Journal ArticleDOI

# Constraints on Cosmological Parameters from the Angular Power Spectrum of a Combined 2500 deg$^2$ SPT-SZ and Planck Gravitational Lensing Map

Abstract: We report constraints on cosmological parameters from the angular power spectrum of a cosmic microwave background (CMB) gravitational lensing potential map created using temperature data from 2500 deg$^2$ of South Pole Telescope (SPT) data supplemented with data from Planck in the same sky region, with the statistical power in the combined map primarily from the SPT data. We fit the corresponding lensing angular power spectrum to a model including cold dark matter and a cosmological constant ($\Lambda$CDM), and to models with single-parameter extensions to $\Lambda$CDM. We find constraints that are comparable to and consistent with constraints found using the full-sky Planck CMB lensing data. Specifically, we find $\sigma_8 \Omega_{\rm m}^{0.25}=0.598 \pm 0.024$ from the lensing data alone with relatively weak priors placed on the other $\Lambda$CDM parameters. In combination with primary CMB data from Planck, we explore single-parameter extensions to the $\Lambda$CDM model. We find $\Omega_k = -0.012^{+0.021}_{-0.023}$ or $M_{ u}< 0.70$eV both at 95% confidence, all in good agreement with results that include the lensing potential as measured by Planck over the full sky. We include two independent free parameters that scale the effect of lensing on the CMB: $A_{L}$, which scales the lensing power spectrum in both the lens reconstruction power and in the smearing of the acoustic peaks, and $A^{\phi \phi}$, which scales only the amplitude of the CMB lensing reconstruction power spectrum. We find $A^{\phi \phi} \times A_{L} =1.01 \pm 0.08$ for the lensing map made from combined SPT and Planck temperature data, indicating that the amount of lensing is in excellent agreement with what is expected from the observed CMB angular power spectrum when not including the information from smearing of the acoustic peaks.
Topics: Gravitational lens (55%), Planck temperature (54%), South Pole Telescope (53%), Spectral density (52%)

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Lawrence Berkeley National Laboratory
Recent Work
Title
Constraints on Cosmological Parameters from the Angular Power Spectrum of a Combined
2500 deg
2
SPT-SZ and Planck Gravitational Lensing Map
https://escholarship.org/uc/item/3fz895wq
Journal
Astrophysical Journal, 860(2)
ISSN
0004-637X
Authors
Simard, G
Omori, Y
Aylor, K
et al.
Publication Date
2018-06-20
DOI
10.3847/1538-4357/aac264
Peer reviewed
University of California

Constraints on Cosmological Parameters from the Angular Power Spectrum of a
Combined 2500 deg
2
SPT-SZ and Planck Gravitational Lensing Map
G. Simard
1
, Y. Omori
1,2,3
, K. Aylor
4
, E. J. Baxter
5,6,7
, B. A. Benson
6,7,8
, L. E. Bleem
6,9
, J. E. Carlstrom
6,7,9,10,11
,
C. L. Chang
6,7,9
, H-M. Cho
12
, R. Chown
1
, T. M. Crawford
6,7
, A. T. Crites
6,7,13
, T. de Haan
1,14
, M. A. Dobbs
1,15
,
W. B. Everett
16
, E. M. George
14,17
, N. W. Halverson
16,18
, N. L. Harrington
14
, J. W. Henning
6,9
, G. P. Holder
1,15,19,20
,
Z. Hou
6,7
, W. L. Holzapfel
14
, J. D. Hrubes
21
, L. Knox
4
, A. T. Lee
14,22
, E. M. Leitch
6,7
, D. Luong-Van
21
, A. Manzotti
6,7
,
J. J. McMahon
23
, S. S. Meyer
6,7,10,11
, L. M. Mocanu
6,7
, J. J. Mohr
24,25,26
, T. Natoli
6,10,27
6,7
, C. Pryke
28
,
C. L. Reichardt
14,29
, J. E. Ruhl
30
, J. T. Sayre
16,30
, K. K. Schaffer
6,11,31
, E. Shirokoff
6,7,14
, Z. Staniszewski
30,32
, A. A. Stark
33
,
K. T. Story
2,3,6,10
, K. Vanderlinde
27,34
, J. D. Vieira
19,20
, R. Williamson
6,7
, and W. L. K. Wu
6
1
Department of Physics and McGill Space Institute, McGill University, Montreal, Quebec H3A 2T8, Canada; gholder@illinois.edu
2
Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, 452 Lomita Mall, Stanford, CA 94305, USA
3
Dept. of Physics, Stanford University, 382 Via Pueblo Mall, Stanford, CA 94305, USA
4
Department of Physics, University of California, Davis, CA 95616, USA
5
Center for Particle Cosmology, Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA
6
Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA
7
Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL 60637, USA
8
Fermi National Accelerator Laboratory, MS209, P.O. Box 500, Batavia, IL 60510, USA
9
High Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA
10
Department of Physics, University of Chicago, Chicago, IL 60637, USA
11
Enrico Fermi Institute, University of Chicago, Chicago, IL 60637, USA
12
SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA
13
California Institute of Technology, Pasadena, CA 91125, USA
14
Department of Physics, University of California, Berkeley, CA 94720, USA
15
16
Center for Astrophysics and Space Astronomy, Department of Astrophysical and Planetary Sciences, University of Colorado, Boulder, CO 80309, USA
17
European Southern Observatory, Karl-Schwarzschild-Straße 2, D-85748 Garching, Germany
18
Department of Physics, University of Colorado, Boulder, CO 80309, USA
19
Astronomy Department, University of Illinois at Urbana-Champaign, 1002 W. Green Street, Urbana, IL 61801, USA
20
Department of Physics, University of Illinois Urbana-Champaign, 1110 W. Green Street, Urbana, IL 61801, USA
21
University of Chicago, Chicago, IL 60637, USA
22
Physics Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
23
Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA
24
Faculty of Physics, Ludwig-Maximilians-Universität, D-81679 München, Germany
25
Excellence Cluster Universe, D-85748 Garching, Germany
26
Max-Planck-Institut für extraterrestrische Physik, D-85748 Garching, Germany
27
Dunlap Institute for Astronomy & Astrophysics, University of Toronto, 50 St George St, Toronto, ON, M5S 3H4, Canada
28
Department of Physics, University of Minnesota, Minneapolis, MN 55455, USA
29
School of Physics, University of Melbourne, Parkville, VIC 3010, Australia
30
Physics Department, Center for Education and Research in Cosmology and Astrophysics, Case Western Reserve University, Cleveland, OH 44106, USA
31
Liberal Arts Department, School of the Art Institute of Chicago, Chicago, IL 60603, USA
32
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA
33
Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA
34
Department of Astronomy & Astrophysics, University of Toronto, 50 St George St, Toronto, ON, M5S 3H4, Canada
Received 2017 December 28; revised 2018 April 19; accepted 2018 May 1; published 2018 June 20
Abstract
We report constraints on cosmological parameters from the angular power spectrum of a cosmic microwave
background (CMB) gravitational lensing potential map created using temperature data from 2500 deg
2
of South
Pole Telescope (SPT) data supplemented with data from Planckin the same sky region, with the statistical power
in the combined map primarily from the SPT data. We t the lensing power spectrum to a model including cold
dark matter and a cosmological constant (
LCDM
), and to models with single-parameter extensions to
LCDM
.We
nd constraints that are comparable to and consistent with those found using the full-sky PlanckCMB lensing
data, e.g.,
s
W
8
m
0.2
5
=0.598±0.024 from the lensing data alone with weak priors placed on other parameters.
Combining with primary CMB data, we explore single-parameter extensions to
LCDM
.Wend
W=
k
-
-
+
0.012
0.023
0.021
or
n
M
<0.70 eV at 95% condence, in good agreement with results including the lensing potential as
measured by Planck. We include two parameters that scale the effect of lensing on the CMB:
A
L
, which scales the
lensing power spectrum in both the lens reconstruction power and in the smearing of the acoustic peaks, and
f
f
A
,
which scales only the amplitude of the lensing reconstruction power spectrum. We nd
f
f
A
×
A
L
=1.01±0.08
for the lensing map made from combined SPT and Planckdata, indicating that the amount of lensing is in excellent
agreement with expectations from the observed CMB angular power spectrum when not including the information
from smearing of the acoustic peaks.
Key words: cosmic background radiation cosmological parameters gravitational lensing: weak
The Astrophysical Journal, 860:137 (9pp), 2018 June 20 https://doi.org/10.3847/1538-4357/aac264
1

