 7 3 (_ 1 )m 2^*_m
\
iY£"m" N e„m„Q+Q-
A ei/m » Q - Q + 0Yi"m" A
0
\72 2^ m
2 ^ " to" A \ " m " T Q ~
y
167
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
TV-1
impflmp'
0
0
^
IS2
Y
0
+ “ ^ ( — l ) m _ 2 y ? !- r n
o Y i '/ m " N \ n rni t j 'j '
E
—2 Y i " m "
2 Y i " m " N l „m „ Q + T
—l ) m 2^,~ m
£iirniij'Q +
^
2 Y l" m " N £ " m " T Q ~
0 Y i " m " N e,lrn„ Q + Q +
4 Y l " m " N i „m „ Q + Q
- 4 Y t" m ," N i„m „ Q -Q +
o Y £ " r n i i N £ /, r n „ Q _ Q ^
l" m "
y -2 Y e " m " N l „ r n „ Q _ T
Y,i m
0
\
0
0
^ 2 2 Y(_m,
0
—2
^2
J
(D.60)
Xlm
- 2X 1711
A nd for reference, we re-write our main integral:
c2ir
sind dO sYe*m{0> 4>) s X m '{ 9 , 0) s"Ye»m"(9, 4>) 5S^ +S
,+m+ w + r . I W + l ) ( 2 f + l ) ( 2 f + 1) I
= i-iy
47T
f
<" C
-s
s'
a
1
—m
s"
m! m"
(D.61)
r2iz
/*7T
sind
IQ
sYe,m(9, <fi) sX 'm '{9, 4>) s'X "m "{9 1<$>) (5s+s/+s//)0
JO
r+r'+r"
/ ( ^ A 1)(2^' + 1)(2£" + 1) ^ £
47T
= (-i)
r+r'+r''
s
(2£ + l)(2A + l)(2r + l)
47T
s
£' £" \
s'
^
?
s
s
s"
(
I
f
I” ^
m
m! m"
(D.62)
y
\
/ V
m" m
rr!
y
(D.63)
This allows us to com pute any of the elements of N in harm onic space, as long as we
can calculate arbitrary spin transform s (specifically, transform s for spin: -4, -2, 0, 2, and 4).
168
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
We also need to be able to calculate W igner 3-j symbols, but there exists Fortran code to
do th a t.
D .3
D .3 .1
A few rem ain in g d eta ils on spin
S p in 2 to E B an d b ack a g a in
To convert from spin 2 spherical harmonics to E and B modes, we need equation D.22 and
following.
aE/m
— —2 a^ em ~~
O -B /m
=
a 2,£m
«2im
D .3 .2
- 0 ^ 2 ,t m
~
(D.64)
—
= -aE,em - iaB,em
a 2 ,t,-m
(D.65)
for all m
(D.66)
=
if
m<0
(D.67)
A w ay arou n d th e sp in -4 p ro b lem
Spin 4 spherical harmonics have not yet been implemented in the HEALPix code. However,
there is a way to compute the m = 0 harmonics from the spin 0 harmonics, which have been
implemented. Fortunately, the m = 0 harmonics are the only ones we need.
For integer s, we have
hf _ i
,V W * .
A w
d<- J 9 ) e"""
169
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
<D'68)
sY,,„,(M) = ( - 1 r
e,m*
=
,U„(9,<M
(D.69)
(D.70)
=
(-1 )~m - mYc_-,($,</>) e"* em *
(D.71)
=
oY l - s I
(D.72)
O A )^
where we have specialized to m = 0 in the last equation.
Now, let us decompose an azim uthally symmetric complex function (m = 0) into it’s
spin s spherical harmonics.
N(e,cp) = J 2 N eosyio(0A)
i
=
=
=
j «>?„(«, •WA'(M)
J
J
(D.75)
Yt% ( e , < p ) [ e - ‘“» N ( e , 4 , ) ]
(D.76)
So we can get allthe spin s coefficients of an azim uthally symm etric complex
usual spin 0
(D.74)
Yl% , ( 6 , 4 > ) e - l* N ( e , t )
= [<r‘*JV(0,
doing the
(D.73)
(D.77)
field by
transform of a suitably modified field. Modifying the fieldisfast.
