1 Introduction

The Atacama Cosmology Telescope: Cosmological parameters from three seasons of data.

Abstract

We present constraints on cosmological and astrophysical parameters from high-resolution microwave background maps at 148 GHz and 218 GHz made by the Atacama Cosmology Telescope (ACT) in three seasons of observations from 2008 to 2010. A model of primary cosmological and secondary foreground parameters is fit to the map power spectra and lensing deflection power spectrum, including contributions from both the thermal Sunyaev-ZelÕdovich (tSZ) effect and the kinematic Sunyaev-ZelÕdovich (kSZ) effect, Poisson and correlated anisotropy from unresolved infrared sources, radio sources, and the correlation between the tSZ effect and infrared sources. The power of the thermal SZ power spectrum at 148 GHz is measured to be at while the corresponding amplitude of the kinematic SZ power spectrum has a 95% confidence level upper limit of . Combining ACT power spectra with the WMAP 7-year temperature and polarization power spectra, we find excellent consistency with the LCDM model. We constrain the number of effective relativistic degrees of freedom in the early universe to be , in agreement with the canonical value of for three massless neutrinos. We constrain the sum of the neutrino masses to be eV at 95% confidence when combining ACT and WMAP 7-year data with BAO and Hubble constant measurements. We constrain the amount of primordial helium to be , and measure no variation in the fine structure constant since recombination, with . We also find no evidence for any running of the scalar spectral index, .

Subject headings:
Microwave Telescopes, CMB Observations

1. Introduction

Studies of the cosmic microwave background (CMB) have dramatically progressed over the past two decades (e.g., Smoot et al., 1992; Cheng et al., 1997; Baker et al., 1999; Miller et al., 1999; de Bernardis et al., 2000; Knox & Page, 2000; Hanany et al., 2000; Lee et al., 2001; Romeo et al., 2001; Netterfield et al., 2002; Halverson et al., 2002; Kovac et al., 2002; Carlstrom et al., 2003; Pearson et al., 2003; Scott et al., 2003; Benoît et al., 2003; Spergel et al., 2003; Johnson et al., 2007; Chiang et al., 2010). The current CDM cosmological model provides an excellent fit to the CMB data across a wide range of angular scales, and is supported by complementary observations of large scale structure (e.g., Reid et al., 2012; Ho et al., 2012), Baryon Acoustic Oscillations (BAO e.g., Blake et al., 2011; Anderson et al., 2012; Busca et al., 2013), Type Ia supernovae (e.g., Hicken et al., 2009; Kessler et al., 2009; Conley et al., 2011), galaxy cluster measurements (e.g., Vikhlinin et al., 2009; Mantz et al., 2010; Rozo et al., 2010; Tinker et al., 2012) and observations of gravitational lensing (e.g., Massey et al., 2007; Fu et al., 2008; Schrabback et al., 2010; Suyu et al., 2010; Heymans et al., 2012; Kilbinger et al., 2013). While the CMB on angular scales of greater than has been definitively measured by the Wilkinson Microwave Anisotropy Probe (WMAP, Bennett et al., 2013), a wealth of information in the CMB on smaller angular scales continues to be probed with ever-increasing precision (e.g., Hedman et al., 2002; Kuo et al., 2007; Pryke et al., 2009; Reichardt et al., 2009; Sievers et al., 2009; QUIET Collaboration et al., 2011; Reichardt et al., 2012; Das et al., 2011b; Keisler et al., 2011; QUIET Collaboration et al., 2012). As this paper was being finalized the SPT collaboration released a new set of papers (Story et al., 2012; Hou et al., 2012) and the WMAP team released its final 9-year results (Bennett et al., 2013; Hinshaw et al., 2013). Our analysis does not incorporate these results although in a few places we make direct comparisons. While this paper was under peer review, the Planck satellite released its first cosmological results. We leave a final combined ACT+Planck analysis to future work.

The Atacama Cosmology Telescope (ACT) complements measurements from WMAP by observing from to . This widens the range of data available to constrain both cosmological parameters through the Silk damping tail of the primary CMB (Silk, 1968) and the residual power from secondary sources between us and the surface of last scattering. These sources include galaxy clusters, which are detectable at microwave frequencies through the Sunyaev-Zel’dovich (SZ) effect (Zel’dovich & Sunyaev, 1969; Sunyaev & Zel’dovich, 1970). The thermal SZ (tSZ) effect describes the spectral distortion due to the inverse-Compton scattering of CMB photons to higher frequencies by the hot gas in clusters, while the kinematic SZ (kSZ) effect measures the corresponding temperature shift due to the bulk peculiar motion of the clusters. While these effects produce a diffuse signal at small scales from unresolved clusters, their influence has recently been detected directly through the cross-correlation of the ACT temperature maps with other tracers (Hand et al., 2011, 2012). In addition to emission from clusters via the SZ effect, radio galaxies and dusty star-forming galaxies also contribute to the power on small scales, and indeed dominate the cosmological signal for multipoles and frequencies GHz. Gravitational lensing by structures along the line of sight also generates a microwave background signal, and distorts the primordial CMB.

