Page 1
Draft version May 17, 2011
Preprint typeset using LATEX style emulateapj v. 11/10/09
A MEASUREMENT OF THE DAMPING TAIL OF THE COSMIC MICROWAVE BACKGROUND POWER
SPECTRUM WITH THE SOUTH POLE TELESCOPE
R. Keisler,1,2C. L. Reichardt,3K. A. Aird,4B. A. Benson,1,5L. E. Bleem,1,2J. E. Carlstrom,1,2,5,6,7
C. L. Chang,1,5,7H. M. Cho,8T. M. Crawford,1,6A. T. Crites,1,6T. de Haan,9M. A. Dobbs,9J. Dudley,9
E. M. George,3N. W. Halverson,10G. P. Holder,9W. L. Holzapfel,3S. Hoover,1,2Z. Hou,11J. D. Hrubes,4
M. Joy,12L. Knox,11A. T. Lee,3,13E. M. Leitch,1,6M. Lueker,14D. Luong-Van,4J. J. McMahon,15J. Mehl,1
S. S. Meyer,1,2,5,6M. Millea,11J. J. Mohr,16,17,18T. E. Montroy,19T. Natoli,1,2S. Padin,1,6,14T. Plagge,1,6
C. Pryke,1,5,6,20J. E. Ruhl,19K. K. Schaffer,1,5,21L. Shaw,22E. Shirokoff,3H. G. Spieler,13Z. Staniszewski,19
A. A. Stark,23K. Story,1,2A. van Engelen,9K. Vanderlinde,9J. D. Vieira,14R. Williamson,1,6and O. Zahn24
Draft version May 17, 2011
ABSTRACT
We present a measurement of the angular power spectrum of the cosmic microwave background
(CMB) using data from the South Pole Telescope (SPT). The data consist of 790 square degrees
of sky observed at 150 GHz during 2008 and 2009. Here we present the power spectrum over the
multipole range 650 < ? < 3000, where it is dominated by primary CMB anisotropy. We combine
this power spectrum with the power spectra from the seven-year Wilkinson Microwave Anisotropy
Probe (WMAP) data release to constrain cosmological models. We find that the SPT and WMAP
data are consistent with each other and, when combined, are well fit by a spatially flat, ΛCDM
cosmological model. The SPT+WMAP constraint on the spectral index of scalar fluctuations is
ns= 0.9663 ± 0.0112. We detect, at ∼5σ significance, the effect of gravitational lensing on the CMB
power spectrum, and find its amplitude to be consistent with the ΛCDM cosmological model. We
explore a number of extensions beyond the ΛCDM model. Each extension is tested independently,
although there are degeneracies between some of the extension parameters. We constrain the tensor-
to-scalar ratio to be r < 0.21 (95% CL) and constrain the running of the scalar spectral index to be
dns/dlnk = −0.024 ± 0.013. We strongly detect the effects of primordial helium and neutrinos on
the CMB; a model without helium is rejected at 7.7σ, while a model without neutrinos is rejected
at 7.5σ. The primordial helium abundance is measured to be Yp= 0.296 ± 0.030, and the effective
number of relativistic species is measured to be Neff= 3.85 ± 0.62. The constraints on these models
are strengthened when the CMB data are combined with measurements of the Hubble constant and
the baryon acoustic oscillation feature. Notable improvements include ns= 0.9668±0.0093, r < 0.17
(95% CL), and Neff= 3.86±0.42. The SPT+WMAP data show a mild preference for low power in the
CMB damping tail, and while this preference may be accommodated by models that have a negative
spectral running, a high primordial helium abundance, or a high effective number of relativistic species,
such models are disfavored by the abundance of low-redshift galaxy clusters.
Subject headings: cosmology – cosmology:cosmic microwave background – cosmology: observations –
large-scale structure of universe
rkeisler@uchicago.edu
1Kavli Institute for Cosmological Physics, University of
Chicago, 5640 South Ellis Avenue, Chicago, IL, USA 60637
2Department of Physics, University of Chicago, 5640 South
Ellis Avenue, Chicago, IL, USA 60637
3Department of Physics, University of California, Berkeley,
CA, USA 94720
4University of Chicago, 5640 South Ellis Avenue, Chicago, IL,
USA 60637
5Enrico Fermi Institute, University of Chicago, 5640 South
Ellis Avenue, Chicago, IL, USA 60637
6Department of Astronomy and Astrophysics, University of
Chicago, 5640 South Ellis Avenue, Chicago, IL, USA 60637
7Argonne National Laboratory, 9700 S. Cass Avenue, Ar-
gonne, IL, USA 60439
8NIST Quantum Devices Group, 325 Broadway Mailcode
817.03, Boulder, CO, USA 80305
9Department of Physics, McGill University, 3600 Rue Univer-
sity, Montreal, Quebec H3A 2T8, Canada
10Department of Astrophysical and Planetary Sciences and
Department of Physics, University of Colorado, Boulder, CO,
USA 80309
11Department of Physics, University of California, One
Shields Avenue, Davis, CA, USA 95616
12Department of Space Science, VP62, NASA Marshall Space
Flight Center, Huntsville, AL, USA 35812
13Physics Division, Lawrence Berkeley National Laboratory,
Berkeley, CA, USA 94720
14California Institute of Technology, MS 249-17, 1216 E. Cal-
ifornia Blvd., Pasadena, CA, USA 91125
15Department of Physics, University of Michigan, 450 Church
Street, Ann Arbor, MI, USA 48109
16Department of Physics, Ludwig-Maximilians-Universit¨ at,
Scheinerstr. 1, 81679 M¨ unchen, Germany
17Excellence Cluster Universe, Boltzmannstr. 2, 85748 Garch-
ing, Germany
18Max-Planck-Institut f¨ ur extraterrestrische Physik, Giessen-
bachstr. 85748 Garching, Germany
19Physics Department, Center for Education and Research in
Cosmology and Astrophysics, Case Western Reserve University,
Cleveland, OH, USA 44106
20Department of Physics, University of Minnesota, 116
Church Street S.E. Minneapolis, MN, USA 55455
21Liberal Arts Department, School of the Art Institute of
Chicago, 112 S Michigan Ave, Chicago, IL, USA 60603
22Department of Physics, Yale University, P.O. Box 208210,
New Haven, CT, USA 06520-8120
23Harvard-Smithsonian Center for Astrophysics, 60 Garden
Street, Cambridge, MA, USA 02138
24Berkeley Center for Cosmological Physics, Department of
arXiv:1105.3182v1 [astro-ph.CO] 16 May 2011
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1. INTRODUCTION
Measurements of anisotropy in the temperature of
the cosmic microwave background (CMB) are among
the most informative and robust probes of cosmology.
The acoustic oscillations of the primordial plasma have
been measured on degree scales (? ? 500) with cosmic-
variance-limited precision by the Wilkinson Microwave
Anisotropy Probe (WMAP) (Larson et al. 2010), yield-
ing a wealth of cosmological information (Komatsu et al.
2010). On much smaller scales, ? > 3000, the millimeter-
wave anisotropy is dominated by secondary anisotropies
from the Sunyaev-Zel’dovich (SZ) effects and by emis-
sion from foreground galaxies. The thermal SZ effect
arises from the scattering of CMB photons off the hot
gas in gravitationally collapsed structures (Sunyaev &
Zel’dovich 1972), and thereby encodes information on
the amplitude of matter fluctuations at intermediate red-
shifts. Recently, Lueker et al. (2010) reported the first
statistical measurement of the SZ effect using multi-
frequency South Pole Telescope (SPT) data. This was
followed by a measurement using data from the Atacama
Cosmology Telescope (ACT, Das et al. 2011a; Dunkley
et al. 2010) and an improved SPT measurement (Shi-
rokoff et al. 2010).The angular power spectrum of
millimeter-wave emission from high-redshift, dusty, star-
forming galaxies has also been characterized by SPT,
ACT, and Planck (Hall et al. 2010; Dunkley et al. 2010;
Shirokoff et al. 2010; Planck Collaboration 2011).
On intermediate scales, 500 < ? < 3000, the primary
CMB anisotropy is the dominant source of millimeter-
wave anisotropy, but its power is falling exponentially
with decreasing angular scale. The reduction in CMB
power is due to the diffusion of photons in the primor-
dial plasma and is often referred to as Silk damping
(Silk 1968). This “damping tail” of the primary CMB
anisotropy has been measured by a number of experi-
ments, notably the Arcminute Cosmology Bolometer Ar-
ray Receiver (ACBAR, Reichardt et al. 2009), QUEST at
DASI (QUaD, Brown et al. 2009; Friedman et al. 2009),
and ACT (Das et al. 2011a).
Measurements of the CMB damping tail, in conjunc-
tion with WMAP’s measurements of the degree-scale
CMB anisotropy, provide a powerful probe of early-
universe physics. The damping tail measurements sig-
nificantly increase the angular dynamic range of CMB
measurements and thereby improve the constraints on
inflationary parameters such as the scalar spectral in-
dex and the amplitude of tensor fluctuations. Measure-
ments of the angular scale of the damping can constrain
the primordial helium abundance and the effective num-
ber of relativistic particle species during the radiation-
dominated era. Finally, the damping tail is altered at the
few-percent level by gravitational lensing of the CMB,
and is therefore sensitive to the matter fluctuations at
intermediate redshifts.
The work presented here is a measurement of the CMB
damping tail using data from the SPT. The data were
taken at 150 GHz during 2008 and 2009 and cover ap-
proximately 790 square degrees of sky. This is approx-
imately four times the area used in the preceding SPT
Physics, University of California, and Lawrence Berkeley Na-
tional Labs, Berkeley, CA, USA 94720
Fig. 1.— The beam-deconvolved noise power in the SPT maps
used in this analysis (symbols) compared to theoretical power spec-
tra including CMB only (dashed line) and CMB+foregrounds (solid
line). The precision of the power spectrum measurement is limited
by sample variance rather than detector or atmospheric noise across
most of the 650 < ? < 3000 range.
power spectrum result, Shirokoff et al. (2010). The new
power spectrum spans the multipole range 650 < ? <
3000 (angular scales of approximately 4?< θ < 16?) and
is dominated by primary CMB temperature anisotropy.
The paper is organized as follows. We describe the
SPT, the observations used in this analysis, and the
pipeline used to process the raw data into calibrated
maps in Section 2. We discuss the pipeline used to pro-
cess the maps into an angular power spectrum in Sec-
tion 3. We combine the SPT power spectrum with exter-
nal data, most importantly the seven-year WMAP data
release, to constrain cosmological models in Section 4,
and we conclude in Section 5.
2. OBSERVATIONS AND DATA REDUCTION
The SPT is a 10-meter diameter off-axis Gregorian
telescope located at the South Pole. The current receiver
is equipped with 960 horn-coupled spiderweb bolome-
ters with superconducting transition-edge sensors. The
receiver included science-quality detectors at frequency
bands centered at approximately 150 and 220 GHz in
2008, and at 95, 150, and 220 GHz in 2009. The tele-
scope and receiver are discussed in further detail in Ruhl
et al. (2004), Padin et al. (2008), and Carlstrom et al.
(2011).
2.1. Fields and Observation Strategy
In this work we use data at 150 GHz taken dur-
ing the 2008 and 2009 austral winters.
five fields whose locations, shapes, and effective areas
(i.e. the area of the masks used in the power spec-
trum analysis) are given in Table 1.
tive area is approximately 790 square degrees.25
mean beam-convolved noise power in these fields is ap-
proximated by the sum of a white noise component and
This includes
The total effec-
The
25The ra21hdec-50 and ra21hdec-60 fields overlap slightly.
This reduces the effective total area of the power spectrum analysis.
We have ignored this effect in our simulations, and have therefore
underestimated the SPT bandpower errors by at most 0.4%.
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TABLE 1
SPT Fields from 2008 and 2009
Name
ra5h30dec-55
ra23h30dec-55
ra21hdec-60
ra3h30dec-60
ra21hdec-50
Total
R.A. (◦)
82.5
352.5
315.0
52.5
315.0
Decl. (◦)
-55
-55
-60
-60
-50
∆R.A. (◦)
15.7
18.2
30.5
45.3
30.2
∆Decl. (◦)
10.3
10.1
10.5
10.6
10.5
Effective Area (sq. degrees)
91.6
105.5
156.9
236.0
202.1
792.1
The locations and sizes of the fields observed by SPT in 2008 and 2009. For each field we give the center of the field in Right
Ascension (R.A.) and Declination (Decl.), the extent of the field in Right Ascension and Declination, and the effective field
area.
a component that increases in power with decreasing ?:
C?= (17.9 µK−arcmin)2+3×10−4(
beam-deconvolved noise power is shown in Figure 1.
The fields were observed with two different types
of scan strategies.The scan strategy used for the
ra5h30dec-55field consisted of constant-elevation
scans across the field. After each scan back and forth
in azimuth across the field, the telescope stepped 0.125◦
in elevation. We refer to a complete set of scans covering
the entire field as an observation.
The remaining four fields were observed using a
lead/trail scan strategy. In this strategy each field was
divided into two halves in Right Ascension, and the
two halves were observed sequentially using constant-
elevation scans. Due to the Earth’s rotation, both halves
of the field are observed at the same range of azimuth
angle. This strategy allows for the possible removal of
ground-synchronous signal, although we see no evidence
for such a signal at the angular scales of interest and do
not difference the lead and trail maps.
?
1000)−3.1µK2.26The
2.2. Time-ordered Data to Maps
Each SPT detector measures the sky brightness tem-
perature plus noise, and records this measurement as
the time-ordered data (TOD). The TOD are recorded
at 100 Hz, so we have information in the TOD on sig-
nals up to 50 Hz. For a typical scan speed and eleva-
tion, 50 Hz corresponds to a mode oscillating along the
scan direction at ? ∼ 60,000. This analysis, which only
measures power at ? < 3000, can benefit computation-
ally from using a down-sampled version of the TOD. We
down-sample by a factor of six. Prior to down-sampling,
we low-pass filter the TOD at 7.5 Hz. The combined
effect of the filter and down-sampling is negligibly small
(< 0.1% in power) on the scales of interest and we do
not correct for it.
The TOD are further low-pass filtered at 5 Hz as a
safeguard against high-frequency noise being aliased into
the signal band. The TOD are effectively high-pass fil-
tered by the removal of a Legendre polynomial from the
TOD of each detector on each scan. The order of the
polynomial ranges from 9 to 18 and is chosen to have
approximately the same number of degrees of freedom
(dof) per unit angular distance (∼1.5 dof per degree).
The polynomial fit removes low-frequency instrumental
26Throughout this work, the unit K refers to equivalent fluctu-
ations in the CMB temperature, i.e., the temperature fluctuation
of a 2.73K blackbody that would be required to produce the same
power fluctuation. The conversion factor is given by the derivative
of the blackbody spectrum,dB
dT, evaluated at 2.73K.
and atmospheric noise. Regions of sky within 5 arcmin-
utes of point sources with fluxes S150GHz> 50 mJy are
masked during the polynomial fits.
Correlated atmospheric noise remains in the TOD after
the bandpass filtering. We remove the correlated noise
by subtracting the mean signal across each bolometer
wedge27at each time sample. This subtraction serves as
an approximately isotropic high-pass filter.
The data from each detector receives a weight based
on the power spectral density of its calibrated TOD in
the 1-3 Hz band, which corresponds approximately to the
signal band of the power spectrum analysis presented in
Section 3. We bin the data into map pixels based on
the telescope pointing information. The maps use the
oblique Lambert equal-area azimuthal projection (Sny-
der 1987) with 1?pixel resolution. The power spectrum
analysis presented in Section 3 adopts the flat-sky ap-
proximation, for which wavenumber k is equivalent to
multipole ? and Fourier transforms replace spherical har-
monic transforms. The 150 GHz map of the ra3h30dec-
60 field is shown in Figure 2.
2.3. Beam Functions
The optical response as a function of angle—or beam—
of the SPT must be measured accurately in order to cal-
ibrate the signals in the maps as a function of angular
scale. Due to the limited dynamic range of the detec-
tors, the SPT beams were measured by combining maps
of three types of sources: Jupiter, Venus, and the five
brightest point sources in the CMB fields. The beam
within a radius of 4?is measured on the five brightest
point sources in these fields, and this naturally takes
into account the enlargement of the effective beam due
to random errors in the pointing reconstruction. Outside
of this 4?radius, maps of Jupiter are used to constrain a
diffuse, low-level sidelobe that accounts for roughly 15%
of the total beam solid angle. Maps of Venus are used to
stitch together the outer and inner beams.
We measure the beam function, B?, which is the az-
imuthally averaged Fourier transform of the beam map.
The beam function is normalized to unity at ? = 350. We
express our uncertainty in the beam as an uncertainty in
B?. We account for the uncertainty arising from several
statistical and systematic effects, including residual at-
mospheric noise in the maps of Venus and Jupiter, and
the weak dependence of B?on the choice of radius used
27The SPT array consists of 6 wedge-shaped bolometer modules,
each with 160 detectors. Wedges are configured with a set of filters
that determine their observing frequency (e.g. 95, 150 or 220 GHz).
Page 4
4
Fig. 2.— A map of the ra3h30dec-60 field, which is typical of the fields used in this analysis. The effective area is 236 square degrees.
The structure visible in this map is due to primary CMB anisotropy, not instrumental or atmospheric noise. Modes with ? ? 600 are
strongly suppressed due to the high-pass filtering of the time-ordered data. The map has been multiplied by the apodization and point
source masks described in Section 3.2, such that bright point sources with S150GHz> 50 mJy have been masked. A vertical stripe along
the center of the map has been filtered more strongly than other regions. This stripe lies on the boundary of the lead and trail fields and is
caused by high-pass filtering the time-ordered data by removing polynomial functions. This effect is accounted for in our analysis by using
simulated observations.
Page 5
5
to stitch together the inner and outer beam maps. The
different sources of uncertainty are incorporated into the
power spectrum analysis through the bin-to-bin covari-
ance matrix, as described in Section 3.6. In Figure 3,
we show the beam functions and the quadrature sum of
the different beam uncertainties, which gives a sense of
the total uncertainty. The beam function is uncertain
at the percent level across the multipoles of the power
spectrum presented here. The SPT beams are discussed
in more detail in Lueker et al. (2010) and Schaffer et al.
(in prep.).
Fig. 3.— The 150 GHz beam functions (bold, left scale) and frac-
tional errors (thin, right scale). The beam function is normalized
to one at ? = 350.
2.4. Calibration
The TOD are initially calibrated using a galactic HII
region, RCW38. The final calibration used in this anal-
ysis is calculated by comparing the power spectrum de-
scribed in Section 3 with the seven-year WMAP temper-
ature power spectrum reported in Larson et al. (2010).
The power spectra are compared across the angular range
650 < ? < 1000, where the primary CMB anisotropy
is the dominant source of power. Over this range, the
WMAP and SPT bandpowers share the same binning: 7
bins with δ?= 50. The fractional error for each bin is
calculated according to the ratio of the quadrature sum
of the two experiments’ errors to the WMAP bandpow-
ers. These fractional errors are used to weight each bin’s
contribution to the final calibration and are combined to
estimate the final error budget. This calibration method
is model-independent; it assumes only that the power
measured in the SPT fields is statistically representa-
tive of the all-sky power measured by WMAP. As such,
this calibration method does not bias the constraints on
cosmological parameters described in Section 4. We esti-
mate the uncertainty in the SPT calibration to be 3.1%
in power. This uncertainty is included in the bin-to-bin
covariance matrix as described in Section 3.7.
3. POWER SPECTRUM ANALYSIS
In this section we describe the pipeline used to process
the maps into an angular power spectrum. The method
closely follows the approach used by Lueker et al. (2010)
and Shirokoff et al. (2010). We adopt the flat-sky approx-
imation, in which the angular wavenumber k is equivalent
to multipole ? and Fourier transforms replace spherical
harmonic transforms. The distortion to the power spec-
trum due to adopting the flat-sky approximation on the
SPT maps is negligibly small. We refer to the power in a
given band of angular frequencies as the bandpower. We
report bandpowers in terms of D?, where
D?=?(? + 1)
2π
C?.(1)
3.1. Maps
The power spectrum analysis begins with a set of
150 GHz maps for each field.
to a single bottom-to-top observation of the field. For
fields that were observed using a lead/trail method, we
define a map to be a combination of consecutive lead and
trail maps. We do not subtract the lead and trail maps.
For the ra23h30dec-55 field, which was observed using
large elevation steps, we take this one step further and
define a map as the combination of a pair of consecutive
lead/trail pairs. The composite map has more uniform
coverage than the individual maps due to small elevation
offsets between the individual maps.
Each map corresponds
3.2. Windows
The next step is to calculate the Fourier transform of
each map, ˜ mA, where A is the observation index. All
maps of the same field are multiplied by the same win-
dow W prior to the Fourier transform. This window
is the product of an apodization mask used to avoid
sharp edges at the map borders and a point source mask
used to reduce the power from bright point sources. The
apodization mask is a smoothed version of all pixels that
are observed at least once in each map. We mask all point
sources that we measure to have 150 GHz flux > 50 mJy.
Each point source is masked by a 5?-radius disk, with
a Gaussian taper outside this radius with σtaper = 5?.
Using previous measurements of the mm-wave point
source population (Vieira et al. 2010; Shirokoff et al.
2010), we estimate that the power from residual point
sources below this flux cut is C? ∼ 1.3 × 10−5µK2, or
D? ∼ 18 µK2?
the upper edge of the multipole range of this analysis. A
more aggressive point source cut, say 10 mJy, could have
been used to further reduce the residual power, but the
gains were not considered worth the cost of decreased sky
area and increased mode-coupling.
?
3000
?2. This is approximately equal to
the power from the primary CMB anisotropy at ? = 3000,
3.3. Cross-Spectra
The next step is to cross-correlate maps from different
observations of the same field. The noise in each individ-
ual observation map is assumed to be uncorrelated with
the noise in all other maps, so the resulting cross-spectra
are free from noise bias. We calculate the cross-spectrum
between maps from two different observations, A and B,
and average within ?-bins:
??(? + 1)
?DAB
b
≡
2π
H? ? ?Re[˜ mA
? ? ?˜ mB∗
? ? ?]
?
?∈b
,(2)
Page 6
6
where H? ? ? is a two-dimensional weight array described
below and ? ? ? is a vector in the two-dimensional, grid-
ded Fourier plane of δ? = 10 resolution. We average
?DAB
same weight. The maps are zero-padded prior to the
Fourier transform such that the native ? resolution is
δ? = 10, which allows for clean separation into the final
bins, which have a width of δ? = 50. The lower edge of
the lowest bin is ? = 650, while the upper edge of the
upper bin is ? = 3000.
The noise in the maps used in this analysis is statisti-
cally anisotropic; for a fixed ?, modes that oscillate per-
pendicular to the scan direction (here defined as ?x= 0)
are the most noisy. For this reason, the modes that con-
tribute to a given ?-bin do not necessarily have uniform
noise properties. We construct a two-dimensional weight
array to optimally combine the modes contributing to
each ?-bin,
b
among all pairs of observations A and B, where
A ?= B, to produce?Db. Each observation receives the
H? ? ?∝ (Cth
? + N? ? ?)−2,(3)
where Cth
simulations described in Section 3.5.1 and N? ? ?is the two-
dimensional, calibrated, beam-deconvolved noise power.
We smooth (Cth
?+ N? ? ?) with a Gaussian kernel of width
σ?=425 in order to capture only the broad anisotropy of
the noise power. The weight array is normalized such
that?
bins (? < 2700) and begin to preferentially de-weight
?x ? 0 modes for bins at ? > 2700. We estimate that
this weighting scheme reduces our bandpower errors by
approximately 8% in the highest bin.
?
is the theoretical power spectrum used in the
?∈bH? ? ?= 1 for each bin b. The weights are ap-
proximately uniform for the sample-variance-dominated
3.4. Jackknives
Before proceeding with the rest of the power spectrum
analysis, we apply a set of “jackknife” tests to the band-
powers to search for possible systematic errors.
jackknife test, the data are divided into two halves as-
sociated with potential sources of systematic error. The
two halves are differenced to remove any astronomical
signal, and the resulting power spectrum is compared
to an expectation spectrum. This expectation spectrum
can be non-zero due to mundane observational effects
(e.g. a mid-season adjustment in the scanning strategy,
or unequal weighting of left- and right-going scans). We
use simulations to estimate the expectation spectra and
find that they are small, with D? < 1 µK2at all mul-
tipoles. Significant deviations of the jackknife spectrum
from the expectation spectrum could indicate either a
systematic contamination of the data or a misestimate
of the noise. We construct difference maps (a single map
in one jackknife half subtracted from a single map in
the other half) and measure the jackknife spectrum as
the average cross-spectrum between the difference maps,
in a method analogous to that described in Section 3.3.
The jackknife spectra are calculated for five broad ?-bins.
We perform four jackknife tests based on the following
criteria.
In a
• Time: We split the data into the first and second
halves of observation. This tests for any system-
atic that might be changing on weekly or monthly
timescales.
• Scan Direction:
going and right-going halves. This tests for scan-
synchronous signals or any signal that is not time-
symmetric, such as inaccurate deconvolution of the
detector transfer functions.
We split the data into left-
• Azimuthal Range: We split the data into observa-
tions taken at azimuths that we expect to be more
or less susceptible to ground pickup. We determine
these azimuths by making maps using “ground-
centered” coordinates (Azimuth/Elevation) as op-
posed to “sky-centered” (R.A./Dec.). Although we
detect emission from the ground on large scales
(? ∼ 50) in these ground-centered maps, we do
not expect such emission to bias our measurement
of the sky power, as our observations are spread
randomly in azimuth. We use the azimuth-based
jackknife to test this assertion.
• Moon: We split the data into observations taken
at times when the Moon was either above or below
the horizon. This tests for any significant coupling
to the moon via far sidelobes of the SPT beam.
For each test we calculate the χ2of the jackknife spec-
trum with respect to the expectation model. We cal-
culate the probability to exceed (PTE) this χ2for five
degrees of freedom for each test and find PTE = 0.38,
0.05, 0.92, 0.31 for the Time, Scan Direction, Azimuthal
Range, and Moon tests, respectively. We therefore find
no significant evidence for systematic contamination of
the SPT bandpowers.
3.5. Unbiased Spectra
The spectra calculated in Section 3.3 are biased esti-
mates of the true sky power. The unbiased spectra are
Db≡?K−1?
where b?is summed over. The K matrix accounts for the
effects of the beams, TOD filtering, pixelization, window-
ing, and band-averaging. It can be expanded as
?M???[W]F??B2
where ? and ??are summed over. Q?b and Pb? are the
binning and re-binning operators (Hivon et al. 2002).
B?is the beam function described in Section 2.3. F?is
the transfer function due to TOD filtering and map pix-
elization, and is described in Section 3.5.1. The “mode-
coupling matrix” M???[W] is due to observing a limited
portion of the sky and is calculated analytically from the
known window W, as described in Lueker et al. (2010).
At the multipoles considered in this work, the elements
of the mode-coupling kernel depend only on the distance
from the diagonal.
bb??Db?
(4)
Kbb? = Pb?
???Q??b?
(5)
3.5.1. Transfer Function
The transfer function F?is calculated from simulated
observations of 1500 sky realizations (300 per field) that
have been smoothed by the appropriate beam. These
simulations are also used to calculate the sample variance
Page 7
7
described in Section 3.6, and it is therefore important
that the power spectrum used to generate the simulated
skies be consistent with previous measurements and with
the power spectrum measured in this work. The simu-
lated skies are Gaussian realizations of the primary CMB
from the best-fit lensed WMAP+CMB ΛCDM model
from the seven-year WMAP release28combined with con-
tributions from randomly distributed point sources, clus-
tered point sources, and the Sunyaev-Zel’dovich (SZ) ef-
fect. The random point source component uses C? =
12.6 × 10−6µK2(D?=3000 ≡ D3000 = 18.1 µK2). The
clustered point source component uses D?= 3.5 µK2f?,
where f? = 1 for ? < 1500 and f? = (
? ≥ 1500. This shape is designed to approximate the
shape of the clustering power in both the linear and non-
linear regimes (Shirokoff et al. 2010; Millea et al. 2011).
The SZ component uses the thermal SZ template of Se-
hgal et al. (2010a), which has a shape similar to the
templates from more recent models (Trac et al. 2010;
Shaw et al. 2010; Battaglia et al. 2010), normalized to
D? = 5.5 µK2at ? = 3000.
nents are consistent with the measurements of Shirokoff
et al. (2010) and Vieira et al. (2010), and the total power
is consistent with the power spectrum presented in this
analysis.
The simulated skies are observed using the SPT point-
ing information, filtered identically to the real data, and
processed into maps. The power spectrum of the sim-
ulated maps is compared to the known input spectrum
to calculate the effective transfer function (Hivon et al.
2002) using an iterative scheme. The initial estimate is
??D?
where the superscript (0) indicates that this is the first
iteration in the transfer function estimates. We approxi-
mate the coupling matrix as diagonal for this initial esti-
mate. The factor w2=?dxW2is a normalization factor
mate using the mode-coupling matrix:
?
1500)0.8for
The foreground compo-
F(0)
?
=
?
sim
w2B?2Dth
?
,(6)
for the area of the window. We then iterate on this esti-
F(i+1)
?
= F(i)
?
+
??D?
?
sim− M???F(i)
w2B?2Dth
?? B??2Dth
??
?
(7)
where ??is summed over. The transfer function estimate
has converged after the second iteration, and we use the
fifth iteration.
3.6. Bandpower Covariance Matrix
The bandpower covariance matrix describes the bin-to-
bin covariance of the unbiased spectrum and has signal
term and a noise term. The signal term is estimated us-
ing the bandpowers from the signal-only simulations de-
scribed in Section 3.5.1 and is referred to as the “sample
variance.” The noise term is estimated directly from the
data using the distribution of the cross-spectrum band-
powers DAB
b
between observations A and B, as described
in Lueker et al. (2010), and is referred to as the “noise
28http://lambda.gsfc.nasa.gov/
variance.” The covariance is dominated by sample vari-
ance at low multipoles and noise variance at high multi-
poles, with the two being equal at ? ∼ 2700.
The initial estimate for the bandpower covariance ma-
trix is poor for off-diagonal elements. We expect some
statistical uncertainty,
?
(Cij− ?Cij?)2?
=C2
ij+ CiiCjj
nobs
.(8)
This uncertainty is significantly higher than the true co-
variance for almost all off-diagonal terms due to its de-
pendence on the large diagonal covariances. We reduce
the impact of this uncertainty by “conditioning” the co-
variance matrix in the following manner. First we intro-
duce the correlation matrix
ρ ρ ρij=
Cij
?CiiCjj
.(9)
The shape of the correlation matrix is determined by
the mode-coupling matrix and is a function only of the
distance from the diagonal. We calculate the conditioned
correlation matrix by averaging the off-diagonal elements
at a fixed separation from the diagonal:
?
We set ρ ρ ρ?
a distance ? > 250 from the diagonal. The conditioned
covariance matrix is then
C?
ij
ρ ρ ρ?
ii? =
i1−i2=i−i?ρ ρ ρi1i2
?
i1−i2=i−i?1
.(10)
ij= 0 for all off-diagonal elements that are
ij= ρ ρ ρ?
?CiiCjj.(11)
We must also consider the bin-to-bin covariance due to
the uncertainties in the beam function B?as described in
Section 2.3. We construct a “beam correlation matrix”
for each of the sources of beam uncertainty described in
Section 2.3:
?δDi
where
δDi
Di
ρ ρ ρbeam
ij
=
Di
??δDj
Dj
?
(12)
= 1 −
?
1 +δBi
Bi
?−2
.(13)
The combined beam correlation matrix is the sum of
the beam correlation matrices due to each of the sources
of uncertainty. The beam covariance matrix is then
Cbeam
ij
= ρ ρ ρbeam
ij
DiDj.(14)
We calculate the beam covariance matrix for each year
and add them to the bandpower covariance matrix as
described in Section 3.7.
3.7. Combining Different Fields
We have five sets of bandpowers and covariances, one
set per field, which must be combined into a single set of
bandpowers and covariances. In the limit that the noise
properties of all fields are identical, or in the limit that
the precision of the power spectrum is limited by sam-
ple variance on all scales of interest, each field would be
Page 8
8
weighted by its effective area (i.e. the area of its window).
While neither of these conditions is exactly true for our
fields (the ra21hdec-50 field has higher noise than the
other fields, and the power spectrum is dominated by
noise variance at ? > 2700), they are both nearly true,
and we use the area-based weights. The field-averaged
bandpowers and covariance are then
?
Cbb? =
i
Db=
i
Di
bwi
(15)
?
Ci
bb?(wi)2
(16)
where
wi=
Ai
iAi
?
(17)
is the area-based weight. We introduce the beam covari-
ances by first calculating the covariance for each year,
then adding in the beam covariance for that year, and
finally combining the covariances of the two years. The
last step is to add the covariance due to the SPT cal-
ibration uncertainty, Ccal
ij
= ?2DiDj, where ? = 0.031
corresponds to the 3.1% uncertainty in the SPT power
calibration discussed in Section 2.4.
The final bandpowers are listed in Table 2 and shown
in Figure 4.
3.8. Bandpower Window Functions
In order to allow for a theoretical power spectrum Cth
to be compared to the SPT bandpowers Cb, we calculate
the bandpower window functions Wb
Cth
?
?/?, defined as
b = (Wb
?/?)Cth
?. (18)
Following the formalism described in Section 3.5, we
can write this as
Cth
b = (K−1)bb?Pb???M???F?B2
which implies that
?Cth
?,(19)
Wb
?/? = (K−1)bb?Pb???M???F?B2
We calculate the bandpower window functions to be
used with the final spectrum as the weighted average of
the bandpower window functions from each field.
?.(20)
4. COSMOLOGICAL CONSTRAINTS
The SPT power spectrum29described in the previous
section should be dominated by primary CMB anisotropy
and can be used to refine estimates of cosmological model
parameters. In this section we constrain cosmological
parameters using the SPT power spectrum in conjunc-
tion with data from the seven-year WMAP data release
(WMAP7, Larson et al. 2010)30and, in some cases, in
conjunction with low-redshift measurements of the Hub-
ble constant H0using the Hubble Space Telescope (Riess
29Several of the data products presented in this work will be
made available at http://pole.uchicago.edu/public/data/keisler11
.
30We note that there is a small covariance between the SPT
and WMAP bandpowers due to common sky coverage, but that it
is negligibly small. The composite error has been underestimated
by < 1% across the overlapping ? range.
et al. 2011) and the baryon acoustic oscillation (BAO)
feature using SDSS and 2dFGRS data (Percival et al.
2010). In the analyses that follow, the label “H0+BAO”
implies that the following Gaussian priors have been ap-
plied: H0= 73.8±2.4 km s−1Mpc−1; rs/DV(z = 0.2) =
0.1905±0.0061; and rs/DV(z = 0.35) = 0.1097±0.0036;
where rsis the comoving sound horizon size at the baryon
drag epoch, DV(z) ≡ [(1+z)2D2
is the angular diameter distance, and H(z) is the Hubble
parameter. The inverse covariance matrix given in Eq.
5 of Percival et al. (2010) is used for the BAO measure-
ments.
A(z)cz/H(z)]1/3, DA(z)
4.1. Cosmological Model
We fit the bandpowers to a model that includes four
components:
• Primary
parameter, spatially flat, ΛCDM cosmological
model to predict the power from primary CMB
anisotropy. The six parameters are the baryon den-
sity Ωbh2, the density of cold dark matter Ωch2, the
optical depth of reionization τ, the angular scale of
the sound horizon at last scattering θs, the ampli-
tude of the primordial scalar fluctuations (at pivot
scale k0= 0.002 Mpc−1) ∆2
dex of the scalar fluctuations ns. The effects of
gravitational lensing on the power spectrum of the
CMB are calculated using a cosmology-dependent
lensing potential (Lewis & Challinor 2006).
CMB.Weuse thestandard,six-
R, and the spectral in-
• “Poisson” point source power. Our model includes
a term to account for the shot-noise fluctuation
power from randomly distributed, emissive galax-
ies.This term is constant in C? and goes as
DPS
?
∝ ?2.
• “Clustered” point source power.
cludes a term to account for the clustering of emis-
sive galaxies. For this clustering contribution we
use the template DCL
?
? < 1500, and f? = (
shape is designed to approximate the shape of the
clustering power in both the linear and non-linear
regimes (Shirokoff et al. 2010; Millea et al. 2011).
Our model in-
∝ f?, where f? = 1 for
?
1500)0.8for ? ≥ 1500. This
• SZ power. Our model includes a term to account
for power from the thermal and kinetic SZ ef-
fects. At the angular scales considered here, the
two effects are expected to have similar shapes in
?-space. We therefore adopt the thermal SZ tem-
plate provided in Sehgal et al. (2010a), which has
a shape similar to the templates predicted by more
recent models (Trac et al. 2010; Shaw et al. 2010;
Battaglia et al. 2010), to account for the total SZ
power.
For the purposes of this analysis, the primary CMB en-
codes the cosmological information, while the last three
components, the “foreground” terms, are nuisance pa-
rameters. The foreground terms are used only when cal-
culating the SPT likelihood; they are not used when cal-
culating the WMAP likelihood. In our baseline model,
we apply a Gaussian prior on the amplitude of each of
Page 9
9
TABLE 2
SPT Bandpowers and Bandpower Errors
?center
675
725
775
825
875
925
975
1025
1075
1125
1175
1225
1275
1325
1375
1425
The SPT bandpowers and associated errors in units of µK2. The
errors do not include uncertainty in the SPT beam or calibration.
D?
1710
2010
2530
2560
2150
1600
1160
1100
1190
1250
1130
946
839
696
813
869
σ(D?)
95
98
110
110
93
69
51
43
46
47
43
35
29
27
27
28
?center
1475
1525
1575
1625
1675
1725
1775
1825
1875
1925
1975
2025
2075
2125
2175
2225
D?
739
612
489
407
388
424
396
343
286
234
247
241
233
210
182
142
σ(D?)
25
21
17
14
14
13
13
11
9.8
8.8
8.2
8.2
7.7
7.0
6.2
5.4
?center
2275
2325
2375
2425
2475
2525
2575
2625
2675
2725
2775
2825
2875
2925
2975
D?
143
132
133
110
108
102
86.6
83.6
83.4
83.8
71.9
69.5
63.1
58.9
62.0
σ(D?)
5.4
5.1
5.0
4.5
4.4
4.7
3.8
3.8
4.1
4.0
3.6
3.9
4.1
3.5
4.1
the foreground terms. The prior on the Poisson power
is DPS
from sources with S150GHz < 6.4 mJy, as measured in
Shirokoff et al. (2010), and the power from sources with
6.4 mJy < S150GHz < 50 mJy, as measured in Vieira
et al. (2010) and Marriage et al. (2010). The priors on the
clustered power and SZ power are DCL
and DSZ
et al. (2010). The widths of these priors span the mod-
eling uncertainties in the relevant papers. Finally, we
require the foreground terms to be positive.
that our constraints on cosmological parameters do not
depend strongly on these priors, as discussed below.
One foreground that we have not explicitly accounted
for is the emission from cirrus-like dust clouds in our
Galaxy. Using a procedure similar to that described in
Hall et al. (2010), we cross-correlate the SPT maps with
predictions for the galactic dust emission at 150 GHz
in the SPT fields using model 8 of Finkbeiner et al.
(1999). We detect the galactic dust in the cross-
correlation and estimate the power from galactic dust
in the field-averaged SPT spectrum to be D? = (1.4 ±
0.4)?
the χ2by 0.04 when all other parameters are kept fixed
and by a negligible amount if the other foreground pa-
rameters are allowed to move by amounts that are small
compared to the widths of their priors. We conclude that
galactic dust does not significantly contaminate the SPT
power spectrum.
The total number of parameters in our baseline model
is nine: six for the primary CMB and three for fore-
grounds. The nine-dimensional space is explored using
a Markov Chain Monte Carlo (MCMC) technique. We
use the CosmoMC31software package (Lewis & Bridle
3000= 19.3 ± 3.0 µK2and is based on the power
3000= 5.0±2.5 µK2
3000= 5.5 ± 3.0 µK2, as measured in Shirokoff
We find
?
1000
?−1.2µK2. This is small compared to the SPT
bandpower errors; subtracting this component changes
31http://cosmologist.info/cosmomc/
2002), which itself uses the CAMB32software package
(Lewis et al. 2000) to calculate the lensed CMB power
spectra. The CMB temperature and polarization spec-
tra are calculated by CAMB for each cosmology. These
spectra are passed to the likelihood software provided by
the WMAP team33to calculate the WMAP likelihood.
The SPT likelihood is calculated using the bandpowers,
covariances, and bandpower window functions described
in Section 3. The age of the universe is required to be be-
tween 10 and 20 Gyr, and the Hubble constant is required
to be 0.4 < h < 1.0 where H0= h 100 km s−1Mpc−1.
We assume that neutrinos are massless.
Before combining the SPT and WMAP likelihoods, we
first check that they independently give consistent con-
straints on the six cosmological parameters. Because the
scalar amplitude ∆2
Rand the optical depth τ are com-
pletely degenerate given only the high-? SPT bandpow-
ers, we fix τ = 0.088 for the cosmological model that is
constrained using only SPT data. We find that WMAP
alone gives {100Ωbh2,Ωch2,100θs,ns,109∆2
0.056,0.112±0.0054,1.039±0.0027,0.971±0.014,2.42±
0.11}, while SPT alone gives {2.19 ± 0.18,0.110 ±
0.013,1.043 ± 0.0022,0.953 ± 0.048,2.49 ± 0.49}. Thus
the two likelihoods are consistent with each other, and
we proceed to combine them.
The best-fit model from the joint SPT+WMAP like-
lihood is shown in Figure 5. The baseline model pro-
vides a good fit to the SPT data. The χ2/dof is 35.5/38
(PTE=0.58) if the six cosmological parameters are con-
sidered free and 35.5/44 (PTE=0.82) if the six cosmo-
logical parameters are considered to be essentially fixed
by the WMAP data.
R} = {2.24±
32http://camb.info/ . We use RECFAST Version 1.5.
33We use Version 4.1 of the WMAP likelihood code, avail-
able at http://lambda.gsfc.nasa.gov/ .
tent with our SPT+WMAP Markov chains, we recalculate all
WMAP-only Markov chains rather than use those available at
http://lambda.gsfc.nasa.gov/ .
In order to be consis-
Page 10
10
Fig. 4.— The SPT power spectrum is shown in the left panel. The peak at ? ∼ 800 is the third acoustic peak. For comparison we show
in the right panel other recent measurements of the CMB damping tail from ACBAR (Reichardt et al. 2009), QUaD (Friedman et al. 2009;
Brown et al. 2009), ACT (Das et al. 2011a), and SPT (Shirokoff et al. 2010). The bandpower errors shown in these panels do not include
beam or calibration uncertainties. The ACT spectrum extends to ? = 10,000. The previous SPT spectra, from Lueker et al. (2010) and
Shirokoff et al. (2010), spanned the angular range 2000 < ? < 10,000 and targeted secondary CMB anisotropy.
Page 11
11
The marginalized likelihood distributions for the six
cosmological parameters are shown in Figure 6. The ad-
dition of the SPT data improves the constraints on Ωbh2
and ns by ∼25% and the constraint on θs by nearly a
factor of two. The parameter constraints for the base-
line model are summarized in Table 3, and constraints
on this model using SPT+WMAP+H0+BAO are given
in Table 4.
The scalar spectral index ns is less than one in sim-
ple models of inflation (Linde 2008). Recent measure-
ments of ns come from WMAP7, ns = 0.967 ± 0.014
(Komatsu et al. 2010), ACBAR+QUaD+WMAP7, ns=
0.966+0.014
−0.013(Komatsu et al. 2010), and ACT+WMAP7,
ns
= 0.962 ± 0.013 (Dunkley et al. 2010).
SPT+WMAP constraint is
The
ns= 0.9663 ± 0.0112.(21)
This is a 3.0σ preference for ns< 1 over the Harrison-
Zel’dovich-Peebles index, ns= 1. This constraint is not
significantly altered if we double the width of the priors
on the foreground terms, in which case ns= 0.9666 ±
0.0112. The constraint is also robust to doubling our
uncertainties on the SPT beam functions or SPT cali-
bration, in which cases ns= 0.9671 ± 0.0113 and ns=
0.9661 ± 0.0111, respectively. When the H0 and BAO
data are included, the constraint is ns= 0.9668±0.0093,
a 3.6σ preference for ns< 1.
4.2. Model Extensions
In this section we consider extensions to our baseline
model. These models continue to use a spatially flat,
ΛCDM cosmological model, but allow a previously fixed
parameter—the strength of gravitational lensing, the am-
plitude of tensor fluctuations, the running of the spectral
index, the primordial helium abundance, or the number
of relativistic species—to vary freely. The structure of
this section closely follows the clear presentation and dis-
cussion of the ACT+WMAP constraints on parameter
extensions by Dunkley et al. (2010), and therefore allows
a straightforward comparison of these similar datasets.
We summarize constraints on these extension parame-
ters using recent CMB datasets in Table 5.
We note that we have also considered extensions with
a free dark energy equation of state w or with massive
neutrinos, and found that the addition of SPT data did
not significantly improve upon the constraints on these
models from WMAP alone.
4.2.1. Gravitational Lensing
The paths of CMB photons are distorted by the gravity
of intervening matter as they travel from the surface of
last scattering to us, a process referred to as gravitational
lensing. The typical deflection angle is a few arcminutes,
and the deflections are coherent over degree scales. Lens-
ing encodes information on the distribution of matter at
intermediate redshifts, and this information can be par-
tially recovered using correlations of the CMB temper-
ature and polarization fields (Bernardeau 1997; Zaldar-
riaga & Seljak 1999; Hu 2001; Hu & Okamoto 2002).
Previous efforts to detect lensing of the CMB include
∼3σ detections from correlating quadratic reconstruc-
tions of the lensing field from the CMB with other mass
tracers (Smith et al. 2007; Hirata et al. 2008), ? 3σ detec-
tions from the temperature power spectrum (Reichardt
et al. 2009; Calabrese et al. 2008; Das et al. 2011a), a 2σ
detection from the four-point function of the tempera-
ture field (Smidt et al. 2011), and a 4σ detection from the
four-point function of the temperature field (Das et al.
2011b).
The CMB temperature power spectrum is altered by
lensing at the few percent level.
structure is smoothed, and power is preferentially added
to smaller angular scales. The SPT bandpowers accu-
rately measure the acoustic peaks in the damping tail and
should be sensitive to these effects. The effect of lensing
on the CMB temperature power spectrum is largely cap-
tured by considering large-scale structure in the linear
regime (Lewis & Challinor 2006). The corrections due
to non-linear structures are small compared to the SPT
bandpower errors and are not considered in this analysis.
As a simple measure of the preference for lensing, we
fit the SPT+WMAP bandpowers to a model in which
the CMB is not lensed but that is otherwise identical
to our baseline model. All parameters are free in this
no-lensing model.We compare the likelihood of the
best-fit no-lensing model to the likelihood of the best-fit
lensed model. We find ∆χ2= 2(lnLlens− lnLno lens) =
23.6, corresponding to a 4.9σ preference for the lensed
model. This preference remains if we double the widths
of the priors on the foreground parameters, in which case
∆χ2= 23.0, or if we double the SPT beam uncertainty,
in which case ∆χ2= 22.9. We note that the distortion to
the power spectrum caused by adopting the flat-sky ap-
proximation on the SPT maps is small compared to the
effect of gravitational lensing on the power spectrum. We
estimate that correcting for this distortion would change
the SPT likelihood by ∆χ2< 0.04.
To better quantify the strength of lensing preferred by
the data, we consider the parameter AL which rescales
the lensing potential power spectrum,
The acoustic peak
Cφφ
?
→ ALCφφ
?.(22)
As in all of our models, Cφφ
dependent manner at each point in the Markov chain.
The ALparameter is phenomenological and we allow it
to be negative. In the standard scenario, AL= 1, while
AL= 0 corresponds to no lensing.
This parameter has been constrained using measure-
ments of the CMB damping tail in conjunction with
WMAP data, and more recently by measuring the lens-
ing signal encoded in the CMB temperature four-point
function. Reichardt et al. (2009) used ACBAR data
in conjunction with five-year WMAP data (Nolta et al.
2009) and found AL= 1.60+0.55
(2008) found AL= 3.0±0.9 using the same datasets. Das
et al. (2011a) used power spectra from ACT+WMAP7
to measure AL= 1.3 ± 0.5, and Das et al. (2011b) used
the four-point function of the ACT temperature maps to
measure AL= 1.16 ± 0.29.
The constraints on ALfrom SPT+WMAP7 are shown
in Figure 7. While the constraint on ALis non-Gaussian
in shape, we find that the constraint on A0.65
has an expectation value of 1) is approximately Gaussian.
?
is calculated in a cosmology-
−0.26, while Calabrese et al.
L
(which still
Page 12
12
Fig. 5.—
(CMB+foregrounds) lines. The bandpower errors do not include beam or calibration uncertainties.
The SPT bandpowers, WMAP bandpowers, and best-fit ΛCDM theory spectrum shown with dashed (CMB) and solid
Fig. 6.— The one-dimensional marginalized constraints on the six cosmological parameters in the baseline model. The constraints from
SPT+WMAP are shown by the blue solid lines, while the constraints from WMAP alone are shown by the orange dashed lines.
Page 13
13
TABLE 3
Constraints on Cosmological Parameters using SPT+WMAP
ΛCDMΛCDM
+ AL
2.22 ± 0.044
0.112 ± 0.0050
1.04 ± 0.0016
0.9655 ± 0.0114
0.0853 ± 0.014
2.44 ± 0.11
0.94 ± 0.15
—
—
(0.2478 ± 0.0002)
(3.046)
(0.816 ± 0.025)
7506.2
ΛCDM
+r
2.24 ± 0.045
0.109 ± 0.0050
1.04 ± 0.0016
0.9743 ± 0.0128
0.0860 ± 0.014
2.36 ± 0.11
—
< 0.21
—
(0.2479 ± 0.0002)
(3.046)
(0.805 ± 0.025)
7506.4
ΛCDM
+ dns/dlnk
2.21 ± 0.043
0.117 ± 0.0059
1.04 ± 0.0017
0.9732 ± 0.0120
0.0909 ± 0.016
2.37 ± 0.11
—
—
−0.024 ± 0.013
(0.2477 ± 0.0002)
(3.046)
(0.832 ± 0.027)
7503.6
ΛCDM
+ Yp
2.26 ± 0.048
0.114 ± 0.0051
1.04 ± 0.0020
0.9793 ± 0.0140
0.0884 ± 0.015
2.39 ± 0.10
—
—
—
0.296 ± 0.030
(3.046)
(0.837 ± 0.029)
7504.4
ΛCDM
+ Neff
2.27 ± 0.054
0.125 ± 0.012
1.04 ± 0.0019
0.9874 ± 0.0193
0.0883 ± 0.015
2.37 ± 0.11
—
—
—
(0.2579 ± 0.008)
3.85 ± 0.62
(0.859 ± 0.043)
7505.5
Primary
Parameters
100Ωbh2
Ωch2
100θs
ns
τ
109∆2
A0.65
L
r
dns/dlnk
Yp
Neff
σ8
χ2
min
2.22 ± 0.042
0.112 ± 0.0048
1.04 ± 0.0016
0.9663 ± 0.0112
0.0851 ± 0.014
2.43 ± 0.10
—
—
—
(0.2478 ± 0.0002)
(3.046)
(0.814 ± 0.024)
7506.5
R
Extension
Parameters
Derived
The constraints on cosmological parameters using SPT+WMAP7. We report the mean of the likelihood distribution and the symmetric 68%
confidence interval about the mean. We report the 95% upper limit on the tensor-to-scalar ratio r.
TABLE 4
Constraints on Cosmological Parameters using SPT+WMAP+H0+BAO
ΛCDMΛCDM
+ AL
2.22 ± 0.039
0.112 ± 0.0029
1.04 ± 0.0016
0.9659 ± 0.0095
0.0852 ± 0.014
2.44 ± 0.085
0.95 ± 0.15
—
—
(0.2478 ± 0.0002)
(3.046)
(0.818 ± 0.019)
7510.6
ΛCDM
+r
2.24 ± 0.040
0.112 ± 0.0030
1.04 ± 0.0015
0.9711 ± 0.0099
0.0842 ± 0.014
2.39 ± 0.088
—
< 0.17
—
(0.2478 ± 0.0002)
(3.046)
(0.816 ± 0.019)
7510.7
ΛCDM
+ dns/dlnk
2.23 ± 0.040
0.114 ± 0.0031
1.04 ± 0.0016
0.9758 ± 0.0111
0.0934 ± 0.016
2.35 ± 0.095
—
—
−0.020 ± 0.012
(0.2478 ± 0.0002)
(3.046)
(0.824 ± 0.020)
7507.8
ΛCDM
+ Yp
2.27 ± 0.044
0.114 ± 0.0032
1.04 ± 0.0020
0.9814 ± 0.0126
0.0890 ± 0.015
2.39 ± 0.085
—
—
—
0.300 ± 0.030
(3.046)
(0.841 ± 0.024)
7508.0
ΛCDM
+ Neff
2.26 ± 0.042
0.129 ± 0.0093
1.04 ± 0.0017
0.9836 ± 0.0124
0.0859 ± 0.014
2.41 ± 0.084
—
—
—
(0.2581 ± 0.005)
3.86 ± 0.42
(0.871 ± 0.033)
7507.4
Primary
Parameters
100Ωbh2
Ωch2
100θs
ns
τ
109∆2
A0.65
L
r
dns/dlnk
Yp
Neff
σ8
χ2
min
2.23 ± 0.038
0.112 ± 0.0028
1.04 ± 0.0015
0.9668 ± 0.0093
0.0851 ± 0.014
2.43 ± 0.082
—
—
—
(0.2478 ± 0.0002)
(3.046)
(0.818 ± 0.019)
7510.7
R
Extension
Parameters
Derived
The constraints on cosmological parameters using SPT+WMAP7+H0+BAO. We report the mean of the likelihood distribution and the
symmetric 68% confidence interval about the mean. We report the 95% upper limit on the tensor-to-scalar ratio r.
TABLE 5
Constraints on Model Extensions using Recent CMB Datasets
WMAP7
< 0.7
[-0.084, 0.020]
< 0.51
> 2.7
ACBAR+QUaD+WMAP7
< 0.33
[-0.084, 0.003]
0.326 ± 0.075
—
ACT+WMAP7
< 0.25
−0.034 ± 0.018
0.313 ± 0.044
5.3 ± 1.3
SPT+WMAP7
< 0.21
−0.024 ± 0.013
0.296 ± 0.030
3.85 ± 0.62
r
dns/dlnk
Yp
Neff
The constraints on cosmological parameters in certain model extensions using recent CMB
datasets. We use WMAP7 (Larson et al. 2010; Komatsu et al. 2010), ACBAR (Reichardt
et al. 2009), QUaD (Brown et al. 2009), ACT (Das et al. 2011a), and SPT (this work). All
upper and lower limits and all two-sided limits (shown in brackets) are 95%.
Page 14
14
With SPT+WMAP7, we find34
A0.65
L
= 0.94 ± 0.15.(23)
Thus the SPT+WMAP7 data reject a non-lensed CMB
and are consistent with the expected level of lensing.
This provides a consistency check on the standard pic-
ture of large-scale structure formation. Ongoing and fu-
ture measurements of CMB lensing will move beyond this
consistency check and constrain parameters that affect
the growth of large-scale structure, such as the proper-
ties of dark energy and the sum of the neutrino masses
(Smith et al. 2006; Sherwin et al. 2011).
4.2.2. Tensor Perturbations
Inflation is expected to produce primordial tensor per-
turbations (i.e. gravitational waves). These perturba-
tions imprint potentially detectable effects onto the CMB
temperature and polarization spectra. The amplitude of
the tensor spectrum is often given in terms of the the
tensor-to-scalar ratio, r = ∆2
scale k0= 0.002 Mpc−1.35A detection of r would pro-
vide an extremely interesting window onto the energy
scale of inflation.
Measurements of the B-mode polarization at low multi-
poles will ultimately provide the strongest constraints on
r. To date, the best constraint on r from B-mode polar-
ization comes from the BICEP experiment (Chiang et al.
2010), giving r < 0.7 (95% CL). Stronger constraints
are currently placed using WMAP’s measurement of the
temperature and polarization spectra (Komatsu et al.
2010), which give r < 0.36 (95% CL).
The constraint on r can be improved indirectly with
the addition of small-scale CMB measurements.
CMB power at low multipoles increases as r increases,
but this effect can be partially cancelled by increasing
nsand decreasing ∆2
ments help to break these degeneracies, as demonstrated
in Komatsu et al. (2010), which found r < 0.33 (95%
CL) using ACBAR+QUaD+WMAP7, and in Dunkley
et al. (2010), which found r < 0.25 (95% CL) using
ACT+WMAP7. The SPT+WMAP7 data constrain r
to be
h(k0)/∆2
R(k0) with pivot
The
R. The small-scale CMB measure-
r < 0.21 (95% CL).(24)
When the H0and BAO data are added, the constraint
improves to r < 0.17 (95% CL). Figure 8 shows the one-
dimensional marginalized constraint on r and the two-
dimensional constraint for r and the spectral index ns.
We show the predictions for r and ns from chaotic in-
flationary models (Linde 1983) with inflaton potential
V (φ) ∝ φpand N = 60, where N is the number of e-
folds between the epoch when modes that are measured
by SPT and WMAP exited the horizon during inflation
and the end of inflation. These models predict r = 4p/N
and ns= 1 − (p + 2)/2N. Models with p ≥ 3 are disfa-
vored at more than 95% confidence for N ≤ 60.
34Note that this constraint, along with the other constraints
on A0.65
L
listed in this work, implicitly assumes a uniform prior
on AL. However, the result does not change significantly if we
modify the prior to be uniform in A0.65
A0.65
L
= 0.93 ± 0.15.
35We assume that the spectral index of the tensor perturbations
is nt= −r/8.
L
instead, in which case
4.2.3. Running of the Spectral Index
The power spectrum of primordial scalar fluctuations
is typically parametrized as a power law,
∆2
R(k) = ∆2
R(k0)
?k
k0
?ns−1
. (25)
In this section we allow the power spectrum to depart
from a pure power law. The “running” of the spectral
index is parametrized as36
∆2
R(k) = ∆2
R(k0)
?k
k0
?ns−1+1
2ln(k/k0)dns/dlnk
.(26)
The running parameter dns/dlnk is predicted to be
small by most inflationary theories, and a detection of
a non-zero dns/dlnk could provide information on the
inflationary potential (Kosowsky & Turner 1995). Re-
cent CMB constraints on the running include −0.084 <
dns/dlnk < 0.020 (95% CL) from WMAP7 (Komatsu
et al. 2010), −0.084 < dns/dlnk < 0.003 (95% CL)
from ACBAR+QUaD+WMAP7 (Komatsu et al. 2010),
and dns/dlnk = −0.034 ± 0.018 from ACT+WMAP7
(Dunkley et al. 2010).The SPT+WMAP7 data con-
strain dns/dlnk to be
dns/dlnk = −0.024 ± 0.013.
The data mildly prefer, at 1.8σ, a negative spectral run-
ning.The constraint is dns/dlnk = −0.020 ± 0.012,
a 1.7σ preference for negative running, when the H0
and BAO data are added. As discussed in Section 4.3
and shown in Table 6, the constraint is dns/dlnk =
−0.017 ± 0.012, a 1.4σ preference for negative running,
when information from local galaxy clusters is added.
Figure 9 shows the one-dimensional marginalized con-
straint on dns/dlnk and the two-dimensional constraint
for dns/dlnk and the spectral index ns.37
(27)
4.2.4. Primordial Helium Abundance
When the universe cools to T ∼ 0.1 MeV, light nuclei
begin to form, a process known as big bang nucleosyn-
thesis (BBN) (Schramm & Turner 1998; Steigman 2007).
The primordial abundance (mass fraction) of4He is de-
noted as Ypand is a function of baryon density and the
expansion rate during BBN (Simha & Steigman 2008):
Yp= 0.2485+0.0016[(273.9Ωbh2−6)+100(S−1)], (28)
where
S2= 1 + (7/43)(Neff− 3.046).
The S2
factor generically accounts for any non-
standard expansion rate prior to and during BBN, here
parametrized in terms of the effective number of rela-
tivistic particle species Neff.
(29)
We calculate Yp in this
36The factor of 1/2 is due to considering the effective change in
ns in dln∆2
R/dlnk.
37The estimates for dns/dlnk and ns are highly correlated for
the typical pivot scale, k0 = 0.002 Mpc−1. As in Dunkley et al.
(2010), we have calculated the spectral index at a new, less corre-
lated pivot scale k0= 0.015 Mpc−1, where ns(k0= 0.015 Mpc−1)
= ns(k0= 0.002 Mpc−1) + ln(0.015/0.002)dns/dlnk.
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15
Fig. 7.— The one-dimensional marginalized constraint on the gravitational lensing parameter AL. This parameter rescales the lensing
potential power spectrum as Cφφ
??
.
→ ALCφφ
Fig. 8.— The one-dimensional marginalized constraint on the tensor-to-scalar ratio r (left) and the two-dimensional constraint on r
and the spectral index ns (right). The dashed line shows predictions for r and ns from chaotic inflationary models with inflaton potential
V (φ) ∝ φpand N = 60, where N is the number of e-folds between the epoch when modes that are measured by SPT and WMAP exited
the horizon during inflation and the end of inflation. The two-dimensional contours show the 68% and 95% confidence intervals.
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