1. Introduction
Gravitational lensing of the cosmic microwave background
(CMB) has emerged as a useful cosmological tool. CMB
lensing, which probes all structure along a given line of sight,
provides complementary information to the primary CMB
uctuations which measure structure at z1100. The
sensitivity of CMB lensing peaks at intermediate redshifts
(z3), making it complementary to large-scale structure
surveys, the sensitivity of which typically peaks at lower
redshifts, and with very different sources of possible systematic
errors. Lensing of the CMB was rst detected in cross-
correlation with galaxies a decade ago (Smith et al. 2007); high
signal-to-noise internal detections have now been achieved by
many experiments (Das et al. 2011b; van Engelen et al. 2012;
Planck Collaboration 2014b; POLARBEAR Collaboration
2014; BICEP2 Collaboration et al. 2016). For a review of
CMB lensing, see Challinor & Lewis (2005).
The uctuations in the CMB lensing potential form a nearly
Gaussian projected eld on the sky, with statistical properties
determined by the geometry and the history of structure
formation in the universe. Because the eld is nearly Gaussian,
essentially all the information is encoded in the angular power
spectrum. The most precise CMB lensing power spectrum
measurements to date are from the Planckexperiment (Planck
Collaboration 2016b).
Cosmological parameter ts that include CMB lensing
information are broadly consistent with expectations from the
primary CMB measurements alone (Planck Collaboration
2016a). There are, however, mild but interesting tensions
(2σ) between constraints on cosmology from Planckprimary
CMB measurements and other cosmological probes. Speci-
cally related to lensing, the amplitude of the matter power
spectrum on galaxy scales (σ
8
) inferred from Planckprimary
CMB data is slightly higher than that determined from cosmic
shear measurements (Hildebrandt et al. 2017; Joudaki et al.
2017; Troxel et al. 2017). Further, specically related to
lensing of the CMB, the amount of lensing inferred from
the measured smearing of the acoustic peaks is higher than
that inferred from the direct measurement of the lensing-
induced mode-coupling (Planck Collaboration 2016a). The
amplitude of lensing is expected to be a powerful probe of
neutrino masses (Abazajian et al. 2015), so discordance in
measurements of lensing amplitudes is important for under-
standing the utility of these measurements as probes of particle
physics.
This paper is a companion to Omori et al. (2017), referred to
as O17 hereafter. In that work, we obtained a CMB temperature
map by combining 150 GHz SPT and 143 GHz Planckdata in
the 2500deg
2
South Pole Telescope (SPT)-SZ survey region,
and we used the resulting temperature map to produce a map of
the projected gravitational lensing potential. In this paper, we
present a cosmological parameter analysis of the CMB lensing
power spectrum derived in O17. This spectrum is shown in
Figure 1, along with other recent measurements, including the
full-sky Plancklensing power spectrum.
This work is divided as follows: in Section 2 we review
gravitational lensing of the CMB and reconstruction of the
lensing potential; in Section 3 we describe the CMB
temperature data and simulations used for the O17 analysis
and for this work; in Section 4 we describe how the lensing
likelihood is constructed, including linear corrections for the
unknown true CMB and lensing potential power spectra; in
Section 5 we present the primary result of this paper:
constraints on cosmological parameters; we close with a
discussion.
Throughout this work, we use the Planck
TT + LOWP +
LENSING
cosmology
35
(Planck Collaboration 2016a) as a
ducial model. This ducial cosmology is used for generating
the simulated data necessary for the lensing reconstruction. All
CMB temperature and lensing potential power spectra used in
the present analysis have been computed with the
CAMB
Boltzmann code
36
(Lewis et al. 2000).
2. Lensing Reconstruction Framework
In this section, we build the theoretical framework for the
lensing likelihood, presenting selected elements from the
lensing reconstruction pipeline. A more complete description
of the procedure can be found in O17.
2.1. Lensing of CMB Temperature Fluctuations
Gravitational lensing remaps CMB uctuations in position
space (Lewis & Challinor 2006):
f=+(
ˆ
)(
ˆ
(
ˆ
)) ( )nnnTT ,1
LU
where
f
(
ˆ
)n
is the projected gravitational lensing potential and
superscripts L and U refer to the lensed and unlensed
temperature elds respectively. To gain intuition, Equation (1)
can be Taylor expanded as
f=++¼(
ˆ
)(
ˆ
(
ˆ
)()nn nTT T .2
LU U
From the second term, it can be seen that the observed lensed
temperature has a component that is the gradient of the
unlensed eld modulated by the lensing deection f .Ifwe
transform to harmonic space, Equation (2) would have the
second term on the right-hand side written as a weighted
convolution of the temperature eld and the lensing potential,
where the harmonic transform for any particular mode for the
lensed eld could involve a sum over all of the modes of the
unlensed eld. Lensing thus introduces non-zero off-diagonal
elements in the covariance of observed temperature elds in
harmonic space (Okamoto & Hu 2003):
å
f
ñ
=-
-
f
() ()
TT
ℓℓ L
mm M
W1,3
ℓm m
LM
M
ℓℓL
LM
12
12
11 2 2
12
where T
m
are the spherical harmonic expansion coefcients of
the temperature elds and f
LM
the coefcients of the projected
lensing potential. The weight
p
=-
+++
´
+-
-
´+++«
f
++
⎜⎟
()( )()
()
()()( ) ()
W
ℓℓ L
C
ℓℓ L
LL
212 121
4
11
2
10 1
11 4
ℓℓL
TT
ℓℓL
12
12
11 1 2
12
1
12
characterizes the mode coupling induced by lensing (i.e., the
effect of the convolution in Equation (2)).
35
base_plikHM_TT_lowTEB_lensing.
36
http://camb.info (2016 May version).
2
The Astrophysical Journal, 860:137 (9pp), 2018 June 20 Simard et al.

