THE ATACAMA COSMOLOGY TELESCOPE: COSMOLOGICAL PARAMETERS FROM THE 2008 POWER SPECTRUM

The ACT calibration is described in Hajian et al. (2010). The 148 GHz maps are calibrated using WMAP, resulting in a 2% map calibration error in temperature units, at effective ℓ = 700. The 218 GHz maps are calibrated using observations of Uranus, with a 7% calibration error at ℓ = 1500. The two calibration errors have negligible covariance and are treated as independent errors. For analyses using 148 GHz data alone we marginalize over the calibration uncertainty analytically following Ganga et al. (1996) and Bridle et al. (2002). For joint analyses with the 148 GHz and 218 GHz data we explicitly sample the calibration parameters y(ν_i) with Gaussian priors of y(148) = 1.00 ± 0.02 and y(218) = 1.00 ± 0.07. We check that the analytic and numerical marginalization methods give the same results for 148 GHz.

The beam window functions are described in Das et al. (2011) and are estimated using maps of Saturn. The maps are made with an independent pipeline to the initial ACT beam estimates made in Hincks et al. (2010), but produce consistent results. The uncertainties on the beam window functions are of order 0.7% for the 148 GHz band and 1.5% at 218 GHz. Uncertainties in the measured beams are incorporated using a likelihood approximation described in Appendix A; the magnitude of the derived uncertainties is consistent with Hincks et al. (2010) and the uncertainties used in the parameter estimation in Fowler et al. (2010).

The thermal SZ template $\mathcal{B}^{\rm tSZ}_{0,\ell }$ describes the power from tSZ temperature fluctuations from all clusters, normalized for a universe with amplitude of matter fluctuations σ₈ = 0.8. There is uncertainty in the expected shape and amplitude of this signal, arising due to incomplete knowledge of the detailed gas physics that affects the integrated pressure of the clusters. Previous cosmological studies, e.g., the ACBAR and CBI experiments, have used templates derived from hydrodynamical simulations (Bond et al. 2005). The analysis for WMAP used the analytic Komatsu–Seljak (K-S) spectrum derived from a halo model (Komatsu & Seljak 2001). Recent studies for SPT have considered simulations and analytic templates from Sehgal et al. (2010) and Shaw et al. (2009).

In this analysis we consider four thermal SZ templates, from Sehgal et al. (2010), Trac et al. (2011), Battaglia et al. (2010), and Shaw et al. (2010). Trac et al. (2011) constructed several templates by processing a dark matter simulation to include gas in dark matter halos and in the filamentary intergalactic medium. Their "standard" model, which was first described in Sehgal et al. (2010), is referred to here as "TBO-1." It is based on the gas model in Bode et al. (2009), with the hot gas modeled with a polytropic equation of state and in hydrostatic equilibrium, with star formation and feedback calibrated against observations of local clusters. This is the template considered in the ACT analysis by Fowler et al. (2010) and has a similar amplitude and shape to the K-S spectrum.

Recent state-of-the-art simulations described in Battaglia et al. (2010) have been used to predict the SZ spectrum (referred to as "Battaglia"). Full smoothed hydrodynamical smoothed particle hydrodynamics (SPH) simulations were made of large-scale cosmic structure, including radiative cooling, star formation, feedback from active galactic nuclei (AGNs) and supernovae, and non-thermal pressure support. The predicted spectrum has 2/3 the power compared to the TBO-1 spectrum and is more consistent with SZ measurements from SPT (Lueker et al. 2010). It is also compatible with predictions made that AGN heating would decrease the expected SZ power (Roychowdhury et al. 2005).

These hydrodynamical simulations pre-date the SPT observations, but there have also been recent developments in simulating and modeling the expected SZ effect in light of the SPT results in Lueker et al. (2010), and motivated by recent observations of the intracluster medium (see Trac et al. 2011 for a discussion). In a second model described in Trac et al. (2011), the nonthermal20 model referred to here as "TBO-2," there is 20% non-thermal pressure support, with increased star formation and reduced feedback, which has the effect of lowering the predicted SZ power. The "Shaw" model, described in Shaw et al. (2010), takes an analytic halo model approach, assuming hydrostatic equilibrium and a polytropic equation of state, with star formation, feedback from supernova and AGN, and energy transfer from dark matter to gas during mergers. It includes non-thermal pressure support. The spectra from each of these models are shown in Figure 1, with all models normalized to σ₈ = 0.8. We study constraints on all four templates. Apart from the TBO-1 template, all have similar amplitudes of $\mathcal{B}_{3000} \approx 5\hbox{--}6 \,\mu {\rm K}^2$ at ℓ = 3000, although the models have different amounts of star formation, feedback, and non-thermal support.

