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Cosmology from cosmic microwave background and large -scale structure

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COSMOLOGY FROM COSMIC MICROWAVE
BACKGROUND AND LARGE SCALE STRUCTURE
YONGZHONG XU
A Dissertation
in
PHYSICS AND ASTRONOMY
Presented to the Faculties of the University of Pennsylvania in Partial
Fulfillment of the Requirements for the Degree of Doctor of Philosophy.
2003
Max Tegmark
Supervisor of Dissertation
Randall D. Kamien
G raduate Group Chair
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DEDICATION
To my wife Peifang and my two daughters Anqi and Yueling
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ACKNOW LEDGEM ENTS
I have many people to thank for their influence and help in the journey of finishing
this dissertation. Special thanks to my advisor, Max Tegmark, for his guidance
through this whole process, without whom I could never have accomplished this.
W ith Max's help, I have improved my research abilities and developed the necessary
skills to be an independent researcher and for that I will always be thankful.
I am also grateful to everyone in the astro group here at PENN, especially Mark
Devlin, Charles Alcock, Angelica de Oliveira-Costa and Bhuvnesh Jain to whom I
could always turn for com m ents and opinions whenever needed. I would also like to
thank Lyman Page at Princeton University for very useful comments, Andrew J. S.
Hamilton at University of Colorado for his consistent help with the Powerline package
and many helpful discussions about various topics and Sergei F. Shandarin and Hume
A. Feldman at University of Kansas for their brilliant ideas on non-Gaussianity test.
No graduate school experience would be complete without a cast of fellow grad
students. I am thankful to my office mates Xiaomin Wang, Reiko Nakajima, Peter
Allen, Kelle Cruz, Derek Dolney and Taryn Nihei for insightful dismissions and for
providing this friendly environment. I have had a wonderful time here.
Last, but certainly not least, I devote my thanks to my family. I thank my
parents for their encouragement and support. I give my special thanks to my wife
Peifang for her love and support through all these years — we have learned a lot
from this long journey together — and to my wonderful daughters Anqi and Yueling
for bringing so much happiness and such a wonderful new world to our lives.
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ABSTRACT
COSMOLOGY FROM COSMIC MICROWAVE BACKGROUND AND LARGE
SCALE STRUCTURE
Yongzhong Xu
Advisor. Max Tegmark
This dissertation consists of a series of studies, constituting four published papers,
involving the Cosmic Microwave Background and the large scale structure, which
help constrain Cosmological parameters and potential systematic errors.
First, we present a method for comparing and combining maps with different res­
olutions and beam shapes, and apply it to the Saskatoon, QMAP and COBE/DMR
data sets. Although the Saskatoon and QMAP maps detect signal at the 21<r and
40a levels, respectively, their difference is consistent with pure noise, placing strong
limits on possible systematic errors. In particular, we obtain quantitative upper
limits on relative calibration and pointing errors. Splitting the combined data by
frequency shows similar consistency between the Ka- and Q-bands, placing limits
on foreground contamination. The visual agreement between the maps is equally
striking. Our combined QMAP+Saskatoon map, nicknamed QMASK, is publicly
available at xvww.hep.upenn.edu/~xuyz/qmask.html together with its 6495 x 6495
noise covariance matrix. This thoroughly tested data set covers a large enough area
(648 square degrees — at the time, the largest degree-scale map available) to allow a
statistical comparison with COBE/DMR, showing good agreement. By band-passfiltering the QMAP and Saskatoon maps, we are also able to spatially compare them
scale-bv-scale to check for beam- and pointing-related systematic errors. Using the
QMASK map, we then measure the cosmic microwave background (CMB) power
spectrum on angular scales £ ~ 30 —200 (1° —6°), and we test it for non-Gaussianity
using morphological statistics known as Minkowski functionals. We conclude th at
the QMASK map is neither a very typical nor a very exceptional realization of a
Gaussian random field. At least about 20% of the 1000 Gaussian Monte Carlo maps
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differ more than the QMASK map from the mean morphological parameters of the
Gaussian fields.
Finally, we compute the real-space power spectrum and the redshift-space dis­
tortions of galaxies in the 2dF 100k galaxy redshift survey using pseudo-KarhunenLoeve eigenmodes and the stochastic bias formalism. Our results agree well with
those published by the 2dFGRS team, and have the added advantage of produc­
ing easy-to-interpret uncorrelated minimum-variance measurements of the galaxygalaxy, galaxy-velocity and velocity-velocity power spectra in 27 fc-bands, with nar­
row and well-behaved window functions in the range 0.01 h/M pc < k < 0.8/i/M pc.
We find no significant detection of baryonic wiggles. We measure the galaxy-matter
correlation coefficient r > 0.4 and the redshift-distortion parameter 8 = 0.49 ± 0.16
for r = 1.
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Contents
D E D IC A T IO N
in
ACKNOW LEDGEM ENTS
v
1 In trod u ction
1
1.1
Hot Big Bang M o d e l..................................................................................
1
1.2
An Overview of the Cosmic Microwave B a c k g ro u n d ...........................
3
1.2.1
Discovery of the Cosmic Microwave B ack g ro u n d .....................
3
1.2.2
CMB measurements
.....................................................................
3
1.2.3
Properties of the C M B ..................................................................
4
1.3
Large Scale Structure and the SloanDigital SkyS u rv e y ........................
6
1.4
Motivation for this D iss e rta tio n ...............................................................
7
2 C om paring and com b ining th e Saskatoon, Q M A P and C O BE C M B
m aps
11
2.1
IN T R O D U C T IO N .....................................................................................
13
2.2
M E T H O D .....................................................................................................
15
2.3
R E S U L T S .....................................................................................................
16
2.3.1
Saskatoon D a t a ...............................................................................
16
2.3.2
Combining QMAP with S a s k a to o n ............................................
17
2.3.3
Combining QMASK with COBE
...............................................
22
2.3.4
Comparing QMAP with S a s k a to o n ............................................
22
be
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2.4
2.3.5
Comparing QMASK with C O B E .................................................
30
2.3.6
Foreground c o n stra in ts....................................................................
30
2.3.7
The coldest spot
.............................................................................
33
D ISC U SSIO N ...............................................................................................
36
3 T he C M B pow er sp ectru m a t t = 30 —200from Q M A SK
4
3.1
INTRODUCTION
3.2
Combining the SASK and QMAP Experiments
..................................
42
3.3
The Angular Power S p e c tru m ..................................................................
44
3.4
D iscussion.....................................................................................................
45
3.4.1
A method for scale-by-scale comparison of two m a p s ..............
50
3.4.2
Results of comparing QMAP and Saskatoon seal e-by-scale . .
50
3.4.3
C o n clu sio n s.......................................................................................
53
3.4.4
Comparison with other experim ents..............................................
54
39
M orphological M easures o f non -G au ssian ity in C M B M aps
57
4.1
In tro d u ctio n ..................................................................................................
58
4.2
QMASK M a p ...............................................................................................
62
4.3
Mock M a p s ...................................................................................................
63
4.4
Morphological S t a t i s t i c s ............................................................................
63
4.4.1
Global Minkowski fu n c tio n a ls .......................................................
64
4.4.2
Percolating region..............................................................................
65
4.4.3 Numerical te c h n iq u e ........................................................................
66
4.4.4 Parameterization by the level of AT ...............................................
69
4.4.5 Parameterization by the total area A ..............................................
70
4.4.6 Cross-correlations..............................................................................
74
D iscussion......................................................................................................
75
4.5
5
.....................................................................................
39
T he P ow er S p ectru m o f G alaxies in th e 2dF 100k R ed sh ift Survey 87
5.1
In tro d u ctio n ...................................................................................................
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88
5.2 D ata modeling
5.3
...................................................................................
91
5.2.1 The basic angular m a s k ................................................................
92
5.2.2 Angular selection fu n c tio n .............................................................
96
5.2.3 The radial selection fu n ctio n ..........................................................
99
Method and basic a n a ly s is ............................................................................100
5.3.1 Step 1: Finger-of-god compression....................................................101
5.3.2 Step 2: Pseudo-KL p ix e liz a tio n .......................................................102
5.3.3 Step 3: Expansion into true KL m o d e s .......................................... 108
5.3.4 W hat we wish to measure: three power spectra, not one
. . . 109
5.3.5 Step 4: Quadratic compression into band powers.......................... I l l
5.3.6 Step 5: Fisher decorrelation and flavor disentanglement . . . .
113
5.4 R e su lts............................................................................................................... 117
5.4.1 The three power s p e c t r a ................................................................... 117
5.4.2 Constraints on redshift space d is to rtio n s ....................................... 118
5.4.3 The galaxy-galaxy power spectrum a lo n e ....................................... 135
5.5 How reliable are our re su lts? ...........................................................................136
5.5.1 Validation of method and software
.................................................136
5.5.2 Robustness to method d e t a i l s .......................................................... 143
5.5.3 Tests for problems with data m o d e lin g .......................................... 147
5.5.4 Non-linearity issu e s............................................................................. 151
5.5.5 Bias issu e s.............................................................................................154
5.6 Discussion and conclusions.............................................................................. 155
5.6.1 Comparison with other s u r v e y s ....................................................... 155
5.6.2 Cosmological c o n stra in ts....................................................................158
5.6.3 O u tlo o k ................................................................................................ 161
A C om bining m aps
177
B P lo ttin g m aps
181
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C
C om paring m aps
183
D
D econ volvin g m aps
185
D .l Why is it u s e f u l ? ............................................................................................185
D.2 How does it work?
........................................................................................ 186
D.3 T e s t s ..................................................................................................................188
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List o f Figures
2.1
The three maps that we will compare and combine are shown in Galactic coordi­
nates. QMAP location in COBE map is shown in black.............................................
2.2
12
Wiener-filtered Saskatoon map. The CMB temperature is shown in coordinates
where the north celestial pole is at the center of a circle of 16° diameter, with R.A.
being zero at the top and increasing clockwise. In addition to the data included
in the map of (Tegmark et al., 1996), “RING” data is included here. Note that
the orientation of this and all following maps is different from that in Figure 2.1.
2.3
Wiener-filtered QMAP map. The coordinates are the same as in the previous
figure....................................................................................................................................
2.4
19
Wiener-filtered map combining the QMAP and Saskatoon experiments. The co­
ordinates are the same as in Figure 2.2..........................................................................
2.5
18
20
Wiener-filtered map of the combined QMASK and COBE data. The coordinates
are the same as in the previous figure. COBE adds only large-scale information.
For example, the upper left region is brightened somewhat........................................
2.6
21
Comparison of QMAP (left) and Saskatoon (right). The upper panel shows both
maps Wiener filtered with the same weighting in the overlap region. The lower
panel shows the number of standard deviations (“sigmas”) at which the difference
map Xqmap ~ rxsASK is inconsistent with mere noise. Note th at this is only for
the overlap region...............................................................................................................
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23
2.7
Same as previous figure, but comparing Saskatoon with various subsets of the
QMAP data. The curves correspond to Ka-band (dotted), Q-band (solid), flight
1 (short-dashed), flight 2 (long-dashed), £2Kal2(dot-long-dashed), f2Ql2 (dotshort-dashed) and f2Q34 (short-dashed-long dashed)..................................................
2.8
25
Test of the relative pointing of QMAP and Saskatoon. The curves show the
number of “sigmas” at which the difference map is inconsistent with noise when
the QMAP map is shifted vertically and horizontally. Starting from the inside,
the contours are at 1, 2, 3, 4, 5, and 6cr respectively. Cross indicates no shift. The
pixels are squares of side 0.3125°....................................................................................
28
2.9 Comparison of QMASK (left) with COBE (right). The upper panel shows both
maps Wiener filtered with the same weighting in the overlap region, using equa­
tion (2.4). The lower panel shows the number of standard deviations (“sigmas” )
at which the difference map
x q m a s k
~
rx c o B E
is inconsistent with mere noise,
and illustrates that the visual discrepancy at “2 o’clock” is consistent with a fluc­
tuation in the (correlated) noise......................................................................................
29
2.10 Comparison of QMASK at two different frequencies, Ka-band (left) and Q-band
(right). The upper panel shows both maps Wiener filtered with the same weighting
in the overlap region that was observed at both frequencies. Arrows indicate the
coldest spot discussed below. The lower panel shows the number of standard
deviations (“sigmas” ) at which the difference map xk» —ricq is inconsistent with
mere noise.
.....................................................................................................................
32
2.11 The temperature towards RA=3/*20m, DEC=84°55 at different frequencies. Er­
rors bars correspond to detector noise alone. Note that these points cannot be
interpreted as a spectrum of this sky region, since the COBE points have much
lower resolution and the two QMASK points have been Wiener filtered with dif­
ferent weights, pushing them closer to zero than the underlying sky temperature.
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34
3.1
Wiener-filtered QMASK map combining the QMAP and Saskatoon
experiments. The CMB tem perature is shown in coordinates where
the north celestial pole is at the center of the dashed circle of 16°
diameter, with R.A. being zero at the top and increasing clockwise.
This map differs from the one published in (Xu et al., 2001) by the
erasing of QMAP information for t £ 200 described in the text.
3.2
...
41
Angular power spectrum ST = [f(/ + 1)C</2«,]1/’2 (uncorrelated) in 20 bands of
CMB anisotropy from the combined QMASK data in case of p = 0(no additional
noise at all, just simple combination of SASK and QMAP),1, 3, 5, 7 and 1000 (no
QMAP information, only SASK information). For comparison, we also plot the a
recent “concordance” model (Tegmark, Zaldarriaga and Hamilton, 2001) and the
power measurements from MAXIMA and BOOMERanG...........................................
3.3
46
Uncorrelated measurements of the CMB power spectrum ST = [l{l + l)C //2ir]1/2
from the combined QMASK data. For comparison, we also plot the measurements
from COBE/DMR, MAXIMA. DASI and BOOMERanG. These error bars do not
include calibration uncertainties of 10% (QMASK), 10% (BOOMERanG), 4%
(DASI) and 4% (MAXIMA). As described in the text, we suggest using only
the first four points (I < 200), not the two dashed ones, for cosmological model
constraints..........................................................................................................................
3.4
48
Comparison of the QMAP and Saskatoon experiments on different angular scales,
corresponding to the multipole ranges shown in square brackets. The curves
show the number of standard deviations (“sigmas” ) at which the difference map
xqmap
—rxsASK is inconsistent with mere noise. Note that this is only for the
spatial region observed by both experiments.
...........................................................
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52
4.1
One-dimensional illustration of a non-Gaussian field $ (dashed line) having exactly
same number of peaks (which is analogous to genus in 2D) in the level parameter­
ization as the Gaussian field <t>(solid line). Compare the number of peaks in two
fields at two levels marked by the dotted and dash-dotted horizontal lines. The
length parameterization in ID is analogous to the area parameterization in 2D. In
order to compare the number of peaks in the length parameterization one has to
count the peaks of the two fields at different levels. The marked levels are chosen
in such a way that the total length of the excursion set of the non-Gaussian field
at ♦ = 1. (the sum of four heavy segments) equals the total length of the the
excursion set of the Gaussian field at <t>= 1.95 (the sum of two heavy segments). .
4.2
78
Grey scale QMASK map. Light color correspond to higher temperatures. Note
that there are three clearly distinct white regions in the map where data are absent. 79
4.3
An illustration of the numerical technique. The solid line shows the elliptical
contour corresponding to the certain threshold u = uth- Triangles mark the sites
of the lattice: solid with u > u th and empty with u < u(*. Solid circles mark the
contour points satisfying the condition u = uth (the roots of eq. 4.8); the dashed
line is the resulting contour. The true area of the region is the area within the
solid ellipse. We approximate it by the area within the dashed contour while in
most works it is approximated by the sum of areas of the elementary squares (solid
squares). We approximate the perimeter by the length of the dashed contour while
in other works it is often approximated by the sum of the external edges of the
elementary squares. One can easily see that this approximation gives the value
of the perimeter of the large dotted rectangle. As the lattice constant approaches
zero it converges to the perimeter of the large solid rectangle, while our perimeter
converges to the true value of the perimeter of the solid ellipse..................................
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80
4.4 Minkowski functionals as functions of u = A T/ ctat for the excursion set (left
hand side column) and for the percolating region (right hand side column). Top
row; the area fraction A and Ap, middle row : contour length C and Cp, bottom
row; genus G and Gp. The contour lengths are given in mesh units. The genus
is “number of regions” — “number of holes” . The solid lines show the parameters
of the QMASK map, heavy dashed lines show the median Gaussian values, thin
dashed lines show 68% and 95% ranges.........................................................................
81
4.5 The number of regions N c as a function of the level u = AT / o^ t (left hand panel)
and as a function of .4 (right hand panel).Other notation is as inPig. 4.4
4.6
Illustration of the transformation from A T to .4parameterization.
. . .
82
The thick
solid lines show C = C {A T ) (top left panel) C = C(.4) (top right panel) and .4 =
.4(AT) (bottom left panel) for the QMASK map. The thin solid lines illustrate
the transformation. The dashed lines show similar transformation for a randomly
chosen Gaussian map........................................................................................................
83
4.7 The global Minkowski functionals of the parent Gaussian field u and derived nonGaussian field w = expu are shown in two left hand side panels as functions of
the level. Both the perimeter and genus remain the same for both fields if they
are parameterized by the total area (two bottom right hand side panels). All
information about the non-Gaussianity of the
-field is stored in the cumulative
10
probability function (dashed lines in the top panels). The solid lines in the top
panels show the
G a u s s ia n
cumulative probability function.........................................
84
4.8 The figure is similar to Fig. 4.4, except that all morphological parameters are
functions of .4 = A ss/A m , where
A s s and -4m = A b s {—<x>) is the area ofthe
excursion set and that of the whole map, resp ectiv ely .............................................
5.1
85
The upper half shows the 59832 2dF galaxies in our baseline sample, in equatorial
1950 coordinates. The lower half shows the corresponding angular mask, the
relative probabilities that galaxies in various directions get included.............................. 90
5.2 Number of galaxies surviving as a function of uniform magnitude cut..........................120
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5.3
The redshift distribution of the galaxies in our sample is shown both as a his­
togram (top) and relative to the expected distribution (bottom), in comoving
coordinates assuming a flat ftm = 0.3 cosmology. The curves correspond to the
the radial selection function n(r) employed in our analysis (solid) and by C01 (dot­
ted). The four vertical lines indicate the redshift limits employed in our analysis
(10/»~lMpc < r < 650/i-1 Mpc) and where spectral type subsamples are available
(3 3 h -, M p c < r < 5 3 8 h - 1Mpc)......................................................................................... 121
5.4
The effect of our Fingers-of-god (FOG) removal is shown in the southern slice
<5 = —27.7°, —35° < RA < 53°. The slice has thickness 2° and has been rotated
to lie in the plane of the page. FYom left to right, the panels show all 15,055
galaxies in the slice, the 6,211 that are identified as belonging to FOGs (with
density threshold 100) and the same galaxies after FOG compression, respectively. 122
5.5
A sample of four angular pseudo-KL (PKL) modes are shown in Hammer-Aitoff
projection in equatorial coordinates, with grey representing zero weight, and
lighter/darker shades indicating positive/negative weight, respectively. FYom top
to bottom, they are angular modes 1 (the mean mode), 3, 20 and 106, and are
seen to probe successively smaller angular scales............................................................. 123
5.6
A sample of six pseudo-KL modes are shown in the plane of the southern 2dF
slice with S = —27.7°, —35° < R A < 53°. Grey represents zero weight, and
lighter/darker shades indicate positive/negative weight, respectively. FYom left to
right, top to bottom, these are modes 1 (the mean mode), 14, 104, 148, 58 and
178, and are seen to probe successively smaller scales. Those in the middle panel
are examples of purely radial (left) and purely angular (right) modes......................... 124
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5.7
The triangles show the 4,000 elements Xi of the data vector x (the pseudo-KL
expansion coefficients) for the baseline galaxy sample. If there were no clustering
in the survey, merely shot noise, they would have unit variance, and about 68% of
them would be expected to lie within the blue/dark grey band. If our prior power
spectrum were correct, then the standard deviation would be larger, as indicated
by the shaded yellow/light grey band................................................................................ 125
5.8
The triangles show the 3999 uncorrelated elements ju of the transformed data vec­
tor y = B x (the true KL expansion coefficients) for the baseline galaxy sample.
If there were no clustering in the survey, merely shot noise, they would have unit
variance, and about 68% of them would be expected to lie within the blue/dark
grey band. If our prior power spectrum were correct, then the standard devi­
ation would be larger, as indicated by the shaded yellow/light grey band. The
green/grey curve is the rms of the data points
averaged in bands of width 25,
and is seen to agree better with the yellow/light grey band than the blue/dark
grey band................................................................................................................................ 126
5.9
The 147 quadratic estimators qt, normalised so that their window functions equal
unity and with the shot noise contribution /< (dashed curve) subtracted out. They
c a n n o t be directly interpreted as power spectrum measurements, since each point
probes a linear combination of all three power spectra over a broad range of scales,
typically centered at a fc-value different than the nominal k where it is plotted.
Moreover, nearby points are strongly correlated, causing this plot to overrepresent
the amount of information present in the data. The solid curves show the prior
power spectrum used to compute the error bars.............................................................. 127
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5.10 The rows of the gg-portion of the Fisher matrix F . The ith row typically peaks at
the itk band, the scale k th at the band power estimator ft was designed to probe.
All curves have been renormalized to unit area, so the highest peaks indicate
the scales the the window functions obtained are narrowest. The turnover in the
envelope at k —0.1 h/M pc reflects our running out of information due to omission
of modes probing smaller scales. For comparison with the next figure, these are
the rows of W when M is diagonal....................................................................................128
5.11 The window functions (rows of the gg-portion of W ) are shown using decorrelated
estimations. The ith row of W typically peaks at the ith band, the scale k that
the band power estimator p, was designed to probe. Comparison with figure 5.10
shows that decorrelation makes all windows substantially narrower.............................129
5.12 The window function for our measurement of the 25th band of the galaxy-galaxy
power is shown before (left) and after (right) disentanglement. Whereas unwanted
leakage of gv and w power is present initially, these unwanted window functions
both average to zero afterward. The success of this method hinges on the fact
that since the three initial functions (left) have similar shape, it is possible to take
linear combinations of them that almost vanish (right).................................................. 130
5.13 Decorrelated and disentangled measurements of the galaxy-galaxy power spec­
trum (top), the galaxy-velocity power spectrum (middle) and the velocity-velocity)
power spectrum (bottom) for the baseline galaxy sample. Red points represent
negative values — since the points are differences between two positive quantities
(total power minus expected shot noise power), they can be negative when the
signal-to-noise is poor. Each points is plotted a t the Ar-value that is the median
of its window function, and 68% of this function is contained within the range of
the horizontal bars. The curves shows our prior power spectrum. Note that most
of the information in the survey is on the galaxy-galaxy spectrum. Band-power
measurements with very low information content have been binned into fewer (still
uncorrelated) bands.............................................................................................................. 131
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5.14 The blue/grey band shows the l<r allowed range for 0, assuming r = 1 and the
shape of the prior PM(/fc) but marginalizing over the power spectrum normaliza­
tion, using FOG compression with density threshold l+<fc = 100. These fits are
performed cumulatively, using all measurements for all wavenumbers < k. From
bottom to top, the five curves show the best fit 0 for FOG thresholds 1+<SC = oo
(no FOG compression), 200, 100 (heavy),50 and 25.......................................................132
5.15 1-dimensional likelihood curves for T, 0 and r are shown after marginalizing over
the power spectrum normalization and the other parameters using our baseline
(l+ £ c = 100) finger-of-god compression. The 68% and 95% constraints are where
the curves intersect the dashed horizontal lines. The dashed curve in the middle
panel shows how the ^-constraints tighten up when assuming r = 1............................ 133
5.16 Constraints in the (0,r) plane are shown for our baseline (1+<5C = 100) finger-ofgod compression, using all measurements with k < 0.3h/Mpc and marginalising
over the power spectrum normalization for fixed spectral shape. The four contours
correspond to A *2 = 1, 2.29, 6.18 and 11.83, and would enclose 39%, 68%, 95%
and 99.8% of the probability, respectively, if the likelihood function were Gaussian. 133
5.17 The decorrelated galaxy-galaxy power spectrum is shown for the baseline galaxy
sample assuming 0 = 0.5 and r = 1. As discussed in the text, uncertainty in 0
and r contribute to an overall calibration uncertainty of order 12% which is not
included in these error bars................................................................................................. 137
5.18 The triangles show the elements x, of the data vector x (the pseudo-KL expansion
coefficients) averaged over 100 Monte-Carlo simulations of the baseline galaxy
sample. If the algorithms and software are correct, then their mean should be
zero and about 68% of them should lie within the shaded yellow/grey region
giving their standard deviation........................................................................................... 139
xxi
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5.19 The triangles show the nns fluctuations of the elements
from 100 Monte-Carlo
simulations. If the algorithms and software are correct, then the expectation value
of this rms is given by the thin blue curve, and most of them should scatter in the
yellow/grey region................................................................................................................. 140
5.20 In this alternative representation of the test from figure 5.19, most of the vertical
lines should intersect the 45° line if the algorithms and software are correct.
. . 141
5.21 The triangles show the rms fluctuations of the elements (B x)i from 100 MonteCarlo simulations. If the algorithms and software are correct, then the expectation
value of this rms is given by the thin blue curve, and most of them should scatter
in the yellow/grey banana-shaped region.......................................................................... 142
5.22 The triangles show the decorrelated and disentangled band-power estimates Pi,
averaged over 100 Monte-Carlo simulations of the baseline galaxy sample. If
the algorithms and software are correct, then this should recover the windowconvolved input power spectrum W p , plotted as a thin blue line. The thin shaded
yellow/grey band indicates the expected scatter. The harmless discontinuity in
the middle panel is an artifact of the disentangled galaxy-velocity windows hav­
ing negative area on the largest scales where there is essentially no information
available.................................................................................................................................. 144
5.23 Same as the previous figure, but testing the error bars Ap* rather than the power
itself. The triangles show the observed rms of the power spectrum estimates from
100 simulations and the solid blue curve shows the predicted curve around which
they should scatter................................................................................................................145
5.24 Numerical convergence. The figure shows for how many of our 4000 PKL modes
the numerical calculations are converged to accurately measure the power up to a
given wavenumber k. FYom left to right, the 12 curves correspond to truncation
at In,. =20, 40, 60, 80, 100, 120, 140, 160, 180, 200, 220 and 240.............................. 148
xxii
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5.25 Constraints on excess power in special modes. Our 2dF power spectrum measure­
ments from figure 5.17 are averaged into fewer bands and compared with measure­
ments using only special (radial, angular and local group) modes and only generic
(the remaining) modes (dashed)..........................................................................................149
5.26 Comparison with other power spectrum measurements. Our 2dF power spectrum
measurements from figure 5.17 are averaged into fewer bands and compared with
measurements from the PSCz (HTPOO) and UZC (this work) redshift surveys as
well as angular clustering in the APM survey (Efstathiou & Moody 2001) and the
SDSS (the points are from Tegmark et al.
2002 for galaxies in the magnitude
range 21 < r' < 22 — see also Dodelson et al. 2002)................................................. 157
5.2 7 Our 2dF power spectrum measurements from figure 5.17 are averaged into fewer
bands and compared with theoretical models. The BBKS model is the wigglefree prior used for our calculation. The flat ACDM “concordance” models from
Wang et al. (2002) and Efstathiou et al. (2002), both renormalized to our 2dF
measurements, are seen to be quite similar. The wigglier curve corresponds to the
best-fit high baryon model in the upper right corner of figure 5.28. Only data to
the left of the dashed vertical line are included in our fits..............................................159
5.28 Constraints in on the m atter density flm and the baryon fraction fi*/f1m from
the linear power spectrum over the range 0.01 h /Mpc < k < 0.3 h f Mpc, after
marginalizing over the power spectrum amplitude. These constraints assumes
a flat, scale-invariant cosmological model with h — 0.72. For comparison with
Percival et al (2001), contours have been plotted at the level for one-parameter
confidence of 68% and two-parameter confidence of 68%, 95% and 99% (i.e., \ 2 —
\ 2 min = 1,2.3,6.0,9.2. Marginalizing over the Hubble parameter h and limiting
the analysis to scales k < 0.15/»/Mpc as in Percival et al (2001) further weakens
the constraints....................................................................................................................... 162
xxiii
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xxiv
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List of Tables
2.1 Summary of map comparisons. The first three lines give the number of
"sigmas" at which map 1, the difference map and map 2, respectively,
are inconsistent with noise. The remaining lines give the best fit value
and limits on the relative calibration r, or, for the Ka vs. Q case, the
spectral index 3 ..............................................................................................
27
3.1 The power spectra ST = [1(1 + l)C //2 x ]1/<2 from the QMASK map
in two cases:p = 0 and p = 3. The tabulated error bars are uncorre­
lated between the twenty measurements, but do not include an overall
calibration uncertainty of 10% for S T .........................................................
47
3.2 The power spectrum ST = [1(1 + l)C</27r]1/'2 from the QMASK map.
The tabulated error bars are uncorrelated between the six measure­
ments, but do not include an overall calibration uncertainty of 10% for
ST. We recommend using only the first four for cosmological model
constraints.......................................................................................................
49
4.1 Percentage of Gaussian maps deviating less than QMASK.....................
77
4.2 Correlations between different non-Gaussianity statistics parametrized
by temperature, AT (below the diagonal) and area, A (above the di­
agonal).............................................................................................................
XXV
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77
Chapter 1
Introduction
Cosmology is the scientific study o f the large scale properties o f the Universe as a
whole. It endeavors to use the scientific method to understand the origin, evolution
and ultimate fate o f the entire Universe. Like any field of science, cosmology involves
the formation o f theories or hypotheses about the Universe which make specific predic­
tions for phenomena that can be tested with observations. Depending on the outcome
o f the observations, the theories will need to be abandoned, revised or extended to ac­
commodate the data.
from WMAP web site http://m ap.gsfc.nasa.gov/rn.uni.htm l
1.1
H ot B ig B an g M odel
The Hot Big Bang (HBB) model is the standard theory, broadly accepted by the
cosmological community, for the origin and evolution of the Universe. It is based
on General Relativity (GR) and the notion that the Universe is homogeneous and
isotropic on the large scales. This means that it can be described by the so called
Robertson-Walker metric:
ds2 = (cdt)2 —a(t)2
+ r2( ^ + sia2 #d<p2)j ,
1
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(1*1)
where r is the comoving radial coordinate, d and f are the angles defining a polar
coordinate system, t is the proper time, a(t) is called the cosmic scale factor and K
is called the curvature parameter, which is equal to 1, 0, or -1 for closed, flat, or
open Universes respectively. There exists a singularity, where a(t) = 0, at a special
time t = 0. This singularity is called the Big Bang Singularity. According to the
HBB theory, the Universe was tiny but extremely hot right after the HBB. The
HBB theory predicts that the Universe is expanding, th at the light elements like
H, He, and Li should have been fused from protons and neutrons during the first
few minutes after the HBB, and that there should be remnant heat left over from
the HBB. These three predictions are often referred to as the expanding Universe,
Big Bang Nucleosynthesis and the Cosmic Microwave Background (CMB) radiation,
respectively. These three predictions are supported by experiments of ever increasing
accuracy.
