FIVE-YEAR WILKINSON MICROWAVE ANISOTROPY PROBE^* OBSERVATIONS: LIKELIHOODS AND PARAMETERS FROM THE WMAP DATA

We continue to evaluate the exact likelihood for the polarization maps at low multipole, ℓ ⩽ 23, as described in Appendix D of Page et al. (2007). The input maps and inverse covariance matrix used in the main analysis are produced by coadding the template-cleaned maps described by Gold et al. (2009). In both cases these are weighted to account for the P06 mask using the method described by Page et al. (2007). In the three-year analysis we conservatively used only the Q and V bands in the likelihood. We are now confident that the Ka band is cleaned sufficiently for inclusion in analyses (see Hinshaw et al. 2009 for a discussion).

We also cross-check the polarization likelihood by using polarization maps obtained with an alternative component-separation method, described by Dunkley et al. (2008). The low-resolution polarization maps in the K, Ka, Q, and V bands, degraded to HEALPix N_side = 8,¹⁵ are used to estimate a single set of marginalized Q and U CMB maps and associated inverse covariance matrix, that can then be used as inputs for the ℓ < 23 likelihood. This is done by estimating the joint posterior distribution for the amplitudes and spectral indices of the synchrotron, dust, and CMB Q and U components, using a Markov Chain Monte Carlo (MCMC) method. One then marginalizes over the foreground amplitudes and spectral indices to estimate the CMB component. A benefit of this method is that errors due to both instrument noise and foreground uncertainty are accounted for in the marginalized CMB covariance matrix.

For a given set of cosmological parameters with theoretical power spectrum C_ℓ, the likelihood function returns p(d|C_ℓ), the likelihood of the observed map d, or its transformed a_lm coefficients. Originally, the likelihood code was written as a hybrid combination of a normal and log-normal distribution (Verde et al. 2003). This algorithm did not properly model the tails of the likelihood at low multipoles (Efstathiou 2004; Slosar et al. 2004; O'Dwyer et al. 2004; Hinshaw et al. 2007), and so for the three-year data the ℓ ⩽ 30 likelihood was computed exactly, using

$$\begin{equation} p(\mathbf d|C_\ell)=\frac{\exp [(-(1/2)\mathbf d^T {\mathbf C}^{-1} \mathbf d]}{\sqrt{\det \mathbf C}}, \end{equation} \tag{ 1 }$$

where C is the covariance matrix of the data including both the signal covariance matrix and noise C(C_ℓ) = S(C_ℓ) + N (e.g., Tegmark 1997; Bond et al. 1998; Hinshaw et al. 2007). This approach is computationally intensive, however, since it requires the inversion of a large covariance matrix each time the likelihood is called.

In Jewell et al. (2004), Wandelt et al. (2004), and Eriksen et al. (2004a), a faster method was developed and implemented to compute p(d|C_ℓ), which we now adopt. It is described in detail in those papers, so we only briefly outline the method here. The method uses Gibbs sampling to first sample from the joint posterior distribution p(C_ℓ, s|d), where C_ℓ is the power spectrum and s is the true sky signal. From these samples, a Blackwell–Rao (BR) estimator provides a continuous approximation to p(C_ℓ|d). When a flat prior, p(C_ℓ) = const, is used in the sampling, we have p(C_ℓ|d) ∝ p(d|C_ℓ), where the constant of proportionality is independent of C_ℓ. The BR estimator can then be used as an accurate representation of the likelihood, p(d|C_ℓ) (Wandelt et al. 2004; Chu et al. 2005).

The first step requires drawing samples from p(C_ℓ, s|d). We cannot draw samples from the joint distribution directly, but we can from the two conditional distributions p(s|C_ℓ, d) and p(C_ℓ|s, d), each a slice through the (N_p × ℓ_max)-dimensional space. Samples are drawn alternately, forming a Markov Chain of points by the Gibbs algorithm. For the case of one C_ℓ parameter and one s parameter the sampling goes as follows. We start from some arbitrary point (Cⁱ_ℓ, sⁱ) in the parameter space, and then draw

$$\begin{equation} \big(C_\ell ^{i+1}, s^{i+1}\big), \big(C_\ell ^{i+2}, s^{i+2}\big),\ldots \end{equation} \tag{ 2 }$$

by first drawing C^{i + 1}_ℓ from p(C_ℓ|sⁱ, d) and then drawing s^{i + 1} from p(s|C^{i + 1}_ℓ, d). Then we iterate many times. The result is a Markov chain whose stationary distribution is p(C_ℓ, s|d). This is extended to vectors for C_ℓ (of length ℓ_max) and s (of length N_p) following the same method:

$$\begin{equation} \mathbf s^{i+1} \leftarrow p\big(\mathbf s^i\big|C_\ell ^i,\mathbf d\big) \end{equation} \tag{ 3 }$$

$$\begin{equation} C_\ell ^{i+1} \leftarrow p\big(C_\ell ^i\big|\mathbf s^{i+1},\mathbf d\big), \end{equation} \tag{ 4 }$$

sampling the joint distribution p(C_ℓ, s|d). This sampling procedure is also described by Jewell et al. (2004), Wandelt et al. (2004), and Eriksen et al. (2007).