170
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 .
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C urriculum V ita e
Office:
University of Illinois at U rbana-Cham paign
D epartm ent of Physics
1110 West Green Street
U rb an a,IL 61801-3080
217-333-2807
dlar son 1@uiuc.edu
W o r k /P o sitio n s H eld
• Postdoctoral position at Johns Hopkins University w ith W M A P collaboration, Septem­
ber 2006.
• Research w ith Prof. Benjamin W andelt, Summer 2002-Sum m er 2006.
• Research Assistant funded through Prof.
W andelt, Summer 2005-Present.
• Discussion Teaching A ssistant for Physics 102 (U ndergraduate Electricity h
Mag­
netism), Spring 2005.
• Research Assistant funded through Prof.
W andelt, Summ er 2003-Fall 2004.
• Lab Teaching A ssistant for Physics 114 (U ndergraduate Q uantum Physics),Spring
2003.
180
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 .
• Grader for Physics 481 (G raduate Q uantum Mechanics), Fall 2002.
• Recipient of a UIUC fellowship, Fall 2001-Sum m er 2002.
• Research Assistant in Prof. Dale Van H arlingen’s condensed m atter lab, UIUC, JunAug 2001.
• U ndergraduate student at the L aboratory for Physical Sciences, College Park, M ary­
land, Jun-Aug 2000 and Jan 2001. Helped to purchase, set up, design, and program
equipment for Dr. K eith Schwab’s quantum com puting lab.
• Vice President, Society for Physics Students, UMCP, Spring 2000.
• U ndergraduate Researcher, Jefferson Labs, VA, Jun-A ug 1999.
Helped design and
construct a system for testing particle detectors.
• U ndergraduate Researcher, H am pton University, VA, Jul-Aug 1998. Program m ed a
user interface for a laser radar system.
E d u cation
• Ph.D. in Physics, Summer 2006. University of Illinois at U rbana-Cham paign (UIUC).
• M.Sc. in Physics, Fall 2003. University of Illinois at U rbana-Cham paign.
• B.Sc. in Physics and M athem atics, May 2001. University of M aryland at College Park
(UMCP).
C o n feren ces/T a lk s
• External Correlations of the CMB and Cosmology, May 25-27, 2006. Fermi National
Accelerator Laboratory.
181
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 .
• US Planck D ata Analysis Review, May 9-10, 2006. Jet Propulsion Laboratory, Cali­
fornia. Talk title: Polarized Gibbs Sampling: W M AP Results.
• US Planck D ata Analysis Review, November 18-19, 2005. Jet Propulsion Labora­
tory, California. Talk title: Polarized MAGIC (progress report on development of the
MAGIC code).
• SF05 Cosmology Summer Workshop, July 5-22, 2005. Saint Jo h n ’s College, Santa Fe,
New Mexico.
h ttp ://t8 w e b .Ianl.gov/people/salm an/sf05/.
• April APS Meeting, April 16-19, 2005. Tampa, Florida. Talk title: Hot and Cold Spot
Tests for non-G aussianity in the Wilkinson Microwave A nisotropy Probe (W MAP)
Cosmic Microwave Background (CMB) data.
• US Planck D ata Analysis Review, November 9-10, 2004. Jet Propulsion Laboratory,
California. Talk title: H otspot Analysis for Planck D ata.
• SF04 Cosmology Summer Workshop, July 5-23, 2004. Saint Jo h n ’s College, Santa Fe,
New Mexico.
h ttp ://t8w eb.lanl.gov/people/salm an /sf0 4 /. Talk title: The Local Extrem a of the
CMB.
• Prelim inary Exam ination. April 26, 2004. University of Illinois at Urbana-Cham paign.
Talk title: Testing the Standard Model of Cosmology.