However, the HBB theory has limitations: it does not explain what physical
process produced the initial fluctuations in the density of m atter or why the Universe
is so uniform on very large scales. The inflationary model answers these questions.
Inflation theory takes into account quantum fluctuations during the evolution of a
scalar field right after the HBB. As long as the scalar potential in any patch of the
Universe is large, uniform and approximately static, the Universe will automatically
expand and exponentially “inflate”. The quantum fluctuations of this scalar field in
the very early Universe cause the variations of m atter density, which are revealed by
the large scale structure and the CMB fluctuations.
2
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1.2
A n O verview o f th e C osm ic M icrow ave B ack­
ground
1.2.1
D iscovery o f the Cosmic Microwave Background
The Cosmic Microwave Background (CMB) radiation was predicted by the HBB
theory (Gamow, 1946). Gamow and his group believed that a HBB would leave
the Universe with a calculable, non-zero temperature (Partridge, 1995). A couple of
years later, Alpher and Herman (1949) explicitly pointed out that the present value
of this non-zero temperature, To, should be about equal to 7°K. At that time, one
believed that it was impossible to experimentally measure this 'background tempera­
ture' due to technological difficulties. The CMB was observationally discovered in an
amusing way. By the early 1960s, radio receiver technology had improved dram at­
ically. Bell Telephone Laboratories developed some sensitive receivers which could
make measurements with a precision of a few tenths of one Kelvin, and Penzias and
Wilson found that they detected some ‘excess noise’ with a tem perature about 3.5°K
whenever and wherever they were pointed at the sky (Partridge, 1995; Penzias and
Wilson, 1965). This ‘problem’ haunted them for a couple of years until they called
the Dicke group a t Princeton University, and Robert Dicke convinced them that this
‘excess noise’ was the relic of the HBB.
1.2.2
CM B m easurem ents
As soon as the CMB was discovered (and even before, in the case of the Dicke group),
cosmologists began building many CMB experiments. There are two crucial tests
for experimentally confirming th at the measured signal is the relic of the HBB:
1. It should have thermal (Planck) black-body spectrum.
2. It should be approximately isotropic (intensity independent on pointing direc­
tion).
3
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Both tests were passed (Penzias and Wilson, 1967; Howell and Shakeshaft, 1966,
1967; Wilkinson and Partridge, 1967; Partridge and Wilkinson, 1967) soon after the
first discovery of the CMB.
Although many significant and interesting papers on CMB were published during
the two decades after 1967, the spectacular results of the Cosmic Background Ex­
plorer (COBE) satellite constituted a major milestone in study of the CMB. COBE
not only precisely measured the CMB tem perature over a large range of frequency,
but also detected the large-scale CMB anisotropy in all sky for the first time (Smoot
et al., 1992; Bennett et al., 1996).
During the 1990s and the very early 2000s, many balloon-borne and groundbased CMB experiments measured CMB anisotropy on small angular (degree-scale
and below) scales with comparatively longer observation time. Many of these ex­
periments successfully measured the first acoustic peak (£ ~ 220) of the angular
power spectrum of CMB anisotropy, which suggests the space is flat, t.e., K = 0 in
eq. (1.1). Some of them also located the position of the second peak and gave hints
about the positions of the third or fourth peaks (Devlin et al., 1998; Netterfield et
al., 1997; Hanany et al., 2000: Jaffe et al., 2000; Halverson et al., 2001; Padin et
al., 2001; Mason et al., 2002; Grainge et al., 2002; Kuo et al., 2002; Benoit et al.,
2003). The superb CMB anisotropy data recently released by Wilkinson Microwave
Anisotropy Probe (WMAP) team measured the angular power spectrum on angu­
lar scales i < 900 extremely accurately, and placed very strict constraints on many
cosmological parameters (Bennett et al., 2003; Spergel et al., 2003; Hinshaw et al.,
2003), bringing CMB measurements into a new era.
1.2.3
Properties o f th e CM B
The measured CMB tem perature is extremely uniform or isotropic across the sky,
and its spectrum fits th at of blade body radiation with tem perature of T — 2.725K
amazingly well, but we can also see tiny tem perature fluctuations (at the level
4
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5 T /T ~ 10~5). Expanding the CMB signal in spherical harmonics, one finds a
dipole moment (( = 1) of a few mK and other moments with amplitudes of fiK on
smaller scales (£ > 2). The generally accepted explanation of the dipole moment is
the Doppler effect due to the E arth’s motion. Due to the Doppler effect, we see that
the CMB photons will be blueshifted to higher energy in the direction of our motion,
and that redshifted to lower energy in the opposite direction of his motion. Peebles
and Wilkinson (1968) showed that this dipole is
sm
= * -± ,
(1.2)
where v is our velocity relative to the cosmic rest frame.
The remaining tem perature variations, with jjK amplitudes, are caused mainly
by the m atter density fluctuations at the surface of last scattering, contaminated by
emission from foreground galaxies including our Milky Way. The surface of last scat­
tering is the opaque surface from which CMB photons emanated when the Universe
had a tem perature around 3,000°K. Before this time, there were no atoms, only
free electrons and nuclei. The CMB photons were easily scattered by free electrons.
Therefore, the CMB photons were just wandering through the hot early Universe,
trapped in the ionized plasma. In others words, the m atter was opaque to CMB
photons. W*hen the Universe cooled below 3,000°K, most nuclei and free electrons
recombined to form neutral atoms and the Universe became transparent, and the
CMB photons escaped from this last scattering surface. By detecting temperature
variations on the surface of last scattering with very sensitive instruments, we can
understand the content, shape, origin and evolution of the Universe.
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1.3
Large S cale S tru ctu re and th e Sloan D igital
Sky S u rvey
As mentioned in subsection 1.2.3, m atter density fluctuations cause CMB fluctu­
ations. Similarly, the currently observed Large scale structure (LSS) is caused by
m atter density fluctuations. At the epoch of generation of CMB, about 380,000 years
after Big Bang, the m atter density fluctuations are only about 10-5. Gravity made
these variations larger and larger. When the Universe was about one billion years
old (its size was about one fifth of its present size), it is speculated that the cores of
galaxies were assembled. At this time, the typical m atter density in such a galaxy
core was a few hundred times higher than the overall average m atter density. Those
early galaxies are not distributed randomly throughout the Universe, but form struc­
tures like clusters, filaments, voids, bubbles and sheets. Over the last two decades,
Galaxy redshift surveys have revealed such structures on very large scales, up to 100
Mpc and beyond.
In 1985, the first set of observations for the CfA Redshift Survey (1,100 galaxies)
revealed large scale structures in a strip on the sky 6 degrees wide and about 130
degrees long. After that, many other redshift survey projects have been carried
out, e.g., the Updated Zwicky Catalog (UZC) (about 11,000 galaxies), the Point
Source Catalog Redshift Survey (PSCz) (about 15,000 galaxies), the two-degree Field
Galaxy Redshift Survey (2dFGRS) (more than 200,000 galaxies), and the Sloan
Digital Sky Survey (SDSS). SDSS is the most ambitious sky survey project to date,
and it will bring this modem practice of comprehensive and quantitative mapping
to cosmography, the science of mapping the Universe and determining our place in
it ( see SDSS website at http://www.sdss.org). The goal of the SDSS is to observe
about 100 million of photometric celestial objects, about half of which are galaxies
and half are stars, and to measure spectra for about one million of galaxies, covering
a sky area of about ten thousand square degrees (about one-quarter of the entire
6
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sky). The SDSS will also measure redshifts for about 100,000 quasars, the most
distant objects known, giving us an unprecedented hint at the distribution of m atter
to the edge of the visible universe.This superb data set of SDSS will have significant
impacts on many areas, for example, large scale structure, the evolution of galaxies,
the relation between m atter and luminosity, and the structure of our own Galaxy.
By combining CMB and galaxy clustering data with other cosmological probes
(the Lyman-o forest, weak-lensing etc.), our constraints on cosmological models and
their free parameters should greatly improve in the years ahead.
1.4
M otiva tion for th is D isserta tio n
As mentioned above, combining CMB data and LSS data, especially WMAP and
SDSS, can help us understand the m atter budget of the Universe, the shape and
age of the Universe, and the values of other cosmological parameters like the Hubble
constant h, the spectral index na, the amplitude of m atter fluctuations cr8, the optical
depth t etc. This dissertation consists of studies in these two areas, CMB and
LSS. In the following four chapters, which correspond to four published papers, the
motivation of each one will be addressed in detail. Here I just give a brief summary.
The CMB experimental teams generally test for systematic errors at many steps
in their d ata analysis pipeline, from data acquisition, cleaning, calibration and point­
ing reconstruction to mapmaking and power spectrum estimation. In addition, nu­
merous detailed comparisons have been made between the angular power spectra
Ci measured by different experiments to determine whether they are all consistent
(Scott et al., 1995; Lineweaver, 1998; Tegmark, 1999a; Dodelson and Knox, 1999;
Tegmark and Zaldarriaga, 2000b; Jaffe et al., 2000). However, such comparisons use
only a very small fraction of the information at hand: average band powers, not the
spatial phase information. A more powerful test (in the statistical sense of being
more likely to discover systematic errors) involves a direct comparison of the sky
7
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maps from experiments th at overlap in both spatial and angular coverage. However,
such a comparison is non-trivial for maps with nonidentical resolution and beam
shape (see details in Xu et al. (2001) and chapter 2). Chapter 2 derives a technique
for simplifying this task in practice, and applies it to compare and combine the three
overlapping d a ta sets shown in figure 2.1: COBE/DMR, Saskatoon and QMAP.
It is always im portant to improve measurements of the CMB angular power
spectrum, both for measuring cosmological parameters and to cross-validate differ­
ent experiments against potential systematic errors. This is one of the main goals of
chapter 3. The QMASK data was extensively tested for systematic errors in Chap­
ter 2, with the conclusion that the QMAP and Saskatoon experiments agree well
overall. However, for the present power spectrum analysis, it is important to perform
additional systematic tests to see if there is evidence of scale-dependent problems in
any of the maps (see also in Xu, Tegmark, and de Oliveira-Costa (2001)). This is
another goal of Chapter 3.
The issue of Gaussianity of CMB maps plays a crucial role in testing assump­
tions about the early Universe. Gaussianity tests are very helpful to understand
non-Gaussian perturbation, predicted by assumed cosmic strings or topological de­
fects, and undetected foreground contamination. The first study of Gaussianity on
degree scale showed the consistency of the QMASK map with the assumption of the
Gaussianity (Park et al., 2001). However, this study tested only the Gaussianity of
the topology of the map (the so-called Genus test), and Genus test detects some
special types of non-Gaussianity and is not sensitive to the others (see details in
Shandarin et al. (2001)). Therefore, Chapter 4 will test non-Gaussianity in CMB
maps using more general morphological statistics known as Minkowski functionals.
Large scale structure plays a key role in cosmology by providing non-CMB pre­
cision measurements. Three-dimensional power spectrum measurements of galaxy
redshift surveys place powerful constraints on cosmological models. Previous large
redshift surveys like UZC (Huchra et al., 1990; Falco et al., 1999), LCRS (Shechtman
8
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et al. 1996), and PSCz (Saunders et al., 2000) have more than ten thousand galax­
ies with redshift. In 2001, 2dfGRS team publicly released their catalog with about
one hundred thousand redshift galaxies. Given the huge effort involved in creating
this state-of-the-art sample, it is clearly worthwhile to subject it to an independent
power spectrum analysis with improvements in two aspects: we are able to produce
uncorrelated measurements of the linear power spectrum with minimal error bars
and quite narrow window functions: we measure independently not one power spec­
trum but three real-space power spectra, encoding clustering anisotropy (see details
in Tegmark, Hamilton and Xu (2002)). This is the main purpose of Chapter 5.
Chapters 2-5 correspond to the following four peer-reviewed papers:
• "Comparing and combining the Saskatoon. QMAP, and COBE CMB maps” ,
Xu, Yongzhong; Tegmark, Max; de Oliveira-Costa, Angelica; Devlin, Mark
J.; Herbig, Thomas; Miller, Amber D.; Netterfield, C. Barth; Page, Lyman.
Physical Review D, vol. 63, Issue 10, id. 103002 (2001/5)
• “CMB power spectrum at I = 30 — 200 from QMASK ” , Xu, Yongzhong ;
Tegmark, Max; de Oliveira-Costa, Angelica, Physical Review D, vol. 65, Issue
8. id. 083002 (2002/4)
• “Morphological Measures of Non-Gaussianity in Cosmic Microwave Background
Maps” , Shandarin, Sergei F.; Feldman, Hume A.; Xu, Yongzhong; Tegmark,
Max, The Astrophysical Journal Supplement Series, Volume 141, Issue 1, pp.
1-11 (2002/7)
• “The power spectrum of galaxies in the 2dF 100k redshift survey” , Tegmark,
Max; Hamilton, Andrew J. S.; Xu, Yongzhong, Monthly Notices of the Royal
Astronomical Society, V335, No. 4, p. 887 (2002/10)
These four parts will be fully discussed in Chapters 2, 3, 4 and 5 respectively. In
Chapter 2, we present a method to combine and compare CMB maps, and use this
9
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method on QMAP and Saskatoon CMB d ata sets. In Chapter 3, we measure the
angular power spectrum of the combined QMASK data, and do systematic tests
on dependence on different scales. In Chapter 4, we do non-Gaussianity tests on
QMASK using Minkowski functionals. In Chapter 5, we measure the 3-D power
spectrum of 2dFGRS, and place constraints on some cosmological parameters.
In all four cases, the order of author lists above reflects the relative contributions
of the different authors. I did most of the calculations and analysis in chapters 2
and 3. I provided the QMASK data and generated one thousand mock QMASK
data sets for the non-Gaussianity tests in chapter 4, including the complex noise
and smoothing properties of the Wiener-filtered maps. For chapter 5, I worked
extensively on testing and applying the power spectrum estimation software. I also
wrote software for handling the extremely complicated angular mask of the 2dF
survey, using set-theory to subdivide it into a set of thousands of non-intersecting
polygons.
10
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Chapter 2
Comparing and combining th e
Saskatoon, QM AP and COBE
CMB maps
In this chapter, we present a method for comparing and combining maps with dif­
ferent resolutions and beam shapes, and apply it to the Saskatoon, QMAP and
COBE/DMR data sets. Although the Saskatoon and QMAP maps detect signal
at the 21(7 and 40<7 levels, respectively, their difference is consistent with pure
noise, placing strong limits on possible systematic errors.
In particular, we ob­
tain quantitative upper limits on relative calibration and pointing errors. Splitting
the combined data by frequency shows similar consistency between the Ka- and Qbands, placing limits on foreground contamination. The visual agreement between
the maps is equally striking. Our combined QMAP+Saskatoon map, nicknamed
QMASK, is publicly available at wxmDMep.upenn.edu/~xuyz/qmask.html together
with its 6495 x 6495 noise covariance matrix. This thoroughly tested data set cov­
ers a large enough area (648 square degrees — currently the largest degree-scale
map available) to allow a statistical comparison with COBE/DMR, showing good
agreement.
11
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SASKATOON
COBE
DMR
/
a
Figure 2.1 The three maps th at we will compare and combine are shown in Galactic coordinates.
QMAP location in COBE map is shown in black.
12
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2.1
IN T R O D U C T IO N
The cosmic microwave background (CMB) field is currently enjoying a bonanza of
new high-quality data (Gawiser and Silk, 2000; de Bemardis et al., 2000; Hanany
et al., 2000), which has triggered a surge of new papers about the implications for
cosmological parameters (Lange et al., 2000; Tegmark and Zaldarriaga, 2000a: Balbi
et al., 2000; Bridle et al., 2000; Hu et al., 2000; Jaffe et al., 2000; Kinney et al., 2000;
Tegmark, Zaldarriaga and Hamilton, 2001). There is currently such wide interest in
these cosmological results that it is tempting to temporarily ignore underlying as­
sumptions. However, it is nonetheless important to consolidate these gains by careful
study of the many technical analysis steps upon which they rest. This can be done
at many levels. The experimental teams generally test for systematic errors at many
steps in their data analysis pipeline, from data acquisition, cleaning, calibration and
pointing reconstruction to mapmaking and power spectrum estimation. In addition,
numerous detailed comparisons have been made between the angular power spectra
Ci measured by different experiments to determine whether they are all consistent
(Scott et al., 1995; Lineweaver, 1998; Tegmark, 1999a; Dodelson and Knox, 1999;
Tegmark and Zaldarriaga, 2000b; Jaffe et al., 2000). However, such comparisons use
only a very small fraction of the information at hand: average band powers, not the
spatial phase information. A more powerful test (in the statistical sense of being
more likely to discover systematic errors) involves a direct comparison of the sky
maps from experiments th at overlap in both spatial and angular coverage.
Such a comparison is straightforward for maps with identical resolution and beam
shape, simply testing whether the difference map is consistent with pure detector
noise. Such tests have been successfully performed for the COBE/DMR maps (Ben­
nett et al., 1996). Unfortunately, comparisons are usually complicated by angular
resolution differences between channels. Some experiments probe the sky in an even
more complicated way, with, e.g., elliptical beams, double beams, triple beams, interferometric beams or complicated elongated software-modulated beams. Correlated
13
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noise further complicates the problem. Despite these difficulties, accurate compar­
isons between different experiments are crucial. Some of the best testimony to the
quality of CMB maps comes from the success of such comparisons in the past —
between FIRS and DMR (Ganga et al., 1993), Tenerife and DMR (Lineweaver et al.,
1995), MSAM and Saskatoon (Netterfield et al., 1997; Knox et al., 1998), two years
of Python data (Ruhl et al., 1995), three years of Saskatoon d ata (Tegmark et al.,
1996), two flights of MSAM (Inman et al., 1997) and different channels of QMAP
(Devlin et al., 1998; Herbig et al., 1998; de Oliveira-Costa et al., 1998), Boomerang
(de Bernardis et al., 2000) and Maxima (Hanany et al., 2000).
General methods have been developed for both comparing (Knox et al., 1998;
Tegmark, 1999b) and combining (Tegmark, 1999b) arbitrarily complicated experi­
ments. In this paper we will derive a technique for simplifying this task in practice,
and apply it to compare and combine the three overlapping data sets shown in Fig­
ure 2.1: COBE/DMR, Saskatoon and QMAP. Our motivation is threefold:
• To test and provide methods th at can be used by experimental groups in the
future.
• To search for systematic problems that may be relevant to ongoing and future
experiments.
• To quality-test and publicly release the largest degree-scale map to date.
We stress th at combining maps is not just a m atter of making pretty pictures. Power
spectra from different experiments are routinely combined as if their sample variance
were independent. However, since this approximation breaks down whenever the
underlying maps overlap in spatial and angular coverage, the only correct way to
compute their combined power spectrum is to extract it from the combined map.
We present our methods in Section 2.2, present our results in Section 2.3 and
summarize our conclusions in Section 2.4.
14
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2.2
M ETHOD
We use the methods for combining and comparing maps presented in (Tegmark,
1999b), which are most easily expressed with matrix notation. Given two data sets
represented by the vectors yi and y 2, we write
y x = A ,x + n i ,
y 2 = A 2x -I- n 2.
(2 . 1)
Here the vector x contains the temperature of the true sky at various locations
(pixels). A t and A 2 are two known matrices incorporating the pointing strategy
and beam shape of each experiment, n i and
112
are two random noise vectors with
zero mean and known covariance matrices N i = (n in ‘) and N 2 = (n2n^). It is
convenient to define larger matrices and vectors
( 2 .2 )
and to write the full noise covariance m atrix as
(2.3)
We review the mathematical details of combining, filtering and comparing maps
in Appendices A, B and C, with some explicit details added (beyond Tegmark
(1999b)) th at are useful when implementing these methods in practice.
We derive a new deconvolution method in Appendix D which substantially sim­
plifies our calculations by eliminating the A-matrices above. In the generic case,
deconvolution is strictly speaking impossible, since the matrix A is not invertible
and certain pieces of information about x are simply not present in y. It is common
practice to find approximate solutions to such under-determined problems using sin­
gular value decomposition or other techniques, but our goal is different: we need a
deconvolved sky m ap x th a t can be analyzed as a true sky map with A = I without
approximations, shifting all complications into the new noise covariance matrix. The
15
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method derived in Appendix D is found to be stable numerically, and can be used
both for "unsmoothing" low-resolution maps and to deconvolve more complicated
oscillatory beam patterns such as that of the Saskatoon experiment.
2.3
R ESU L T S
In this section, we combine the QMAP, Saskatoon and COBE maps.
We then
perform a battery of tests for systematic errors by comparing the maps with each
other, paying particular attention to possible calibration, pointing and foreground
problems.
2.3.1
Saskatoon D ata
The Saskatoon data set is very different from other data sets such as QMAP and
COBE since it does not contain simple sky tem perature measurements. Instead,
the 2970 Saskatoon measurements are different linear combinations of the sky tem­
peratures with rather complicated weight functions reminiscent of caterpillars —
examples are plotted in (Netterfield et al., 1997; Tegmark et al., 1996). These mea­
surements probe a circular sky patch with about 16° diameter, centered on the
the North Celestial Pole (NCP). In addition to the 2590 weight functions used in
(Tegmark et al., 1996), which are all oriented like spokes of a wheel, we include the
380 “RING” data measurements, linear combinations in the perpendicular direction
going around the periphery of the observing region.
We pixelize this sky region into 2016 pixels in the same coordinate system as
QMAP, i.e., a simple square grid in gnomonic equal area projection, and define x
to be the true sky convolved with a Gaussian beam of FWHM 0.68°. We compute
the 2970 x 2016 matrix A using the software from the original Saskatoon analysis
(Netterfield et al., 1995, 1997). The rows of A have a vanishing sum since the
beam functions are all insensitive to the monopole — they are normalized so that
16
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the absolute values sum to two.
Using equations (D .l) and (D.2), we obtain a
deconvolved Saskatoon CMB map
xsask
and its corresponding covariance m atrix
E sask- Figure 2.2 shows the Wiener-filtered Saskatoon map, which is visually almost
identical to th at in (Tegmark et al., 1996) except for the additional information from
the RING data near the border.
2.3.2
Com bining Q M AP w ith Saskatoon
The QMAP data consists of Ka-band and Q-band measurements with angular reso­
lution 0.89° and 0.68°, respectively. We first deconvolve the Ka-band data to 0.68°
resolution using the method of Appendix D. We then produce a unified QMAP data
set (xqmap? Eqm ap) with a single resolution, 0.68°, by combining the result with
the Q-band as described in Appendix A, using equation (A .ll) to compute the final
noise covariance H q m ap since the overlap between the two bands is only partial.
This combined QMAP map has 5396 pixels covering a sky area of about 538 square
degrees.
Combining QMAP and SASK is now a straightforward task, since the data sets
xqmap
and Xsask have the same angular resolution and pixelization scheme. Since
the spatial overlap is only partial, we once again use equation (A .ll) to compute the
combined noise covariance matrix. There are 917 overlapping pixels, so the combined
map consists of 6495 pixels, covering a sky area of about 648 square degrees. We
will nickname the combined data set “QMASK” .
The main improvement in the combined is not the area covered (the QMASK
map is only 20% larger than QMAP), but the signal-to-noise and the topology.
SASK has excellent signal-to-noise in the region th at it covers, which overlaps the
most sensitive region of QMAP. Indeed, the two maps have comparable sensitivity in
the overlap region, so both of them have substantial impact on the spatial features
seen in Fig. 3. Filling in the “hole” in the map is also useful for comparing with
lower resolution maps like COBE and for potential future applications, e.g., genus
17
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Figure 2.2 Wiener-filtered Saskatoon map. The CMB temperature is shown in coordinates where
the north celestial pole is at the center of a circle of 16° diameter, with R.A. being zero a t the top
and increasing clockwise. In addition to the data included in the map of (Tegmark et al., 1996),
“RING"’ data is included here. Note that the orientation of this and all following maps is different
from that in Figure 2.1.
18
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Figure 2.3 W iener-filtered QMAP map. The coordinates are the same as in the previous figure.
19
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Figure 2.4 W iener-filtered map combining the QMAP and Saskatoon experiments. The coordinates
are the same as in Figure 2.2.
20
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Figure 2.5 Wiener-filtered map of the combined QMASK and COBE data. The coordinates are the
same as in the previous figure. COBE adds only large-scale information. Fbr example, the upper
left region is brightened somewhat.
21
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statistics, where a large contiguous area is desirable.
2.3.3
Com bining QM ASK w ith COBE
The COBE data (Smoot et al., 1992; Bennett et al., 1996) has much lower angular
resolution than QMASK (about 7.08°), and the pixel size of COBE is much bigger
than that of QMASK as well (about 2.6° x 2.6°). In total, there are 6144 pixels
in the whole COBE sky map. We select those pixels whose centers are within the
QMASK map and at least 3° away from the perimeter. Only 58 COBE pixels satisfy
these criteria.
We first deconvolve this COBE data to the QMASK angular resolution, using
the method of Appendix D with A i being a 58 x 6495 matrix with a Gaussian
COBE beam on each row, normalized to sum to unity. As input, we use the inversevariance weighted average of the 53 and 90 GHz COBE/DMR channels (Bennett et
al., 1996). By construction, our resulting COBE and QMASK maps overlap each
other perfectly, so we obtain our combined map by simply using equations (A.5)
and (A.6).
At first glance, Figure 2.4 and Figure 2.5 look very similar. However, inspecting
them more carefully reveals that although the small scale patterns are the same,
the upper part in Figure 2.5 is brighter than that in Figure 2.4 (this is related to
the QMASK cold spot that we will discuss subsection 2.3.5). In other words, since
COBE contains only large scale information, this is precisely what it has added in
Figure 2.5, leaving the small scale structure unaffected.
2.3.4
Com paring QM AP w ith Saskatoon
As mentioned above, there are 917 pixels overlapping between the QMAP and Saska­
toon data sets. After extracting these pixels and their two noise covariance matrices
from the full maps, we can compare the two experiments using the null-buster test
described in Appendix C.
22
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QMAP
a
C 30
o
a
>
u
Q 20
SASK
■o
u
a
.
T3
c
«
co
0.001
0.01
0.1
1
10
100
1000
R e la tiv e N o r m a liz a tio n r
Figure 2.6 Comparison of QMAP (left) and Saskatoon (right). The upper panel shows both maps
Wiener filtered with the same weighting in the overlap region. The lower panel shows the number
of standard deviations (“sigmas”) at which the difference map xqmap —»"Xsask is inconsistent with
mere noise. Note that this is only for the overlap region.
23
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V isu al c o m p a riso n
Before delving into statistical details, it is useful to compare the two maps visually.
Comparing plots of the raw maps x is rather useless, since they are so noisy. Unfor­
tunately, comparing plots of the two Wiener-filtered maps like figures 2.2-2.5 is not
ideal either: since the noise matrices N i and N 2 are different, this would entail com­
paring apples and oranges, since the two Wiener-filtered maps would be smoothed
and weighted differently. For instance, if a prominent spot in one map is invisible
in the other, this could either signal a problem or be due to that particular region
being very noisy in the second map and therefore suppressed by the Wiener filtering.
To circumvent this problem, we Wiener-filter both maps exactly in the same way,
x,*” = S[S + N x + N a]"1^ ,
i = 1,2.
(2.4)
In other words, we replace the individual noise covariance matrices by their sum,
so that the map will only show information that is accurately measured by both
experiments. These maps x “ and xJT are compared in the upper part of Figure 2.6,
and are seen to look encouragingly similar.
T ests fo r s y s te m a tic a n d c a lib ra tio n e rro rs
The lower plot in Figure 2.6 shows the results of applying the null-buster test to
the difference map
x q m a p
—r x
S/\ s K
for different values of the constant r.
(The
corresponding noise covariance matrix N = N i + r 2^ . ) The left part of the curve
where r <
1 is dominated by information from QMAP, and we see that QMAP
alone (the r = 0 case) is inconsistent with noise at about the 40cr-level. Similarly,
we see th at the Saskatoon map alone (the case r =
00 )
is inconsistent with noise at
about the 2Ckr-level. Note th at these significance levels are still higher for the full
maps — here we are limiting ourselves to the sky region where they overlap.
In summary, both the QMAP and Saskatoon maps contain plenty of signal. Is
this signal consistent between the two maps? The answer is given by the most
24
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40
co
C
O
.5
ju 20
Q
TJ
u
QMAP
SASK
.
CO
T3
C
<0
0.001
0.01
0.1
1
10
100
1000
R elative N o r m a liz a tio n r
Figure 2.7 Same as previous figure, but comparing Saskatoon with various subsets of the QMAP
data. The curves correspond to Ka-band (dotted), Q-band (solid), flight 1 (short-dashed), flight
2 (long-dashed), f2Kal2(dot-long-dashed), f2Ql2 (dot-short-dashed) and QQ34 (short-dashed-long
dashed).
25
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interesting point on the curve, where the relative normalization value r = 1. In the
absence of systematic or calibration errors, the corresponding difference map should
contain pure noise. In our case, when r = 1, the difference map xqmap —xsask is
seen to be consistent with pure noise, i.e., less than 2a away from zero. The strong
signal seen in both maps therefore appears to be a true sky signal, with no evidence
for significant systematic errors in either QMAP or Saskatoon.
The QMAP experiment consists of two flights, each with two frequencies (Ka and
Q-band) and three slightly different observing regions. We label these six sub-maps
flK al2, flQ2 and flQ34 (from flight 1) and f2Kal2, f2Ql2 and f2Q34 (from flight 2).
The pointing and calibration analyses for these two flights were completely separate,
and the map-making algorithm was applied separately for these six sub-maps (Devlin
et al., 1998; Herbiget al., 1998; de Oliveira-Costa et al., 1998). To investigate possible
problems with these individual sub-maps that may have been averaged away in the
combined analysis, we repeat the comparison with Saskatoon separately for each
one. The results are shown in Figure 2.7. None of these curves show any evidence
for systematic or calibration errors. The f2Q34 map is seen to contain the strongest
signal, inconsistent with noise at the 38<r level. This is because f2Q34 contains a
striking cold spot — we will return to this in more detail in subsection 2.3.7.