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The shape and amplitude of the expected kinetic SZ power spectrum is highly uncertain. We use the kSZ template described in Sehgal et al. (2010), also shown in Figure 1. The corresponding template for the nonthermal20 model in Trac et al. (2011) has a similar amplitude. It is also consistent with predictions from second-order perturbation theory (Hernández-Monteagudo & Ho 2009). In this analysis the contamination of the SZ signal by point sources is neglected, which is shown in Lin et al. (2009) to be a good approximation for radio galaxies. Lueker et al. (2010) show it is also a good assumption for IR sources for the current levels of sensitivity.

The shape and amplitude of the power spectrum of clustered dusty galaxies are not yet well characterized (Knox et al. 2001; Fernandez-Conde et al. 2008; Viero et al. 2009; Hall et al. 2010), although there have been measurements made by Viero et al. (2009) from the BLAST experiment. In Fowler et al. (2010) we adopted a simple power-law model, with $\mathcal{B}_\ell \propto \ell$. In this analysis we move beyond this simple parameterization to consider two alternative model templates. The first, "Src-1," is obtained from the infrared source model described in Section 2.5.2 of Sehgal et al. (2010). This model assumes that the IR emission traces star formation in halos at z < 3, and that the number of IR galaxies in a given halo is proportional to its mass. For the spectral energy distribution (SED) of the galaxies, the model uses an effective graybody law in which the dust temperature is a function of the CMB temperature, but its value at z = 0 is a free parameter. The dust emissivity spectral index and the typical IR luminosity and characteristic masses of the halos hosting IR galaxies are free parameters of the model. The model includes only the two-halo power spectrum given in Equation (24) of Sehgal et al. (2010), with contributions from pairs of galaxies in different halos. The parameters of the model have been updated from Sehgal et al. (2010) to better fit the observed BLAST power spectra at 250–500 μm.³³ This updated template is shown in Figure 1, normalized to unity at ℓ = 3000. The shape is similar to the clustered model used in the SPT analysis by Hall et al. (2010), peaking at ℓ ≃ 1000.

We also consider the effect on our results of using an alternative template, "Src-2" that has both one-halo and two-halo power, following a halo model prescription similar to Viero et al. (2009). Dark matter halos are populated using galaxy source counts from the source model presented in Lagache et al. (2003), and halo occupation distribution parameters are tuned to fit the BLAST power spectra. This normalized template is also shown in Figure 1. At large scales this has a similar shape to the Src-1 template, but at small scales tends approximately to the $\mathcal{B}_\ell \propto \ell$ scaling adopted in Fowler et al. (2010), which was motivated by observations of galaxy clustering at small angles with typical correlation function C(θ)∝θ^−0.8 (Peebles 1980). The two templates differ at ℓ > 3000, but at these angular scales the Poisson power is expected to dominate over the clustering term.

To summarize the methods, an analysis with the "148+218" likelihood follows these steps to return the ACT likelihood for a given model.

• 1.  
  Select primary cosmological parameters, and compute a theoretical lensed CMB power spectrum $\mathcal{B}_\ell ^{\rm {CMB}}$ using the CAMB numerical Boltzmann code (Lewis et al. 2000).
• 2.  
  Select values for secondary parameters Θ = {A_tSZ, A_kSZ, A_d, A_c, A_s, α_d, α_s}, and compute the total theoretical power spectra $\mathcal{B}_\ell ^{\rm {th,ij}}$ for 148 × 148, 148 × 218 and 218 × 218 using Equation (5).
• 3.  
  Compute the bandpower theoretical power spectra C^{th, ij}_b = w^ij_bℓC^{th, ij}_ℓ.
• 4.  
  Select values for the calibration factors for 148 GHz and 218 GHz, and compute the likelihood using Equation (14) for 500 < ℓ < 10, 000 for 148 × 148 and 1500 < ℓ < 10, 000 for 148 × 218 and 218 × 218.
• 5.  
  Add the likelihood term due to beam uncertainty, described in Appendix A.

A large part of our analysis uses only the 148 GHz spectrum. An analysis done with this "148-only" likelihood follows these steps.

