Detection of the power spectrum of cosmic microwave background lensing by the Atacama Cosmology Telescope.
ABSTRACT We report the first detection of the gravitational lensing of the cosmic microwave background through a measurement of the four-point correlation function in the temperature maps made by the Atacama Cosmology Telescope. We verify our detection by calculating the levels of potential contaminants and performing a number of null tests. The resulting convergence power spectrum at 2° angular scales measures the amplitude of matter density fluctuations on comoving length scales of around 100 Mpc at redshifts around 0.5 to 3. The measured amplitude of the signal agrees with Lambda cold dark matter cosmology predictions. Since the amplitude of the convergence power spectrum scales as the square of the amplitude of the density fluctuations, the 4σ detection of the lensing signal measures the amplitude of density fluctuations to 12%.
arXiv:1103.2124v1 [astro-ph.CO] 10 Mar 2011
The Atacama Cosmology Telescope: Detection of the Power Spectrum of
Sudeep Das,1,2,3Blake D. Sherwin,2Paula Aguirre,4John W. Appel,2J. Richard Bond,5C. Sofia Carvalho,6,7
Mark J. Devlin,8Joanna Dunkley,9, 2,3Thomas Essinger-Hileman,2Joseph W. Fowler,10,2Amir Hajian,5,3,2
Mark Halpern,11Matthew Hasselfield,11Adam D. Hincks,2Ren´ ee Hlozek,9Kevin M. Huffenberger,12
John P. Hughes,13Kent D. Irwin,10Jeff Klein,8Arthur Kosowsky,14Robert H. Lupton,3Tobias A. Marriage,15,3
Danica Marsden,8Kavilan Moodley,16Michael D. Niemack,10,2Michael R. Nolta,5Lyman A. Page,2Lucas Parker,2
Erik D. Reese,8Benjamin L. Schmitt,8Neelima Sehgal,17Jon Sievers,5David N. Spergel,3Suzanne T. Staggs,2
Daniel S. Swetz,8,10Eric R. Switzer,18,2Robert Thornton,8,19Katerina Visnjic,2and Ed Wollack20
1BCCP, Dept. of Physics, University of California, Berkeley, CA, USA 94720
2Dept. of Physics, Princeton University, Princeton, NJ, USA 08544
3Dept. of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ USA 08544
4Departamento de Astronom´ ıa y Astrof´ ısica, Pontific´ ıa Univ. Cat´ olica, Casilla 306, Santiago 22, Chile
5CITA, University of Toronto, Toronto, ON, Canada M5S 3H8
6IPFN, IST, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
7Academy of Athens, RCAAM, Soranou Efessiou, 11-527 Athens, Greece
8Dept. of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA, USA 19104
9Dept. of Astrophysics, Oxford University, Oxford, UK OX1 3RH
10NIST Quantum Devices Group, 325 Broadway Mailcode 817.03, Boulder, CO, USA 80305
11Dept. of Physics and Astronomy, University of British Columbia, Vancouver, BC, Canada V6T 1Z4
12Dept. of Physics, University of Miami, Coral Gables, FL, USA 33124
13Dept. of Physics and Astronomy, Rutgers, The State University of New Jersey, Piscataway, NJ USA 08854-8019
14Dept. of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA, USA 15260
15Dept. of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD 21218-2686
16Astrophysics and Cosmology Research Unit, Univ. of KwaZulu-Natal, Durban, 4041, South Africa
17KIPAC, Stanford University, Stanford, CA, USA 94305-4085
18Kavli Institute for Cosmological Physics, 5620 South Ellis Ave., Chicago, IL, USA 60637
19Dept. of Physics , West Chester University of Pennsylvania, West Chester, PA, USA 19383
20Code 553/665, NASA/Goddard Space Flight Center, Greenbelt, MD, USA 20771
We report the first detection of the gravitational lensing of the cosmic microwave background
through a measurement of the four-point correlation function in the temperature maps made by
the Atacama Cosmology Telescope. We verify our detection by calculating the levels of potential
contaminants and performing a number of null tests. The resulting convergence power spectrum at
2-degree angular scales measures the amplitude of matter density fluctuations on comoving length
scales of around 100 Mpc at redshifts around 1 to 3. The measured amplitude of the signal agrees
with Lambda Cold Dark Matter cosmology predictions. Since the amplitude of the convergence
power spectrum scales as the square of the amplitude of the density fluctuations, the 4-sigma de-
tection of the lensing signal measures the amplitude of density fluctuations to 12%.