The null-buster curves are interesting at more than just the points r = 0,1,
and oo; the entire region near r = 1 places limits on calibration errors. A relative
calibration error of say 10% would shift the minimum of the curve sideways to 0.9
or 1.1, depending on whether the QMAP or Saskatoon map was too high. We can
therefore place limits on calibration errors by reading off the r-values where noise is
ruled out at say 2 sigma. For instance, the QMAP-Saskatoon comparison constrains
the relative calibration error to be less than 20% (3<r), and the r-values for different
significance levels are shown in table 2.1. This method may prove quite useful for
upcoming experiments th at have higher sensitivity.
26
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f*mm(2o’)
QMAP vs SK
40
1.97
21
0.79
0.95
r bcst
1.1
7"max(l^)
f max (2<r)
fjnaxi. 3<^)
1.2
1.48
i/(r = 0)
v(r = 1)
u{r = oo)
r mrn (3<t)
QMASK vs COBE
62
-0.64
3
0.09
0.13
0.2
1.25
11.7
20
63
Ka vs Q band
40
-0.53
26
-2.73*
-2.2*
-1.7*
0.*
1.42*
1.9*
2.2*
* values of the power spectral index 8 in subsection 2.3.6
Table 2.1 Summary of map comparisons. The first three lines give the number of
“sigmas” at which map 1, the difference map and map 2, respectively, are inconsistent
with noise. The remaining lines give the best fit value and limits on the relative
calibration r, or, for the Ka vs. Q case, the spectral index p.
P oin tin g te sts
Our comparison method can also be used to test for relative pointing errors, as
a complement to the standard lower-level pointing tests that are routinely made
using point sources etc. Although an overall sideways shift of a single map will not
affect the measured power spectrum, such errors can become disastrous if the map
is combined with another one.
As an illustration of such a test, we compare the f2Q34 map with the Saskatoon
map with the null-buster test (setting r = 1) after shifting it vertically and horizon­
tally by an integer number of pixels. Figure 2.8 shows that there is no evidence for
pointing error although we cannot give a strong constraint. Just as the calibration
test, this pointing test based on CMB maps alone is likely to be useful for upcoming
high-sensitivity experiments.
27
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tn
vx
CL
C
.c
cn
ao
-4
-4
-2
0
2
Horizonal shift in pixels
4
Figure 2.8 Test of the relative pointing of QMAP and Saskatoon. The curves show the number of
“sigmas’' at which the difference map is inconsistent with noise when the QMAP map is shifted
vertically and horizontally. Starting from the inside, the contours are at 1, 2, 3, 4, 5, and (xr
respectively. Cross indicates no shift. The pixels are squares of side 0.3125°.
28
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60
QMASK
a
V)
C
o
a
’>
03
a
T3
t-
(0
TJ
C
-ma<0-J
COBE
0.001
0.01
0.1
1
10
100
.
1000
R e la tiv e N o r m a liz a tio n r
Figure 2.9 Comparison of QMASK (left) with COBE (right). The upper panel shows both maps
Wiener filtered with the same weighting in the overlap region, using equation (2.4). The lower panel
shows the number of standard deviations (“sigmas”) at which the difference map xqmask —rxcoBE
is inconsistent with mere noise, and illustrates th at the visual discrepancy at “2 o’clock” is consistent
with a fluctuation in the (correlated) noise.
29
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2.3.5
Comparing QM ASK w ith COBE
Our QMASK map and our deconvolved COBE map cover the exact same sky region,
so there are 6495 pixels in the overlap maps. Generating Figure 2.9, which compares
these two maps, therefore involved a m arathon computer run, processing the 6495 x
6495 matrices of equation (C .l) for each r-value.1 As can be seen in the lower
panel of Figure 2.9, the QMASK data is inconsistent with noise at about the 62(7level, whereas the COBE d a ta is inconsistent with noise slightly above the Ikr-level.
Since neither the QMAP nor the Saskatoon experiments were designed to probe such
large angular scales, it is quite encouraging that the QMASK-COBE difference map
(r = 1) is seen to be consistent with pure noise. Since the minimum of the curve is
so broad, however, we obtain no interesting constraints on calibration errors.
The upper panel in Figure 2.9 shows that the two maps look fairly similar con­
sidering the weak (3<r) COBE signal, with the notable exception of the upper right
part of the Saskatoon disk. Here COBE shows a hot spot whereas QMASK shows
a cold spot. We will return to this issue in more detail below, in subsection 2.3.7.
When generating these two maps, we used the same equal-weighting Wiener filtering
method that was described in subsubsection 2.3.4. This is particularly im portant
here, since the QMASK and COBE have such dramatically different angular reso­
lutions — in contrast, a visual comparison of the Wiener filtered COBE map with
the normal Wiener filtered QMASK map from Figure 2.4 is rather useless, since the
latter is dominated by small-scale fluctuations.
2.3.6
Foreground constraints
The previous two sections used map comparisons to test for calibration and pointing
errors. Here we will compare maps at different frequencies to constrain the spectrum
'I f CPU time had been an issue, this calculation could have been accelerated by binning the
QMASK pixels into larger ones. This would give essentially the same answer, since the null-buster
test gives statistical weight only to modes where both maps are sensitive — in this case, to large
scale modes only.
30
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of the detected sky signal.
The presence of foreground contamination (see Tegmark et al., 2000b, for a recent
review) has been quantified for both the Saskatoon (de Oliveira-Costa et al., 1997)
and QMAP (de Oliveira-Costa et al., 2000) experiments by cross-correlating the maps
with various foreground templates. The dominant foreground emission is expected
to be due to synchrotron radiation, free-free emission and (vibrational and spinning)
dust emission, from both the Milky Way (seen as diffuse emission) and other galaxies
(seen as point sources). These cross-correlation analyses concluded that foregrounds
played only a subdominant role in Saskatoon and QMAP.
By comparing the Ka- and Q-band maps, we are able to place a direct constraint
on the frequency dependence of the signal. Both Saskatoon and QMAP observe in
both of these frequency bands. We therefore repeat the analysis described above
(Saskatoon deconvolution, merging with QMAP, etc.) separately for each of the two
bands. The upper panel of Figure 2.10 shows equal-weighting W*iener-filtered maps
for Ka-band and Q-band in the sky region th at was observed at both frequencies,
showing that they visually agree well.
If we fit the frequency dependence by a power law ST(u) ex u0 over the narrow
frequency range in question (i^k* ss 30 GHz, i/k* ~ 40 GHz), then characteristic
spectral indices are 8 ~ —2.8 for synchrotron, 8 ~ —2.15 for free-free emission,
3 ~ —3 for spinning dust and
~ 2 for vibrating dust. By definition, 8 = 0 for
CMB. If the sky signal in our maps obeyed ST{u) cx i/^, then the difference map
XRa —t*xq would contain pure noise when r was such that
a = I. g, ^lgr.
.■
/ t 'Q )
(2.5)
The lower panel of Figure 2.10 is therefore plotted with 8 rather than r on the
horizontal axis. Insisting th at the difference map not be inconsistent with noise at
more than ler gives the spectral index constraint 8 —
which is inconsistent
with the signal being any single one of the foregrounds mentioned above.
31
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Q Band
m
e
o
w
C
Ka Band
*o>
Q
20
w
c
e
-20
-10
0
10
20
S p e c tr a l in d e x 0
Figure 2.10 Comparison of QMASK a t two different frequencies, Ka-band (left) and Q-band (right).
The upper panel shows both maps Wiener filtered with the same weighting in the overlap region
th a t was observed a t both frequencies. Arrows indicate the coldest spot discussed below. The lower
panel shows the number of standard deviations ( “sigmas”) a t which the difference map Xk» —rx q
is inconsistent with mere noise.
32
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2.3.7
The coldest spot
Up until now, we have presented a battery of tests for systematic errors and other
problems, all of which passed. However, our maps did turn up one somewhat anoma­
lous feature: an unusually cold spot around “two o’clock” in the Saskatoon disk. The
spot’s coldest pixel in the QMASK map is located at RA=3fc20ro, DEC=84°55\ Here
the Wiener-filtered Q-band map in Figure 2.10 gives 8T « —230/iK. For compar­
ison, the expected rms fluctuations in this map are 27/iK from detector noise and
49jiK from CMB fluctuations (for the “concordance” power spectrum of Tegmark,
Zaldarriaga and Hamilton (2001)), summing to 56/jK in quadrature. Taken at face
value, this would indicate that the spot is a —4.1<r fluctuation. For completeness,
this section describes a number of additional tests performed in an attem pt to clarify
its nature.
It is unlikely th at the cold spot is due entirely to systematic error, since it is
clearly detected by both QMAP (covered by the Q3 and Q4 detectors from in flight
2) and Saskatoon. The Saskatoon experiment even detected this spot independently
in each of its three observing seasons (Netterfield et al., 1997; Tegmark et al., 1996).
Indeed, the reason th at the f2Q34 map shows the most spectacular agreement with
Saskatoon in Figure 2.7 is that f2Q34 is the only QMAP map that covers the area
containing this spot.
We have plotted all available microwave measurements of this region in Fig­
ure 2.11 as a function of frequency, including both the Ka- and Q-band measure­
ments from QMASK and the COBE/DMR observations at 31.5, 53 and 90 GHz
(Bennett et al., 1996). Unfortunately, Figure 2.11 is not an actual spectrum of the
spot. It is more of a comparison of apples and oranges, since the measurements
differ dramatically in angular resolution (7° for COBE) and the QMASK maps are
Wiener-filtered. Since Wiener filtering always pushes the signal towards zero when
noise is present, the QMASK points should be interpreted as lower limits on |#T| —
the true sky tem perature is likely to be even colder.
33
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100
— -100
▲QMAP&SASK■ COBE
-200
-300
20
40
60
80
100
F r e q u e n c y v [GHz]
Figure 2.11 The temperature towards RA=3fc20TO, DEC=84°55 at different frequencies. Errors
bars correspond to detector noise alone. Note that these points cannot be interpreted as a spectrum
of this sky region, since the COBE points have much lower resolution and the two QMASK points
have been Wiener filtered with different weights, pushing them closer to zero than the underlying
sky temperature.
34
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On the seven degree scale probed by COBE, no evidence is seen for a cold spot
(this can also be seen in the map of Figure 2.9), and the COBE spectrum of this
region appears consistent with CMB, *.e., 3 = 0. One possible interpretation is that
a small cold region resolved by QMAP and Saskatoon is partly smoothed away by
COBE.
Unfortunately, this sky patch is not covered by QMAP in Ka-band, so the Kainformation comes from Saskatoon alone and is therefore noisier than the Q-band
measurement. This means th at the Wiener-filtering has suppressed the Ka-band
more, as well as lowered its resolution by more aggressive smoothing. However,
the expected rms CMB fluctuations in the Wiener-filtered maps are not nearly as
different as the data points in Figure 2.11 (—31/iK at Ka-band and —230/iK at Qband), indicating that the low Q-band temperature does not persist fully down to
Ka-band. This argues against both a CMB origin and a thermal SZ-origin, which
would cause a cold spot that was essentially frequency-independent for u < 100 GHz.
All other known microwave foregrounds produce hot rather than cold spots. Some
sort of absorption process also appears unlikely, since the absorbing medium would
have to be colder than 3K. No relevant foreground emission or X-ray cluster is found
in radio, infrared or X-ray maps of the region. Some adjacent dust emission is seen
in the IRAS 100/xm map (de Oliveira-Costa et al., 1997), which could potentially
make this region look cold in contrast (since none of our maps are sensitive to the
monopole mode and measure merely relative temperatures), but only at the level of
about 10 /iK.
In Q-band, the spot is observed by QMAP only in flight two and only in the
Q3 and Q4 channels. The latter dominates statistically, and has a 13% calibra­
tion uncertainty. However, Saskatoon also observes the spot in Q-band, and both
experiments measure around —200/xK in their individual Wiener filtered maps.
The explanation is fairly likely to be something mundane, since a 4-sigma fluc­
tuation (which should happen about once for every 16,000 independent pixels) is
35
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not extremely unlikely when many pixels are considered — if the effective number
of independent regions in the Wiener filtered map is 102 taking the smoothing into
account, the significance level gets downgraded to (1 —1/16000)100 ~ 99% (Bromley
and Tegmark, 1999).
In summary, the cold spot is definitely out there at some level, but we have no
single simple interpretation of what is causing it. Its unusual spectrum argues against
a non-Gaussian CMB fluctuation, foreground contamination and an SZ signal. The
most likely remaining explanation is a confluence of a less extreme CMB cold spot,
noise fluctuations, calibration uncertainty and perhaps some small systematic error.
We have described this spot in such detail simply to ensure that no hints of problems
with the data get swept under the rug.
The MAP satellite should resolve this puzzle next year, observing the spot with
high sensitivity and resolution at 22, 30, 40, 60 and 90 GHz.
2.4
D IS C U S S IO N
We have presented methods for comparing and combining CMB maps and applied
them to the QMAP, Saskatoon and COBE DMR data sets. We found th at these
methods were able to place interesting constraints on calibration and pointing prob­
lems, foreground contaminations and systematic errors in general. This should make
them quite useful for ongoing and upcoming high-precision experiments.
The d ata sets passed our entire battery of consistency tests, placing strong limits
on systematic errors. Although the Saskatoon and QMAP maps detect signal at the
21o and 40(7 levels in the overlap region, respectively, their difference is consistent
with pure noise. Our combined QMAP + Saskatoon map, nicknamed QMASK,
covers a large enough area to allow a statistical comparison with COBE/DMR,
showing good agreement.
The one surprise that our battery of tests turned up is th at a small region around
36
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(RA=3h20m,DEC=84°55 ) appears unusually cold, mainly in Q-band. Its unusual
frequency dependence argues against non-Gaussian CMB fluctuations, SZ-signal and
known foregrounds.
The QMASK map presented here has been made publicly available at
www.hep.upenn.edu/^xuyz/qmask.html together with its 6495 x 6495 noise covariance
matrix. W ith its 648 square degrees, this thoroughly tested d ata set is currently the
largest degree-scale map available, detecting signal at a level exceeding 60a.
37
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38
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Chapter 3
The CMB power spectrum at
i = 30
—
200 from QMASK
In this chapter, we measure the cosmic microwave background (CMB) power spec­
trum on angular scales £ ~ 30 —200 (1° —6°) from the QMASK map, which combines
the data from the QMAP and Saskatoon experiments. Since the accuracy of recent
measurements leftward of the first acoustic peak is limited by sample-variance, the
large area of the QMASK map (648 square degrees) allows us to place among the
sharpest constraints to date in this range, in good agreement with BOOMERanG
and (on the largest scales) COBE/DMR. By band-pass-filtering the QMAP and
Saskatoon maps, we are able to spatially compare them scale-bv-scale to check for
beam- and pointing-related systematic errors.
3.1
IN T R O D U C T IO N
After the discovery of large-scale Cosmic Microwave Background (CMB) fluctuations
by the COBE satellite (Smoot et al., 1992), experimental groups have forged ahead to
probe ever smaller scales. Now th at TOCO (Torbet et al., 1999; Miller et al., 1999),
BOOMERanG (de Bernardis et al., 2000) and Maxima (Hanany et al., 2000) have
39
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convincingly measured the location and height of the first acoustic peak, attention
is shifting to still smaller scales to resolve outstanding theoretical questions. For
instance, the structure of the second and third peaks constrains the cosmic m atter
budget1. However, it remains important to improve measurements on larger angular
scales as well, both for measuring cosmological parameters and to cross-validate
different experiments against potential systematic errors. This is the goal of the
present paper.
Since the accuracy of recent measurements leftward of the first acoustic peak is
limited by sample-variance rather than instrumental noise, we will use the largest
area CMB map publicly available to date with degree scale angular resolution. This
map, nicknamed QMASK (Xu et al., 2001), is shown in Figure 3.1 and combines
the data from the QMAP (Devlin et al., 1998; Herbig et al., 1998; de Oliveira-Costa
et al., 1998) and Saskatoon (Netterfield et al., 1995, 1997; Tegmark et al., 1996)
(hereafter SASK sometimes) experiments into a 648 square degree map around the
North Celestial Pole. This map has been extensively tested for systematic errors (Xu
et al., 2001), with the conclusion that the QMAP and Saskatoon experiments agree
well overall, as well as analyzed for non-Gaussianitv (Park et al., 2001; Shandarin
et al., 2001). However, for the present power spectrum analysis, it is im portant to
perform additional systematic tests to see if there is evidence of scale-dependent
problems in any of the maps. In particular, pointing problems, beam uncertainties,
sampling and pixelization effects can smear the maps in a way that changes the
shape of the power spectrum, suppressing small-scale fluctuations (and under some
circumstances boosting the fluctuations as well).
'A fter this paper was first submitted, accurate new measurements of these higher peaks were
reported by the Boomerang (Netterfield et al., 2001), DASI (Halverson et al., 2001) and Maxima
(lee et al., 2001) teams, so we have added comparisons with these results below.
40
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Figure 3.1 Wiener-filtered QMASK map combining the QMAP and Saskatoon experiments. The
CMB temperature is shown in coordinates where the north celestial pole is at the center of the
dashed circle of 16° diameter, with R-A. being zero at the top and increasing clockwise. This map
differs from the one published in (Xu et al-, 2001) by the erasing of QMAP information for I > 200
described in the text.
41
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The rest of this paper is organized as follows. In Section 3.2, we present a tech­
nique for erasing this type of unreliable information, and apply it to produce a new
combined QMASK map where all the statistical weight on small scales comes from
Saskatoon. We compute the power spectrum of this combined map in Section 3.3.
We discuss systematic errors in section 3.4, finding good agreement but a hint of
suppressed QMAP power for £ £ 200, and conclude with a rather conservatively cut
data set that we believe to be reliable as a starting point for cosmological parameter
analysis.
3.2
C om bining th e SA SK and Q M A P E xperim en ts
As will be discussed in more detail in Section 3.4, pointing problems, beam uncer­
tainties, pixelization and sampling effects can suppress small-scale fluctuations in
the CMB power spectrum. The QMAP results may therefore only be valid on large
angular scales (Devlin et al., 1998; Herbig et al., 1998; de Oliveira-Costa et al., 1998).
Both pointing inaccuracies and pixelization effects could have effectively smoothed
the QMAP map, suppressing small-scale power. Flight 1 of QMAP (Devlin et al.,
1998) was sampled at a relatively low rate, causing the effective beam shapes to
be elongated along the scan direction. Since this ellipticity was not modeled in
the mapmaking algorithm (de Oliveira-Costa et al., 1998), the resulting smoothing
would again be expected to suppress small-scale power. Miller A.et al. (2001) present
a detailed discussion of these issues, and conclude th at the QMAP measurements
are likely to be unreliable for £ £ 200. Although attem pts can be made to model
and correct for some of these effects, an accurate treatm ent of the undersampling
problem in particular would be way beyond the scope of the present paper, requir­
ing the entire maps to be regenerated from the time-ordered d ata with an order of
magnitude more pixels to be able to resolve the beam ellipticity (the ellipticity is
not uniform in direction, since the sky rotated relative to the scan axis during the
42
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flights). Such an analysis would hardly be worthwhile anyway, since the strength of
QMAP compared to subsequent experiments lies on large scales, not on small scales.
For these reasons, we adopt a more conservative approach, combining the QMAP
and Saskatoon maps in such a way that the small-scale QMAP information can be
optionally erased as a precaution. By this we do not mean removing the signal
(smoothing the map), which would just lead to further underestimation of the true
power. Rather, we mean removing the information, i.e., doing something that causes
subsequent analysis steps (like combining with Saskatoon or measuring the power
spectrum) to give negligible statistical weight to the small-scale QMAP signal. We
achieve this by creating a random map with very large small-scale noise and adding
it to the QMAP map, modifying its noise covariance matrix N accordingly.
In practice, we start by generating a white noise map xWhite which has the follow­
ing properties: it covers the same sky region as QMAP, and each pixel temperature
is drawn independently from a Gaussian distribution with zero mean and standard
deviation a, giving it a noise covariance matrix
= a2I. This makes its an­
gular power spectrum Cg independent of £. We then apply the Laplace operator
V2 to the mock map. Since it is pixelized on a square grid, we do this in practice
by multiplying by a matrix L that subtracts each pixel from the average of its four
nearest neighbors. The transformed map Xbiue = Lx white thereby obtains a very blue
power spectrum Cg 2S £*, since Laplace transformation corresponds to multiplying
by £{£ + 1 ) in the Fourier (multipole) domain. We choose the normalization factor a
such th at the blue noise starts dominating the noise and signal of the QMAP map
around £ ~ 200. Since the CMB power falls off as Cg 2S £~2, the result is that the
added noise is negligible for I < 200 and dominates completely for £ > 300. Finally,
we add this blue noise map to the QMAP map, obtaining
X nem
=
Snco =
L*white»
(3.1)
X qmAP + Sblue = E qMAP -f-^LLV
(3.2)
X q m aP +
X b lu e =
*Q M A P +
To quantitatively assess the effect of this procedure, we perform a series of numerical
43
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experiments where we rescale the noise level by a factor p = 0,1,3,5,7,1000. This
corresponds to multiplying Xbiue by p and multiplying Ebiue by p2. p = 0 means
adding no noise, i.e., that Xnm, retains all QMAP information; p = oo (or p = 1000
in practice) means that Xnew contains essentially pure noise. Below we will see that
the choice p = 3 erases information on small scales I > 200 very well, while doing
little harm on larger scales. After erasing with p = 3, we combine the resulting maps
with the Saskatoon data as in Xu et al. (2001). The result is shown in Figure 3.1,
and looks almost unchanged (compare Figure 4 in Xu et al. (2001)) since Wiener
filtering suppresses noisy modes and the smallest scales were noisy to start with.
3.3
T h e A ngular Pow er Spectru m
In this section, we compute the angular power spectrum of the QMASK m ap pro­
duced in the previous section, shown in Figure 3.1. It contains 6495 pixels and covers
a 648 square degree sky region. We calculate the angular power spectrum using the
quadratic estimator method of (Tegmark, 1997; Bond, Jaffe and Knox, 2000), im­
plemented as described in Padmanabhan, Tegmark, and Hamilton (2001); Tegmark
and de Oliveira-Costa (2001). This method involves the following steps: (i) S/N
compression of data and relevant matrices by omitting Karhunen-Loeve (KL) eigenmodes with very low signal-to-noise ratio, (ii) computation of Fisher m atrix and raw
quadratic estimators, (iii) decorrelation of data points. We compute the power in 20
bands (listed in table 3.1) from £ = 2 to 400 of width &£ = 20, which takes about a
week on a workstation. We repeated this calculation in its entirity for the following
values of the p-parameten 0 (no information erased), 1, 3, 5, 7 and 1000 (Saskatoon
only, QMAP frilly erased), as shown in Figure 3.2.
In the progression of power spectra shown in this figure, the l-values beyond which
QMAP is effectively erased is seen to shift to the left as p increases. For p = 3, this
scale is seen to be of order 200 in the sense th at the leftmost points are almost
44
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indistinguishable from the p = 0 case whereas those with decent signal-to-noise for
£
200 (say points number 10 and 11) are similar to the p = 1000 case. Adopting
this cutoff scale as our baseline calculation (we further discuss this choice in the
next section), we then average these rather noisy p = 3 measurements into six (still
uncorrelated) measurements listed in table 3.2. The first four probe angular scales
where the above-mentioned systematics are likely to be negligible, incorporating the
first 9 f-bands (up to £ = 180), and are shown in Figure 3.3. The horizontal bars
show the mean and rms width of the corresponding window functions.
3 .4
D iscussion
We have measured the CMB power spectrum from the combined QMAP and Saska­
toon data sets, obtaining significant detections in the range £ ~ 30 —300. The key
question that we need to address in this section is how reliable these measurements
are.
The •‘usual suspects” as far as systematics and non-CMB contamination are
concerned tend to have power spectra that are either redder or bluer than that of
the CMB, and therefore naturally split into two categories:
• Problems on large angular scales can be caused by contamination from dif­
fuse foregrounds (synchrotron, free-free and spinning dust emission), and sys­
tem atic errors related to atmospheric contamination, scan-synchronous offsets,
atmospheric contamination, etc.
• Problems on small angular scales can be caused by contamination from point
source foregrounds and systematic errors related to pointing problems, beam
uncertainties, pixelization and sampling effects.
The foreground contamination (Tegmark et al., 2000b) has been previously quan­
tified for both Saskatoon (de Oliveira-Costa et al., 1997) and QMAP (de OliveiraCosta et al., 2000), by cross-correlating the maps in question with a foreground
45
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I 1 I | 1 I I I [ m I ljl J LWI l " M M I ♦ M
1
P=3
' 4 t f e L 5 fc -JL p = i o o o _ r J t S j r r i L I
100
200
300
M ultipole 1
0
100
200
300
M ultipole 1
Figure 3.2 Angular power spectrum ST = [/(/ + l)C</2x]1/2 (uncorrelated) in 20 bands of CMB
anisotropy from the combined QMASK data in case of p = 0(no additional noise at all, just
simple combination of SASK and QMAP),1, 3, 5, 7 and 1000 (no QMAP information, only SASK
information). For comparison, we also plot the a recent “concordance” model (Tegmark, Zaldarriaga
and Hamilton, 2001) and the power measurements from MAXIMA and BOOMERanG.
46
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p = 0
= 3
I
ST} [/iATJi]
£
ST} [pK*]
24 ± 11
49 ± 12
70 ± 9
90 ± 10
110 ± 10
130 ± 11
150 ± 11
170 ± 1 1
190 ± 11
210 ± 13
231 ± 17
250 ± 20
270 ± 21
291 ± 27
309 ± 2 7
325 ± 31
339 ± 3 7
349 ± 4 7
348 ± 6 5
268 ± 7 9
641 ± 7 6 3
1821 ± 559
2327 ± 551
1543 ± 606
2646 ± 698
2386 ± 815
2083 ± 955
3358 ± 1108
4717 ± 1266
2030 ± 1430
5791 ± 1586
2918 ± 1728
7116 ± 1865
1408 ± 2009
3969 ± 2182
7263 ± 2421
1475 ± 2776
2530 ± 3278
5842 ± 3985
3430 ± 5187
24 ± 11
49 ± 12
70 ± 9
90 ± 1 0
109 ± 10
130 ± 11
150 ± 11
170 ± 11
190 ± 11
211 ± 13
231 ± 17
253 ± 20
273 ± 21
291 ± 27
309 ± 2 7
323 ± 31
336 ± 3 7
346 ± 4 7
345 ± 65
271 ± 79
474 ± 769
1459 ± 570
2056 ± 567
1492 ± 629
2403 ± 738
2737 ± 885
1872 ± 1071
3652 ± 1278
5265 ± 1487
749 ± 1689
7007 ± 1861
3964 ± 1991
7952 ± 2101
3207 ± 2 2 1 1
6268 ± 2352
7903 ± 2563
241.4 ± 2897
2540 ± 3378
6800 ± 4 0 6 4
3167 ± 5237
Table 3.1 The power spectra ST = [£{£ + l)C//27r]1/2 from the QMASK map in two
casesrp = 0 and p = 3. The tabulated error bars are uncorrelated between the
twenty measurements, but do not include an overall calibration uncertainty of 10%
for ST.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
80 4.
ft
60 -
CM
+
£
40 QMASK
x COBE
Maxima
^ B o o m e ra n g
□ DASI
20
I
I
2 510
■ i I m u l m i i i m l m i ................. i i i i l i i i m i i i m i m m l .......
40
100
200
400
M ultip ole 1
600
800
Figure 3.3 Uncorrelated measurements of the CMB power spectrum ST = [/(£ + l)C//2ir]1/r2 from
the combined QMASK data. For comparison, we also plot the measurements from COBE/DMR,
MAXIMA, DASI and BOOMERanG. These error bars do not include calibration uncertainties of
10% (QMASK), 10% (BOOMERanG), 4% (DASI) and 4% (MAXIMA). As described in the text,
we suggest using only the first four points (I < 200), not the two dashed ones, for cosmological
model constraints.
48
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1
40 ± 16
79 ± 13
125 ± 19
178 ± 15
259 ± 46
335 ± 5 8
8TI [fiK2]
1308 ± 452
1899 ± 410
2436 ± 471
4295 ± 869
4265 ± 726
2596 ± 1795
Table 3.2 The power spectrum 8T = [£(£ + 1)Ci /2 tt]1^2 from the QMASK map. The
tabulated error bars are uncorrelated between the six measurements, but do not
include an overall calibration uncertainty of 10% for ST. We recommend using only
the first four for cosmological model constraints.
templates tracing synchrotron, dust and free-free emission, point sources, etc. The
conclusion was that foregrounds contribute at most a few percent to the angular
power spectrum reported here, mainly on the largest angular scales. The errors re­
ported on the power spectrum assume th at the underlying CMB signal is Gaussian,
which is supported by two recent Gaussianity analyses of the QMASK map (Park et
al., 2001; Shandarin et al., 2001).
As opposed to foregrounds, which are true features on the microwave sky, the
remaining problems listed above are experiment-specific effects which would be ex­
pected to affect QMAP and Saskatoon differently. This provides us with a powerful
tool with which to test for their presence, which we will now employ: comparing the
QMASK and Saskatoon maps where they overlap.
Previous comparisons between CMB data sets have been done either spatially
or in terms of power spectra (discarding phase information). Since two inconsistent
maps can have identical power spectra, one should be able to obtain still stronger
tests for systematic errors by comparing power with phase information. For instance,
one could imagine band-pass filtering the two maps to retain only a particular range
of multipoles I, and then testing whether these two filtered maps were consistent.
We will now describe a simpler way of implementing a test in this spirit.
49
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3.4.1
A m ethod for scale-by-scale com parison o f two maps
Numerous map comparisons have been performed with the “null-buster” test (Tegmark,
1999b)
_ z*N-1S N -1z - tr {N -1S}
"
p t r f N ^ S N - ^ S }]1/2 ’
{
’
where v can be interpreted as the number of “sigmas” at which the difference map
z is inconsistent with pure noise. If the two maps are stored in vectors x t and
x 2and have noisecovariance matrices N t and N 2, then a weighteddifference map
z = x t —rx 2willhave noise covariance N = N i + r 2N 2- The
matrix S tells the test
which modes (linear combinations of the pixels z) to pay most attention to, and can
be chosen arbitrarily. The choice S = N gives a standard x2-test- It can be shown
(Tegmark, 1999b) that the null hypothesis that z is pure noise (that (zz‘) = N ) is
ruled out with maximal significance on average if S is chosen to be the covariance of
the expected signal in the map, i.e., S = (zz1) —N . In our case, we choose S to be
the CMB covariance matrix corresponding to a power spectrum
6T, = \l" K
I 0 nK
i f < € I*™*-*”*“ ];
(34)
otherwise.
ensuring that the test only uses information in the multipole interval [fmn» ftnax]The overall normalization of S is irrelevant, since it cancels out in equation (3.3).