2.2. Lensing Map Reconstruction
The lensing potential can be estimated from observed CMB
maps by measuring the lensing-induced mode coupling of
Equation (3) between pairs of modes in the observed
temperature eld (Zaldarriaga & Seljak 1999; Hu & Okamoto
2002). In general, it is best to use pairs in harmonic space that
have good signal-to-noise for measuring lensing. For this
purpose, it is useful to work with a ltered map:
º
¯
TFT
ℓm ℓm ℓm
,
with the lter
º+
-
()FCN
ℓm ℓm
1
for a given CMB power
spectrum C
and an anisotropic (m-dependent) noise power
spectrum N
m
.
A formally optimal estimator (at rst order) which
maximizes signal to noise in the estimated lensing potential
(Hu & Okamoto 2002) is
å
f
=
-
-
f
¯
()
()
ℓℓ L
mm M
WTT
1
2
.5
LM
M
ℓm
ℓm
ℓℓL
ℓm m
,
,
12
12
11
22
12
11 2 2
We use Equation (5) as our f estimator for this analysis.
There are other choices (e.g., Namikawa et al. 2013) for how to
weight the mode pairs that sacrice some signal-to-noise but
reduce foreground contamination. Lensing reconstruction is
done with the
QUICKLENS code.
37
The relationship between the ltered estimate of the lensing
potential resulting from Equation (5) and the true potential can
be written as
ffº
f
¯
(),6
LM LM LM
dening a response function
LM
that in general depends on
both L and M. As outlined in O17, this response function has
been calibrated using simulations. We estimate it by measuring
the cross-spectrum of simulated lensing potential outputs with
the input lensing potential maps and normalizing by the
autospectrum of the inputs.
The true amplitude of mode coupling in the CMB
temperature eld induced by lensing is sensitive to the true
(unknown) temperature power spectrum, as can be seen in
Equations (3) and (4). What is measured in the data is some
amount of mode coupling; to turn this into an estimate of the
the typical amplitudes of the modes being coupled. The
response function thus depends on the assumed cosmological
parameters. To explore this cosmological dependence, we use
an isotropic approximation to the full anisotropic response
function and its dependence on cosmology. In the case where
both the signal and noise are isotropic (i.e., the CMB signal and
noise only depend statistically on and not m), the response
function can be written as
å
=
+
fff
()
L
WWFF
1
21
,7
L
ℓℓ
ℓℓL
t
ℓℓL
f
ℓℓ
,
,,
12
12 12
12
where we have indicated extra superscripts on the weight
functions for either the true amount of mode coupling (t) or the
assumed amount for our ducial cosmology ( f ). The lters F
are calculated for the ducial cosmology. We use Equation (7)
and its dependence on cosmology to determine the cosmology-
dependent corrections to the simulation-based response
function.
noise all violate statistical stationarity in the data, and
consequently they introduce mode coupling that can bias the
lensing reconstruction. The result is that the lensing reconstruc-
tion has a non-zero mean signaleven in the absence of true
lensing signalthat depends on the survey geometry, mask,
and noise properties. This mean eld
¯
LM
M
is calculated using
simulations and removed.
Figure 1. SPT + Plancklensing bandpowers from O17 along with earlier lensing estimates from the SPT-SZ survey (van Engelen et al. 2012) and recent lensing
bandpowers obtained from temperature and polarization measurements from
SPT
POL
(Story et al. 2015). Also plotted are the most recent lensing autospectrum
measurements from
BICEP2+KECK ARRAY (BICEP2 Collaboration et al. 2016), Planck(Planck Collaboration 2016b), POLARBEAR (POLARBEAR
Collaboration 2014) and
ACT
POL
(Sherwin et al. 2017), and a prediction for the lensing power spectrum using the best-t cosmological parameters from the
Planck
TT + LOWP + LENSING cosmology (Planck Collaboration 2016a).
37
http://github.com/dhanson/quicklens
3
The Astrophysical Journal, 860:137 (9pp), 2018 June 20 Simard et al.

