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RAFT VERSION
A
PRIL
22, 2022
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A
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A Simulation-Based Method for Correcting Mode Coupling in CMB Angular Power Spectra
J. S.-Y. L
EUNG
,
1, 2
J. H
ARTLEY
,
3
J. M. N
AGY
,
4, 5
C. B. N
ETTERFIELD
,
1, 3
J. A. S
HARIFF
,
6
P. A. R. A
DE
,
7
M. A
MIRI
,
8
S. J. B
ENTON
,
9
A. S. B
ERGMAN
,
9
R. B
IHARY
,
10
J. J. B
OCK
,
11, 12
J. R. B
OND
,
6
J. A. B
ONETTI
,
12
S. A. B
RYAN
,
13
H. C. C
HIANG
,
14, 15
C. R. C
ONTALDI
,
16
O. D
ORÉ
,
11, 12
A. J. D
UIVENVOORDEN
,
9
H. K. E
RIKSEN
,
17
M. F
ARHANG
,
18, 6, 1
J. P. F
ILIPPINI
,
19, 20
A. A. F
RAISSE
,
9
K. F
REESE
,
21, 22
M. G
ALLOWAY
,
17
A. E. G
AMBREL
,
23
N. N. G
ANDILO
,
24
K. G
ANGA
,
25
R. G
UALTIERI
,
26
J. E. G
UDMUNDSSON
,
22
M. H
ALPERN
,
8
M. H
ASSELFIELD
,
27
G. H
ILTON
,
28
W. H
OLMES
,
12
V. V. H
RISTOV
,
11
Z. H
UANG
,
6
K. D. I
RWIN
,
29, 30
W. C. J
ONES
,
9
A. K
ARAKCI
,
17
C. L. K
UO
,
29
Z. D. K
ERMISH
,
9
S. L
I
,
9, 31
D. S. Y. M
AK
,
16
P. V. M
ASON
,
11
K. M
EGERIAN
,
12
L. M
ONCELSI
,
11
T. A. M
ORFORD
,
11
M. N
OLTA
,
6
R. O’B
RIENT
,
12
B. O
SHERSON
,
19
I. L. P
ADILLA
,
1, 32
B. R
ACINE
,
17
A. S. R
AHLIN
,
33, 23
C. R
EINTSEMA
,
28
J. E. R
UHL
,
10
M. C. R
UNYAN
,
11
T. M. R
UUD
,
17
E. C. S
HAW
,
19
C. S
HIU
,
9
J. D. S
OLER
,
34
X. S
ONG
,
9
A. T
RANGSRUD
,
11, 12
C. T
UCKER
,
7
R. S. T
UCKER
,
11
A. D. T
URNER
,
12
J. F.
VAN DER
L
IST
,
9
A. C. W
EBER
,
12
I. K. W
EHUS
,
17
S. W
EN
,
10
D. V. W
IEBE
,
8
AND
E. Y. Y
OUNG
29, 30
1
Department of Astronomy and Astrophysics, University of Toronto, 50 St. George Street, Toronto, ON M5S 3H4 Canada
2
Dunlap Institute for Astronomy and Astrophysics, University of Toronto, 50 St. George Street, Toronto, ON M5S 3H4 Canada
3
Department of Physics, University of Toronto, 60 St. George Street, Toronto, ON M5S 3H4 Canada
4
Department of Physics, Washington University in St. Louis, 1 Brookings Drive, St. Louis, MO 63130, USA
5
McDonnell Center for the Space Sciences, Washington University in St. Louis, 1 Brookings Drive, St. Louis, MO 63130, USA
6
Canadian Institute for Theoretical Astrophysics, University of Toronto, 60 St. George Street, Toronto, ON M5S 3H8, Canada
7
School of Physics and Astronomy, Cardiff University, The Parade, Cardiff, CF24 3AA, UK
8
Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada
9
Department of Physics, Princeton University, Jadwin Hall, Princeton, NJ 08544, USA
10
Physics Department, Case Western Reserve University, 10900 Euclid Ave, Rockefeller Building, Cleveland, OH 44106, USA
11
Division of Physics, Mathematics and Astronomy, California Institute of Technology, MS 367-17, 1200 E. California Blvd., Pasadena, CA 91125, USA
12
Jet Propulsion Laboratory, Pasadena, CA 91109, USA
13
School of Electrical, Computer, and Energy Engineering, Arizona State University, 650 E Tyler Mall, Tempe, AZ 85281, USA
14
Department of Physics, McGill University, 3600 Rue University, Montreal, QC, H3A 2T8, Canada
15
School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa
16
Blackett Laboratory, Imperial College London, SW7 2AZ, London, UK
17
Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029 Blindern, NO-0315 Oslo, Norway
18
Department of Physics, Shahid Beheshti University, 1983969411, Tehran Iran
19
Department of Physics, University of Illinois at Urbana-Champaign, 1110 W. Green Street, Urbana, IL 61801, USA
20
Department of Astronomy, University of Illinois at Urbana-Champaign, 1002 W. Green Street, Urbana, IL 61801, USA
21