3.4.2
R esults o f com paring Q M AP and Saskatoon scale-byscale
We will now use the method described above to compare the QMAP and Saskatoon
data scale-by-scale, ue., in different multipole intervals.
The QMASK map has been shown to be inconsistent with noise at the 62cr level
(Xu et al., 2001). In the region where the QMAP and Saskatoon maps overlap, they
were found to detect signal at 40a and 21a, respectively, while the difference map
50
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was consistent with pure noise. Which angular scales are contributing most of this
information, and how well do the two maps agree scale-by-scale?
To answer these questions, Figure 3.4 shows the result of comparing QMAP
with Saskatoon in the four multipole intervals [2,100], [101,200], [201,300], and
[301,400]. QMAP is seen to detect signal in the overlap region at 47a, 18<t, 12<r and
6a, respectively, whereas the corresponding numbers for Saskatoon are l&r, 14<7, l \ a
and 7a. In other words, QMAP dominates on large scales, whereas Saskatoon has
a t least comparable information content on small scales because of superior angular
resolution.
Although QMAP and Saskatoon both detect significant CMB signal in all four
bands, this signal is seen to be common to both maps since the difference maps z
for r = 1 are consistent with noise. Furthermore, there is no evidence of relative
calibration errors for the [2,100] or [101,200] bands, since the minima of these two
curves are at r ss 1. However, the situation is less clear on smaller scales: the best-fit
amplitude of QMAP is only 63% of the amplitude of Saskatoon for £ € [201,300],
and even lower for £ € [301,400]. None of these departures of the minimum from
r = I are statistically significant — we cannot determine whether this is a a problem
or not simply because the amount of information in the maps drops sharply on small
scales where detector noise and beam dilution become important. However, whereas
the Saskatoon information was extracted from highly oversampled calculations of
the relevant beam patterns on the sky, and should be reliable on small scales, there
are a number of reasons why the QMAP d ata may only be valid on larger {£ ^ 200)
angular scales (Devlin et al., 1998; Herbig et al., 1998; de Oliveira-Costa et al., 1998):
1. The QMAP pointing solution was only accurate to this level (Herbig et al.,
1998), and small residual pointing uncertainties could have effectively smoothed
the map, suppressing power for £ ^ 200 relative to Saskatoon just as Figure 3.4
indicates.
2. The QMAP maps were generated by subdividing the sky into square pixels of
51
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40
30
QMAP
20
Saskatoon:
+ +«■
QMAP
Saskatooni
>10
QMAP
; = [2 0 1 .3 0 0
Saskatoon:
-o
QMAP
= [3 0 1 ,4 0 0 ]
co
0.001
0.01
0.1
1
Saskatoon:
10
100
1000
R ela tiv e n o r m a liz a tio n r
Figure 3.4 Comparison of the QMAP and Saskatoon experiments on different angular scales, cor­
responding to the multipole ranges shown in square brackets. The curves show the number of
standard deviations (“sigmas”) at which the difference map
—r x s A S K is inconsistent with
mere noise. Note that this is only for the spatial region observed by both experiments.
x q m
a p
52
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side 9 = 0.3125°, and the effect of this pixelization may well become im portant
on angular scales substantially exceeding £ ~ 1/9 « 200, suppressing power on
these scales.
3. Flight 1 of QMAP (Devlin et al., 1998), which dominates the sky coverage
in Figure 3.1 was sampled at a relatively low rate, causing the effective beam
shapes to be elongated along the scan direction. Since this ellipticity was
not modeled in the mapmaking algorithm (de Oliveira-Costa et al., 1998),
the resulting smoothing would again be expected to suppress power on scales
t
£ 200.
Miller A.et al. (2001) present a detailed discussion of these issues, and conclude that
the QMAP measurements are likely to be unreliable for I ^ 200. This justifies our
p — 3 choice in Section 2, effectively discarding this suspect small-scale information
from QMAP.
3.4.3
Conclusions
We have measured the CMB power spectrum (see values in table 3.1 from the com­
bined QMAP and Saskatoon data sets, and performed a number of tests for potential
systematic errors. On large angular-scales, £ & 200, our results appear very solid:
foreground contamination has been quantified to contribute no more than a few
percent to the power spectrum, and we find beautiful internal consistency between
the QMAP and Saskatoon components of our map, both in terms of power ampli­
tude and in terms of spatial phase information. On smaller scales £ ^ 200, there
are good a priori reasons to discard the QMAP information, and we have therefore
done so. Our internal scale-by-scale comparison between QMAP and Saskatoon in­
dependently suggests that there may be systematic problems on these scales. Since
this unfortunately leaves us with no independent way of validating our Saskatoon
53
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results in this small-scale regime, e.g., testing our Saskatoon deconvolution proce­
dure, we strongly recommend the conservative approach of using only the first four
band power measurements we have reported in table 3.2 and Figure 3 — this is also
where QMASK is most sensitive relative to other experiments. These four measure­
ments, spanning the range £ ~ 30 —200, provide among the sharpest and best tested
constraints to date on these scales, provide a good starting point for constraining
cosmological models.
3.4.4
Com parison w ith other experim ents
Figure 3.2 shows that our results agree well with the “concordance” model of Tegmark,
Zaldarriaga and Hamilton (2001). They are also consistent with BOOMERanG (Netterfield et al., 2001), DASI (Halverson et al., 2001) and Maxima (Hanany et al., 2000)
once calibration uncertainties are taken into account. The QMAP and Saskatoon
calibration uncertainties are 6% —10% and
10%,
respectively (we have corrected the
original ST results (Devlin et al., 1998; Netterfield et al., 1997) by a factor 1.05 using
the latest Cassiopeia A data (Mason et al., 1999) as in Gawiser and Silk (2000)). Al­
though the uncorrelated components of this may average down somewhat when the
two maps are combined, we quote a
10%
uncertainty on our result to be conservative.
Our results are also consistent with those obtained from QMAP and Saskatoon alone.
Our leftmost data point agrees well with the last point measured from COBE/DM R
(Bennett et al., 1996) by Tegmark (1997).
The original BOOMERanG98 power spectrum was based on a map area of 436
square degrees (de Bemardis et al., 2000), so one might expect our error bars on
large scales to be a factor (648/436)1/2 as 1.2 smaller. O ur actual error bars on STf
are only about 10% smaller on the best QMASK scales after adjusting for bandwidth
differences, which is because of BOOMERanG’s lower noise levels (the scan strategy
and l//-n o ise of QMAP introduced a non-negligible amount on noise even on the
largest scales). We note th at the substantial reduction in error bars relative to the
54
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original Saskatoon analysis (Netterfield et al., 1997) is due to additional information
not only from QMAP, but from Saskatoon as well. This is because our present
method extracts all the information present, whereas that employed in Netterfield et
al. (1997) was limited to information along radial scans, not using phase information
between scans.
In conclusion, we have measured the CMB power spectrum on angular scales £ ~
30 —200 from the QMASK map, placing among sharpest and best tested constraints
to date on the shape of the CMB power spectrum as it rises towards the first acoustic
peak. Our window functions, the combined map and its noise covariance matrix are
available at www.hep.upenn.edu/ ~ xuyzfqmask.html.
55
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^
I1-*>
C
Ip U h (o-
56
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Chapter 4
M orphological M easures of
non-G aussianity in CMB Maps
In this chapter, we discuss the tests of nonGaussianity in CMB maps using morpho­
logical statistics known as Minkowski functionals. As an example we test degreescale cosmic microwave background (CMB) anisotropy for Gaussianity by studying
the QMASK map th at was obtained from combining the QMAP and Saskatoon data.
We compute seven morphological functions Mi (AT), i = 1,..., 7: six Minkowski func­
tionals and the number of regions .Vc at a hundred A T levels. We also introduce
a new parameterization of the morphological functions A/, (A) in terms of the total
area .4 of the excursion set. We show that the latter considerably decorrelates the
morphological statistics and makes them more robust because they are less sensi­
tive to the measurements at extreme levels. We compare these results with those
from 1000 Gaussian Monte Carlo maps with the same sky coverage, noise properties
and power spectrum, and conclude that the QMASK map is neither a very typical
nor a very exceptional realization of a Gaussian field. At least about 20% of the
1000 Gaussian Monte Carlo maps differ more than the QMASK map from the mean
morphological parameters of the Gaussian fields.
57
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4.1
In tro d u ctio n
The issue of Gaussianity of CMB maps plays a crucial role in testing assumptions
about the early Universe. The simplest inflation models strongly favor Gaussianity
of the primordial inhomogeneities (see Turner, 1997, and references therein), whereas
other scenarios assuming cosmic strings or topological defects (see Vilenkin and Shellard, 1994, and references therein) predict non-Gaussian perturbations. Gaussianity
is also a key underlying assumption of all experimental power spectrum analyses to
date, entering into the computation of error bars (Tegmark, 1997; Bond and Jaffe,
1998), and therefore needs to be observationally tested. A third reason for studying
Gaussianity of CMB maps is that it may reveal otherwise undetected foreground
contamination.
Numerous tests of Gaussianity in the COBE maps (Colley, Gott, and Park, 1996:
Kogut et al., 1996; Ferreira, Magueijo, and Gorski, 1998; Pando, Yalls-Gabaud, and
Fang , 1998; Bromley and Tegmark, 1999; Novikov, Feldman, and Shandarin, 1999;
Banday, Zaroubi, and Gorski , 2000; Barreiro et al., 2000; Magueijo, 2000; Mukherjee, Hobson; Aghanim, Fomi, and Bouchet, 2001; Phillips and Kogut, 2001) have
resulted in the general agreement that all non-Gaussian signals were of noncosmological origin. This was not unexpected because of COBE’s low (7° deg) angular
resolution.
Testing Gaussianity on smaller scales may bring much more interesting results.
The first study of Gaussianity on degree scale showed the consistency of the QMASK
map with the assumption of the Gaussianity (Park et al., 2001). However, this study
tested only the Gaussianity of the topology of the map. The study of the MAXIMA1
anisotropy d ata also showed the consistency with the Gaussianity in the range
between 10 arc-minutes and 5 degrees (Wu et al., 2001). A total of 82 tests for
Gaussianity were made in this study. Although performing many tests is always
better than few it is not clear how independent these tests were.
58
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The question of statistical independence of different non-Gaussianity tests is com­
plicated. However, in some simple cases some of the tests can be statistically dis­
entangled. For instance, consider the probability function, which is perhaps the
simplest test for non-Gaussianity. It easy to transform the probability function to
any given form simply by relabeling the contours using a monotonic function. This
relabeling obviously
does not have any effect on the morphology of the field since
every contour line remains the same as well as the order of levels due to monotonicity
of the transformation. However, this transformation may strongly affect the whole
hierarchy of the n-point correlation functions. This is of course due to well known
connection of the probability function to all n-point functions.
Genus represents only one statistic sensitive to non-Gaussianity. It may detect
some types of non-Gaussianity and be not sensitive to the others. Consider, for
instance, the following mapping of a Gaussian field <t>(x, y ) into <£(.Y, V')
4>(X,V')=<*>(x,y),
(4.1)
where
-Y
=
x + tu { x , y),
Y
=
y + Tv{ x, y) .
(4.2)
Equation 4.1 guarantees that every level label remains unchanged, while the level
lines are shifted and deformed according to eqs. 4.2. Functions u(x, y ) and v(x, y )
can be any smooth functions including random functions. At small r before caus­
tics have occurred, the mapping (x, y) —►(.Y, Y) is single valued and therefore the
contour map of $(-Y, Y ) is one-to-one image of 4>(x. y ) , although geometrically dis­
torted. The genus of the #(JY, V') field as a function of the level remains exactly
the same (t.e. Gaussian) as th at of the 0 (x, y) field because the mapping is continous
and nonsingular and therfore preserves toplogy. On the other hand, the mapping
(x, y) —> (X, Y ) can be neither isometric (conserving the lengths) nor area-preserving
and therefore both the areas and the contours of the excursion sets may be distorted.
59
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As a result, the $ (A , Y) can be a strongly non-Gausssian field having exactly the
Gaussian genus in the level parameterization. Perhaps, it is worth noting th at a par­
ticular case of the above example (u(x, y) =
y ) / d x and v(x, y) = drp(x, y)/dy)
corresponds to viewing the Gaussin contour map
0 (x,
y) through a glass layer with
varying thickness rp(x,y) (see e.^.Zel’dovich, Mamaev, and Shandarin (1983); Shandarin and Zel’dovich (1989)).
A one-dimensional illustration of the above field is shown in Fig. 4.1. A realiza­
tion of the Gaussian field <t>(x) is shown by the solid line and of the non-Gaussian
field $(x ) by the dashed line. The total area of the excursion set in 2D is analogous
to the total length of the excursion set in ID and the genus in 2D is analogous to
the number of peaks in ID. The level parameterization means the comparison of the
number of peaks in the two fields at the same level. Both fields have the same num­
ber of peaks at every level as illustrated by the dotted and dash-dotted horizontal
lines. However, the total lengths occupied by the peaks of the <t>(x) and $(x) fields
are different at a given level. The length parameterization in ID means th at the
comparison of the numbers of peaks must be done at the same total length of the ex­
cursion sets which generally happens at different levels for the <£(x) and <f»(x) fields.
For instant, the length of the excursion set of the Gaussian field at
0 (x)
= 1.95 (the
sum of two heavy line segments at level = 1.95) equals the length of the excursion
set of the non-Gaussian field a t 4>(x) =
level =
1)
1
(the sum of four heavy line segments at
. One can see at this total length the number of peaks in the G aussian
field is two while in the non-Gaussian field it is four. Thus, the number of peaks
parameterized by the total length does detect this type of non-Gaussianity while the
number of peaks parameterized by the level does not. Similarly, the area parame­
terization of the genus in two dimensions does detect this type of non-Gaussianity
while the level parameterization of the genus statistic does not.
If the mapping (eq. 4.2) is area-preserving than the genus of the <J>(A\ F ) field
remains exactly Gaussian also in the area parameterization. Thus, a detection of
60
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such non-Gaussianity requires additional tests.
We use two different parameterization of the measured morphological charac­
teristics and show th at the parameterization by the total area of the excursion set
(x.e.the cumulative probability function) has an advantage over parameterization by
the tem perature because it gives considerably smaller correlations between different
measures. In particular, the total area parameterization detects non-Gaussianity
by the genus statistic in the above example while the tem perature parameterization
does not. For additional discussion of this issue, see Shandarin (2002).
Along with the QMASK map (Xu et ai.2001), we analyze a thousand refer­
ence maps with the same sky coverage, noise properties and power spectrum as the
QMASK map. For each map we compute seven morphological functions at a hundred
tem perature/area levels. These functions are:
1. the total area of the excursion set .4(A!T),
2. the total contour length C(A!T) and C[A) ,
3. the genus of the excursion set G(AT) and G(.4) ,
4. the area of the largest (by area) region .4P(AT) and .4P(.4) ,
5. its perimeter CP(AT) and Cp(-4),
6.
genus GP(AT) and Gp(-4), and finally
7. the number of regions JVC(A!T) and :VC(.4) .
The first three are the global Minkowski functionals and the following three are
the Minkowski functionals of the largest (by area) region. The largest region at
some threshold spans the whole map, or “percolates” . Its Minkowski functionals are
the best indicators of the percolation phase transition (Shandarin, 1983; Yess and
Shandarin, 1996). Our notations reflect some of the jargon used in cluster analysis:
61
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Ap, Cp, Gp for the Minkowski functionals of the percolating region and Nc for the
number of regions, often referred to as “clusters” in cluster analysis.
We then calculate the mean for each of these fourteen functions (Mj(AT’) and
Mi(A), i = l ,- - - ,7 ) computed from a thousand Gaussian maps and quantify the
differences of every function for every map with respect to the mean functions in
quadratic measure. Finally, we compute these differences for the QMASK map and
test the Gaussianity hypothesis by comparing them to the thousand reference maps.
The rest of the paper is organized as follows. In Sec. 2 and 3 we briefly describe
the QMASK map and the Gaussian simulations thereof. In Sec. 4 we define the
morphological functions, their parameterization and the method of quantifying the
departure from the Gaussianity. In Sec. 5 we summarize the results.
4.2
Q M A SK M ap
The QMASK map, described in (Xu et al., 2001), combines all the information from
the QMAP (Devlin et al., 1998; Herbig et al., 1998; de Oliveira-Costa et al., 1998)
and Saskatoon (Netterfield et al., 1995, 1997; Tegmark et al., 1996) experiments into
a single map at 30-40 GHz covering about 648 square degrees around the North Ce­
lestial Pole. The map consists of sky temperatures in 6495 sky pixels, conveniently
grouped into a 6495-dimensional vector x, with a FWHM angular resolution of 0.68°.
As detailed in Xu et al. (2001), all the complications of the map making and deconvolution process are encoded in the corresponding 6495 x 6495 noise covariance
matrix N . The map has a vanishing expectation value (x) = 0 and a covariance
matrix given by
C = (xx1) = N + S,
(4.3)
where the signal covariance m atrix S is given by
S# - E
e=2
(4.4)
%7r
62
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the Pi are Legendre polynomials, Ct is the angular power spectrum, r, is a unit vector
pointing towards the Ith pixel, and 6 =FW H M /V 8 In 2 is the rms beam width. When
computing S in practice, we use the smooth power spectrum of (Xu, Tegmark, and de
Oliveira-Costa, 2001) that fits the observed QMASK power spectrum measurements.
Since the raw map has a large and complicated noise component, we work with
the Wiener-filtered version of the map in this paper, shown in Fig. 4.2 and defined
as
Xu, = W x,
W = S(N + S )-1.
(4.5)
This approach was also taken in Park et al (2001) and Wu et al (2001).
4 .3
M ock M aps
In order to quantify the statistical properties of our Minkowski functionals, we need
large numbers of simulated QMASK maps. We therefore generate one thousand
mock Gaussian maps as follows. The covariance matrix C„, of the Wiener-filtered
map is given by
Cw = (xu,j4> = ((W x) (W x)‘) = W (x x t)W t = W C W 1.
(4.6)
and we Cholesky-decompose it as C w = L L ‘, where the matrix L lower triangular.
It is straightforward to show that if z is a vector of independent Gaussian random
variables with zero mean and unit variance (which is trivial to generate numerically),
i.e., (zz‘) = I, the identity matrix, then the mock maps defined x^, = Lz will have
a multivariate Gaussian distribution with (x^x^) = Cw.
4 .4
M orphological S ta tistic s
In this paper we use a set of morphological statistics based on Minkowski functionals.
As suggested by (Novikov, Feldman, and Shandarin, 1999), one can use both global
63
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and partial Minkowski functionals in studies of CMB maps. In this study we use the
global Minkowski functionals and those of the largest region by area (the so-called
percolating region).
4.4.1
Global Minkowski functionals
Global Minkowski functionals were introduced into cosmology as quantitative mea­
sures of CMB anisotropy by Gott et al. (1990) although without reference to the
Minkowski functionals and thus without stressing their unique role in differential
and integral geometry. Mecke, Buchert, and Wagner (1994) were the first to place
the studies of morphology of the large scale structure in the context of differen­
tial and integral geometry. In particular, they emphasized a powerful theorem by
(Hadwiger, 1957) stating that with rather broad restrictions, the set of Minkowski
functionals provides a complete description of the morphology (for further discussion
see Kerscher, 1999).
Global Minkowski functionals describe the morphology of the excursion set at
a chosen threshold: .4(AT), C( AT) and G(A T) are the total area, contour length
and genus of the excursion set, respectively. The Minkowski functionals of Gaussian
fields are known analytically as functions of the threshold. Assuming that the field
is normalized, i.e.u = (A T — < A T >)/cr.yr so that < u > = 0, < u2 > = 1, the
Minkowski functionals are
(4.7)
where R = \/2/<T\ is the characteristic scale of fluctuations in the field; cr\ is the rms
of the first derivatives (in statistically isotropic fields both derivatives d u / d x and
d u f d y have equal rms). -4(u) is the fraction of the area in the excursion set and thus
64
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equals the cumulative probability function: A(u) =
p(u’)du', where p(u) is the
probability function. C(u) and G(u) are the length of the contour and the genus of
the excursion set per unit area respectively.
In this study we use somewhat different normalization and units. We measure
the total contour length in the mesh units and we define G as the number of isolated
regions minus the number of holes in the excursion set.
4.4.2
Percolating region
The morphology of every isolated region in the excursion set can be characterized
by three partial Minkowski functionals : the area, boundary contour length and
genus of the region. In order to describe the morphology of the field in more de­
tail, Novikov, Feldman, and Shandarin (1999) used the distribution functions of the
partial Minkowski functionals at several level thresholds. Bharadwaj et al. (2000)
used the averaged shape parameters in the study of the morphology in the LCRS
slices. Here we choose to utilize only the Minkowski functionals of the largest region
(AP(A T ),C P(A T ),G P(AT)) in addition to the global Minkowski functionals . This
is dictated by the relatively small size of the QMASK map: the maximum num­
ber of isolated regions is about 30 and therefore the distribution functions are very
noisy. The Minkowski functionals of the largest, t.e., percolating region of Gaussian
fields are not known in an analytic form but can be easily measured in Monte Carlo
simulations (Shandarin, 2002).
An additional comment may be useful. A naive idea of percolation invokes an
image of the region connecting the oposite sides of the map. Thus, the question may
arise: what does percolation mean in a map with complex boundaries and holes like
the QMASK map (Fig. 4.2). In particular, which points must be connected a t the
percolation threshold in a map with irregular boundaries? The fact is th at percola­
tion is a phase transition which takes place regardless of the shape and topology of
65
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the map (see e. (/.Stauffer and Aharony (1992)). It can be studied and relatively eas­
ily measured in maps of arbitrary shapes including holes. The Minkowski functionals
of the largest region are particularly useful while the direct detection of the connec­
tion between the oposite sides is much more unstable even in simple square maps
(Dominik and Shandarin, 1992). At percolation transition the area and perimeter
of the largest region experience a rapid growth while the genus rapidly decreases
(Shandarin, 2002). Similarly to many other statistics the percolation transition is
affected mostly by the size of the map and its effective dimensions and much less
by its shape or presence of holes. For example, if a three-dimensional map has the
shape of a thin wedge then percolation becomes two-dimensional (Shandarin and
Yess, 1998). Likewise, a two-dimensional map having the shape of a narrow strip
effectively percolates as a one-dimensional map. More accurately, if the smallest
size of the map is smaller than the scale of the field then the percolation transition
effectively reduces the dimensions. A simple visual inspection of Fig. 4.2 shows that
nothing of the kind of the problems discussed above is present in the QMASK map.
The m ajor drawback of the map is its size not the shape.
In addition, we wish to stress that we do not compare the percolation curves mea­
sured for the QMASK map with the standard Gaussian curves. Instead we compare
them with the curves measured in the reference maps having exactly same irregular
shape and holes as the QMASK map. Therefore, the question of accuracy of re­
production of the standard percolation properties in two-dimensional maps becomes
largely irrelevant.
4.4.3
Num erical technique
The numerical technique used for measuring the Minkowski functionals is described
in detail in (Shandarin, 2002). Here we outline the main features th at may differ
from numerical methods used by others (Coles, 1988; Gott et al., 1990; Schmalzing
and Buchert, 1997; Winitzki and Kosowsky, 1997; Schmalzing and Gorski, 1998;
66
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Park et al., 2001; Wu et al., 2001).
First, we use neither Koendemik invariants (Koenderink, 1984) nor Croftons’s
formulae (Crofton, 1968) which are often used in other studies. A pixelized map is
considered to be an approximation to a continuous field. This approach was used
by Novikov, Feldman, and Shandarin (1999) but here we use a different and more
refined version of the code computing Minkowski functionals . The major effect of
this improvement is the numerical efficiency of the code.
The major features of our technique are illustrated by Fig. 4.3. Suppose we
use a regular square lattice and the true contour corresponding to u =
is a
simple ellipse shown by the solid line in Fig. 4.3. The sites satisfying the threshold
condition u > tith are shown by filled triangles, while the sites with u < Uth are
shown by empty triangles. The union of elementary squares (shown by heavy solid
lines) placed on each site satisfying the threshold condition is widely used as an
approximation to the region. One can easily count the number of elementary squares,
number of edges of the elementary squares and number of vertices in the union.
Then by using Croftonrs formulae one computes the area, perimeter and the Euler
characteristic of the region. For instance, the area in this approximation is simply
the sum of areas of the elementary squares. The perimeter is the number of external
edges of the elementary squares. While the area converges to the right value as the
lattice constant approaches zero, the perimeter does not (for a detailed discussion of
this effect see e. <7.Winitzki and Kosowsky (1997)). The perimeter of the set of the
elementary squares obviously equals the perimeter of the rectangular region shown
by the dotted line. As the lattice parameter approaches zero the perimeter of the
region approches the perimeter of the large rectangle shown by the solid line which
obviously is a wrong value. Assuming the isotropy of the map one can show th at
the mean length of a segment is 4 /x times larger than its true value (Winitzki and
Kosowsky, 1997) and can be corrected only statistically. Our method is completely
free of this drawback and does not need such a correction.
67
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In our approach we begin with the construction of the contour points for each
isolated region of the excursion set. Suppose th at u(t, k) < t/y, but u(i +
Uth then the contour point lies somewhere between (t, k) and (i +
1,
1, k)
>
k) sites. Its
coordinates (or, t/*) can be obtained by solving the linear interpolation equation
....
u ( i + l , k ) —u (i.k ),
.. ...
= “(1’ k) + *(■ + 1, k) k ) (X - I ( l - * »
(4'8)
for the coordinate x. We construct the contour points by linear interpolation of the
field between the sites in both horizontal and vertical directions. These points are
marked by the solid circles in Fig. 4.3. Then, computing the perimeter we assume
th at the contour points are connected by the segments of a straight line (dashed line).
The contours made by this method converges to the right value of the perimeter as
the lattice constant approaches zero.
The other major difference consists in allowing a direct linkage of diagonal grid
sites in some cases depending on the values in the four neighboring sites. Consder a
case when two diagonally adjacent pixels satisfy the threshold condition: u(i, k) >
Uth and u(i + l,fc +
1)
> uy, but the sites u(i + 1,k) < uy, and u ( i ,k +
1)
< ny*.
One usually chooses either consider the (i, k) and (i + 1, k + 1) sites always directly
linked or always not linked implying that the difference between the two alternatives
disappears with vanishing the lattice parameter. Our definition of the direct linkage
of two diagonally adjacent pixels is different from both. The diagonally adjacent
pixels are directly linked if titneon = (u(i, fc)+ u(i+ l, k)+u(i, fc + l)+ u (t+ l, fc+ i))/4 >
Uth &nd directly not linked otherwise. In the latter case they of course are allowed to
be linked by the friend of friend procedure if they have proper friends. This method
affects the number of regions, their sizes and the number of holes in the regions
and therefore the values of all Minkowski functionals (including genus). The visual
inspection of Fig. 4.4 and Fig. 4 in Park et al. (2001) reveals small but noticeable
differences in the genus curves. This treatm ent of the direct diagonal linkage also
considerably reduces the intrinsic anisotropy of a rectangular grid. Since the QMASK
m ap occupies a small part of the sky, we ignore the intrinsic curvature of the map
68
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when computing the areas and perimeters.
If a part of the region boundary is the boundary of the map mask, then it does
not contribute to the contour length of the region. Similarly, the holes in the map
mask contribute neither into the genus nor the contour length.
The analysis of one realization of a Gaussian random field at a hundred levels
takes about 0.7s, 2.9s, 12.3s, and 73s for 642, 1282, 2562, and 5122 maps respectively
on HP C240 workstation. It corresponds approximately to the t cx iVp** ln^V,**)
dependence of the computational time on the size of the map.
4.4.4
Param eterization by th e level o f AT.
In addition to six Minkowski functionals , we computed the number of isolated
regions N e (Novikov, Feldman, & Shandarin 1999). Thus, in total we compute seven
morphological parameters
Mi (AT) = [.4(AT), C (A T ), G(AT), .4p(AT), Cp(AT), GP(AT), .VC(AT)],
i = 1 , .. ., 7
(4.9)
at a hundred threshold levels AT for the QMASK map and the thousand reference
maps. The six panels of Fig. 4.4 show six Minkowski functionals measured for the
QMASK map (solid lines) and the mean values for a thousand Gaussian maps (heavy
dashed lines). The thin dashed lines show the
68%
and 95% intervals.
The genus curve (left bottom panel) is in good qualitative agreement with that
obtained by Park et al. (2001). However, it is worth noting that we use a different
normalization and a somewhat different variable on the horizontal axis: we use the
level normalized to the rms u = A T /<7a t- while Park et o/.use a more complex
quantity uA related to the excursion set of the Gaussian field. Park et a/.describe
their parameterization as follows: “the area fraction threshold level uA is defined
to be the tem perature threshold level at which the corresponding isotemperature
contour encloses a fraction of the survey area equal to th at at the tem perature level
vA for a Gaussian field” (the bottom of p.586 ). Some quantitative differences may
69
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be also ascribed to a different linkage scheme used by Park et a/, (see the previous
section). We wish to stress that the parameterization by uA has similar properties
(including the noise correlation properties) with the parameterization by the area
(.4) discussed in the following section. The only difference between them is the
geometrical interpretaion.
The number of isolated regions is shown in Fig. 4.5 for both the temperature
and global area parameterizations. The latter is described in detail in Sec. 4.5.
In order to quantify the
and compute
degree of non-Gaussianity of the QMASK map,we define
a functional that quantifies the “distance”betweentwo functions in
the functional space of measured statistics. The distance is defined by the integral
D 'i 71
=
i =
<i(AT)}'/2 .
1,..., 7;
fc = l,...,1000
(4.10)
where
1
Mi(AT) = —
1000
£
M ,(A T )
(4.11)
is the mean value of the statistic as a function of A T. The integral in eq. (4.10)
is approximated by the stun over a hundred levels of AT. We also compute the
distances of the each morphological curve from the Gaussian mean for the QMASK
map. The third column of Table 1 shows the percentile of the QMASK map distances
from the mean for each morphological statistic parameterized by AT. The highest
departure from the mean is shown by the -4(AT) curve which is the cumulative
probability function. It shows that only about 27% of Gaussian curves are closer to
the Gaussian mean than the QMASK curve. Other parameters are even closer to
the corresponding Gaussian mean.
4.4.5
Param eterization by th e total area A.