After removing the mean eld and correcting for the
response function, the nal estimate of the lensing potential is
f
ff
=
-
f
ˆ
¯¯
().8
LM
LM
LM
LM
MF
2.3. Lensing Autospectrum Estimation
To estimate the angular power spectrum of the CMB lensing
map obtained in the previous section, we multiply the estimate
f
ˆ
by the survey mask (including point source and galaxy
38
(Szapudi et al. 2001;
Chon et al. 2004) to compute the spectrum of the masked map.
The resulting power spectrum is a biased estimate of the true
lensing power spectrum. Known sources of bias include a
straightforward noise bias,
()
N
L
0
, that comes from taking an
autospectrum of data with noise in it (where noise here
includes the Gaussian part of the CMB temperature eld and
any other sky signal), and a bias that arises from ambiguity in
exactly which lensing modes are being measured in the power
spectrum,
()
N
L
1
(Kesden et al. 2003). The superscript denotes
the order of the lensing power spectrum involved:
()
N
L
0
is
independent of the true lensing power and only depends on the
instrument noise and sky power, while
()
N
L
1
has a linear
dependence on the lensing power. As detailed in O17,we
calculate these biases using simulations and subtract them from
the measured power spectrum
=--
ff
ff
ˆ
()
ˆˆ
() ()
CCNN.9
L
LLL
01
We use a realization-dependent
()
N
L
0
estimate that takes into
account the power in the particular realization but does not
depend on the assumed cosmology (Namikawa et al. 2013).
The
()
N
L
1
bias depends linearly on the lensing power and will
therefore depend on cosmological parameters. In the at-sky
limit (Kesden et al. 2003; Das et al. 2011b; Planck
Collaboration 2014b) and assuming isotropic noise and
ltering, the bias is

òò
pp
=
´
´--
+--
ff
ff
ff
ff
ff
ff
-
-
() ()
() ()
[()()
()()] ()
()
∣∣
∣∣
ℓℓ
ℓℓ
ℓℓ
ℓℓ
N
dd
FFFF W W
CW W
CW W
1
22
,,
,,
,,,10
ℓℓ
ℓℓ
L
LL
ℓℓ
ff
tt
tt
1
2
1
2
2
3
2
,
12
,
34
,
13
,
24
,
14
,
23
123 4
13
14
where the weight
f
()ℓℓ
W
,
12
is the at-sky version of
Equation (4).
There is a dependence on both the true CMB power (just as
for
f
L
) and the lensing power. To explore this cosmological
dependence (below), we will use Equation (10) to determine
the cosmology-dependent corrections to the
()
N
L
1
that is derived
from simulations.
The next-order
()
N
L
2
bias is largely removed by using the
lensed theory temperature power spectrum rather than the
unlensed spectrum when constructing the lensing estimator
(Hanson et al. 2011). There are other biases, such as the
()
N
L
32
bias ( Böhm et al. 2016), which are small at the precision of the
current work, and will be neglected.
We estimate uncertainties on the lensing power spectrum by
averaging over N
s
=198 simulations:
å
D=
-
ñ
ff ff ff
=
(
ˆ
)(
ˆˆ
)()C
N
CC
1
1
.11
L
s
i
N
Li L
N
2
1
;
2
s
s
This procedure could be used to generate a full covariance
matrix, but for this analysis we assume that uncertainties are
uncorrelated between bins. This is expected for the relatively
large bins that we use and the realization-dependent removal of
the
()
N
L
0
bias that strongly reduces the off-diagonal elements
of the covariance matrix ( Schmittfull et al. 2013). From
simulations, we measured the correlation between bins to be no
more than 5%.
3. Lensing Data
The binned CMB lensing angular power spectrum (or
lensing bandpowers)
f
f
ˆ
C
L
b
computed in O17, using the
methods described in that work and summarized in the
previous section, is shown in Figure 1 (along with other recent
measurements from the literature), and the bin ranges and
bandpower values and uncertainties are listed in Table 1.
39
We
will hereafter refer to this as the
SPT + Planck lensing
measurement.
The higher angular resolution of the SPT greatly increases the
lensing signal-to-noise per pixel over Planckfrom the larger
number of available small-scale modes which can be used
for measuring the lensing-induced mode coupling. Combining
the Planckand SPT temperature maps strongly reduces the
uncertainties, in particular on small scales (higher L) as
compared to using only the SPT data. This happens because
the lensing map only uses modes in the temperature map
extending to =3000, to minimize possible foreground
contamination. The high-L lensing modes require probing
correlations in the temperature angular modes that are widely
separated in harmonic space. By using the Planckdata to
recover the low- modes, there is an increased number of large-
separation mode pairs.
As shown in O17, the
SPT + Planckmeasurements over
the 2500 deg
2
SPT-SZ survey area are more precise than the
Planck-only full-sky constraints for L1000. From the
relative sky coverage, the Planck-only uncertainties using only
the SPT region would be more than three times larger than the
Planck-only full-sky constraints. The combined
SPT +
Planckmeasurements are thus nearly statistically independent,
Small-scale lensing measurements are most susceptible to
foreground contamination, as shown in van Engelen et al.
(2014). In that work, it was found that foreground contamina-
tion increased dramatically beyond L2000 for CMB map
ltering choices similar to those adopted in O17. For the
cosmological parameter estimation in this work, we therefore
use the
SPT + Plancklensing measurements only below
L=2000.
A comparison of the O17 bandpowers with the prediction
from the best-t Planckcosmology is shown in Figure 2. The
ratio is shown with and without a correction for foreground
contamination, based on van Engelen et al. (2014). The
estimated contamination is small, never exceeding more than
5% of the uncertainty in any L bin, but not completely
38
http://www2.iap.fr/users/hivon/software/PolSpice
39
https://pole.uchicago.edu/public/data/simard18
4
The Astrophysical Journal, 860:137 (9pp), 2018 June 20 Simard et al.