• Cosmology, Particles and Strings, June 30-July 11, 2003. In stitu te for Advanced Study,
Princeton, New Jersey, http://w w w .adm in.ias.edu/pitp/archive2003.htm l
• Great Lakes Cosmology VII, May 15-18, 2003. University of Michigan, Ann Arbor.
• Great Lakes Cosmology VI, 2002, Chicago, Illinois.
182
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 .
H onors and A w ard s
• One year UIUC fellowship (2001-2002).
• Phi B eta K appa Honor Society.
• Golden Key N ational Honor Society.
• Society of Physics Students.
• N ational Society of Collegiate Scholars.
• Gemstone Program at University of M aryland.
• President’s Scholarship, 1997-2001 ($4500 per year).
• Distinguished Scholarship, 1997-2000 ($3000 per year).
C om p u ter E x p erien ce
• Parallel programming at National Center for Supercom puting Applications (NCSA).
• Languages: experience with Fortran 90, C + + , C, Perl, M athem atica, Html, Bash,
Ruby, OpenGL, Labview VI, Java, QBasic, M atlab, Turbo Pascal.
P rofession al S o cieties
• American Physical Society
183
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 .
P u b lica tio n s
1. D. L. Larson, H. K. Eriksen, B. D. W andelt, K. M. Gorski, Greg Huey, J. B. Jewell,
I. J. O ’Dwyer.
E stim ation of Polarized Power Spectra by Gibbs sampling,
astro-
ph/0608007, August 2006. Subm itted to Astrophysical Journal.
2. H. K. Eriksen, Greg Huey, R. Saha, F. K. Hansen, J. Dick, A. J. Banday, K. M.
Gorski, P. Jain, J. B. Jewell, L. Knox, D. L. Larson, I. J. O ’Dwyer, T. Souradeep,
B. D. W andelt A re-analysis of th e three-year W M AP tem perature power spectrum
and likelihood, astro-ph/0606088, June 2006. Subm itted to Astrophysical Journal.
3. David L. Larson and Benjamin D. W andelt. A Statistically R obust 3-Sigma Detection
of non-Gaussianity in the W M AP D ata Using Hot and Cold Spots, astro-ph/0505046,
May 2005. Subm itted to Physical Review D.
4. M. Chu, H. K. Eriksen, L. Knox, K. M. Gorski, J. B. Jewell, D. L. Larson, I. J.
O ’Dwyer, and B. D. W andelt. Cosmological Param eter C onstraints as Derived from
the W ilkinson Microwave Anisotropy Probe D ata via Gibbs Sampling and th e BlackwellRao Estim ator. Physical Review D. 71(103002), 2005. astro-ph/0411737.
5. H. K. Eriksen, I. J. O ’Dwyer, J. B. Jewell, B. D. W andelt, D. L. Larson, K. M. Gorski,
S. Levin, A. J. Banday, and P. B. Lilje.
Power Spectrum Estim ation from High-
Resolution Maps by Gibbs Sampling. Astrophysical Journal Supplement, 155:227-241,
2004. astro-ph/0407028.
6. I. J. O ’Dwyer, H. K. Eriksen, B. D. W andelt, J. B. Jewell, D. L. Larson, K. M. Gorski,
A. J. Banday, S. Levin, and P. B. Lilje.
First-Year W M AP D ata.
Bayesian Power Spectrum Analysis of the
Astrophysical Journal Letters, 617:L99-102, 2004.
astro-
ph/0407027.
7. David L. Larson and Benjamin D. W andelt. The Hot and Cold Spots in the Wilkinson
184
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 .
Microwave A nisotropy Probe D a ta are Not Hot and Cold Enough.
Astrophysical
Journal Letters, 613:L85-L88, 2004. astro-ph/0404037.
8. Benjamin D. W andelt, David L. Larson, and A run Lakshm inarayanan. Global, Exact
Cosmic Microwave Background D a ta Analysis Using Gibbs Sampling. Physical Review
D, 70(083511), 2004. astro -p h /0 3 10080.
185
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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