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New approaches to map-making for cosmic microwave background observations

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NEW APPROACHES TO MAP-MAKING FOR
COSMIC MICROWAVE BACKGROUND OBSERVATIONS
BY
CHARMAINE ROSE ARMITAGE-CAPLAN
B.Sc, University of British Columbia, 2003
M.S., University of Illinois at Urbana-Champaign, 2005
M.S., University of Illinois at Urbana-Champaign, 2007
DISSERTATION
Submitted in partial fulfillment of the requirements
for the degree of Doctor of Philosophy in Physics
in the Graduate College of the
University of Illinois at Urbana-Champaign, 2009
Urbana, Illinois
Doctoral Committee:
Associate Professor Susan Lamb, Chair
Associate Professor Benjamin Wandelt, Director of Research
Associate Professor Robert Brunner
Professor Aida El-Khadra
UMI Number: 3391874
All rights reserved
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© 2009 by Charmaine Rose Armitage-Caplan. All rights reserved.
Abstract
In this thesis, I examine systematic effects due to beams and foregrounds that will
effect precision temperature and polarization measurements of the Cosmic Microwave
Background (CMB) by satellite instruments. I have developed a unique, pixel-free solution to the general CMB map-making problem, which addresses the impact of beam
asymmetries through beam deconvolution. The resulting algorithm, called PReBeaM,
solves for the map directly in spherical harmonic space, avoiding pixelization artifacts.
The most promising experimental means for testing the widely popular inflationary model of the early Universe is through the detection of primordial gravitational
waves (or tensor perturbations) and the resulting imprint of B-mode polarization.
As we enter the PLANCK era and become more sensitive to the CMB polarization
signal on both large and small scales, the possible detection of inflationary B-modes
seems achievable. Detection of this putative, minute signal will most certainly depend
on our understanding of, and control over, sources of uncertainty due to systematic
effects and foregrounds. My PhD research has been focused on developing a novel
CMB map-making technique for removing systematics as a result of asymmetries in
the beam. I demonstrate that my deconvolution method produces maps and spectra
with less contamination when compared to methods that do not account for beam
asymmetries in both temperature and polarization.
I show that the true sky is recovered with greatly enhanced accuracy via the
deconvolution method for temperature measurements in realistic experiments with
asymmetric beams and far-side lobe Galactic foreground contamination. I also illustrate that standard map-making is unable to remove artifacts due to optical systematics in these cases. I then extend my method to operate on polarization data and
apply my technique to the actual PLANCK 30 GHz simulated CMB data. In an actual
CMB experiment, we are faced not only with challenges due to beam asymmetries,
but also noise and astrophysical foregrounds. I study the noise properties of CMB
maps from PReBeaM for PLANCK and include important foreground signals from
dust, synchrotron and free-free emission.
ii
Extragalactic point sources are an additional foreground contaminant in CMB
maps; proper identification and removal of these sources is a crucial step in the analysis pipeline. Prom the standpoint of Extragalactic Astrophysics, point source data
from the WMAP instrument can provide important information about source variability on a range of different time scales. I establish a program to search the public
time-ordered WMAP datasets for under-exploited multi-band data on the variability
and evolutionary properties of point sources catalogued by the WMAP team.
hi
For David
IV
Acknowledgments
There are many people deserving of acknowledgement for helping to bring me to this
point in my academic career. I owe my advisor, Ben Wandelt, many thanks and my
sincerest appreciation for guiding me so well through my PhD research. Many times,
I have been inspired by Ben's excitement about CMB research and benefited from his
breadth of knowledge.
I credit my two summers spent doing undergraduate research at the Canadian
Institute for Theoretical Astrophysics for starting me on the path of Cosmology and
CMB research. It was there I was supervised by Dick Bond and Carlo Contaldi, both
of whom were excellent mentors for someone who knew nothing about CMB power
spectra or Fortran programming.
During my time at Illinois, I have had the pleasure to work with many friendly and
bright colleagues and I have received help from countless people. I thank David Larson, Chad Fendt, Amit Yadav, Rahul Biswas, and Rishi Khatri for helpful discussions
and for sharing Ben's time and attention with me.
I am grateful to many people at the NASA Jet Propulsion Lab who worked with
me on the PLANCK collaboration. They include, but are not limited to, Charles
Lawrence, Kris Gorski, Gary Prezeau, Jeff Jewell, and Graca Rocha. I am also
thankful to Mark Ashdown at Cambridge for early conversations on map-making.
I thank the team at the Lawrence Berkley Lab — Julian Borrill, Chris Canteloupo,
and Ted Kisner — for being experts at computing and helping me with many of the
issues I had with analysing the simulated PLANCK data.
I was very fortunate to have been able to spend three weeks in June of 2007 at the
Max Planck Institute for Astrophysics. I thank MPA for their generous hospitality.
While I was there, and in many instances before and after, I received invaluable help
and advice from Martin Reinecke on all things convolution and interpolation related.
Without this help, it surely would have taken twice as long to debug my code.
I would like to acknowledge the following sources of funding for this research: the
National Center for Supercomputing Applications (NCSA), the Center for Advanced
v
Studies, NASA contract JPL1236748 by the National Computational Science Alliance
under AST300029N and the University of Illinois, NASA contract JPL1371158, and
NSF grant AST-05-07676. Additionally, I used resources at the National Energy
Research Scientific Computing Center (NERSC), which is supported by the Office of
Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.
Some of the results from Chapter 3 were computed on the NCSA computer Copper.
Many of the results contained in my thesis, and particularly those in Chapter 4,
were calculated on the NERSC computers Bassi and Seaborg. I thank the NERSC
help consultants for being so prompt and thorough with all of my help requests.
Additionally, I am indebted to the entire team who worked to produce the PLANCK
simulated data on which I ran my code to produce the results in Chapter 4.
Some of the work that I did in Chapter 5 was completed in collaboration with
Torsti Poutanen at the University of Helsinki, who was always willing to answer my
questions with prompt replies.
None of my work on the WMAP variable point sources in Chapter 6 would have
been possible without vital help from some members of the WMAP team: Gary
Hinshaw, Nils Odegard, and Bob Hill. I also acknowledge extensive use of the Legacy
Archive for Microwave Background Data Analysis (LAMBDA) site for WMAP timeordered data files, beam profiles, masks, and the point source catalogue.
I thank my committee members: Susan Lamb, Robert Brunner and Aida ElKhadra for their interesting questions and thoughtful comments. I am indebted to
Wendy Wimmer for answering dozens of my questions about dissertating and defending, for quelling my anxiety in the last weeks of my studenthood, and for striving to
make my experience at Illinois as smooth and painless as possible.
I thank Xiaoyue Guan for being an excellent office-mate, and for many funny
conversations during long work days. I thank fellow physicists, Keiko Kircher and
Madalina Colci-O'Hara for being good friends and for sharing the unique experience
of entering motherhood during graduate school.
It is
Caplan.
frequent
much to
without a doubt that I owe my deepest gratitude to my husband, David
I can't imagine how I would have done this without his love, support, and
pep talks. Finally, I thank my son, Emmanuel Caplan, for giving me so
smile and laugh about every single day for this last year and a half.
VI
Table of Contents
List of Tables
x
List of Figures
xi
Chapter 1
Introduction
1
Chapter 2 CMB Theory
2.1 Introduction
2.2 The Standard Cosmological Model
2.3 Models for the formation of perturbations
2.3.1 Fourier analysis of density fluctuations
2.3.2 Inflationary models
2.3.3 The cyclic model
2.4 Inflationary gravitational waves
2.5 Formation of Cosmic Microwave Background Anisotropics
2.6 Theory of CMB polarization
2.7 Review of Stokes parameters
2.8 Harmonic analysis for temperature and polarization anisotropics . . .
2.9 Conclusions
5
5
5
7
7
8
10
10
11
12
15
15
16
Chapter 3 Deconvolution Map-Making
3.1 Summary of chapter
3.2 Introduction
3.3 The map-making problem
3.3.1 Maximum-likelihood estimator of the true sky
3.4 Standard map-making methods
3.5 Deconvolution Map-Making
3.5.1 Regularization technique
3.5.2 Fast convolution on the sphere
3.5.3 Binning and simulation
3.6 Solving the Deconvolution Equations
3.6.1 Preconditioning
3.7 Standard map-maker for comparison
3.8 Test Cases
3.9 Results and Discussion
18
18
18
19
21
22
23
24
25
26
27
28
29
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31
vii
3.10 Point source tests with elliptical beams
3.11 Conclusions
37
37
Chapter 4 Polarized map-making for Planck
4.1 Summary of Chapter
4.2 Introduction
41
41
41
4.3
PLANCK
42
4.3.1 The map-making challenge for PLANCK
PReBeaM Method
Comparison to standard map-making and other map-making methods
Fast all-sky convolution for polarimetry measurements
PReBeaM Implementation
4.7.1 Polynomial Interpolation and Zero-Padding
4.7.2 Parallelization Description
4.8 Simulations and Beams
4.9 Results and Discussion
4.9.1 Computational Considerations
4.10 Conclusion
4.4
4.5
4.6
4.7
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Chapter 5 Noise and Diffuse Foregrounds
5.1 Noise model for PLANCK
5.2 Testing PReBeaM on CMB + white noise
5.3 Knowledge of polarized foregrounds
5.3.1 Synchrotron emission
5.3.2 Dust emission
5.4 Testing PReBeaM on polarized foregrounds
5.5 Conclusions
66
66
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69
70
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Chapter 6 Variable Point Sources
6.1 WMAP findings on point sources
6.2 Description of data extraction and processing
6.2.1 Loss imbalance parameter
6.2.2 Differencing assembly parameters
6.2.3 Beam normalization
6.2.4 Dipole corrections
6.3 Chi-square analysis
6.3.1 Null test
6.4 Results and discussion
6.5 Conclusions and further work
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81
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Chapter 7 Conclusions
7.1 Summary of Progress
7.2 Future work with PReBeaM
7.2.1 Improving the computational requirements
7.2.2 Interface with destriper for noise treatment
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93
93
93
94
viii
7.3
7.2.3 Re-analysis of the WMAP data
7.2.4 Analysis of PLANCK data with far sidelobes
Future work with the convolution routines
References
94
95
95
97
Vita
102
IX
List of Tables
3.1
Fractional RMS error for each of the six main test cases
32
4.1
The four detectors of the 30GHz PLANCK channel and their FWHM,
ellipticity, angle of polarization measurement ippoi, and orientation of
beam major axis tpeii
54
6.1
6.2
Beamsize (9b), loss imbalance parameter (xjm), U>A,B, and noise term
(ao) for each of the differencing assemblies (DA)
Point sources that we have identified as being variable. Sources are
listed, following the WMAP convention, by their 5 GHz ID. Variability
observed in either the A or B side (or both) is indicated by a checkmark
for the K and Ka band separately. The peak K and Ka band flux is
listed in columns 6 and 7, respectively. The type (G for galaxy, and
QSO for quasar) is listed in column 8. v WMAP-identified as probable
variable: x 2 > 36.7. v WMAP-identified as variable: \2 > 100
x
80
86
List of Figures
2.1 This plot shows a model for the dynamics of the scalar field during
inflation and the transformation of fluctuations in the scalar field into
density fluctuations. Accelerated expansion of the universe occurs during the slow-roll phase. Inflation ends when the field reaches the minimum of the potential, oscillates and decays into ordinary radiation and
matter. Points on the potential perturbed by a fluctuation S(j) finish
inflation at different times 5t, inducing a density fluctuation 5 = HSt.
Reprinted with permission from [1]
2.2 Current measurements of the polarized CMB signal. The TE measurements (grey) are from the first-year WMAP data. The E-mode measurements (coloured) are from CAPMAP, CBI, DASI and Boomerang.
The B-mode spectra is shown in red and blue for two cases, r = 0.30
and r = 0.01, respectively. B-mode due to lensing is shown in green.
The black curves are the predicted one-sigma sensitivity for WMAP
and PLANCK (neglecting correction for foreground emission). Figure
reproduced from [2]
On the left is a figure showing the orbit of the PLANCK satellite in
the Earth-Sun system. The figure on the right shows the main beam
of the telescope sweeping out large circles on the sky, while Galactic
straylight is entering the detectors through the far sidelobe, seen at 90°
from the main beam. Figure modified from [3]
3.2 A map of the spatial structure of the far sidelobes for one 30 GHz
channel of PLANCK. The main beam would sit in the region at the pole.
3.3 Measured WMAP focal plane co- and cross-polar beams, shown on the
left and right, respectively. The contours are spaced by 3 dB and the
maximum value of the gain in dB is given next to the individual beams.
Figure reproduced from [4] by permission of the A AS
3.4 Ratios of the spectra of the residual map to the spectrum of the input
map for each of the beam models and both scanning strategies. The
BSP results are plotted in the left column and the WSP results are
plotted in the right column. Results for the sidelobe, elliptical and
two-beam beam are shown in the top, middle, and bottom panels,
respectively. The solid lines correspond to deconvolved spectra and
the dashed lines correspond to the standard spectra
9
14
3.1
xi
20
20
21
33
3.5
Residuals after the first 100 iterations are shown in the first three rows.
The figures on the left are for the basic scan path, and the ones on the
right are for the WMAP-like scan path. First, second, and third rows
correspond to the sidelobe, elliptical and two-beam beams, respectively.
The true sky is shown in the fourth row. Note that solutions of the
two-beam BSP and, to a lesser extent, the elliptical beam BSP test
cases have not converged to sufficient accuracy. We chose to present
the results for all cases after a fixed number of iterations to show the
impact of scanning strategy and beam pattern on the condition number
of the map-making equations
3.6 Convergence rates of the preconditioned conjugate gradient solver for
each test case. The left panel refers to the basic scan path and the right
panel to the WMAP-like scan path. The solid lines correspond to the
sidelobe beam, dotted lines to the elliptical beam, and dot-dashed to
the two-beam model
3.7 Deconvolving the effects of a large sidelobe in simulated observations
of the WMAP Ka band map, using the coarsened WMAP scanning
strategy described in the text. The top map is the input sky map, the
middle map is the standard map-making result, and the bottom map
is the deconvolved result
3.8 The top plot shows the distribution of point sources in the upper left
quadrant of the CMB sky (note that the upper end of the color scale
has been cut-off at 93.6^K; the original scale terminates at 374fJ,K).
The middle and bottom plots show the difference maps between the
input and output skies for m — 0 (middle) and m = 4 (bottom). It
can be seen that the overall residual in the m — 4 case is roughly five
times smaller than the m = 0 case. We also find that point sources in
regions of the sky that are observed by the beam at a wide range of
orientations are reconstructed with smaller errors
3.9 Examing the effect of beam ellipticity and subsequent deconvolution
on point sources in the CMB. The actual point source (top) is scanned
with an elliptical beam and reconstucted using deconvolution mapmaking with the asymmetry parameter set t o m = 0 (bottom left) and
m — 4 (bottom right)
4.1
4.2
Comparison of one segment of the sky from a binned noiseless map
(left) and the PReBeaM temperature map (right), both at a Healpix
resolution of nside 1024. At this resolution, the binned map contains
a number of unobserved pixels, of which three are visible in this frame.
The PReBeaM map contains no unobserved pixels
Forward interpolation from ring set to TOD element and transpose
interpolation from TOD element to ring set
xii
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35
36
38
39
47
50
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
Depiction of hybrid parallelization scheme used in PReBeaM. Rectangles represent work done on a node, ellipses represent data products,
and arrows represent transfer of data. The work done within a node
(convolution, interpolation and their transpose operations) is parallelized using OpenMP. This figure shows a slice of two head nodes,
however, the algorithm may operate on many more nodes
Plot showing the footprint of the LFI main beams on PLANCK as seen
along the optical axis looking towards the satellite. The pair of horns
comprising the 30 GHz channel are labeled 27 and 28 in the plot. The
scan direction and polarization orientations for the co-polar beams are
also depicted. Figure reproduced from [5] with permission
Contour plots in the wu-plane of the PLANCK 30 GHz main beam
detectors. This coordinate system permits the beam to be mapped
from a spherical surface to a plane. From left to right, the beams are:
LFI-27a, LFI-27b, LFI-28a, and LFI-28b. Figure reproduced from [5]
with permission
EE, TE, and BB spectra of smoothed input map (black curve), binned
map (blue curve) and PReBeaM (red curve). Note that the red and
black curves are nearly on top of each other. The EE and TE spectra
show the effect of temperature-to-polarization cross-coupling seen in
the binned map spectra as shifts in the peaks and valleys and absent
from the PReBeaM spectra. The input BB spectra is absent from the
BB plot since the input B-modes were zero. For comparison, we show
a theoretical B-mode spectrum (dashed curve) from a cosmological
model with a 10% tensor-to-scalar ratio. TT spectra are omitted since
differences in the three spectra are not apparent in this representation.
Fractional difference in power spectrum for PReBeaM (red, blue, and
cyan curves) and the binned map (black curve) for TT, EE, and TE.
Spectra for PReBeaM are shown as a function of number of iterations
to demonstrate convergence
Power spectra of the difference map for the binned map (black) and
for PReBeam (red). For comparison, we calculate the power spectra
from the difference in a^m's (aemout — aemin) for the PReBeaM method
(blue). In the case of the BB spectra, the blue curve lies exactly on
top of the red curve since the input afm were zero
The absolute value of the beam aem coefficients for m = 0 (solid),
m = 2 (dotted), m = 4 (dashed), m = 6 (dash-dot), and m = 8
(dash-dot-dot-dot). We plot \ajm\ (left) and \afm\ (right) (we omit the
B component plot since \afm\ = \afm\)
BB power spectra as a function of asymmetry parameter mmax for
m
max = 2 (blue curve) , 4 (cyan curve), and 6 (red curve). The input
BB spectra was zero so the smallest output BB spectra is most desirable. In this run, the PReBeaM input parameters interpolation order
and zero-padding were set to 1 and 2, respectively
xin
52
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58
59
60
61
4.11 Worst case bias in estimation of cosmological parameters due to errors
in the power spectra of PReBeaM (dashed curve) and due to the errors
in the power spectra of the binned map (solid curve). These plots
estimate the bias due to the 30 GHz channel only, and should not be
taken as representative of PLANCK as a whole
4.12 The residuals between the input reference sky and PReBeaM output
(left column) and the residual between the input reference sky and the
binned map (right column) for Temperature (T), and the Stokes Q,
and U parameters
5.1
5.2
5.3
5.4
5.5
TE spectrum of CMB and white noise for Springtide (blue curve) and
PReBeaM (red curve). The smoothed input map is shown in black.
Following the example in [5], we reduce £ to £ variation by filtering
the spectra by a sliding average (A£ = 20). In this run, the PReBeaM input parameters interpolation-order and zero-padding were set
to 1 and 4, respectively. While PReBeaM performs at least as well
as Springtide in the TT, EE, and BB spectra, we omit these spectra
since the detailed differences are difficult to assess without an in-depth
Monte Carlo study
Simulated map for synchrotron polarization amplitude at 30 GHz. . .