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Journal ArticleDOI
Nabila Aghanim1, Yashar Akrami2, Yashar Akrami3, Yashar Akrami4  +202 moreInstitutions (63)
Abstract: We present measurements of the cosmic microwave background (CMB) lensing potential using the final Planck 2018 temperature and polarization data. Using polarization maps filtered to account for the noise anisotropy, we increase the significance of the detection of lensing in the polarization maps from 5σ to 9σ . Combined with temperature, lensing is detected at 40σ . We present an extensive set of tests of the robustness of the lensing-potential power spectrum, and construct a minimum-variance estimator likelihood over lensing multipoles 8 ≤ L ≤ 400 (extending the range to lower L compared to 2015), which we use to constrain cosmological parameters. We find good consistency between lensing constraints and the results from the Planck CMB power spectra within the ΛCDM model. Combined with baryon density and other weak priors, the lensing analysis alone constrains (1σ errors). Also combining with baryon acoustic oscillation data, we find tight individual parameter constraints, σ 8 = 0.811 ± 0.019, , and . Combining with Planck CMB power spectrum data, we measure σ 8 to better than 1% precision, finding σ 8 = 0.811 ± 0.006. CMB lensing reconstruction data are complementary to galaxy lensing data at lower redshift, having a different degeneracy direction in σ 8 − Ωm space; we find consistency with the lensing results from the Dark Energy Survey, and give combined lensing-only parameter constraints that are tighter than joint results using galaxy clustering. Using the Planck cosmic infrared background (CIB) maps as an additional tracer of high-redshift matter, we make a combined Planck -only estimate of the lensing potential over 60% of the sky with considerably more small-scale signal. We additionally demonstrate delensing of the Planck power spectra using the joint and individual lensing potential estimates, detecting a maximum removal of 40% of the lensing-induced power in all spectra. The improvement in the sharpening of the acoustic peaks by including both CIB and the quadratic lensing reconstruction is detected at high significance.

228 citations

Journal ArticleDOI
Abstract: Next-generation cosmic microwave background (CMB) experiments will have lower noise and therefore increased sensitivity, enabling improved constraints on fundamental physics parameters such as the sum of neutrino masses and the tensor-to-scalar ratio r. Achieving competitive constraints on these parameters requires high signal-to-noise extraction of the projected gravitational potential from the CMB maps. Standard methods for reconstructing the lensing potential employ the quadratic estimator (QE). However, the QE performs suboptimally at the low noise levels expected in upcoming experiments. Other methods, like maximum likelihood estimators (MLE), are under active development. In this work, we demonstrate reconstruction of the CMB lensing potential with deep convolutional neural networks (CNN) - ie, a ResUNet. The network is trained and tested on simulated data, and otherwise has no physical parametrization related to the physical processes of the CMB and gravitational lensing. We show that, over a wide range of angular scales, ResUNets recover the input gravitational potential with a higher signal-to-noise ratio than the QE method, reaching levels comparable to analytic approximations of MLE methods. We demonstrate that the network outputs quantifiably different lensing maps when given input CMB maps generated with different cosmologies. We also show we can use the reconstructed lensing map for cosmological parameter estimation. This application of CNN provides a few innovations at the intersection of cosmology and machine learning. First, while training and regressing on images, we predict a continuous-variable field rather than discrete classes. Second, we are able to establish uncertainty measures for the network output that are analogous to standard methods. We expect this approach to excel in capturing hard-to-model non-Gaussian astrophysical foreground and noise contributions.