The parameterization of the morphological parameters by the level of the field u has
serious drawbacks mentioned in Introduction. Although the parameterization by vA
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used in most studies of the genus is free of these drwbacks it seems to be more nat­
ural for morphological studies to use the total area A ( A T ) = A es (AT) / A m (where
A es (-AT) is the area of the excursion set at A T and Am = A es (—o o ) is the total
area of the map) as an independent parameter (Yess and Shandarin, 1996). It is
one of the Minkowski functionals measured for the map, it represents the cumulative
distribution functions of the field, and in addition it has a very simple geometrical interpretaaiton. It also has an additional advantage of decorrelating different statistics
as discussed bellow. The uA parameterization must have similar properties because
it is related to .4 by a monotonic transform (the first eq. of 4.7).
The transformation from the level parameterization to the area parameterization
is a highly nonlinear procedure illustrated in Fig. 4.6. The thick lines correspond
to the QMASK map (solid) and a randomly chosen Gaussian map (dashed). The
transformation from C ( A T ) to C(.4) involves the A ( A T ) function shown in the lower
panel, which is different for each map. The thin solid lines show the transformation
for the QMASK map and dashed lines for the Gaussian map. Note that in order to
illustrate this transformation, we reversed the .4 axis in the C = C(.4) plot.
As an illustration of the difference between the level and area parameterization
consider a simple example of a strong nonlinear field. Let the field w(x, y) is the
exponent of a Gaussian field u(x. y): w = exp(cm). The global Minkowski functionals
of such a field can be easily derived from eq. 4.7
(4.12)
Fig. 4.7 illustrates the difference between the Minkowski functionals of the Gaus­
sian field u and non-Gaussian field w (with a =
1)
if both are parameterized by
the level. The percolation curves differ strongly as well. On the other hand both
the perimeter and genus as the functions of the area remain identical for the both
71
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fields because the transformation simply relabels the levels but does not affect the
geometry of the contours. This is an example of a trivial non-Gaussian field when
all non-Gaussian information is stored in the one-point probability density function.
The further discussion of such fields can be found in Shandarin (2002).
Using the area parameterization, we compute seven new morphological functions
for each map:
Mi(A) = [ A T ( ^ ) ,C M ) ,G M ) ,^ U ) ,C p(^),C?p(A),iVe(^)],
i = 1,...,7.
(4.13)
Figure 4.8 shows the measured quantities as functions of A. Similarly to Fig. 4.4 six
panels show six Minkowski functionals measured for the QMASK map (solid lines)
and the mean values for a thousand Gaussian maps (heavy dashed lines). The thin
dashed lines show the
68 %
and 95% intervals.
The difference of the QMASK curves from the Gaussian mean is quantified simi­
larly to that described before by the distance between curves in the functional space:
D“ ’ = U ' ['v/“ (-4) - ;V7'(-4>]2
i = 1,..., 7;
k = 1,..., 1000
dA}n
(4.14)
where
i
1000
M.M) =
E
(4.15)
There isan im portant difference between the two distances defined by eq. 4.10 and
4.14. Using dA = —p*(AT)d(AT) where p*(AT) is the estimate of the probability
function derived from the fc-th map one can rewrite eq. 4.14 as an integral over AT:
4fc4) =
=
{ j f [A M .4)-A 7,(.4)]2 dA j 1/2
{ / ^ [M*(AT) - Mi (AT )]2pk( AT) d(A T ) } 1/2
(4.16)
Comparing eq. (4.10) and (4.16), one notices that the distance between the curves
computed in the area space (eq. 4.14) depends more on the bulk of the probability
function and less on the tails of the probability function, since it is weighted by
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the probability function. This obviously makes the measure more robust because
the measurements at extreme levels are subject to large fluctuations. It is worth
stressing th at the calculation of all morphological parameters Mik themselves (eq. 4.9
and 4.13) is independent of the type of parameterization because Mik {A) = Mifc(AT)
provided th at A is measured at corresponding A T : A = .4(AT). The only quantities
th at depend on the parameterization are the distances from the Gaussian expectation
functions (eqs. 4.10 and 4.14) that are used for ranking the maps.
The second column of Table 1 shows the percentile of the QMASK distances
from the mean in the area parameterization. Comparing the values corresponding
to different parameterizations, one can easily see that the parameterization by A T
suggests that the QMASK map is a more typical example of a Gaussian field than
th at using .4 as an independent parameter. The parameterization by the level sug­
gests that the most exceptional deviation of the QMASK map from Gaussianity is
given by the .4(AT) curve, but more than 70% of Gaussian maps differ from the
mean more than the QMASK map. Thus, the QMASK map looks like a very typical
realization of the Gaussian field. Parameterizing by the total area, one may conclude
th at the Gaussian P(.4) and -Vc(.4) curves differ more than the QMASK map in only
about 21% and 18% of the cases respectively. Thus, this parameterization reveals
th at the QMASK map is not quite a typical realization of the Gaussian field, but is
not very unusual either, quite consistent with Gaussianity.
It is perhaps worth noting that all statistics but the genus of the percolating
region, Gp show a greater difference of the QMASK map with respect to the corre­
sponding expectation value of the set of Gaussian maps in the A parameterization
than that in A T parameterization. Since the difference between two parameteriza­
tions for Gp is not large ( 12% and
21%)
may note th a t two columns of Table
1
it maybe a statistical fluctuation. One also
appear anti-correlated. As we have discussed
before the quantities shown in two columns of Table 1 have different sensitivity to
the measurements at extreme A T levels both highest and lowest. The numbers in
73
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the first column corresponding to the A parameterization are more robust.
4.4.6
Cross-correlations
The parameterization of the morphological statistics by the total area makes the set
of parameters Mi (A) (eq. 4.13) less correlated than A/j(AT) (eq. 4.9). Table 2 shows
the correlation coefficients of the distances
1000
£ (A* - Di)(Djk - Dj)
Tij = -------------J
r /i o o o
_
\ / iooo
_
\ i 1/2
(4.17)
[ ( E ( ^ - A ) 2) f e d * - * , ) ’)]
for all 21 pairs of morphological statistics. The values above the diagonal show the
correlations when all statistics were parameterized by the total area of the excursion
set. .4, while the values below the diagonal were computed using the parameterization
by the level. AT.
The correlation coefficient shows for each pair of statistics how well the separation
from the mean Gaussian curve in one statistic A/, can be predicted from the other
one Mj. The closer rtJ to unity, the less independent are corresponding statistics.
In this respect the AT and .4 parameterizations are very different. For instance, the
lowest correlation in the class of the level parameterization is 0.82 between GP(AT)
and iVc(AT) distances and 16 out of 21 cross-correlations are greater than 0.9. In
the class of the area parameterization, only one correlation coefficient is greater
than 0.82: th at between G(.4) and iVc(.4) distances (0.85). Seventeen out of 21
pairs correlate at a level lower than 0.5 thirteen of which at the level lower than
0.3. Although absence of correlation does not prove the statistical independence of
two sets of numbers, a correlation coefficient approaching unity indicates a strong
statistical dependence.
74
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4 .5
D iscu ssion
We present the results of a study of Gaussianity in the QMASK map. In agreement
with the first study of topology of the degree-scale CMB anisotropy (Park et al.,
2001), which was limited to the genus of QMASK, we conclude that the QMASK
map is compatible with the assumption of Gaussian AT / T fluctuations. In this
study, we used six morphological statistics in addition to the genus, G : the areas
.4 and Ap of the excursion set and the largest (i.e.percolating) region, the contour
lengths or perimeters C and Cp of the excursion set and the largest region; the genus
Gp of the percolating region and the number of isolated regions Nc. According to its
morphology, the QMASK map is not a very typical example of a Gaussian field, but
not very exceptional eith er about 20% of a thousand Gaussian maps have greater
differences with at least one mean function (Table 1).
We show that the parameterization of the morphological statistic is very im­
portant. A naive parameterization by the temperature threshold AT results in a
highly correlated (almost degenerate) set of morphological characteristics: 16 out of
21 cross-correlation coefficients are greater than 0.9 and all the rest are greater than
0.82 (Table 2, bellow the diagonal). Representing the same morphological parame­
ters as functions of the total area of the excursion set .4 helps to decorrelate them:
now 13 pairs correlate less than 0.3, four more correlate less than 0.5 and the rest
less than 0.85 (Table 2, above the diagoanl).
Our measurements of the deviations of the morphological functions from the
Gaussian mean strongly rely on the bulk of the probability function (see eq. 4.16)
and therefore are robust to spurious effects.
In the A parameterization the strongest deviations of the QMASK map from the
Gaussian expectation values were shown by the number of peaks, NC(A) (82% *.e.the
QMASK map deviates from the Gaussian expectation number of peaks more than
82% of Gaussian maps), total contour length of the excursion set, C(.4) (79%) and
area of the percolating region, Ap(A) (55%) while in the AT parameterization by
75
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the total area of the excursion set *.e.the cumulative probability function, .4(AX)
(27%), genus of the percolating region, GP(AT) (21%) and contour length of the
percolating region, CP(AX) (20%).
The techniques that we have described here are computationally efficient (ex
0 { N vtx \nNpix)), and should be useful also for much larger upcoming datasets such
as that from the MAP satellite. Another promising area for future applications
is computing Minkowski functionals from foreground maps and topological defect
simulations, thereby enabling quantitative limits to be placed on the presence of
such structures in CMB maps.
76
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Table 4.1.
Percentage of Gaussian maps deviating less than QMASK.
P a r a m e t e r i z e d b y A r e a , .4
L e v e l, A T
A r e a , .4
P e rim e te r, C
G enus, G
A r e a ( P e r c ) , .4 P
P e rim e te r (P e rc ), C p
G en us (h e re ), G p
N u m b e r o f re g io n s , Nc
P a r a m e te r i z e d b y L e v e l, A T
2 8 .2 %
2 7 .2 %
6 .1 %
2 .0 %
1 2 .1 %
1 9 .7 ^ 6
2 1 .4 %
0 .4 %
7 8 .8 %
4 8 .3 %
5 4 .6 %
2 4 .3 %
1 1 .6 %
8 2 .5 %
Table 4.2. Correlations between different non-Gaussianity statistics parametrized
by temperature, A T (below the diagonal) and area, .4 (above the diagonal).
H
II
A T
C
G
cr
GV
ffc
A
C
G
~Ap
~Cp
Gp
~Nc
0 .9 8
0 .9 2
0 .9 7
0 .8 9
0 .8 8
0 .9 2
0 .0 8
1
0 .9 6
0 .9 7
0 .9 2
0 .9 1
0 .9 4
0 .1 5
0 .6 6
1
0 .9 3
0 .9 1
0 .9 3
0 .9 6
0 .0 6
0 .1 3
0 .3 0
1
0 .9 1
0 .9 0
0 .9 2
0 .1 0
0 .2 3
0 .2 2
0 .5 8
1
0 .9 4
0 .8 4
0 .1 3
0 .3 4
0 .4 4
0 .3 7
0 .5 6
1
0 .8 2
0 .1 6
0 .4 9
0 .8 5
0 .3 0
0 .0 8
0 .1 2
1
77
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2
1
e
0
0
-1
-2
Figure 4.1 One-dimensional illustration of a non-Gaussian field ♦ (dashed line) having exactly same
number of peaks (which is analogous to genus in 2D) in the level parameterization as the Gaussian
field 0 (solid line). Compare the number of peaks in two fields at two levels marked by the dotted
and dash-dotted horizontal lines. The length parameterization in ID is analogous to the area
parameterization in 2D. In order to compare the number of peaks in the length parameterization
one has to count the peaks of the two fields at different levels. The marked levels are chosen in
such a way th a t the total length of the excursion set of the non-Gaussian field at ♦ = 1. (the sum
of four heavy segments) equals the total length of the the excursion set of the Gaussian field at
0 = 1.95 (the sum of two heavy segments).
78
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100
-1 0 0
Figure 4.2 Grey scale QMASK map. Light color correspond to higher temperatures. Note that
there are three clearly distinct white regions in the map where data are absent.
79
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A
A
A
A
Figure 4.3 An illustration of the numerical technique. The solid line shows the elliptical contour
corresponding to the certain threshold u = u(/,. Triangles mark the sites of the lattice: solid with
u > uth and empty with u < u(*. Solid circles mark the contour points satisfying the condition
u = uth (the roots of eq. 4.8); the dashed line is the resulting contour. The true area of the region
is the area within the solid ellipse. We approximate it by the area within the dashed contour while
in most works it is approximated by the sum of areas of the elementary squares (solid squares).
We approximate the perimeter by the length of the dashed contour while in other works it is often
approximated by the sum of the external edges of the elementary squares. One can easily see
th a t this approximation gives the value of the perimeter of the large dotted rectangle. As the
lattice constant approaches zero it converges to the perimeter of the large solid rectangle, while our
perimeter converges to the true value of the perimeter of the solid ellipse.
80
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£ 0 .5
1500
1000
2 io o o
Figure 4.4 Minkowski functionals as functions of u = A TI<t&t for the excursion set (left hand side
column) and for the percolating region (right hand side column). Top row: the area fraction .4 and
Ap, middle row : contour length C and Cp, bottom row: genus G and Gp. The contour lengths
are given in mesh units. The genus is “number of regions” — “number of holes”. The solid lines
show the parameters of the QMASK map, heavy dashed lines show the median Gaussian values,
thin dashed lines show 68% and 95% ranges.
81
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e
o3>
40
v\
> \\.
N\ V
^VVX
"
xVvv
C 20
z
L \\
-4
-2
0
4 0
Level
0.2
0.4 0.6
Area
Figure 4.5 The number of regions N e as a function of the level u —
as a function of .4 (right hand panel). Other notation is as in Fig. 4.4
0.8
1
(left hand panel) and
82
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1500
1000
CL
500
0.6
0.2
0.8
0.6
0.4
0
2
0
2
Level
Figure 4.6 Illustration of the transformation from A T to .4 parameterization. The thick solid lines
show C = C(A T) (top left panel) C = C(A) (top right panel) and A = .4(AT) (bottom left panel)
for the QMASK map. The thin solid lines illustrate the transformation. The dashed lines show
similar transformation for a randomly chosen Gaussian map.
83
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1
10
0.8
1
A
« 0.6
«
<
~
-
•I
0 .4
"
- \
- \
\
0.2
5
-
;
0
—
N
0.6
2
4>
0.6
■r
^ '
+ \
0 .4
E
:
t"
f
t
« 0.2
-
■
\
0.2
“
v
-
0
0
0 .0 4
• /
- /
- /
3
V
C9
\
0 .4
0 .0 4
0.02
C
_
^
0.02
-
0
0
’!
0.02
;
-
i
-0 .0 4
-2
0
2 0
0.02
- 0 .0 4
2
4
6
8 10
0
u
0 .5
1
Area
Figure 4.7 The global Minkowski functionals of the parent Gaussian field u and derived nonGaussian field w — exp u are shown in two left hand side panels as functions of the level. Both the
perimeter and genus remain the same for both fields if they are parameterized by the total area
(two bottom right hand side panels). All information about the non-Gaussianity of the to-field is
stored in the cumulative probability function (dashed lines in the top panels). The solid lines in
the top panels show the Gaussian cumulative probability function.
84
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H
I I I |
I I 1 |
M
I | I I I
|
I I I
ou
v
CO
0)
0 .5
u
<
:t'H +4+-n
t|l M
U
I[
\ \ \ [ f f -t | \ I I | I
1500
1000
"1000
F 500
-I i i i I i i i I i i i I i i i I i i i F
0 .2 0 .4 0.6 0.8
Area
0
0 .2 0.4 0.6 0 .8
Area
1
Figure 4.8 The figure is similar to Fig. 4.4, except that all morphological parameters are
of .4 = A cs/A m , where A b s and Am = A es(—oo) is the area of the excursion set and that of
whole map, respectively.
85
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Chapter 5
The Power Spectrum of Galaxies
in the 2dF 100k Redshift Survey
In this chapter, we compute the real-space power spectrum and the redshift-space
distortions of galaxies in the 2dF 100k galaxy redshift survey using pseudo-KarhunenLoeve eigenmodes and the stochastic bias formalism. Our results agree well with
those published by the 2dFGRS team, and have the added advantage of produc­
ing easy-to-interpret uncorrelated minimum-variance measurements of the galaxygalaxy, galaxy-velocity and velocity-velocity power spectra in 27 A:-bands, with nar­
row and well-behaved window functions in the range 0.01 h/M pc < k < 0.8 /i/M pc.
We find no significant detection of baryonic wiggles, although our results are con­
sistent with a standard flat
= 0.7 “concordance” model and previous tantalizing
hints of baryonic oscillations. We measure the galaxy-matter correlation coefficient
r > 0.4 and the redshift-distortion parameter ft = 0.49 ± 0.16 for r = 1 (0 =
0.47±0.16 without finger-of-god compression). Since this is an apparent-magnitude
limited sample, luminosity-dependent bias may cause a slight red-tilt in the power
spectum. A battery of systematic error tests indicate th at the survey is not only
impressive in size, but also unusually clean, free of systematic errors at the level
to which our tests are sensitive.
Our measurements and window functions are
87
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available at http : / / www.hep.upenn.edu/ ~ m a x /2df.htm l together with the survey
mask, radial selection function and uniform subsample of the survey that we have
constructed.
5.1
In tro d u ctio n
Three-dimensional maps of the Universe provided by galaxy redshift surveys place
powerful constraints on cosmological models, which has motivated ever more ambi­
tious observational efforts such as the the CfA/UZC (Huchra et al. 1990; Falco et al.
1999), LCRS (Shechtman et al. 1996) and PSCz (Saunders et al. 2000) surveys, each
well in excess of 104 galaxies. This has been an exciting year in this regard, with
early results released from two even more ambitions projects; the AAT two degree
field galaxy redshift survey (2dFGRS; Colless et al. 2001) and the Sloan Digital
Sky Survey (SDSS; York et al. 1999), which aim for 250,000 and
1
million galaxies,
respectively.
Analysis of the first 147,000 2dFGRS galaxies (Peacock et al. 2001; Percival
et al. 2001; Norberg et al. 2001a; Madgwick et al. 2001) and the first 29,000 SDSS
galaxies (Zehavi et al. 2002) have supported a flat dark-energy dominated cosmology,
as have angular clustering analyses of the parent catalogs underlying the 2dFGRS
(Efetathiou & Moody 2001) and SDSS (Scranton et al. 2002; Connolly et al. 2002;
Tegmark e£ al. 2002; Szalay e£ al. 2002; Dodelson et al. 2002). Tantalizing evidence
for baryonic waggles in the galaxy power spectrum has been discussed (Percival
et al. 2001; Miller et al. 2001), and cosmological models have been constrained in
conjunction with cosmic microwave background (CMB) data (Efstathiou et aL 2002).
The 2dFGRS team has kindly made their first 102,000 redshifts publicly available.
Given the huge effort involved in creating this state-of-the-art sample, it is clearly
worthwhile to subject it to an independent power spectrum analysis. This is the
purpose of the present paper, focusing on large (k < 0.3 h f Mpc) scales. Since the
88
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cosmological constraints from galaxy surveys are only as accurate as our modeling
of bias, extinction, integral constraints, geometry-induced power smearing and other
real-world effects, we will employ a number of recently developed techniques for
tackling these issues. Compared with the solid and thorough analysis by the 2dFGRS
team in Peacock et a i (2001) and Percival et al. (2001), our main improvements will
be in the following areas:
• By using an approach based on information theory, involving pseudo-KarhunenLoeve eigenmodes, quadratic estimators and Fisher matrix decorrelation, we
are able to produce uncorrelated measurements of the linear power spectrum
with minimal error bars and quite narrow window functions. This allows the
power spectrum to be plotted in an easy-to-interpret model-independent way
and, because of the narrow windows, minimizes aliasing from non-linear scales
when fitting to linear models.
• Using the stochastic bias formalism, we measure independently not one power
spectrum but three, encoding clustering anisotropy. On large scales where
redshift distortions are linear (Kaiser 1987), these three curves are the realspace power spectra of the galaxies, their velocity divergence (related to the
m atter density) and the cross-correlation between the two. On smaller scales,
the information they encode can be extracted using simulations.
The rest of this paper is organized as follows. In section 5.2, we describe the
2dFGRS data used and construct an easy-to-interpret subsample th at is strictly
magnitude limited after taking various real-world complications into account. We
perform our basic analysis in section 5.3 and report the results in section 5.4. In
section 5.5, we test for a variety of systematic errors in section 5.5 and quantify the
effect of non-linearity and non-Gaussianity on our measurements. In section 5.6, we
discuss our results, fit to cosmological models and compare our results with those in
the literature.
89
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3
A
s
o
&*
a
eg
C
230
220
210
200
190
180
170
R ight A scen sion (d eg )
30
20
10
0
-10
160
-20
ISO
-3 0
140
-4 0
-5 0
R igh t A sc e n sio n (d e g )
J
230
220
k
200
190
180
170
R ight A scen sio n (d eg )
1 40
■
— I—1— I— 1— I—'
30
20
10
0
R igh t A sc e n sio n (d e g )
Figure 5.1 The upper half shows the 59832 2dF galaxies in our baseline sample, in equatorial 1950
coordinates. The lower half shows the corresponding angular mask, the relative probabilities that
qq
galaxies in various directions get included.
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-50
5.2
D a ta m od elin g
The 2dFGRS is described in detail in Colless et al. (2001, hereafter C01). The
publicly released 2dFGRS sample consists of 102,426 unique objects (excluding
duplicates), of which 93,843 have survey quality redshifts (quality factor > 3 ) . Of
these 5,131 objects have heliocentric redshifts z < 0.002 and are therefore probable
stars, while a further 240 galaxies lie outside the defined angular boundaries of the
survey (usually inside a hole in one of the parent UKST fields, occasionally marginally
outside one of the 381 surveyed 2° fields). This leaves a sample of 88,472 galaxies
with survey quality redshifts. To do full justice to the quality of this data set in
a power spectrum analysis, it is crucial to model accurately the three-dimensional
selection function n(r), which gives the expected (not observed) number density of
galaxies as a function of 3D position. This is the goal of the present section.
As will be described in section 5.3, our method for measuring the power spectrum
requires, in its current implementation, that the selection function be separable into
the product of an angular part and a radial part:
n ( r ) = n ( ? ) f i( r ) ,
(5.1)
where r = t t and r is a unit vector. The angular part n(f) may take any value
between
0
and
1,
and gives the completeness as a function of position, *.e., the
fraction of all survey-selected galaxies for which survey quality redshifts are actually
obtained, while n(r) gives the radial selection function. Although it would be possible
to generalize the method to a non-separable selection function (by breaking up the
selection function into a sum of piece-wise separable parts), we have chosen to stick
to the simple case of a separable selection function, for two reasons. First, although
the selection function of the 2dF 100k release is not separable, it is nearly so (the
survey was originally designed so that it would be), and the gain from allowing a nonseparable selection function has seemed insufficient to justify the extra complexity.
Second, as described in Sections 5.5.3 and 5.5.3, we wish to be able to test for
91
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possible systematic effects arising from a misestimate of extinction, which would
cause a purely angular modulation of density fluctuations, or from a misestimate of
the radial selection function, which would cause a purely radial modulation of the
density. Such tests are facilitated if the selection function is separable.
There are two complications that cause slight departures from such separability
(C01):
(i). The magnitude limit varies slightly across the sky, because both the photo­
metric calibration of the parent UKST fields, and the extinction correction at
each angular position was improved after the survey had begun.
(ii). Seeing issues lead to lower completeness for faint galaxies, and weather varia­
tions therefore cause the magnitude-dependent completeness fraction to vary
in different 2° fields.
Below we will eliminate both of these complications with appropriate cuts on the
d ata set, obtaining a uniform subsample with a separable selection function as in
equation (5.1).
5.2.1
The basic angular mask
In this subsection, we describe our modeling of the angular mask n(f) for the full
sample. In subsequent subsections, we will shrink and re-weight this mask slightly
to eliminate the above-mentioned complications, obtaining the final result shown in
figure 5.1.
Once the 2dFGRS is complete, it will contain a total of 1192 circular 2° fields,
including 450 fields in a 75° x 10° strip near the North Galactic Pole, 643 fields in
an 85° x 15° strip near the South Galactic Pole, and a further 99 fields distributed
randomly around the Southern strip. The various intersections of these fields with
each other yield 7189 non-overlapping intersection regions, referred to as sectors.
Parts of sectors are excluded if they fall outside the boundaries of the 314 rectangular
92
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UKST plates of the parent APM survey, or inside one of the holes excised from the
plates in order to eliminate e.g. bright stars and satellite trails. The data release
specifies 2024 holes, of which 1670 lie within, or overlap, those parts of the UKST
plates designated as part of the 2dF survey.
The 100k release is a subset of the survey, containing data from 381 circular 2°
fields, including 39 random fields. Eventually, when the survey is done, the observed
region will be complete, but in the interim the released fields are variably incomplete,
with a different completeness fraction n(f) in each sector, as described in C01.
As part of the 2dFGRS data release, Peder Norberg and Shaun Cole provide
software that evaluates n(r) in each of approximately 2.5 million 3' x 3' pixels, taking
all the various complications into account. However, we wish to adopt a different
angular mask that admits a separable selection function, and we also wish to be
able to compute the spherical harmonics of the angular mask using the fast, analytic
method described in Appendix A of Hamilton (1993). We therefore use a more
explicit geometric (not pixellized) specification of the mask, described immediately
below.
All field, plate and hole boundaries are simple arcs on the celestial sphere,
corresponding to the intersection of the sphere with some appropriate plane. This
means th at any spherical polygon (a field, plate, hole, sector, etc.) can be defined
as the intersection of a set of caps, where a cap is the set of directions r satisfying
a -f >
6
for some unit vector a and some constant b 6 [—1,1]. For instance, a 2° field
is a single cap. and a rectangular plate is the intersection of four caps. We define
masks such as th at the one plotted in figure 5.1 as a list of non-overlapping polygons
such th at n(r) is constant in each one. We construct the basic 2dFGRS mask as
follows:
(i). We generate a list of 8903 polygons comprised of 7189 sectors and 1670 holes,
plus 44 polygons defining boundaries of UKST plates.
(ii). Whenever two polygons intersect, we split them into non-intersecting parts,
93
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thereby obtaining a longer list of 12066 non-overlapping polygons. Although
slightly tricky in practice, such an algorithm is easy to visualize: if you draw
all boundary lines on a sphere and give it to your child as a coloring exercise,
using four crayons and not allowing identically colored neighbors, you would
soon be looking at such a list of non-overlapping polygons.
(iii). We compute the completeness n(r) for each of these new polygons, originally
using the Norberg-Cole software, but subsequently using our own computa­
tions, described in the following subsections.
(iv). We simplify the result by om itting polygons with zero weight and merging
adjacent polygons th at have identical weight.
W ith the original Norberg and Cole completenesses, the result is a list of 3765
polygons, with a total (unweighted) area of 983 square degrees, and an effective
(weighted) area / n(f)dQ of of 537 square degrees.
W ith the revised completenesses described in Section 5.2.2, there are 3614 poly­
gons, with an (unweighted) area of 711 square degrees, and an effective (weighted)
area / h(r)dQ of 431 square degrees. This angular mask, and the polygons into which
it resolves, are illustrated in Figure 5.1.
Section 5.2.2 explains how we eliminate the two above-mentioned complications,
the variations in the magnitude limit, and the variations in the weather, so as
to produce an angular mask with the same radial selection function at all points.
The reader uninterested in such details can safely skip all this, jumping straight to
section 5.2.3, remembering only the simple bottom line: we create a uniform sample
with 64,844 galaxies over 711 square degrees th at is complete down to bj magnitude
19.27.
94
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C u ttin g to a uniform m agnitud e lim it
The 2dFGRS aimed to be complete to a limiting bj magnitude m = 19.45 after
correction for extinction. However, the actual limiting magnitude varies slightly
across the sky as described in C01. This is because after the survey began, there
have been improvements in both the photometric calibrations of the underlying
parent catalog (Maddox et al 2001, in preparation) and in the extinction corrections
(Schlegel, Finkbeiner & Davis 1998). We eliminate this complication by creating a
sub-sample that is complete down to a slightly brighter limiting magnitude m .,
applying the following two cuts:
(i). Reject all galaxies whose extinction-corrected magnitude bj is fainter than m ..
(ii). Reject all sectors whose extinction-corrected magnitude is brighter than m ,.
The magnitude limit of a sector is defined in the most conservative possible
fashion: it is the brightest among all the magnitude limits at the position of
each galaxy and of each Norberg-Cole pixel within the sector. The extinction
at each position is evaluated using the extinction map of Schlegel, Finkbeiner
&c Davis (1988).
Figure 5.2 shows the number of surviving galaxies as a function of m». As we increase
m ., the first cut eliminates fewer galaxies whereas the second cut eliminates more
galaxies. The result is seen to be a rather sharply peaked curve, taking its maximum
for m , = 19.27, for which 66,050, or 75 percent, of the 88,472 galaxies survive.
The choice m , = 19.27 turns out to maximize not only the number of galaxies,
but the effective survey volume as well. As the flux cut F . is made fainter, the depth
of the survey (cx F T 1/2 in the Euclidean limit) increases, but the area decreases
because there are fewer sectors complete down to F„. Therefore the survey volume
55(area)
x F T 3/2, and this also happens to peak for F . corresponding to magnitude
19.27.
95
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5.2.2
Angular selection function
M odeling th e w eather
One of the more time-consuming aspects of our analysis was modeling another
departure from uniformity in the 2dFGRS: spatial dependence of the magnitudedependent incompleteness. As described by C01, although the success-rate P for
measuring reliable redshifts (quality > 3) for targeted galaxies is in general quite
high, it depends on weather. The poorer the seeing is when a given field is observed,
the lower the success rate. Moreover, this weather modulation affects fainter galaxies
more than bright ones. C01 found the success rate to be well fit by an expression of
the form
P (F ) = 7(1 - (* //F )* ],
where F is the observed flux from the galaxy,
7
(5-2)
= 0.99, a = 2.5/ ln(10) ~ 1.086 and
is a parameter that is fit for separately for each observed field / , interpretable
as the faintest observable flux. Note that since this observational selection effect
depends only on magnitude and weather, this issue can be analyzed and resolved in
terms of apparent magnitudes alone, without explicitly involving redshifts.
The ^/-values computed by the 2dFGRS team were not part of the public release,
but it is straightforward to generate values from the data provided. Whereas C01
estimated $ / from the observed completeness fraction for each field, we performed a
maximum-likelihood fit over the fluxes of all objects (galaxies and stars) targeted for
observation in each of the 381 field-nights, the likelihood being a product of terms
P(Fi) for all successful observations (those yielding a survey-quality redshift), and
terms [1 —P(Fj)] for all unsuccessful observations. Maximizing over 382 parameters
(<£/ for each of 381 distinct field-nights / , and a global value of a, with
7
fixed equal
to 1), we obtain a best fit exponent a = 0.96±0.04. Since the exponent is consistent
with unity, we set a = 1 for simplicity. We repeated the analysis with a perm itted
to vary separately in each field, but the likelihood is consistent with constant values.