Figure 1.— Data used in this cosmological analysis. The data from the WMAP 7-year data release (Larson et al., 2011; Komatsu et al., 2011) are combined with the ACT data. In this figure we show a weighted co-added spectrum from the equatorial and southern patches at 148 GHz. The full ACT likelihood, however, considers independent southern and equatorial spectra for both 148 GHz and 218 GHz (given in Table 3 of Das et al., 2013, and online on LAMBDA) and their cross-frequency spectrum, which are shown in Figure 2. The solid line indicates the best-fit cosmological model including foreground emission, while the dashed line shows the best-fit primordial CMB spectrum. This binning was selected for the common analysis of the equatorial and southern data. See Das et al. (2013) for alternative binnings.

This paper forms part of a set of papers presenting the 3-year analysis of the ACT data; the ACT temperature and deflection power spectra are presented in Das et al. (2013), while Dunkley et al. (2013) presents the likelihood used in this analysis. Hasselfield et al. (2013b) presents a catalogue of SZ-detected clusters from the ACT data, and interprets them. This paper contains the parameter estimation of both primary (cosmological) and secondary (foreground) parameters. We outline the data used in this analysis in Section 2 and describe the methodology and likelihood in Section 3. We present constraints on primary cosmological parameters in Section 4 and on secondary parameters in Section 5. We conclude in Section 6, after which we provide an appendix of analysis tests.

2. Data

This paper presents results from a combination of observations at two frequencies, 148 GHz and 218 GHz, of multiple fields taken over three years. The southern fields (ACT-S) varied over different seasons, with the 2008 season containing a 292 deg patch and the 2009 and 2010 seasons focusing on a smaller 146 deg footprint. Equatorial data (ACT-E) were taken only over the 2009 and 2010 seasons; we use a 300 deg patch for the 3-year analysis. We follow a similar procedure for going from maps to the temperature power spectra as was used in Das et al. (2011b), using the power spectrum estimation procedure presented in Das et al. (2009).

The data and map-making procedure are described in Dünner et al. (2013); the power spectrum method and systematic tests are presented in Das et al. (2013). Using the spectrum presented in Das et al. (2013), we have constructed two likelihoods; these likelihoods that are also presented in Dunkley et al. (2013). The multi-frequency likelihood parameterizes the foreground emission using additional parameters which we list in Section 3, while the CMB-only likelihood marginalizes over these foregrounds. For the analysis presented here we use the multi-frequency likelihood; however we show in Appendix D that the two likelihoods give equivalent results.

The data used in this analysis are the multi-frequency temperature spectra estimated from the ACT maps with combined spectra from both ACT-S and ACT-E including their covariance. The spectra are presented in Das et al. (2013). For the results presented in Figure 4 we use the marginalized CMB-only likelihood. In addition, we include the measurement of the power spectrum of the lensing deflection angle, (Das et al., 2013).

The lensing spectrum is estimated from the ACT temperature maps using an optimal quadratic estimator (Hu & Okamoto, 2002). Only data from the ACT equatorial patches are used to measure the deflection power, as the signal-to-noise of the southern patch was much lower than that of the equatorial data. The covariance between the lensing power spectrum and the temperature power spectrum is small. When adding in the deflection data, we will use the abbreviation ‘ACTDefl’.

The maps are cross-correlated with WMAP7 maps in order to obtain a calibration factor in multipole space. The cross-correlation calibration method is described in Hajian et al. (2012); details of the calibration of the ACT 3-year data are given in Das et al. (2013).

2.1. Beam and calibration errors

Understanding the beam profiles is essential for interpreting the high- aspects of the power spectrum. At and 148 GHz, the window function is its value at . The beams are estimated independently for each array and season (Dünner et al., 2013) from observations of Saturn and Uranus. The beams vary slightly with season due to changes in the telescope focus. We include a contribution in the likelihood to the full covariance matrix from the covariance of the beams for each season. The beam error includes contributions from the uncertainty in the pointing variation of the telescope. At , the pivot point for the beam and calibration uncertainties, the effective calibration error is 2% for the 148 GHz maps and 2.6% for the 218 GHz maps. Different seasons have somewhat different calibration uncertainties and so these numbers should be considered as representative of the effective combined calibration. While the absolute calibration is performed from cross-correlations to WMAP data, the telescope pointing solution, beams, and detector responsivity are characterized independently in each observing season and thus the calibration uncertainties of ACT-E and ACT-S are relatively independent.