Introduction.— The large-scale distribution of matter
deflects the paths of microwave background photons by
roughly 3′, a scale larger than the ? 1.4′angular reso-
lution of the Atacama Cosmology Telescope (ACT). This
gravitational lensing imprints a distinctive non-Gaussian
signature on the temperature pattern of the microwave
sky . Since the cosmic microwave background (CMB)
temperature fluctuations are very nearly Gaussian 
with a power spectrum now well characterized by WMAP
 and ground-based experiments [5, 6], measurements of
the distinctive four-point correlation function due to lens-
ing yield a direct determination of the integrated mass
fluctuations along the line of sight . CMB lensing mea-
surements complement optical lensing measurements by
probing larger scales and higher redshifts, and have the
advantage of a precisely known source redshift.
Previous analyses have detected the lensing signature
on the microwave sky through cross-correlations of large-
scale structure tracers with WMAP data [7, 8], or seen
the signature of lensing in the temperature power spec-
trum at ? 3 σ [5, 6]. Here, we report the first measure-
ment of the lensing signature using only the CMB tem-
perature four-point function and constrain the amplitude
of the projected matter power spectrum.
Data.— ACT is a six-meter telescope operating in the
Atacama Desert of Chile at an altitude of 5200 meters.
The telescope has three 1024-element arrays of super-
conducting transition-edge sensing bolometers, one each
operating at 148 GHz, 218 GHz, and 277 GHz.
vious ACT team publications describe the instrument,
observations, and data reduction and initial scientific re-
sults [6, 9–11]. The analysis presented here is done on a
324-square-degree stripe of average noise level ≃ 23 µK-
arcmin, made from three seasons of 148 GHz observations
of the celestial equator. The region is cut into six equally
sized (3×18 degree) patches on which we perform lensing
reconstruction separately, and then combine the results
with inverse variance weighting.
The ACT temperature maps (made as in ) are fur-
ther processed in order to minimize the effects of atmo-
spheric noise and point sources. Temperature modes be-
low ℓ = 500 as well as a ‘stripe’ of width ℓ = 180 along the
Fourier axis corresponding to map declination are filtered
out to reduce the effects of non-white atmospheric noise
and scan-synchronous noise respectively .
point sources with a signal-to-noise (S/N) greater than 5
are identified in a match-filtered map . An ACT beam
template scaled to the peak brightness of each of these
sources is subtracted from the raw data. Using an algo-
rithm inspired by the CLEAN algorithm , we repeat
this filtering, source identification, and subtraction un-
til there are no S/N> 5 identifications. Because the 148
GHz data also contains temperature decrements from the
thermal Sunyaev-Zel’dovich(SZ) effect in galaxy clusters,
the entire subtraction algorithm is also run on the neg-
ative of the map. The effect of unresolved point sources
is minimized by filtering out all data above ℓ = 2300.
Methods.— Gravitational lensing remaps the CMB
˜T(ˆ n + α(ˆ n)), where α(ˆ n) is the deflection field and un-
lensed quantities are denoted by a tilde. In this paper,
we compute the power spectrum of the convergence field,
2∇·α, using an optimal quadratic estimator :
on thesky:T(ˆ n)=
(2π)2δ(L − L′)ˆCκκ
L = |Nκ(L)|2
T∗(ℓ) T∗(L − ℓ) T(ℓ′) T(L′− ℓ′)
−?T∗(ℓ) T∗(L − ℓ) T(ℓ′) T(L′− ℓ′)?Gauss
where ℓ,ℓ,L,L′are coordinates in Fourier space (we are
using the flat-sky approximation), g defines filters that
can be tuned to optimize signal-to-noise, N is a normal-
ization, and the second term is the Gaussian part of the
four-point function. We will refer to the second term as
the “Gaussian bias”, as it is a Gaussian term one sub-
tracts from the full four-point function to obtain the non-
Gaussian lensing signal. Note that the convergence and
lensing potential φ are related by: κ(L) = φ(L)L2/2. See
 for the properties of this estimator.
While the optimal quadratic estimator has the advan-
tage of maximizing the signal-to-noise, an experimental
measurement of its amplitude involves subtracting two
large numbers (the full four-point function and the bias).