• 1.  
  Select primary cosmological parameters and compute a theoretical lensed CMB power spectrum $\mathcal{B}_\ell ^{\rm {CMB}}$ using CAMB.
• 2.  
  Select values for secondary parameters Θ = {A_SZ, A_p, A_c} and compute the total theoretical power spectrum $\mathcal{B}_\ell ^{\rm {th}}$ at 148 GHz using Equation (5).
• 3.  
  Compute the bandpower theoretical power spectrum C^th_b = w_bℓC^th_ℓ.
• 4.  
  Compute the likelihood using Equation (14) for 500 < ℓ < 10, 000 for 148 GHz.
• 5.  
  Add the likelihood term due to beam uncertainty, described in Appendix A, and analytically marginalize over the calibration uncertainty.

For most of the investigation with the 148+218 likelihood we fix the primary cosmological parameters to the best-fit ΛCDM parameters estimated from WMAP, as our goal is to characterize the small-scale power observed by ACT, and check the goodness of fit of this simple model. To map out the probability distribution for these parameters we use an MCMC method. This uses the Metropolis algorithm to sample parameters (Metropolis et al. 1953), following the methodology described in Dunkley et al. (2005).

There are seven possible secondary parameters, but we do not allow them all to vary freely. The radio sources detected at 148 GHz, described in Marriage et al. (2011), are observed to have typical spectral index S(ν)∝ν^−0.5 in flux units. By fitting a scaled source model from Toffolatti et al. (1998) to the detected sources, and using it to extrapolate to fainter sources, Marriage et al. (2011) predict a residual radio source power of C_ℓ = 2.8 ± 0.3 nK². Converting units, we use these measurements to impose a Gaussian prior of A_s = 4.0 ± 0.4 μK², and we fix α_s = −0.5. We also fix A_kSZ = A_tSZ, as the kSZ component is subdominant at 148 GHz and the SZ models predict them to be the same for a given cosmology. The other parameters (A_tSZ, A_d, A_c, and α_d) have uniform priors with positivity imposed on the amplitudes. Parameter results are quoted using the means and 68% confidence limits of the marginalized distributions, with 95% upper or lower limits given when the distribution is one-tailed. We also quote derived parameters to indicate the power in different components at 148 GHz and 218 GHz, for example the total power in SZ at ℓ = 3000, $\mathcal{B}_{3000}^{\rm SZ} \equiv (\mathcal{B}^{\rm kSZ}+\mathcal{B}^{\rm tSZ})_{3000}$.

In order to explore the probability distributions for a set of cosmological models, we use the 148-only likelihood for parameter estimation. The focus is on using the 1000 < ℓ < 3000 spectrum to improve constraints on primary cosmological parameters. It is important that the SZ and foreground contribution be marginalized over, but we exclude the more contaminated 218 GHz data given the current uncertainties in the foreground model. To map out the distribution for cosmological parameters we use MCMC methods to explore the probability distributions for various cosmological models.

We parameterize cosmological models using

$$\begin{equation} \big\lbrace \Omega _b h^2, \Omega _c h^2, \Omega _\Lambda, \Delta _{\mathcal{R}}^2,n_s,\tau \big\rbrace. \end{equation} \tag{ 15 }$$

These are the basic ΛCDM parameters, describing a flat universe with baryon density Ω_bh², cold dark matter (CDM) density Ω_ch², and a cosmological constant $\Omega _\Lambda$. Primordial perturbations are assumed to be scalar, adiabatic, and Gaussian, described by a power law with spectral tilt n_s, and amplitude $\Delta _{\mathcal{R}}^2$, defined at pivot scale k₀ = 0.002 Mpc⁻¹. We assume "instantaneous" reionization, where the universe transitions from neutral to ionized over a redshift range Δz = 0.5, with optical depth τ. Reionization likely takes place more slowly (e.g., Gnedin 2000; Trac et al. 2008), but current CMB measurements are insensitive to this choice (Larson et al. 2011). We also consider an additional set of primary parameters

$$\begin{equation} \lbrace dn_s/d\ln k,r, N_{\rm eff}, Y_P, G\mu \rbrace \end{equation} \tag{ 16 }$$

that describe primordial perturbations with a running scalar spectral index dn_s/dln k, a tensor contribution with tensor-to-scalar ratio r, a varying number of relativistic species N_eff, varying primordial Helium fraction Y_P, and cosmic strings with tension Gμ, using the Nambu string template described in Battye & Moss (2010). These parameters are added individually to the ΛCDM model in order to look for possible deviations from the concordance cosmology. Apart from Gμ these parameters all take uniform priors, with positivity priors on r, N_eff, and Y_P. The tensor spectral index is fixed at n_t = −r/8, and both the index and ratio are defined as in, e.g., Komatsu et al. (2009). The CMB power spectrum from cosmic strings is expected to scale as (Gμ)², so we follow Sievers et al. (2009) and Battye & Moss (2010) by parameterizing the string power using q_str∝(Gμ)². Limits on Gμ are then derived from q_str.