In the cosmological analysis, we apply a calibration prior for the 148 GHz spectra obtained from the WMAP-ACT cross-correlation calibration procedure described above. In the chains, we allow for a small error in the overall calibration of the spectrum by marginalizing over independent calibration factors for the south and equatorial spectra, at both 148 GHz and 218 GHz. This extra calibration allows for the overall spectra to adjust themselves at the 1% level and has a negligible effect on the cosmological parameters, as discussed in Appendix C. Similarly, in the same appendix, we test for the dependence of the cosmological parameters on the beam’s assumed uncertainty, and find that beam error has a negligible effect on the parameters of interest.

2.2. Additional data

We use temperature and polarization data from the seven year data release of the WMAP satellite (Larson et al., 2011; Komatsu et al., 2011) in addition to the measurements of the microwave temperature from ACT. We include measurements of the Baryon Acoustic Oscillations (BAO) from the Six-degree Field Galaxy Redshift Survey (6dFGRS, Beutler et al., 2011) and the Sloan Digital Sky Survey Data Releases 7 (SDSS DR7, Percival et al., 2010) and 9 (SDSS DR9, Anderson et al., 2012), measured at redshifts: and In addition, we supplement our data with a measurement of the Hubble constant of (Riess et al., 2011), although Freedman et al. (2012) find In some cases we also include a prior on from skewness measurements of the tSZ effect from ACT (Wilson et al., 2012).

Unless explicitly specified, the ACT 3-year data are combined with WMAP7 data. For some model constraints, such as those on and the secondary parameters, we also include the ’low-’ and ’high-’ spectrum measurements from SPT (Keisler et al., 2011; Reichardt et al., 2012), following the prescription in Dunkley et al. (2013). We show the ACT and WMAP7 data in Figure 1. The best-fit model for the various frequency components is shown in Figure 2.

3. Methodology

We use Markov Chain Monte Carlo (MCMC) methods to determine parameters associated with a variety of models. The basic cosmological CDM model consists of 6 parameters describing a flat universe. These include the physical baryon density (where is the dimensionless Hubble parameter), cold dark matter (CDM) density , and , the ratio of the acoustic horizon to the angular diameter distance at decoupling. This parameter is sensitive to the dark energy density, but less degenerate with other parameters (Kosowsky et al., 2002). The value of is then a derived parameter. We assume the primordial perturbations to be scalar, adiabatic, and Gaussian and parametrize them via a spectral tilt , and amplitude , defined at pivot scale Mpc. We assume that the universe transitioned from a neutral to an ionized state over a small redshift range, with optical depth The reionization history of the universe can be probed by small-scale CMB measurements through the impact of reionization on the kSZ effect (Ostriker & Vishniac, 1986; Gruzinov & Hu, 1998; Knox et al., 1998; Zahn et al., 2012), although care must be taken to allow for correlations between the tSZ effect and the microwave emission from unresolved dusty galaxies (Mesinger et al., 2012; Addison et al., 2012a).

We express the CDM set of parameters as

(1)

In this CDM model, the number of effective relativistic degrees of freedom is assumed to be with the abundance of primordial helium fixed at The likelihood used in the ACT analysis is described in Dunkley et al. (2013), which we briefly summarize here. We fit a model of secondary emission to the ACT multi-frequency power spectra that includes an additional nine parameters when considering ACT data in combination with WMAP7 data. For the 148 GHz data we use modes while for the 218 GHz data we restrict ourselves to The theoretical spectrum for frequency bands and is

(2)

where and is the lensed primary CMB power spectrum. The secondary spectra components are modeled as

(3)

with contributions from the tSZ and kSZ effects, CIB sources, the cross-correlation between the tSZ and CIB signals (tSZ-CIB), radio galaxies (rad), and residual Galactic dust (Gal). In addition to the six primary parameters, we add the following nine parameters

(4)

The parameter parameterizes the amplitude of the tSZ power; the kSZ amplitude; and the Poisson and clustered Cosmic Infrared Background (CIB) power and model the residual Galactic dust anisotropy in the southern and equatorial survey regions. All the parameter amplitudes are dimensionless, and are defined for a template spectrum normalized to at , and frequency GHz. The frequency dependence of the correlated and Poisson CIB power is given in flux density units by the product of modified blackbodies with effective temperature 9.7 K and emissivity index , following Addison et al. (2012b), and as described in Equations (8) and (9) of Dunkley et al. (2013). The radio source power has an amplitude and a spectral index fixed to , based on the assumption that the index obtained for brighter sources from ACT and SPT source catalogs (Vieira et al., 2010; Marriage et al., 2011) holds for fainter sources.