96
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As a cross-check, we repeated the entire analysis sector-by-sector instead of fieldby-field, obtaining reassuringly similar results.
R andom sam pling to a sharp m agnitud e lim it
As mentioned, our power spectrum analysis requires a selection function of the
separable form of equation (5.1). Yet the discussion above shows th at the shape
of the radial selection function n(r) varies across the sky, since the success rate
P/ (F) is different for each of the 381 field-nights / , as given by equation (5.2).
We remedy this problem by sparse-sampling the galaxies in such a way th at the
shape of the success rate P( F) (as opposed to its amplitude) becomes the same for all
fields. The amplitude variations can then be absorbed into the angular mask n(r),
restoring separability. There are clearly infinitely many ways of doing this — we
wish to find the way that maximizes the effective volume of the survey for measuring
large scale power.
If we throw away galaxies at random, keeping galaxies in a given field / with a
probability pj that depends on their observed flux F , then the original success rate
Pf (F) for the field from equation (5.2) gets replaced by P/ (F)pf (F). Our goal then
becomes to choose these probabilities p /(F ) such that
Pf(F)pf{F) = wf P.{F),
(5-3)
where P .(F ) is the desired uniform, global success rate and the weights Wf are are
scaling factors that will be absorbed into the angular mask. Since the functions
Pf (F) are known, equation (5.3) immediately specifies how we should choose the
probabilities once the function P. and the weights have been fixed. To maximize the
number of surviving galaxies, we want to make p/ and hence wj as large as possible.
Since probabilities cannot exceed unity, this implies that the best weights are
Wf = nun
Pf(F)
P .(F )
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(5.4)
It remains to choose the target success rate P. (F ). Since we are interested in large
scale power, our aim is to maximize not so much the number of galaxies, but rather
the effective volume of the su rvey, and we must accomplish this goal by adjusting a
function P .(F ) of apparent flux F. The way to do this is to choose P .(F ) so as to
retain all galaxies at the faint limit of the survey, and then to make P .(F ) as large
as possible at all other fluxes. Given that the original P{F) decreases monotonically
to fainter fluxes for all values of the weather parameter $ / , and th at 4>/ includes
cases of perfectly observed fields ( $ / = oo), the solution is simply to choose P .(F )
to be constant, which can be taken to equal
1 without
loss of generality, at all values
brighter than the flux limit.
This is delightfully simple and convenient: it means that the best choice is a pure
magnitude-limited sample with no magnitude incompleteness to keep track of! The
corresponding weights are
wj = m in P /(F ) = F /(F .) =
1
- $ //F .
(5.5)
where F. is the flux limit. The scheme thus keeps all galaxies at the flux limit F „
and discards a progressively larger fraction of the brighter galaxies in each sector so
as to cancel exactly the magnitude-dependence of the incompleteness.
The magnitude limit 19.27 arrived at in the previous subsection turns out to
maximize the number of galaxies not only before sparse-sampling, but also after
sparse-sampling.
The final result is a list of 3614 polygons with associated weights, available at
http://w w w .hep.upenn.edu/~m ax/ 2df.html together with the uniform galaxy sam­
ple and our power spectrum measurements. The total area is 711 square degrees,
and the effective area / n(r)d£l is 432 square degrees.
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5 .2 .3
T h e radial selectio n fu n ction
After the modeling of angular effects above, it remains to measure the radial selection
function n(r) for the uniform sample. It is important to do this as accurately as
possible, since errors in the selection function translate into spurious large scale
power.
The radial selection function n(r) that results from the analysis described imme­
diately below is shown in figure 5.3.
In addition to imposing a faint magnitude limit of bj = 19.27, we follow the advice
of the 2dFGRS team (Matthew Colless 2001, private communication) in cutting the
survey to a bright limit of bj = 15. We use a maximum likelihood method based on
the C~ method of Lynden-Bell (1971), which assumes that luminosity is uncorrelated
with position. We generate an initial approximation to the selection function using
a continuous version of the Turner (1979) method, which yields the exact maximum
likelihood solution for the case of a survey with a sharp faint flux limit. The Turner
method has the merit of being exceedingly fast (less than one CPU second), but it
works only if the survey is flux-limited at one end (e.g. the faint end). Starting from
the Turner solution, we use an iterative method designed to converge towards the
exact maximum likelihood solution for the selection function, which can be shown
(Hamilton & Tegmark 2002) to be a step function with steps at the limiting distance
of each of the ~ 60,000 galaxies in the sample. To implement the Bayesian prejudice
th at the selection function should be smooth, we interpolate the resulting 60,000point function at ~ 500 points, through which we pass a cubic spline.
We follow the 2dFGRS team in assuming a flat Q.\ = 0.7 cosmology when
converting redshifts to comoving distances r. We transform the galaxy positions
into the Local Group frame assuming th at the solar motion relative to the Local
Group is 306 km /s toward I = 99°, b = —4° (Courteau & van den Bergh 1999).
We model fc-corrections and luminosity evolution (e-corrections) together as a power
law luminosity evolution oc (1 ■+• z)K with exponent k = —0.7. This exponent was
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chosen so as to make the comoving density shown in the lower panel in figure 5.3 as
flat as possible, :.e., by assuming minimal evolution in the comoving number density
of galaxies. Similar results have been reported by Cole ef al. (2001), C01, Cross
et al. (2001) and Madgewick et al. (2001) and Norberg et al. (2001b). The slight
differences between our n(r) and th at of C01 seen figure 5.3 are due to our different
methods for estimating this function from the data, and below we find that they do
not have a major impact on the final power spectrum.
We truncate the sample radially by eliminating objects with r < 10h- l Mpc
(to eliminate stellar contamination) and r > 650/i_1Mpc (where figure 5.3 shows
evidence of incompleteness). This leaves 59832 galaxies in the sample.
5.3
M eth od an d basic an alysis
In this section, we analyze the uniform galaxy sample described in the previous
section, measuring the power spectrum and redshift space distortions of the galaxy
density field. We adopt the matrix-based approach described in Tegmark et al. (1998,
hereafter THSVS98), using the mode expansion of Hamilton & Culhane (1996) and
including the stochastic bias formalism. Our analysis consists of the following five
steps:
(i). Finger-of-god compression
(ii). Pseudo-Karhunen-Loeve compression
(iii). True Karhunen-Loeve expansion
(iv). Quadratic band-power estimation
(v). Fisher decorrelation and flavor disentanglement
We will now describe these steps in more detail. We will see that step (iii) is not
required in practice, and we use it only for systematics tests.
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5.3.1
S tep 1: F in ger-of-god com pression
Since our analysis uses the linear Kaiser approximation for redshift space distortions,
it is crucial that we are able to empirically quantify our sensitivity to the socalled finger-of-god (FOG) effect whereby radial velocities in virialized clusters make
them appear elongated along the line of sight. We therefore start our analysis by
compressing (isotropizing) FOGs, as illustrated in figure 5.4. The FOG compression
involves a tunable threshold density, and in section 5.5.4 below we will study how
the final results change as we gradually change this threshold to include lesser or
greater numbers of FOGs.
We use a standard friends-of-friends algorithm, in which two galaxies are con­
sidered friends, therefore in the same cluster, if the density windowed through an
ellipse
10
times longer in the radial than transverse directions, centered on the pair,
exceeds a certain overdensity threshold. To avoid linking well-separated galaxies in
deep, sparsely sampled parts of the survey, we impose the additional constraint that
friends should be closer than r iraax = 5 h -1Mpc in the transverse direction. The two
conditions are combined into the following single criterion: two galaxies separated
by r|| in the radial direction and by r x in the transverse direction are considered
friends if
[(r||/10 )2 + r i ] 1/2 < [(3 / 47r)7m (l + 5C) +
1/3
(5.6)
where n is the selection function (geometrically averaged) at the position of the
pair, and Sc is an overdensity threshold. Note that Sc represents not the overdensity
of the pair as seen in redshift space, but rather the overdensity of the pair after
their radial separation has been reduced by a factor of 10. In other words, Sc is
intended to approximate the threshold overdensity of a cluster in real space, not the
overdensity of the elongated FOG seen in redshift space. Having identified a cluster
by friends-of-friends in this fashion, we measure the dispersion of galaxy positions
about the center of the cluster in both radial and transverse directions. If the 1dimensional radial dispersion exceeds the transverse dispersion, then the cluster is
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deemed a FOG, and the FOG is then compressed radially so th at the cluster becomes
round, that is, the transverse dispersion equals the radial dispersion. We perform
the entire analysis five times, using progressively more aggressive compression with
density cutoffs 1+<SC = oo, 200, 100, 50 and 25, respectively. The infinite threshold
1+<5C= oo corresponds to no compression at all.
figure 5.4 illustrates FOG compression with threshold density 1+<5C= 100, which
is the baseline case adopted in this paper. It corresponds to fairly aggressive FOG
removal since the overdensity of a cluster is around
200
at virialization and rises as
the Universe expands and the background density continues to drop.
5.3.2
Step 2: Pseudo-K L pixelization
The raw d ata consists of N&i = 59,832 three-dimensional vectors r Q, a = 1,...,
giving the measured positions of each galaxy in redshift space. As in THSVS98, we
define the density in Nx “pixels” x,, i =
1,...,
N x by
(5.7)
for some set of functions v* and work with the :Vx-dimensional data vector x instead
of the the 3 x
numbers r a . Although these are perhaps more aptly termed
“modes” since we will choose quite non-local functions t/)j, we will keep referring to
them as pixels to highlight the useful analogy with CMB map analysis.
Galaxies are (from a cosmological perspective) delta-functions in space, so the
integral in equation (5.7) reduces to a discrete sum over galaxies. We do not rebin
the galaxies spatially, which would artificially degrade the resolution. It is convenient
to isolate the mean density into a single mode t/h(r) = n(r), with amplitude
(5.8)
and to arrange all other modes to have zero mean
(5.9)
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The covariance m atrix of the vector x of amplitudes is a sum of noise and signal
terms
(A xA x‘> = C = N + S,
(5.10)
where the shot noise covariance m atrix is given by
/B M s W r f .r
J
n(r)
(5.11)
and the signal covariance m atrix is
S* = / * ( k ) « j ( k ) * P ( t ) ^ j
(5.12)
in the absence of redshift-space distortions. Here hats denote Fourier transforms and
n is the three-dimensional selection function described in section 5.2, *.e., h(r)dV is
the expected (not the observed) number of galaxies in a volume d V about r. P(k) is
the power spectrum, which for a random field of density fluctuations <J(r) is defined
by (l(k )-l(k ')) = (27r)3<5Dirac(k - 10.
As our functions % (r), we use the pseudo-Karhunen-Loeve (PKL)
defined
eigenmodes
inHamilton, Tegmark & Padm anabhan (2000;hereafter “HTPOO” ). To
provide an intuitive feel for the nature of these modes, a sample is plotted in
figure 5.5 and figure 5.6. We use these modes because they have the following
desirable properties:
(i). They form a complete set of basis functions probing successively smaller scales,
so that a finite number of them (we use the first 4,000) allow essentially all
information about the density field on large scales to be distilled into the vector
x.
(ii). They allow the covariance matrices N and S to be fairly rapidly computed.
(iii). They are the product of an angular and a radial part, *.e.. take the separable
form 0j(r) = Vi(r)tpi(r), which accelerates numerical computations.
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(iv). A range of potential sources of systematic problems are isolated into special
modes th at are orthogonal to all other modes. This means that we can test
for the presence of such problems by looking for excess power in these modes,
and immunize against their effects by discarding these modes.
We have four types of such special modes:
(i). The very first mode is the mean density, Vi(r) = n(r). The mean mode is
used in determining the maximum likelihood normalization of the selection
function, but is then discarded from the analysis, since it is impossible to
measure the fluctuation of the mean mode. The idea of solving the so-called
integral constraint problem by making all modes orthogonal to the mean goes
back to Fisher et al. (1993).
(ii). Modes 2-5 are associated with the motion of the Local Group through the
Cosmic Microwave Background a t 622 km /s towards (B1950 FK4) RA = 162°,
Dec = —27° (Lineweaver et al. 1996; Courteau & van den Bergh 1999). In the
angular direction, these Local Group modes are monopole and dipole modes
multiplied by the angular mask, while in the radial direction they take the form
specified by equation (4.42) of Hamilton (1997c). Mode 2 is a pure monopole
mode (multiplied by the angular mask), and is present because the survey is not
all-sky. The other three Local Group modes are dipole modes with admixtures
of the Local Group monopole mode 2, such as to make them orthogonal to the
mean mode 1.
(iii). Purely radial modes (for example mode 104 in figure 5.6) are to first order the
only ones affected by mis-estimates of the radial selection function n(r).
(iv). Purely angular modes (for example mode 148 in figure 5.6) are to first order
the only ones affected by misestimates of the angular selection function n(r),
as may result from inadequate corrections for, e.g., extinction, the variable
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magnitude limit, the variable magnitude completeness or photometric zeropoint offsets.
As described in HTPOO, the modes w are computed in the logarithmic spheri­
cal wave basis (Hamilton Sc Culhane 1996), which are orthonormal eigenfunctions
Zutmit) = (27r)_l/'2e -<3/,2+u‘,)lnrYftn(r) of the complete set of commuting Hermitian
operators
Slightly better numerical behavior is obtained by expanding not ipi(r) itself but
rather t£i(r)/h (r ) 1/2 (the denominator is the square root of the radial part of the
selection function only, not the angular part) in logarithmic spherical waves, since
this mitigates some difficulties that arise from the fact that the radial selection
function n(r) varies by orders of magnitude. The merits of working in a basis of
spherical harmonics were first emphasized by Fisher, Scharf Sc Lahav (1994) and by
Heavens Sc Taylor (1995). The advantages of working with logarithmic radial waves
e-( 3/ 2+iw)inr compared for example to spherical Bessel functions, are both numerical
and physical:
(i). Numerically, the logarithmic radial wave basis permits rapid transformation
between real, u>, and Fourier space using Fast Fourier Transforms. The transfor­
mation is mathematically equivalent to the Fast Fourier-Hankel-Bessel Trans­
form FFTL og described in Appendix B of Hamilton (2000).
(ii). Physically, logarithmic radial waves are well matched to real galaxy surveys
like the 2dFGRS, which are finely sampled nearby, and coarsely sampled far
away.
(iii). The linear redshift distortion operator is diagonal in this basis (Hamilton Sc
Culhane 1996).
The logarithmic radial wave basis discretizes naturally on to a logarithmically
equispaced grid (in both real and Fourier space), and is periodic over a logarithmic
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interval. To avoid potential problems of aliasing between small and large scales, we
embed the survey inside a suitably large logarithmic interval of depths, extending
in real space from
10- 2 h _1Mpc
to 104 /i_IMpc. As remarked in Section 5.2.3, we
truncate the survey to radial depths 10-650 h_1Mpc within this interval.
The dimensionless log-frequency ui in the radial eigenmode e -( 3/ 2+iw)inr jg a r ^ j a i
analogue (in a precise mathematical sense) of the dimensionless angular harmonic
number i. Similar resolution in the radial and angular directions is secured by choos­
ing the maximum log-frequency to be about equal to the the maximum harmonic
number, wmax ~ f max. The maximum log-frequency is related to the radial resolution
A In r by u;max = 7r / A In r . We adopt a maximum harmonic number of £max = 40, and
a radial resolution of 32 points per decade, so A In r = (In 10)/32, giving 01^
= 43.7
(the same as in HTP00). These choices ensure comparable effective resolutions in
radial and angular directions.
A maximum angular harmonic number of £max = 40 gives (£„»* + l )2 = 1681
spherical harmonics, while 32 points per radial decade over
6
decades gives 192
radial modes. Thus there is a potential pool of 412 x 192 ~ 320,000 modes from
which we would like to construct Karhunen-Loeve (KL) modes. The usual way to
construct such modes would be to diagonalize a 320,000 x 320,000 matrix, but this
is evidently utterly intractable numerically.
How do we build the PKL modes in practice? To make the problem tractable,
we instead proceed hierarchically, first constructing angular PKL modes, and then
constructing a set of radial PKL modes associated with each angular KL mode.
The procedure is possible because we have required the selection function to be
separable into angular and radial parts, equation (5.1). We refer to the resulting
modes as pseudo Karhunen-Loeve (PKL) modes. The PKL basis contains almost as
much information as a true KL basis, but it circumvents the need to diagonalize an
impossibly huge matrix. O ur procedure is the same as that of HTPOO. A different,
but similar in spirit, hierarchical approach to the KL problem has been proposed by
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Taylor et al. (2001).
As we proceed from angular PKL mode to angular PKL mode, extending each
into 3D PKL modes by computing associated radial functions, we retain only the
Nx = 4000 PKL modes with the highest expected signal-to-noise. As detailed below,
we make this truncation both to render the various N x x jVx matrices numerically
tractable and to limit sensitivity to small, nonlinear scales. As the signal-to-noise
of the angular PKL mode decreases, fewer and fewer of the associated radial PKL
modes make the cut into the pool of PKL modes. We stop when 10 successive angular
PKL modes have contributed no new PKL mode. In practice only 140 of the angular
PKL modes actually contribute to the PKL modes. The reduction from 1681 to 140
angular modes with little information loss is possible because the spherical harmonics
are overcomplete and redundant on the modest fraction of the sky actually covered
by the 2dFGRS.
The orthogonality of the PKL modes to the mean and the properties of the
"special” modes are enforced in the construction of the modes. We perform the
PKL decomposition after selecting out the special modes (rather than doing the
KL decomposition and then making them orthogonal to the special modes), since
we find that this makes better PKL modes. We do this as decribed in Appendix
B of THSVS98, with the complication that we make the non-special modes exactly
orthogonal to the masked mean and the masked LG modes, not merely orthogonal
up to the finite order of the discrete matrices.
The pixelized data vector x is shown in figure 5.7. This data compression step
has thus distilled the large-scale information about the galaxy density field from
:Vgai = 59,832 galaxy position vectors into 4,000 PKL-coefficients. The functions
are normalized so that N tl =
1,
i.e., so th at the shot noise contribution to
their variance is unity. If there were no cosmological density fluctuations in the
survey, merely Poisson fluctuations, the PKL-coefficients x, would thus have unit
variance, and about
68%
of them would be expected to lie within the blue/dark grey
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band. Figure 5.7 shows th at the fluctuations are considerably larger than Poisson,
especially for the largest-scale modes (to the left), demonstrating that cosmological
density fluctuations are present, as expected.
5.3.3
Step 3: Expansion into true KL m odes
Karhunen-Loeve (KL) expansion (Karhunen 1947) was first introduced into largescale structure analysis by Vogeley & Szalay (1996). It has since been applied to the
Las Campanas redshift survey (Matsubara ef al. 1999), the UZC survey (PTH01)
and the SDSS (Szalay et aL 2002; Tegmark et al. 2002) and has been successfully
applied to Cosmic Microwave Background data as well, first by Bond (1995) and
Bunn (1995).
Given x, N and S from the previous section, it is straightforward to compute the
true Karhunen-Loeve (KL) coefficients. They are defined by
y = B ‘x ,
(5.14)
where b, the columns of the matrix B, are the Nx eigenvectors of the generalized
eigenvalue problem
Sb = ANb,
(5.15)
sorted from highest to lowest eigenvalue A and normalized so that b*Nb = I. This
implies that
(y»ib) = M i + A*)>
(5-16)
which means that the transformed data values y have the desirable property of
being uncorrelated. In the approximation that the distribution function of x is a
multivariate Gaussian, this also implies th at they are statistically independent —
then y is merely a vector of independent Gaussian random variables. Moreover,
equation (5.15) shows that the eigenvalues A* can be interpreted as a signal-to-noise
ratio S / N . Since the m atrix B is invertible, the final data set y clearly retains all the
information that was present in x. In summary, the KL transformation partitions the
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information content of the original data set x into Nx chunks that are mutually exclu­
sive (independent), collectively exhaustive (jointly retaining all the information), and
sorted from best to worst in terms of their information content. In most applications,
the chief use of KL-coefficients is for data compression, discarding modes containing
almost no information and thereby accelerating subsequent calculations. The KLcoefficients for our dataset are plotted in figure 5.8, and it is seen that even the worst
coefficients still have non-negligible signal-to-noise, bearing numerical testimony to
the quality of the PKL-modes we have used.
This means that KL-compression
would not accelerate our particular analysis, and we will indeed work directly with
the uncompressed data x in the following subsections.
Rather, the reason we have computed KL-coefficients is as an additional check
against systematic errors and incorrect assumptions, to verify that we modeled not
only the diagonal terms in C correctly (as seen in figure 5.7), but the off-diagonal
correlations as well.
As discussed in many of the above-mentioned KL-papers,
inspection of the KL-coefficients as in figure 5.8 provides yet another opportunity to
detect suspicious outliers and to check whether the variance predicted by the prior
power spectrum is consistent with the data. We will provide a detailed test based
on the KL-coefficients in section 5.5.1.
5.3.4
W hat we w ish to measure: three power spectra, not
one
Before analyzing the x-vector in the following subsections, let us first discuss precisely
what we want to measure. Cosmological constraints based on galaxy power spectrum
measurements are only as accurate as our understanding of biasing. We will therefore
perform our analysis in a way that avoids making any assumptions about the relation
between galaxies and m atter, as described in Tegmark (1998) and HTPOO.
Unfortunately, bias is complicated. The commonly used assumption th at the
m atter density fluctuations <$(r) and the galaxy number density fluctuations g(r)
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obey
g{ r) = 6<S(r)
(5.17)
for some constant b (the bias factor) appears to be violated in a number of ways.
It has been long known (Davis & Geller 1976; Dressier 1980) that b must depend
on galaxy type. However, there is also evidence th at it depends on scale (see e.g.
Mann et al. 1997; Blanton et al. 1999; Hamilton & Tegmark 2002 and references
therein) and on time (Fry 1996; Tegmark & Peebles 1998; Giavalisco et al. 1998).
Finally, there are good reasons to believe that there is no deterministic relation that
can replace equation (5.17), but that bias is inherently somewhat stochastic (Dekel
& Lahav 1999) — this has been demonstrated in both simulations (Blanton et al.
2000) and real d ata (Tegmark & Bromley 1999). The term stochastic does of course
not imply any randomness in the galaxy formation process, merely that additional
factors besides density may be important (gas tem perature, say).
The good news for our present analysis is that, restricting attention to second
moments, all the information about stochasticity is contained in a single new function
r(k) (Pen 1998; Tegmark &c Peebles 1998). Grouping the fluctuations into a twodimensional vector
(5.18)
and assuming nothing except translational invariance, its Fourier transform x(k) =
/ e *k px ( r )<Pr obeys
(x(k)x(k')f) = (27r)3<5D(k - k')
(5.19)
for some 2 x 2 power spectrum matrix that we will denote P (k ). Here P is the
conventional power spectrum of the mass distribution, Pa is the power spectrum of
the galaxies, and P x is the cross spectrum. It is convenient to rewrite this covariance
m atrix as
1
6(k)r(k)
6(k)r(k)
6(k)2
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(5.20)
where b = (Pgg/P)1/2 is the bias factor (the ratio of galaxy and total fluctuations)
and the new function r = P x/(PPgg)1/2 is the dimensionless correlation coefficient
between galaxies and m atter. Note that both b and r generally depend on scale
k. The Schwarz inequality shows that the special case r = 1 implies the simple
deterministic equation (5.17), and the converse is of course true as well.
On large scales where linear perturbation theory is valid, redshift distortions
(Kaiser 1987; Hamilton 1998) conveniently allow all three of these functions to be
measured. Specifically, the correlation between the observed densities at any two
points depends linearly on these three power spectra:
galaxy-galaxy power : Pgg(fc) = b(k)2P(k)
galaxy-velocity power : Pp,(k) = r(k)b(k)fP (k)
(5-21)
velocity-velocity power : Pw(fc) = f 2P(k)
Here / a s
is the dimensionless growth rate for linear density perturbations (see
Hamilton 2001).
More correctly, the 'velocity’ here refers to minus the velocity
divergence, which in linear theory is related to the mass (not galaxy) overdensity S
b y / J + V - v = 0 , where V denotes the comoving gradient in velocity units. Note that
PgV (k )
= f P x (fc) and that the parameter / is conveniently eliminated by defining
3{k) = f/b (k ) , which gives
P ^k)
=
Pw(fc) =
5.3.5
3(k)r(k)Pa {k),
8(k)2Pa (k).
(5.22)
Step 4: Quadratic com pression into band powers
In this step, we perform a much more radical data compression by taking certain
quadratic combinations of the data vector that can easily be converted into power
spectrum measurements.
We parametrize the three power spectra Pffi(fc), Pgv(k) and Pvv(k) as piecewise
constant functions, each with 49 “steps” of height Pi, which we term the band powers.
I ll
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To avoid unnecessarily jagged spectra, we take k l5P rather than P to be constant
within each band. We group these 3 x 49 numbers into a 147-dimensional vector
p. We choose our 49 fc-bands to be centered on logarithmically equispaced Ar-values
fa = lO^*1 h / Mpc, i = 1,..., 49, i.e., ranging from 0.00316 h/M pc to 3.16/i/Mpc.
For instance, Pgg(fc) = (k/ki)~l5pi for | lgfc — lgk,| < 1/32. This should provide
fine enough fc-resolution to resolve any baryonic wiggles and other spectral features
th at may be present in the power spectrum. For instance, baryon wiggles have a
characteristic scale of order A k ~ 0.1, so we oversample the first one around fc ~ 0.1
by a factor Afc/(&26 — k^s) ~ 16/ In 10 ~ 7.
This parametrization means that we can write the pixel covariance matrix of
equation (5.10) as
147
C = 5 > iC ,,,
(5.23)
«=o
where the derivative matrix C ,j = dC /dpi is the contribution from the Ith band.
For notational convenience, we have included the noise term in equation (5.23) by
defining C,o = N , corresponding to an extra dummy parameter po = 1 giving the
shot noise normalization. As in Hamilton & Tegmark (2000) and HTP00, we in
practice redefine the parameters pi to be ratio of the actual band power to the prior
band power. As long as the prior agrees fairly well with the measured result, this has
the advantage of giving better behaved window functions, as described in Hamilton
&c Tegmark (2000).
Our quadratic band power estimates are defined by
ft = i x ‘C _1C ,jC _1x,
(5.24)
i — 0,..., 147. These numbers are shown in figure 5.9, and we group them together in
a 148-dimensional vector q. Note that whereas x (and therefore C) isdimensionless,
p has units of power, t.e., volume. Equation (5.24) therefore shows th at q has
units of inverse power, t.e., inverse volume. It is not immediately obvious th at the
vector q is a useful quantity. It is certainly not the final result (the power spectrum
112
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estimates) th at we want, since it does not even have the right units. Rather, it
is a useful intermediate step. In the approximation th at the pixelized data has a
Gaussian probability distribution (a good approximation in our case because of the
central limit theorem, since :Vgai is large) q has been shown to retain all the power
spectrum information from the original data set (Tegmark 1997, hereafter “T97” ).
The numbers $ have the additional advantage (as compared with, e.g., maximumlikelihood estimators) that their properties are easy to compute: their mean and
covariance are given by
(q) =
Fp,
(qq‘> - (q)(q)£ = f .
(5.25)
(5.26)
where F is the Fisher information matrix (Tegmark et al. 1997)
Fi, = J t r [c-'C jC -'C j] .
(5.27)
Quadratic estimators were first derived for galaxy survey applications (Hamilton
1997ab). They were accelerated and first applied to CMB analysis (T97: Bond, Jaffe
& Knox 2000).
In conclusion, this step takes the vector x and its covariance matrix C from
figure 5.7 and compressesit into the smaller vector q and itscovariancematrix F,
illustrated in figure 5.9 and figure 5.10. Although equation (5.25) shows that we can
obtain unbiased estimates of the true powers p by computing F -1q, there are better
options, as will be described in the next subsection.
5.3.6
Step 5: Fisher decorrelation and flavor disentangle­
m ent
Let us first eliminate the shot-noise dummy parameter po, since we know its value.
We define
f
to be the 0** column of the Fisher matrix defined above (/, = F<o) and
restrict the indices i and j to run from 1 to 147 from now on, so f , q and p are
113
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147-dimensional vectors and F is a 147 x 147 matrix. Since po = 1, equation (5.25)
then becomes (q) = F p + f.
We now define a vector of shot noise corrected band power estimates
p = M (q —f),
(5.28)
where M is some matrix whose rows are normalized so that the rows of M F sum to
unity. Using equations (5.25) and (5.26), this gives the mean and covariance
(p> =
S
=
W p,
(5.29)
<pp‘>-(P><P>‘ = M F M ‘,
(5.30)
where W = M F . We will refer to the rows of W as window functions, since they
sum to unity and equation (5.29) shows that pi probes a weighted average of the
true band powers pj, the ith row of W giving the weights.
C o rre la te d , a n tic o rre la te d a n d u n c o rre la te d b a n d pow ers
For the purpose of fitting models p to our measurements p, we are already done —
the last two equations tell us how to compute \ 2 f°r any given p, and the result
x2 = ( p - ( p » £S - 1( p - ( p ) ) t
(5.31)
is independent of the choice of M . However, since one of the key goals of our analysis
is to provide model-independent measurement of the three power spectra, the choice
of M is crucial. Ideally, we would like both uncorrelated error bars (diagonal E )
and well-behaved (narrow, unimodal and non-negative) window functions W th at
do not mix the three power spectra.
There are a number of interesting choices of M th at each have their pros and cons
(Tegmark & Hamilton 1998; Hamilton & Tegmark 2000). The simple choice where M
is diagonal gives the “best guess” estimates in the sense of having minimum variance
(Hamilton 1997a; T97; Bond, Jaffe & Knox 2000), and also has the advantage
114
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of being independent of the number of bands used in the limit of high spectral
resolution. It was used for figure 5.9 and figure 5.10. Here the window functions are
simply the rows of the Fisher matrix, and are seen to be rather broad. All entries of
F are guaranteed to be positive as proven in PTH01, which means not only that all
windows are positive (which is good) but also that all measurements are positively
correlated (which is bad).
Another interesting choice is (T97) M = F -1, which gives W = I. In other
words, all window functions are Kronecker delta functions, and p gives completely
unbiased estimates of the band powers, with (pi) = p, regardless of what values the
other band powers take. This gives an answer similar to the maximum-likelihood
method (THSVS98), and the covariance matrix of equation (5.30) reduces to F _ l. A
serious drawback of this choice is that that if we have sampled the power spectrum
on a scale finer than the inverse survey size in an attem pt to retain all information
about wiggles etc., this covariance matrix tends to give substantially larger error
bars (Api = M*/2 = [(F-1)^]1^2) than the first method, anti-correlated between
neighboring bands.
