NINE-YEAR WILKINSON MICROWAVE ANISOTROPY PROBE (WMAP) OBSERVATIONS: COSMOLOGICAL PARAMETER RESULTS

For the most part, the structure of the likelihood code remains as it was in the seven-year WMAP data release. However, instead of using the Monte Carlo Apodised Spherical Transform EstimatoR (MASTER) estimate (Hivon et al. 2002) for the l > 32 TT spectrum, we now use an optimally estimated power spectrum and errors based on the quadratic estimator from Tegmark et al. (1997), as discussed in detail in Bennett et al. (2013). This l > 32 TT spectrum is based on the template-cleaned V- and W-band data, and the KQ85y9 sky mask (see Bennett et al. (2013) for an update on the analysis masks). The likelihood function for l > 32 continues to use the Gaussian plus log-normal approximation described in Bond et al. (1998) and Verde et al. (2003).

The l ⩽ 32 TT spectrum uses the Blackwell–Rao estimator, as before. This is based on Gibbs samples obtained from a nine-year one-region bias-corrected Internal Linear Combination map described in (Bennett et al. 2013) and sampled outside the KQ85y9 sky mask. The map and mask were degraded to HEALPix r5,¹⁸ and 2 μK of random noise was added to each pixel in the map.

The form of the polarization likelihood is unchanged. The l > 23 TE spectrum is based on a MASTER estimate and uses the template-cleaned Q-, V-, and W-band maps, evaluated outside the KQ85y9 temperature and polarization masks. The l ⩽ 23 TE, EE, and BB likelihood retains the pixel-space form described in Appendix D of Page et al. (2007). The inputs are template-cleaned Ka-, Q-, and V-band maps and the HEALPix r3 polarization mask used previously.

As before, the likelihood code accounts for several important effects: mode coupling due to sky masking and non-uniform pixel weighting (due to non-uniform noise), beam window function uncertainty, which is correlated across the entire spectrum, and residual point source subtraction uncertainty, which is also highly correlated. The treatment of these effects is described in Verde et al. (2003), Nolta et al. (2009), and Dunkley et al. (2009).

2.2.1. Small-scale CMB Measurements

Since the time when the seven-year WMAP analyses were published, there have been new measurements of small-scale CMB fluctuations by the Atacama Cosmology Telescope (ACT; Fowler et al. 2010; Das et al. 2011b) and the South Pole Telescope (SPT; Keisler et al. 2011; Reichardt et al. 2012). They have reported the angular power spectrum at 148 and 217 GHz for ACT, and at 95, 150, and 220 GHz for SPT, to 1' resolution, over ∼1000 deg² of sky. At least seven acoustic peaks are observed in the angular power spectrum, and the results are in remarkable agreement with the model predicted by the WMAP seven-year data (Keisler et al. 2011).

Figure 1 shows data from ACT and SPT at 150 GHz, which constitutes the extended CMB data set used extensively in this paper (subsequently denoted "eCMB"). We incorporate the SPT data from Keisler et al. (2011), using 47 band-powers in the range 600 < l < 3000. The likelihood is assumed to be Gaussian, and we use the published band-power window functions and covariance matrix, the latter of which accounts for noise, beam, and calibration uncertainty. Following the treatment of the ACT and SPT teams, we account for residual extragalactic foregrounds by marginalizing over three parameters: the Poisson and clustered point source amplitudes, and the SZ amplitude (Keisler et al. 2011). For ACT we use the 148 GHz power spectrum from Das et al. (2011b) in the multipole range 500 < l < 10000, marginalizing over the same clustered point source and SZ amplitudes as in the SPT likelihood, but over a separate Poisson source amplitude. See Sections 2.3 and 3.2 for more details.

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In addition to the temperature spectra, both ACT and SPT have estimated the deflection spectra due to gravitational lensing (Das et al. 2011a; van Engelen et al. 2012). These measurements are consistent with predictions of the ΛCDM model fit to WMAP. When we incorporate SPT and ACT data in the nine-year analysis, we also include the lensing likelihoods provided by each group¹⁹ to further constrain parameter fits.

New observations of the CMB polarization power spectra have also been released by the QUIET experiment (QUIET Collaboration 2011, 2012); their TE and EE polarization spectra are in excellent agreement with predictions based primarily on WMAP temperature fluctuation measurements. These data are the most recent in a series of polarization measurements at l ≳ 50. However, high-l polarization observations do not (yet) substantially enhance the power of the full data to constrain parameters, so we do not include them in the nine-year analysis.

2.2.2. Baryon Acoustic Oscillations

The acoustic peak in the galaxy correlation function has now been detected over a range of redshifts from z = 0.1 to z = 0.7. This linear feature in the galaxy data provides a standard ruler with which to measure the distance ratio, D_V/r_s, the distance to objects at redshift z in units of the sound horizon at recombination, independent of the local Hubble constant. In particular, the observed angular and radial baryon acoustic oscillation (BAO) scales at redshift z provide a geometric estimate of the effective distance,

$$\begin{equation} D_V(z) \equiv \left[(1+z)^2 \, D_A^2(z) \, cz \, /H(z) \right]^{1/3}, \end{equation} \tag{ 1 }$$

where D_A(z) is the angular diameter distance and H(z) is the Hubble parameter. The measured ratio D_V/r_s, where r_s is the co-moving sound horizon scale at the end of the drag era, can be compared to theoretical predictions.

Since the release of the seven-year WMAP data, the acoustic scale has been more precisely measured by the Sloan Digital Sky Survey (SDSS) and SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS) galaxy surveys, and by the WiggleZ and 6dFGS surveys. Previously, over half a million galaxies and luminous red galaxies from the SDSS-DR7 catalog had been combined with galaxies from 2dFGRS by Percival et al. (2010) to measure the acoustic scale at z = 0.2 and z = 0.35. (These data were used in the WMAP seven-year analysis; see also Kazin et al. 2010). Using the reconstruction method of Eisenstein et al. (2007), an improved estimate of the acoustic scale in the SDSS-DR7 data was made by Padmanabhan et al. (2012), giving D_V(0.35)/r_s = 8.88 ± 0.17, and reducing the uncertainty from 3.5% to 1.9%. More recently the SDSS-DR9 data from the BOSS survey has been used to estimate the BAO scale of the CMASS sample. They report D_V(0.57)/r_s = 13.67 ± 0.22 for galaxies in the range 0.43 < z < 0.7 (at an effective redshift z = 0.57; Anderson et al. 2012). This result is used to constrain cosmological models in Sánchez et al. (2012).

The acoustic scale has also been measured at higher redshift using the WiggleZ galaxy survey. Blake et al. (2012) report distances in three correlated redshift bins between 0.44 and 0.73. At lower redshift, z = 0.1, a detection of the BAO scale has been made using the 6dFGS survey (Beutler et al. 2011). These measurements are summarized in Table 1, and plotted as a function of redshift in Figure 19 of Anderson et al. (2012), together with the best-fit ΛCDM model prediction from the WMAP seven-year analysis (Komatsu et al. 2011). The BAO data are consistent with the CMB-based prediction over the measured redshift range.

Table 1. BAO Data Used in the Nine-year Analysis

Redshift	Data Set	rs/DV(z)	Ref.
0.1	6dFGS	0.336 ± 0.015	Beutler et al. (2011)
0.35	SDSS-DR7-rec	0.113 ± 0.002a	Padmanabhan et al. (2012)
0.57	SDSS-DR9-rec	0.073 ± 0.001a	Anderson et al. (2012)
0.44	WiggleZ	0.0916 ± 0.0071	Blake et al. (2012)
0.60	WiggleZ	0.0726 ± 0.0034	Blake et al. (2012)
0.73	WiggleZ	0.0592 ± 0.0032	Blake et al. (2012)

Note. ^aFor uniformity, the SDSS values given here have been inverted from the published values: D_V(0.35)/r_s = 8.88 ± 0.17, and D_V(0.57)/r_s = 13.67 ± 0.22.

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For the nine-year analysis, we incorporate these data into a likelihood of the form

$$\begin{equation} -2\ln L = ({\bf x-d})^T{\bf C}^{-1}({\bf x-d}), \end{equation} \tag{ 2 }$$

where

$$\begin{eqnarray} {\bf x-d} &=& [r_s/D_V(0.1) - 0.336, \, D_V(0.35)/r_s - 8.88, \nonumber\\ && D_V(0.57)/r_s - 13.67, \nonumber \\ && r_s/D_V(0.44) - 0.0916, \, r_s/D_V(0.60) - 0.0726, \nonumber\\ && r_s/D_V(0.73) - 0.0592] \end{eqnarray} \tag{ 3 }$$

and

$$\begin{equation} { {\bf C}^{-1} = \left(\begin{array}{@{}rrrrrr@{}}4444.4 &\! 0 &\! 0 &\! 0 &\! 0 &\! 0 \\ 0 &\! 34.602 &\! 0 &\! 0 &\! 0 &\! 0 \\ 0 &\! 0 &\! 20.661157 &\! 0 &\! 0 &\! 0 \\ 0 &\! 0 &\! 0 &\! 24532.1 &\! -25137.7 &\! 12099.1 \\ 0 &\! 0 &\! 0 &\! -25137.7 &\! 134598.4 &\! -64783.9 \\ 0 &\! 0 &\! 0 &\! 12099.1 &\! -64783.9 &\! 128837.6 \\ \end{array} \right). } \end{equation} \tag{ 4 }$$

The model distances are derived from the ΛCDM parameters using the same scheme we used in the WMAP seven-year analysis (Komatsu et al. 2011).

2.2.3. Hubble Parameter

2.2.4. Type Ia Supernovae

The first direct evidence for acceleration in the expansion of the universe came from measurements of luminosity distance as a function of redshift using Type Ia supernovae as standard candles (Riess et al. 1998; Schmidt et al. 1998; Perlmutter et al. 1999). Numerous follow-up observations have been made, extending these early measurements to higher redshift. After the seven-year WMAP analysis was published, the Supernova Legacy Survey analyzed their three-year sample ("SNLS3") of high redshift supernovae (Guy et al. 2010; Conley et al. 2011; Sullivan et al. 2011). They measured 242 Type Ia supernovae in the redshift range 0.08 < z < 1.06, three times more than their first-year sample (Astier et al. 2006). The SNLS team combined the 242 SNLS3 supernovae with 123 SNe at low redshift (Hamuy et al. 1996; Riess et al. 1999; Jha et al. 2006; Hicken et al. 2009; Contreras et al. 2010), 93 SNe from the SDSS supernovae search (Holtzman et al. 2008), and 14 SNe at z > 1 from HST measurements by Riess et al. (2007) to form a sample of 472 SNe. All of these supernovae were re-analyzed using both the SALT2 and SiFTO light curve fitters, which give an estimate of the SN peak rest-frame B-band apparent magnitude at the epoch of maximum light in that filter.

Sullivan et al. (2011) carry out a cosmological analysis of this combined data, accounting for systematic uncertainties including common photometric zero-point errors and selection effects. They adopt a likelihood of the form

$$\begin{equation} -2\ln L = \left({\bf m}_B - {\bf m}_{B}^{\rm th} \right)^T {\bf C}^{-1} \left({\bf m}_B - {\bf m}_{B}^{\rm th} \right), \end{equation} \tag{ 5 }$$

where m_B is the peak-light apparent magnitude in B-band for each supernova, ${\bf m}_{B}^{\rm th}$ is the corresponding magnitude predicted by the model, and C is the covariance matrix of the data. Their analysis assigns three terms to the covariance matrix, C = D_stat + C_stat + C_sys, where D_stat contains the independent (diagonal) statistical errors for each supernova, C_stat includes the statistical errors that are correlated by the light-curve fitting, and C_sys has eight terms to track systematic uncertainties, including calibration errors, Milky Way extinction, and redshift evolution. The theoretical magnitude for each supernova is modeled as

$$\begin{equation} m_B^{\rm th} = 5\log [d_L(z)/\rm Mpc] - \alpha (s-1) + \beta C + \cal M, \end{equation} \tag{ 6 }$$

where ${\cal M}$ is the empirical intercept of the $m_B^{\rm th}{--}z$ relation. The parameters α and β quantify the stretch–luminosity and color–luminosity relationships, and the statistical error, C_stat, is coupled to both parameters. Assuming a constant w model, Sullivan et al. (2011) measure α = 1.37 ± 0.09, and β = 3.2 ± 0.1. Including the term C_sys has a significant effect: it increases the error on the dark energy equation of state, σ_w from 0.05 to 0.08 in a flat universe.