The clustered and Poisson (both CIB and radio) templates vary with scale as and , respectively. We allow for a correlation between the tSZ effect and CIB sources, with scale dependence given by the template calculated by Addison et al. (2012a), and a frequency-independent correlation coefficient, , which is defined in Equation (11) of Dunkley et al. (2013), and is restricted to lie in the range . The parameters in our foreground model are summarized in Table 1.

Symbol Description
120 Thermal Sunyaev-Zel’dovich (SZ) power amplitude.
121 Kinematic SZ power amplitude.
122 Poisson Cosmic Infrared Background (CIB) power amplitude.
123 Clustered CIB power amplitude.
124 Residual galactic emission amplitude for the ACT-S spectrum.
125 Residual galactic emission amplitude for the ACT-E spectrum.
Emissivity index of the clustered CIB power.
126 Radio Poisson power amplitude.
127 tSZ-CIB correlation amplitude.
128
Table 1Secondary parameters in the ACT 3-year foreground model.

The spatially variable Galactic emission has been masked, leaving only a small residual component. To determine the amplitude and spectrum of the residual component we cross correlate with the IRIS map (Miville-Deschênes & Lagache, 2005) as described in detail in Das et al. (2013). We then marginalize over this dust component in the likelihood with separate amplitudes in the equatorial and southern regions. There is roughly double the amount of dust in the equatorial region than in the south. In addition, there are some bright clouds. Investigations were performed to identify possible residual dusty clouds that were missed by our treatment, but no clear evidence for them was found. The likelihood marginalization described in Dunkley et al. (2013) was found to be the most general and parsimonious treatment.

We mask all sources above a detection threshold of 15 mJy. The source detection algorithm is described in Marriage et al. (2011). The 148 and 218 GHz source samples in the southern map will be presented in Marsden et al. (2013), and source catalogs for the full data set will be presented in Gralla et al. (2013). Sources are masked to a flux level of 6.4 mJy in the SPT analysis presented in Reichardt et al. (2012), resulting in a source amplitude at 150 GHz of Sources are masked to a level of 50 mJy in Keisler et al. (2011). The estimated difference in unresolved radio source power is Hence we include a separate amplitude for radio emission measured by the South Pole telescope, when including the SPT data in our analysis, whereas the other parameters in the secondary model, apart from the Galactic dust parameters, are common to both data sets. Thus we first subtract an amplitude of radio Poisson power of from the Keisler et al. (2011) ’low-’ spectrum.

For the parameter analysis, we use the publicly available CosmoMC code, which includes version 1.5 of the Recfast code (Seager et al., 1999, 2000; Wong et al., 2008; Switzer & Hirata, 2008; Ali-Haïmoud & Hirata, 2011; Chluba & Thomas, 2011).

Figure 2.— The ACT multi-frequency data. Each solid line indicates the best-fit total model for each cross spectrum, while the dashed lines show the radio power; the dot-dashed line indicates the power from CIB sources (both Poisson and correlated CIB); the gray lines show the total SZ power. The solid and dotted gray lines are for those spectra including 218 GHz and consist of only the kSZ contribution, while the dashed gray line is for the 148x148 GHz spectrum. The dotted lines show the (negative of) the (SZ-CIB) correlation. In all cases the models are for the best fit in a CDM scenario.

We combine measurements of the three independent lensed cross spectra: 148x148 GHz, 148x218 GHz and 218x218 GHz made from the ACT-S and ACT-E fields described in Section 2. In addition, we use the measurement of the lensing deflection field by ACT, presented in (Das et al., 2013). We investigate four types of fits in this analysis:

  • We apply the full ACT likelihood to the data in combination with WMAP7 data and obtain constraints on both the primary and secondary parameters, with a full 15, 16 or 17 parameter model (and four additional calibration nuisance parameters).

  • We estimate bandpowers marginalized over the secondary foregrounds, and obtain constraints on primary parameters based on these marginalized bandpowers.

  • We combine ACT and SPT to check for consistency between these small-scale experiments.

  • We consider the ACT data alone without WMAP7 data, while fixing (or placing priors on) the spectral index and the optical depth .

The models obtained from fitting only the ACT-S and ACT-E spectra are consistent with the models fit to the combined data, the best-fit spectra from each region agreeing to within 4%. Moreover, the best-fit theoretical spectrum from the current 3-year ACT data agrees with the spectrum derived from the 1-year ACT data to the 1% level. We discuss the consistency of the ACT spectrum in Appendix A.

.

Figure 3.— Parameter constraints on the CDM model for the combined ACT and WMAP 7-year power spectra. For each parameter, the full likelihood is marginalized over the other parameters. Dotted curves for the primary parameters are for WMAP7 data only. The solid curves are for ACT plus WMAP7, and the dashed lines add priors from measurements of the baryon acoustic peak (Anderson et al., 2012; Percival et al., 2010; Beutler et al., 2011) and a direct measurement of (Riess et al., 2011). The constraints are summarized in Tables 2,4 and 5. Since the parameter is unconstrained and set only by the prior, it is not shown in the tables.