We generate the lensed theoretical CMB spectra using CAMB,³⁴ and for computational efficiency set the CMB to zero above ℓ = 4000 where the contribution is subdominant, less than 5% of the total power. To use the 148-only ACT likelihood there are three secondary parameters, A_SZ, A_p, and A_c. For this part of the analysis we use the TBO-1 and Src-1 SZ and clustered source templates, checking the effect on the primary parameters of substituting alternative templates. We also impose positivity priors on these parameters. We do not use any information explicitly from the 218 GHz spectrum in this part of the analysis, using just the 148-only likelihood, although results are checked using the 148+218 likelihood. The ACT likelihood is combined with the seven-year WMAP data and other cosmological data sets. We use the MCMC code and methodology described in Appendix C of Dunkley et al. (2009), with the convergence test described in Dunkley et al. (2005). A subset of results are cross-checked against the publicly available CosmoMC code.

To place constraints on cosmological parameters we use the seven-year WMAP data in combination with ACT, using the WMAP likelihood package v4.1 described in Larson et al. (2011). WMAP measures the CMB over the full sky to 02 scales. All WMAP-only results shown for comparison use MCMC chains from LAMBDA,³⁵ described in Larson et al. (2011). While small-scale CMB data from ACBAR and QUAD provide higher signal-to-noise on ℓ < 1600 scales (Reichardt et al. 2009; Brown et al. 2009), we do not include them in this analysis, to avoid combining data from multiple experiments and to better interpret results derived from ACT. We follow the methodology described in Komatsu et al. (2011) to consider the addition of distance measurements from astrophysical observations, on the angular diameter distances measured from Baryon Acoustic Oscillations (BAO) at z = 0.2 and 0.35, and on the Hubble constant. The Gaussian priors on the distance ratios, r_s/D_V(z = 0.2) = 0.1905 ± 0.0061 and r_s/D_V(z = 0.35) = 0.1097 ± 0.0036, are derived from measurements from the Two-Degree Field Galaxy Redshift Survey (2dFGRS) and the Sloan Digital Sky Survey Data Release 7 (SDSS DR7), using a combined analysis of the two data sets by Percival et al. (2010). The parameter r_s is the comoving sound horizon size at the baryon drag epoch, and D_V(z) ≡ [(1 + z)²D²_A(z)cz/H(z)]^1/3 is the effective distance measure for angular diameter distance D_A, and Hubble parameter H(z). The inverse covariance matrix is given by Equation (5) of Percival et al. (2010). The Gaussian prior on the Hubble constant, H₀ = 74.2 ± 3.6 km s⁻¹ Mpc⁻¹, comes from the magnitude–redshift relation from Hubble Space Telescope (HST) observations of 240 low-z Type Ia supernovae at z < 0.1 by Riess et al. (2009). The error includes both statistical and systematic errors.

We constrain a possible deviation from power-law primordial fluctuations using the running of the index, dn_s/dln k, with curvature perturbations described by

$$\begin{equation} \Delta ^2_{\mathcal{R}}(k) = \Delta ^2_{\mathcal{R}}(k_0)\left(\frac{k}{k_0}\right)^{n_s(k_0)-1+\frac{1}{2}\ln (k/k_0)dn_s/d\ln k}. \end{equation} \tag{ 21 }$$

The spectral index at scale k is related to the index at pivot point k₀ by

$$\begin{equation} n_s(k)=n_s(k_0)+ \frac{dn_s}{d\ln k} \ln \left(\frac{k}{k_0}\right). \end{equation} \tag{ 22 }$$