The 472 Type Ia supernovae used in the SNLS3 analysis are consistent with the ΛCDM model predicted by WMAP (Sullivan et al. 2011), thus we can justify including these data in the present analysis. However, the extensive study presented by the SNLS team shows that a significant level of systematic error still exists in current supernova observations. Hence we restrict our use of supernova data in this paper to the subset of models that examine the dark energy equation of state. When SNe data are included, we marginalize over the three parameters α, β, and M. α and β are sampled in the Markov Chain Monte Carlo (MCMC) chains, while M is marginalized analytically (Lewis & Bridle 2002).

As with previous WMAP analyses, we use MCMC methods to evaluate the likelihood of cosmological parameters. Aside from incorporating new likelihood codes for the external data sets described above, the main methodological update for the nine-year analysis centers on how we marginalize over SZ and point source amplitudes when analyzing multiple CMB data sets (i.e., "WMAP+eCMB"). We have also incorporated updates to the Code for Anisotropies in the Microwave Background (CAMB; Lewis et al. 2000), as described in Section 2.3.1.

SZ amplitude. When combining data from multiple CMB experiments (WMAP, ACT, SPT) we sample and marginalize over a single SZ amplitude, A_SZ, that parameterizes the SZ contribution to all three data sets. To do so, we adopt a common SZ power spectrum template, and scale it to each experiment as follows. Battaglia et al. (2012b) compute a nominal SZ power spectrum at 150 GHz for the SPT experiment (their Figure 5, left panel, blue curve). We adopt this curve as a spectral template and scale it by a factor of 1.05 and 3.6 to describe the relative SZ contribution at 148 GHz (for ACT) and 61 GHz (for WMAP), respectively. The nuisance parameter A_SZ then multiplies all three SZ spectra simultaneously.

The above frequency scaling assumes a thermal SZ spectrum. For WMAP we assume an effective frequency of 61 GHz, even though the WMAP power spectrum includes 94 GHz data. We ignore this error because WMAP data provide negligible constraints on the SZ amplitude when analyzed on their own. In the SPT and ACT frequency range, the thermal SZ spectrum is very similar to the kinetic SZ spectrum, so our procedure effectively accounts for that contribution as well.

Clustered point sources. We adopt a common parameterization for the clustered point source contribution to both the ACT and SPT data, namely l(l + 1)C_l/2π = A_cps l^0.8 (Addison et al. 2012). Both the ACT and SPT teams use this form in their separate analyses at high l. (At low l, the SPT group adopts a constant spectrum, but this makes a negligible difference to our analysis.) By using a common amplitude for both experiments, we introduce one additional nuisance parameter.

Poisson point sources. For unclustered residual point sources we adopt the standard power spectrum C_l = const. Since the ACT and SPT groups use different algorithms for identifying and removing bright point sources, we allow the templates describing the residual power to have different amplitudes for the two experiments. This adds two additional nuisance parameters to our chains.

2.3.1. CAMB

Table 3 gives the best-fit ΛCDM parameters (mean and standard deviation, marginalized over all other parameters) for selected nine-year and seven-year data combinations. In the case where only WMAP data are used, we evaluate parameters using both the C⁻¹-weighted spectrum and the MASTER-based one. For the case where we include BAO and H₀ priors, we use only the C⁻¹-weighted spectrum for the nine-year WMAP data, and we update the priors, as per Section 2.2. The seven-year results are taken from Table 1 of Komatsu et al. (2011).

Table 3. WMAP Seven-year to Nine-year Comparison of the Six-parameter ΛCDM Model^a

Parameter	WMAP-onlyb			WMAP+BAO+H0b
	Nine-year	Nine-year (MASTER)c	Seven-year	Nine-year	Seven-year
Fit parameters
Ωbh2	0.02264 ± 0.00050	0.02243 ± 0.00055	$0.02249^{+0.00056}_{-0.00057}$	0.02266 ± 0.00043	0.02255 ± 0.00054
Ωch2	0.1138 ± 0.0045	0.1147 ± 0.0051	0.1120 ± 0.0056	0.1157 ± 0.0023	0.1126 ± 0.0036
$\Omega _\Lambda$	0.721 ± 0.025	0.716 ± 0.028	$0.727^{+0.030}_{-0.029}$	0.712 ± 0.010	0.725 ± 0.016
$10^9 \Delta _{\cal R}^2$	2.41 ± 0.10	2.47 ± 0.11	2.43 ± 0.11	$2.427^{+ 0.078}_{- 0.079}$	2.430 ± 0.091
ns	0.972 ± 0.013	0.962 ± 0.014	0.967 ± 0.014	0.971 ± 0.010	0.968 ± 0.012
τ	0.089 ± 0.014	0.087 ± 0.014	0.088 ± 0.015	0.088 ± 0.013	0.088 ± 0.014
Derived parameters
t0 (Gyr)	13.74 ± 0.11	13.75 ± 0.12	13.77 ± 0.13	13.750 ± 0.085	13.76 ± 0.11
H0 (km s−1 Mpc−1)	70.0 ± 2.2	69.7 ± 2.4	70.4 ± 2.5	69.33 ± 0.88	70.2 ± 1.4
σ8	0.821 ± 0.023	0.818 ± 0.026	$0.811^{+0.030}_{-0.031}$	0.830 ± 0.018	0.816 ± 0.024
Ωb	0.0463 ± 0.0024	0.0462 ± 0.0026	0.0455 ± 0.0028	0.0472 ± 0.0010	0.0458 ± 0.0016
Ωc	0.233 ± 0.023	0.237 ± 0.026	0.228 ± 0.027	$0.2408^{+ 0.0093}_{- 0.0092}$	0.229 ± 0.015
zreion	10.6 ± 1.1	10.5 ± 1.1	10.6 ± 1.2	10.5 ± 1.1	10.6 ± 1.2

Notes. ^aComparison of six-parameter ΛCDM fits with seven-year and nine-year WMAP data, with and without BAO and H₀ priors. ^bThe first three data columns give results from fitting to WMAP data only. The last two columns give results when BAO and H₀ priors are added. As discussed in Section 2.2, these priors have been updated for the nine-year analysis. The seven-year results are taken directly from Table 1 of Komatsu et al. (2011). ^cUnless otherwise noted, the nine-year WMAP likelihood uses the C⁻¹-weighted power spectrum whereas the seven-year likelihood used the MASTER-based spectrum. The column labeled "Nine-year (MASTER)" is a special run for comparing to the seven-year results.

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3.1.1. WMAP Data Alone

We first compare seven-year and nine-year results based on the MASTER spectra. Table 3 shows that the nine-year ΛCDM parameters are all within 0.5σ of each other, with Ω_ch² having the largest difference. We note that the combination $\Omega _mh^2+\Omega _\Lambda$ is approximately constant between the two models, reflecting the fact that this combination is well constrained by primary CMB fluctuations, whereas $\Omega _m-\Omega _\Lambda$ is less so due to the geometric degeneracy. Turning to the C⁻¹-weighted spectrum, we note that the nine-year ΛCDM parameters based on this spectrum are all within ∼0.3σ of the seven-year values. Thus we conclude that the nine-year model fits are consistent with the seven-year fit.

Next, we examine the consistency of the two ΛCDM model fits, derived from the two nine-year spectrum estimates. As seen in Table 3, the six parameters agree reasonably well, but we note that the estimates for n_s differ by 0.75σ, which we discuss below. To help visualize the fits, we plot both spectra (C⁻¹-weighted and MASTER), and both models in Figure 2. As noted in Bennett et al. (2013), the difference between the two spectrum estimates is most noticeable in the range l ∼ 30–60 where the C⁻¹-weighted spectrum is lower than the MASTER spectrum, by up to 4% in one bin. However, the ΛCDM model fits only differ noticeably for l ≲ 10 where the fit is relatively weakly constrained due to cosmic variance.

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To understand why these two model spectra are so similar, we examine parameter degeneracies between the six ΛCDM parameters when fit to the nine-year WMAP data. In Figure 3 we show the two largest degeneracies that affect the spectral index n_s, namely $10^9 \Delta _{\cal R}^2$ and Ω_bh². The contours show the 68% and 95% CL regions for the fits to the C⁻¹-weighted spectrum while the plus signs show the maximum likelihood points from the MASTER fit. Note that the C⁻¹-weighted fits favor lower $10^9 \Delta _{\cal R}^2$ and higher Ω_bh², both of which push the C⁻¹-weighted fit toward higher n_s. Given the consistency of the fit model spectra, we conclude that the underlying data are quite robust and in subsequent subsections, we look to external data to help break any degeneracies that remain in the nine-year data.

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We conclude this subsection with a summary of some additional tests we carried out on simulations to assess the robustness of the C⁻¹-weighted spectrum estimate in general, and the n_s fits in particular. The simulation data used were the 500 "parameter recovery" simulations developed for our seven-year analysis, described in detail in Larson et al. (2011). These data include yearly sky maps for each differencing assembly, where the maps include simulated ΛCDM signal (convolved with the appropriate beam) using the parameters given in Appendix A of Larson et al. (2011), and a model of correlated instrument noise appropriate to each differencing assembly. For each realization in the simulation, we computed both the C⁻¹-weighted spectrum and the MASTER spectrum using the same prescription as was used for the flight data. We found that both spectrum estimators were unbiased to within the standard spectrum errors divided by $\sqrt{500}$, i.e., to within the sensitivity of the test.

We next evaluated a number of difference statistics, but the one that was deemed most pertinent to understanding the n_s fit was the average power difference between l = 32–64 (this is admittedly a posterior choice of l range). When the parameter recovery simulations were analyzed with the conservative KQ75y9 mask (Bennett et al. 2013), more than one-third of the simulated spectrum pairs had a larger power difference (C⁻¹ −MASTER, in the l = 32–64 bin) than did the flight data. This result indicates nothing unusual for that choice of mask. However, when the same analysis was performed with the smaller KQ85y9 mask, only 2 out of 500 simulation realizations had a larger difference than did the flight data, which was 4% in this bin. Since the flight data appear to be unusual at the 0.4% level (2/500) with the smaller mask, we suspected that (significant) residual foreground contamination might be present in the flight data, and that the two spectrum estimators might be responding to this differently.

To test this, we amended the CMB-only parameter recovery simulations with model foreground signals that we deemed to be representative of both the raw foreground signal outside the KQ85y9 mask, and an estimate of the residual contamination after template cleaning. The simulated foreground signals were based on the modeling studies described in Bennett et al. (2013); in particular, the full-strength signal was based on the "Model 9" foreground model in Bennett et al. (2013), while the residual signal after cleaning was estimated from the rms among the multiple foreground models studied. With these foreground-contaminated simulations, we repeated the comparison of the two spectrum estimates considered above. Both estimates showed slightly elevated power in the l = 32–64 bin (a few percent), with the MASTER estimate being slightly more elevated. However, the distribution of spectrum differences was not significantly different than with the CMB-only simulations: only 1% of the simulated, foreground-contaminated difference spectra exceeded the difference seen in the flight data. In the end, we attribute the spectrum differences to statistical fluctuations and we adopt the C⁻¹-weighted spectrum for our final analysis because it has lower uncertainties (Bennett et al. 2013) and because it was more stable to the introduction of foreground contamination in our simulations. Nonetheless, we report ΛCDM parameter fits for both spectrum estimates in Table 3 and Figure 3 to give a sense of the potential systematic uncertainty in these parameters.

To conclude the seven-year/nine-year comparison, we note that the remaining 5 ΛCDM parameters changed by less than 0.3σ indicating very good consistency. The overall effect of the nine-year WMAP data is to improve the average parameter uncertainty by about 10%, with Ω_ch² and $\Omega _\Lambda$ each improving by nearly 20%. The latter improvement is a result of higher precision in the third acoustic peak measurement (Bennett et al. 2013) which gives a better determination of Ω_ch². This, in turn, improves $\Omega _\Lambda$, which is constrained by flatness (or in non-flat models, by the geometric degeneracy discussed in Section 4.5). The overall volume reduction in the allowed six-dimensional ΛCDM parameter space in the switch from seven-year to nine-year data is a factor of two, the majority of which derives from switching to the C⁻¹-weighted spectrum estimate.

3.1.2. WMAP Data with BAO and H₀

From the standpoint of astrophysics, primary CMB fluctuations, combined with CMB lensing, arguably provide the cleanest probe of cosmology because the fluctuations dominate Galactic foreground emission over most of the sky, and they can (so far) be understood in terms of linear perturbation theory and Gaussian statistics. Thus we next consider parameter constraints that can be obtained when adding additional CMB data to the nine-year WMAP data. Specifically, we examine the effects of adding SPT and ACT data (see Section 2.2.1): the best-fit parameters are given in the "+eCMB" column of Table 4.