4. Constraints on primordial parameters

The CDM model continues to fit the ACT data well, when combined with the independent WMAP7 data. Figure 3 illustrates the constraints on the CDM model with the additional secondary parameters for the ACT+WMAP7 data combination. In addition, we plot the constraints from the WMAP7 power spectrum alone. ACT extends the angular range measured by WMAP, but the parameters from the joint fit are consistent with those from WMAP alone. In addition, the plot shows that the six parameters are robust to the presence of low levels of foreground emission that can be identified and extracted by ACT because of its higher resolution and different frequency coverage.

We start by constraining the parameters in the CDM model. Our constraint on the scalar spectral index is using ACT data in combination with WMAP7 data, BAO and measurements. We discuss the constraints on in Section 4.5. In addition, we improve the constraints on the baryon density using only CMB data, to which is due to the fact that the ACT spectrum places tight limits on the positions of the higher order peaks below The models are summarized in Tables 2 and 4 for the ACT data in combination with WMAP7. Table 5 shows the constraints when adding BAO and measurements to the ACT power spectrum data. In Appendix D we show that the CDM parameters derived from the CMB-only likelihood agree with the full likelihood to within . 129

Figure 4.— Constraints on the CDM cosmological model using the likelihood of marginalized CMB bandpowers using ACT+ACTDefl data, compared to the constraints from ACT+ACTDefl+WMAP7, where ACTDefl is the reconstructed deflection power spectrum from ACT data, and WMAP7 alone. The top six panels are fitted parameters, while the bottom two rows are derived parameters. For the ACT-only parameters, two cases are shown: one where we fix the optical depth , and another where the scalar spectral index is fixed. Fixing tightens the constraint on the amplitude of scalar perturbations (the ACT+ACTDefl, fixed curve lies on the ACT+ACTDefl+WMAP7 curve), but does not cause significant shifts in the other primary parameters, relative to leaving free. The independent CMB data sets, ACT and WMAP7, are consistent with the CDM model.

The greatest power of ACT comes when quantifying models beyond the standard cosmological model because the temperature power spectrum contains little additional information on the simple CDM parameters at angular scales smaller than the third acoustic peak, (e.g., Kosowsky et al., 2002). The damping of the higher-order acoustic peaks relative to the baseline model (Komatsu et al., 2009) provides constraints on a variety of non-standard models. The ACT 1-year spectrum data (Das et al., 2011b) showed a slight excess of damping at small scales relative to the baseline model. Evidence for this slight excess is not seen in the ACT 3-year data set. More data resulted in a spectrum with smaller error bars, providing tighter constraints on parameters such as the baryon density, while the best-fit theoretical spectra are consistent between the two results at the 1% level. The consistency between the results presented here and those presented in Dunkley et al. (2011) is discussed in Section D of the appendix.

Figure 5.— Marginalized one-dimensional likelihood of the for ACT data in combination with other probes. Left panel: The improvement in constraints of the ACT in combination with WMAP7 data relative to WMAP7 data on its own. Right panel: The likelihood for WMAP7+ACT data and a variety of other probes. The constraints are consistent with , the value in the model with three neutrino species. The slight shifts in the central value can be understood in the context of the changes in the value of the Hubble parameter, as illustrated in Figure 6.

Figure 6.— Marginalized two-dimensional () contours in the plane. The two parameters are highly correlated, with larger values of leading to higher values of the effective number of relativistic degrees of freedom. Here BAO refers to the combined 6dF and SDSS DR9 BAO measurements, while HST is the measurement from Riess et al. (2011). In contrast, the analysis of Chen & Ratra (2011) prefers where the 95% error bar includes both systematic and statistical errors. When combining both BAO and HST measurements, is pushed to higher values than in either of the individual cases, as tries to reconcile the differences in the distance ratio (where is the co-moving sound horizon at recombination and is an effective geometric distance) between the two probes. A similar trend is seen for the matter density, .

Various authors have explored possible models which lead to excess damping of the Silk damping tail (e.g., Galli et al., 2011; Calabrese et al., 2011a, b; Hasenkamp, 2012; Hamann, 2012; Menestrina & Scherrer, 2012; Foot, 2012; Abazajian et al., 2011; Menegoni et al., 2012; Farhang et al., 2013). In this analysis we broaden our standard picture with other parameters and interpret the damping tail of the ACT data in this context.