The simplest inflationary models predict that the running of the spectral index with scale should be small (see, e.g., Kosowsky & Turner 1995; Baumann et al. 2009), and the detection of a scale dependence would provide evidence for alternative models for the early universe. Cosmological constraints on deviations from scale invariance have been considered recently by, e.g., Easther & Peiris (2006), Kinney et al. (2006), Shafieloo & Souradeep (2008), Verde & Peiris (2008), and Reichardt et al. (2009), using various parameterizations. With CMB data alone, the seven-year WMAP data show no evidence for significant running, with dn_s/dln k = −0.034 ± 0.026 and −0.041 ± 0.023 when combined with ACBAR and QUAD data (Komatsu et al. 2011). With the measurement of the power spectrum at small scales by ACT, we find

$$\begin{equation} dn_s/d\ln {k} = -0.034\pm 0.018\ (68\%\ {\rm CL}) \end{equation} \tag{ 23 }$$

and dn_s/dln k =−0.024 ± 0.015 including BAO+H₀. The estimated parameters are given in Tables 4 and 5. Parameters are sampled using a pivot point k₀ = 0.002 Mpc⁻¹ for the spectral index and are reported in the tables using this pivot to allow easy comparison with previous analyses, including Komatsu et al. (2011). This choice of pivot point results in the index being strongly anti-correlated with the running, with n_s(0.002) = 1.032 ± 0.039. In Figure 8, we show the index and its running at a decorrelated pivot point k₀ = 0.015/Mpc, chosen to minimize the correlation between the two parameters (Cortês, Liddle, & Mukherjee 2007). This illustrates the strength of the ACT data to constrain the primordial spectral index independently of its scale dependence. To allow conversion between pivot points, the relation between the index at these two scales is

$$\begin{eqnarray} n_s(k_0=0.015\ {{\rm Mpc}^{-1}})&=&n_s(k_0=0.002\ { {\rm Mpc}^{-1}}) \nonumber \\ &&+ \ln (0.015/0.002)\frac{dn_s}{d\ln k}, \end{eqnarray} \tag{ 24 }$$

with other cosmological parameters unchanged. The running prefers a negative value at 1.8σ, indicating enhanced damping at small scales, but there is no statistically significant deviation from a power-law spectral index. The running is weakly anti-correlated with the SZ and clustered source amplitudes, since a more negative running decreases the power at small scales, and can be partly compensated by a ∼25% increase in the upper limit on the SZ and clustered source power for this model compared to ΛCDM, as shown in Table 4. To fit the data at the smallest scales leads to a ∼0.5σ decrease in the Poisson source power.

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Given that dn_s/dln k is more sensitive to the small-scale spectrum, and may be affected by the modeling of the point source and SZ contributions, we choose this model to investigate the sensitivity of the constraints to choices made in the likelihood. We find less than 0.1σ variation in primordial parameters if we substitute alternative shapes for the SZ and clustered source templates, or limit the analysis to the ℓ < 5000 data. The beam is measured sufficiently well over the angular range of interest that results are not changed if the beam uncertainty is neglected. These tests are described further in Appendix B, and give us confidence that the errors are not dominated by systematic effects.

The concordance ΛCDM model assumes purely scalar fluctuations. Tensor fluctuations can also be seeded at early times, propagating as gravitational waves. They contribute to the CMB temperature and polarization anisotropy, polarizing the CMB with both an E-mode and B-mode pattern (e.g., Kamionkowski et al. 1997; Zaldarriaga & Seljak 1997). The tensor fluctuation power is quantified using the tensor-to-scalar ratio $r =\Delta _h^2(k_0)/\Delta _{\mathcal{R}}^2(k_0)$, where Δ²_h is the amplitude of primordial gravitational waves, with pivot scale k₀ = 0.002 Mpc⁻¹. Inflationary models predict tensor fluctuations, with amplitude related to the potential of the inflaton field (see, e.g., Baumann et al. 2009 for a recent review.)

Direct B-mode polarization measurements from the BICEP experiment provide limits of r < 0.7 (95% CL; Chiang et al. 2010). Temperature and E-mode fluctuations over a range of scales currently provide a stronger indirect constraint on r, with r < 0.36 (95% CL) from the WMAP data (Komatsu et al. 2011). Models with a large value for r have increased power at large scales, which can be partly compensated by increasing the spectral index of scalar fluctuations and reducing the scalar amplitude. This "n_s − r" degeneracy can be partly broken with lower-redshift observations that limit r < 0.24 (95% CL) from WMAP+BAO+H₀ (Komatsu et al. 2011). It can also be broken by measuring temperature fluctuations at ℓ > 1000. The tightest CMB-only constraints so far have come from WMAP CMB data combined with ACBAR and QUAD small-scale CMB data (Brown et al. 2009), with r < 0.33 (95% CL). With ACT combined with WMAP, the improved measurement of the primordial power at 1500 < ℓ < 2500 scales gives