Depending on the quality of data and the relevant length
scales, this Gaussian four-point bias term can be up to an
order of magnitude larger than the lensing convergence
spectrum. As the size of the Gaussian bias term depends
sensitively on the CMB temperature power spectrum,
foregrounds and noise, calculating it to sufficient accu-
racy using the standard simulation or theory approach is
very difficult, and can lead to large discrepancies. Smidt
et al.  use this standard approach for an analysis of
the WMAP data and, without a detailed treatment of
the covariance between temperature power spectrum and
lensing reconstruction constraints, report a detection sig-
nificance larger than expected from Fisher information
theory. An alternative approach that does not require
this subtraction is presented in .
In this paper, we use the data themselves to obtain a
first approximation to the Gaussian bias part of the four-
point function, then compute a small correction using
Monte Carlo simulations. To approximate the bias with
our new method, we first generate multiple randomized
versions of the original data map. The Fourier modes of
these randomized maps have the same amplitude as the
original map, but with their phases randomized. This
phase randomization destroys any non-Gaussian lensing
correlation between modes, yet approximately preserves
the Gaussian part of the four point function we wish to
model. By then averaging the Gaussian biases calculated
for many realizations of randomized maps, we obtain a
good estimate of the second term in Eq. (1). The small
correction we subtract from our estimator (a “null bias”
at high ℓ due to spatially varying noise and window func-
tions) is easily calculated from Monte-Carlo simulations.
Simulations.— We test our lensing reconstruction
pipeline by analyzing a large number of simulated lensed
and unlensed maps. The simulated maps are obtained
by generating Gaussian random fields with the best
fit WMAP+ACT temperature power spectrum [6, 10],
which includes foreground models, on maps with the
ACT pixelization. We then generate lensed maps from
these unlensed maps by oversampling the unlensed map
to five times finer resolution, and displacing the pixels
according to Gaussian random deflection fields realized
from an input theory.Finally, we convolve the maps
with the ACT beam, and add simulated noise with the
same statistical properties as the ACT data, seeded by
season-split difference maps .
We apply our lensing estimator to 480 simulations of
the equatorial ACT temperature map. For each simu-
lated map we estimate the full four-point function and
subtract the Gaussian and null bias terms obtained from
15 realizations of the random phase maps (we found that
15 realizations were sufficient for the bias error to be-
come negligible). We thus obtain a mean reconstructed
lensing power spectrum, Eq. (1), as well as the standard
error on each reconstructed point of the power spectrum.
The red points in Fig. 1 show the estimated mean con-
vergence power spectrum from the lensed simulations; it
can be seen that the input (theory) convergence power
spectrum is reconstructed accurately by our pipeline.
Results.— Fig. 2 shows the lensing convergence power
spectrum estimated from the ACT equatorial data, using
FIG. 1. Mean convergence power spectrum (red points) from
480 simulated lensed maps with noise similar to our data. The
solid line is the input lensing power spectrum, taken from
the best-fit WMAP+ACT cosmological model.
correspond to the scatter of power spectrum values obtained
from individual maps.
FIG. 2. Convergence power spectrum (red points) measured
from ACT equatorial sky patches. The solid line is the power
spectrum from the best-fit WMAP+ACT cosmological model
with amplitude AL = 1, which is consistent with the measured
points. The error bars are from the Monte Carlo simulation
results displayed in Fig. 1. The best-fit lensing power spec-
trum amplitude to our data is AL = 1.16 ± 0.29
the value of the Gaussian term as well as the null bias
obtained from the Monte Carlo simulations previously
described. The error bars are obtained from the scatter
of simulations shown in Fig. 1.
Here, we introduce the parameter ALas a lensing con-
vergence power spectrum amplitude, defined such that
AL= 1 corresponds to the best-fit WMAP+ACT ΛCDM
model (with σ8= 0.813). The reconstructed points are
consistent with the theoretical expectation for the con-
vergence power spectrum. From our results we obtain
a value of AL = 1.16 ± 0.29, a 4-σ detection.
restrict our analysis to the first three points, we find
AL= 0.96 ± 0.31. Fitting our five points to the theory,
we calculate χ2/dof = 6.4/4. Since the lensing kernel
FIG. 3. Convergence power spectrum for simulated thermal
and kinematic SZ maps and point source maps  which
are a good fit to the ACT data. Note that we only show
the non-Gaussian contribution, as the Gaussian part which
is of similar negligible size is automatically included in the
subtracted bias generated by phase randomization. The solid
line is the convergence power spectrum due to lensing in the
best-fit WMAP+ACT cosmological model.
has a broad peak at z ≃ 2 and a conformal distance of
≃ 5000 Mpc, our 4-σ detection is a direct measurement
of the amplitude of matter fluctuations at a comoving
wavenumber k ∼ 0.02Mpc−1around this redshift.