Simulated map for dust polarization amplitude at 30 GHz
Fractional difference in power spectrum for PReBeaM (red curve) and
Madam (black curve) for TT, EE, BB, and TE
Output spectra from PReBeaM (red curve) and Madam (black curve)
for free-free emission only. Since free-free emission is temperatureonly, all of the resulting power in the polarization modes is a result of
leakage from temperature to polarization through beam effects. Note
that the PReBeaM TT spectra nearly exactly overlays the binned map
TT spectra
Map of the 390 point sources detected by the WMAP team. The
shaded region shows the mask used to exclude extended foreground
emission. The size of the plotted points indicates the flux of the source:
the area of the dot scales like the maximum flux over the 5 WMAP
bands plus 4 Jy. Image courtesy of the WMAP science team
6.2 Innermost parts of the normalized profiles for all 10 channels of the
WMAP instrument
6.3 The 24 most variable point sources found on the A side of the K band.
The distribution of the x2 probabilities for the null points plotted as a
histogram for the A side (blue) and B side (green). The x2 probability,
p, for the actual point sources are overplotted for the A side (dotted)
and B side (dashed). This figure shows the point sources ordered in
increasing p for side A. Note that the dotted and dashed lines for
sources J0423-0120, J1229+0202, J1256-0547, J2258-2758, and J05384405 are difficult to see as they sit nearly exactly at zero
63
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xiv
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4
The 24 most variable point sources found on the B side of the K band.
The distribution of the x 2 probabilities for the null points plotted as a
histogram for the A side (blue) and B side (green). The x 2 probability,
p, for the actual point sources are overplotted for the A side (dotted)
and B side (dashed). This figure shows the point sources ordered in
increasing p for side B. Note that the dotted and dashed lines for
sources J0423-0120, J1229+0202, J1256-0547, J2258-2758, and J05384405 are difficult to see as they sit nearly exactly at zero
5 The 24 most variable point sources found on the A side of the Ka band.
The distribution of the x 2 probabilities for the null points plotted as a
histogram for the A side (blue) and B side (green). The x 2 probability,
p, for the actual point sources are overplotted for the A side (dotted)
and B side (dashed). This figure shows the point sources ordered in
increasing p for side A. Note that the dotted and dashed lines for
sources J1258-0547, J1229+0202, J2258-2758, J0423-0120, J0538-4405,
and J1014-4508 are difficult to see as they sit nearly exactly at zero. .
6 The 24 most variable point sources found on the B side of the Ka band.
The distribution of the x 2 probabilities for the null points plotted as a
histogram for the A side (blue) and B side (green). The x 2 probability,
p, for the actual point sources are overplotted for the A side (dotted)
and B side (dashed). This figure shows the point sources ordered in
increasing p for side B. Note that the dotted and dashed lines for
sources J1258-0547, J0423-0120, J0538-4405, J1229+0202, J2258-2758,
and J1014-4508 are difficult to see as they sit nearly exactly at zero. .
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Chapter 1
Introduction
Ever since its discovery by Penzias and Wilson in 1965 [6], the CMB has yielded a
wealth of cosmological information. Immediately following the serendipitous discovery
of the CMB, researchers at Princeton University led by Robert Dicke interpreted this
excess antenna temperature to be the relic radiation of the Big Bang [7]. Twentyseven years later, in 1992, the Cosmic Background Explorer (COBE) mission first
reported a measurement of the tiny fluctuations in the temperature of the CMB [8]
as predicted by General Relativity and the theory of inflation. Most recently, five
years of measurements of both the temperature and polarization anisotropies by the
Wilkinson Microwave Anisotropy Probe (WMAP) 1 with 45 times the sensitivity of
COBE and 33 times the angular resolution have constrained the model of our universe
to fantastic precision[9].
By measuring the statistical properties of the fluctuations in the CMB, cosmologists have been able to refine a standard model of cosmology. This standard model
consists of a flat universe in which roughly 74% of the total energy density is dark
energy and 22% is dark matter, with the remainder in the baryon density [9]. The
standard model also predicts nearly scale-invariant adiabatic Gaussian fluctuations.
Cosmologists are now interested in answering new questions about our universe, including: Did inflation seed the primordial fluctuations? And, if so, what is the correct
model for inflation?
The inflationary model of the universe was first described in 1981 by Alan Guth
as a means of solving two major puzzles in the standard model of the early universe
[10]. The first puzzle is known as the horizon problem, in which causally disconnected
regions of the universe are found to be homogeneous [10]. The second puzzle, referred
to as the flatness problem, indicates that the critical value of the energy density of the
universe today (which gives us a flat universe) requires fine-tuning of initial conditions
to incredible precision.
While current models of inflation (see [11] and [12] for recent reviews) are incredi1
http://map. gsfc.nasa.gov
1
bly successful at describing the observed universe, a distinct class of models has been
developed which also predicts these same conditions. This new class of theories is
called the ekpyrotic [13] or cyclic model [14], and is based on a multi-dimensional
framework motivated by string theory.
Fortunately, there exists a very promising means of experimentally verifying the
standard inflationary cosmology model through the detection of the presence of a
gravitational wave background (GWB). If present, a GWB will leave a unique signature, known as B-modes, in the polarization anisotropics of the CMB. Thus, a
detection of B-mode polarization in the CMB would be a direct probe of inflationary
gravitational waves.
One of the most exciting prospects for current and up-coming CMB satellite missions, such as PLANCK 2 and CMBPol, is their potential to measure the B-mode
polarization in the CMB. Measurement of this signal will depend on two things.
First, it will depend on the actual amplitude of the B-mode power spectrum, which
is uncertain and related to the energy scale of inflation. Second, it will depend on a
successful treatment of the data in which the overwhelming signal from intervening
foregrounds is removed and the systematics are corrected.
PLANCK is a satellite mission of the European Space Agency that is due to be
launched in May 2009. PLANCK is imaging the sky in nine frequency channels from 30
GHz to 857 GHz with angular resolution down to 5'. While PLANCK is not specifically
designed to detect primordial B-modes, it does have the potential to do so. A better
chance for B-mode detection will come with the future CMBPol mission, [15] designed
specifically to measure this elusive signal.
The process by which CMB data are collected suffers from both instrumental
and astrophysical systematic effects. Instrumental effects include asymmetries of the
beam in the main focal plane, and the pick-up of large foreground signals far from
the focal plane in the side lobes. We use the term beam throughout this dissertation
to describe the response of the detector instruments to the photons being measured.
Astrophysical systematic effects include all Galactic and extra-galactic signals appearing in the frequencies where the CMB is measured. In an optimal treatment of CMB
data, these effects must be corrected during the "map-making" step, the method by
which time-ordered data (TOD) are turned into a sky map. Power spectra are generated from maps, which are then used to estimate cosmological parameters, such as
the density of baryons fl^h2, the density of cold dark matter Qch2, and the amplitude
of curvature perturbations A27£ . Systematic effects that are left untreated in the
2
http://astro.estec.esa.nl/Planck
2
maps may be propogated through the data analysis pipeline and may lead to errors
in estimated parameters.
In this dissertation, we describe the development of a novel technique for making
maps of the CMB from satellite-generated instruments that fully addresses issues due
to beams, optics, and foregrounds. The chapters are organized both by chronological
development of our technique, and also by the increasing complexity of the problem
being examined.
In order to provide motivation for the quest for precision CMB polarization measurements, we review the main principles of CMB physics and inflationary theory
in Chapter 2. We briefly compare inflationary models for the formation of metric
perturbations in the universe with cyclic models. This is followed by a review of
the theory, mathematics and formalism for the formation of CMB anisotropies and
polarization.
Chapter 3 describes in detail the map-making problem, some standard solutions,
and our unique approach to the solution, known as deconvolution map-making. We
compare the results from our approach to those from a standard map-making method
in a variety of test cases that examine effects of scan strategies, beams, and foregrounds. Results from this chapter are restricted to temperature-only measurements
of the CMB.
In Chapter 4 we extend our description of deconvolution map-making from the
previous chapter to include polarization measurements of the CMB. We describe
the development of our technique under the framework of analysis for the PLANCK
mission and we adopt the name PReBeaM for our tool. We introduce the objectives
and parameters of the PLANCK mission and discuss the beams and simulated data.
We describe interpolation and parallelization features that we implemented in order
to meet the challenges that are faced when manipulating such large data sets and
when attempting to make accurate polarization maps.
Following the successful performance of PReBeaM in the cases of temperatureonly and temperature-plus-polarization-only CMB data, we introduce white noise and
foregrounds into the data in Chapter 5. We compare the performance of our code to
a map-making method currently employed by the PLANCK Algorithm Development
group in the US.
Point sources are an important foreground in CMB maps and some extraction
methods rely on a complete physical characterization. In Chapter 6, we describe a
program to identify point source variability in the public WMAP time-ordered data.
An implicit assumption in our work with the convolution method is that neither
3
the sky nor the beam changes over the course of the survey. Variable sources are
one contributor to time variation in the sky that have received little attention in the
literature to date. This study of point source variability will help to provide important
information about the structure of extragalactic radio sources over a wide frequency
range.
We conclude the thesis in Chapter 7 with a summary of the results and a plan for
future work with the PReBeaM code.
4
Chapter 2
CMB Theory
2.1
Introduction
In this chapter we introduce the standard model of cosmology and describe the formation of primordial density perturbations. The physics of the inflationary model for
the beginning of the Universe is examined in detail. Following that, we discuss similarities and differences between the inflationary model and the cyclic class of models
in terms of their specific predictions about the formation of scalar perturbations and
gravitational waves.
Next, we discuss the formation of temperature and polarization anisotropies in
the CMB and the relationship between CMB polarization and gravitational waves.
We conclude this chapter by reviewing the Stokes parameters, standard description
for a polarized field and the application of harmonic analysis to temperature and
polarization anisotropies.
Over 40 years of study of the CMB has led to a very sound understanding of
the underlying physical principles. There are many texts that present in great detail
descriptions of the formation of initial perturbations, and temperature and polarization anisotropies. For a more detailed look at CMB theory than is presented in this
chapter, we recommend the following texts: [16, 17, 1, 18].
2.2
The Standard Cosmological Model
Standard hot big bang cosmology describes a universe expanded from a hot dense
state, dominated by thermal blackbody radiation, and evolved under the laws of Einstein's general theory of relativity. The Friedmann-Robertson-Walker (FRW) model
[19, 20, 21, 22, 23] was developed as the simplest possible model obeying the basic
assumptions that (i) the universe is the same at all points (homogeneous) and (ii) all
spatial directions at a point are equivalent (isotropic). In this work, we assume the
5
FRW framework for the universe, the metric for which is given by
ds2 = dt2 - R2(t)[,
dr2
, „ + r2(d92 + sin2(e)d(f,2}.
1 — kr2
(2.1)
Note that the speed of light c has been set to 1 throughout. This equation for the line
element ds is characterized by the cosmological scale factor R(t) which determines
proper distances in terms of the comoving coordinates, and the constant A; which
describes the curvature of the universe. By a rescaling of the radial coordinate, we
can restrict k to be either 0, -1 or 1 corresponding to a flat, open or closed universe
respectively. R(t) is commonly normalized by R0 = R(t0), the value of R at the
present epoch, defmining the dimensionless scale factor a(t) = R(t)/R0. The rate of
change of the scale factor describes the background evolution of the universe and is
known as the Hubble parameter H(t) = R(t)/R(t).
Prom Einstein's field equations, we can define two cosmological equations of motion that govern the expansion and acceleration of the universe:
and
R
R
A
3
4irG
3 * + %>)•
(2 3
' >
In the above equations, G is the gravitational constant, p is the energy density, A is
the cosmological constant, and p is the isotropic pressure.
In addition to the Hubble parameter, it is useful to define a handful of other
cosmological parameters. A critical density can be derived from Equation 2.2 by
requiring k = 0 when A = 0:
AW - 8 ^ -
(2-4)
The cosmological density parameter Qtot is defined as the ratio of the energy density
to the critical density
Oot(t) = P(t)/Pc(t)
(2.5)
and we often consider the individual contributions of densities as Clx(t) — px(t)/pc(t).
Today, our best estimates for the composition of our universe are: cold dark matter,
Qc = 0.228±0.013; dark energy, 0 A = 0.726±0.015; and baryons, Qb = 0.0456±0.0015
[24].
In this section, we have described the background evolution of the universe un-
6
der the assumptions that it is homogeneous and isotropic on large scales. However,
observations from telescopes show structures on a wide range of scales, from single
galaxies to superclusters of galaxies. The existance of these structures tells us something about the initial conditions of the Big Bang. We review these special initial
conditions for the formation of density fluctuations in the next section.
2.3
Models for the formation of perturbations
In this section, we review a Fourier analysis of cosmological density fields. Then we
describe the inflationary scenario and its predictions for the formation of the initial
perturbations. We compare and contrast these predictions with those of the cyclic
class of models for the early universe.
2.3.1
Fourier analysis of density
fluctuations
Gravitational instability is understood to leave an imprint of inhomogeneity and structure in the universe as a result of a primordial field of density fluctuations, <5(x). We
briefly summarize the statistical measures used to study <5(x). The dimensionless
density perturbation field is defined as
*(x) ^ * * ^ M .
(2.6)
An important feature of 8 is that it occupies a universe that is isotropic and homogenous in its large scale properties. This suggests that the statistical properties of 5
should be similar to those of a stationary random process.
We often expand 5 as a Fourier superposition and assume periodic boundary
conditions in a cube of some large volume:
5(x) = ^ 4 e -
t k x
(2.7)
where k is the wavenumber of the fluctuations. Then power spectrum is defined as
ensemble-average of 5^
V(k) = (|4| 2 )-
7
(2.8)
2.3.2
Inflationary models
Inflationary models [10, 25, 26] are based on the notion that there was a beginning
of the universe in both space and time, with nearly infinite temperature and density.
At a time of order 10~35 seconds (corresponding to an energy scale of 1016GeV),
the universe experienced a period of accelerated expansion. In the simplest models,
inflation is driven by a scalar field and excites fluctuations in all available modes.
This period of accelerated expansion inflates quantum fluctuations in the scalar field
to sizes much larger than the Hubble radius, after which gravitational instability leads
to fluctuations in the density of the universe. Quantum fluctuations in the metric
itself give rise to a spectrum of gravitational waves or tensor perturbations. We now
look more closely at the inflationary scenario.
Considering Equation 2.2, we find that each term on the right hand side has a
different time-dependence. At high redshift, space curvature and the cosmological
constant are negligible compared to the mass density so the cosmological equation
for the expansion rate becomes
h
/8
\1/2
H{t) = -a= (JTTGPJ
•
(2.9)
Local energy conservation implies that
p=-3-(p
+ p).
(2.10)
a
Inflation resolves the puzzle that the observable universe is homogeneous by assuming
that there was a time when the net pressure was negative,
V < -P/3
(2.11)
and the universe underwent a period of accelerated expansion.
The negative pressure in the inflation epoch is driven by a scalar field 4>, sometimes
called the inflaton, with Langrangian density
£ = \^4>jj
~ V{<j>).
(2.12)
In the limit that the field is nearly spatially homogenous, the energy density and
8
Figure 2.1: This plot shows a model for the dynamics of the scalar field during inflation
and the transformation of fluctuations in the scalar field into density fluctuations.
Accelerated expansion of the universe occurs during the slow-roll phase. Inflation
ends when the field reaches the minimum of the potential, oscillates and decays into
ordinary radiation and matter. Points on the potential perturbed by a fluctuation 5(f)
finish inflation at different times St, inducing a density fluctuation S = H5t. Reprinted
with permission from [1].
effective pressure in the field are
P4>
= <j>2/2 + V,
P(j)
= <j>2/2-V.
(2.13)
Thus, the condition for accelerated expansion dominated by the scalar field 0 is
(d(/)/dt)2 < V(<p). This corresponds to the case when the kinetic energy of the field
is much smaller than the potential energy, a condition which is referred to as the
slow-roll of the scalar field. We depict a simple inflation scenario in Figure 2.1.
Inflation ends as (p approaches a minimum of the potential and oscillates rapidly.
Couplings between the scalar field and matter fields will produce particles, leading to
what is called reheating.
The key inflationary observables are the level of anisotropy due to scalar perturbations and the level of anisotropy due to tensor perturbations. The exact relationship
between the level of anisotropy due to scalar and tensor perturbations and the inflaton
potential depends upon the cosmological model (see, for example, [27]). If these can
be measured, along with the power-law index, ns, that characterizes the density perturbations, then the scalar-field potential that drove inflation can be reconstructed.
These parameters are described in Section 2.4.
9
2.3.3
The cyclic model
The cyclic model [13, 14] predicts that the universe is infinite in space and time,
undergoing endless cycles of "big bangs" and "big crunches". Each cycle begins with
a bang, which is a transition from a contracting phase to an expanding phase, in which
the density and temperature of the universe do not diverge. This transition results
from a collision of two three-dimensional domain walls known as 'branes', and results
in a period of radiation-domination and a subsequent period of matter-domination.
Following this is a dark energy dominated epoch, where the dark energy is due to a
scalar field (f> rolling down a shallow potential V(<p). Acting as dark energy, <fi causes a
period of slow acceleration, converts the acceleration into deceleration, contracts into
a "big crunch" and begins the cycle again. The dark energy phase is responsible for
making the universe flat and homogeneous and provides the source for the presently
observed dark energy and cosmic acceleration.
Expansion slows down and reverses to contraction when <j> rolls to values where
V{4>) is negative. Contraction occurs very slowly at first, and during this period
quantum fluctuations have time to cause spatial variations in the rate of contraction.
It has been shown that the resulting spectrum is scale-invariant [28], and evolves to
become a spectrum of temperature and density perturbations after the bang. Current calculations [29] estimate that string-theory motivated models do not produce
measurable gravitational waves.
2.4
Inflationary gravitational waves
Inflationary gravitational waves are tensor perturbations in the background spacetime metric and were first proposed by [30]. Quantum fluctuations in the scalar field
driving inflation lead to scalar metric perturbations, while quantum fluctuations in
the metric itself lead to tensor metric perturbations. In a spatially-flat, FriedmannRobertson-Walker universe, the perturbations are described by the line element,
ds2 = a2[-dr2 + (8i:j + hij)dxidxj]
(2.14)
where r is the conformal time, x% are comoving spatial coordinates, and hij is the
tensor metric perturbation.
The scalar and tensor power spectra predicted by most inflationary models ap-
10
proximately follow power laws
A (/c)
* = Sy{llZkl2)
and
2k3
Al(k)^j^y2(\h+k\*
K kns+l
+ \hxk\2)<xkn>
(2 15)
"
(2.16)
where A^ and A\ are the variance due to the scalar and tensor modes respectively.
Here, 1Z is the curvature perturbation, and h+ and hx are the two polarization states
of the primordial tensor perturbation. They are defined by
<TC2> =
/ f
A^(fc)
(2.17)
T*l(k).
(2.18)
rlk
/
The scalar spectral index ns and the tensor spectral index nt are nearly scale invariant
(i.e., ns = 1 and nt = 0). The current best estimate on ns from WMAP with
constraints from Type la Supernovae and Baryon Acoustic Oscillations (BAO) is
ns = 0.960 ± 0.013 [9].
The tensor contribution, quantified by the tensor-to-scalar ratio r and evaluated
at A;0 = 0.002Mpc_1, is
The combination of the most recent WMAP results + Type la supernovae events +
BAO puts an upper limit on the tensor-to-scalar ratio, r < 0.22 (95% CL) [24].
Since gravitational waves are small in amplitude they evolve according to linear
theory. Thus, we can predict their evolution with high precision. Gravitational
waves are a useful probe of the early universe as they imprint essentially unperturbed
information on the CMB polarization anisotropy.