$$\begin{equation} r<0.25\ (95\%\ {\rm CL}) \end{equation} \tag{ 25 }$$

for the CMB temperature anisotropy power spectrum alone, comparable to constraints from combined cosmological data sets (r < 0.19 at 95% CL for ACT+WMAP+BAO+H₀). The parameter estimates are given in Tables 4 and 5, and the dependence of the tensor amplitude on the spectral index is shown in Figure 9. For this model the secondary parameters are not strongly correlated with the tensor amplitude. For chaotic inflationary models with inflaton potential V(ϕ)∝ϕ^p and N e-folds of inflation, the predicted tensor-to-scalar ratio is r = 4p/N, with n_s = 1 − (p + 2)/2N. The CMB data exclude p ⩾ 3 at more than 95% confidence for N = 60 e-folds.

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The CMB is sensitive to the number of relativistic species at decoupling. Changing the effective number of species affects the evolution of perturbations by altering the expansion rate of the universe. Neutrinos also stream relativistically out of density fluctuations, with additional species suppressing the CMB peak heights and shifting the acoustic peak positions (Ma & Bertschinger 1995; Jungman et al. 1996; Bashinsky & Seljak 2004).

The standard model of particle physics has three light neutrino species, consistent with measurements of the width of the Z boson, giving N_ν = 2.984 ± 0.008 (Particle Data Book). Three neutrino species contribute about 11% of the energy density of the universe at z ≈ 1100, with ρ_rel = [7/8(4/11)^4/3N_eff]ρ_γ. Cosmological data sets are sensitive to ρ_rel, which can be composed of any light particles produced during the big bang that do not couple to electrons, ions, or photons; or any additional contribution to the energy density of the universe such as gravitational waves. Three light neutrino species correspond to N_eff = 3.04. Any deviation would indicate either additional relativistic species or evidence for non-standard interactions or non-thermal decoupling (Bashinsky & Seljak 2004), or possibly a contribution of gravitational waves to the spectrum (Smith et al. 2006).

Recent constraints on the number of relativistic species have been explored with CMB data from WMAP combined with low redshift probes by, e.g., Spergel et al. (2007), Ichikawa et al. (2007), Mangano et al. (2007), Hamann et al. (2007), Dunkley et al. (2009), Komatsu et al. (2011), and Reid et al. (2010). With WMAP data a detection was made of relativistic species with N_eff >2.7 (95% CL), but the upper level was unconstrained. By combining with distance measures, Komatsu et al. (2011) limit the number of species to N_eff =4.34 ± 0.88, and Reid et al. (2010) added optical cluster limits and luminous red galaxy power spectrum measures to find N_eff =3.77 ± 0.67. Mantz et al. (2010a) include X-ray cluster gas fraction and cluster luminosity measurements from ROSAT and Chandra to estimate N_eff =3.4^+0.6 _{− 0.5}, improving limits by constraining the matter power spectrum at low redshift. BBN observations limit N_eff to 3.24 ± 0.6 (Cyburt et al. 2005).

By combining the ACT power spectrum measurement with WMAP, the effective number of species is estimated from the CMB to be

$$\begin{equation} N_{\rm eff} = 5.3\pm 1.3 (68\%\, {\rm CL}). \end{equation} \tag{ 26 }$$

A universe with no neutrinos is excluded at 4σ from the CMB alone, with the marginalized distribution shown in Figure 10. We can now put an upper bound on N_eff from the CMB alone using ACT. This improved measurement comes from the third to seventh peak positions and heights. The right panel of Figure 10 shows the redshift of equality, z_eq, as a function of the number of species. The relation of z_eq to the number of species is given in Equation (53) of Komatsu et al. (2011). With large-scale measurements the observable quantity from the third peak height is just z_eq, leading to a strong degeneracy between N_eff and Ω_ch². With small-scale information the CMB data allow a measure of N_eff in addition to z_eq due to the additional effects of anisotropic stress on the perturbations. As an example, a model with N_eff =10 that fits the WMAP data is shown in Figure 6. With a large N_eff the higher peaks are damped, and slightly shifted to larger multipoles. The model is excluded by the ACT spectrum in the 1000 < ℓ < 2500 regime.