We estimate potential contamination by point sources
and SZ clusters by running our reconstruction pipeline on
simulated patches which contain only IR point sources or
only thermal or kinetic SZ signal , while keeping the
filters and the normalization the same as for the data
run. Fig. 3 shows that the estimated spurious conver-
gence power is at least two orders of magnitude below the
predicted signal, due partially to our use of only temper-
ature modes with ℓ < 2300. We also verified that recon-
struction on simulated maps containing all foregrounds
(unresolved point sources and SZ) and lensed CMB was
unbiased. We found no evidence of artifacts in the recon-
structed convergence power maps.
Null Tests.— We compute a mean cross-correlation
power of convergence maps reconstructed from neighbor-
ing patches of the data map, which is expected to be
zero as these patches should be uncorrelated. We find
a χ2/dof = 5.8/4 for a fit to zero signal (Fig. 4, upper
panel). For the second null test we construct a noise map
for each sky patch by taking the difference of maps made
from the first half and second half of the season’s data,
and run our lensing estimator on them. Fig. 4, lower
panel, shows the mean reconstructed convergence power
spectrum for these noise-only maps. Fitting to null we
calculate χ2= 5.7 for 4 degrees of freedom. The null test
is consistent with zero, which shows that the contamina-
tion of our lensing reconstruction by noise is minimal. We
also tested our phase randomization scheme by random-
izing the phases on a map, using this map to reconstruct
FIG. 4. Upper panel: Mean cross-correlation power spectrum
of convergence fields reconstructed from different sky patches.
The result is consistent with null, as expected. Lower panel:
Mean convergence power spectrum of noise maps constructed
from the difference of half-season patches, which is consistent
with a null signal. The error bars in either case are determined
from Monte Carlo simulations, and those in the lower panel
are much smaller as they do not contain cosmic variance.
a convergence map, and cross correlating it with a recon-
struction from the same map but with a different phase
randomization; our results were consistent with null as
Implications and Conclusions.— We have reported a
first detection of the convergence power spectrum of the
cosmic microwave background due to gravitational lens-
ing. The inferred amplitude of the lensing signal is con-
sistent with theoretical expectations, providing another
independent test of the basic cosmological model.
Data from the Planck satellite should provide a much
more precise measurement of the lensing power spectrum
. A detection is also anticipated from the South Pole
Telescope team. CMB polarization measurements with
ACTPol, SPTPol, PolarBear and other next generation
experiments  will yield even more accurate measure-
ments of CMB lensing. Such measurements are also an
important goal for a future polarization satellite mission
. The work reported here is the first step of an excit-
ing research program.
This work was supported by the U.S. NSF through
awards AST-0408698 for the ACT project, and PHY-
0355328, AST-0707731 and PIRE-0507768.
was also provided by Princeton Univ.
of Pennsylvania, RCUK Fellowship(JD), NASA grant
NNX08AH30G (SD, AH and TM), NSERC PGSD schol-
arship (ADH), NSF AST-0546035 and AST-0807790
and the Univ.
(AK), NSF Physics Frontier Center grant PHY-0114422
(ES), KICP Fellowship (ES), SLAC no.
76SF00515 (NS), and the BCCP (SD). Computations
were performed on the GPC supercomputer at the SciNet
HPC Consortium. We thank B. Berger, R. Escribano, T.
Evans, D. Faber, P. Gallardo, A. Gomez, M. Gordon, D.
Holtz, M. McLaren, W. Page, R. Plimpton, D. Sanchez,
O. Stryzak, M. Uehara, and the Astro-Norte group for
assistance with ACT observations. We thank Thibaut
Louis, Oliver Zahn and Duncan Hanson, and are grateful
to Kendrick Smith for discussions and draft comments.
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