2.5
Formation of Cosmic Microwave Background
Anisotropies
In both the inflationary and cyclic model, our universe began as a hot, dense plasma of
photons, baryons and electrons. This ionized matter underwent Thomson scattering,
which effectively coupled radiation to matter. As the universe expanded, it cooled
11
adiabatically. At a temperature of roughly 3000K, it had cooled sufficiently so that
protons and electrons could combine to form hydrogen. As a result, the opacity of
the universe from Thomson scattering was reduced and radiation decoupled from
matter. This event occurred about 380,000 years after the big bang and is known as
last scattering. The CMB photons that we detect today have been redshifted by the
changing scale-factor of the universe, thus the CMB now has a mean temperature of
about 2.725K.
The level at which the CMB is anisotropic is about one part in 100,000, or fluctuations on the microKelvin scale. These anisotropics are generated through well
understood physics of the evolution of linear perturbations with a background FRW
cosmology. The three mechanisms for the formation of primary anisotropies can be
roughly separated by the range of scales that they each operate on: the Sachs-Wolfe
effect (£ < 100); the acoustic oscillations (100 < £ < 1000); and the damping tail
(*>1000).
Photons which last-scattered from an overdense region (or potential well) lose
energy from climbing out of the well and are redshifted, becoming cooler. Photons
which last-scattered from an underdense region, gain energy and are blue-shifted and
become hotter. This is the Sachs-Wolfe effect [31, 32], which contributes mainly to
anisotropies on scales larger than about one degree, or £ < 100.
On sub-degree scales, anisotropy is due to acoustic oscillations in the baryonradiation fluid at decoupling [33]. Perturbations in the gravitational field drive oscillations in the photon-baryon fluid. Regions of compressed fluid contain hot photons,
while regions of rarefied fluid contain cooler photons.
A damping of the anisotropies occurs at the highest £s because recombination is
not instantaneous. The photon-baryon fluid exhibits imperfect couplings, and the
amplitude of the oscillations decreases with time. This damping tail, also known as
Silk damping [34], corresponds to scales smaller than that subtended by the thickness
of the last scattering surface.
2.6
Theory of CMB polarization
Polarization of the CMB is generated when an anisotropic radiation field Thomson
scatters off free electrons. For incident radiation with a quadrupole moment, linear
polarization results from the scattering [35, 36]. These polarization fluctuations will
be an order of magnitude smaller than temperature fluctuations as the scattering will
occur near the end of recombination, when there are fewer scattering sites.
12
Polarization is also generated by scattering at later times when the universe reionizes. Therefore, CMB polarization can also provide information about the reionization history of the universe and may shed light on the nature of polarized foregrounds,
such as our Galaxy and thermal emission from dust.
The contribution to anisotropy which leads to polarization may either be due
to primordial density or gravitational wave perturbations. The primordial density
fluctuations are scalar and can only generate E-modes, a curl-free pattern classified
by polarization directions that are either purely radial or purely tangential to the
hot and cold temperature spots. On the other hand, the spin-2 nature of tensor
perturbations (or inflationary gravitational waves) permits both the generation of
E-modes and B-modes [37, 38, 39], a divergence-free polarization pattern.
The amplitude of B-modes depends linearly upon the amplitude of gravitational
waves during inflation and is parameterized by the tensor-to-scalar power ratio (r),
as shown in Figure 2.2.
In 2002, CMB polarization was first detected at sub-degree angular scales by the
DASI team [40, 41]. These results were confirmed and extended by results from CBI
[42], CAPMAP [43], and Boomerang [44]. More recently, with the release of the
third-year WMAP results, the first large-scale detection of E-mode polarization has
been made [45].
On the largest angular scales, CMB polarization should also reveal information
about the first stars that formed in our universe. In addition, gravitational lensing by
clustered matter at low redshifts should be detected in polarization and temperature
anisotropics.
Though a measurement of primordial B-mode polarization in the CMB would be
direct evidence of gravitational waves from inflation, a measurement of B-mode due
to lensing may be more likely. E- and B-modes can mix through a weak lensing shear.
The effect on B-modes due to this mixing is much more pronounced than for E-modes
since the amplitude of E is much greater. Figure 2.2 demonstrates that primordial
B-modes dominate on large scales, but B-modes due to lensing dominates on small
scales. While the measurement of the power spectrum of B-modes due to weak-lensing
would reveal important information about the intervening large-scale structure of the
universe, only a primordial gravitational wave detection would confirm an inflationary
epoch and set a corresponding energy scale.
13
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2.7
Review of Stokes parameters
In this section, I will review the standard description of CMB polarization following
the conventions in [38, 46]. A monochromatic electromagnetic wave propagating in
the z-direction with frequency ui0 has electric field components
Ex
=
ex{t) cos[ui0t - 9x(t)},
(2.20)
Ey
=
ey(t) cos[v0t - 6y(t)}
(2.21)
Linear polarization can be decomposed into two components, the Q and U Stokes
parameters, denned as
Q = (el)-(e2y),
(2.22)
U = {2exeycos(9x-ey)).
(2.23)
It can be shown [39, 38], that, for a coordinate system rotation of a, the polarization
field is now described by
Q' = Qcos(2a) + f/sin(2a),
(2.24)
U' = -Qsin(2a) + f/cos(2a)
(2.25)
We will show the connection between the Stokes Q and U parameters and the E and
B modes in the next section.
2.8
Harmonic analysis for temperature and
polarization anisotropies
The CMB is described as a field of temperature fluctuations on the sky, AT/T(n),
which can be expanded as a sum over the spherical harmonic coefficients ajm:
AT
— (n) = ^ a j m y , m ( n ) .
(2.26)
We say that the a^m are band-limited if they are zero for some £ > £max- The
temperature power spectrum is given by
C7 = (|aLl2>.
15
(2-27)
Similarly, Q and U can be expanded in the spin-2 basis:
(Q + U)(n)
= J2a^rn2Yem{n),
(2.28)
(Q~U)(h)
=
(2.29)
J]a_2,,m_2^m(n).
Now it is convenient to introduce the linear combinations of a 2 / m and a_2,em'afm
= -(a2,em + a-2,em)/2
(2.30)
afm
= i(a2,em - a_ 2 / m )/2
(2.31)
It follows that afm and afm can be constructed from the Q and U Stokes parameters,
afm ± iaBm = -J
dh±2Y;m(h)(Q ± iU).
(2.32)
For real space calculations it is convenient to define two scalar quantities E(h)
and B(h) corresponding to the divergence and curl parts of linear polarization
a,EmY,m(n), B(n) = £ af m Y, m (n).
E(n) = £
(2.33)
The existence of linear polarization leads to six possible cross power spectra. The
general form
{almat'm<)
= Ce
Su'Smm'
(2.34)
for X, X' e {T, E, B}, which gives:
CjT = (\aJJ),
B
2
Cf
Q
CfE = (KEm|2)
E
= (\afm\ ),
=
iaemaem)i
(2.35)
CJ = (ajma^)
(2.36)
Ce
(2.37)
= {aernae£)
Two of the spectra, CjB and CfB, are eliminated by parity.
2.9
Conclusions
I have reviewed some of the physics of the formation of CMB temperature and polarization anisotropics and have shown the connection between inflationary induced
primoridal gravitation waves and the unique B-mode polarization signal. In the next
16
chapter I will present the map-making problem and show how beam asymmetries and
foregrounds may induce systematics that will distort maps.
17
Chapter 3
Deconvolution Map-Making
3.1
Summary of chapter
We describe a new map-making code for cosmic microwave background (CMB) observations. It implements fast algorithms for convolution and transpose convolution
of two functions on the sphere [47]. Our code can account for arbitrary beam asymmetries and can be applied to any scanning strategy. We demonstrate the method
using simulated time-ordered data for three beam models and two scanning patterns,
including a coarsened version of the WMAP strategy. We quantitatively compare our
results with a standard map-making method and demonstrate that the true sky is
recovered with high accuracy using deconvolution map-making.1
3.2
Introduction
In Chapter 2, we reviewed the formation of the CMB and provided motivation for
seeking out precision measurements of the temperature and polarization anisotropies
in the CMB. We focused, in particular, on precison polarization measurements, and
the significance of a B-mode detection for inflationary physics. We will revisit polarization of the CMB and B-modes in Chapter 4. In this chapter, we will focus on some
of the challenges faced when taking measurements of the temperature anisotropies of
the CMB. We will discuss the problem of going from time-ordered data to maps of
the CMB and some standard procedures for this data-processing step. Finally, we
will present our deconvolution solution to this problem.
We describe the general map-making problem for satellite instruments measuring
the temperature anisotropies of the CMB in Section 3.3. A review of some standard
map-making approaches is given in Section 3.4. Section 3.5 details our deconvolution approach to the map-making problem. Some of the algorithmic details of the
deconvolution method are outlined in Section 3.6. In Section 3.7, we create a sim1
This chapter contains material previously published in [48].
18
pie binning map-maker to use as our standard of comparison. The test cases that
we examine are listed in Section 3.8, followed by the results in Section 3.9 We show
the results in Section 3.10 from an addtional test using point sources observed with
elliptical beams. We summarize the chapter in Section 3.11.
3.3
The map-making problem
Real microwave telescopes collect distorted information about the CMB anisotropics
due to asymmetries in the beam shape [49] and stray light from sources such as the
Galaxy [50, 51]. To correct for these systematic errors we must be able to remove the
detector response at all orientations of the telescope over the whole sky. In an optimal
treatment, this correction must be applied during the map-making step of the CMB
data analysis pipeline, before the angular power spectrum can be reconstructed. The
problem becomes increasingly important as new generations of CMB observations
probe for ever fainter signals in the CMB sky, and especially as we are preparing to
measure the polarization of the CMB with high sensitivity.
A microwave telescope scans the CMB sky according to some scanning strategy,
recording time-ordered scan data. The scanning of the satellite and the line-of-sight
of the detector aboard the satellite is depicted in Figure 3.1 for the PLANCK satellite
(here, we make brief mentions of the PLANCK satellite; we will be speaking in much
more detail about the PLANCK instrument in Chapter 4). The process by which the
data recorded by the detector is "wrapped" back on to the sphere to create an image
of the CMB sky is known as map-making.
The map-making process is made extremely difficult by the combination of two
major effects: distorted beam shapes and large foreground signals. In an ideal experiment, a telescope would observe the CMB sky away from undesired microwave
sources, with a perfect delta-function beam. In reality, while the gain of the optical
system does drop very rapidly away from the main beam line-of-sight, the signal from
the Galactic foreground can be 1000 times brighter than the fluctuations in the CMB.
This Galactic straylight enters the detectors through the far sidelobes, as depicted in
Figure 3.1 and also in Figure 3.2. It is therefore important to characterize the beam,
and use that information during map-making to deconvolve beam effects.
An ideal linearly polarized detector would have only a co-polar response and no
cross-polar response to signals from the orthogonal linear polarization. In reality,
optics will induce a cross-polar response in the detector. This has the effect of mixing
the Q and U signal, and therefore the mixing of the much larger E-mode into the
19
Figure 3.1: On the left is a figure showing the orbit of the PLANCK satellite in
the Earth-Sun system. The figure on the right shows the main beam of the telescope
sweeping out large circles on the sky, while Galactic straylight is entering the detectors
through the far sidelobc, seen at 90° from the main beam. Figure modified from [3].
Figure 3.2: A map of the spatial structure of the far sidelobes for one 30 GHz channel
of PLANCK. The main beam would sit in the region at the pole.
20
A side * por* measured
I
A
l
i
l
L_j
-2
1
1
1
0
1
i
i
I
2
Cress-polar response
i
i__j
I
4t
l__i
-4
i
i
I
-2
l
l
i
I
C-
i
i
i
I
2
t
i
i
I
4
Cross-e'evation (deg)
Figure 3.3: Measured WMAP focal plane co- and cross-polar beams, shown on the
left and right, respectively. The contours are spaced by 3 dB and the maximum value
of the gain in dB is given next to the individual beams. Figure reproduced from [4]
by permission of the AAS.
B-mode.
The co- and cross-polarization beam profiles measured for the WMAP
experiment are shown in Figure 3.3, clearly depicting a non-Gaussian beam response
and a nonzero cross-polar component.
3.3.1
Maximum-likelihood estimator of t h e t r u e sky
Now we will review the path from observations to maps. The TOD vector is effectively
a convolution of the true sky with a beam function. Mathematically, we represent
this as the matrix multiplication of the observation matrix A with the npj xe rvector
containing the true sky s returning a vector d, containing the UTOD samples of the
time-ordered data,
A s = d,
(3.1)
The matrix A can encode information about both the scanning strategy and the optics
of the scanning instrument, in which case it is called a convolution operator, or it
can simply be a pointing matrix in the case of non-deconvolution methods. Each row
of the pointing matrix has only one nonzero element corresponding to the direction
of the observed TOD. The least-squares estimate of the true sky, s, is given by the
normal equation
ATAs = ATd
21
(3.2)
where A T is the transpose convolution operator or transpose pointing matrix.
A realistic model of the data should also include noise
As + n = d,
(3.3)
where n is a time-ordered vector containing the noise. For the case where the noise
is stationary and uncorrelated in the time-domain, then Equation 3.2 is exact. For
more general noise cases, the normal equation is modified as
ATN-1As = ATN-M
(3.4)
where N = (nnT) is the noise correlation matrix.
3.4
Standard map-making methods
Standard map-making methods generally employ one of two basic algorithms for
making maps from TOD. The first is an implementation of the maximum-likelihood
solution (as described in the previous section) and the second is an approximate solver
known as the destriping method. This second method is motivated by the presence of
low-frequency noise in the data, which, if left untreated, leads to stripes in the final
map. Destriping requires no prior knowledge of the characteristics of the instrument
noise and is able to provide an estimate of the low-frequency part of the instrument
noise.
Although the map-making process is but one part of the data-analysis pipeline
for CMB experiments, it has now been recognized as a critical step which requires
efficient and exact computational methods [52, 53]. A search of the literature reveals at least six unique algorithms developed and tested for application to PLANCK
data by international groups: Madam [54], MapCUMBA [55], ROMA [56], Polar
[53], MADmap [53], and Springtide [53]. MapCUMBA, ROMA, and MADmap use
optimal algorithms, computing the minimum-variance map in the case of Gaussiandistributed, stationary detector noise. All three codes solve the maximum-likelihood
map-making equations using the iterative conjugate gradient method and FFT techniques. In order to produce accurate results, these codes require a robust estimate of
the noise power spectrum. Springtide and Madam are destripers and produce equivalent results to the optimal algorithms in the limit of short baseline length (baseline
parameters are a feature of destripers that allow for tuning the accuracy of the output map and computational resources required). Springtide achieves reduced memory
22
requirements by compressing the data into binned ring maps
Early work on the map-making problem have relied on the brute force method of
direct matrix inversion. However, current and future CMB experiments, like WMAP
and the PLANCK satellite, return enormous data sets that render the brute force
method intractable. More recent advancements include map-making methods applicable to the latest experiments; however, many treat the beam like a perfect deltafunction (e.g. [55, 57]) or assume a symmetric beam profile (e.g. [58]), and thereby
relegate the problem of treating a non-Gaussian radial response of the beam to subsequent stages in the data analysis [59, 60, 55, 61]. In this class, special techniques
exist to deal with differential measurements like that of the Differential Microwave
Radiometer (DMR) on the Cosmic Background Explorer (COBE) satellite [62] or
WMAP [63, 64]. A Fourier method has been developed [65] to perform deconvolution but only for nonrotating asymmetric beams, which renders it inapplicable to the
PLANCK data. Lastly, [66] present a method to remove the main beam distortion over
patches of the sky for asymmetric, rotating beams but operate in pixel-space which
is computationally more expensive than spherical-harmonic-space algorithms (for the
same level of accuracy [47, 67]) and too time-consuming for full-sky high resolution
maps such as those from PLANCK.
3.5
Deconvolution M a p - M a k i n g
We call our approach deconvolution map-making, a generalization of existing CMB
map-making techniques to solve the maximum likelihood map-making problem for
arbitrary beam shapes. For sufficiently high signal-to-noise data this technique allows
super-resolution imaging of the CMB from time-ordered scans. We implement our
method using the exact algorithms for the convolution and transpose convolution of
two arbitrary function on the sphere — in this case the sky and the beam — as detailed
by Wandelt and Gorski in [47]. These fast methods for convolution and transpose
convolution are efficient because they make use of the Fast Fourier Transform (FFT)
algorithm. They are guaranteed to work to numerical precision for band-limited
functions on the sphere.
A microwave telescope scans the CMB sky according to some scanning strategy,
effectively convolving the true sky with a beam function, and returns a vector d,
containing the UTOD samples of the TOD as described in Equation 3.1.
We call our matrix A the convolution operator; A encodes both the scanning
strategy and some level of information about the optics of the CMB instrument.
23
Each sample of the TOD is modeled as the scalar product of a row of the matrix A
with the sky s. Each of the UTOD rows of A contains a rotated map of the beam. In
a given row the beam rotation corresponds to the orientation of the antenna at the
point in time when the sample is taken. We will assume the beam shape and pointing
of the satellite to be known.
The observation matrix A generalizes the notion of the pointing matrix, which
is often used in expositions of map-making algorithms by including both optics and
scanning strategy. This generalization is necessary for any map-making method that
accounts for beam functions with azimuthal structure.
Equation 3.2 is exact if the noise is stationary and uncorrelated in the time-ordered
domain. For a more general noise covariance matrix in the time-ordered domain N,
the normal equation is modified to Equation 3.4. We proceed, considering only white
noise in this chapter; however, it is straightforward to generalize to non-white noise
as indicated in equation 3.4. Indeed, the matrix-vector operations required for this
generalization have already been implemented in publically available map-making
codes (e.g., MADmap 2 ).
3.5.1
Regularization technique
The least-squares estimate of the true sky, s, is given by Equation 3.2. The coefficient matrix in this system of equations, A T A , is a smoothing matrix and hence
ill-conditioned. Inverting it to solve Equation 3.2 therefore poses a problem.
We describe here a regularization technique for dealing with this problem. We split
off the ill-conditioned part of A by factoring the convolution operator into A = BG
where G is a simple Gaussian smoothing matrix, represented in harmonic-space by
Ge = exp I
1 ,
(3.5)
where a = FWHM/V81n2.
Substituting the factorization into Equation 3.2, we get
G T B T BGs
=
GTBTd
(3.6)
BTBx
=
BTd
(3.7)
where we are solving for x = Gs so as not to reconstruct the sky at higher resolution
2
http://crd.lbl.gov/~cmc/MADmap/doc/
24
than that of the instrument. The method that we will employ to solve for x (discussed
below) requires a symmetric coefficient matrix. We have constructed the symmetric
coefficient matrix by removing the left-most factor, G T , on both sides of Equation
3.7.
3.5.2
Fast convolution on the sphere
We now review the relevant formalism for general convolutions on the sphere and refer
the reader to [47] for the full details of the fast convolution and transpose convolution
of two functions in the spherical harmonic basis. The convolution of a band-limited
beam function 6(7) with the sky 3(7) is given by the following integral over all solid
angles
T ( $ 2 , e , $ ! ) = y'dfy[jD($ 2 ,e,$ 1 )&](7)M7)
(3-8)
where D is the operator of finite rotations 3 , Db is the rotated beam, and the asterisk
denotes complex conjucation. T($ 2 , @, $1) represents the beam-convolved signal for
each beam orientation (<3>i,0,$2)Following a geometrically-motivated argument, the evaluation of Equation 3.8 is
simplified by factorizing the rotation into two parts
Z>(<E>2, e , $1)
= D(4>E, eE, o)£>(0, e, u)
(3.9)
where the various angles (4>E,9E,<fi,9,u) are defined in [47]. Then, Equation 3.8 is
rewritten as
T(4>E,<I>,U)
= f dn^\b{(f>E,eE^)b{<i>,d,u))b]^Ys{fi
(3.10)
for 9 and 9E fixed. The three-dimensional Fourier transform of T(<fiE, 0, u>) is defined
as
1
/*27r
Tmm'm"
= J^y
J
d<f>Ed<f>du}T {<j>E, 0 ,
w
)e^B-im'*-im''W)
( 3 > n )
and we find that
Tmm'm"
, s l m ^ m m ' (^g)^m'm" (^)^m"-
= /
(3.12)
e
The D-function, also known as the Wigner D-matrix, is related to the function dl
3
,
Our Euler angle convention is denned as active right-hand rotations about the z, y, and z axes
by $ 2 , 0 , $ i , respectively
25
by
D^ m ,( a ,/?, 7 ) = e - i m a O ( ^ ) e - i m ' 7 .