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The central value for N_eff preferred by the ACT+WMAP data is 1.7σ above the concordance value, with increased damping over the ΛCDM model; improved measurements of the spectrum will help refine this measurement. This is not interpreted as a statistically significant departure from the concordance value; the best-fit χ² is only 1.3 less than for N_eff = 3.04. The degeneracy between N_eff and Ω_ch² results in a higher mean value for σ₈, 0.906 ± 0.059, with all estimated parameters given in Table 4. An increase in neutrino species, leading to enhanced damping, is weakly correlated with an increase in the upper limits on SZ and clustered source power, and corresponding decrease in the Poisson power, at the ∼1 μK² level. By adding the BAO and H₀ data the N_eff−Ω_ch² degeneracy is further broken, with N_eff = 4.56 ± 0.75 (68% CL). This central value is higher than from joint constraints including X-ray and optical cluster measurements (Reid et al. 2010; Mantz et al. 2010a); improved CMB and low redshift measurements will allow further constraints and consistency checks.

In the standard BBN model, light nuclides are synthesized in the first few minutes after the big bang. Measurements of the abundance of helium are therefore sensitive to the expansion rate of the universe during this time (Peebles 1966; Steigman et al. 1977). In standard BBN, the expected ⁴He mass fraction, Y_P, is related to the baryon density, Ω_bh², and the number of neutrino (or relativistic) species, N_eff, by

$$\begin{equation} Y_P = 0.2485 + 0.0016[(273.9\, \Omega _bh^2 -6) + 100(S-1)], \end{equation} \tag{ 27 }$$

where S² = 1 + (7/43)(N_eff − 3) (see, e.g., Kneller & Steigman 2004; Steigman 2007; Simha & Steigman 2008). For the ΛCDM model, with baryon density 100 Ω_b h² = 2.214 ± 0.050 and N = 3.04 effective species, the predicted helium fraction is Y_P = 0.2486 ± 0.0006, with error dominated by the 0.02% uncertainty on the linear fit in Equation (27) (Steigman 2010). When the neutrino species are allowed to vary (as in Section 4.3.1), the current prediction from ACT+WMAP+BAO+H₀ is Y_P = 0.267 ± 0.009. For comparison, the prediction from the baryon density derived from deuterium abundance measurements is Y_P = 0.2482 ± 0.0007 (see Steigman 2010, for a review). A measurement of any deviation from this prediction could point the way to non-standard models, in particular those that affect the timing of BBN (Steigman et al. 1977; Boesgaard & Steigman 1985; Jedamzik & Pospelov 2009). This includes modifications to the Hubble expansion rate during BBN, energy injection due to annihilation or decay of heavy particles, particle catalysis of BBN reactions, and time variations in fundamental constants (see, e.g., Peimbert 2008 and Jedamzik & Pospelov 2009 for discussions).

The ⁴He abundance estimated from observations of metal poor extragalactic regions (see Steigman 2007; Peimbert 2008, for example) is Y_P = 0.252 ± 0.004 and 0.252 ± 0.001 (Izotov et al. 2007), although a higher measurement of Y_P = 0.2565 ± 0.0010(stat) ± 0.0050(syst) has recently been made (Izotov & Thuan 2010). There are systematic uncertainties in the astrophysically derived abundances, as helium is depleted in stars.

The CMB provides an alternative probe of the helium abundance when the universe was ≃400,000 years old. Helium recombines earlier than hydrogen, at z ≈ 1800 rather than z ≈ 1100, reducing the number density of electrons at recombination to n_e = n_b(1 − Y_P), where n_b is the baryon number density (Hu et al. 1995). It affects the CMB at small scales by modifying the recombination process. A larger Y_P decreases the electron number density, increasing the mean free path of Compton scattering. This leads to decreased power on small scales, due to enhanced Silk damping, as shown in Trotta & Hansen (2004) and Komatsu et al. (2011).

For CMB analysis the primordial helium abundance is usually assumed to be Y_P = 0.24. Constraints on a varying abundance from the CMB have been presented in Trotta & Hansen (2004), Huey et al. (2004), Ichikawa & Takahashi (2006), Ichikawa et al. (2008), Dunkley et al. (2009), and Komatsu et al. (2011), with a >3σ detection of Y_P = 0.33 ± 0.08 reported in Komatsu et al. (2011) for the seven-year WMAP data combined with small-scale CMB observations from ACBAR (Reichardt et al. 2009) and QUAD (Pryke et al. 2009). We now find

$$\begin{equation} Y_P=0.313\pm 0.044\ (68\%\ {\rm CL}) \end{equation} \tag{ 28 }$$