Table 4. Six-parameter ΛCDM Fit: WMAP Plus External Data^a

Parameter	WMAP	+eCMB	+eCMB+BAO	+eCMB+H0	+eCMB+BAO+H0
Fit parameters
Ωbh2	0.02264 ± 0.00050	0.02229 ± 0.00037	0.02211 ± 0.00034	0.02244 ± 0.00035	0.02223 ± 0.00033
Ωch2	0.1138 ± 0.0045	0.1126 ± 0.0035	0.1162 ± 0.0020	0.1106 ± 0.0030	0.1153 ± 0.0019
$\Omega _\Lambda$	0.721 ± 0.025	0.728 ± 0.019	0.707 ± 0.010	0.740 ± 0.015	$0.7135^{+ 0.0095}_{- 0.0096}$
$10^9 \Delta _{\cal R}^2$	2.41 ± 0.10	2.430 ± 0.084	$2.484^{+ 0.073}_{- 0.072}$	$2.396^{+ 0.079}_{- 0.078}$	2.464 ± 0.072
ns	0.972 ± 0.013	0.9646 ± 0.0098	$0.9579^{+ 0.0081}_{- 0.0082}$	$0.9690^{+ 0.0091}_{- 0.0090}$	0.9608 ± 0.0080
τ	0.089 ± 0.014	0.084 ± 0.013	$0.079^{+ 0.011}_{- 0.012}$	0.087 ± 0.013	0.081 ± 0.012
Derived parameters
t0 (Gyr)	13.74 ± 0.11	13.742 ± 0.077	13.800 ± 0.061	13.702 ± 0.069	13.772 ± 0.059
H0 (km s−1 Mpc−1)	70.0 ± 2.2	70.5 ± 1.6	68.76 ± 0.84	71.6 ± 1.4	69.32 ± 0.80
σ8	0.821 ± 0.023	0.810 ± 0.017	$0.822^{+ 0.013}_{- 0.014}$	0.803 ± 0.016	$0.820^{+ 0.013}_{- 0.014}$
Ωb	0.0463 ± 0.0024	0.0449 ± 0.0018	0.04678 ± 0.00098	0.0438 ± 0.0015	0.04628 ± 0.00093
Ωc	0.233 ± 0.023	0.227 ± 0.017	0.2460 ± 0.0094	0.216 ± 0.014	$0.2402^{+ 0.0088}_{- 0.0087}$
zeq	$3265^{+ 106}_{- 105}$	3230 ± 81	3312 ± 48	3184 ± 70	3293 ± 47
zreion	10.6 ± 1.1	10.3 ± 1.1	10.0 ± 1.0	10.5 ± 1.1	10.1 ± 1.0

Notes. ^aΛCDM model fit to WMAP nine-year data combined with a progression of external data sets. A complete list of parameter values for this model, with additional data combinations, may be found at http://lambda.gsfc.nasa.gov/.

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With the addition of the high-l CMB data, the constraints on the energy density parameters Ω_bh², Ω_ch², and $\Omega _\Lambda$ all improve by 25% over the precision from WMAP data alone. The improvement in the baryon density measurement is due to more precise measurements of the Silk damping tail in the power spectrum at l ≳ 1000; the improvements in Ω_ch² and $\Omega _\Lambda$ are due in part to improvements in the high-l TT data, but also to the detection of CMB lensing in the SPT and ACT data (Das et al. 2011a; van Engelen et al. 2012), which helps to constrain Ω_m by fixing the growth rate of structure between z = 1100 and z = 1–2 (the peak in the lensing kernel). Taken together, CMB data available at the end of the WMAP mission produce a 1.6% measurement of Ω_bh² and a 3.0% measurement of Ω_ch².

The increased k-space lever arm provided by the high-l CMB data improves the uncertainty on the scalar spectral index by 25%, giving n_s = 0.9646  ±  0.0098, which implies a non-zero tilt in the primordial spectrum (i.e., n_s < 1) at 3.6σ. We examine the implications of this measurement for inflation models in Section 4.1.

If we assume a flat universe, which breaks the CMB's geometric degeneracy, then CMB data alone now provide a 2.3% measurement of the Hubble parameter, H₀ = 70.5 ± 1.6 km s⁻¹ Mpc⁻¹, independent of the cosmic distance ladder. As discussed in Section 3.4, this is consistent with the recent determination of the Hubble parameter using the cosmic distance ladder: H₀ = 73.8 ± 2.4 km s⁻¹ Mpc⁻¹ (Riess et al. 2011); we explore the effect of adding this prior in Section 3.4. We relax the assumption of flatness in Section 4.5.

We conclude by comparing our results for the ACT and SPT foreground "nuisance" parameters to those found by the ACT and SPT teams. For example, we find $A_{\rm Poisson}^{\rm ACT} = 14.8^{+ 2.3}_{- 2.4}$ while the ACT team finds $A_{\rm Poisson}^{\rm ACT} = 12.0 \pm 1.9$. (Note that we do not expect perfect agreement because we use nine-year WMAP data and we fit the clustered source amplitude jointly with SPT data, unlike the ACT team's treatment.) The ACT team concluded that the ΛCDM cosmological model (fit to) the 148 GHz spectrum (and the seven-year WMAP data), marginalized over SZ and source power is a good fit to the data (Dunkley et al. 2011). The complete set of foreground parameters fit to the ACT and SPT data may be found at http://lambda.gsfc.nasa.gov/ for all the models reported in this paper.

Acoustic structure in the large-scale distribution of galaxies is manifest on a co-moving scale of 152 Mpc, where the evolution of matter fluctuations is largely within the linear regime. A number of authors have studied the degree to which the acoustic structure could be perturbed by nonlinear evolution (e.g., Seo & Eisenstein 2005, 2007; Jeong & Komatsu 2006, 2009; Crocce & Scoccimarro 2008; Matsubara 2008; Taruya & Hiramatsu 2008; Padmanabhan & White 2009), and the effects are well below the current measurement uncertainties. Because it is based on the same well-understood physics that governs the CMB anisotropy, we consider measurements of the BAO scale to be the next-most robust cosmological probe after CMB fluctuations. The ΛCDM parameters fit to CMB and BAO data are given in the "+eCMB+BAO" column of Table 4.

Measurements of the tangential and radial BAO scale at redshift z measure the effective distance D_V(z), given in Equation (1), in units of the sound horizon r_s(z_d). This quantity is primarily sensitive to the total matter and dark energy densities, and to the current Hubble parameter. Since the BAO scale is relatively insensitive to the baryon density, Ω_bh², this parameter does not improve significantly with the addition of the BAO prior. However, the low-redshift distance information imposes complementary constraints on the matter density and Hubble parameter, improving the precision on Ω_ch² from 3.0% to 1.6%, and on H₀ from 2.3% to 1.2%. In the context of standard ΛCDM these improvements lead to a measurement of the age of the universe with 0.4% precision: t₀ = 13.800 ± 0.061 Gyr.

The addition of the BAO prior helps to break some residual degeneracy between the primordial spectral index, n_s, on the one hand, and Ω_ch² and H₀ on the other. Figure 4 shows the two-dimensional parameter likelihoods for (n_s,Ω_ch²) and (n_s, H₀) for the three data combinations considered to this point. With only CMB data (black and blue contours) there remains a weak degeneracy between n_s and the other two. When the BAO prior is added (red), it pushes Ω_ch² toward the upper end of the range allowed by the CMB, and vice versa for H₀. Both of these results push n_s toward the lower end of its CMB-allowable range; consequently, with the BAO prior included, the marginalized measurement of the primordial spectral index is n_s = $0.9579^{+ 0.0081}_{- 0.0082}$ which constitutes a 5σ measurement of tilt (n_s < 1) in the primordial spectrum. We discuss the implications of this measurement for inflation models in Section 4.1.

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Measurements of the Hubble parameter using the cosmic distance ladder have a long history, and are subject to a variety of different systematic errors that have been steadily reduced over time. However, an accurate, direct measurement of the current expansion rate is vital for testing the validity of the ΛCDM model because the value derived from the CMB and BAO data is model-dependent. Measurements of H₀ provide an excellent complement to CMB and BAO measurements. The H₀ prior considered here has a precision that approaches the ΛCDM-based value given above. Consequently, we next consider the addition of the H₀ prior discussed in Section 2.2.3, without the inclusion of the BAO prior. The ΛCDM parameters fit to CMB and H₀ data are given in "+eCMB+H₀" column of Table 4.

Two cosmological quantities that significantly shape the observed CMB spectrum are the epoch of matter radiation equality, z_eq, which depends on Ω_ch², and the angular diameter distance to the last scattering surface, d_A(z_*), which depends primarily on H₀. As illustrated in Figure 5 (see also Section 4.3.1), the CMB data still admit a weak degeneracy between Ω_ch² and H₀ that the BAO and H₀ priors help to break. The black contours in Figure 5 show the constraints from CMB data (WMAP+eCMB), the red from CMB and BAO data, and the blue from CMB with the H₀ prior. While these measurements are all consistent, it is interesting to note that the BAO and H₀ priors are pushing toward opposite ends of the range allowed by the CMB data for this pair of parameters. Given this minor tension, it is worth examining independent sets of constraints that do not share common CMB data. A simple test is to compare the marginalized constraints on the Hubble parameter from the CMB+BAO data (H₀ = 68.76 ± 0.84 km s⁻¹ Mpc⁻¹), to the direct, and independent, measurement from the distance ladder (H₀ = 73.8 ± 2.4 km s⁻¹ Mpc⁻¹). In our Markov Chain that samples the ΛCDM model with the WMAP+eCMB+BAO data, we found that only 0.1% of the H₀ values in the chain fell within the 1σ range of the Hubble prior, but that 45% fell within the 2σ range of 73.8 ± 4.8 km s⁻¹ Mpc⁻¹. Based on this, we conclude that these measurements do not disagree, and that they may be combined to form more stringent constraints on the ΛCDM parameters.

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We conclude this subsection by noting that measurements of the remaining ΛCDM parameters are modestly improved by the addition of the H₀ prior to the CMB data, with Ω_ch² improving the most due to the effect discussed above and illustrated in Figure 5.

Given the consistency of the data sets considered above, we conclude with a summary of the ΛCDM fits derived from the union of these data. The marginalized results are given in the "+eCMB+BAO+H₀" column of Table 4. The matter and energy densities, Ω_bh², Ω_ch², and $\Omega _\Lambda$ are all now determined to ∼1.5% precision with the current data. The amplitude of the primordial spectrum is measured to within 3%, and there is now evidence for tilt in the primordial spectrum at the 5σ level.

At the end of the WMAP mission, the nine-year data produced a factor of 68,000 decease in the allowable volume of the six-dimensional ΛCDM parameter space, relative to the pre-WMAP measurements (Bennett et al. 2013). Specifically, the allowable volume is measured by the square root of the determinant of the 6 × 6 parameter covariance matrix, as discussed in Larson et al. (2011). The pre-WMAP volume is determined from chains run with the data compiled by Wang et al. (2003). When these data are combined with the eCMB+BAO+H₀ priors, we obtain an additional factor of 27 over the WMAP-only constraints. As an illustration of the predictive power of the current data, Figure 6 shows the 1σ range of high-l power spectra allowed by the six-parameter fits to the nine-year WMAP data. As shown in Figure 1, this model has already predicted the current small-scale measurements. If future measurements of the spectrum, for example by Planck, lie outside this range, then either there is a problem with the six-parameter model, or a problem with the data.

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Remarkably, despite this dramatic increase in precision, the six-parameter ΛCDM model still produces an acceptable fit to all of the available data. Bennett et al. (2013) present a detailed breakdown of the goodness of fit to the nine-year WMAP data. In the next section we place limits on parameters beyond the six required to describe our universe.

As noted in Section 3.5, the nine-year WMAP data, when combined with eCMB, BAO and H₀ priors, exclude a scale-invariant primordial power spectrum at 5σ significance. For a power-law spectrum of primordial curvature perturbations,

$$\begin{equation} \Delta ^2_{\cal R}(k) = \Delta ^2_{\cal R}(k_0) \left(\frac{k}{k_0}\right)^{n_s-1}, \end{equation} \tag{ 7 }$$

with k₀ = 0.002 Mpc⁻¹, we find n_s = 0.9608 ± 0.0080. This result assumes that tensor modes (gravitational waves) contribute insignificantly to the CMB anisotropy.