(3-13)
The explicit expression for the d-function is given in [68] as
dmmW)-^
l
) (e + m-t)l(l-m'-t)\(t
+ m'-m)\
x(cosp/2)2i+m-m'-2t(sinP/2)2t+m'-m
{
j
(3.15)
where the sum is taken over all values of t which lead to non-negative factorials.
In practice, the d-matrices are more conveniently computed by making use of their
recursion properties, as we do in our algorithm [69].
Following an analogous derivation as for the convolution, the transpose convolution of T($ 2 , Q, $i) is given by
y*(7)
= y , d<Med$ 1 [D($ 2) e,$ 1 )&]( 7 y:r($ 2 ,e,$ 1 ).
(3.16)
In spherical harmonic space, this becomes
ylm = 2 ^ dmml(0E)dm,mll(6)b}m„Tmm>m».
(3.17)
m'm"
An important feature of our approach is that it economizes the computational
effort if the beam is nearly azimuthally symmetric. The parameter of the method
that sets the degree to which asymmetries of the beam are taken into account is
"i.max, the maximum m" in Equations 3.17 and 3.12. For m max = 0 we recover the
computational cost of simple spherical harmonics transforms, (^(^ax)- Since m max
is bounded from above by £max, the computational cost of the method never scales
worse than C(^max)- For a mildy elliptical beam, we anticipate that just including
the m max = 0 and m max = 2 terms will suffice, since the mmax = 1 term vanishes by
symmetry.
3.5.3
Binning and simulation
Now we will show how the fast convolution techniques of the previous section are
implemented in the deconvolution map-making algorithm. For clarity, we rewrite
Equation 3.7 in the compact spherical-harmonic basis (summing over repeated in-
26
dices)
BVM>mm'm"^'rnmlm"LM^LM
= B L ; M'mm'm" -^mrn'm">
(3.18)
where B T acting on Tmm>m» is given by Equation 3.17 and B acting on xLM is given
by Equation 3.12.
To make matters even more concrete, we now explicitly describe the steps required
to simulate time-ordered data d from a map — our "simulation" step. First, we
convolve the beam bem with the map aem to obtain Tmm>m» following Equation 3.12.
Then we inverse Fourier transform the Tmm/m» to get T($ 2 ) 0 , $i). Next, we must
account for the scan path ($2(2), ©(*)> $i(0)> where $ 2 and G specify the position on
the sphere and <&! specifies the orientation of the beam. This is achieved by extracting
those values in T($ 2 , ©, $1) which fall on the scan path of the instrument.
As a second example, we describe how to compute the right hand side of Equation
3.7. Start with the TOD d. For each sample in d, the scanning strategy specifies
the orientation ( $ 2 , 0 , $1). The sampled temperature is added into the element
of an initially empty array which is identical in size and shape to the array which
stored T($ 2 , ©, $i)- We have effectively binned the TOD d, according to the position
and orientation of the beam on the sky. Let us therefore refer to this operation as
"binning". In order to minimize discreteness effects due to the gridded representation
of T ( $ 2 , 0 , $ ! ) , more sophisticated interpolation techniques could be implemented.
Additionally, the resolution of the grid into which the data is binned may be increased.
These additional features will be revisited when we add in polarization measurements
in Chapter 4.
3.6
Solving the Deconvolution Equations
To obtain the optimal map estimate, we numerically solve the linear system of equations defined in Equation 3.7 for x^m. We have a choice between direct and iterative
solution methods. An iterative method is advantageous compared to a direct method
(such as Cholesky inversion) if the cost per iteration times the number of iterations
required to converge to sufficient accuracy is less than the cost of the direct method.
For the problem sizes of current and upcoming CMB missions, where the map contains a number of pixels npix ~ 106-107, direct solution methods would be prohibitive
for two reasons. First, the required number of floating point operations scales as r?pix.
Second, the amount of space required to store the coefficient matrix and its inverse
scales as riiix. Therefore, the direct solution exceeds the capabilities of modern super-
27
computers by several orders of magnitude. For the PLANCK mission, direct solution
would require of order 1021 floating point operations and hundreds of Terabytes of
random access memory.
We therefore advocate using an iterative technique, the Conjugate Gradient (CG)
method [70, 71, 72]. The CG method is well suited to this problem, as it solves
linear systems with a symmetric positive definite coefficient matrix and has advantageous convergence properties compared to other iterative methods such as the Jacobi
method [70, 71, 72]. In order to apply the CG method, we must be able to apply the
coefficient matrix on the left hand side of Equation 3.7 to our current guess for the
solution x. In order to do so, we simply perform the two operations of "simulation"
and "binning" in succession. The fast convolution and transpose convolution algorithms allow computing the action of the coefficient matrix on a map without ever
having to store the matrix coefficient in memory.
3.6.1
Preconditioning
It is desirable to minimize the number of iterations the CG method requires to converge to a given level of accuracy. This can be done by preconditioning the system
of equations. Preconditioning amounts to multiplying on both sides with an approximation of the inverse of the coefficient matrix and solving this modified system. As
long as the preconditioner is nonsingular the solution will be the same for the original
and the preconditioned systems, but for a well-chosen preconditioner the number of
iterations can be reduced significantly. A natural choice of the preconditioning matrix
that we used to obtain the results in this paper is the diagonal matrix [diag(A T A)] _1 ,
which, for a delta-function beam, is just the inverse of the number of observations
per pixel.
At every iteration we have an approximate solution x for Equation 3.7. We assess
convergence by computing the ratio of L2 norms
L2[BTd]
where Z^fx] = \ / | x • x|.
28
lJ
Uj
3.7
Standard map-maker for comparison
In order to compare our results from deconvolution map-making to more traditional
techniques we also implemented a map-making code that solves the normal equation
(Equation 3.2) assuming an azimuthally symmetric beam. In this implementation the
observation matrix A becomes the pointing matrix, containing only a single entry on
each row corresponding to the direction in which the main beam lobe is pointing at
the time of sampling. Standard map-making therefore reconstructs a map which is
smoothed by an effective beam whose shape varies as function of position on the map.
This variation depends on the scanning strategy. More precisely, at any given position
on the estimated map the effective beam shape depends on the various orientations
of the beam as it passed through this position during the scan.
For uncorrelated noise and an azimuthally symmetric beam the solution of the
normal equation is simple to compute: bin the TOD into discrete sky pixels, summing
over repeated hits, and dividing through by the number of hits per pixel. Numerical
implementations of this algorithm and its generalization to correlated noise have been
described in the literature [59, 61, 60, 55]. However, all of these treatments assume
azimuthally symmetric beams. For experiments with highly asymmetric beams, and
where contamination from the Galaxy is picked up in the sidelobes, we expect that
this method will not fare well against our deconvolution method that also removes
artifacts due to these optical systematics.
In Section 3.9 we compare our deconvolution method to a standard method. We
use the same TOD and scan path as for our deconvolution method. For the standard
method, the data is binned into pixelized maps, rather than into the T($2, 0, $i)
grid. Unless otherwise stated we use the HEALPix pixelization scheme [73] with
resolution parameter nside = 64. The angular scale of a pixel is therefore just under
1°. Recall that our regularization method returns a smoothed map with an effective,
azimuthally symmetric Gaussian beam. Thus, in order to compare the two methods
we must make a similar modification to our standard map-making. We read out the
resulting a^m of our standard map (using the HEALPix anaf a s t routine), after the
binning step, and modify them in the following way
where B( is the beam power spectrum and Ge is given in Equation 3.5.
29
3.8
Test Cases
In this section we detail our tests and comparisons of the deconvolution and standard
map-making methods. For the purposes of testing our method, we create several
mock beam models, b^m. We test three possible beam shapes that break azimuthal
symmetry at progressively stronger levels, two scanning patterns, and skies with and
without Galactic emission.
We test our algorithms on a simulated foreground- and Galaxy-free sky using a
standard ACDM power spectrum and simulated spherical harmonic multipoles a(m
up to £ = 128. We also use the first-year WMAP Ka-band (33 GHz) temperature
map as our true sky containing Galactic emission.
The first beam is a simple model of a sidelobe; it is composed of a Gaussian beam
of FWHM = 1800' rotated at 90° to another Gaussian beam of FWHM = 180'. Both
the main beam and the sidelobe are normalized such that they integrate to one. The
second beam models a (somewhat exaggerated) elliptical shape, composed of two
identical Gaussian beams with FWHM = 180' whose centers are on both sides of the
optical axis, separated by 180'. The third beam is composed of two identical Gaussian
beams (FWHM = 180') rotated at 140° from each other; we refer to this as the twobeam model. This case is motivated by the design of the WMAP satellite, which
makes a differential measurement from two horns separated by 140° [74]. We set the
asymmetry parameter m max for our three cases (sidelobe, elliptical, and two-beam)
to 8, 38, and 128, respectively.
Following Wandelt & Gorski (2001) [47], we first considered a basic scan path
(BSP) in which the beam scans the full sky on rings of constant longitude with no
rotation about its outward axis. To be clear, for the case of the sidelobe beam, the
smaller beam follows this ringed-scan while the larger beam remains fixed at the
equatorial longitude. Similarly, in the two-beam model, one beam follows the ringscan while the other rotates in smaller circles 140° away. The central lobe therefore
covers the whole sky, while the offset beam remains within a band of ±50° centered on
the Ecliptic plane. The elliptical beam simply follows the ring-scan, and is oriented
such that its long axis remains perpendicular to the lines of longitude.
A more realistic observational strategy has a beam that revisits locations on the
sky in different orientations. Therefore, we model the one-year WMAP scan path
followed by one horn. The WMAP scan strategy also covers the full sky and includes
a spin modulation of 0.464 revolutions per minute and a spin precession of one revolution per hour [74]. We used a scaled-down model of the WMAP scan in which
30
the spin modulation is 0.00232 revolutions per minute and a step size of about 46
seconds (roughly 562 samples per period). This produces a pattern very similar to the
spirograph-type pattern shown in Figure 4 of [74]. We refer to this as the WMAP-like
scan path (WSP). The WSP has about six times as many samples as the BSP. For
this strategy, the spirograph pattern is followed by the small beam of the sidelobe,
the elliptical beam, and both beams of the two-beam model. In the two-beam case,
both beams are offset from the spin axis of the satellite to mimic the WMAP scanning
geometry. However, it is not differential in nature, since both beams have positive
weight.
We test each beam (sidelobe, elliptical, and two-beam) with both scanning patterns (BSP and WSP) on a sky without Galactic emission. We refer to these as the
six main test cases.
In reality, CMB experiments will also pick up a signal from the Galaxy. We use
the first-year WMAP Ka-band temperature map, degraded to an nside of 64 and
smoothed with a Gaussian beam of FWHM = 180', as our model of the true sky
with Galactic emission. For our last test case, we convolve this sky with the sidelobe
beam.
For each test case, we assume that the beamshapes of the instrument are known
and we use the deconvolution method to deconvolve the map with the same beam
that was originally used to convolve our true sky. We attempt to recover features in
the map corresponding to the smallest scale features of our test beams. We therefore
set the width of our regularization kernel, represented by the matrix G in Equations
3.5 and 3.6, to FWHM = 180' in every case. We compare our map estimates to the
true sky, smoothed by the regularization kernel. When we refer to the "true" sky in
the following we mean this kernel-smoothed input sky.
3.9
Results and Discussion
We present the results of the deconvolution algorithm for the six main test cases in the
form of residual maps. We compare these results to the results from standard mapmaking by examining their power spectra and by calculating the root mean square
(RMS) difference between the estimated and true sky. For the tests that include the
Galactic signal we show the actual map estimates.
In Figure 3.4 we plot ratios of the power spectra (ACe/Ce) of the residual maps
(both standard and deconvolved) and the power spectrum of the input map. The
BSP (WSP) results are plotted in the left (right) column. The solid (dashed) lines
31
represent the relative difference in Ci between the deconvolved (standard) map and
true sky map. The standard map-making algorithm failed to give meaningful results
for the two-beam test. We therefore excluded this case from the plot.
For all cases we chose to present the results after a fixed number of iterations to
show the impact of scanning strategy and beam pattern on the condition number of
the map-making equations. We find that the deconvolution algorithm outperforms
standard map-making by orders of magnitude in accuracy.
For a fixed number of iterations, the BSP tests performed less well than the WSP
tests. The two-beam BSP and, to a lesser extent, the elliptical beam BSP test cases
have not converged to sufficient accuracy.
There are several possible causes for this behaviour. The BSP leads to an extremely non-uniform sky coverage. Also, the BSP visits each pixel in a narrow range
of beam orientations. Further, the number of sky samples is smaller for the BSP case
than for the WSP case (as noted in Section 3.8). All of these aspects can contribute
to increasing the condition number of the normal equation, which in turn leads to
smaller error decay per iteration of the preconditioned CG solver.
In Table 1 we summarize the RMS difference between the reconstructed and true
sky. The RMS values are computed using the standard deviations the residual and
true maps:
stdev(residual map)
(3.21)
RMS =
stdev(true map)
where
residual map = estimated map — true map.
(3.22)
The RMS values reflect the trends seen in the spectra in Figure 3.4.
Beam
Sidelobe
Elliptical
Two-beam
BSP
Standard
0.257467
0.186262
N/A
Deconvolved
0.000216367
0.0213657
0.102778
WSP
Standard
0.178828
0.129715
N/A
Deconvolved
3.13741e-07
2.92714e-05
1.08579e-06
Table 3.1: Fractional RMS error for each of the six main test cases.
The residual maps are shown in Figure 3.5. In order that the scale of the axes on
the residual maps are meaningful, we also show the true sky map.
Achieving a stably converging iterative solution method for the deconvolution
problem is a success of our regularization technique. The convergence of our iterative
solver as function of iteration number is plotted in Figure 3.6.
32
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Figure 3.5: Residuals after the first 100 iterations are shown in the first three rows.
The figures on the left are for the basic scan path, and the ones on the right are for
the WMAP-like scan path. First, second, and third rows correspond to the sidelobe,
elliptical and two-beam beams, respectively. The true sky is shown in the fourth row.
Note that solutions of the two-beam BSP and, to a lesser extent, the elliptical beam
BS.P test cases have not converged to sufficient accuracy. We chose to present the
results for all cases after a fixed uumber of iterations to show the impact of scanning
strategy and beam pattern on the condition number of the map-making equations.
34
BSP
1CT
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1 I' I I
irr
~1
1—I—I I I I I I
1
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10
number of iterations
100
100
Figure 3.6: Convergence rates of the preconditioned conjugate gradient solver for
each test case. The left panel refers to the basic scan path and the right panel to the
WMAP-like scan path. The solid lines correspond to the sidelobe beam, dotted lines
to the elliptical beam, and dot-dashed to the two-beam model.
In order to be able to compare the performance of our method for different beam
patterns and scanning strategies we make the deliberate choice of limiting the number
of iterations to 100 and comparing the best results obtained up to this point. Since
our error estimate continues to drop stably (except in two cases, where we reach the
single precision numerical accuracy floor after ~ 25 and ~ 70 iterations) it is clear
that the accuracy of the reconstruction can be improved by allowing the system to
iterate further, or by choosing a more sophisticated pre-conditioner.
Our final test case consists of a model sky with Galaxy emission convolved with
the sidelobe beam over the WSP. We present the output maps of both the standard
and deconvolution methods in Figure 3.7, where the maps are shown in Ecliptic
coordinates. One can see that the standard map contains a distorted image of the
Galaxy and that the deconvolved map is virtually identical to the true map.
35
True sky with Galaxy
Standard map
Dcconvolution method
Figure 3.7: Deconvolving the effects of a large sidelobe in simulated observations of
the WMAP Ka band niap. using the coarsened WMAP scanning strategy described
in the text. The top map is the input sky map, the middle map is the standard
map-making result, and the bottom map is the deconvolved result.
36
3.10
Point source tests with elliptical beams
At small angular scales, extragalactic radio sources are an important contamination
in CMB maps. The characterization and removal of all foregrounds, including point
sources, is a critical step in CMB data processing. In this section we will show that
point sources can be used as tools to see how the effect of the beam manifests itself
in the map. The reconstructed image of a point source in a map essentially shows an
image of the effective beam at that location in the sky. We will revisit point sources in
Chapter 6 when we examine the variability of identified radio sources in the WMAP
data.
In this test, we placed 13 artificial point sources on a grid in one quadrant of the
sky as shown in Figure 3.8. We then scanned this simulated sky using the WSP and
elliptical beam, described in Section 3.8, to generate a set of TOD. Deconvolution
map-making was then run on the simulated TOD to generate skies with reconstructed
point sources. We were interested to see how the reconstruction of the point sources
depended on their positions in the sky. For scans like the WSP, some regions of the sky
are revisited by the scan strategy with the beam having a wider range of orientations
than other regions. This has the effect of symmetrizing the effective beam profile over
those region. We were also interested to examine the effect of reconstruction on the
asymmetry parameter m.
The results from our first, low-resolution run at nside = 64 are shown in Figure
3.8. It can be seen that the overall residual in the m = 4 case is roughly five times
smaller than the m = 0 case. We also find that point sources in regions of the sky
which are observed by the beam at a wide range of orientations are reconstructed
with smaller errors.
We then increased our resolution to nside = 1024 and examined the reconstruction
of an individual point source for m = 0 and m = 4. The results are shown in Figure
3.9.
3.11
Conclusions
In this chapter, we have presented a deconvolution map-making method for temperature data from scanning CMB telescopes. Our method removes artifacts due to
beam asymmetries and far sidelobes. We compare our technique with the standard
map-making method and demonstrate that the true sky is recovered with greatly enhanced accuracy via our deconvolution method. Deconvolution map-making recovers
37
Origin**] map
r
•
•
•
•*
,1'
•
•0
•'.
\
Cs
Diffrrence map, m - 0
Difference map, jn=4
Figure 3.8: The top plot shows the distribution of point sources in the upper left
quadrant of the CMB sky (note that the upper end of the color scale has been cut-off
at 93.6/i/v'; the original scale terminates at 374/xK). The middle and bottom plots
show the difference maps between the input and output skies for m = 0 (middle)
and m — 4 (bottom). It can be seen that the overall residual in the m = 4 case is
roughly live times smaller than the ?n = 0 case. We also find that point sources in
regions of the sky that are observed by the beam at a wide range of orientations are
reconstructed with smaller errors.
38
fT
Figure 3.9: Examing the effect of beam ellipticity and subsequent deconvolution on
point sources in the CMB. The actual point source (top) is scanned with an elliptical beam and reconstucted using deconvolution map-making with the asymmetry
parameter set to m = 0 (bottom left) and m = 4 (bottom right).
39
features of the CMB sky on the smallest scale of the beam, thereby achieving a form
of super-resolution imaging. This extracts more of the information content in CMB
data sets.
One of the key difficulties encountered in deconvolution problems is that the systems of linear equations we need to solve are very nearly singular. We solve this
problem by introducing a regularization method that allows us to solve the systems
stably and recover maps at a uniform resolution and with an effective beam that is
azimuthally symmetric and has a Gaussian profile.
We tested the convergence speed of two particular scanning strategies and found
that the WMAP-like scan is superior to the basic scan in both rate of convergence
and true-sky recovery. We hypothesize that this is due to the nature of the BSP,
where the poles are the location of the only beam crossings and thus receive many
more hits than the rest of the sky. In addition, our implementation of the BSP had
a smaller number of samples overall than our implementation of the WSP.
We have also shown the relevance of this algorithm to the WMAP mission by
demonstrating its operation using a WMAP-like scanning strategy and a two-beam
model, which, while not differential, resembles the telescope orientations of the WMAP
spacecraft. Our results underline the qualities of the WMAP scanning strategy compared to a BSP strategy for deconvolution map-making.
Low-resolution and high-resolution tests with point sources and elliptical beams
were performed. The results from these tests are another important way to examine
the effect of beams on the maps and the effective symmetrization of asymmetric beams
due to the scanning strategy.
In order to decouple from issues that are not directly related to the optical performance of CMB instruments, we did not consider the effects of noise in our simulations.
For a realistic assessment of the performance of our methods on real data this needs to
be added. In particular, the choice of scale for the regularization kernel will depend on
weighing the benefits of increased resolution against increased high-frequency. Noise
issues will be revisited in Chapter 5.
In the next chapter, we will extend our description of the deconvolution mapmaking method to include both temperature and polarization measurements. We
will compare our results from polarized map-making with those from the PLANCK
algorithm development group, using actual simulated PLANCK data.
40
Chapter 4
Polarized map-making for Planck
4.1
Summary of Chapter
We describe a maximum likelihood, regularized beam deconvolution map-making algorithm for data from high resolution, polarization sensitive instruments, such as the
PLANCK satellite. The resulting algorithm, which we call PReBeaM, is pixel-free and
solves for the map directly in spherical harmonic space, avoiding pixelization artifacts. While Fourier methods like ours are expected to work best when applied to
smooth, large-scale asymmetric beam systematics (such as far-side lobe effects), we
show that our m-truncated spherical harmonic representation of the beam results in
negligible reconstruction error — even for m as small as 4 for a polarized elliptically
asymmetric beam. We describe a hybrid OpenMP/MPI parallelization scheme which
allows us to store and manipulate the time-ordered data from satellite instruments
with a typical full-sky scanning strategy. Finally, we apply our technique to noisy
data and show that it succeeds in removing visible power spectrum artifacts without
generating excess noise on small scales.1
4.2
Introduction
As discussed in Chapter 2, the inflationary model of the universe can be tested by
searching for B-mode polarization in the CMB radiation. The detectors on PLANCK
will scan the full sky measuring the polarization of the CMB. The raw data will be
processed into maps and then into power spectra. If present, the tiny primordial
B-mode signal will be polluted by foreground contamination that leaks into the detector through the far sidelobes, and it will be mixed with the much larger E-mode
signal by asymmetric beams with cross-polar responses. We continue work on the
crucial step of processing data into maps, where we will use the deconvolution tool to
remove systematic effects due to beam asymmetries. We will model the full PLANCK
1
This chapter contains material previously published in [75].
41
simulation for a single channel with the largest beam asymmetries and investigate
the effect on map-reconstruction due to beams. In the next chapter, we will examine
the effect of noise and diffuse foregrounds on map reconstruction.
In Section 4.3 we review the PLANCK satellite mission and science goals. In
Section 4.4 we describe the deconvolution map-making algorithm for PReBeaM. Some
standard methods for polarized map-making are described in Section 4.5, and we also
detail the routine that we use as a baseline for comparison for the tests done in this
chapter. Fast convolution for polarimetry measurements are described in Section
4.6. Special features that were implemented in our code in order to boost accuracy
are explained in Section 4.7. The simulated data and beams are detailed in Section
4.8. We present results in Section 4.9 showing the effectiveness of PReBeaM in
removing systematic effects due to beam asymmetry and we discuss computational
considerations. We finish the chapter with our conclusions from this study in Section
4.10.
4.3
Planck
The PLANCK 2 satellite will be launched in May 2009 and will measure the CMB
anisotropics over the full sky in nine frequency bands from 30 GHz to 857 GHz.
The resolution of the instrument in CMB-dominated bands will be 5 — 15' and the
sensitivity will be A T / T ~ 2xl0~ 6 [76]. The PLANCK satellite is designed to extract
essentially all of the information in the primordial temperature anisotropies, and to
measure the polarization anisotropies to high accuracy for 2 < £ < 2500.
The major deliverables of PLANCK include all-sky CMB maps at each frequency
and all-sky foreground component maps of Galactic synchrotron, free-free, and dust
emission. In addition, PLANCK aims to determine the parameters that describe the
geometry and composition of our universe to about one percent. The scientific performance of PLANCK depends, in part, on the behavior of systematic effects which
may distort the signal.
One of the most exciting prospects for the upcoming PLANCK satellite is its capability to measure the polarization anisotropies of the CMB over the entire sky in nine
frequency channels. The potential rewards from these measurements are many, and
include tighter constraints on cosmological parameters, determination of the reionization history of the universe, and detection of signatures left by primordial gravitational
waves generated during inflation [76].
2
http://astro.estec.esa.nl/Planck
42
4.3.1
The map-making challenge for Planck
Measurement of the CMB polarization signal presents a great experimental challenge
as it is an order of magnitude smaller than the temperature signal, and it is especially
susceptible to distortions due to optical systematics and foreground contaminants. If
left untreated, leakage from the much stronger temperature signal will contaminate
the polarization maps. Maps and spectra will also suffer from leakage from E-mode
polarization to B-mode polarization, jeopardizing the potential detection of inflationary B-modes. At the resolution and sensitivity of the next generation of experiments,
including the PLANCK mission, studies of primordial non-Gaussianity may also be
sensitive to beam-induced systematics.
A primary objective of PLANCK is to produce all-sky CMB maps at each frequency.
The process by which the satellite's time-ordered data (TOD) is wrapped back on
to the sphere to create an image is known as map-making. The map-making process
becomes difficult due to a number of challenges: distortions in the beam, foreground
contamination through far-side lobes, size of the data, and correlated noise effects. It
is of critical importance to fully characterize the beam, and use this information during
map-making to deconvolve beam effects. We will be most interested in studying the
map-making problem for the 30 GHz channel of PLANCK, as this will be the most
asymmetric beam.
Within the PLANCK collaboration, the CTP working group has developed five
map-making methods and compared their results using the simulated 30 GHz data in
what is known as the Trieste paper [5]. The Trieste paper assessed the impact of beam
asymmetries on the PLANCK spectra without attempting to treat the problem of beam
asymmetry at the map-making level (an angular power spectrum correction method
was developed based on simplifying assumptions). We will revisit some standard
map-making methods in Section 4.5.
4.4
P R e B e a M Method
We have previously described, in Chapter 3, a powerful map-making algorithm that
implements the beam deconvolution technique for temperature measurements. In this
section, we will extend that description to include polarization measurements. We
refer to this new technique as PReBeaM: Polarized Regularized Beam deconvolution
Map-making. While we focus on reconstructing the map with a uniform effective
beam and realize corrections to the power spectrum as a consequence, other works
43
by [77] and [78] have focused on deriving corrections to the power spectrum due
to asymmetric (noncircular) beam effects. First, we review the standard setup to
the map-making problem for a solution of the least-squares (or maximum-likelihood)
type.
The TOD generated by a detector is effectively a convolution of the true CMB
sky with a beam function. If we consider the sky as a pixelized vector, it will have
length nPiXei x npoi where npoi = 3 for the / (total intensity), Q, and U Stokes components. The nToD-length TOD vector d is the result of a matrix multiplication of
the observation matrix A with the sky s plus the noise n
d = As + n.
(4.1)
In our implementation of the maximum-likelihood solution, we refer to A as the
convolution operator. A encodes information about both the scanning strategy and
the optics of the scanning instrument. The least-squares estimate of the true sky, s,
is given by the normal equation
ATAs = ATd
(4.2)
where A T is the transpose convolution operator. Equation 4.2 holds if the noise
is stationary and uncorrelated in the time-ordered domain. The generalization to
nonwhite noise is as follows
ATN"1As = A ^ - M
(4.3)
where N is a noise covariance matrix. In this chapter we consider CMB only and we
will examine CMB plus white noise in chapter 5.
We modify the normal equation by introducing a regularization technique in order
cope with the ill-conditioned nature of the coefficient matrix A T A. We split off the
ill-conditioned part of A by factoring it into two parts: A = BG. The factor G is
what we refer to as the regularizer in PReBeaM. In our study, we choose G to be a
Gaussian smoothing matrix, defined in harmonic space as
O- = exp ( ^ | ± I > )
or
= exp
(-w+D-4))
44
m
where a = FWHM/\/81n2. The superscript / refers to the intensity, and G and C
refer to the gradient and curl components in the typical linear polarization decomposition.
In general, the width of the regularizer can be set so as to reconstruct the sky at
any target resolution. In practice, one would not want to choose a regularizer that is
smaller than or close to the size of the sample spacing as this would likely introduce
sampling effects into the map. We use the width of the angle-averaged detector beam
and suggest this as a rule-of-thumb for choosing the regularizer's width. The choice
of a regularizer will certainly affect the noise properties of the reconstructed map and
power spectrum as it acts to smooth away noise on small scales. PreBeaM estimates
the beam-corrected and regularized atm and the map is simply a visualization of
this anm solution. If the sampling points were distributed in a way that aliased two
different a^m, then the reconstruction would be singular. Our regularization scheme
ensures that we are robust to the details of sampling as long as the sample space is
finer than the scale set by the width of the regularizer.
Our modified normal equation becomes
BTBx = BTd
(4.5)
where we are solving for x = Gs. In this way, we are attempting to construct an
image of the sky at the angular resolution of the chosen regularizer.
A complete characterization of the beam includes both the main beam and the farside lobes. Sidelobes are located as far away as 90° from the main focal plane beam,
and therefore require a large mmax parameter for a complete harmonic description. In
[48] we demonstrated the full potential of our method using far-side lobes and maps
with foreground signals. Here, we show the usefulness of PReBeaM for deconvolving
main-beam distortions. In fact, we find that it makes sense to use PReBeaM for main
beam effects since only a small mmax parameter is needed to capture the azimuthal
structure of the main beam. In this way, we profit from the computational advantage
of our method in the case of small mmax, allowing for the unified treatment of main
beam and side lobe effects.
45
4.5
Comparison to standard map-making and
other map-making methods
In standard pixel-based optimal mapmaking (see, for example, [63, 79, 55, 58, 56]), in
which one assumes that the observing beam is spherically symmetric, A is a sparsely
filled pointing matrix. In the case of a single dish experiment, each row of A contains
only three nonzero elements. The deconvolution map-making approach does not
assume spherically symmetric beams, instead allowing for arbitrary beam shapes. We
achieve this added complexity primarily by solving the normal equation in spherical
harmonic space in order to make use of fast and exact algorithms for the convolution
and transpose convolution of two arbitrary functions on the sphere [47, 80]. These
algorithms are described in abbreviated form in Section 4.6.
In addition to PReBeaM, another deconvolution map-making technique for PLANCK
has been established by Harrison et al. [81]. Both methods allow for arbitrary beam
shapes, and, in both cases, the asymmetry of the beam is parameterized by an asymmetry parameter mmax that can vary between 0 and £max- Our method scales computationally as 0 ( ^ a a . m m a x ) ; this is advantageous when large gains in accuracy can be
achieved with small increases in mmax. In contrast, the Harrison approximate method
scales as 0 ( ^ a 2 . ) thereby incurring a fixed computational expense for arbitrarily large
mrnax and effectively setting a limit to the maximum £ at which the analysis can be
done. Excluding TOD operations we achieve a speed-up of imax/f^max, which is of
O(l02) for the case we examine herein. The Harrison method takes advantage of the
PLANCK scanning strategy to condense the full TOD into phase-binned rings, thereby
achieving a significant reduction in processing time.
A secondary advantage of operating entirely in harmonic space is that artifacts
due to pixelization (such as uneven sampling of the pixel) are completely avoided.
Issues with undersampled pixels can be problematic for a pixel-based method — as in
the case of pixels with less than three non-degenerate observations — and can result
in pixels which are excluded from the map-making completely. Our approach is less
sensitive to degenerate pixels resulting from poor sampling and does not result in
excised pixels. The assumption of a band-limited signal, justified by finite resolution,
regularizes the solution for degenerate pixels based on neighboring observations in
the time stream. We demonstrate the difference between a pixel and harmonic-based
method by comparing a PReBeaM temperature map at the Healpix [73] resolution
of nside 1024 with a binned noiseless temperature map at the same resolution. A
binning of the TOD from the 30GHz channel into an nside 1024 pixel map results
46
Binned Temperature Map
PReBeaM Temperature Map
*H
#
(8.0, 23.0) Ecliptic
(6.0, 23.0) Ecliptic
Figure 4.1: Comparison of one segment of the sky from a binned noiseless map (left)
and the PReBeaM temperature map (right), both at a Healpix resolution of nside
1024. At this resolution, the binned map contains a number of unobserved pixels,
of which three are visible in this frame. The PReBeaM map contains no unobserved
pixels.
in a number of unobserved pixels. The analogous PReBeaM map at nside 1024
contains no unobserved pixels. Figure 4.1 shows identical regions of the sky from the
binned noiseless map and from PReBeaM. The difference between pixel and harmonic
methods is even more distinct if we consider polarization maps. We quote results from
[82] in which polarization maps were made from the 30GHz data at nside 1024. In
section 3.7 of [82] it is noted that the nside 1024 binned polarization maps have
912,968 unobserved pixels (where they reject not only missed pixels, but also those
with a condition number less than 0.01 — which indicates a degenerate pixel). In
contrast, a PReBeaM polarization map at the same resolution has no missing pixels.
While it is true that our approach would suffer in the case of gross undersampling or
huge chunks of missing sky, we are not expecting to deal with these sorts of issues
with CMB satellite missions such as PLANCK.
Our method does have its own unique sampling issues that can be understood by
examining the problem in harmonic space rather than in pixel space. The beam a^m
used to simulate the TOD set a natural band limit where they approach zero. We
choose an £max cutoff at which to reconstruct the sky aem. If we choose £max much less
than the beam band limit then we may introduce small-scale features such as ringing
47
(of particular importance when we include foregrounds such as point sources). If we
choose £max much larger than the £ where the beam effectively bandlimits the signal,
then all agm for £ higher than that are noise dominated, and the regularized solution
will therefore be driven to 0 for those large £.
4.6
Fast all-sky convolution for polarimetry
measurements
For a full presentation of the formalism for convolution of an instrument beam with
a sky signal, the reader is referred to [80].
In compact spherical harmonic basis, Equation (4.2) is written as
•^•L'M'mm'm"J^mm'm"LMsLM
= AL,M'mm'm"^mm'm"
(4-6)
where SLM is the spherical harmonic representation of the sky and Tmmim" is defined as
the result of a convolution of a band-limited function b with the sky s. The PLANCK
Level-S software [83] nomenclature refers to T mm ' m " as a ring set. This is written in
harmonic space as
T
-X^t^J
h1* -L e G hG*
S
•l-mm'm" — / , l'2p f a % ' "•" btm°e.M'
t
+s?mb?M>)dernM(9E)deMM,(8)
(4.7)
where (6E, 0) are fixed parameters that define the scanning geometry.
In Equation 4.7, demM(6E) and deMM,(8) are related to the Wigner D-matrices by
BL'mfa 0, VO = e - ^ m ^ e - * " * -
(4-8)
Analogously, the transpose convolution in harmonic space is given by
Vem = 2 ^ dmm/(0£)dm,m,,(0)6£7£,,Tmm/m//
m'm"
where P = I,G, C.
48
(4.9)
4.7
PReBeaM Implementation
Now we outline the algorithmic steps taken to make a map from a TOD vector by
PReBeaM.
First we construct the right-hand side of Equation 4.6 in two steps: converting
TOD to an Tmm'm" array and applying A T . Tmm'm" is constructed by transpose
interpolating the TOD vector d. The transpose interpolation of the TOD vector onto
the Tmm'm>i grid is akin to a binning step, where each element of the TOD is mapped,
via interpolation weights, to several elements of the Tmm'm" cube according to the
orientation and position of that data point in the scanning strategy. The interpolation
scheme is described in greater detail in Section 4.7.1. Next, we transpose convolve
the beam coefficients b(m with Tmm'm" according to Equation 4.9.
Once the right-hand side has been computed, we use the conjugate gradient iterative method to solve equation (4.2). With each iteration, the coefficient matrix A T A
is applied using the following procedure:
1. Apply the convolution operator, A, to project the sky a^m onto the convolution
grid
Tmm'm".
2. Inverse Fourier transform over the first two indices of Tmm'm" to get T<s>2,Q,m» (we
omit the transform over m" as it is incorporated in the interpolation scheme).
3. Forward interpolate from T$2)eim» to a TOD vector.
4. Transpose interpolate from the TOD vector to a new ring set T$
5. Fourier transform over the first two indices of T$2 e r n „ to get
e
„.
Tmm,m„.
6. Apply the transpose convolution operator, A T , to project the ring set Tmrn,m„
back into a new sky a^m vector.
4.7.1
Polynomial Interpolation and Zero-Padding
PReBeaM uses the same polynomial interpolation as implemented in the Level-S
software used to generate the simulation TODs as described in [83]. The objective
of forward interpolation is to construct a TOD element at a particular co-latitude,
longitude, and beam orientation using several elements of the ring set T and their corresponding weights. Transpose interpolation operates in exactly the opposite manner
as the forward interpolation: distributing a single element in the TOD to multiple
49
elements of the ring set. This is done using the same weights calculated for the forward interpolation. The entire operation of interpolation and transpose interpolation
from ring set to TOD and back again is depicted in Figure 4.2
Ring Set
•
/
/
/
_ ^ A_'
/
V
/ / ,
/ / / // /
/
/ / / / / ,
/ /
/
/
Ring Set
7
Transpose
Interpolation
•1
•/ ;j~~ \
-
y
mr >
\
\
\
\
vs \ \ \ \ \ \ \ \ \
\ \ \ \ \ \
Forward
J9
\
s
:i••
k,
\
\
k
/
i
>i
i
Figure 4.2: Forward interpolation from ring set to TOD element and transpose interpolation from TOD element to ring set.
PReBeaM also includes the option to zero-pad during the Fast Fourier transform
(FFT) and inverse FFT steps. This means that the working array (either Tmm'm»
or T$2ie,m») is enlarged and padded with zeros out to H.max,pad > (-max- This has the
effect of decreasing the sampling interval. We found that the combined effects of
small-order polynomial interpolation (order 1 or 3) and zero-padding of 2 x £max or
4 x £max dramatically reduced the residuals in our maps.
4.7.2
Parallelization Description
PReBeaM employs a hierarchical parallelization scheme using both shared-memory
(OpenMP) and distributed-memory (MPI) types of parallelization. The map-making
was performed on the NERSC computer Bassi3. Bassi processors are distributed
among compute nodes, with eight processors per node. OpenMP tasks occur within
a node and MPI tasks occur between nodes.
We show a diagram of our hybrid parallelization scheme in Figure 4.3. The full
TOD and pointings are divided equally between the nodes for input and storage of
pointings. Within an iteration loop, four head nodes are designated to perform the
convolutions. All other tasks are shared by all nodes, including the head nodes. Since
convolution is minimally time consuming, we are not concerned with the small amount
of time the nonhead nodes spend idling while convolution is performed. The number
of head nodes needed corresponds to the number of detectors; for the low-frequency
3
https://www.nersc.gov/nusers/systems/bassi/
50
instrument (LFI) 30GHz channel, this is four. Each of these four nodes performs the
convolution of the sky with one of the four detectors. The resulting arrays are then
distributed to all nodes for interpolation over the segment of data stored locally and
then gathered back onto the designated nodes for the transpose convolution. Finally,
the aim are summed, using the MPI task mpi_reduce, into a single aem on a single
node; this is the new estimate for the sky vector.
This particular scheme was devised so that the four distinct convolutions that
must occur (one sky with four different beams) can take place simultaneously, while
the pointings are distributed among as many nodes as possible for maximum speed
in interpolation. Both convolution and interpolation and their transpose operations
make use of all processors on a node by using OpenMP directives.
4.8
Simulations and Beams
The simulated PLANCK data on which PReBeaM was run were generated by the
PLANCK CTP working group for the study of the performance and accuracy of five
map-making codes summarized in the Trieste paper [5]. The 30 GHz channel is
composed of two horns, labelled "27" and "28", each of which is comprised of two
detectors tuned to orthogonal linear polarizations as shown in Figure 4.4. PLANCK
will spin at a rate of approximately 1 rpm, with an angle between the spin axis and
the optical axis of ~ 85°. We used the cycloidal scan strategy in which the spin axis
follows a circular path with a period of six months, and the angle between the spin
axis and the anti-Sun direction is 7 ? 5. TODs were generated for 366 days for the
four 30 GHz LFI detectors. At a sampling frequency of 32.5 Hz, this corresponds to
1.028 x 109 samples per detector, for a total of over 65 Gb of data and pointings. The
simulated data also included the effects of variable spin velocity and nutation (the
option to include the effects of a finite sampling period was not included).
The data were simulated with elliptical beams having a geometric mean FWHM
of 32' 2352 and 32' 1377, and an ellipticity (maximum FWHM divided by minimum
FWHM) of 1.3562 and 1.3929 for each pair of horns. The full details of the four
detectors which make up the 30 GHz channel are given in Table 4.8 and contour plots
of the main beam co-polar components are shown in Figure 4.5. The contour beams
are plotted in a uv-plane (described in [5]) where u = sin(9)cos(p), v = sin(6)sin((p)
and 9 and ip are the polar and azimuth angles of the spherical polar coordinates
of the beam coordinate system. The quantity ifjpoi is the angle of the polarization
measurement, defined with the beam in a fiducial orientation. The orientation of the
51
TOD
convolution
with beam 1
convolution
with beam 2
mpi broadcast
of ring array
interpolation and
transpose
interpolation on
TOD segment
interpolation and
transpose
interpolation on
TOD segment
mpi reduction
of ring array
transpose
convolution
with beam 2
transpose
convolution
with beam 1
mpi reduction
of aim
Figure 4.3: Depiction of hybrid parallelization scheme used in PReBeaM. Rectangles
represent work done on a node, ellipses represent data products, and arrows represent
transfer of data. The work done within a node (convolution, interpolation and their
transpose operations) is parallelized using OpenMP. This figure shows a slice of two
head nodes, however, the algorithm may operate on many more nodes.
52
LFI Main B e a m s
,
,
,
.
!
,
,
,
1
25
0.05
i
i
*"-"- J 22
j
S
C
A
N
-24
D
I
R
E
C
T
I
O
N
V
^ ^ 2 0
18
1
»
1
iK
"I
U
•—|
0.00
^}
i
-
i
i
1
_28
1
0.05
1'
26
i
i
i
1
i
-0.05
i
i
i
i
0.00
i
J
...
i
i
, ,L
1
0.05
Figure 4.4: Plot showing the footprint of the LFI main beams on PLANCK as seen
along the optical axis looking towards the satellite. The pair of horns comprising the
30 GHz channel are labeled 27 and 28 in the plot. The scan direction and polarization
orientations for the co-polar beams are also depicted. Figure reproduced from [5] with
permission.
53
Figure 4.5: Contour plots in the uu-plane of the PLANCK 30 GHz main beam detectors. This coordinate system permits the beam to be mapped from a spherical
surface to a plane. From left to right, the beams are: LFI-27a, LFI-27b, LFI-28a,
and LFI-28b. Figure reproduced from [5] with permission.
beam major axis is given by ipeu. The widths and orientations of the beams were
different; this was referred to as beam mismatch in the Trieste paper. In spherical
harmonic space, the simulation beams were described up to a beam mmax of 14. The
same beams were used in PReBeaM to solve for the map, although we allowed the
beam asymmetry parameter mmax to vary.
Detector
27a
27b
28a
28b
FWHM
32f2352
32^1377
32^2352
32:1377
Ellipticity
1.3562
1.3929
1.3562
1.3929
fj'pol
tpell
0?2
89?9
-0?2
-89?9
101?68
100?89
78? 32
79? 11
Table 4.1: The four detectors of the 30GHz PLANCK channel and their FWHM,
ellipticity, angle of polarization measurement Vv> a n d orientation of beam major
axis ijjeii.
4.9
Results and Discussion
For our tests, we make temperature and polarization maps from simulated one-year
observations of the four 30 GHz detectors of the PLANCK LFI. We examine the case of
a CMB signal only and relegate the case of CMB plus uncorrected (white) noise and
foregrounds to Chapter 5. The data from the 30 GHz channel was an optimal choice
for this analysis. These receivers sit farthest away from the center of the focal plane
and therefore have the strongest main beam ellipticities of any PLANCK channel,
while the low sampling rate and resolution minimize the data volume.
PReBeaM operates entirely in harmonic space, solving for and producing as output, a set of a(m. For visualization purposes, maps were made from a^m's out to
54
Pmax 512 and at the Healpix resolution of nside 512 (~ 7" pixel size). Most of the
results presented here were attained with an FFT zero-padding of factor 4, an interpolation order of 3, and an asymmetry parameter of m max = 4 (we will note where
the parameters differ from this). To compare with the input signal, a reference map
representing the true sky was created by smoothing the input a^m with a Gaussian
beam of FWHM = 32/1865. Similarly, our regularizer G (in Equation 4.4) was set
to have a FWHM of 32/1865 to match this smoothing. As noted in Section 4.8, the
same data we use here have been processed by five map-making codes in [5]. We have
chosen to compare our results with the analogous results from Springtide, one of the
codes in this study. Springtide was chosen, out of the five codes, because it is the
map-making code installed in and used by the PLANCK Data Processing Centers for
the HFI and LFI instruments. It is sufficient to compare with Springtide only as no
significant differences in accuracy were found between codes (with similar baselines
and in the absence of noise; [5]). Springtide is a destriping algorithm designed to
remove low-frequency correlated noise from the TOD by fitting for, and subtracting,
offsets. While Springtide does fit for offsets even if the noise is absent or uncorrelated,
the results in this case are not significantly different from that which one would derive
from a simple binning algorithm. Because we are using Springtide to represent all
non-beam-deconvolution methods we will refer to the Springtide maps as the binned
maps.
We begin by examining the spectra in the binned map, PReBeaM map and the
smoothed input map shown in Figure 4.6. The effect of the beam mismatch is clearly
seen where the peaks and valleys of the binned map spectra have been shifted toward
higher multipoles. The detectors measure different Stokes / parameters, which translates to artifacts in the polarization map. Deconvolution suppresses leakage from
temperature to polarization as evidenced by the PReBeaM spectra that overlays the
input spectra. This shift is expected to remain apparent in the TE spectra of nonbeam-deconvolved maps even in the presence of noise, because of a larger temperature
signal and the temperature-to-polarization cross-coupling.
The fractional difference in the angular power spectrum (denned as (Ceout —
Cein)/Cein) of the input and output maps is shown in Figure 4.7. We show the
fractional difference spectra for the TT, EE, and cross-correlation TE signals, omitting the BB spectra since CfB is zero in the simulation of the CMB map. The results
for PReBeaM are shown at three intervals: the 25th, 50th and 75th iterations. This
shows the behavior of the power spectra as PReBeaM converges on the solution. The
beam mismatch effect is also seen in Figure 4.7, where the fractional difference in the
55
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56
PReBeaM spectra lies closer to zero than the binned map spectra over the full range
of multipole moments for EE and TE.
Next, we examine the residuals by looking at the power spectra of the difference
maps and comparing that with the power spectra of the difference in a^m's. The
difference in aem can be calculated for PReBeaM as we have the option to output
either a^m or maps, while the comparison method produces only maps. Figure 4.8
shows the power spectra of the difference map for the binned map and PReBeaM map.
We overplot the power spectra calculated from the difference in a^s {flimout ~~ atmin)
for PReBeaM. It can be seen that the spectra from the difference in a^m is smaller on
small scales than the spectra of the difference maps. This is likely due to inaccuracies
introduced during the transformation from pixel-map to afm-space to Q's. By-passing
this extra step is another advantage of a harmonic-based method.
As described earlier, PReBeaM allows for variation in the asymmetry parameter
iTT'max- We examined the performance of PReBeaM as a function of mmax, setting it
to 0, 2, 4, 6, and 8. In conjunction with this, we plot the beam a^m coefficients as
a function of the first few m in Figure 4.9. A remarkable improvement in the power
spectra was found by increasing mmax from 2 to 4, while an increase from 4 to 6
only resulted in marginal improvements. This effect is best seen in the BB power
spectra as shown in Figure 4.10. There is a simple explanation for the fact that
the PReBeaM a^m become very close to the input a^m at mmax 4, and thus do not
change much beyond mmax 4: there is an order of magnitude drop in the beam aem
coefficients from mmax = 4 to mmax = 6 for £ = 0 to £ — 512 as seen in Figure 4.9.
Thus, while the input TOD was simulated with a beam having an mmax cutoff of 14,
PReBeaM operates optimally at an mmax of just 4, thereby allowing us to capitalize
on the computational property that PReBeaM scales as mmax.
We define a quantity called na that models the expected bias due to beam asymmetry systematics to the x 2 statistic
nai =
£
\ £/=2
^
.
,4,0,
Planck^/
We use na to quantify the maximum, or worst-case bias beam systematics could induce
in a cosmological parameter that happened to be degenerate with that parameter.
The quantity apianck is the expected la errors for the LFI 30 GHz channel, computed
as the diagonal elements of the covariance matrix for the simulated input spectra,
assuming a sky fraction of 0.65. The na values are plotted in Figure 4.11 and show
57
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PReBeaM, mmax=2
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Figure 4.10: BB power spectra as a function of asymmetry parameter mmax for
mm(tx = 2 (blue curve) , 4 (cyan curve), and 6 (red curve). The input BB spectra
was zero so the smallest output BB spectra is most desirable. In this run, the PReBeaM input parameters interpolation order and zero-padding were set to 1 and 2,
respectively.
61
that PReBeaM reduces the worst case bias due to untreated beam systematics by 1
or 2 orders of magnitude.
We examine the resulting temperature and polarization (Q and U) maps. The output map for both PReBeaM and the binned map was subtracted from the smoothed
input map at the same resolution to make the residual maps shown in Figure 4.12.
PReBeaM residuals were plotted on the same color scale as the binned map, showing
that PReBeaM attained smaller residuals for both temperature and polarization.
4.9.1
Computational Considerations
The computational costs and advantages of our method should be noted. To perform
a convolution up to £max requires 0(£^ o:r ra max ) for the general case. Since mmax
is bounded by (,max, the cost never scales worse than C?(^ aa .) and is only 0(1^^)
for the symmetric beam case. By comparison, a brute force computation in pixel
space would require 0 ( ^ a x ) . In this study, data were simulated with beams having
an asymmetry parameter of mmax = 14, but maps were made using a cutoff value
of mmax — 4 in PReBeaM. We have demonstrated that computational cost can be
minimized while still achieving the benefits of beam deconvolution
It was found that an increase in the zero-padding factor from two to four produced
superior results over an increase in the interpolation order from one to three. An
optimal run of PReBeaM will therefore include the largest zero-padding possible,
given machine memory constraints, in conjunction with a polynomial interpolation
of order one or three. This is advantageous since the time spent in an FFT is nearly
negligible and affected only minimally with an increase in zero-padding. In contrast,
time for interpolation scales as the interpolation-order-squared, and, as this is a TODhandling step, it dominates over any cost incurred by convolutions. In the case of
the results shown here, interpolation steps consume more than 90% of the wall-clock
time per iteration.
The results produced here were generated using 12 nodes on the NERSC computer
Bassi (making use of all eight processors per node) and was complete in about 29 wallclock hours, for a total of 2797-CPU hours. The maximum task memory was 20 GB
on a single node.
62
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PKeifruM Residual T
Sprinylide Kesidua! T
PKI'IIPPM K M d n a l Q
Springtide Residue! Q
I'Hrijt'dM Residuwl U
Springtide Residual U
Figure 4.12: The residuals between the input reference sky and PReBeaM output
(left column) and the residual between the input reference sky and the binned map
(right column) for Temperature (T), and the Stokes Q, and U parameters.
64
4.10
Conclusion
We have found that PReBeaM has outperformed the standard binned noiseless map
using two measures: spectra and residual maps. We examined the fractional differences in the spectra and found markedly smaller differences in the PReBeaM spectra versus the binned map spectra across a range of multipole moments. We find
that map-making codes that do not deconvolve beam asymmetries lead to significant
systematics in the polarization power spectra measurements. The temperature-topolarization cross-coupling due to beam asymmetries is manifested as shifts in the
peaks and valleys of the spectra. These shifts are absent from the PReBeaM spectra.
We translated the errors found in the power spectra to an estimate of the statistical significance of the errors in a parameter estimation resulting from these spectra,
which we call na. This analysis showed that the worst case parameter bias due to
beam-induced power spectrum systematics could be tens of sigma while PReBeaM
reduces the risk of parameter bias due to beam systematics to much less than la.
We also found the /, Q, and U component residual maps to be smaller for PReBeaM
than for the binned map, implying smaller map-making errors.
We have discussed some advantages of using a harmonic-based map-making routine such as the avoidance of artifacts due to pixelization. An additional feature of
our method is the direct generation of sky a^m. Analysis of CMB data often requires
calculation of the power spectrum or bispectrum from the sky aem. A pixel-based
map-making method generates a map which must first be transformed into a set of
aim before computing the spectra. This extra step may introduce inaccuracies which
will carry forward in the analysis done on the spectra.
We have presented here the first results from PReBeaM for the straightforward
test case of CMB only, and including only the effects of beams in the main focal plane.
However, there is great potential for using PReBeaM to remove or assess systematics
due to the combination of foregrounds and beam side lobes. Systematics introduced
by side lobes will appear on the largest scales, potentially impeding the detection
of primordial B-modes on the scales where they are most likely to be measured.
We have already shown for temperature measurements [48] that our deconvolution
technique can be used to remove effects due to side lobes. In the next chapter, we will
examine the noise properties of PReBeaM maps and will include foregrounds from
extragalactic sources and diffuse Galactic emission.
65
Chapter 5
Noise and Diffuse Foregrounds
In the previous chapter, using the ideal case of CMB signal only, we showed the
results from our polarized map-making algorithm. In this chapter we will examine
the noise properties of CMB maps, and we will include foregrounds from diffuse
Galactic emission.
This chapter is organized as follows. In §5.1 we review some of the general noise
properties of PLANCK. In §5.2 we test PReBeaM on CMB plus white noise and
contrast these results with Springtide, our standard of comparison. Some of the diffuse
Galactic foreground emissions that are expected to contaminate CMB measurements
are described in §5.3. Results from a PReBeaM run on a map with only polarized
foregrounds are presented in §5.4, where we also look at a temperature-to-polarization
leakage case. We summarize our conclusions for this chapter on noise and foreground
issues in §5.5.
5.1
Noise model for Planck
Instrumental effects include not only beam effects but also noise, which can be white
(uncorrelated) noise or correlated (1//) noise. White noise refers to random, uncorrected instrument noise. The expected white noise contribution to the RMS total error
is proportional to \J(l/N0bs). We use the white noise from the Trieste [5] round of
simulations. Its nominal white noise standard deviation per time sample integration
time is a = 1350/J,K (CMB scale). Correlated 1 / / noise is characterized by a power
spectrum that rises at low temporal frequencies.
We test PReBeaM on white noise only, omitting correlated noise and noise due
to temperature fluctuations in the instrument cooling system. We refer the reader to
[5] for more details about 1 / / and cooler noise and also for tests that were done on
other map-making routines with the full PLANCK noise model.
66
5.2
Testing PReBeaM on CMB + white noise
We run PReBeaM on TOD containing CMB signal and white noise and compare
the results to the smoothed input CMB spectrum and the analogous results from
Springtide (in this case we refer to Springtide directly since this is not simply a binned
map). The level of the uncorrelated noise is specified in the detector database, and
has a nominal standard deviation per sample time of a — 1350/LJK [5]. PReBeaM
achieves a noticeably superior fit to the input spectrum compared with Springtide
from t ~ 150 to ~ 250. We acknowledge that the reconstructed power spectrum
is sensitive to the choice of regularizer. Without exploring the full parameter space
for this variable, we found that our choice for the regularizer fortuitously led to a
recovered power spectrum free of excess noise on small scales. Assessing the relative
performance of PReBeaM and Springtide in more detail would require performing
Monte Carlo averages. We focus on the TE spectrum, shown in Figure 5.1, since the
improvement is visible even for a single simulation. For the other spectra, PReBeaM
performs as least as well as Springtide, but the detailed difference are more difficult
to assess without a Monte Carlo study.
5.3
Knowledge of polarized foregrounds
Foreground emission fluctuations are a limiting factor for precise CMB temperature
and polarization anisotropy measurements. Foregrounds include any emissions that
obscure the primordial CMB signal after last scattering. These may include extragalactic discrete point sources, synchrotron emission, dust emission, free-free (or
Bremsstrahlung) emission, spinning dust emission, and atmospheric emission. Among
these sources, synchrotron and dust emission are expected to be significantly polarized. The level at which these foregrounds contaminate the CMB signal depends on
both the observing frequency and angular scale. In this section, we review what is
currently known about synchrotron and polarized dust emission, and also highlight
areas of limited understanding.
At 30 GHz, a dominant source of extragalactic emission is radio point sources.
We will study some properties of these sources in the next chapter.
5.3.1
Synchrotron emission
Highly polarized synchrotron emission is the result of cosmic-ray electrons accelerated
by the magnetic field of the Galaxy (e.g. [84]). Synchrotron emission is the main
67
TE S p e c t r a
60
of
CMB +
White
Noise
-T-r-r-r-r-i-pr
Smoothed input
40
Springtide
PReBeaM
-40
-60
100
300
200
400
Figure 5.1: TE spectrum of CMB and white noise for Springtide (blue curve) and
PReBeaM (red curve). The smoothed input map is shown in black. Following the
example in [5], we reduce £ to £ variation by filtering the spectra by a sliding average
(A£ = 20). In this run, the PReBeaM input parameters interpolation-order and zeropadding were set to 1 and 4, respectively. While PReBeaM performs at least as well
as Springtide in the TT, EE, and BB spectra, we omit these spectra since the detailed
differences are difficult to assess without an in-depth Monte Carlo study.
68
contaminant of the polarized CMB at frequencies below about 80 GHz.
The intensity, frequency scaling, and intrinsic polarization fraction of the synchrotron emission depends on the properties of the cosmic rays. The density of the
electrons follow a power law of index p, N(E) oc E~p and the synchrotron frequency
dependence also follows a power law,
S(u) oc vPs
(5.1)
where (3s = — (p + 3)/2. The spectral index of Galactic synchrotron emission, (3s,
varies with frequency and position on the sky. The polarization fraction,
/. = ^ + E
( , 2)
/ s = 3(p + l)/(3p + 7).
(5.3)
goes as
For a typical value p = 3, the spectral index (3s = — 3 and fs = 0.75. A polarization
fraction this large is not observed in practice as variations in both the polarization
angle along the line-of-sight and beam averaging effects tends to depolarize the signal.
Observational constraints from WMAP find an index averaged over regions of high
signal-to-noise of (3s = —3.02±0.04 [85]. However, there is still significant uncertainty
in the exact variability of the synchrotron spectral index.
5.3.2
Dust emission
Thermal emission from warm (10-100K) interstellar dust grains dominates Galactic
emission in the 100-6000 GHz frequency range. The thermal emissions originate from
thermal fluctuations in the electric dipole moment of the grains. Observations of the
polarization of starlight by dust grains suggests that elongated grains are partially
aligned with the Galactic magnetic field, although the exact alignment mechanisms
are not well understood (see, e.g., [86]).
Interstellar dust grains are composed of many different collections of matter particles, which prohibits the construction of a single theoretical emission law for dust.
However, an emission law can be fit to observational data. For a frequency range
typical of CMB observations, it has been shown that dust emission intensity can be
modelled by emission from a dual-component mixture of silicate and carbon grains
[87]. The thermal emissions spectrum is modelled as a modified grey-body emission
69
with intensity Iv as Bu(T)ua
temperature T.
5.4
where B is the Planck function at frequency v and
Testing PReBeaM on polarized foregrounds
In this section we describe a test in which we make noiseless foreground maps using the simulated data for the Trieste work [5]. Our motivation for separating out
the foreground signal from the CMB signal in this test is to examine how well PReBeaM can recover the polarization amplitude and angle of these foregrounds. We will
compare to Madam [54], one of the map-making routines developed by the PLANCK
algorithm group for pre-launch data analysis tests. Madam is a destriping algorithm
as described in Chapters 3 and 4. It is expected that Madam, which assumes an effective symmetric beam that varies from pixel to pixel, will recover a map and power
spectra that distort the polarization parameters of the foreground signals.
In this test, PReBeaM and Madam are run on TOD of the synchrotron, dust, and
free-free emission (temperature only). There is no CMB signal in this test. The input
map of the polarization amplitude of the synchrotron and dust emission are shown
in Figures 5.2 and 5.3, respectively. These simulated maps were developed under
the PLANCK Sky Model (PSM). The PSM is a simulation tool built by the PLANCK
collaboration in preparation for data analysis [88].
Our input TOD also included free-free (bremsstrahlung) emission due to electronelectron scattering in the warm ionized gas of the interstellar medium. The spectrum
of free-free emission follows a power-law with a spectral index of -2.1. Scattering directions are random, therefore free-free emission is intrinsically unpolarized. However,
secondary polarization can arise at the edges of bright free-free areas such as HII regions (clouds of dust and plasma, undergoing star formation and rich in ionised atomic
hydrogen) from Thomson scattering, causing substantial scattering in the Galactic
plane. Far from the Galactic plane, at high latitudes, the expected residual polarization is < 1%. Thus, free-free emission is not expected to be a major foreground
contaminant for polarization measurements.
The fractional difference in the angular power spectrum (defined as {Ctout —
Cein)/Cein) of the input and output maps is shown in Figure 5.4. We show the
fractional difference spectra for the TT, EE, BB, and cross-correlation TE signals.
The beam-mismatch effect described in Section 4.9 is clearly seen in these spectra
where the fractional difference is measured to be as high as 30% for Madam at small
scales in the EE spectra. On the same plot, the fractional difference in the PReBeaM
70
Synchrotron P (30 GHz)
D - 2 . 2 Log (K)
Figure 5.2: Simulated map for synchrotron polarization amplitude at 30 GHz.
Dust P (30 GHz)
] - 3 . 9 Log (K)
Figure 5.3: Simulated map for dust polarization amplitude at 30 GHz.
71
spectra remain closer to zero at all scales and remain under 10% for small scales.
Similary, the BB plot shows that the PReBeaM residual remains under 1% for the
full range of scales that we examined, while the Madam residual reaches 6% at high
£.
As a further test, we constructed PReBeaM maps of the free-free emission only.
Because free-free emission contains no polarization signal, an examination of the Q
and U maps produced by this test will show the effect of leakage from temperature
to polarization. We compare to the Madam maps for this case in Figure 5.5. We
refer to the Madam results as the binned maps because in the absense of noise, the
destriping algorithm is akin to a TOD binning algorithm. We find that the PReBeaM
algorithm iterates to a map containing stripe features in the Q component. This
translates to some noise in the EE and TE spectra. Excluding a small region at
t < 20 in the BB spectra, the PReBeaM polarization spectra have smaller residuals
than the binned map spectra by up to three orders of magnitude. This test of the
free-free emission only case demonstrates that PReBeaM's beam deconvolution aids
in reducing temperature-to-polarization leakage due to the beam mistmatch effect.
5.5
Conclusions
In Chapter 4, we examined the results of our PReBeaM algorithm for the case of
temperature and polarization signal only for CMB measurements. In this chapter,
we separated out noise and foregrounds in order to look at their effects in our maps
and spectra.
First, we described the noise model for PLANCK and showed that PReBeaM
achieved a superior fit to the input TE spectra compared to Springtide, for the case
of CMB plus white noise.
Next, we briefly described the current state of the knowledge about polarized
foregrounds including synchrotron and dust emission. We showed that PReBeaM
reconstructed spectra with smaller residuals than Madam for TOD containing synchrotron, dust, and free-free emission. We then separated out the free-free emission,
which contains no polarization component, in order to examine the leakage from temperature to polarization. We showed that PReBeaM polarization spectra contain less
leakage from temperature than Madam polarization spectra over a wide range of £'s.
A class of map-making techniques called destriping methods has been developed
to deal with the low-frequency noise component [53]. This approach calculates a series
of offsets which are subtracted from the TOD. A binned map can then be made from
72
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with a destriping algorithm in which the noisy TOD is pre-treated with a destriper
before being processed by PReBeaM to remove beam effects.
75
Chapter 6
Variable Point Sources
In their most recent release of data, the WMAP team has identified 390 bright point
sources in their maps [89]. It is important to study these sources from the CMB
standpoint in which extragalactic flat-spectrum radio sources are the most important
foreground at small scales. If these sources are not removed properly, they can affect
estimations of power spectra and tests of Gaussianity from CMB data [90]. The aim of
extraction techniques is to identify and remove these sources as accurately as possible;
a greater understanding of these sources and their variability will aid in that goal (see,
for example, [91] and [92] for reviews of point source removal methods). Moreover, it
is interesting to investigate these sources as astrophysical objects. Sources have been
identified to be interesting astrophysical objects such as quasars, Active Galactic
Nuclei (AGN), and BL Lac-type objects. The WMAP data allows for a study of the
variability of extragalactic sources over a new, largely unstudied, frequency regime
from 23-100 GHz. Variability measurements will provide important information on
the structure of these sources.
In this chapter, we establish a program to search the public WMAP TOD datasets
for under-exploited data on the variability and evolutionary properties of point sources
cataloged by the WMAP team. The chapter is organized as follows. We review
work by the WMAP team on point source variability in Section 6.1. In Section
6.2, the procedure for data extraction and processing is described. Our method
for determining excess variability through a x2 analysis is discussed in Section 6.3.
Finally, we show our current results and discuss the implications in Section 6.4 and
conclude with plans for future work in Section 6.5.
6.1
W M A P findings on point sources
In this section, we briefly review the techniques used by the WMAP team to identify
point sources in their maps and their findings on point source variability. Further
details may be found in the five-year point source catalog paper [89].
76
Figure 6.1: Map of the 390 point sources detected by the WMAP team. The shaded
region shows the mask used to exclude extended foreground emission. The size of
the plotted points indicates the flux of the source: the area of the dot scales like the
maximum flux over the 5 WMAP bands plus 4 Jy. Image courtesy of the WMAP
science team.
The WMAP team finds point sources by searching their maps for bright spots that
approximate the beam profile. They perform this search in order to generate masks
that will be used to discard regions around the brightest sources before proceeding
with data analysis.
Figure 6.1 is a map showing the location of the 390 point sources that the WMAP
team found by searching the individual band maps. The WMAP team cross-correlate
detected sources with external surveys. The GB6 [93], PMN [94], and Kiihr et al.
[95] catalogs are used to identify 5 GHz counterparts; we will use the same 5 GHz
IDs to distinguish point sources.
Of the 390 sources, 141 have been optically identified as quasars, 29 as galaxies,
19 AGNs, 19 as BL Lac-type objects, and one as a planetary nebulae [96].
In [89], the WMAP team looks at the variability of sources by forming fluxes from
individual year maps. They measure the variability of a source by subtracting the
five-year average map from each individual year. They then fit their data points to an
arbitrary spectrum that is constant in time and perform a chi-square goodness-of-fit
test. They find that 137 of the 390 sources have a x 2 > 37.6 and are therefore variable
at greater than 99% confidence. It is found that 54 sources have x 2 > 100 and 15
sources have x 2 > 450.
77
Our analysis of source varability will be more in depth, as we will be looking not
at the average year-to-year variation in sources, but rather at all the data from five
years worth of observations for each point source. In some cases, this amounts to tens
of thousands of data points. We will cross-check our results with the sources that the
WMAP team found to be variable.
6.2
Description of data extraction and processing
Fortunately, WMAP has made their data available for download on a public web
page1. All of the data that we use, as well as the beam profiles, source catalogs, and
masks, can be found on their site.
We also made extensive use of the WMAP Interactive Data Language (IDL) Library 2 . The library contains a suite of procedures and routines to assist with reading
and manipulating the data products.
The WMAP instrument is composed of 10 differencing assemblies (DAs), covering
5 frequency bands from 23-94 GHz. The DAs are identified as follows: 1 DA at 23
GHz (Kl), 1 DA at 33 GHz (Kal), 2 DAs at 41 GHz (Q1,Q2), 2 DAs at 61 GHz
(V1,V2), and 4 DAs at 94 GHz (W1-W4). Each DA is formed from two differential
radiometers.
We use the calibrated TOD files in which gain and baseline factors have been
removed. Each TOD fits file contains 1875 records making up 24 hours of measuring
time. One record has 30 major science frames of length 1.536 seconds. One major
science frame consists from 12 measurements (for the K band) up to 30 measurements
(for the W band).
The calibrated archives have been coadded into two radiometer channels for each
measurement so we read two vectors of data from each TOD fits file. The general
flags and differencing assembly flags for each record are also read. Flags may indicate
quality issues within the science frame, such as the sun, moon or a planet is visible
in one of the beams. All records with any nonzero flags are rejected.
6.2.1
Loss imbalance parameter
WMAP is a differential instrument consisting of two back-to-back Gregorian telescopes. The instrument records a TOD measurement as the difference between the
1
2
http://lambda.gsfc.nasa.gov/product/map/dr3/m_products.cfm
http://lambda. gsfc.nasa.gov/product/map/current/m_sw.cfm
78
A-side and B-side sky signals. We will be interested in separating out the differential measurement into that signal which is seen by the A-side and that seen by the
B-side. However, we will need to account for the fact that the output of each WMAP
radiometer does not produce a perfect differential response. Power transmission coefficients, a A and aB, of the optics and waveguide components couple the sky signals
from the A and B side beams to the radiometer. In reality, these coefficients are not
equal. The output of a radiometer, S, when the beams observe regions of the sky
with temperatures TA and TB becomes
S = G{aATA-aBTB\
(6.1)
where G is the gain of the radiometer. Jarosik et al [97] define the loss imbalance
parameter as
xwhere (3C = G(aA —
6.2.2
OLB)/2
(6.2)
=&-
and (3d = G(aA +
«B)/2.
Differencing assembly parameters
For each DA, we define the following quantities:
• 0b = beamsize (we choose beamsize to be the point at which the normalized
beam profile falls to 75%)
• Xim,i,2 — l° s s imbalance parameters in radiometer 1 and 2
• UJA = 1 + Xim
• U>B = 1 - Xim
• (To = noise per observation to ~ 0.1% uncertainty (in mK)
The values for each of these parameters for each DA are given in Table 6.2.2.
6.2.3
Beam normalization
In the five-year analysis, an azimuthally symmetrized radial beam profile is calculated
by binning individual A- and B-side hybridized Jupiter observations. As such, the
beam profiles are presented in units of mK with a constant bin size of 0.25'. For
79
DA
Kl
Kal
Ql
Q2
VI
V2
Wl
W2
W3
W4
8b (deg)
•Eim
0.2521
0.0028
0.1979 0.00245
0.1521
0.0057
0.0089
0.1563
0.1063 0.00685
0.0094
0.1063
0.0688 0.00575
0.0646 0.00935
0.0646
0.0059
0.0688 0.02135
U1A
UJB
1.0028
1.00245
1.0057
1.0089
1.00685
1.0094
1.00575
1.00935
1.0059
1.02135
0.9972
0.99755
0.9943
0.9911
0.99315
0.9906
0.99425
0.99065
0.9941
0.97865
Table 6.1: Beamsize (8b), loss imbalance parameter (xim),
for each of the differencing assemblies (DA).
a0 (mK)
1.436
1.470
2.254
2.141
3.314
2.953
5.899
6.565
6.926
6.761
U>A,B,
and noise term (<70)
the purposes of our analysis, we require a normalized beam profile. We calculate the
normalization factor, Tpeak, following the description in Section 4.1 of the five-year
beam paper [98],
•'-peak
=
1 Jupiter TZ.
^ ^beam
l^-"/
The relevant values for Tjupiter, flfid, and f2&eam can be found in Tables 4 and 5 of
[98]. We plot the inner parts of the normalized profiles in Figure 6.2. We show here
only the region of the beams that we use for our analysis. Data points which fall at
a distance from the point source where the beam profile has fallen to less than 75%
are omitted. The peak is noisy because the inner region of the beam is directly from
Jupiter data in the five year analysis. Some of the differencing assemblies have a peak
in the innermost bin and some a dip.
6.2.4
Dipole corrections
For unbalanced differential data, the TOD at time t is given by
d(t) = UJAKPA) -
UBKPB)
+ 0(5xim)
(6.4)
where \(PA) is the Stokes parameter I at pixel pA and \(PB) is the Stokes parameter
I at pixel PBWe accept only TOD for which the following two conditions hold: (i) the distance
between the pointing for the TOD element and the point source is less than 8b; (ii)
the opposite horn is outside of a region that masks a point sources with a disc of
radius 0.6°.
80
If the above conditions hold, then we calculate the variation for a point source
observed in a single horn as
_ dit) - {uAIA{pA) - LUBIB(PB))
x—
-
- dTf - dTv
(o.oj
Jbeam
where IA is the Stokes parameter I from the uncleaned WMAP map at pixel pA, dTf
is the fixed dipole component, dTv is the velocity dipole term, and fbeam is the value
of the normalized beam profile at the pointing distance from the point source.
The fixed dipole modulation term, dTf, due to the solar system barycenter's (SSB)
motion with respect to the CMB is
dTf = uAdA — u>BdB-
(6.6)
Here, dAtB are the values of the fixed dipole component at the A- and B-side, found
by dotting the fixed dipole vector (given as d = [-0.233, -2.222,2.504] mK in [24])
with the position vectors for the A- and B-side.
The dipole modulation term, dTv, due to WMAP's velocity with respect to the
solar system barycenter (SSB) is calculated as
T0
dTv = —v • (uAnA - w B n B )
(6.7)
c
where T0 is the CMB mean temperature, c is the speed of light, v is the WMAP
velocity with respect to the SSB, and UA, fiB are the line-of-sight vectors for the Aand B-sides, respectively [24]. The error on x is calculated as
(6.8)
<yx = -pJ beam
where 0.75 < fbeam < 1-0.
6.3
Chi-square analysis
Given xA, XB, PX,A, and ax<B from the previous section, we calculate the chi-square
minimum and chi-square probability following [99].
For n0bs repeated measurements (XJ, <7j) of the same quantitiy x, the x2 IS defined
as:
n
obs /
81
~\2
X2 is minimized by
E(iM2)
(6.10)
Therefore, the chi-square minimum is given by
Xmin = x V ) -
(6.11)
We compute xmin a n d compare with x2(n) t o determine the goodness of fit where
n is the number of independent degrees of freedom. We find p, the probability that, in
a x2 distribution with n degrees of freedom, a random variable is greater than Xmin'p = Prob{ X 2 (n) > xLn>
(6-12)
where, in our case, n = n0bs — 1, with one degree of freedom used to find x.
6.3.1
Null test
Next, we do a null test as a means of determining a detection of excess variability in
the actual point sources. For the null test we choose 1000 points outside of the WMAP
mask, which are expected to contain CMB-only signal, and perform same analysis
described above on the null points as we do for the actual point sources. This gives
a set of x2 probabilities for the null points. We then histogram the probabilities and
identify those actual point sources having a p that fall below some cutoff value on
the null histogram. In this way, we may still be confident in our detection of excess
variability even in the event that we have under or overestimated the noise term.
6.4
Results and discussion
The x2 analysis described above was performed on the 390 actual point sources and
1000 null points using the K (23 GHz) and Ka (33 GHz) band data as a first test.
The distribution of the x2 probabilities for the null points is plotted as a histogram,
seen in Figures 6.3, 6.4, 6.5, and 6.6. The value of the x2 probability for those actual
point sources whose value falls below the 5% cutoff on the null histogram is then
overplotted as a vertical line. The results are summarized in Table 6.2, in which
points are listed as being observed to be variable in either the A- side or B-side or
both. Point sources are identified, as in the WMAP analysis, according to their 5
GHz ID. We also list the K and Ka band flux in Janskies (Jy), the type of object
82
as identified by [96] (if available), and a note which indicates the level of variability
found in the WMAP analysis.
K
5 GHz ID
J0003-4752
PMN J0006-0623
PMN J0025-2602
GB6 J0029+0554B
A
B
PMN J0204-1701
GB6 J0221+3556
PMN J0403-3605
Uy 0403+76
PMN J0423-0120
GB6 J0424+0226
PMN J0442-0017
PMN J0457-2324
PMN J0538-4405
PMN J0540-5418
GB6 J0542+4951
GB6 J0555+3948
PMN J0609-1542
PMN J0633-2223
PMN J0634-2335
GB6 J0646+4451
K
Flux (Jy)
/
/
2.3
0.9
1.1
/
0.7
1.0
1.9
1.4
1.1
1.8
0.8
3.8
1.2
1.3
0.7
1.2
3.4
1.0
8.2
1.2
0.9
2.4
5.6
1.4
1.7
3.0
3.7
0.5
0.6
2.9
/
/
PMN J0038-2459
PMN J0050-0928
GB6 J0108+0135
GB6 J0108+1319
PMN J0125-0005
PMN J0132-1654
PMN J0133+5159
GB6 J0136+4751
GB6 J0152+2206
GB6 J0204+1514
Ka
A B
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
Continued on next page
83
Ka
Flux (Jy)
0.7
2.3
0.7
1.3
0.8
1.0
1.9
1.1
1.2
1.8
1.1
3.8
1.3
1.3
1.2
3.8
0.7
8.3
1.0
0.8
2.5
5.9
1.4
1.3
2.4
3.3
0.6
0.7
2.4
Type
Note
G
V
QSO
V
QSO
V
V
QSO
V
QSO
V
QSO
QSO
V
V
V
QSO
V
V
V
Table 6.2 - continued from previous page
K
Ka
K
Ka
5 GHz I D
GB6 J0720+0404
GB6 J0753+5353
GB6 J0825+0309
GB6 J0841+7053
PMN J0847-0703
GB6 J0909+0121
GB6 J0914+0245
GB6 J0920+4441
GB6 J0927+3902
GB6 J0957+5522
PMN J1014-4508
PMN J1018-3123
GB6 J1022+4004
GB6 J1043+2408
J1053+8109
GB6 J1058+0133
PMN J1118-1232
PMN J l 147-3812
GB6 J1159+2914
PMN J1215-1731
GB6 J1219+0549
GB6 J1229+0202
GB6 J1230+1223
PMN J1246-2547
PMN J1256-0547
GB6 J1332+0200
GB6 J1333+2725
PMN J1337-1257
PMN J1427-3306
GB6 J1436+6336
GB6 J1443+5201
GB6 J1549+5038
A
B
A
B
Flux (Jy)
Flux (Jy)
0.9
1.0
1.6
1.7
0.9
2.1
1.4
1.3
6.7
0.9
1.1
0.9
0.9
0.8
0.9
4.6
1.0
2.1
0.7
1.0
1.9
1.7
1.0
2.0
1.6
1.4
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
2.0
1.4
2.7
20.0
19.7
1.3
17.1
1.4
0.8
5.8
1.0
0.5
0.8
0.9
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
Continued on next page
84
5.8
0.9
0.8
0.9
0.9
0.8
0.9
4.4
0.9
2.3
2.2
1.2
2.2
18.4
15.5
1.4
17.9
1.4
0.9
6.0
1.4
0.9
0.8
Type
Note
V
QSO
V
QSO
V
V
QSO
V
QSO
V
V
QSO
V
QSO
G
QSO
QSO
V
V
QSO
V
V
V
Table 6.2 - continued from previous page
K
5 GHz I D
A
GB6 J1608+1029
J1633+8226
GB6 J1635+3808
GB6 J1642+3948
GB6 J1657+5705
GB6 J1658+0741
/
GB6 J1727+4530
Uy 1803+78
PMN J1819-6345
PMN J1834-5856
GB6 J1848+3219
PMN J1923-2104
PMN J1924-2914
PMN J1939-6342
GB6 J1952+0230
PMN J2056-4714
GB6 J2123+0535
PMN J2131-1207
PMN J2134-0153
GB6 J2148+0657
GB6 J2202+4216
GB6 J2212+2355
PMN J2218-0335
PMN J2225-0457
GB6 J2253+1608
PMN J2258-2758
GB6 J2322+5057
GB6 J2330+1100
GB6 J2349+3849
PMN J2358-6054
/
B
Ka
A B
K
Flux (Jy)
Ka
Flux (Jy)
2.0
2.0
1.5
4.3
6.0
0.6
1.5
1.0
1.7
1.5
1.1
1.3
3.9
6.5
0.5
1.4
0.9
1.8
1.7
1.1
0.7
2.3
12.3
0.9
0.8
2.2
2.2
2.7
2.0
8.0
3.4
1.3
2.3
5.2
7.4
5.2
0.9
1.0
0.8
1.9
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
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/
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Continued on next page
85
0.8
2.5
12.0
0.7
0.6
2.5
1.8
2.4
1.9
7.7
3.5
1.5
2.0
4.9
7.5
5.2
0.8
1.0
0.7
1.4
Type
Nol
V
QSO
QSO
V
V
V
V
V
V
QSO
QSO
V
V
V
QSO
QSO
V
V
V
QSO
QSO
QSO
QSO
V
V
V
V
Table 6.2 - continued from previous p a g e
K
Ka^
K
Ka^
5 GHz ID
A B A B Flux (Jy) Flux (Jy) Type Note
Table 6.2: Point sources that we have identified as being variable. Sources are listed,
following the WMAP convention, by their 5 GHz ID. Variability observed in either the
A or B side (or both) is indicated by a checkmark for the K and Ka band separately.
The peak K and Ka band flux is listed in columns 6 and 7, respectively. The type (G
for galaxy, and QSO for quasar) is listed in column 8.
v
WMAP-identified as probable variable: x2 > 36.7.
v
WMAP-identified as variable: x2 > 100.
Several observations can be drawn from Figures 6.3, 6.4, 6.5, 6.6 and Table 6.2.
First, it was expected that the null points should have a uniform distribution of probabilities between 0 and 1. Rather, we find that they are peaked near p = 1 indicating
that the null points are, on average, less variable than we expected. This suggests
the possibility that the noise term has been overestimated. We find that reducing the
noise term by 1.5% does indeed produce a uniform x2 probability distribution for the
null points. This finding is in direct conflict with the results from a recent Bayesian
analysis of the white noise levels in the WMAP data by Groeneboom et al. [100], who
find an indication that the noise levels in the maps are underestimated by 0.5-1.0%.
A second feature that one observes in Table 6.2 is the overall discrepancy between
the x2 probabilities measured in the A and B side for the same point source. Of the 63
sources that we identify as being variable in either the A or B side for the K band, only
eight are observed to be variable in both horns. Similarly, of the 43 sources identified
as variable in the Ka band data, only seven are found to be variable in both horns.
These sources that are seen to be variable in both horns also tend to be the brightest
sources as indicated by the fluxes in Table 6.2. As we have only examined the data
from the K and Ka band, it is possible that we have not reached the threshold of
detectability for many of these sources. The test that we have performed to identify
variability in point sources in each of the bands and horns individually is much more
stringent than the WMAP test which combines all of the data.
86
6.5
Conclusions and further work
Using the raw, calibrated TOD from the K and Ka band DAs of the WMAP instrument, we identify 10 point sources that display excess variability in both the A- and
B-side, and in either the K band, Ka band, or both. These sources, and their optical identifications, are: PMN J0403-3605 (quasar), PMN J0423-0120 (quasar), PMN
J0538-4405 (quasar), PMN J1014-4508, GB6 J1229+0202 (quasar), GB6 J1230+1223
(radio galaxy) PMN J1246-2547 (quasar), PMN J1256-0547 (quasar), PMN J13371257 (quasar), and PMN J2258-2758 (quasar). All of these sources (except PMN
J1014-4508) have also been flagged as variable or probable-variable by the WMAP
team.
Subsequent work on the variability of point sources in the WMAP data will include
an analysis of the data from the remaining differencing assemblies (Ql, Ql, VI, V2,
Wl, W2, W3, and W4). We will be interested to see if the differing noise properities
in these channels will produce results which are distinct from those that we found
here for the K and Ka band analysis.
87
profile
Ko1
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o
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"
^^v.
•a 0.9
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0.00 0.05 0.10 0.15 0.20 0.25 0.30
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Q1
1.00
0.95
0.8
i
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Q2
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:
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1.00
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\
'
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W4
-.
1.00
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:
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\
.
\^ -
0.00
0.02
0.04
0.06
0.08
Distance from beam center (deg)
Figure 6.2: Innermost parts of the normalized profiles for all 10 channels of the
WMAP instrument.
88
. I i i i I 111 I i i i I i • H
J1256-0547Q
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Figure 6.3: The 24 most variable point sources found on the A side of the K band.
The distribution of the x 2 probabilities for the null points plotted as a histogram
for the A side (blue) and B side (green). The x2 probability, p, for the actual point
sources are overplotted for the A side (dotted) and B side (dashed). This figure shows
the point sources ordered in increasing p for side A. Note that the dotted and dashed
lines for sources J0423-0120, J1229+0202, J1256-0547, J2258-2758, and J0538-4405
are difficult to see as they sit nearly exactly at zero.
89
rJ1266-6547
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Figure 6.4: The 24 most variable point sources found on the B side of the K band.
The distribution of the x2 probabilities for the null points plotted as a histogram
for the A side (blue) and B side (green). The x2 probability, p, for the actual point
sources are overplotted for the A side (dotted) and B side (dashed). This figure shows
the point sources ordered in increasing p for side B. Note that the dotted and dashed
lines for sources J0423-0120, J1229+0202, J1256-0547, J2258-2758, and J0538-4405
are difficult to see as they sit nearly exactly at zero.
90
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Figure 6.5: The 24 most variable point sources found on the A side of the Ka band.
The distribution of the x2 probabilities for the null points plotted as a histogram
for the A side (blue) and B side (green). The \2 probability, p, for the actual, point
sources are overplotted for the A side (dotted) and B side (dashed). This figure shows
the point sources ordered in increasing p for side A. Note that the dotted and dashed
lines for sources J1258-0547, J1229+0202, J2258-2758, J0423-0120, J0538-4405, and
J1014-4508 are difficult to see as they sit nearly exactly at zero.
91
TTTTTTTTTTTTTTTTTm
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0.2 0.4 0.8 0.8
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0
iluli I i l i i i l i i i l l n i l
0.2 0.4 0.8 0.8 I
P
Figure 6.6: The 24 most variable point sources found on the B side of the Ka band.
The distribution of the x2 probabilities for the null points plotted as a histogram
for the A side (blue) and B side (green). The x2 probability, p, for the actual point
sources are overplotted for the A side (dotted) and B side (dashed). This figure shows
the point sources ordered in increasing p for side B. Note that the dotted and dashed
lines for sources J1258-0547, J0423-0120, .10538-4405, J1229+0202, J2258-2758, and
J1014-4508 are difficult to see as they sit nearly exactly at zero.
92
Chapter 7
Conclusions
7.1
Summary of Progress
In this chapter, I summarize the achievements that were completed during the thesis
work described in this dissertation.
I developed deconvolution map-making for temperature observations; this included tests of several beam models and scanning strategies, and included Galactic foregrounds. This work was done at a relatively low resolution with beams of
3° or larger. I showed that deconvolution map-making was successful at removing
systematic effects in CMB maps due to beam asymmetries.
I then extended the deconvolution map-making technique to include polarization
observations (PReBeaM) and tested the method on actual PLANCK simulated data.
This study was done at a resolution of about 30', or about 0.5°. Realistic PLANCK
beams and scan strategy were used in the tests and we showed that PReBeaM corrected shifts in the CMB power spectra due to beam asymmetry effects. Final tests
of PReBeaM included white noise and polarized foreground signals.
Finally, I developed a program to search public WMAP data archives for point
source variability on a wide range of time scales.
7.2
Future work with PReBeaM
There is still much that can be done to improve the PReBeaM algorithm and much
that can be done to research the effect of beam systematics in CMB maps with the
aid of our technique. Here I list a few areas of possible future work.
7.2.1
Improving the computational requirements
The work described in this dissertation has resulted in the development of a mapmaking code which has been shown to be able to remove a significant level of sys-
93
tematic beam effects from maps and power spectra for the PLANCK satellite. Up to
this point, the main focus of my PhD research has been to develop this code and to
demonstrate its effectiveness at beam deconvolution, rather than to optimize the computational requirements of the algorithm. While efforts have been made to streamline
the code, there is certainly room for improving both the memory requirements and
the computational runtime.
One such improvement that we have identified is to turn the entire operation
of interpolation and transpose interpolation of all the TOD elements (described in
Chapter 4) into a matrix operation. This has the dual advantage of omitting the
need to store a TOD-sized object (less memory required) and removing the steps in
which we run through each TOD element for every iteration (faster computational
runtime).
7.2.2
Interface with destriper for noise treatment
In Chapter 5, we presented the results from a PReBeaM run on PLANCK simulated
data containing CMB and white noise. In reality, PLANCK and other satellite instrument data will contain correlated types of noise.
A class of map-making techniques called destriping methods has been developed
to deal with the low-frequency noise component [53]. This approach calculates a
series of offsets which are subtracted from the TOD. A binned map can then be made
from the resulting TOD. There is great potential for PReBeaM to be interfaced with
a destriping algorithm in which the noisy TOD is pre-treated by the destriper before
being processed by PReBeaM to remove beam effects.
7.2.3
Re-analysis of the W M A P data
Through our involvement in the PLANCK collaboration, we have developed PReBeaM
to run on simulated PLANCK data. However, the algorithm is easily adaptable to run
on data from other CMB satellite instruments.
A recent paper by Wehus et al. [101] has assessed the impact of asymmetric beams
on cosmological parameters estimated from the WMAP temperature data. They find
shifts of 0.4a in the amplitude of scalar perturbations and the physical density of
cold dark matter, and a shift of 0.3<r in the spectral index of scalar perturbations.
Wehus et al. point out that shifts this size should not be neglected as they are of the
same order of magnitude or larger than marginalization over the Sunyaev-Zeldovich
effect or unresolved point sources. It would be of interest to re-analyze the WMAP
94
data, which is freely available, using PReBeaM and the actual measured asymmetric
WMAP beams in order to examine the level at which beam deconvolution can correct
these observed shifts. It would be of great interest to repeat the full analysis on the
polarization data which also suffers from beam effects and confusion due to leakage
from the much larger temperature signal.
7.2.4
Analysis of Planck data with far sidelobes
For the PLANCK instrument, a complete characterization of the beam includes both
the main beam and the far sidelobes. Sidelobes are located as far away as 90° from
the main focal plane and may induce systematics on the largest scales, potentially
inhibiting the detection of B-modes. A major strength of the deconvolution technique
is the ability to account for significant beam asymmetries such as sidelobes with
relative ease using the harmonic method. In Chapter 3, we tested deconvolution mapmaking for temperature observations at low resolution with a simulated far-side lobe
beam. We would like to repeat the PLANCK 30 GHz temperature and polarization
analysis using the full beam model, that is main beam plus sidelobe. This gives rise
to the unresolved question of how to best combine the benefits of a local, pixel-based
approach for the innermost part of the main beam, with the advantages of harmonic
space convolution for the remaining beam pattern.
7.3
Future work with the convolution routines
At its most fundamental level, the standard model of cosmology assumes a homogeneous and isotropic universe. However, there have been numerous studies of the
WMAP data resulting in claims of evidence for a preferred direction in the universe
(e.g., most recently [102, 103, 104]). A preferred direction is defined as a direction in
which local features in the CMB tend to be oriented towards.
Recalling that the temperature field on the sphere can be expanded in harmonic
coefficients aem and assuming an isotropic Gaussian process, each a^m is an independent Gaussian random variable and is fully characterized by the power spectrum CeOne finds no evident shape to the multipole if Ce is randomly and uniformly distributed among all ms. Therefore, the identification of an orientation for which an
excess amount of power is concentrated in a particular m-mode suggests deviation
from isotropy.
Some of the numerical techniques that we develop for deconvolution map-making
95
can be naturally applied to this problem, giving an immediately science return when
applied to the WMAP data. The convolution algorithm that we use was developed
to compute the detector response at every pointing of the telescope, in other words,
to perform the convolution of the beam with the sky for all orientations of the beam.
We can use our fast convolution algorithm to search for correlations in the CMB
anisotropies. These techniques will allow us to quickly and efficiently do this search
out to high £ and for all orientations.
In addition to searching for non-Gaussian signatures such as filaments and preferred directions, we can use our tool to expose systematic effects left in the data.
For example, scan-synchronous effects may leave correlated noise stripes in the data
products. Given full sky maps of Galactic dust emission and other astrophysical
emissions, we can use the same formalism to search for anomalous structure in these
signals.
96
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101
Vita
CHARMAINE ARMITAGE-CAPLAN
Contact Information
Department of Physics
University of Illinois at Urbana-Champaign
1110 West Green Street
Urbana, IL 61801 USA
Office: (217) 333-2807
E-mail: carmitag@illinois.edu
www. astro. uiuc. edu/ ~ car mit ag
Education
University of Illinois at Urbana-Champaign, Urbana, Illinois USA
• Ph.D., Physics, expected October 2009
• M.Sc, Finance, December 2007
• M.Sc, Physics, August 2005
University of British Columbia, Vancouver, British Columbia Canada
• B.Sc, Combined Honours in Physics and Astronomy, May, 2003
Research Experience
• Postdoctoral researcher in theoretical cosmology at the University of Oxford,
Oxford, UK. Begins October 2009.
• Graduate research assistant, Department of Physics, UIUC. Advised by Benjamin Wandelt. Summer 2003 - Summer 2009.
• Undergraduate research for honours thesis, UBC, Vancouver, Canada. Advised
by Douglas Scott. 2002-2003.
102
• Summer research assistant at the Canadian Institute for Theoretical Astrophysics (CITA), Toronto, Canada. Supervised by Richard Bond and Carlo
Contaldi. Summer 2002.
• Summer research assistant at CITA, Toronto, Canada. Supervised by Richard
Bond and Carlo Contaldi. Summer 2001.
Honors and Awards
• UIUC: Ranked as Excellent Teacher, 2003
• CITA: Natural Sciences and Engineering Research Council (NSERC) Award,
2002
• CITA: NSERC Award, 2001
• UBC: Outstanding Student Initiative Award, 1997-2001
Teaching Experience
• Teaching Assistant, How Things Work, UIUC, Spring 2005
• Teaching Assistant, Thermal and Quantum Physics, UIUC, Spring 2004 & Fall
2003
• Teaching Assistant, Physics & Astronomy Department, UBC, Spring 2002
Talks and Conferences
• UIUC Center for Theoretical Astrophysics Seminar. Urbana, IL, 2006. Talk
title: Probing the Early Universe through CMB Polarization Maps.
• Preliminary Examination. Urbana, IL, April 10, 2006. Talk title: New Approaches to CMB Map-Making and Astrophysical Anomaly Detection.
• American Physical Society April Meeting, Dallas, TX, 2006. Talk title: In
Search of B-modes.
• SF05 Cosmology Summer Workshop, July 5-22, 2005. Saint John's College,
Santa Fe, New Mexico.
103
• American Physical Society April Meeting, Tampa, FL, 2005. Talk title: Deconvolution Map-Making for CMB Observations.
• Planck HFI/LFI Joint Consortium Meeting, January 2005. Poster title: Deconvolution Map-Making for CMB Observations.
• US Planck Algorithm Development Group Meeting, Pasadena, CA, October
27,2004. Talk title: Deconvolution Map-Making for CMB Observations.
Publications
• Armitage-Caplan, C. and Wandelt, B.D. "PReBeaM for Planck: A Polarized
Regularized Beam Deconvolution Map-Making Method", ApJS 181, No.2, 533542 (2009).
• Armitage, C , and Wandelt, B.D. "Generalized Beam Deconvolution MapMaking for Cosmic Microwave Background Observations", Physical Review D
70, 123007 (2004).
Appearances in the Media
• Interviewed on Channel 15 Sunrise This Morning News (Champaign, IL, 01/24/2007)
to advertise the Saturday Astrophysics Honors Program.
Computer Skills
• Languages: Fortran, C++, Perl, OpenMP, MPI, IDL.
• Operating Systems: Unix/Linux, Mac OS X, Windows.
• Five years of experience in Linux-based high-performance parallel computing
environment
Professional Societies
• American Physical Society
• Union for Concerned Scientists
104
• Federation of American Scientists
105
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