The two above-mentioned choices for M both tend to produce correlations be­
tween the band power error bars. The minimum-variance choice generally gives
positive correlations, since the Fisher matrix cannot have negative elements, whereas
the unbiased choice tends to give anticorrelation between neighboring bands. The
choice (Tegmark Sc Hamilton 1998; Hamilton Sc Tegmark 2000) M = F -l/2 with
the rows renormalized has the attractive property of making the errors uncorrelated,
with the covariance matrix of equation (5.30) diagonal. The corresponding window
functions W are plotted in figure 5.11, and are seen to be quite well-behaved, even
narrower than those in figure 5.10 while remaining positive.1 This choice, which
is the one we make in this paper, is a compromise between the two first ones: it
'T he reader interested in mathematical challenges will be interested to know th at it remains a
mystery to the authors why this F 1/ 2 method works so well. We have been unable to prove that
F 1/2 has no negative elements (indeed, counterexamples can be contrived), yet the method works
like magic in practice in all LSS and CMB applications we have tried.
115
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narrows the minimum variance window functions at the cost of only a small noise
increase, with uncorrelated noise as an extra bonus. The minimum-variance band
power estimators are essentially a smoothed version of the uncorrelated ones, and
their lower variance was paid for by correlations which reduced the effective number
of independent measurements.
D ise n tan g lin g th e th r e e p ow er s p e c tra
The fact that we are measuring three power spectra rather than one introduces an
additional complication. As illustrated by figure 5.12, an estimate of the power in
one of the three spectra generally picks up unwanted contributions ( “leakage”) from
the other two, making it complicated to interpret. Although the above-mentioned
F -1 -method in principle eliminates leakage completely, the cost in terms of increased
error bars is found to be prohibitive. We therefore follow HTPOO in adopting the
following procedure for disentangling this three power spectra: For each of the 49
fc-bands, we take linear combinations of the gg, gv and w measurements such that
the unwanted parts of the window functions average to zero. This procedure is
mathematically identical to that used in Tegmark & de Oliveira-Costa (2001) for
separating different types of CMB polarization, so the interested reader is referred
there for the explicit equations. The procedure is illustrated in figure 5.12, and by
construction has the property that leakage is completely eliminated if the true power
has the same shape (not necessarily the same amplitude) as the prior. We find that
this method works quite well (in the sense that the unwanted windows do not merely
average to zero) for the most accurately measured band powers, in particular the
central parts of the gg-spectrum, with slightly poorer leakage elimination for bands
with larger error bars.
The window functions plotted in figure 5.11 are the gg-windows after disentan­
glement. Note that although our disentanglement introduces correlations between
116
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the gg, gv and w measurements for a given fc-band, different fc-bands remain uncorrelated.
5.4
R esu lts
5.4.1
The three power spectra
Our basic results are shown in figure 5.13. The single most striking feature of
this plot is clearly that the 2dFGRS is an amazing d ata set with unprecedented
constraining power. The window functions in Figure 5.11 are seen to be quite narrow
despite the complicated survey geometry. The galaxy-galaxy power is constrained
to 20% or better over an order of magnitude in length scale, in about a dozen
uncorrelated bands centered around k ~ 0.1 h / Mpc. Whereas the increase in error
bars on large scales reflects the finite survey volume, the lack of information on
small scales is caused by our analysis being limited to the first 4000 PKL-modes.
Comparing figure 5.13 with figure 5.9 serves as a sobering reminder of the importance
of decorrelating and disentangling the measurements to avoid a misleadingly rosy
picture of how well one can do.
Whereas Pgg(fc) is well measured, figure 5.13 shows th at the information about
Pgv{k) is quite limited and that on P„(fc) almost nonexistent. To avoid excessive
cluttering in figure 5.13, band-power measurements with very low information con­
tent have been binned into fewer (still uncorrelated) bands. The main cause of these
large error bars is that the information on Pw and Pffi comes from the quadrupole
and hexadecapole moments of the clustering anisotropy, which are intrinsically small
and hence poorly constrained quantities. However, the problem may be exacerbated
by the lack of large contiguous angular regions in the current data, impeding accurate
comparisons of angular and radial clustering (the situation is similar for the SDSS;
Zehavi et al. 2002), and should improve as the survey nears completion and gets more
filled in. This effect is evident from a comparison with the results from the much
117
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more contiguous PSCz survey (HTPOO): the error bars on Pgg are appreciably larger
for PSCz than 2dFGRS, but those on redshift distortions (say 3) are comparable.
In the remainder of this paper, we will address two separate issues in turn:
redshift-space distortions/biasing (/3,r) and the detailed shape of the galaxy-galaxy
power spectrum (model fits, evidence for baryonic wiggles, etc.).
5.4.2
C onstraints on redshift space distortions
As seen from Figure 5.13, the constraints on Pgv{k) and Pw(k) from 2dFGRS
are too weak to allow fi(k) and r(fc) to be measured reliably as a function of
scale.
As data on Large Scale Structure improve, it should become possible to
accomplish such a measurement, and thereby to establish quantitatively the scale
dependence of biasing at linear scales. In the meantime we limit ourselves to the
less ambitious goal of measuring overall parameters j3 and r, simply treating them
as scale-independent constants. This has not been previously done for the case of
r. Such scale-independence of bias on linear scales is a feature of local bias models
(Coles 1993; Fry and Gaztanaga 1993; Scherrer k Weinberg 1998; Coles, Melott k
Munshi 1999; Heavens, Matarrese k Verde 1999).
For our redshift-distortion analysis, we employ a simple scale-invariant power
spectrum Pre(fc) of the BBKS form (Bardeen et al. 1986), parametrized by an am­
plitude as and a “shape param eter’ T that on a log plot shifts the curve vertically and
horizontally, respectively. We will use more physically motivated power spectra with
baryon wiggles etc. in section 5.6.2 — we tried various alternative parametrizations,
and found th at the detailed form had essentially no effect on the (r, /3)-constraints,
since they come from the ratios of the three spectra, not from their shapes. Our
model for the underlying band power vector p thus depends on four parameters
(r, <7g, 3, r).
We map out the likelihood function L = e - *2/2 using equation (5.31) on
a fine grid in this param eter space, and compute constraints on individual parameters
by marginalizing over the other parameters as described in Tegmark k Zaldarriaga
118
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(2000), maximizing rather than integrating over them. The results are plotted
figures 5.14, 5.15 and 5.16.
119
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«
m
10000
0
14
15
1»
16
Magnitude limit m.
18
IB
20
Figure 5.2 Number of galaxies surviving as a function of uniform magnitude cut.
120
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10*
0.1
100
1000
r [h- Mpc]
to*
Figure 5.3 The redshift distribution of the galaxies in our sample is shown both as a histogram
(top) and relative to the expected distribution (bottom), in comoving coordinates assuming a flat
flm = 0.3 cosmology. The curves correspond to the the radial selection function h(r) employed
in our analysis (solid) and by C01 (dotted). The four vertical lines indicate the redshift lim it s
employed in our analysis (10/i- , Mpc < r < 650/i_IMpc) and where spectral type subsamples are
available (33h- I Mpc < r < 538h-1Mpc).
121
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Comoving y [ i r ‘ Mpc]
400
200
0
-200
-4 0 0
0
200
400 600 800 0
200 400 600 800 0
200 400
Comoving x [h-1 Mpc] (tow ards ra = 9 ', d e c = —27.7*)
600
Figure 5.4 The effect of our Fingers-of-god (FOG) removal is shown in the southern slice S = —27.7°,
—35° < R A < 53°. The slice has thickness 2° and has been rotated to lie in the plane of the page.
From left to right, the panels show all 15,055 galaxies in the slice, the 6,211 that are identified as
belonging to FOGs (with density threshold 100) and the same galaxies after FOG compression,
respectively.
122
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800
Figure 5.5 A sample of four angular pseudo-KL (PKL) modes are shown in Hammer-Aitoff
projection in equatorial coordinates, with grey representing zero weight, and lighter/darker shades
indicating positive/negative weight, respectively. From top to bottom, they are angular modes 1
(the mean mode), 3, 20 and 106, and are seen to probe successively smaller angular scales.
123
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Figure 5.6 A sample of six pseudo-KL modes are shown in the plane of the southern 2dF slice with
S = —27.7°, —35° < R A < 53°. Grey represents zero weight, and lighter /darker shades indicate
positive/negative weight, respectively. R om left to right, top to bottom, these are modes 1 (the
mean mode), 14, 104, 148, 58 and 178, and are seen to probe successively smaller scales. Those in
the middle panel are examples of purely radial (left) and purely angular (right) modes.
124
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-20
1000
2000
3000
P se u d o -K L m o d e n u m b e r i
4000
Figure 5.7 The triangles show the 4,000 elements of the data vector x (the pseudo-KL expansion
coefficients) for the baseline galaxy sample. If there were no clustering in the survey, merely shot
noise, they would have unit variance, and about 68% of them would be expected to lie within the
blue/dark grey band. If our prior power spectrum were correct, then the standard deviation would
be larger, as indicated by the shaded yellow/light grey band.
125
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t
I
i
i
j
i
i
i
i
i
i
i
i
2000
i
1
i
i
3000
i
r
4000
KL m ode n u m b e r i
Figure 5.8 The triangles show the 3999 uncorrelated elements y, of the transformed data vector
y = B x (the true KL expansion coefficients) for the baseline galaxy sample. If there were no
clustering in the survey, merely shot noise, they would have unit variance, and about 68% of them
would be expected to lie within the blue/dark grey band. If our prior power spectrum were correct,
then the standard deviation would be larger, as indicated by the shaded yellow/light grey band.
The green/grey curve is the rms of the data points x*, averaged in bands of width 25, and is seen
to agree better with the yellow/light grey band than the blue/dark grey band.
126
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10*
•
i
a
I
10*
10*
10*
"m
I*
10*
10*
m
I
a
_
104
10*
N ,
JB
3“
r*
10*
10*
10*
m
Eh
a
104
10*
JC
3
I*
Ns
10*
10*
10- *
%
10-»
k [h M p c-1]
Figure 5.9 The 147 quadratic estimators qt , normalized so th at their window functions equal unity
and with the shot noise contribution / , (dashed curve) subtracted out. They c a n n o t be directly
interpreted as power spectrum measurements, since each point probes a linear combination of all
three power spectra over a broad range of scales, typically centered at a fc-value different than the
n o m in a l k where it is plotted. Moreover, nearby points are strongly correlated, causing this plot
to overrepresent the amount of information present in the data. The solid curves show the prior
power spectrum used to compute the error bars.
127
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0.01
0.1
1
W avenumber k [h/M pc]
Figure 5.10 The rows of the jg-portion of the Fisher matrix F . The ith row typically peaks at the
ith band, the scale k that the band power estimator (ft was designed to probe. All curves have
been renormalized to unit area, so the highest peaks indicate the scales the the window functions
obtained are narrowest. The turnover in the envelope at k ~ 0.1 h/M pc reflects our running out of
information due to omission of modes probing smaller scales. For comparison with the next figure,
these are the rows of W when M is diagonal.
128
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0.01
0.1
1
W avenumber k [h /lip c ]
Figure 5.11 The window functions (rows of the 35 -portion of W ) are shown using decorrelated
estimations. The itk row of W typically peaks at the itk band, the scale k that the band power
estimator p, was designed to probe. Comparison with figure 5.10 shows that decorrelation makes
all windows substantially narrower.
129
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111 III
ie 0.4
| 0.3
gg
B efore
gg
A fter
g v
B efore
gv
After
B efore
W
A fter
1 0.2
3
o .i
0.4
| 0.3
l
| 0.2
* 0.1
3
8 0.4
0.3
0.2
=- vv
=-
0.1
0.01
0.1
0.01
1
V m a w ta r k [h /k p c]
0.1
1
V M n u n b v k [h/M pe]
Figure 5.12 The window function for our measurement of the 25th band of the galaxy-galaxy power
is shown before (left) and after (right) disentanglement. Whereas unwanted leakage of gv and w
power is present initially, these unwanted window functions both average to zero afterward. The
success of this method hinges on the fact that since the three in itia l functions (left) have s im ila r
shape, it is possible to take linear combinations of them that almost va n is h (right).
130
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10*
*
10»
*
10»
10 - *
io-«
1
k [h lip c-1]
Figure 5.13 Decorrelated and disentangled measurements of the galaxy-galaxy power spectrum
(top), the galaxy-velodty power spectrum (middle) and the velocity-velocity) power spectrum
(bottom) for the baseline galaxy sample. Red points represent negative values — since the points
are differences between two positive quantities (total power minus expected shot noise power), they
can be negative when the signal-to-noise is poor. Each points is plotted at the fc-value that is
the median of its window function, and 68% of this function is contained within the range of the
horizontal bars. The curves shows our prior power spectrum. Note that most of the information in
the survey is on the galaxy-galaxy spectrum. Band-power measurements with very low information
content have been binned into fewer (still uncorrelated) bands.
131
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0.1
1
Maximum k - b a n d included
Figure 5.14 The blue/grey band shows the l<r allowed range for 0, assuming r = 1 and the shape of
the prior Pa (k) but marginalizing over the power spectrum normalization, using FOG compression
with density threshold l+ d c = 100. These fits are performed cumulatively, using all measurements
for all wavenumbers < k. From bottom to top, the five curves show the best fit 0 for FOG thresholds
1-h£c = oo (no FOG compression), 200 , 100 (heavy), 50 and 25.
132
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■o
at
r
r
Figure 5.15 1-dimensional likelihood curves for T, & and r are shown after marginalizing over the
power spectrum normalization and the other parameters using our baseline (1-h5c = 100) finger-ofgod compression. The 68% and 95% constraints are where the curves intersect the dashed horizontal
lines. The dashed curve in the middle panel shows how the d-constraints tighten up when assuming
r = 1.
1.0
0 .5
i-
0 .0
- 0 .5
-
1.0
0
1
2
3
4
Figure 5.16 Constraints in the (/?, r) plane are shown for our baseline (1-Hfc = 100) finger-ofgod compression, using all measurements with k < 0.3h/Mpc and marginalizing over the power
spectrum normalization for fixed spectral shape. The four contours correspond to A*2 = 1, 2.29,
6.18 and 11.83, and would enclose 39%, 68%, 95% and 99.8% of the probability, respectively, if the
likelihood function were Gaussian.
133
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Figure 5.14 assumes T = 0.14, r = 1 (the best fit values) and explores how
the results change as we include information from smaller and smaller scales. As
will be discussed in more detail in section 5.6, non-linear effects invalidate the
Kaiser approximation for redshift space distortions on small scales. A smoking gun
signature of such nonlinearities is r and hence the best-fit 3 dropping and ultimately
going negative, as nonlinear “fingers of god” (FOGs) reverse the effect of linear
redshift distortions. The fact that figure 5.14 does not show this effect is reassuring
evidence that little small-scale information is present in our data. This is of course by
design, since our PKL-modes contain contributions only from £ < 40, corresponding
to a comoving distance around 20 h - l Mpc at the characteristic survey depth of
400 /i-1Mpc. This lack of small-scale information in our PKL-modes is also reflected
in the error bars on 3, which are seen to stop decreasing around k ~ 0.2 h/M pc.
Figure 5.14 also shows how the results depend on the FOG removal described
in section 5.3.1. The curves are seen to diverge markedly around k ~ 0.2/i/M pc,
with the FOG-related uncertainty becoming as large as the statistical error bars for
k ~ 1 h/M pc. We will return to these nonlinearity issues in section 5.5.4 below.
Figure 5.15 shows the constraints on T, 3 and r after marginalizing over the
other parameters. The best fit model is T = 0.14, 3 = 0.50, r = 1, cr8 = 0.99. The
reason that the constraints on 3 are so weak is illustrated in figure 5.16: there is a
degeneracy with r. Figure 5.13 shows that our information about redshift distortions
is coming predominantly from Pgy(fc), not from the poorly constrained Pvv(fc), so we
are to first order measuring the combination /?r rather than 3 and r individually.
Imposing the prior r = 1, as was implicitly done in Peacock et al. and almost all
prior work, therefore tightens the upper limit on 3 substantially, as shown by the
dashed curve in figure 5.15.
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5.4.3
The galaxy-galaxy power spectrum alone
The previous subsection discussed the 2dFGRS constraints on redshift space distor­
tions, essentially the ratios of the power spectra Pgg(k), Pgv(k) and Pvv(k), without
regard to their shape. Let us now do the opposite, and focus on the shape of the
galaxy power spectrum Pgg(k). The success of the disentanglement scheme illustrated
in figure 5.12 implies th at the galaxy power spectrum plotted in figure 5.13 is robust,
essentially independent of what the power spectra Pgv (k ) and Pvv(k) are doing.
However, this robustness came at a price in terms of increased error bars. Assuming
th at all three power spectra have essentially the same shape, but not the same
amplitudes, we compute a more accurate estimate of Pm (k) as follows.
We first assume some fixed values for 8 and r. This allows us to eliminate
P g v (fc )
and Pw(k) using equation (5.22), reducing the size of our param eter vector p from
3 x 49 = 147 to 49 and our Fisher matrix to size 49 x 49, and gives 49 decorrelated
estimators of Pgg(k). The result assuming 3 = 0.5, r = 1 (our best fit values) is
shown in figure 5.17. We perform no binning here except averaging the noisy bands
with k < 0.02 and k > 0.8 into single bins to reduce clutter. We then repeat this
exercise for a range of values of 3 and r consistent with our analysis in the previous
subsection to quantify the uncertainty these parameters introduce. We find these
uncertainties to be quite small, as expected considering the small initial leakage of
gv and w power (see figure 5.12), and can therefore quantify the added uncertainty
SP„ to first order as
S\nP g£(k) =
d in Pa (k)
s m
d{0r)
+
5 In PK (k)
6(32).
a m
(5.32)
Numerically, we find these two derivatives to be approximately —0.2 and —0.04,
respectively, essentially independent of k. This scale-independence is not surprising
in the sm all-angle limit, where these derivatives would involve simply various average
moments of /*, the angle between the k-vector and the line of sight. Assuming uncer­
tainties S3 = 0.15 and and St = 0.5, equation (5.32) thus gives S In Pa (k) ~ 0.12, the
135
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second term being negligible relative to the first. In conclusion, the uncertainties in
figure 5.17 induced by uncertainties about 8 and r can be summarized as simply
an overall multiplicative calibration error of order 12% for the measured power
spectrum.
5.5
H ow reliab le are our resu lts?
How reliable are the results presented in the previous section? In this section, we
perform a series of tests, both of our software and algorithms and of potential
systematic errors. We also discuss the underlying assumptions that are likely to
be most important for interpreting the results.
5.5.1
V alidation o f m ethod and software
Since our analysis consists of a number of numerically non-trivial steps, it is impor­
tant to test both the software and the underlying methods. We do this by generating
-Nmonte = 100 Monte Carlo simulations of the 2dFGRS catalog with a known power
spectrum, processing them through our analysis pipeline and checking whether they
give the correct answer on average and with a scatter corresponding to the predicted
error bars. We found this end-to-end testing to be quite useful in all phases of
this project — indeed, we had to run the pipeline 43 times until everything finally
worked...
T he m ock su rvey generator
Standard N-body simulations would not suffice for our precision test, because of a
slight catch-22 situation: the true non-linear power spectrum of which an N-body
simulation is a realization (with shot noise added) is not known analytically, and is
usually estimated by measuring it from the simulation — but this is precisely the
step that we wish to test. We therefore generate realizations th at are firmly in the
136
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1000
100
0.01
Figure 5.17 The decorrelated galaxy-galaxy power spectrum is shown for the baseline galaxy sample
assuming 0 = 0.5 and r = 1. As discussed in the text, uncertainty in 0 and r contribute to an
overall calibration uncertainty of order 12% which is not included in these error bars.
137
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linear regime, returning to nonlinearity issues below. We do this as described in
PTH01, with a test power spectrum of the simple Gaussian form P(fc) oc e_(Rfc)2/r2
with R = 32 h- l Mpc, normalized so that the rms fluctuations (S2) 1^2 = 0.2.
T estin g t h e P K L p ix e liz a tio n
Figure 5.18 shows the result of processing the Monte Carlo simulations through
the first step of the analysis pipeline, i.e., computing the corresponding Pseudo-KL
expansion coefficients x*. This is a sensitive test of the mean correction given by
equation (5.9), which can be a couple of orders of magnitude larger than the scatter
in figure 5.18 for some modes.
A number of problems with the radial selection
function integration and the spherical harmonic expansion of the angular mask in
our code were discovered in this way. After fixing these problems, the coefficients x,
became consistent with having zero mean as seen in the figure.
Figure 5.18 also shows that the scatter in the modes is consistent with the
predicted standard deviation cq = (C u /N moate)l/2 (shaded region), with most of the
the fluctuations being localized to modes probing large scales (with i being small).
A more sensitive test of this scatter is shown in Figure 5.19, which shows that the
theoretically predicted variance for each mode agrees with what is observed in the
100 Monte Carlo realizations. Since crowding makes it hard to verify all modes in
this plot, am alternative representation of this test is shown in figure 5.20.
Although these tests verify th at the mean and variance of each mode come out
as they should, they are not sensitive to errors in the off-diagonal elements of the
covariance m atrix C , i.e., to incorrect correlations between the mode coefficients.
To close this loophole, figure 5.21 shows the scatter in the true KL-modes (y = B x),
illustrating agreement with the theoretical variance prediction even in this alternative
basis where all coefficients y,- should be uncorrelated. Note th at the expected variance
decreases monotonically here, as opposed to in figure 5.19, since the true KL-modes
are strictly sorted by decreasing variance.
138
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‘
•** / . * ** -M
- 4* .V * *•.
*4. V
a - 0 .5
uv
>
<
-1
-
200
400
600
P seu d o —KL m o d e n u m b e r i
800
1000
Figure 5.18 The triangles show the elements x< of the data vector x (the pseudo-KL expansion
coefficients) averaged over 100 Monte-Carlo simulations of the baseline galaxy sample. If the
algorithms and software are correct, then their mean should be zero and about 68 % of them should
lie within the shaded yellow/grey region giving their standard deviation.
139
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J
I
I
I
200
I
I
1
I
I
1
I
I
I
I
400
600
P seu d o —KL m ode n u m b e r i
I
I
1-------- 1---------1--------
800
1000
Figure 5.19 The triangles show the rms fluctuations of the elements x, from 100 Monte-Carlo
simulations. If the algorithms and software are correct, then the expectation value of this rms is
given by the thin blue curve, and most of them should scatter in the yellow/grey region.
140
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I
T
I
I
T
r
T
i
r
11^
6 eo
o©
*4
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>
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2
4
Mode rms predcted theoretically
Figure 5.20 In this alternative representation of the test from figure 5.19, most of the vertical lines
should intersect the 45° line if the algorithms and software are correct.
141
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8
0)
c
JO
£
3
£
"w
oo
E
ou
X
ta
tn
E
a;
6
4
2
0
200
400
600
800
1000
KL m o d e n u m b e r i
Figure 5.21 The triangles show the rms fluctuations of the elements (Bx)i from 100 Monte-Carlo
simulations. If the algorithms and software are correct, then the expectation value of this rms is
given by the thin blue curve, and most of them should scatter in the yellow/grey banana-shaped
region.
142
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T estin g th e quadratic com p ression , F isher d ecorrelation and d isen tan gle­
m ent
Figures 5.22 and 5.23 show the result of processing the Monte Carlo simulations
through the remaining steps of the analysis pipeline, i.e., computing the raw quadratic
estim ator vector q and, from it, the decorrelated and disentangled band-power vector
p. The mean recovered power spectra are seen to be in excellent agreement with
the Gaussian prior used in the simulations (figure 5.22) convolved with the window
functions, and the observed scatter is seen to be consistent with the predicted error
bars (figure 5.23). These two figures therefore constitute an end-to-end test of our
data analysis pipeline, since errors in any of the many intermediate steps would
have shown up here at some level. Since information from large numbers of modes
contributes to each p^ the scatter is seen to be small. Therefore, even quite subtle
bugs and inaccuracies can be (and were!) discovered and remedied as a result of this
test.
5.5.2
R obustness to m ethod details
Our analysis pipeline has a few “knobs” that can be set in more than one way. This
section discusses the sensitivity to such settings.
E ffect o f changin g th e prior
The analysis method employed assumes a "prior” power spectrum via equation (5.23),
both to compute band power error bars and to find the galaxy pair weighting that
minimizes them. As mentioned, an iterative approach was adopted starting with
a simple BBKS model, then shifting it vertically and horizontally to better fit the
resulting measurements and recomputing the measurements a second time. To what
extent does this choice of prior affect the results? On purely theoretical grounds
{e.g., Tegmark, Taylor &c Heavens 1997), one expects a grossly incorrect prior to
143
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7 2000
1000
3000
7 2000
1000
3000
1000
io-»
1
k [h Mpc~l]
Figure 5.22 The triangles show the decorrelated and disentangled band-power estimates pi, averaged
over 100 Monte-Carlo simulations of the baseline galaxy sample. If the algorithms and software
are correct, then this should recover the window-convolved input power spectrum W p , plotted as
a thin blue line. The thin shaded yellow/grey band indicates the expected scatter. The harmless
discontinuity in the middle panel is an artifact of the disentangled galaxy-vekxdty windows having
negative area on the largest scales where there is essentially no information available.
144
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- 1500 r
S.
“ 1000 r
m
?
500 i
2500 ■
2000
-
1500
-
1500
.c
“
J*
1000
I* 500
io-«
Figure 5.23 Same as the previous figure, but testing the error bars Ap^ rather than the power itself.
The triangles show the observed rms of the power spectrum estimates from 100 simulations and
the solid blue curve shows the predicted curve around which they should scatter.
145
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give unbiased results but with unnecessarily large variance. If the prior is too high,
the sample-variance contribution to error bars will be overestimated and vice versa.
This hypothesis has been extensively tested and confirmed in the context of power
spectrum measurements from both the Cosmic Microwave Background (e.^., Bunn
1995) and galaxy redshift surveys (PTH01), confirming that the correct result is
recovered on average even when using a grossly incorrect prior. In our case, the prior
by construction agrees quite well with the actual measurements (see figure 5.13), so
the quoted error bars should be reliable as well.
E ffect o f changing th e num ber o f PK L m odes
We have limited our analysis to the first N = 4000 PKL modes whose angular
part is spanned by spherical harmonics with I < 40. This choice was a tradeoff
between the desire to capture as much information as possible about the galaxy
survey and the need to stay away from small scales where non-linear effects invalidate
the Kaiser approximation to redshift distortions. To quantify our sensitivity to these
choices, we repeated the entire analysis using 500, 1000, 2000 and 4000 modes. Our
power spectrum measurements on the very largest scales were recovered even with
merely 500 modes. As we added more and more modes (more and mode small-scale
information), the power measurements converged to those in figure 5.13 for larger
and larger k. The rising part of the envelope in figure 5.10 remained essentially
unchanged, merely continuing to grow further as more modes were added, so the
turnover of this envelope directly shows the fc-scale beyond which we start running
out of information. The version of Figure 5.10 shown in this paper indicates that
our 4000 PKL modes have captured essentially all cosmological information from the
2dFGRS for k £ 0.1.
146
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N u m erical issu es
The computation of the matrices P , involves a summation over multipoles I th at
should, strictly speaking, run from I = 0 to I = oo, since the angular mask itself
has sharp edges involving harmonics to £ = oo. In practice, this summation must
of course be truncated at some finite multipole
To quantity the effect of this
truncation, we plot the diagonal elements of the P-m atrices as a function of
and study how they converge as 4m increases. We define a given PKL-mode as
having converged by some multipole if subsequent f-values contribute less than 1%
of its variance. Figure 5.24 plots the number of usable PKL-modes as a function of
wavenumber k, defining a mode to be usable for our analysis only if it is converged
for all smaller wavenumbers k' < k for all three power flavors (Pgg, Pgv and Pw)We use fcut = 260 in our final analysis, since this guarantees that all 4000 modes
are usable for wavenumbers k in the range 0 —0.5 h j Mpc, i.e., comfortably beyond
the large scales 0 —0.3/i/M pc that are the focus of this paper. W ith this cutoff,
the computation of the P-matrices (which scales as
asymptotically), took about
a week on a SunBladelOOO workstation. Our power spectrum estimates are likely
to remain fairly accurate as far out as we plot them, i.e., to k ~ 1 h/M pc, since
figure 5.24 shows most modes remaining usable out to this scale, and since we find
th at even the ones that do not meet our strict 1% convergence criterion at every
single band are generally fairly accurately treated. Indeed, we repeated our entire
analysis with l e t = 120 and obtained almost indistinguishable power spectra.
5.5.3
Tests for problem s w ith data m odeling
In section 5.2, we performed detailed modeling of the way in which the 2dFGRS
d ata was selected, and produced a uniform galaxy sample fully characterized by a
selection function ii(r) of the separable form of equation (5.1). Let us now assess how
sensitive our results are to potential mis-estimates of ft, both angularly and radially,
by discarding purely angular and radial modes from our analysis.
147
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4000
3000
C
B
«
■ou
E
_e 2000
3
(0
ca
=i
1000
0.01
0.1
1
lg k - b a n d
Figure 5.24 Numerical convergence. The figure shows for how many of our 4000 PKL modes the
numerical calculations are converged to accurately measure the power up to a given wavenumber
k. From left to right, the 12 curves correspond to truncation at icui =20, 40, 60, 80, 100, 120, 140,
160, 180, 200, 220 and 240.
148
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■ All m odes
* G eneric m odes
* S p e c ia l m o d e s
SB
O
CL
s
I
Q.
1000
100
0.01
0.1
1
k [h llp c- *]
Figure 5.25 Constraints on excess power in special modes. Our 2dF power spectrum measurements
from figure 5.17 are averaged into fewer bands and compared with measurements using only special
(radial, angular and local group) modes and only generic (the r e m a in in g ) modes (dashed).
149
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R ob u stn ess to angular problem s
A n g u lar modulations caused by dust extinction tend to have a power spectrum
rising sharply toward the largest scales (Vogeley 1998), and is therefore of particular
concern for the interpretation of our leftmost bandpower estimates. The galaxy
magnitudes are extinction corrected by the 2dFGRS team, using extinction map
produced by Schlegel, Finkbeiner & Davis (1998), so any inaccuracies in this extinc­
tion model would masquerade as excess large-scale power. Inaccuracies in zero-point
offsets or in the magnitude dependent completeness correction that we applied in
section 5.2.2 could also introduce spurious angular power.