At this time, the most sensitive limits on tensor modes are still obtained from the shape of the temperature power spectrum, in conjunction with additional data. For example, Story et al. (2012) report r < 0.18 (95% CL), where

$$\begin{equation} r \equiv \frac{\Delta ^2_h(k_0)}{\Delta ^2_{\cal R}(k_0)} = \frac{P_h(k_0)}{P_{\cal R}(k_0)}. \end{equation} \tag{ 8 }$$

Due to confusion from density fluctuations, the lowest tensor amplitude that can be reliably detected from temperature data is r ≲ 0.13 (Knox 1995). Several recent experiments are beginning to establish comparable limits from non-detection of B-mode polarization anisotropy, e.g., Chiang et al. (2010) report r < 0.7 (95% CL) from BICEP and the QUIET Collaboration (2012) reports r < 2.8 (95% CL). A host of forthcoming experiments are targeting B-mode measurements that have the potential to detect or limit tensor modes at significantly lower levels than can be achieved with temperature data alone. (Note that E-mode polarization, like temperature anisotropy, is dominated by scalar fluctuations, and since the E-mode signal is more than an order of magnitude weaker than the temperature signal, it contributes negligibly to constraints on tensor fluctuations.)

In Table 5, we report limits on r from the nine-year WMAP data, analyzed alone and jointly with external data; the tightest constraint is

$$\begin{eqnarray*} &&r < 0.13\ {\rm (95\% CL)} \quad {\rm {{\rm WMAP}}+eCMB+BAO+}H_0. \end{eqnarray*}$$

This is effectively at the limit one can reach without B-mode polarization measurements. The joint constraints on n_s and r are shown in Figure 7, along with selected model predictions derived from single-field inflation models. Taken together, the current data strongly disfavor a pure Harrison–Zel'dovich spectrum, even if tensor modes are allowed in the model fits.

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Table 5. Primordial Spectrum: Tensors and Running Scalar Index^a

Parameter	WMAP	+eCMB	+eCMB+BAO	+eCMB+BAO+H0
Tensor mode amplitudeb
r	<0.38 (95% CL)	<0.17 (95% CL)	<0.12 (95% CL)	<0.13 (95% CL)
ns	0.992 ± 0.019	0.970 ± 0.011	0.9606 ± 0.0084	0.9636 ± 0.0084
Running scalar indexb
dns/dln k	−0.019 ± 0.025	$-0.022^{+ 0.012}_{- 0.011}$	−0.024 ± 0.011	−0.023 ± 0.011
ns	1.009 ± 0.049	1.018 ± 0.029	1.020 ± 0.029	1.020 ± 0.029
Tensors and running, jointlyb
r	<0.50 (95% CL)	<0.53 (95% CL)	<0.43 (95% CL)	<0.47 (95% CL)
dns/dln k	−0.032 ± 0.028	−0.039 ± 0.016	−0.039 ± 0.015	−0.040 ± 0.016
ns	1.058 ± 0.063	1.076 ± 0.048	$1.068^{+ 0.045}_{- 0.044}$	1.075 ± 0.046

Notes. ^aA complete list of parameter values for these models, with additional data combinations, may be found at http://lambda.gsfc.nasa.gov/. ^bThe tensor mode amplitude and scalar running index parameter are each fit singly, and then jointly. In models with running, the nominal scalar index is quoted at k₀ = 0.002 Mpc⁻¹.

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4.1.1. Running Spectral Index

Some inflation models predict a scale dependence or "running" in the (nearly) power-law spectrum of scalar perturbations. This is conveniently parameterized by the logarithmic derivative of the spectral index, dn_s/dln k, which gives rise to a spectrum of the form (Kosowsky & Turner 1995)

$$\begin{equation} \Delta ^2_{\cal R}(k) = \Delta ^2_{\cal R}(k_0) \left(\frac{k}{k_0}\right)^{n_s(k_0) - 1 + \frac{1}{2} \ln (k/k_0) dn_s/d\ln {k}}. \end{equation} \tag{ 9 }$$

We do not detect a statistically significant deviation from a pure power-law spectrum with the nine-year WMAP data. The allowed range of dn_s/dln k is both closer to zero and has a smaller confidence range with the nine-year data, dn_s/dln k = −0.019  ±  0.025. However, with the inclusion of the high-l CMB data, the full CMB data prefer a slightly more negative value, with a smaller uncertainty, $dn_s/d\ln {k} = -0.022^{+ 0.012}_{- 0.011}$. While not significant, this result might indicate a trend as the l-range of the data expand. The inclusion of BAO and H₀ data does not affect these results.

If we allow both tensors and running as additional primordial degrees of freedom, the data prefer a slight negative running, but still at less than 3σ significance, and only with the inclusion of the high-l CMB data. Complete results are given in Table 5.

In addition to adiabatic fluctuations, where all species fluctuate in phase and therefore produce curvature fluctuations, it is possible to have isocurvature perturbations: an over-density in one species compensates for an under-density in another, producing no net curvature. These entropy, or isocurvature perturbations have a measurable effect on the CMB by shifting the acoustic peaks in the power spectrum. For CDM and photons, we define the entropy perturbation field

$$\begin{equation} \mathcal {S}_{c,\gamma } \equiv \frac{\delta \rho _c}{\rho _c} - \frac{3 \delta \rho _\gamma }{4 \rho _\gamma } \end{equation} \tag{ 10 }$$

(Bean et al. 2006; Komatsu et al. 2009). The relative amplitude of its power spectrum is parameterized by α,

$$\begin{equation} \frac{\alpha }{1 - \alpha } \equiv \frac{P_\mathcal {S}(k_0)}{P_\mathcal {R}(k_0)}, \end{equation} \tag{ 11 }$$

with k₀ = 0.002 Mpc⁻¹.

We consider two types of isocurvature modes: those which are completely uncorrelated with the curvature modes (with amplitude α₀), motivated by the axion model, and those which are anti-correlated with the curvature modes (with amplitude α₋₁), motivated by the curvaton model. For the latter, we adopt the convention in which anti-correlation increases the power at low multipoles (Komatsu et al. 2009).

The constraints on both types of isocurvature modes are given in Table 6. We do not detect a significant contribution from either type of perturbation in the nine-year data, whether or not additional data are included in the fit. With WMAP data alone, the limits are slightly improved over the seven-year results (Larson et al. 2011), but the addition of the new eCMB data improves limits by a further factor of ∼2. Adding the BAO data (Section 2.2.2) and H₀ data (Section 2.2.3) further improves the limits, to

$$\begin{eqnarray*} \begin{array}{llr}\alpha _{-1} & < 0.0039\ {\rm (95\% CL)} & \\ [2mm] \alpha _{0} & < 0.047\ {\rm (95\% CL)} &\end{array} \quad {\rm {{WMAP}}+eCMB+BAO+}H_0, \end{eqnarray*}$$

due to the fact that these data help to break a modest degeneracy in the CMB anisotropy between the isocurvature modes and the ΛCDM parameters given in Table 6.

Table 6. Isocurvature Modes^a

Parameter	WMAP	+eCMB	+eCMB+BAO	+eCMB+BAO+H0
Anti-correlated modesb
α−1	<0.012 (95% CL)	<0.0076 (95% CL)	<0.0035 (95% CL)	<0.0039 (95% CL)
Ωch2	0.1088 ± 0.0050	0.1097 ± 0.0037	0.1160 ± 0.0020	0.1151 ± 0.0019
ns	0.994 ± 0.017	0.977 ± 0.011	$0.9631^{+ 0.0087}_{- 0.0088}$	$0.9662^{+ 0.0085}_{- 0.0087}$
σ8	$0.807^{+ 0.025}_{- 0.024}$	0.802 ± 0.018	$0.823^{+ 0.014}_{- 0.013}$	$0.821^{+ 0.014}_{- 0.013}$
Uncorrelated modesc
α0	<0.15 (95% CL)	<0.061 (95% CL)	<0.043 (95% CL)	<0.047 (95% CL)
Ωch2	0.1093 ± 0.0056	0.1115 ± 0.0036	0.1161 ± 0.0020	0.1152 ± 0.0019
ns	0.994 ± 0.021	0.970 ± 0.011	$0.9608^{+ 0.0086}_{- 0.0085}$	$0.9639^{+ 0.0085}_{- 0.0084}$
σ8	0.805 ± 0.027	0.805 ± 0.018	0.821 ± 0.014	0.819 ± 0.014

Notes. ^aA complete list of parameter values for these models, with additional data combinations, may be found at http://lambda.gsfc.nasa.gov/. ^bThe anti-correlated isocurvature amplitude comprises one additional parameter in the ΛCDM fit. The remaining parameters in this table section are given for trending. ^cThe uncorrelated isocurvature amplitude comprises one additional parameter in the ΛCDM fit. The remaining parameters in this table section are given for trending.

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4.3.1. The Number of Relativistic Species and the CMB Power Spectrum

Let us write the energy density of relativistic particles near the epoch of photon decoupling, z ≈ 1090, as

$$\begin{equation} \rho _r \equiv \rho _\gamma + \rho _\nu + \rho _{\rm er}, \end{equation} \tag{ 12 }$$

where, in natural units, $\rho _\gamma = ({\pi ^2}/{15}) T_\gamma ^4$ is the photon energy density, $\rho _\nu =({7}/{8})({\pi ^2}/{15})N_\nu T_\nu ^4$ is the neutrino energy density, and ρ_er denotes the energy density of "extra radiation species." (The factor of 7/8 in the neutrino density arises from the Fermi–Dirac distribution.) In the standard model of particle physics, N_ν = 3.046 (Dicus et al. 1982; Mangano et al. 2002), while in the standard thermal history of the universe, T_ν = (4/11)^1/3 T_γ (e.g., Weinberg 1972).

Since we do not know the nature of an extra radiation species, we cannot specify its energy density or temperature uniquely. For example, ρ_er could be comprised of bosons or fermions. Nevertheless, it is customary to parameterize the number of extra radiation species as if they were neutrinos, and write

$$\begin{equation} \rho _\nu + \rho _{\rm er} \equiv \frac{7\pi ^2}{120}\, N_{\rm eff} \, T_\nu ^4, \end{equation} \tag{ 13 }$$

where N_eff is the effective number of neutrino species, which does not need to be an integer. With this parameterization, the total radiation energy density is

$$\begin{equation} \rho _r = \rho _\gamma \left[1 + \frac{7}{8}\left(\frac{4}{11}\right)^{4/3} N_{\rm eff}\right] \simeq \rho _\gamma (1+0.2271\,N_{\rm eff}). \end{equation} \tag{ 14 }$$

While photons interact with baryons efficiently at z ≳ 1090, neutrinos do not interact much at all for z ≪ 10¹⁰. As a result, one can treat neutrinos as free-streaming particles. Here, we also treat extra radiation species as free-streaming. With this assumption, one can use the measured $C_l^{{\rm TT}}$ spectrum to constrain N_eff (Hu et al. 1995, 1999; Bowen et al. 2002; Bashinsky & Seljak 2004). Section 6.2 of Komatsu et al. (2009) and Section 4.7 of Komatsu et al. (2011) discuss previous attempts to constrain N_eff from the CMB and provide references. More recently, Dunkley et al. (2011) and Keisler et al. (2011) constrain N_eff using the seven-year WMAP data combined with ACT and SPT data, respectively. In this paper, we assume the sound speed and anisotropic stress of any extra radiation species are the same as for neutrinos. See Archidiacono et al. (2011, 2012) and Smith et al. (2012) for constraints on other cases.

Neutrinos (and ρ_er) affect the power spectrum, $C_l^{{\rm TT}}$, in four ways. To illustrate and explain each of these effects, Figure 8 compares models with N_eff = 3.046 and N_eff = 7, adjusted in stages to match the two spectra as closely as possible.