with the ACT+WMAP data combination, a significant detection of primordial helium from the CMB alone. The mean value is higher than predicted from ΛCDM, but consistent at the 1.5σ level. A universe with no primordial helium is ruled out at 6σ. The distributions for Y_P and its correlation with the spectral index are shown in Figure 11, with statistics in Table 4. Figure 6 shows how a higher helium fraction consistent with WMAP data (Y_P = 0.5) is ruled out by ACT's determination of Silk damping at small scales. There is still some uncertainty in the exact details of recombination (e.g., Wong & Scott 2007; Chluba et al. 2007; Switzer & Hirata 2008; Fendt et al. 2009; Chluba & Sunyaev 2010). A recent refinement of the numerical code for recombination used for this analysis (Recfast 1.5, by Seager et al. 1999, updated to match Rubiño-Martín et al. 2010), gives a 2% change in the spectrum at ℓ = 2000. This is subdominant to the 7% shift from a 1σ change in Y_P, so these effects are not expected to significantly affect current constraints, although will become more important as the data improve. If we consider the possible variation of both the primordial helium fraction and the number of relativistic species, the constraints on each parameter are weakened as there is some degeneracy between the two effects. The joint marginalized distribution for these parameters is shown in Figure 12, together with the predicted relation between N_eff and Y_P assuming standard BBN. The concordance N_eff = 3.04, Y_P = 0.25 model lies on the edge of the two-dimensional 68% CL, and a model with N_eff = 0, Y_P = 0, is excluded at high significance.

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Observations of the acoustic peaks in the CMB power spectrum have ruled out defects from phase transitions as the dominant mechanism for seeding cosmic structure (see, e.g., Vilenkin & Shellard 2000). However, certain inflation models predict string perturbations of similar amplitudes to the inflationary perturbations (Linde 1994; Dvali & Tye 1999). Using the CMB one can constrain the string tension, Gμ, and therefore the energy scale at which the strings are formed. Unfortunately there is significant uncertainty in the predicted power spectrum from cosmic string-generated anisotropies, due to difficulties in modeling the string network. Most approaches model the network as an ensemble of string segments with constant average properties, with string loops produced that decay into radiation. The equations of motion are solved using either the Nambu or Abelian-Higgs (AH) method, described in, e.g., Bennett & Bouchet (1990), Pogosian & Vachaspati (1999), Bevis et al. (2007), and Battye & Moss (2010).

The small-scale CMB provides a unique probe of cosmic strings, with simulations and forecasts by Fraisse et al. (2008) and Bevis et al. (2010) predicting a power-law behavior that could dominate over the Silk damping tail of the inflationary inhomogeneities, consistent with analytic predictions by Hindmarsh (1994). Constraints have been placed on the cosmic string tension from recent CMB and other cosmological data (Lo & Wright 2005; Wyman et al. 2005, 2006; Battye et al. 2006; Fraisse 2007; Bevis et al. 2007; Urrestilla et al. 2008; Sievers et al. 2009). Most recently, Battye & Moss (2010) report limits of Gμ < 2.6 × 10⁻⁷ (95% CL) for Nambu strings using five-year WMAP data combined with large-scale structure and BBN data. They find a significant dependence of this limit on the chosen string model, with up to a factor of three variation. For a simple comparison, we consider just the Nambu cosmic string template used in Battye & Moss (2010), extended to scales ℓ > 3000 with a power law, $\mathcal{B}_\ell \propto \ell ^{-1}$ (Fraisse et al. 2008). The template is held fixed for all cosmological models. Assuming this model, we find limits from ACT combined with WMAP of q_str < 0.025 (95% CL), which corresponds to a tension of

$$\begin{equation} G\mu < 1.6\times 10^{-7} (95\%\, {\rm CL}). \end{equation} \tag{ 29 }$$

The spectrum corresponding to this 95% upper limit is shown in Figure 13, compared to the upper limit pre-ACT given in Battye & Moss (2010) which overpredicts the power measured by ACT in the range 1500 < ℓ < 4000. The joint constraint on the string tension and the scalar spectral index may also limit the class of hybrid inflation models that produce cosmic strings at the end of inflation, and which typically predict a unity scalar spectral index. Battye et al. (2006) and Bevis et al. (2008) demonstrated that these models provided a good fit to the data, but more recently Battye et al. (2010) found that minimal D-term models are now ruled out at a 4σ level with CMB combined with SDSS and BBN data, and that minimal F-term models are increasingly disfavored. The preference from CMB data alone for a red spectrum with n_s = 0.963 ± 0.013 (ACT+WMAP), marginalized over a string contribution, provides further evidence against these hybrid models.

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