Department of Physics, University of Texas, 2515 Speedway, C1600, Austin, TX 78712, USA
22
The Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm University, AlbaNova, SE-106 91 Stockholm, Sweden
23
Kavli Institute for Cosmological Physics, University of Chicago, 5640 S Ellis Avenue, Chicago, IL 60637 USA
24
Steward Observatory, 933 North Cherry Avenue, Tucson, AZ, 85721, USA
25
APC, Univ. Paris Diderot, CNRS/IN2P3, CEA/Irfu, Obs de Paris, Sorbonne Paris Cité, France
26
High Energy Physics Division, Argonne National Laboratory, Argonne, IL, USA 60439
27
Department of Astronomy and Astrophysics, Pennsylvania State University, 520 Davey Lab, University Park, PA 16802, USA
28
National Institute of Standards and Technology, 325 Broadway Mailcode 817.03, Boulder, CO 80305, USA
29
Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford, CA 94305, USA
30
SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA
31
Department of Mechanical and Aerospace Engineering, Princeton University, Engineering Quadrangle, Princeton, NJ 08544, USA
32
Department of Physics and Astronomy, Johns Hopkins University, 3701 San Martin Drive, Baltimore, MD 21218 USA
33
Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, IL 60510-5011, USA
34
Max-Planck-Institute for Astronomy, Konigstuhl 17, 69117, Heidelberg, Germany
ABSTRACT
Corresponding author: Jason S.-Y. Leung
leung@astro.utoronto.ca
arXiv:2111.01113v2 [astro-ph.CO] 21 Apr 2022
2
S
PIDER
C
OLLABORATION
Modern cosmic microwave background (CMB) analysis pipelines regularly employ complex time-domain
filters, beam models, masking, and other techniques during the production of sky maps and their corresponding
angular power spectra. However, these processes can generate couplings between multipoles from the same
spectrum and from different spectra, in addition to the typical power attenuation. Within the context of pseudo-
C
`
based,
MASTER
-style analyses, the net effect of the time-domain filtering is commonly approximated by a
multiplicative transfer function,
F
`
, that can fail to capture mode mixing and is dependent on the spectrum of the
signal. To address these shortcomings, we have developed a simulation-based spectral correction approach that
constructs a two-dimensional transfer matrix,
J
``
′
, which contains information about mode mixing in addition
to mode attenuation. We demonstrate the application of this approach on data from the first flight of the S
PIDER
balloon-borne CMB experiment.
1.
INTRODUCTION
Producing well-characterized maps of the cosmic mi-
crowave background (CMB) from time-ordered data requires
accurately accounting for the impact of instrumental effects
and any signal processing on the underlying astrophysical
signal (
e.g.
, Jarosik et al. 2007; Planck Collaboration et al.
2016). These processing techniques typically also have a
nontrivial impact on the fidelity of the cosmological signal
in ways that are spatially anisotropic and inhomogeneous.
These effects must be precisely and accurately characterized
in order to avoid biasing the estimation of the CMB angular
power spectra, and therefore the inference of cosmological
parameters.
The impact of timestream processing and instrument re-
sponse is commonly approximated in the harmonic-domain
with a filter window function unique to the instrument, de-
rived from the analysis of large ensembles of signal simula-
tions. In its simplest formulation, the processing pipeline is
modeled as a power attenuation mechanism in each multi-
pole,
i.e.
, the filter window function is a transfer function –
a simple one-dimensional vector of ratios of output to input
power.
This paper addresses the shortcomings of the one-
dimensional model and proposes an alternative approach
through the construction of a two-dimensional transfer ma-
trix.
This simulation-based spectral correction approach
takes into account both the mode mixing and attenuation
from instrumental and data processing effects including
beams, filtering, and masking. Section 2 of this paper de-
scribes the theoretical motivation and compares the transfer
matrix to the one-dimensional transfer function approach. A
concrete example is presented in Section 3 using data from
the first flight of S
PIDER
, a balloon-borne telescope designed
to measure CMB polarization on roughly degree angular
scales (S
PIDER
Collaboration 2022). This section explores
different techniques for constructing the transfer matrix to
reduce the computational demand including using binned
power spectra and performing Fourier-space interpolation.
Section 4 presents several comparisons of these techniques,
including tests of signal recovery on spectra with different
shapes. While all tested approaches were found to accurately
recover a target spectrum identical to that used for the trans-
fer matrix construction, the performance varied when applied
to a different target spectrum. As discussed in Section 5, this
has implications for increasingly sensitive CMB polarization
measurements where the cosmological signals are heavily
obscured by Galactic foregrounds. As the foreground power
spectra are less well constrained and vary substantially be-
tween different sky regions, understanding the signal depen-
dence of potential analysis techniques becomes even more
important.
2.
THEORETICAL DESCRIPTION
Maps of the CMB temperature (
T
) and linear polarization
(
Q
,
U
) anisotropies over the partial sky can be decomposed
into linear combinations of spherical harmonics:
T
(
r
)
W
(
r
) =
∑
`
,
m
̃
a
T
`
m
Y
`
m
(
r
)
,
(1)
[
Q
(
r
)
±
iU
(
r
)
]
W
(
r
) =
∑
`
,
m
±
2
̃
a
`
m
±
2
Y
`
m
(
r
)
,
(2)
where
W
(
r
) is the window representing the relative weights
of the partial-sky mask. To avoid using spin-weighted spher-
ical harmonic components, the
±
2
̃
a
`
m
coefficients are fre-
quently expressed as a combination of scalar and fixed parity
E
and
B
components
±
2
̃
a
`
m
=
−
(
̃
a
E
`
m
±
i
̃
a
B
`
m
)
,
(3)
where the sign convention follows Zaldarriaga & Seljak
(1997). For each ̃
a
`
m
, the pseudo-power spectrum
̃
C
`
is de-
fined as
̃
C
`
=
1
2
`
+
1
∑
m
|
̃
a
`
m
|
2
.
(4)
Also known as a pseudo-
C
`
(PCL), it is related to the angular
power spectrum specified by the theory of primordial pertur-
bations,
C
`
, via
〈
̃
C
X
`
〉
=
∑
`
′
,
X
′
K
XX
′
``
′
C
X
′
`
′
,
(5)
where
K
XX
′
``
′
is a mode coupling kernel (or “mixing matrix”)
that accounts for the mixing within and between the observ-
ables
X
∈{
TT
,
EE
,
BB
,
TE
,
EB
,
TB
}
due to the partial-sky
M
ODE
-C
OUPLING
C
ORRECTION OF
CMB P
OWER
S
PECTRA
3
mask, and the brackets
〈·〉
denote an ensemble average over
infinite spectrum realizations. Because
K
XX
′
``
′
is entirely de-
termined by the chosen pixel weighting and the geometry of
the cut sky (Hivon et al. 2002), in the absence of instrumen-
tal effects and noise, PCL estimators can use the (finite) set
of measured
̃
C
`
to solve for the underlying power spectrum
C
`
. Examples of PCL estimators include
MASTER
(Hivon
et al. 2002),
NaMaster
(Alonso Monge et al. 2019), and
PolSpice
(Chon et al. 2004).
A major challenge in interpreting CMB data is that the in-
struments cannot directly probe the true sky signal; the in-
coming signals are inevitably altered by instrumental sys-
tematics and noise. Therefore, as described in Section 1, an
experiment’s raw datastreams must be processed to remove
many different types of spurious signals. The act of observ-
ing and time-domain filtering both distort the signal estimate
and present additional sources of mode coupling. Assuming
that the coupling is homogeneous, isotropic, and linear, the
impact of the experiment is captured by introducing another
coupling matrix,
F
XX
′
``
′
, such that
〈
̃
C
X
`
〉
=
∑
`
′′
,
X
′′
∑
`
′
,
X
′
K
XX
′
``
′
F
X
′
X
′′
`
′
`
′′
C
X
′′
`
′′
.