4.1. ACT data alone

In Figure 4, we compare the ACT and WMAP7 data both separately and together with the constraints on the CDM model. This provides an important cross-check of the consistency of the two data sets. A key parameter which is primarily constrained with polarization data on the largest scales, such as those probed by WMAP, is the optical depth Hence, when considering the constraints from the ACT data alone, without including WMAP7, we impose a prior on the optical depth, In Figure 4 we show two cases, one in which the scalar spectral index is fixed at and one in which is allowed to vary freely. Fixing tightens the bound on the amplitude of fluctuations, while other parameters are largely insensitive to the effect. The agreement shows that the same model that describes the WMAP7 data for independently fits the damping tail measurement from ACT of The same behavior is observed in Story et al. (2012).

Figure 7.— Marginalized two-dimensional () contours in the plane. The two parameters are strongly correlated, with larger values of leading to higher values of the effective number of relativistic degrees of freedom. This figure is based on the 16 parameter fit of the standard cosmology and including variations in .

4.2. Effective number of relativistic species

The standard cosmological model has three neutrino species, all of which have negligible mass and contribute to , the effective number of relativistic species at recombination.130 Precision electroweak measurements place tight constraints on the number of light neutrino species with standard-model couplings (The ALEPH Collaboration et al., 2006) through Z production in collisions:

(5)

Relativistic species (whether neutrinos or other early relativistic species) change the expansion rate of the universe through their energy density and impact the perturbations in the early universe, affecting the damping tail of the primary CMB spectrum (Bowen et al., 2002; Bashinsky & Seljak, 2004; Hou et al., 2013). In the case of neutrinos, the energy density is lower than that of photons by a factor

(6)

where is 3.046 in the standard CDM model.

Extra relativistic energy density damps the small-scale CMB power – see Hou et al. (2013) for a concise recent review, and the discussions in Hu & Dodelson (2002); Hu et al. (2001); Bashinsky & Seljak (2004); Tegmark (2005); Lesgourgues & Pastor (2006); Hannestad (2010).

Figure 5 and Table 2 illustrate the constraints on the from ACT in combination with various probes. Previous analyses (Dunkley et al., 2011; Keisler et al., 2011) suggested a slight excess in the . This preference for more damping from extra relativistic degrees of freedom is no longer present when analyzing the ACT 3-year data in combination with WMAP7 data. The change is consistent with the improved statistics of the ACT 3-year data. We find

(7)

The improvement in this value relative to the WMAP-only constraints is shown in the top panel of Figure 6.

The result in this analysis was obtained by imposing the consistency relation between the primordial helium fraction at Big Bang Nucleosynthesis (BBN) and the number of effective relativistic degrees of freedom (Trotta & Hansen, 2004; Kneller & Steigman, 2004; Steigman, 2007; Simha & Steigman, 2008, see Section 4.9):

Hence, in the present analysis, the helium fraction is a determined parameter given and the baryon density, rather than remaining fixed at the standard value of The value presented in Dunkley et al. (2011) was higher at as we did not impose this relation; imposing this constraint on the previous ACT-S data would yield a modified value of consistent at with the value of 3.046 expected in standard CDM.

The value of obtained when using only the ACT-S data is closer to the value reported in Dunkley et al. (2011). The ACT-E data prefer a lower value for , leading to a combined result which is lower than presented in Dunkley et al. (2011).

Including the recent BAO data does not change the constraints on the relativistic species, while adding in the SPT data shifts the mean value to slightly higher values of but still consistent with the WMAP7+ACT data. The highest value for comes from the addition of the BAO and Hubble constant data, yielding

(9)

This is due to the degeneracy between (and therefore the Hubble constant in a flat universe) and the relativistic species, shown in Figure 6. In a flat universe, higher leads to lower which increases the power on medium to small scales (as the radiation driving of the acoustic oscillations is reduced), leading in turn to a larger value of needed to damp power in the tail of the spectrum. The mild tension between the inferred through BAO distance measurements and the Hubble constant measurements leads to a value of which is higher than the constraint when only the Hubble constant is added to the CMB data. A similar trend was seen in the recently released SPT results (Hou et al., 2012). Smaller values of the Hubble constant prefer lower values of (e.g., Chen & Ratra, 2011; Calabrese et al., 2012). The results are summarized in Table 3.

In addition, the correlation between the scalar spectral index and the is shown in Figure 7. Decreasing power at small scales (through increasing is compensated by increasing the scalar spectral index, which increases small-scale power.

Finally, we also consider an additional constraint from the recent ACT measurement of the skewness induced by the tSZ effect (Wilson et al., 2012). The tSZ skewness signal is more sensitive to than any other cosmological parameter (scaling approximately as ), allowing for a tight constraint with few degeneracies. Using theoretical calculations similar to those in Wilson et al. (2012), we find that the most significant degeneracy is with , for which the tSZ skewness scales approximately as . Thus, we include the constraint from Wilson et al. (2012) in the following form

(10)

where and 11.1 is the fiducial scaling of the tSZ skewness value with In addition to its correlation with the effective number of relativistic degrees of freedom is strongly correlated with : as increases, so does hence this prior lowers the effective number of relativistic degrees of freedom to

The fact that ACT resolves the higher order peaks of the CMB spectrum allows for comparison with models that allow for departure from pure free-streaming (e.g., Cyr-Racine & Sigurdson, 2013). The effect on the small-scale power of a model with dark photons which are initially coupled to dark matter and hence only start free-streaming after they decouple implies that the phase shift and amplitude suppression associated with the free-streaming of radiation will not be uniform across all multipoles. We leave the testing of such models to future work.