50 citations

Journal ArticleDOI
S. Samuroff1, Jonathan Blazek2, Jonathan Blazek3, Michael Troxel3  +73 moreInstitutions (33)
Abstract: We perform a joint analysis of intrinsic alignments and cosmology using tomographic weak lensing, galaxy clustering and galaxy-galaxy lensing measurements from Year 1 (Y1) of the Dark Energy Survey. We define early- and late-type subsamples, which are found to pass a series of systematics tests, including for spurious photometric redshift error and point spread function correlations. We analyse these split data alongside the fiducial mixed Y1 sample using a range of intrinsic alignment models. In a fiducial Nonlinear Alignment Model (NLA) analysis, assuming a flat \lcdm~cosmology, we find a significant difference in intrinsic alignment amplitude, with early-type galaxies favouring $A_\mathrm{IA} = 2.38^{+0.32}_{-0.31}$ and late-type galaxies consistent with no intrinsic alignments at $0.05^{+0.10}_{-0.09}$. We find weak evidence of a diminishing alignment amplitude at higher redshifts in the early-type sample. The analysis is repeated using a number of extended model spaces, including a physically motivated model that includes both tidal torquing and tidal alignment mechanisms. In multiprobe likelihood chains in which cosmology, intrinsic alignments in both galaxy samples and all other relevant systematics are varied simultaneously, we find the tidal alignment and tidal torquing parts of the intrinsic alignment signal have amplitudes $A_1 = 2.66 ^{+0.67}_{-0.66}$, $A_2=-2.94^{+1.94}_{-1.83}$, respectively, for early-type galaxies and $A_1 = 0.62 ^{+0.41}_{-0.41}$, $A_2 = -2.26^{+1.30}_{-1.16}$ for late-type galaxies. In the full (mixed) Y1 sample the best constraints are $A_1 = 0.70 ^{+0.41}_{-0.38}$, $A_2 = -1.36 ^{+1.08}_{-1.41}$. For all galaxy splits and IA models considered, we report cosmological parameter constraints that are consistent with the results of Troxel et al. (2017) and Dark Energy Survey Collaboration (2017).

49 citations

Posted Content
Abstract: This paper discusses the science case for a sensitive spectro-polarimetric survey of the microwave sky. Such a survey would provide a tomographic and dynamic census of the three-dimensional distribution of hot gas, velocity flows, early metals, dust, and mass distribution in the entire Hubble volume, exploit CMB temperature and polarisation anisotropies down to fundamental limits, and track energy injection and absorption into the radiation background across cosmic times by measuring spectral distortions of the CMB blackbody emission. In addition to its exceptional capability for cosmology and fundamental physics, such a survey would provide an unprecedented view of microwave emissions at sub-arcminute to few-arcminute angular resolution in hundreds of frequency channels, a data set that would be of immense legacy value for many branches of astrophysics. We propose that this survey be carried-out with a large space mission featuring a broad-band polarised imager and a moderate resolution spectro-imager at the focus of a 3.5m aperture telescope actively cooled to about 8K, complemented with absolutely-calibrated Fourier Transform Spectrometer modules observing at degree-scale angular resolution in the 10-2000 GHz frequency range. We propose two observing modes: a survey mode to map the entire sky as well as a few selected wide fields, and an observatory mode for deeper observations of regions of specific interest.

30 citations

Journal ArticleDOI
Abstract: A constant early dark energy (EDE) component contributing a fraction ${f}_{\mathrm{EDE}}({z}_{c})\ensuremath{\sim}10%$ of the energy density of the universe around ${z}_{c}\ensuremath{\simeq}3500$ and diluting as or faster than radiation afterwards, can provide a simple resolution to the Hubble tension, the $\ensuremath{\sim}5\ensuremath{\sigma}$ discrepancy---in the $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ context---between the ${H}_{0}$ value derived from early- and late-universe observations. However, it has been pointed out that including Large-Scale Structure (LSS) data, which are in $\ensuremath{\sim}3\ensuremath{\sigma}$ tension with $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ and EDE cosmologies, might break some parameter degeneracy and alter these conclusions. We reassess the viability of the EDE against a host of high- and low-redshift measurements, by combining LSS observations from recent weak lensing (WL) surveys with CMB, baryon acoustic oscillation (BAO), growth function (FS) and Supernova Ia (SNIa) data. Introducing a model whose only parameter is ${f}_{\mathrm{EDE}}({z}_{c})$, we report in agreement with past work a $\ensuremath{\sim}2\ensuremath{\sigma}$ preference for nonzero ${f}_{\mathrm{EDE}}({z}_{c})$ from Planck CMB data alone, while the tension with the local ${H}_{0}$ measurement from sh0es is reduced below $2\ensuremath{\sigma}$. Adding BAO, FS and SNIa does not affect this conclusion, while the inclusion of a prior on ${H}_{0}$ from sh0es increase the preference for EDE over $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ to the $\ensuremath{\sim}3.6\ensuremath{\sigma}$ level. After checking the EDE nonlinear matter power spectrum as predicted by standard semi-analytical algorithms via a dedicated set of $N$-body simulations, we test the 1-parameter EDE cosmology against WL data. We find that it does not significantly worsen the fit to the ${S}_{8}$ measurement as compared to $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$, and that current WL observations do not exclude the EDE resolution to the Hubble tension. We also caution against the interpretation of constraints obtained from combining statistically inconsistent datasets within the $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ cosmology. In light of the CMB lensing anomalies, we show that the lensing-marginalized CMB data also favor nonzero ${f}_{\mathrm{EDE}}({z}_{c})$ at $\ensuremath{\sim}2\ensuremath{\sigma}$, predicts ${H}_{0}$ in $1.4\ensuremath{\sigma}$ agreement with sh0es and ${S}_{8}$ in $1.5\ensuremath{\sigma}$ and $0.8\ensuremath{\sigma}$ agreement with kids-viking and des respectively. There still exists however a $\ensuremath{\sim}2.5\ensuremath{\sigma}$ tension with the joint results from kids-viking and des. With an eye on Occam's razor, we finally discuss promising extensions of the EDE cosmology that could allow us to fully restore cosmological concordance.