• 1.  
  Peak locations. The extra radiation density increases the early expansion rate via the Friedmann equation, H² = (8πG/3)(ρ_m + ρ_r). As a result, increasing N_eff from 3.046 to 7 reduces the comoving sound horizon, r_s, at the decoupling epoch, from 146.8 Mpc to 130.2 Mpc. The expansion rate after matter-radiation equality is less affected, so the angular diameter distance to the decoupling epoch, d_A, is only slightly reduced (by 2.5%). Therefore, increasing N_eff reduces the angular size of the acoustic scale, θ_* ≡ r_s/d_A, which determines the peak positions. A change in θ_* can be absorbed by rescaling l by a constant factor. In the top left panel of Figure 8, we have rescaled l for the N_eff = 7 model by a factor of 0.890, the ratio of θ_* for these two models ($\theta _\ast =0{\rm {\rlap.}^\circ }5961$, $0{{\rm \rlap.}^\circ }5306$ for N_eff = 3.046, 7, respectively). This rescaling brings the peak positions of these models into agreement, except for a small additive shift in peak positions; see Bashinsky & Seljak (2004).
• 2.  
  Early Integrated Sachs–Wolfe effect. Extra radiation density delays the epoch of matter-radiation equality and thus enhances the first and second peaks via the early Integrated Sachs–Wolfe (ISW) effect (Hu & Sugiyama 1995). This effect can be compensated by increasing the CDM density in the N_eff = 7 model from Ω_ch² = 0.1107 to 0.1817, which brings the matter-radiation equality epoch back into agreement. (We do not change Ω_bh², as that changes the first-to-second peak ratio.) The top right panel of Figure 8 shows the spectra after making this adjustment. Note that changing Ω_ch² also changes θ_*, so the l axis is rescaled by 0.957 for the N_eff = 7 model in this panel.
• 3.  
  Anisotropic stress. Relativistic species that do not interact effectively with themselves or with other species cannot be described as a (perfect) fluid. As a result, the distribution function, f(x, p, t), of free-streaming particles has a non-negligible anisotropic stress,

  $$\begin{equation} \pi _{ij} \equiv \int \frac{d^3p}{(2\pi)^3} p\left(\hat{p}_i\hat{p}_j-\frac{1}{3}\delta _{ij}\right)f(\mathbf {x},\mathbf {p},t), \end{equation} \tag{ 15 }$$

  as well as higher-order moments. The energy density, pressure, and momentum are obtained from the distribution function by ρ = (2π)⁻³∫d³p p f, $P = (2\pi)^{-3} \int d^3p \,\frac{p}{3} \, f$, and uⁱ = (2π)⁻³∫d³p pⁱ f, respectively. This term alters metric perturbations during the radiation era (via Einstein's field equations) and thus temperature fluctuations on scales l ≳ 130, since those scales enter the horizon during the radiation era. On larger scales, fluctuations enter the horizon during the matter era and are less affected by this term. Temperature fluctuations on these scales are given by the Sachs–Wolfe formula, $\delta T/T = -{\cal R}/5$, while those on smaller scales (ignoring the effect of baryons) are given by $\delta T/T = -\left(1+4f_\nu /15\right)^{-1}{\cal R}\cos (kr_s)$ (Hu & Sugiyama 1996), where f_ν is the fraction of the radiation density that is free-streaming,

  $$\begin{equation} f_\nu (N_{\rm eff}) \equiv \frac{0.2271 N_{\rm eff}}{1+0.2271 N_{\rm eff}}. \end{equation} \tag{ 16 }$$

  The small-scale anisotropy is enhanced by a factor of 5(1 + 4f_ν/15)⁻¹ due to the decay of the gravitational potential at the horizon crossing during the radiation era. Since the anisotropic stress alters the gravitational potential (via the field equations), it also alters the degree to which the small-scale anisotropy is enhanced relative to the large-scale anisotropy. Therefore, the effect of anisotropic stress can be removed by multiplying $C_l^{{\rm TT}}(l \gtrsim 130)$ by (1 + 4f_ν/15)². In the bottom left panel of Figure 8, we have multiplied $C_l^{{\rm TT}}$ at all l by [1 + 4f_ν(7)/15]²/[1 + 4f_ν(3.046)/15]², where f_ν(7) = 0.6139 and f_ν(3.046) = 0.4084. The two models now agree well, but the N_eff = 7 model is greater than the standard model at l ≲ 130 because the anisotropic stress term does not affect these multipoles.
• 4.  
  Enhanced damping tail. While the increased expansion rate reduces the sound horizon, r_s, it also reduces the diffusion length, r_d, that photons travel by random walk. The mean free path of a photon is λ_C = 1/(σ_Tn_e). Over the age of the universe, t, photons diffuse a distance $r_d \approx \sqrt{3ct/\lambda _C}\,\lambda _C \propto \sqrt{\lambda _C/H}$, and fluctuations within r_d are exponentially suppressed (Silk damping; Silk 1968). Now, while the sound horizon is proportional to 1/H, the diffusion length is proportional to $1/\sqrt{H}$, due to the random walk nature of the diffusion, thus, $r_d/r_s \propto \sqrt{H}$. As a result, increasing the expansion rate increases the diffusion length relative to the sound horizon, which enhances the Silk damping of the small-scale anisotropy (Bashinsky & Seljak 2004). Note that r_d/r_s also depends on the mean free path of the photon, $r_d/r_s \propto \sqrt{H\lambda _C} \propto \sqrt{H/n_e}$, thus one can compensate for the increased expansion rate by increasing the number density of free electrons. One way to achieve this is to reduce the helium abundance, Y_p (Bashinsky & Seljak 2004; Hou et al. 2013): since helium recombines earlier than the epoch of photon decoupling, the number density of free electrons at the decoupling epoch is given by n_e = (1 − Y_p) n_b, where n_b is the number density of baryons (Hu et al. 1995; see also Section 4.8 of Komatsu et al. 2011). In the bottom right panel of Figure 8, we show $C_l^{{\rm TT}}$ for the N_eff = 7 model after reducing Y_p from 0.24 to 0.08308, which preserves the ratio r_d/r_s. The solid and dashed model curves now agree completely (except for l ≲ 130 where our compensation for anisotropic stress was ad hoc).

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4.3.2. Measurements of N_eff and Y_He: Testing Big Bang Nucleosynthesis

Using the five-year WMAP data alone, Dunkley et al. (2009) measured the effect of anisotropic stress on the power spectrum and set a lower bound on N_eff. However, BAO and H₀ data were still required to set an upper bound due to a degeneracy with the matter-radiation equality redshift (Komatsu et al. 2009). This was unchanged for the seven-year analysis (Komatsu et al. 2011). Now, with much improved measurements of the enhanced damping tail from SPT and ACT (Section 2.2.1), CMB data alone are able to determine N_eff (Dunkley et al. 2011; Keisler et al. 2011). Using the nine-year WMAP data combined with SPT and ACT, we find

$$\begin{eqnarray*} &&N_{\rm eff} = 3.89\pm 0.67 (68\%\ {\rm CL}) \quad {\rm {{WMAP}}+eCMB; $Y_{\rm He}$ fixed}. \end{eqnarray*}$$

The inclusion of lensing in the eCMB likelihood helps this constraint because the primary CMB fluctuations are still relatively insensitive to a combination of N_eff and Ω_mh², as described above. CMB lensing data help constrain Ω_mh² by constraining σ₈. The measurement is further improved by including the BAO and H₀ data, which reduces the degeneracy with the matter-radiation equality redshift. We find

$$\begin{eqnarray*} && N_{\rm eff} = 3.84\pm 0.40 (68\% \ {\rm CL}) \quad {\rm {{WMAP}}+eCMB+BAO+H_0;}\nonumber\\ &&\quad\quad {\rm Y_{\rm He} fixed}, \end{eqnarray*}$$

which is consistent with the standard model value of N_eff = 3.046. We thus find no evidence for the existence of extra radiation species (see also Calabrese et al. 2012).

As noted above, this measurement of N_eff relies on the damping tail measured by ACT and SPT, which is also affected by the primordial helium abundance, Y_He. Figure 9 shows the joint, marginalized constraints on N_eff and Y_He using the above two data combinations. As expected, these two parameters are anti-correlated when fit to CMB data alone (black contours). When BAO and H₀ measurements are included, we find

$$\begin{eqnarray*} \begin{array}{lr}N_{\rm eff} = 3.55^{+ 0.49}_{- 0.48} & \\ [2mm] Y_{\rm He} = 0.278^{+ 0.034}_{- 0.032} &\end{array} (68\% \ {\rm CL})\quad {\rm {{WMAP}}+eCMB+BAO+}H_0. \end{eqnarray*}$$

When combined with our measurement of the baryon density, both of these values are within the 95% CL region of the standard BBN prediction (Steigman 2012), shown by the green curve in Figure 9. Our measurement provides strong support for the standard BBN scenario. Table 7 summarizes the nine-year measurements of N_eff and Y_He.

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Table 7. Relativistic Degrees of Freedom and Big Bang Nucleosynthesis^a

Parameter	WMAP	+eCMB	+eCMB+BAO	+eCMB+BAO+H0
Number of relativistic speciesb
Neff	>1.7 (95% CL)	3.89 ± 0.67	3.55 ± 0.60	3.84 ± 0.40
ns	0.988 ± 0.027	$0.985^{+ 0.018}_{- 0.019}$	0.969 ± 0.015	0.975 ± 0.010
Primordial helium abundanceb
YHe	<0.42 (95% CL)	0.299 ± 0.027	0.295 ± 0.027	0.299 ± 0.027
ns	0.973 ± 0.016	0.982 ± 0.013	0.973 ± 0.011	0.977 ± 0.011
Big bang nucleosynthesisc
Neff	⋅⋅⋅	2.92 ± 0.79	2.58 ± 0.67	$3.55^{+ 0.49}_{- 0.48}$
YHe	⋅⋅⋅	$0.302^{+ 0.038}_{- 0.039}$	$0.311^{+ 0.036}_{- 0.037}$	$0.278^{+ 0.034}_{- 0.032}$
ns	⋅⋅⋅	0.978 ± 0.019	0.965 ± 0.015	0.980 ± 0.011

Notes. ^aA complete list of parameter values for these models, with additional data combinations, may be found at http://lambda.gsfc.nasa.gov/. ^bThe parameters N_eff and Y_He comprise one additional parameter each in these table sections. ^cThe parameters N_eff and Y_He are fit jointly in this section.

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The mean energy of a relativistic neutrino at the epoch of recombination is 〈E〉 = 0.58 eV. In order for the CMB power spectrum to be sensitive to a non-zero neutrino mass, at least one species of neutrino must have a mass in excess of this mean energy. If one assumes that there are N_eff = 3.046 neutrino species with degenerate mass eigenstates, this would suggest that the lowest total mass that could be detected with CMB data is ∑m_ν ∼ 1.8 eV. Using a refined argument, Ichikawa et al. (2005) argue that one could reach ∼1.5 eV. When we add ∑m_ν = 93 eV (Ω_νh²) as a parameter to the ΛCDM model we obtain the fit given in Table 8, specifically

$$\begin{eqnarray*} &&\sum m_\nu < 1.3\ {\rm eV}\ {\rm (95\% CL)} \quad {\rm {{WMAP}}\ only}, \end{eqnarray*}$$

which is at the basic limit just presented.

Table 8. Neutrino Mass^a

Parameter	WMAP	+eCMB	+eCMB+BAO	+eCMB+BAO+H0
New parameter
∑mν (eV)b	<1.3 (95% CL)	<1.5 (95% CL)	<0.56 (95% CL)	<0.44 (95% CL)
Related parameters
σ8	$0.706^{+ 0.077}_{- 0.076}$	$0.660^{+ 0.066}_{- 0.061}$	$0.750^{+ 0.044}_{- 0.042}$	0.770 ± 0.038
Ωch2	$0.1157^{+ 0.0048}_{- 0.0047}$	0.1183 ± 0.0044	0.1133 ± 0.0026	0.1132 ± 0.0025
$\Omega _\Lambda$	$0.641^{+ 0.065}_{- 0.068}$	$0.586^{+ 0.080}_{- 0.076}$	0.695 ± 0.013	0.707 ± 0.011
$10^9 \Delta _{\cal R}^2$	2.48 ± 0.12	2.59 ± 0.12	$2.452^{+ 0.075}_{- 0.074}$	2.438 ± 0.074
ns	0.962 ± 0.016	0.947 ± 0.014	0.9628 ± 0.0086	$0.9649^{+ 0.0085}_{- 0.0083}$

Notes. ^aA complete list of parameter values for these models, with additional data combinations, may be found at http://lambda.gsfc.nasa.gov/. ^bIn the standard model, ∑m_ν = 93.14 eV (Ω_νh²), when neutrino heating is taken into account.

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Table 9. Non-flat ΛCDM Constraints^a

Parameter	WMAP	+eCMB	+eCMB+BAO	+eCMB+H0	+eCMB+BAO+H0
New parameter
Ωk	$-0.037^{+ 0.044}_{- 0.042}$	−0.001 ± 0.012	$-0.0049^{+ 0.0041}_{- 0.0040}$	0.0049 ± 0.0047	$-0.0027^{+ 0.0039}_{- 0.0038}$
Related parameters
Ωtot	$1.037^{+ 0.042}_{- 0.044}$	1.001 ± 0.012	$1.0049^{+ 0.0040}_{- 0.0041}$	0.9951 ± 0.0047	$1.0027^{+ 0.0038}_{- 0.0039}$
Ωm	0.19 < Ωm < 0.95 (95% CL)	0.273 ± 0.049	0.292 ± 0.010	0.252 ± 0.017	$0.2855^{+ 0.0096}_{- 0.0097}$
$\Omega _\Lambda$	$0.22<\Omega _\Lambda <0.79\ {\rm (95\% CL)}$	0.727 ± 0.038	0.713 ± 0.011	0.743 ± 0.015	0.717 ± 0.011
H0 (km s−1 Mpc−1)	38 < H0 < 84 (95% CL)	71.2 ± 6.5	68.0 ± 1.0	$73.4^{+ 2.2}_{- 2.3}$	$68.92^{+ 0.94}_{- 0.95}$
t0 (Gyr)	14.8 ± 1.5	13.71 ± 0.65	13.99 ± 0.17	13.46 ± 0.24	13.88 ± 0.16

Note. ^aA complete list of parameter values for these models, with additional data combinations, may be found at http://lambda.gsfc.nasa.gov/.