4.3. Massive neutrinos

In the previous subsection, we estimated the number of effective relativistic species, . If is assumed to arise solely from neutrinos impacting the CMB, these are also assumed to be massless in our fiducial model. However, the CMB is sensitive to the sum of neutrino masses, which is related to the energy density of massive neutrinos via

(11)

In general there will be a degeneracy between the mass of a neutrino species and increased relativistic degrees of freedom at early times.

Figure 8.— Marginalized one-dimensional likelihood for the sum of the neutrino mass, in the case of the ACT+WMAP7 data, and with the addition of the BAO and constraint. In addition, we add a constraint on from ACT skewness measurements (Wilson et al., 2012).

After the neutrinos become non-relativistic, neutrino free streaming washes out structure on small scales. In Figure 8 we show the constraints on the sum of the neutrino masses assuming is fixed to 3.046. For the ACT in combination with WMAP7 data (and keeping the number of neutrinos fixed at ), we find

(12)

Adding in distance measurements through the BAO and Hubble constant prior (which breaks the degeneracy between and ) improves the constraint to

(13)

Imposing the constraint on from the ACT skewness measurement given in Eq. 10 yields

Equations (12) and (13) show that adding the ACT data strengthens the constraints by factors of 1.5 and 1.8, respectively, compared to Komatsu et al. (2011).

Hou et al. (2012) find a 3 preference for non-zero neutrino mass, eV when combining WMAP7 and SPT CMB data with BAO, HST and SPT cluster constraints. This preference is driven by a combination of factors, including mild tension between the SPT and WMAP7 values of , between SPT and BAO constraints, and between the CMB and SPT cluster constraints on , in the massless neutrino case, which reduces the error by a factor of 1.6 without changing the central value. The lack of evidence for non-zero neutrino mass in our analysis reflects the agreement between the ACT CMB, WMAP7, BAO and ACT cluster measurements when neutrino mass is fixed to zero. A more quantitative comparison is deferred to further work in light of recent Planck results. The constraint on from ACT clusters alone in combination with WMAP7 data (and BAO+ data) for the CDM+ model (Hasselfield et al., 2013b) is , for which

Parameter131 CDM CDM CDM CDM CDM
+ + + +
Primary
CDM
Extended
Derived
Secondary
Calibration
132 8147 8147 8148 8147 8146
133134135
Table 2CDM and extended ‘damping tail’ parameters confidence limits from the CMB data alone: ACT data combined with seven-year WMAP data.

Figure 9.— Marginalized two-dimensional () contours in the plane. The number of effective relativistic degrees of freedom is not strongly correlated with neutrino mass using CMB data alone; however, adding in complementary data sets helps tighten constraints. The star indicates the standard point where 0.045 eV is the minimum neutrino mass.

Figure 9 illustrates the marginalized and contours for the sum of the neutrino masses and the when both are varied simultaneously. Adding in the Hubble constant and BAO data pushes to slightly higher values, while the prior on from ACT tSZ skewness measurements lowers the effective number of degrees of freedom. In all cases the sum of the neutrino masses is consistent with zero.

4.4. Early dark energy

The small-scale damping seen in the ACT data can also be interpreted as arising from a non-negligible amount of dark energy at decoupling. The early dark energy (EDE) component may be specified through its density parameter (relative to the energy required for a flat universe) and an equation of state given by Wetterich (1988); Doran & Robbers (2006)

Data set
WMAP7+ACT
WMAP7+ACT+SPT
WMAP7+ACT+BAO
WMAP7+ACT+HST
WMAP7+ACT+BAO+HST
Table 3Constraints on from the ACT data combined with seven-year WMAP data and other data.

Figure 10.— One-dimensional marginalized likelihood for the early dark energy density, . Adding in deflection measurements (shown by the inner-most curve) tightens the constraints on the allowed energy density in early dark energy.