28 citations

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Journal ArticleDOI
Eiichiro Komatsu1, Kristine M. Smith2, Jo Dunkley3, Charles L. Bennett4  +17 moreInstitutions (10)
Abstract: The combination of seven-year data from WMAP and improved astrophysical data rigorously tests the standard cosmological model and places new constraints on its basic parameters and extensions. By combining the WMAP data with the latest distance measurements from the baryon acoustic oscillations (BAO) in the distribution of galaxies and the Hubble constant (H0) measurement, we determine the parameters of the simplest six-parameter ΛCDM model. The power-law index of the primordial power spectrum is ns = 0.968 ± 0.012 (68% CL) for this data combination, a measurement that excludes the Harrison–Zel’dovich–Peebles spectrum by 99.5% CL. The other parameters, including those beyond the minimal set, are also consistent with, and improved from, the five-year results. We find no convincing deviations from the minimal model. The seven-year temperature power spectrum gives a better determination of the third acoustic peak, which results in a better determination of the redshift of the matter-radiation equality epoch. Notable examples of improved parameters are the total mass of neutrinos, � mν < 0.58 eV (95% CL), and the effective number of neutrino species, Neff = 4.34 +0.86 −0.88 (68% CL), which benefit from better determinations of the third peak and H0. The limit on a constant dark energy equation of state parameter from WMAP+BAO+H0, without high-redshift Type Ia supernovae, is w =− 1.10 ± 0.14 (68% CL). We detect the effect of primordial helium on the temperature power spectrum and provide a new test of big bang nucleosynthesis by measuring Yp = 0.326 ± 0.075 (68% CL). We detect, and show on the map for the first time, the tangential and radial polarization patterns around hot and cold spots of temperature fluctuations, an important test of physical processes at z = 1090 and the dominance of adiabatic scalar fluctuations. The seven-year polarization data have significantly improved: we now detect the temperature–E-mode polarization cross power spectrum at 21σ , compared with 13σ from the five-year data. With the seven-year temperature–B-mode cross power spectrum, the limit on a rotation of the polarization plane due to potential parity-violating effects has improved by 38% to Δα =− 1. 1 ± 1. 4(statistical) ± 1. 5(systematic) (68% CL). We report significant detections of the Sunyaev–Zel’dovich (SZ) effect at the locations of known clusters of galaxies. The measured SZ signal agrees well with the expected signal from the X-ray data on a cluster-by-cluster basis. However, it is a factor of 0.5–0.7 times the predictions from “universal profile” of Arnaud et al., analytical models, and hydrodynamical simulations. We find, for the first time in the SZ effect, a significant difference between the cooling-flow and non-cooling-flow clusters (or relaxed and non-relaxed clusters), which can explain some of the discrepancy. This lower amplitude is consistent with the lower-than-theoretically expected SZ power spectrum recently measured by the South Pole Telescope Collaboration.

10,928 citations

Journal ArticleDOI
Peter A. R. Ade1, Nabila Aghanim2, Monique Arnaud3, M. Ashdown4  +334 moreInstitutions (82)
Abstract: This paper presents cosmological results based on full-mission Planck observations of temperature and polarization anisotropies of the cosmic microwave background (CMB) radiation. Our results are in very good agreement with the 2013 analysis of the Planck nominal-mission temperature data, but with increased precision. The temperature and polarization power spectra are consistent with the standard spatially-flat 6-parameter ΛCDM cosmology with a power-law spectrum of adiabatic scalar perturbations (denoted “base ΛCDM” in this paper). From the Planck temperature data combined with Planck lensing, for this cosmology we find a Hubble constant, H0 = (67.8 ± 0.9) km s-1Mpc-1, a matter density parameter Ωm = 0.308 ± 0.012, and a tilted scalar spectral index with ns = 0.968 ± 0.006, consistent with the 2013 analysis. Note that in this abstract we quote 68% confidence limits on measured parameters and 95% upper limits on other parameters. We present the first results of polarization measurements with the Low Frequency Instrument at large angular scales. Combined with the Planck temperature and lensing data, these measurements give a reionization optical depth of τ = 0.066 ± 0.016, corresponding to a reionization redshift of . These results are consistent with those from WMAP polarization measurements cleaned for dust emission using 353-GHz polarization maps from the High Frequency Instrument. We find no evidence for any departure from base ΛCDM in the neutrino sector of the theory; for example, combining Planck observations with other astrophysical data we find Neff = 3.15 ± 0.23 for the effective number of relativistic degrees of freedom, consistent with the value Neff = 3.046 of the Standard Model of particle physics. The sum of neutrino masses is constrained to ∑ mν < 0.23 eV. The spatial curvature of our Universe is found to be very close to zero, with | ΩK | < 0.005. Adding a tensor component as a single-parameter extension to base ΛCDM we find an upper limit on the tensor-to-scalar ratio of r0.002< 0.11, consistent with the Planck 2013 results and consistent with the B-mode polarization constraints from a joint analysis of BICEP2, Keck Array, and Planck (BKP) data. Adding the BKP B-mode data to our analysis leads to a tighter constraint of r0.002 < 0.09 and disfavours inflationarymodels with a V(φ) ∝ φ2 potential. The addition of Planck polarization data leads to strong constraints on deviations from a purely adiabatic spectrum of fluctuations. We find no evidence for any contribution from isocurvature perturbations or from cosmic defects. Combining Planck data with other astrophysical data, including Type Ia supernovae, the equation of state of dark energy is constrained to w = −1.006 ± 0.045, consistent with the expected value for a cosmological constant. The standard big bang nucleosynthesis predictions for the helium and deuterium abundances for the best-fit Planck base ΛCDM cosmology are in excellent agreement with observations. We also constraints on annihilating dark matter and on possible deviations from the standard recombination history. In neither case do we find no evidence for new physics. The Planck results for base ΛCDM are in good agreement with baryon acoustic oscillation data and with the JLA sample of Type Ia supernovae. However, as in the 2013 analysis, the amplitude of the fluctuation spectrum is found to be higher than inferred from some analyses of rich cluster counts and weak gravitational lensing. We show that these tensions cannot easily be resolved with simple modifications of the base ΛCDM cosmology. Apart from these tensions, the base ΛCDM cosmology provides an excellent description of the Planck CMB observations and many other astrophysical data sets.