(6)
In general,
F
XX
′
``
′
is unique to the experiment and cannot be
determined analytically.
2.1.
The Filter Transfer Function
To reduce complexity,
F
XX
′
``
′
is often approximated as a di-
agonal matrix whose entries represent the one-dimensional
transfer function
F
XX
′
`
, such that
〈
̃
C
`
〉
=
∑
`
′
K
``
′
F
`
′
C
`
′
(7)
(here and hereafter, the superscripts
X
are suppressed for
brevity). Using a set of pseudo-power spectra
̃
C
`
derived
from an ensemble of simulations of a known power spec-
trum
C
`
, the transfer function
F
`
can be estimated through an
iterative process to avoid inverting
K
``
′
(Hivon et al. 2002;
Dutcher et al. 2021). Note that
F
`
is frequently decomposed
further into a filter component
f
`
and a beam component
b
2
`
,
such that
F
`
=
f
`
b
2
`
; additional instrumental effects can be in-
serted in a similar fashion.
This one-dimensional approximation implicitly assumes
that each
`
-mode remains independent throughout the entire
filtering process. As long as this assumption holds, the map-
ping from input to output is one-to-one:
F
`
is simply the ratio
of output to input power for each
`
. However, in practice, we
expect modes to become coupled with one another, where the
mapping becomes many-to-one and the contributions from
the coupled input modes become impossible to disentangle
using
F
`
alone. The consequence is that changing the input
power spectrum
C
`
also changes the output
̃
C
`
in some non-
trivial way due to this many-to-one mapping. In other words,
F
`
is inextricably tied to the input used to compute it.
More concretely, the one-dimensional transfer function
formulation conflates mode mixing with the in-mode filter
gain. This introduces a sensitivity to the power spectrum of
the simulated sky used to calibrate
F
`
; the final spectrum is
correct only if the simulated sky is statistically similar to the
true sky (
i.e.
, a Gaussian sky realization with the same power
spectra).
2.2.
The Multipole–Multipole Transfer Matrix
To address this shortcoming of the
F
`
formulation, we in-
troduce a two-dimensional linear operator that encodes the
(asymmetric) coupling between each
``
′
pair:
〈
̃
C
`
〉
=
∑
`
′
J
``
′
C
`
′
.
(8)
We refer to this coupling operator
J
``
′
as the “transfer ma-
trix” because it directly relates the input of the true power
spectrum to the output of the spectrum estimator.
This
relation holds as long as the filtering process is approxi-
mately linear. Treating the entire pipeline as a single oper-
ator avoids the diagonal approximation and ensures that all
multipole–multipole couplings induced by time-domain and
map-domain filtering are properly taken into account,
i.e.
, we
are not locked into a specific input spectrum.
Note that these couplings extend to those between the six
power spectra; both temperature-to-polarization (
T
-to-
P
) and
E
-to-
B
leakage are automatically included. Because standard
PCL spectrum estimators provide the
TT
,
EE
,
BB
,
TE
,
EB
,
and
TB
power spectra, any input mode can be readily related
to an output mode from any of these six power spectra, result-
ing in 6
2
coupling matrices. We find it convenient to compile
these individual matrices into a single 6
×
6 block matrix en-
capsulating every
``
′
coupling between the six power spec-
tra. Following the rules of matrix multiplication (Equation
8), we arrange these blocks horizontally according to input
spectra and vertically according to output spectra (see Figure
1). The ordering of the six spectra does not matter as long
as it is consistent between the two axes; likewise, the six
C
`
s
must be concatenated in the same order. We choose the above
ordering based on convenience.
While the approach presented here has similarities to that
used by the BICEP/Keck Collaboration for their CMB polar-
ization analyses, the implementation varies due to key dif-
ferences in the observing strategies and analysis pipelines.
As described in Ade et al. (2016), the simplicity of the BI-
CEP/Keck observing strategy allows for the construction of
an observing matrix that renders large simulation ensembles
more computationally tractable. Having determined their fil-
tering operations to be linear, the observing matrix is used
4
S
PIDER
C
OLLABORATION
0
:
0
0
:
2
0
:
4
0
:
6
0
:
8
1
:
0
0
:
0
0
:
2
0
:
4
0
:
6
0
:
8
1
:
0
SPIDER re-observation pipeline
Arrange
TT
,
EE
,
BB
,
TE
,
EB
,
TB
,
responses in a
single column
−
1
:
0
−
0
:
5
0
:
0
0
:
5
1
:
0
−
0
:
50
−
0
:
25
0
:
00
0
:
25
0
:
50
0
:
75
1
:
00
1
:
25
1
:
50
Input mode
‘
Input spectrum
0
20
40
60
80
0
20
40
60
80
synfast
map
0
20
40
60
80
0
20
40
60
80
Re-obs. map
−
1
:
0
−
0
:
5
0
:
0
0
:
5
1
:
0
−
0
:
50
−
0
:
25
0
:
00
0
:
25
0
:
50
0
:
75
1
:
00
1
:
25
Output response
1
2
3
4
5
TT
EE
BB
TE
EB
TB
Input
1
2
3
4
5
TB
EB
TE
BB
EE
TT
Output
Figure 1.
Procedure to create the transfer matrix
J
``
′
. For each input mode, the six output response spectra (gains) are arranged vertically to
form the columns of the matrix. Each of the 36 blocks within the matrix spans a range of multipoles with
`
min
≤
`
≤
`
max
.
to construct the transfer matrix
J
``
′
, which is then used sim-
ilarly to reconstruct and interpret the on-sky power spectra.
Because the observing matrix formulation is less generally
applicable to experiments at other sites with different observ-
ing strategies, the remainder of this paper is dedicated to es-
timating
J
``
′
through other means.
3.
APPLICATION TO S
PIDER
DATA
To illustrate the application of the transfer matrix, we
present results from the S
PIDER
balloon-borne telescope.
During its first Antarctic long-duration balloon (LDB) flight
in January 2015, S
PIDER
mapped 4
.
8 % of the sky with po-
larimeters operating at 95 and 150 GHz to constrain the
B
-
mode power spectrum from primordial gravitational waves
(S
PIDER
Collaboration 2022). An upcoming flight with ad-
ditional 280 GHz receivers will provide improved characteri-
zation of the polarized Galactic dust foregrounds (Shaw et al.
2020).
S
PIDER
’s processing pipeline is described more fully in
S
PIDER
Collaboration (2022) and S
PIDER
Collaboration
(2022, in prep.); here we briefly highlight the most relevant
steps and note that they are sufficiently linear to allow us-
age of Equation 8. Once features such as cosmic-ray hits,
payload transmitter signals, and thermal transients have been
removed from the raw detector timestreams, they are filtered
to reduce quasi-stationary noise. Null test performance was
used to identify the weakest filter that sufficiently removed
quasi-stationary noise, resulting in a fifth-order polynomial
fit to each detector’s data as a function of azimuth angle
between scan turnarounds. The impact of scanning, filter-
ing, and flagging are determined by applying the entire pro-
cessing pipeline to an ensemble of time-domain signal sim-
ulations in a procedure known as re-observation. The re-
observed timestreams are produced at the full data sample
rate without any downsampling or binning. Unlike the BI-
CEP/Keck experiment, S
PIDER
’s observing strategy does not
allow for the creation of an observing matrix, so obtaining the
transfer matrix
J
``
′
requires producing an appropriate set of
re-observed CMB maps.
S
PIDER
’s measured and simulated timestreams are con-
verted into two-dimensional maps of the sky with a binned
mapmaker. This approach assembles the detector data into
spatial pixels based on the telescope pointing and polariza-
tion sensitivity as described further in S
PIDER
Collaboration
(2022). The computational simplicity of this method is cru-
cial for enabling the generation of large simulation ensem-
bles, including those used in this work.
The main S
PIDER
cosmological results presented in S
PI
-
DER
Collaboration (2022) use two complementary pipelines
for power spectrum estimation. Here we describe results
only from the Noise Simulation Independent (NSI) pipeline,
while the other pipeline is presented in Gambrel et al. (2021).
The NSI pipeline is similar to Xspect (Tristram et al. 2005)