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Table 10. Dark Energy Constraints^a

Parameter	WMAP	+eCMB	+eCMB+BAO+H0	+eCMB+BAO+H0+SNe
Constant equation of state; flat universe
w	−1.71 < w < −0.34 (95% CL)	$-1.07^{+ 0.38}_{- 0.41}$	$-1.073^{+ 0.090}_{- 0.089}$	−1.084 ± 0.063
H0	>50 (95% CL)	>55 (95% CL)	$70.7^{+ 1.8}_{- 1.9}$	$71.0^{+ 1.4}_{- 1.3}$
Constant equation of state; non-flat universe
w	> − 2.1 (95% CL)	⋅⋅⋅	−1.19 ± 0.12	$-1.122^{+ 0.068}_{- 0.067}$
Ωk	$-0.052^{+ 0.051}_{- 0.054}$	⋅⋅⋅	$-0.0072^{+ 0.0042}_{- 0.0043}$	$-0.0059^{+ 0.0038}_{- 0.0039}$
H0	37 < H0 < 84 (95% CL)	⋅⋅⋅	71.7 ± 2.0	70.7 ± 1.3
Non-constant equation of state; flat universe
w0	⋅⋅⋅	⋅⋅⋅	−1.34 ± 0.18	$-1.17^{+ 0.13}_{- 0.12}$
wa	⋅⋅⋅	⋅⋅⋅	0.85 ± 0.47b	$0.35^{+ 0.50}_{- 0.49}$
H0	⋅⋅⋅	⋅⋅⋅	72.3 ± 2.0	71.0 ± 1.3

Notes. ^aA complete list of parameter values for these models, with additional data combinations, may be found at http://lambda.gsfc.nasa.gov/. ^bThe quoted error on w_a from WMAP+eCMB+BAO+H₀ is smaller than that from WMAP+eCMB+BAO+H₀+SNe. This is due to the imposition of a hard prior, w_a < 0.2–1.1w₀, depicted in Figure 10. Without this prior, the upper limit on w_a for WMAP+eCMB+BAO+H₀ would extend to larger values.

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When the mass of individual neutrinos is less than 0.58 eV, the CMB power spectrum alone (excluding CMB lensing) cannot determine ∑m_ν; however, tighter limits can be obtained by combining CMB data with BAO and H₀ data. For a given Ω_ch² and H₀, adding massive neutrinos results in a larger present-day total matter density, Ω_m, giving a smaller dark energy density for a flat universe, and hence a smaller angular diameter distance to the decoupling epoch. This change in distance can be compensated by lowering H₀. By the same token, for a given Ω_ch²+Ω_νh², adding massive neutrinos results in a smaller matter density at the decoupling epoch (since neutrinos are still relativistic then), which produces a larger sound horizon size at that epoch. Both of these effects cause the angular size of the acoustic scale, θ_*, to be larger, shifting the CMB peaks to larger angular scales. Furthermore, a reduced matter density at the decoupling epoch produces an earlier matter-radiation equality epoch giving a larger early ISW effect which, in turn, shifts the first peak position to a larger angular scale. This effect can again be compensated by lowering H₀. Therefore, independent information on H₀ obtained from local distance indicators and from BAO data helps tighten the limit on ∑m_ν (Ichikawa et al. 2005). We find

$$\begin{eqnarray*} &&\sum m_\nu < 0.44\ {\rm eV}\ {\rm (95\% CL)} \quad {\rm {{WMAP}}+eCMB+BAO+H_0}, \end{eqnarray*}$$

which is 25% lower than the bound of 0.58 eV that was set with the seven-year analysis (Komatsu et al. 2011).

Since massive neutrinos have a large velocity dispersion, they cannot cluster on small scales. This means that a fraction of matter density in a low redshift universe (when neutrinos are non-relativistic) cannot cluster, which yields a shallower gravitational potential well, hence a lower value of σ₈. As a result, one sees a clear negative correlation between σ₈ and ∑m_ν (see, e.g., the middle panel of Figure 17 of Komatsu et al. 2009). Therefore, adding independent information on σ₈ obtained from, e.g., the abundance of galaxy clusters (Vikhlinin et al. 2009b; Mantz et al. 2010) helps tighten the limit on ∑m_ν. See Section 5 for a discussion of recent measurements of σ₈ from various cosmological probes such as cluster abundances, peculiar velocities, and gravitational lensing.

The geometric degeneracy in the angular diameter distance to the surface of last scattering limits our ability to constrain spatial curvature, Ω_k, with primary CMB fluctuations alone (Bond et al. 1997; Zaldarriaga et al. 1997). For example, the nine-year WMAP data gives a measurement with 4% uncertainty,

$$\begin{eqnarray*} &&\Omega _k = -0.037^{+ 0.044}_{- 0.042} \quad {\rm {{WMAP}}-only} \end{eqnarray*}$$

(see Table 9). However, with the recent detection of CMB lensing in the high-l power spectrum (Das et al. 2011a; van Engelen et al. 2012), the degeneracy between Ω_m and $\Omega _\Lambda$ is now substantially reduced. This produces a significant detection of dark energy, and tight constraints on spatial curvature using only CMB data: when the SPT and ACT data, including the lensing constraints, are combined with nine-year WMAP data we find

$$\begin{eqnarray*} \begin{array}{lclr}\Omega _\Lambda & = & 0.727\pm 0.038 & \\ [2mm] \Omega _k & = & -0.001\pm 0.012 &\end{array} \quad {\rm {{WMAP}}+eCMB}. \end{eqnarray*}$$

Figure 43 in Bennett et al. (2013) shows the joint constraints on (Ω_m, $\Omega _\Lambda$) (and Ω_k, implicitly) from the currently available CMB data. Combining the CMB data with lower-redshift distance indicators, such H₀, BAO, or supernovae further constrains Ω_k (Spergel et al. 2007). Assuming the dark energy is vacuum energy (w = −1), we find

$$\begin{eqnarray*} &&\Omega _k = -0.0027^{+ 0.0039}_{- 0.0038} \quad {\rm {{WMAP}}+eCMB+BAO+H_0}, \end{eqnarray*}$$

which limits spatial curvature to be no more than 0.4% (68% CL) of the critical density. These (Ω_m,$\Omega _\Lambda$) constraints are also shown in Figure 43 of Bennett et al. (2013). An independent analysis of non-flat models based on time-delay measurements of two strong gravitational lens systems, combined with seven-year WMAP data, give consistent and nearly competitive constraints of Ω_k = $0.003^{+0.005}_{-0.006}$ (Suyu et al. 2013).

The limits on curvature weaken slightly if dark energy is allowed to be dynamical, w ≠ −1. However, with new distance measurements at somewhat higher redshift, where dynamical dark energy starts to become significant, the degradation factor is substantially less than it was in our previous analyses. We revisit this topic in Section 4.6.

The dark energy equation-of-state parameter, w ≡ P_de/ρ_de, where P_de and ρ_de are the pressure and density of dark energy, respectively, governs whether ρ_de changes with time (w ≠ −1) or not (w = −1). CMB data alone (excluding the effect of CMB gravitational lensing) are unable to determine w because dark energy only affects the CMB through (1) the comoving angular diameter distance to the decoupling epoch, d_A(z_*), and (2) the late-time ISW effect. The ISW effect has limited ability to constrain dark energy due to its large cosmic variance. The angular diameter distance to z_* depends on several parameters (Ω_m, Ω_k, $\Omega _\Lambda$, w, and H₀), thus a measurement of the angular diameter distance to a single redshift cannot distinguish these parameters.

Distance measurements to multiple redshifts greatly improve the constraint on w. These include the Hubble constant, H₀, which determines the distance scale in the low-redshift universe; D_V's from BAO measurements; and luminosity distances from high-redshift Type Ia supernovae. Gravitational lensing of the CMB also probes w by measuring the ratio of the angular diameter distance to the source plane (the decoupling epoch) and to the lens planes (matter fluctuations in the range of z ∼ 1–2). Current CMB lensing data do not yet provide competitive constraints on w, though they do improve the CMB-only measurement.

In this section we derive dark energy constraints using the full CMB power spectrum information (as opposed to the simplified "distance posteriors" given in Section 4.6.1), both alone and in conjunction with the data sets noted above (see Table 10). Linear perturbations in dark energy are treated following the "parameterized post-Friedmann" approach, implemented in the CAMB code by Fang et al. (2008; see also Zhao et al. 2005). New measurements of the BAO scale (Section 2.2.2) and H₀ (Section 2.2.3) significantly tighten the 68% CL errors on a constant w for both flat and non-flat models

$$\begin{eqnarray*} w = \left\lbrace \begin{array}{@{}lr}-1.073^{+ 0.090}_{- 0.089} & ({\rm flat}) \\ [2mm] -1.19\pm 0.12 & ({\rm non-flat}) \end{array} \right. \!\!\!\quad {\rm {{WMAP}}+eCMB+BAO+}H_0. \end{eqnarray*}$$

These constraints represent 35% and 55% improvements, respectively, over those from the seven-year WMAP+BAO+H₀ combination (see the fourth column in Table 4 of Komatsu et al. 2011): w = −1.10 ± 0.14 (flat) and w = −1.44 ± 0.27 (non-flat). Adding 472 Type Ia supernovae compiled by Conley et al. (2011) improves these limits to

$$\begin{eqnarray*} { w = \left\lbrace \begin{array}{@{}lr}-1.084\pm 0.063 &\!\! ({\rm flat}) \\ [2mm] -1.122^{+ 0.068}_{- 0.067} &\!\! ({\rm non-flat}) \end{array} \right. \!\!\!\!\!\quad {\rm {{WMAP}}+eCMB+BAO+}H_0{\rm +SNe}, } \end{eqnarray*}$$

where the errors include systematic uncertainties in the supernova data. Note that these limits are somewhat weaker than those reported in Komatsu et al. (2011), Table 4, Column 6, despite the smaller number of supernovae (397) in the "Constitution" sample compiled by Hicken et al. (2009), as that analysis did not include SNe systematic uncertainties in the seven-year analysis.

When w is allowed to vary with the scale factor according to w(a) = w₀ + w_a(1 − a) (Chevallier & Polarski 2001; Linder 2003), we find, for a flat universe²⁰

$$\begin{eqnarray*} \begin{array}{lclr}w_0 & = & -1.17^{+ 0.13}_{- 0.12} & \\ [2mm] w_a & = & 0.35^{+ 0.50}_{- 0.49} &\end{array} \quad {\rm {{WMAP}}+eCMB+BAO+}H_0{\rm +SNe}. \end{eqnarray*}$$

Figure 10 shows the joint, marginalized constraint on w₀ and w_a. A cosmological constant (w₀ = −1 and w_a = 0) is at the boundary of the 68% CL region, indicating that the current data are consistent with a time-independent dark energy density. Comparing this measurement with the seven-year result in Figure 13 of Komatsu et al. (2011), we note that adding the new BAO and H₀ data significantly reduces the allowed parameter space by eliminating w_a ≲ −1.

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4.6.1. WMAP Nine-year Distance Posterior

The "WMAP distance posterior" gives the likelihood of three variables: the acoustic scale, l_A, the shift parameter, R, and the decoupling redshift, z_*. This likelihood is based on, and extends, the original idea put forward by several authors (Wang & Mukherjee 2007; Wright 2007; Elgarøy & Multamäki 2007). It allows one to quickly evaluate the likelihood of various dark energy models given the WMAP data, without the need to run a full MCMC exploration of the likelihood.

Here, we provide an updated distance posterior based on the nine-year data. For details on how to use this simplified likelihood, the definition of the above variables, and the limitation of this approach, see Section 5.5 of Komatsu et al. (2011) and Section 5.4 of Komatsu et al. (2009).

The likelihood is given by

$$\begin{equation} -2\ln L = ({\bf x-d})^T{\bf C}^{-1}({\bf x-d}), \end{equation} \tag{ 17 }$$

where x = (l_A, R, z_*) are the parameter values for the proposed model, and the data vector d has components

$$\begin{eqnarray} && d_1 = l_A^{{{WMAP}}} = 302.40 \end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray} && d_2 = R^{{{WMAP}}} = 1.7246 \end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray} && d_3 = z_*^{{{WMAP}}} = 1090.88. \end{eqnarray} \tag{ 20 }$$

These are the maximum-likelihood values obtained from the nine-year data assuming a constant dark energy equation of state and non-zero spatial curvature (the "OWCDM" model). The elements of the inverse covariance matrix, C⁻¹, are given in Table 11.

Table 11. Inverse Covariance Matrix for the WMAP Distance Posteriors

	lA	R	z*
lA	3.182	18.253	−1.429
R		11887.879	−193.808
z*			4.556

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If the polarization direction on the sky were uniformly rotated by an angle Δα, then some of the E-mode polarization would be converted to B-mode polarization. This can arise from a mis-calibration of the detector polarization angle, but also from a physical mechanism called "cosmological birefringence," in which global parity symmetry is broken on cosmological scales (Lue et al. 1999; Carroll 1998). Such an effect yields non-vanishing TB and EB correlations, hence non-vanishing U_r. A non-detection of these correlations limits Δα.

A rotation of the polarization plane by an angle Δα gives the following transformation

$$\begin{eqnarray} C_l^{\rm TE,obs} &=& C_l^{\rm TE}\cos (2\Delta \alpha), \end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray} C_l^{\rm TB,obs} &=& C_l^{\rm TE}\sin (2\Delta \alpha), \end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray} C_l^{\rm EE,obs} &=& C_l^{\rm EE}\cos ^2(2\Delta \alpha), \end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray} C_l^{\rm BB,obs} &=& C_l^{\rm EE}\sin ^2(2\Delta \alpha), \end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray} C_l^{\rm EB,obs} &=& \frac{1}{2}C_l^{\rm EE}\sin (4\Delta \alpha), \end{eqnarray} \tag{ 25 }$$

where the spectra on the right-hand side are the primordial power spectra in the absence of rotation, while the spectra on the left-hand side are what we would observe in the presence of rotation. We assume there is no B-mode polarization in the absence of rotation, for the full expressions including $C_l^{\rm BB}$ (see Lue et al. 1999; Feng et al. 2005).

The low-l TB and EB data at l ⩽ 23 yield $\Delta \alpha =-0{\rm {\rlap.}^\circ }07\,{\pm}\, 4{\rm {\rlap.}^\circ }82$ (68% CL), while the high-l TB data yield $\Delta \alpha =-0{\rm {\rlap.}^\circ }40\pm 1{\rm {\rlap.}^\circ }30$ (68% CL). The high-l EB data are too noisy to yield a significant limit. Combining all the multipoles, we find

$$\begin{eqnarray*} &&\Delta \alpha =-0{\rm {\rlap.}^\circ }36\pm 1{\rm {\rlap.}^\circ }24 ({\rm stat.})\pm 1{\rm {\rlap.}^\circ }5 ({\rm syst.}) ({\rm 68\% CL}). \end{eqnarray*}$$

Here, we have added the systematic uncertainty of ${\pm } 1{\rm {\rlap.}^\circ }5$, to account for uncertainty in the WMAP detector polarization angle (Page et al. 2003, 2007). The WMAP limit on Δα is now dominated by this systematic uncertainty. The statistical error has modestly improved from $1{\rm {\rlap.}^\circ }4$ with the seven-year data (Komatsu et al. 2011; see also Xia et al. 2010).

Clusters are rare, high-mass peaks in the density field, hence their number counts provide an important probe of the matter fluctuation amplitude and, in turn, cosmology (see Allen et al. 2011 for a recent review). For cosmological studies, the main challenge with clusters is relating the astronomical observable (SZ decrement, X-ray flux, optical richness, etc.) to the mass of the cluster. Since the mass function is so steep, a small error in the zero-point of the mass-to-observable scaling can produce a significant error in the determination of cosmological parameters.

In the past two years, many new SZ-selected clusters have been reported by Planck (Planck Collaboration VIII 2011), ACT (Marriage et al. 2011; Menanteau et al. 2013), and SPT (Williamson et al. 2011; Reichardt et al. 2013), and they are providing new impetus for cosmological studies. Clusters are close to virial equilibrium, so they should exhibit a tight relationship between integrated SZ decrement and mass; however, there are significant sources of non-thermal pressure support that need to be modeled (Trac et al. 2011; Battaglia et al. 2012a, 2012c). Estimates of cluster mass based on X-ray data agree well with estimates based on Planck SZ measurements when one has both X-ray and SZ data for the same cluster (Planck Collaboration IX 2011; Planck Collaboration XI 2011), however, there are intriguing discrepancies between some estimates based on optical and SZ data. This is seen in both the Planck (Planck Collaboration XII 2011) and ACT data (Hand et al. 2011; Sehgal et al. 2013), and several groups are exploring this discrepancy (Angulo et al. 2012; Rozo et al. 2012; Biesiadzinski et al. 2012).

The abundance of optically selected clusters provided early, strong evidence that vacuum energy dominates the universe today. Based on the number density of rich clusters, Fan et al. (1997) measured σ₈ = 0.83 ± 0.15 and Ω_m = 0.3 ± 0.1. A recent analysis by Tinker et al. (2012), combining two observables from the SDSS: the galaxy two-point correlation function and the mass-to-galaxy number ratio within clusters, found $\sigma _8\Omega _m^{0.5} = 0.465 \pm 0.026$, with Ω_m = 0.29 ± 0.03 and σ₈ = 0.85 ± 0.06. Zu et al. (2012) used weak lensing measurements to calibrate the masses of MaxBCG clusters in the SDSS data; they find σ₈(Ω_m/0.325)^0.501 = 0.828 ± 0.049.

X-ray-selected cluster samples also provide constraints on the amplitude of matter fluctuations. X-ray data allow one to both select the sample and calibrate its mass under the assumption of hydrostatic equilibrium. Vikhlinin et al. (2009b) analyze cosmological parameter constraints from their Chandra cluster sample. Fitting ΛCDM parameters to the cluster counts, they find σ₈(Ω_M/0.25)^0.47 = 0.813 with a statistical error of ±0.012 and a systematic error of ±0.02, due to absolute mass calibration uncertainty.

With the new SZ-selected cluster samples, groups are calibrating the SZ-decrement-to-mass scaling using weak-lensing measurements (Marrone et al. 2012; Miyatake et al. 2013; High et al. 2012; Planck Collaboration III 2013), X-ray measurements (Bonamente et al. 2012; Benson et al. 2013), and galaxy velocity dispersions (Sifon et al. 2013). Remarkably, these different groups are reporting fluctuation amplitudes that are consistent with the WMAP ΛCDM fluctuation amplitude. For example, Benson et al. (2013) report σ₈(Ω_m/0.25)^0.30 = 0.785 ± 0.037 based on an analysis of 18 SZ-selected clusters from the SPT survey, while Sehgal et al. (2011) report σ₈ = 0.821 ± 0.044 and w = −1.05 ± 0.20 based on a joint analysis of WMAP seven-year data and 9 optically confirmed SZ clusters.

The n-point correlation function of SZ-selected clusters provides complementary information to cluster counts (the one-point function). The two-point function is a potentially-powerful probe of σ₈ (Komatsu & Seljak 2002); however, it is sensitive to the low-mass end of the Y(M) scaling relation for clusters, which is subject to astrophysical corrections (Shaw et al. 2010; Battaglia et al. 2010, 2012a). Reichardt et al. (2012) use simulations and observations to calibrate the SZ power spectrum (two-point function); they apply this to the SPT data to find σ₈ = 0.807 ± 0.016. Higher-order correlation functions are less sensitive to low-mass clusters, so these moments are less affected by non-thermal processes and more sensitive to the matter fluctuation amplitude (Hill & Sherwin 2013; Bhattacharya et al. 2012). Measurement of the three-point function in the ACT SZ data (Wilson et al. 2012) yields $\sigma _8 = 0.78^{+0.03}_{-0.04}$. All of these measurements are consistent with the WMAP nine-year measurement of σ₈ = 0.821 ± 0.023, assuming ΛCDM.

Galaxy peculiar velocities provide another independent probe of gravitational potential fluctuations. Hudson & Turnbull (2012) combine redshift-space distortion data from BOSS (Reid et al. 2012), 6dFGS (Beutler et al. 2012), and WiggleZ (Blake et al. 2011), with local measurements of the peculiar velocity field, to find Ω_m = 0.259 ± 0.045, σ₈ = 0.748 ± 0.035, and a growth rate of γ ≡ dln D/dln a = 0.619 ± 0.514.

The kinematic Sunyaev–Zel'dovich (kSZ) effect can probe peculiar velocity fields over a wide range of redshift. Hand et al. (2012) report the first kSZ measurements of peculiar velocities at z ∼ 0.35; they detect a signal consistent with predictions from N-body simulations that are based on the WMAP seven-year ΛCDM parameters. Future kSZ measurements should be able to provide precision tests of cosmology.

There are a number of complementary gravitational lensing techniques that measure the amplitude of potential fluctuations:

CMB lensing. Large-scale structure along the line of sight deflects CMB photons and imparts a non-Gaussian pattern on the CMB fluctuation field. While this is most easily detected on scales smaller than those probed by WMAP, Smith et al. (2007) reported the first detection of CMB lensing, by cross-correlating the three-year WMAP data with the NVSS survey (see also Feng et al. 2012). Das et al. (2011a) reported the first detection of CMB lensing using a measurement of the four-point correlation function in the ACT temperature maps. They parameterize the lensing signal by a dimensionless parameter A_L which scales the lensing power spectrum relative to the prediction of the best-fit ΛCDM model. They report an amplitude of A_L = 1.16 ± 0.29, where A_L = 1 is the value predicted by the WMAP seven-year ΛCDM model. (A value of A_L significantly different from 1 would signal a problem with the data and/or the lensing calculation.) Using a similar technique on SPT data, van Engelen et al. (2012) report A_L = 0.90 ± 0.19. Since $A_L \propto \sigma _8^2$, this corresponds to an 8% measurement of σ₈. When these CMB lensing measurements are combined with WMAP seven-year data, they provide strong evidence for dark energy based purely on CMB observations (Sherwin et al. 2011; van Engelen et al. 2012).

Cosmic shear. Measurements of cosmic shear in large optical surveys directly probe matter fluctuations on small scales. Huff et al. (2011) analyzed 168 deg² of co-added equatorial images from the SDSS and found $\sigma _8=0.636^{+0.109}_{-0.154}$ (when other cosmological parameters are fixed to the WMAP seven-year ΛCDM values). Lin et al. (2012) analyzed 275 deg² of co-added imaging from SDSS Stripe 82 and found $\Omega _m^{0.7}\sigma _8 = 0.252^{+0.032}_{-0.052}$. Jee et al. (2013) report Ω_m = 0.262 ± 0.051 and σ₈ = 0.868 ± 0.071 from a cosmic shear study using the Deep Lensing Survey; when their results are combined with the WMAP seven-year data, they find Ω_m = 0.278 ± 0.018 and σ₈ = 0.815 ± 0.020. Semboloni et al. (2011) analyzed both the second and third-order moments of the cosmic shear field in the HST COSMOS data; they found $\sigma _8 (\Omega _M/0.3)^{0.49} = 0.78^{+0.11}_{-0.26}$ using the three-point statistic, in agreement with their result from the two-point statistic: $\sigma _8 (\Omega _M/0.3)^{0.67} = 0.70^{+0.11}_{-0.14}$.

Cross correlation. Correlating the cosmic shear field with the large-scale galaxy distribution measures galaxy bias: the relationship been galaxies and dark matter. Mandelbaum et al. (2013) measured this cross-correlation in the SDSS DR7 data and used the inferred bias to determine cosmological parameters. They report σ₈(Ω_m/0.25)^0.57 = 0.80 ± 0.05, where the errors include both statistical and systematic effects. Cacciato et al. (2013) use combined SDSS measurements of galaxy number counts, galaxy clustering, and galaxy-galaxy lensing, together with WMAP seven-year priors on the scalar spectral index, the Hubble parameter, and the baryon density, to find $\Omega _m = 0.278_{-0.026}^{+0.023}$ and ${\sigma }_8 = 0.763_{-0.049}^{+0.064}$ (95% CL).

Strong lensing. The statistics of lensed quasars probes the amplitude of fluctuations, the shape of galaxy halos, and the large-scale geometry of the universe. Oguri et al. (2012) have analyzed the final data from the SDSS Quasar Lens Search (19 lensed quasars selected from 50,836 candidates). They claim that the number of lensed quasar are consistent with predictions based on WMAP seven-year parameters. Assuming the velocity function of galaxies does not evolve with redshift, they report $\Omega _\Lambda = 0.79^{+0.06}_{-0.07}\,^{+0.06}_{-0.06}$, where the errors are statistical and systematic, respectively.

In the standard model of cosmology based upon adiabatic scalar perturbations, temperature hot spots correspond to potential wells (i.e., over-dense regions) at the surface of last scattering; therefore, matter flows toward these hot spots. A crucial length scale in the CMB is the sound horizon at the epoch of decoupling, r_s(z_*); the angular size of the sound horizon sets the acoustic scale, $\theta _A \equiv r_s/d_A \approx 0{\rm {\rlap.}^\circ }6$. At twice the acoustic scale, the flow of matter is accelerating due to gravity, which creates a radial polarization pattern. At the acoustic scale, the flow is decelerating due to the central photon pressure, which creates a tangential pattern (Baccigalupi 1999; Komatsu et al. 2011). Around cold spots (potential hills), the polarization follows the opposite pattern, with tangential and radial polarization formed at $1{\rm {\rlap.}^\circ }2$ and $0{\rm {\rlap.}^\circ }6$, respectively.

We detected this polarization pattern in the seven-year WMAP data with a statistical significance of 8σ. The measured pattern was fully consistent with that predicted by the standard ΛCDM model (see Section 2.4 of Komatsu et al. 2011). Here we apply the same analysis to the nine-year data and find results that are again fully consistent with the standard model, now with a statistical significance of 10σ.

The small-scale polarization data offer a powerful test of the standard model of cosmology. Once the cosmological parameters are determined by the temperature and large-scale polarization data, one can predict the polarization signal on small angular scales with no free parameters. This simple description is an important test of the standard cosmological model.

The nine-year analysis replicates that of the seven-year data (see Section 2.3 and Appendix B of Komatsu et al. 2011). We first smooth the foreground-reduced temperature maps from differencing assemblies V1 through W4 to a common angular resolution of $0{\rm {\rlap.}^\circ }5$ (FWHM). We combine these maps with inverse-noise-variance weighting, and remove the monopole from the region outside the KQ85y9 mask (Bennett et al. 2013). The locations of the local maxima and minima are obtained using the software hotspot in the HEALPix distribution (Gorski et al. 2005).

As in the seven-year analysis, we cull the hot spot list by removing all local peaks with T < 0, and vice versa. In the 66.34% of the sky outside the union of the KQ85y9 (temperature) mask and the P06 (polarization) mask, we find 11,536 hot spots and 11,752 cold spots remain in the temperature map. These counts are consistent with the expectation for a Gaussian random field drawn from the best-fit nine-year WMAP signal plus noise power spectrum.

The raw polarization maps from differencing assemblies V1 through W4 are combined using inverse-noise-variance weighting. We do not smooth the polarization maps for this analysis. We extract a 5° × 5° square region in the Stokes I, Q, and U maps centered on each hot and cold temperature spot. We combine the extracted temperature images with uniform weighting, while the Q and U images are combined with inverse-noise-variance weighting, excluding pixels masked by either analysis mask. Afterward, we remove the monopole from the co-added Q and U images. There are 625 $0{\rm {\rlap.}^\circ }2$ pixels in each polarization image, which sets the number of degrees of freedom in the χ² analysis.

To make contact with the standard model prediction, we work in a rotated polarization basis, Q_r and U_r, introduced by Kamionkowski et al. (1997). These parameters are related to Q and U by

$$\begin{eqnarray} && Q_r(\boldsymbol {\theta }) = -Q(\boldsymbol {\theta }) \cos 2\phi - U(\boldsymbol {\theta }) \sin 2\phi, \end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray} && U_r(\boldsymbol {\theta }) = Q(\boldsymbol {\theta }) \sin 2\phi - U(\boldsymbol {\theta }) \cos 2\phi, \end{eqnarray} \tag{ 27 }$$

where $\boldsymbol {\theta }=(\theta \cos \phi,\theta \sin \phi)$ is a position vector whose origin is the location of the temperature extremum; see Figure 1 of Komatsu et al. (2011). These Stokes parameters offer a simple test of the standard model, which predicts U_r = 0 everywhere, and Q_r(θ) alternating between positive (radial polarization) and negative (tangential polarization) values.

Figure 12 shows the co-added T and Q_r images from the nine-year data. We clearly see the alternating radial and tangential polarization pattern around the average hot spot, and vice versa around the average cold spot. To test the agreement between data and theory, we fit the Q_r maps to their predicted patterns, and let the amplitude be a free parameter. For the hot-spot Q_r, we find a best-fit amplitude of 0.89 ± 0.14 (68% CL) with Δχ² = −41.3 relative to zero amplitude. For the cold-spot Q_r, we find 1.06 ± 0.13 (68% CL) with Δχ² = −61.4. The combined best-fit amplitude is 0.973 ± 0.096 (68% CL). The data are fully consistent with the standard ΛCDM prediction, and the combined statistical significance of the detection is 10σ, compared to 8σ for the seven-year data.

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The co-added U_r maps are consistent with zero. We fit the measured U_r maps to the predicted Q_r patterns and find best-fit amplitudes of 0.02 ± 0.14 and 0.04 ± 0.13 (68% CL) around the average hot and cold spots, respectively.

As an example of a model that is consistent with the WMAP nine-year data, we examine the predictions of Starobinsky's R² inflation model. In 1980, Starobinsky (1980) showed that one-loop quantum corrections to the Einstein–Hilbert action, which generate fourth-order derivative terms of ${\cal O}(R^2)$, where R is the Ricci scalar, lead to a de-Sitter-type accelerated expansion of the universe. Starobinsky's motivation was not to solve the flatness and homogeneity problems of the standard Big Bang model (this was later done by Guth 1981), but to see whether one-loop corrections eliminate the classical singularity at the beginning of the universe. He was attempting to construct a cosmological model beginning from an initial de Sitter stage (not necessarily expanding) and ending in a radiation-dominated stage with a Friedmann–Robertson–Walker metric, with a mechanism for the graceful exit from inflation and the subsequent reheating phase.

Starobinsky's work motivated Mukhanov and Chibisov in 1981 to consider quantum fluctuations in this model (Mukhanov & Chibisov 1981). They made the remarkable observation that quantum fluctuations generated during the de Sitter expansion are approximately scale invariant with a logarithmic dependence on wave number, and "could have lead to formation of galaxies and galactic clusters" (quoted from the abstract of their paper). In current notation, Mukhanov & Chibisov (1981) show that $\Delta _{\cal R}(k) \propto(1+({1}/{2})\ln({aH}/{k})$ (see their Equation (9)), where H is the Hubble rate during inflation and k is the comoving wave number. In the super-horizon limit, k ≪ aH, the primordial tilt observed in the wave number range accessible to WMAP, k_WMAP, is given by

$$\begin{equation} n_s - 1 = \left.\frac{d\ln \Delta ^2_{\cal R}(k)}{d\ln k}\right|_{k=k_{\rm WMAP}} = -\frac{2}{\ln \frac{k_{\rm WMAP}}{aH}} = -\frac{2}{N}, \end{equation} \tag{ A1 }$$

where N is the number of inflationary e-folds between the epoch when k_WMAP left the horizon and the end of inflation. With N = 50, for example, one obtains n_s = 0.96, which is in agreement with our measurement.

Among the one-loop corrections considered by Starobinsky (1980), a term proportional to R² is found to be sufficient for driving inflation and creating curvature perturbations with the above spectrum; the other terms vanish when the metric is conformally flat (e.g., a flat Friedmann–Robertson–Walker metric). One can obtain this result from the standard slow-roll calculation. As first shown by Whitt (1984), an action containing R and R² is equivalent to an action containing R and a scalar field. To see this, start with

$$\begin{equation} I = \frac{1}{2}\int d^4x \sqrt{-g} (R+\alpha R^2 ), \end{equation} \tag{ A2 }$$

where we have set 8πG = 1. Perform the conformal transformation, $g_{\mu \nu } \rightarrow \hat{g}_{\mu \nu } =(1+2\alpha R)g_{\mu \nu }$, and introduce a canonically normalized scalar field, $\Psi = \sqrt{3/2} \ln (1+2\alpha R)$. Maeda (1988) showed that this system is described by a scalar field Ψ governed by a potential

$$\begin{equation} V(\Psi)=\frac{1}{8\alpha } (1-e^{-\sqrt{2/3}\Psi } )^2, \end{equation} \tag{ A3 }$$

which is quite flat for large Ψ (see also Barrow & Cotsakis 1988; Salopek et al. 1989).

Once V(Ψ) is specified, it is straightforward to compute the slow-roll parameters,

$$\begin{equation} \epsilon \equiv \frac{1}{2}\left(\frac{V^{\prime }}{V}\right)^2,\quad \eta \equiv \frac{V^{\prime \prime }}{V}, \end{equation} \tag{ A4 }$$

where prime denotes derivative with respect to Ψ. The number of inflationary e-folds that occur between the epoch when a given perturbation scale leaves the horizon, t, and the end of inflation, t_e, is, to the leading order in , given by

$$\begin{equation} N\equiv \int _{t}^{t_e} Hdt = \int _{\Psi _e}^{\Psi }\frac{d\Psi }{\sqrt{2\epsilon }}. \end{equation} \tag{ A5 }$$

For V(Ψ) as given in Equation (A3), with Ψ ≫ Ψ_e,

$$\begin{equation} \epsilon = \frac{4}{3}e^{-2\sqrt{2/3}\Psi }, \quad \eta = -\frac{4}{3}e^{-\sqrt{2/3}\Psi }, \quad N = \frac{3}{4}e^{\sqrt{2/3}\Psi }, \end{equation} \tag{ A6 }$$

or

$$\begin{equation} \epsilon = \frac{3}{4N^2}, \qquad \eta = -\frac{1}{N}, \end{equation} \tag{ A7 }$$

i.e., ≪ |η|. In terms of the slow-roll parameters, the primordial tilt is given by (Liddle & Lyth 2000)

$$\begin{equation} n_s-1 = -6\epsilon + 2\eta \simeq -\frac{2}{N}, \end{equation} \tag{ A8 }$$

in agreement with the original result of Mukhanov & Chibisov (1981).

Prior to inventing R² inflation, Starobinsky (1979) calculated the energy spectrum of long-wavelength gravitational waves produced during de Sitter expansion, assuming the Einstein–Hilbert action (also see Grishchuk 1975). Such long-wavelength gravitational waves induce temperature and polarization anisotropy in the CMB (Rubakov et al. 1982; Fabbri & Pollock 1983; Abbott & Wise 1984; Starobinsky 1985). Later, Starobinsky (1983) calculated the specific energy spectrum of gravitational waves from R² inflation (also see Mijić et al. 1986). This can most easily be expressed in terms of the slow-roll results, r = 16 (Liddle & Lyth 2000), giving

$$\begin{equation} r = \frac{12}{N^2}. \end{equation} \tag{ A9 }$$

Thus, while R² inflation predicts a tilted power spectrum, the tensor-to-scalar ratio is much smaller than ${\cal O}(1-n_s) = {\cal O}(1/N)$, as it is of order ${\cal O}(1/N^2)$.

The R² inflation predictions for n_s and r are exactly the same as those of a model based on a scalar field non-minimally coupled to the Ricci scalar (Spokoiny 1984; Accetta et al. 1985; Futamase & Maeda 1989; Salopek et al. 1989; Fakir & Unruh 1990),

$$\begin{equation} I = \frac{1}{2}\int d^4x\sqrt{-g} (1+\xi \phi ^2)R, \end{equation} \tag{ A10 }$$

where ξ = −1/6 for conformal coupling. For inflation occurring in a large-field regime, ξϕ² ≫ 1, with a scalar potential V(ϕ)∝ϕ⁴, the tilt is given by n_s − 1 = −2/N (Salopek 1992; Kaiser 1995), and the tensor-to-scalar ratio is given by (see Equation (5.1) of Komatsu & Futamase 1999)

$$\begin{equation} r=\frac{12}{N^2}\frac{1+6\xi }{6\xi }, \end{equation} \tag{ A11 }$$

which has r → 12/N² for ξ ≫ 1 (also see Hwang & Noh 1998).

This model has attracted renewed attention since Bezrukov & Shaposhnikov (2008) showed that the standard-model Higgs field can drive inflation if the Higgs field is non-minimally coupled to the Ricci scalar with ξ ≫ 1. The observable predictions of the original Higgs inflation are therefore n_s − 1 = −2/N and r = 12/N² (e.g., Bezrukov et al. 2009).