10,334 citations

Journal ArticleDOI
Peter A. R. Ade1, Nabila Aghanim2, C. Armitage-Caplan3, Monique Arnaud4  +324 moreInstitutions (70)
Abstract: This paper presents the first cosmological results based on Planck measurements of the cosmic microwave background (CMB) temperature and lensing-potential power spectra. We find that the Planck spectra at high multipoles (l ≳ 40) are extremely well described by the standard spatially-flat six-parameter ΛCDM cosmology with a power-law spectrum of adiabatic scalar perturbations. Within the context of this cosmology, the Planck data determine the cosmological parameters to high precision: the angular size of the sound horizon at recombination, the physical densities of baryons and cold dark matter, and the scalar spectral index are estimated to be θ∗ = (1.04147 ± 0.00062) × 10-2, Ωbh2 = 0.02205 ± 0.00028, Ωch2 = 0.1199 ± 0.0027, and ns = 0.9603 ± 0.0073, respectively(note that in this abstract we quote 68% errors on measured parameters and 95% upper limits on other parameters). For this cosmology, we find a low value of the Hubble constant, H0 = (67.3 ± 1.2) km s-1 Mpc-1, and a high value of the matter density parameter, Ωm = 0.315 ± 0.017. These values are in tension with recent direct measurements of H0 and the magnitude-redshift relation for Type Ia supernovae, but are in excellent agreement with geometrical constraints from baryon acoustic oscillation (BAO) surveys. Including curvature, we find that the Universe is consistent with spatial flatness to percent level precision using Planck CMB data alone. We use high-resolution CMB data together with Planck to provide greater control on extragalactic foreground components in an investigation of extensions to the six-parameter ΛCDM model. We present selected results from a large grid of cosmological models, using a range of additional astrophysical data sets in addition to Planck and high-resolution CMB data. None of these models are favoured over the standard six-parameter ΛCDM cosmology. The deviation of the scalar spectral index from unity isinsensitive to the addition of tensor modes and to changes in the matter content of the Universe. We find an upper limit of r0.002< 0.11 on the tensor-to-scalar ratio. There is no evidence for additional neutrino-like relativistic particles beyond the three families of neutrinos in the standard model. Using BAO and CMB data, we find Neff = 3.30 ± 0.27 for the effective number of relativistic degrees of freedom, and an upper limit of 0.23 eV for the sum of neutrino masses. Our results are in excellent agreement with big bang nucleosynthesis and the standard value of Neff = 3.046. We find no evidence for dynamical dark energy; using BAO and CMB data, the dark energy equation of state parameter is constrained to be w = -1.13-0.10+0.13. We also use the Planck data to set limits on a possible variation of the fine-structure constant, dark matter annihilation and primordial magnetic fields. Despite the success of the six-parameter ΛCDM model in describing the Planck data at high multipoles, we note that this cosmology does not provide a good fit to the temperature power spectrum at low multipoles. The unusual shape of the spectrum in the multipole range 20 ≲ l ≲ 40 was seen previously in the WMAP data and is a real feature of the primordial CMB anisotropies. The poor fit to the spectrum at low multipoles is not of decisive significance, but is an “anomaly” in an otherwise self-consistent analysis of the Planck temperature data.

6,641 citations

Journal ArticleDOI
Abstract: We present a fast Markov chain Monte Carlo exploration of cosmological parameter space. We perform a joint analysis of results from recent cosmic microwave background ~CMB! experiments and provide parameter constraints, including s 8, from the CMB independent of other data. We next combine data from the CMB, HST Key Project, 2dF galaxy redshift survey, supernovae type Ia and big-bang nucleosynthesis. The Monte Carlo method allows the rapid investigation of a large number of parameters, and we present results from 6 and 9 parameter analyses of flat models, and an 11 parameter analysis of non-flat models. Our results include constraints on the neutrino mass ( mn&0.3 eV), equation of state of the dark energy, and the tensor amplitude, as well as demonstrating the effect of additional parameters on the base parameter constraints. In a series of appendixes we describe the many uses of importance sampling, including computing results from new data and accuracy correction of results generated from an approximate method. We also discuss the different ways of converting parameter samples to parameter constraints, the effect of the prior, assess the goodness of fit and consistency, and describe the use of analytic marginalization over normalization parameters.

3,331 citations

Journal ArticleDOI
Abstract: We implement the efficient line of sight method to calculate the anisotropy and polarization of the cosmic microwave background for scalar and tensor modes in almost-Friedmann-Robertson-Walker models with positive spatial curvature. We present new results for the polarization power spectra in such models.

2,531 citations

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