Of our 4000 modes, 147 are purely angular (see figure 5.6 for an example), and as
described in section 5.3.2, the remaining 3853 are orthogonal to them. This means
that to first order, angular problems affect only these 147 PKL-coefficients x*. We
repeated our entire analysis with these coefficients discarded, and found th at the
error bars became so large for k ^ 0.03 h/M pc that no signal could be detected
there. In other words, the information on the power spectrum on the very largest
scales comes mainly from the purely angular modes. On smaller scales, the measured
power spectrum remained essentially unchanged. Although we have no indication
th at angular problems are actually present, it may be prudent to follow the 2dFGRS
team and discard the information on the very largest scales — to be conservative,
we therefore use only the measurements for k > 0.01 h/M pc to be conservative in
our likelihood analyses (for 8. r and cosmological parameters).
R ob u stn ess to problem s w ith th e radial selectio n fun ction
45 of our 4000 modes are purely radial (see figure 5.6 for an example), and are to first
order the only ones affected by mis-estimates of the radial selection function n(r).
Since accurate ^-corrections and evolution modeling are notoriously challenging to
perform, we repeated our entire analysis with these 45 modes omitted as a precaution.
This resulted in a slight increase in error bars on the largest scales, but much less
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noticeable than when we removed the angular modes as described above. This can
be readily understood geometrically. If we count the number of modes that probe
mainly scales k < fc., then the number of purely radial, purely angular and arbitrary
modes will grow as k,, k? and k*, respectively. This means that “special” modes
(radial and angular) will make up a larger fraction of the total pool on large scales
(at small k), and that the purely radial ones will be outnumbered by the purely
angular ones.
Percival et al. (2001) report that slight changes in n(r) did not have a strong
effect on the recovered 2dFGRS power spectrum, and we confirm this. We repeated
our analysis with a number of different radial selection functions n(r), including the
one from Colless et al. (2001) (the dashed curve in figure 5.3), finding only changes
smaller than the error bars for P (k) on the largest scales and no noticeable changes
for larger k.
A final end-to-end test for problems with any special (angular, radial, or local
group) modes is shown in figure 5.25. Here we have repeated the entire analysis
twice, once excluding all the special modes and once using only the special modes
(except the monopole). The latter is seen to give quite large error bars since only 196
modes are used (4 local group, 147 angular and 45 radial), but all three are seen to
be reassuringly consistent. In contrast, systematic problems with any special modes
would tend to add power to the special modes. This shows that any misestimates of
special modes is having a negligible impact on our final results.
5.5.4
N on-linearity issues
A key assumption (essentially the only one) underlying our analysis is th at the
Kaiser (1987) linear perturbation theory approach to redshift space distortions is
valid. This approximation is known to break down on small scales where nonlinear
effects become important, which is why we have limited our analysis to large scales.
To be more precise, our basic measurement of Pa {k), Pgv(k) and Pvv(fc) assumes
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nothing a t all, and measures the quantities th at reduce to the monopole, quadrupole
and hexadecapole of power in the in small-angle approximation (Hamilton 1998).
However, relating these three measured functions to 8(k) and r(fc) via equation (5.22)
does require the Kaiser approximation to be valid.
Substantial progress has recently been made in quantifying nonlinear effects
on redshift distortions, using both perturbation theory, gravitational iV-body sim­
ulations and semianalytic galaxy formation theory (Hatton & Cole 1997, 1999;
Scoccimarro et al. 1999; Heavens, Matarrese & Verde 1999; Scoccimarro, Zaldarriaga
& Hui 1999; Hamilton 2000; Seljak 2001; Scoccimarro & Sheth in preparation). The
consensus is that nonlinear effects may be important even on scales as large as
k ~ 0.1 —0.3/i/M pc), although the critical scale is sensitive to the type of galaxies
involved via their bias properties (Seljak 2001). Moreover, a generic smoking-gun
signature of nonlinear effects is found to be that the ratio Pgv(k)/Pgg(k) starts
dropping and eventually becomes negative, as nonlinear fingers-of-god reverse the
signature of linear infall. The ratio Pw(fc)/Pgg(fc) increases sharply in this regime.
Ideally, to do full justice to the 2dFGRS data set, one would like to perform a
suite of nonlinear simulations until a realistic biasing scheme is found th at reproduces
all observed characteristics of the data. The fast PTHalos approach (Scoccimarro &
Sheth 2002) suggests that such an ambitious approach may ultimately be feasible. In
the interim, the results obtained with analytic approximations must be interpreted
with great caution. Peacock et al. (2001) use the widespread approach of adding
a nuisance parameter to the Kaiser formula, interpreted as a small-scale velocity
dispersion (cite), and marginalizing over it. This gives 8 = 0.43 ± 0 .0 7 from 141,000
2dFGRS galaxies. Hatton &c Cole (1999) and Scoccimarro & Sheth (in preparation)
argue th a t this is approximation is inaccurate, underestimating the nonlinear cor­
rections (hence underestimating 8) on large scales, and that the approximation of
Hatton & Cole (1999) is preferable.
Given these important uncertainties, we adopt a more empirical approach, using
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the above-mentioned Pgv-drop in the data as an indicator of where to stop trusting
the results. This was also done in the PSCz analysis of Hamilton et al. (2001), where
3 was found to start dropping for k ^ 0.3/i/M pc. Figure 5.13 shows no indication
of Pgv(fc)/Pgg(fc) (basically the quadrupole-to-monopole ratio) dropping, suggesting
th at our linear approximation is not seriously biasing our results on the large scales
probed by our PKL modes (which recover information fully down to fc ~ 0.1 as
described above).
To quantify further the effect of non-linearities empirically, we performed our
entire analysis five times with different levels of finger-of-god (FOG) compression as
described in section 5.3.1. The five curves in figure 5.14 correspond to progressively
more aggressive compression with overdensity cutoffs 1+<5C= oo, 200, 100, 50 and 25,
respectively. This corresponds to 6677, 7820, 8643 and 9124 FOG’s compressed, in­
volving 18544, 24031, 29807 and 36098 galaxies, respectively. Figure 5.14 shows that
more aggressive FOG-compression has an effect with the expected sign, increasing
the best-fit j3-value for k
0.1, and that the effect is reassuringly small compared to
the statistical error bars. Since a cluster is expected to have an overdensity around
200 when it virializes, more later since the background density drops, thresholds
l+ d c < 100 are likely to be overkill — we included the cases 1+<JC= 50 and 25 in the
figure merely to explore an extreme range of remedies. By removing essentially all
structures that are elongated along the line of sight, one of course creates an artificial
excess of flattened structures, leading to an overestimate of 3- In conclusion, we
believe that our estimate 3 = 0.49 ±0.16 is not severely affected by nonlinearities. A
conservative approach would be to take our measurement without FOG compression
and use it merely as a lower limit, giving 3 > 0.26 at 90% confidence.
Non-linearities affect our analysis in a different way as well, leading to slight
underestimates of error bars. Our power spectrum measurements are simply certain
second moments of the data, and remain valid regardless of whether the underlying
density field is Gaussian or not. The power spectrum variance, however, involves
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fourth moments, and we have computed our error bars by making the Gaussian
approximation to calculate these moments. The standard rule of thumb is th at this
approximation underestimates the error bars on the correlation function £(r) by a
factor [1 + £(r)]1/2. Norberg et aL (2001) fit the 2dFGRS correlation function to a
power law £(r) = ( r / r . ) -7 with correlation length r . = 4.9/i-1Mpc and slope
7
=
1.71. Taking k ~ 7r /r, this gives error bar correction factors [1 + (r.fc/ 7r )7)]1/'2 as 2%,
7% and 13% at 0.1, 0.2 and 0.3/i/M pc, respectively. Here £(r) should refer to the
correlation function of the m atter, not of the galaxies, so if the 2dFGRS galaxies are
biased with b > 1, the correction factors will be smaller. In conclusion, although
nonlinear error bar corrections certainly become important on very small scales, they
are likely to be of only minor importance on the large scales k < 0.3 /i/M pc th at are
the focus of this paper.
5.5.5
Bias issues
Although our basic measurement of Pa (k), Pgv(k) and Pw{k) assumes nothing about
biasing, a bias model is obviously necessary before the results can be used to constrain
cosmological models. We therefore comment briefly on the bias issue here.
Substantially larger d ata sets such as the complete SDSS catalog hold the promise
of measuring 3{k) and r(k) with sufficient accuracy to quantity their scale-dependence,
if any. figure 5.13 shows th at our present sample is still not quite large enough to
place strong constraints of this type.
An alternate route to constraining b(k) involves comparing the clustering am­
plitudes of various subsamples, selected by, say, luminosity or spectral type. Such
comparisons can also constrain r directly (Tegmark & Bromley 1999; Blanton 2000).
It has been long known th at bright elliptical galaxies are more clustered than spi­
rals, presumably because the former are more likely to reside in clusters. Recent
subsample analysis of the 2dFGRS (Norberg et aL 2001a) and SDSS (Zehavi et al.
2002 )
have confirmed and further quantified this effect.
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Since recent cosmological parameter analyses using P(fc)-measurements (most
recently Wang et al. 2002 and Efstathiou et al. 2002) have assumed that the bias
factor b is scale-independent on linear scales, it is important to note that slight
scale-dependence of bias is likely to be present in Pre(fc)-measurements from a
heterogeneous galaxy sample such as the 2dFGRS. Most of the information about
Pgg(fc) on large scales comes from distant parts of the survey, where bright ellipticals
are over-represented since dimmer galaxies get excluded by the faint magnitude limit.
This could cause b(k) to rise as A: —* 0. If uncorrected, this effect could masquerade
as evidence for a redder power spectrum, i.e., one with a smaller spectral index n.
Figure 5.17 indeed suggests slightly more 2dFGRS power on the largest scales
than currently favored cosmological models with constant bias would suggest, al­
though this excess may also be caused by the angular or radial issues mentioned
above. Detailed power spectrum analysis of subsamples should settle this issue.
5.6
D iscu ssion and con clu sion s
To place our results in context, we will now briefly discuss how they compare with
other recent power spectrum measurements and with cosmological models.
5.6.1
Com parison w ith other surveys
Figure 5.26 compares our 2dFGRS power spectrum measurements from figure 5.17
(averaged into fewer bands to reduce clutter) with measurements from other recent
surveys. The PSCz and UCZ redshift surveys were analyzed with the same basic
method that we have employed here2, so a direct comparison involves no methodrelated interpretational issues. The 2dFGRS sample is seen to be slightly more biased
than PSCz, but slightly less biased than UZC. Figure 5.26 also suggests th at 2dFGRS
2Since the UZC analysis in PTH 01 did not include redshift space distortions, we performed a
complete reanalysis of that data set for this figure, expanding the 13342 galaxies surviving the cuts
described in PTH01 in 1000 PKL modes.
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may have a slightly redder power spectrum than PSCz. This would also be consistent
with the scale-dependent bias scenario mentioned above — the PSCz survey would
probably be less afflicted than 2dFGRS, since the /HAS-selected galaxies in PSCz
tend to avoid clusters.
Although the 2dFGRS error bars are seen to be small compared the PSCz and
UZC ones, due to the larger sample size and survey volume, the horizontal bars
show that the 2dFGRS window functions are somewhat broader. This is easy to
understand: whereas PSCz and UZC cover large contiguous sky regions, the 2dFGRS
sky coverage is currently fragmented into a multitude of regions of small angular
extent, exacerbating aliasing problems. Indeed, since the characteristic width of
2dFGRS patches in the narrowest direction is more than an order of magnitude
smaller than for PSCz or UZC (of order 2° rather than ~ 60°), the fact th at the
windows are only 2-3 times wider reflects the quality of the 2dFGRS survey design
and the power of the quadratic estimator method.
The remaining two power spectra are interesting since they were measured with­
out use of redshift information and thus without the additional complications in­
troduced by redshift space distortions. The APM points are from the likelihood
analysis of Efstathiou & Moody (2001), using a few million galaxies, and reflect
the full uncertainty even on the largest scales.
Here the vertical bands have a
different interpretation, indicating the bands used in the likelihood analysis. Note
th at although the 2dFGRS galaxies are a subset of the APM galaxies, they need not
have the exact same bias. Since the 2dFGRS subset involves on average brighter
and more luminous galaxies, one might expect them to be slightly more clustered.
The SDSS points (from Tegmark et al. 2002) are for about a million galaxies in the
magnitude range 21 < r ' < 22, and the vertical bars have the same interpretation as
for the 2dFGRS points (redshift information obviously helps tighten up the windows).
In contrast, the parameterized SDSS power spectrum in Dodelson et al. (2002) can
be interpreted like the APM one.
156
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• PSCz
* UZC
c SDSS/angular
ua.
2
1000
100
0 .0 1
0.1
k [h Mpc-1]
Figure 5.26 Comparison with other power spectrum measurements. Our 2dF power spectrum
measurements from figure 5.17 are averaged into fewer bands and compared with measurements
from the PSCz (HTP00) and UZC (this work) redshift surveys as well as angular clustering in the
APM survey (Efstathiou & Moody 2001) and the SDSS (the points are from Tegmark et aL 2002
for galaxies in the magnitude range 21 < r1 < 22 — see also Dodelson et aL 2002).
157
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A direct comparison of our power spectrum results with those reported by the
2dFGRS team (Percival 2001) is unfortunately not possible at this time, since their
window functions are of crucial importance and have not yet been made publicly
available.
However, an indirect comparison is possible as described in the next
section, indicating good agreement. Our ^-constraints are consistent with those
reported in Peacock ef al. (2001).
5.6.2
Cosm ological constraints
Figure 5.27 compares our 2dFGRS measurements with theoretical predictions from
a series of models. No corrections have been made for non-linear evolution or scaledependent bias. The measurements are seen to be in good agreement with both our
simple BBKS prior and the recent concordance model from Efstathiou et al. (2002)
— specifically, this is fit B from their paper, a flat scale-invariant scalar model with
Q \ = 0.71, h = 0.69, baryon density u;b = 0.021 and dark m atter density u/<* = 0.12.
(ub = h2fifc, uid = h2Qd-) Both of these are of course good fits by construction: we
iterated our analysis until we found a prior that was consistent with the data, and
Efstathiou et al. (2002) searched for models fitting both the 2dFGRS power spectrum
and CMB data. However, the fact that the Efstathiou et al. (2002) model fits our
data so well provides an important cross-check between the 2dFGRS team power
spectrum measurement (Percival et al. 2001) and ours, indicating good agreement.
Figure 5.27 also shows the concordance model from Wang et al. (2002), resulting
from a fit to all CMB d ata and the PSCz galaxy power spectrum. It is a flat scalar
model with
jjd
= 0.66, h = 0.64, baryon density
— 0.12 and a slight red-tilt,
ns
= 0.020, dark m atter density
= 0.91, here renormalized to the PSCz data. The
fact th at these pre-2dF and post-2dF concordance models agree so well is a reassuring
indication that such multi-parameter analyses are converging to the correct answer,
and th at the final numbers are not overly sensitive to bias issues or methodological
technicalities.
158
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1000
BBKS prior used
Efstathiou e t al
Wang e t al
Best fit high baryon model
100
*—
0.01
0.1
1
k [h M pc'1]
Figure 5.27 Our 2dF power spectrum measurements from figure 5.17 are averaged into fewer bands
and compared with theoretical models. The BBKS model is the wiggle-free prior used for our
calculation. The flat ACDM “concordance” models from Wang et aL (2002) and Efstathiou et al.
(2002), both renormalized to our 2dF measurements, are seen to be quite similar. The wigglier
curve corresponds to the best-fit high baryon model in the upper right comer of figure 5.28. Only
data to the left of the dashed vertical line are included in our fits.
159
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A full m ultiparam eter analysis of our results along the lines of Wang et al.
(2002) and Efstathiou et al. (2002) is clearly beyond the scope of the present paper.
However, since evidence for baryonic wiggles in the galaxy power spectrum has
generated strong recent interest, first from the PSCz d ata (HTP00) and then more
strikingly from the 2dF data (Percival et al. 2001; Miller et al. 2001), we perform a
limited analysis to address the baryon issue.
We consider fiat scale-invariant scalar models parametrized by the total m atter
content Qm, the baryon fraction Qb/Q m, the hubble param eter h and the spectral
index ns. We map out the likelihood function L = e~x*/2 using equation (5.31) on a
fine grid in this param eter space, and compute constraints on individual parameters
by marginalizing over the other parameters. Figure 5.28 shows the result of fixing
ns = 1 and h = 0.72, the best-fit value from Freedman et al. (2001). Here the axes
have been chosen to facilitate comparison with Figure 5 from Percival et al. (2001)3.
The general agreement between the two figures is seen to be good, both in terms of
the shape and location of the banana-shaped degeneracy track, and in that there are
two distinct favored regions — a low-baryon solution like the concordance models
in Figure 5.27 and a high-baryon solution th at is inconsistent with both Big Bang
Nucleosynthesis (Buries et al. 2001) and CMB constraints. To illustrate the nature
of the banana degeneracy in figure 5.28 , we have plotted the best fit high-baryon
model in Figure 5.27. It has Qm = 0.75 and uib = 0.18, and is seen to provide
a slightly better fit to the data around k = 0.04/t/M pc at the expense of slight
difficulties on smaller scales.
There is, however, one notable difference between figure 5.28 and its twin in
Percival et al. (2001). WTiereas the latter excluded Qb/ Q m = 0, we find no significant
detection of baryons. This is of course not an indication of problems with either anal­
ysis, since the Percival et al. figure excludes zero baryons only at modest significance.
3 As a technical point, Percival et aL included band powers up to a nominal wavenumber k = 0.15
in their figure. Since our window functions are narrower, we have included band powers up to
k = 0.3 in figure 5-28 to ensure th at we do not use less small-scale information.
160
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Most importantly, as emphasized by Efstathiou et al. (2002), the constraints get
much weaker when allowing small variations in other parameters, most strikingly the
spectral index na. We confirm this effect by marginalizing over n , and h with various
priors. This means that the full statistical power of the complete 2dF and SDSS data
sets will be needed to provide unequivocal evidence for baryonic signatures in the
galaxy distribution.
5.6.3
Outlook
We have computed the real-space power spectrum and the redshift-space distor­
tions of the first 105 galaxies in the 2dFGRS using pseudo-Karhunen-Loeve eigenmodes and the stochastic bias formalism, providing easy-to-interpret uncorrelated
power measurements with narrow and well-behaved window functions in the range
0.01 /i/Mpc < k < 1 h/M pc. A battery of systematic error tests indicate that the
survey is not only impressive in size, but also unusually clean.
Galaxy redshift surveys are living up to expectations. The striking early successes
of the 2dFGRS and SDSS projects have firmly established galaxy redshift surveys
as a precision tool for constraining cosmological models. However, it is important
to bear in mind that this is only the beginning, and that many of the most exciting
cosmological applications of these surveys still lie ahead. As discussed above, detailed
comparisons with grids of fast simulations are likely to place information extracted
from redshift distortions on a firmer footing and allow substantially more velocity
information to be extracted from translinear scales. A bivariate analysis of how
clustering depends jointly on both spectral type and luminosity should improve
our quantitative understanding of biasing and allow possibilities such as the abovementioned artificial red-tilt to be quantified and e lim in a te d . W ith such progress
combined with an order-of-magnitude increase in sample size, to more than 106
galaxies from 2dFGRS and SDSS combined, exciting opportunities will abound over
the next few years, from definitive constraints on baryons and neutrinos to things
161
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0.6
w
oa
u
eo
>v
ffi
0.2
0.0
0.0
0.2
0 .4
0.6
0.8
M atter d e n sity Q„
Figure 5.28 Constraints in on the matter density Qm and the baryon fraction
from the
linear power spectrum over the range 0.01 hf Mpc < k < 0.3/i/Mpc, after marginalizing over the
power spectrum amplitude. These constraints assumes a flat, scale-invariant cosmological model
with h = 0.72. For comparison with Percival et al (2001), contours have been plotted at the level
for one-parameter confidence of 68% and two-parameter confidence of 68%, 95% and 99% (i.e.,
\ 2 \ 2 mtn — 1,2.3,6.0, 9.2. Marginalizing over the Hubble parameter h and limiting the analysis
to scales k < 0.15/t/Mpc as in Percival et al (2001) further weakens the constraints.
162
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that have not even been thought of yet.
163
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164
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A ppendix A
C om bining m aps
The combined map x defined by
x = [a ‘N - 1a ]" 1 A ‘N - ‘y
(A.l)
can be shown to be unbiased ((x) = x), to minimize the rms noise in each pixel and,
if the noise properties are Gaussian, to retain all information about the true sky
x that was present in the two original maps Tegmark (1999b). The corresponding
covariance matrix of the noise e = x —x is
E = ( « ‘) = [a ‘N " 1a ] ' 1 .
(A.2)
In all cases treated in this paper, the noise is uncorrelated between different maps
((N 12 = 0 ) , which simplifies these equations to
x
=
E ^ N rV i+ A 'N jV z ] ,
(A.3)
E
=
[ A 'N ^ A i + A £ N J 1A 2]~ 1.
(A.4)
Thanks to the deconvolution technique that will be described in Appendix D, we
will generally face the much simpler case where the two data sets are two sky maps
with the exact same angular resolutions, i.e., the case A t = A 2 = I, reducing the
last two equations to simply
x
=
E fN rV i+ N ^y z],
177
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(A.5)
(A.6)
[ N r ' + N j 1] ' 1.
For the case of only a single pixel, we recognize this as a familiar inverse-variance
weighting. More generally, if the two noise matrices can be simultaneously diagonalized, we see that this combination scheme corresponds to an inverse-variance
weighting eigenmode by eigenmode.
Generally maps overlap only partially, so we need only apply this matrix method
in the common region. Yet care needs to be taken in computing the noise covariance
matrix £ of the final map, since it will contain correlations between the common
region and the rest. Let us split the noise vectors for the two maps as
(A.7)
where the subscript c refers to the common region of the two maps whereas a and
b refer to the regions that only belong to maps 1 and 2, respectively. We write the
corresponding covariance matrices as
(A.9)
Substituting equation (2.1) into equation (A.3), we obtain the noise vector lie for
the combined map in the common region:
He = S c [A ^ N ^ H c, + A ^ N ^ n * ] ,
(A. 10)
where E c is given by equation (A.4) for the common part. The final combined noise
covariance matrix S for the noise vector (n«„ n*, lie) of the combined map is therefore
s =
N„
0
0
N6
^ ([“ X ]‘> ([“ *»£]*}
(n«n£) >
(n6n*)
£cJ
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(A ll)
where
We need
to usethese
(n«n‘) =
(nan ‘1)N cl1A clE c,
(A.12)
<n*n‘> =
(n6n^2)N ^1A c2S c.
(A.13)
expressions repeatedly in this paper to combine
partially
overlapping maps, e.g., combining the different QMAP flights with each other and
combining QMAP with Saskatoon.
179
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A ppendix B
P lo ttin g m aps
Although the map x contains all the sky information from am experiment, plotting it
is not very useful when some modes are much more noisy than others, thereby dom­
inating the visual image. For this reason, it has become standard in the community
to plot the corresponding Wiener filtered map, defined as
x„ = E [E + N ]- l x,
(B.l)
where E is an estimate of the covariance m atrix due to sky signal. Throughout this
paper, we use the E-m atrix corresponding to the “concordance” power spectrum
from Tegmark, Zaldarriaga and Hamilton (2001), which agrees well with all current
CMB measurements.
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182
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A p p en d ix C
C om paring m aps
Here we discuss the issue of how to test whether two data sets are consistent or
display evidence of systematic errors. Specifically, is there some true sky x such that
the data sets yi and y 2 are consistent with equation (2.1)? Let us first consider
the simplest case where the two d ata sets sample the sky in the same way, th at is,
A i = A 2. Consider two hypotheses:
H0: The null hypothesis H0 that there are no systematic errors, so that the differ­
ence map z = yi —y2 consists of pure noise with zero mean and covariance
m atrix (zz‘) = N = N i + N 2.
H\ . The alternative hypothesis that the difference map z consists of some signal
besides noise, *.e., (z) = 0, and (zz‘) = N + E for some signal covariance
m atrix E .
The “null-buster” statistic Tegmark (1999b)
z*N_1E N _1z - tr { N - I E}
v = ------------------------- 1
[ 2 tr { N - 1E N - 1E}]1/2
(C. l )
can be shown to rule out the null hypothesis H0 with the largest average significance
({/) if Hi is true, and can be interpreted as the number of “sigmas” at which H0 is
ruled out Tegmark (1999b). Note that for the special case E ex N , it reduces to
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simply u = (x 2 — n)/y/2n, where \ 2 = z*N- I z is a standard chi-squared statistic.
The null-buster test can therefore be viewed as a generalized x2-test which places
more weight on those particular modes where the expected signal-to-noise is high.
It has proven successful comparing both microwave background maps Devlin et al.
(1998); Herbig et al. (1998); de Oliveira-Costa et al. (1998) and galaxy distribution
Tegmark and Bromley (1999); Seaborne et al. (1999).
To evaluate equation (C.l) in practice, it is useful to Cholesky decompose the
noise matrix as N = LL‘ and compute the m atrix R = L -1£ L - t . The remain­
der of the calculation now becomes trivial, since tr { N _1S } = t r R = £ R ,* and
tr {N -1£ N -1S } = t r R 2 = £ ( R a )2For the general case when Ai ^ A 2, the situation is more complicated, since it
is non-trivial to construct a difference map which is free of sky signal. A technique
involving signal-to-noise eigenmode analysis has been derived for this case Tegmark
(1999b), but it is unfortunately rather complicated and cumbersome to implement.
Below we present a simpler method that eliminates need for this by reducing the
general problem to the simple case A! = A 2 = 1 .
184
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A ppendix D
D econvolving m aps
In this section, we present a method for inverting equation (2.1), i.e., for undoing
the convolution with beam and scanning effects given by the A-matrices.
D .l
W h y is it useful?
As we will see, this simplifies calculations by eliminating all A-matrices, encoding
the corresponding complications and correlations in the noise covariance matrices.
When comparing or combining two maps, it is generally undesirable to smooth the
higher resolution one down to the lower resolution of the other, since this destroys
information. Moreover, this tends to cause numerical instabilities by making the
smoothed noise covariance m atrix poorly conditioned. We will see that, surprisingly,
deconvolution can be better conditioned than convolution/smoothing.
This deconvolution (elimination of A-matrices) is useful not only for comparing
d a ta sets as mentioned above, but for combining them as well. The complication
stems from the fact that the sky is sampled by only a finite number of pixels, so to
avoid problems with undersampling, x must be the “true sky” beam-smoothed map
with some finite angular resolution. If we do not deconvolve, but use equations (A.3)
and (A.4) to combine two data sets with different angular resolutions, say 6\ = 0.89°
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and 02 = 0.68° for the QMAP Ka- and Q-band maps, respectively, it is natural
to define x to have the higher (0.68°) resolution, setting A 2 = I and letting Ax
— Q\ in the lower resolution map. The
incorporate the extra smoothing A 6 =
resulting map will now contain two kinds of pixels: ones with resolution 62 in the
region covered only by map 2 and with the higher resolution 9\ elsewhere. If we
need to combine this with a third map, the relevant smoothing scale unfortunately
becomes undefined near the boundary between the two resolutions. Deconvolution
eliminates all these problems.
D .2
H ow d oes it work?
In the generic case, deconvolution is strictly speaking impossible, since the matrix
A is not invertible1 and certain pieces of information about x are simply not present
in y . It is common practice to find approximate solutions to such under-determined
problems using singular value decomposition or other techniques, but our goal here
is different. We wish to compute a vector x th at can be analyzed as a true sky map.
Specifically, we want analysis of (x, £ ) to give exactly the same results as analysis of
(y, N , A) for all cosmological applications, say Wiener filtering or power spectrum
estimation.
Our basic idea is to accept that certain modes in the map x cannot be recovered,
and to record this information in the noise covariance matrix £ by assigning a huge
variance to these modes. Any subsequent analysis (say Wiener filtering or power
spectrum estimation) will then automatically assign essentially zero weight to these
modes.
In practice, we find it conceptually useful to imagine combining our d ata y i with
a ‘Virtual map” y 2 th at is so noisy th at it contains essentially no information, yet
has the angular resolution 6 that we wish to deconvolve down to, i.e., A 2 = I.
1Specifically, the problem is th at A genetically has a non-zero null space, ue., th a t there are
non-zero vectors x such th a t A x = 0.
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Although, as we will see, this virtual map never enters the calculations in practice,
it is a useful notion for intuitively understanding what the deconvolution technique
does. Specifically, let us take the noise in the virtual map to be uncorrelated, with
noise covariance matrix N 2 = o21 for some very large noise level a. Equations (A.3)
and (A.4) give
X =
E A 'N t ly i +
E
[ A ' N ^ A i +<r- 2l ] ~ l
=
o'
2£ y 2,
.
(D-l)
(D.2)
In the limit a >-»• 00, x will clearly become independent of the virtual temperature
map y 2 except for the “junk modes” which have infinite variance according to E .
For convenience, we therefore set y 2 = 0 in practice.2
This deconvolution method has exactly the property we want as long as a is
orders of magnitude larger than the pixel signal due to CMB. If we were to choose
<7
to be too small, then the virtual map would contribute a non-negligible amount
of information and bias the results. If we were to choose <j to be too large, however,
the matrix E would contain some enormous eigenvalues (since A ‘N i A i is typically
not invertible) and be poorly conditioned, which could cause numerical problems
in subsequent analysis. We performed a series of numerical tests to assess these
problems, and found th at with n $ 104 pixels and double precision arithmetic,
<t = 104/xK was a good compromise th at produced neither of these two problems.
We will therefore use this choice throughout the present paper.
2An alternative approach would be to set y2 equal to a M onte-Carlo generated map of
independent Gaussian random variables with standard deviation a — although this results in
different numerical values in x, it will of course not change the results of any subsequent cosmological
analysis of x , since only the “junk modes” are different.
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D .3
T ests
As a first test of the method, we deconvolve (or “unsmooth” ) a map y! with
resolution 0\ into a map with resolution 02 (62 < 6 \). For this case,
<A ' > , =
<D 3 >
where d{j = cos-1 (r, • ij) is the angular separation between pixels i and j , and
A6 =
—B\ is the extra smoothing to be undone. Specifically, we unsmooth the
QMAP Ka-band data to obtain the same resolution as the Q-band data has, from
0.89° to 0.68°. We then Wiener filtered both the original and unsmoothed versions
of the map, obtaining virtually identical results.
As a second test, we deconvolve the Saskatoon data y into a map x using the
full (and rather complicated) A-m atrix described in Tegmark et al. (1996). We then
Wiener-filter x and obtain a map virtually identical to the one that was computed in
Tegmark et al. (1996) — the latter was computed with a completely different method
which circumvented the map step altogether. The details of these maps have already
been presented above.
Both of these tests thus confirm what we expect theoretically: th at the decon­
volved map (x, S ) contains exactly the same information about the true sky as the
input data ( y .N , A), no more and no less.
188
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