The amounts of dark energy and matter today are given by respectively, and is the scale factor at matter-radiation equality. is the fraction of dark energy allowed at early times, while the present value of the equation of state of this early dark energy is expressed as . The scaling behavior of the equation of state, tracking the dominant component at each cosmic era, gives rise to a radiation-like component at high redshifts, enhancing the damping of the small-scale CMB tail. As the amount of EDE at early times tends to zero, this model approximates the standard CDM cosmological model. We place a prior that This EDE model has been considered by many authors (de Putter et al., 2009; Hollenstein et al., 2009; de Putter et al., 2010; Calabrese et al., 2011, 2011a, 2011b; Reichardt et al., 2012) showing however that current CMB observations combined with large scale structure data have no preference for a non-zero EDE density. We include dark energy clustering as discussed in Calabrese et al. (2011), but we fix the dark energy sound speed and viscosity parameters to one and zero, respectively, as expected for a perfect fluid. Prior to this work, the most recent constraint on this model is from Reichardt et al. (2012) who report an upper limit of at confidence from CMB only data, combining WMAP and SPT. In our analysis of the ACT data in combination with WMAP7 data and the ACT deflection measurement, we obtain the upper bound

(16)

with the bound on the equation of state found to be (see Figure 10). We do not include the combination of early dark energy and , as the combination of these two parameters is largely unconstrained using the current small-scale CMB data.

Parameter136 CDM CDM CDM CDM CDM CDM
+ + + + 137 +
Primary
CDM
Extended
Derived
Secondary
Calibration
8147 8147 8148 8147 8148138 8149
139140141
Table 4CDM and extended single-model inflationary and damping parameters from CMB data alone: ACT data combined with seven-year WMAP data.
Parameter142 CDM CDM CDM CDM CDM
+ + +
Primary
CDM
Extended
Derived
Secondary
Calibration
143 8153 8152 8153 8153 8152
144145
Table 5Extended model parameters confidence limits from the ACT data combined with seven-year WMAP data, BAO and measurements.

4.5. Inflationary parameters

Inflation provides a mechanism for the seeding of cosmological structure through small fluctuations in the early universe. We constrain the spectral index of the initial spectrum of scalar density fluctuations through the measurement of the high tail of the angular power spectrum. The scalar spectrum of curvature perturbations is parameterized via (Kosowsky & Turner, 1995)

(17)

We constrain the amplitude , spectral index and ‘running’ of the spectrum defined at a pivot point The amplitude of scalar perturbations is found to be

(18)

where the factor of is a numerical factor included to ensure robustness to numerical precision errors while sampling.

The scalar spectral index

A generic prediction of inflationary models is a nearly scale invariant spectrum; any deviations provide powerful tests of inflationary models (Mukhanov & Chibisov, 1981; Hawking, 1982; Starobinsky, 1982; Guth & Pi, 1982; Bardeen, Steinhardt, & Turner, 1983; Mukhanov, Feldman, & Brandenberger, 1992). Deviations from scale invariance have been tested in a variety of models and contexts (e.g., Wang et al., 1999; Tegmark & Zaldarriaga, 2002; Bridle et al., 2003; Hannestad, 2003; Martin & Ringeval, 2004a, b; Sealfon et al., 2005; Spergel et al., 2007; Verde & Peiris, 2008; Peiris & Verde, 2010; Vázquez et al., 2012), with the high-resolution of ACT expanding possibilities for direct measurements of the power spectrum (Hlozek et al., 2012). The standard cosmological scenario in which no variation of the spectral index is allowed (i.e. ) provides an excellent fit to the data. We obtain a constraint on the scalar spectral index of

(19)

When is allowed to float, moves to slightly different values as shown in Figure 11. When considering the ACT-S and ACT-E spectra separately (while still in combination with WMAP7), we find

(20)

The ACT-E data specifically prefer a higher value of the ACT-S data, which is consistent with , the value in Dunkley et al. (2011). However, the cosmological models that describe the two statistically independent data sets are consistent as shown in Appendix A. The difference in the marginalized is indicative of residual correlations between parameters. Combining the two data sets increases the value of relative to WMAP7 alone. It is similarly higher than from WMAP7+SPT+BAO+H0 reported in Story et al. (2012).

Including BAO and data yields the constraint

(21)

which rules out a scale-invariant spectrum at a significance of The ACT+WMAP7 value of is higher than the recently released WMAP9 result (Bennett et al., 2013; Hinshaw et al., 2013), however this constraint includes SPT data and the improved WMAP data.

CDM + Running Index

Figure 11.— Two-dimensional marginalized limits (68% and 95%) for the spectral index, , plotted at the pivot point , and the running of the index , for ACT+WMAP7, compared to WMAP7 data alone. The data are consistent with the model

In addition to the spectral index alone, we test for deviations from a perfect power law, in the form of running of the spectral index. Our cosmological models are run around a pivot point of . Figure 11 shows the joint constraints on the spectral index and its running at a de-correlated pivot point , chosen to minimize the correlation between the two parameters (Cortês et al., 2007). We show the constraints for the ACT data combination with WMAP7, compared to the WMAP7 data alone. This relation between the indices at these two pivot points is given by: