# A Measurement of the Damping Tail of the Cosmic Microwave Background Power Spectrum with the South Pole Telescope

**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 < ell <

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, LCDM cosmological model. The SPT+WMAP

constraint on the spectral index of scalar fluctuations is ns = 0.9663 +/-

0.0112. We detect, at ~5-sigma significance, the effect of gravitational

lensing on the CMB power spectrum, and find its amplitude to be consistent with

the LCDM cosmological model. We explore a number of extensions beyond the LCDM

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 7.7-sigma, while a model without neutrinos is rejected at 7.5-sigma. 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...

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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

Page 2

2

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%.

Page 3

3

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. Weusethestandard,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

ΛCDM ΛCDM

+ 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.

Page 15

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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