The first conditional distribution is a multivariate Gaussian with mean Sⁱ(Sⁱ + N)⁻¹d and variance [(Sⁱ)⁻¹ + N⁻¹]⁻¹, so at each step a new signal vector s^{i + 1}, of size N_p, can be drawn. This is computationally demanding, however, as described by Eriksen et al. (2004a) and Wandelt et al. (2004), requiring the solution of a linear system of equations Mv = w, with M = 1 + S^1/2N⁻¹S^1/2. These are solved at each step using the conjugate gradient technique, which is sped up by finding an approximate inverse for M, a preconditioner. This requires computation of the inverse noise matrix, N⁻¹, in spherical harmonic space, which is done by computing the components of N⁻¹ term-by-term using spherical harmonics in pixel space. There are more efficient ways to compute N⁻¹ (Hivon et al. 2002; Eriksen et al. 2004b), but computing the preconditioner is a one-time expense, and it is only necessary to compute harmonics up to ℓ = 30.

The second conditional distribution, p(C_ℓ|s, d), is an inverse Gamma distribution, from which a new C_ℓ vector of size ℓ_max can be rapidly drawn using the method by Wandelt et al. (2004). Sampling from these two conditional distributions is continued in turn until convergence, at which point the sample accurately represents the underlying distribution. This is checked in practice using a jacknife test that compares likelihoods from two different samples.

Finally, once the joint distribution p(C_ℓ, s|d) has been pre-computed, the likelihood for any given model C_ℓ is obtained by marginalizing over the signal s, p(d|C_ℓ) ∝ ∫p(C_ℓ, s|d)ds, which holds for a uniform prior distribution p(C_ℓ). In practice this is computed using the BR estimator,

$$\begin{equation} p(\mathbf d|C_\ell) \propto \int p(C_\ell |\mathbf s)p(\mathbf s|\mathbf d) d\mathbf s \approx \frac{1}{n_{\rm G}} \sum _{i=1}^{n_{\rm G}} p(C_\ell |\mathbf s^i), \end{equation} \tag{ 5 }$$

where the sum is over all n_G samples in the Gibbs chain. Since p(C_ℓ|sⁱ) = p(C_ℓ|σⁱ_ℓ), where σ_ℓ = (2ℓ + 1)⁻¹∑_m|s_ℓm|², and s_ℓm are the spherical harmonic coefficients of s, one only needs to store σⁱ_ℓ at each step in the Gibbs sampling. Then, each time the likelihood is called for a new C_ℓ, one computes ${\cal L}=\sum _{i=1}^{n_{\rm G}} p\big(C_\ell \big|\sigma _\ell ^i\big)/n_G$. This requires only O(ℓ_maxn_G) computations, compared to the full O(N³_p) evaluation of Equation (1). This speed-up also means that the exact likelihood can be used to higher resolution than is feasible with the full evaluation, providing a more accurate estimation.

The code used for WMAP is adapted from the MAGIC Gibbs code described by Wandelt (2003) and Wandelt et al. (2004). The input temperature map is the five-year internal linear combination (ILC) map described by Gold et al. (2009). To produce correct results, the Gibbs sampler requires an accurate model of the data. This means that the signal covariance matrix S(C_ℓ) cannot be approximated to be zero except for multipoles where the smoothing makes the signal much less than the noise. For the full WMAP dataset, this would require sampling out to ℓ ∼ 1000, with N_side = 512. This is computationally expensive, taking more than of order 10⁴ processor hours to converge (O'Dwyer et al. 2004). Instead we reduce the resolution and smooth the data to substantially reduce the required multipole range, speeding up the computation.

The ILC map is smoothed to 5° FWHM, and sampled at N_side = 32. The process of smoothing the data has the side effect of correlating the noise. Properly modeling the correlated noise slows down the Gibbs sampling, as it takes longer to draw a sample from p(s|C_ℓ, d). We therefore add uncorrelated white noise to the map such that it dominates over the smoothed noise. However, the added noise must not be so large that it changes the likelihood of the low-ℓ modes; cosmic variance must remain dominant over the noise (Eriksen et al. 2007), so we add 2 μK of noise per pixel. In Appendix A the noise power spectra are shown.

The Gibbs sampler converges more slowly in regions of low signal-to-noise ratio (S/N). Because of this, we only sample spectra out to ℓ = 51 and fix the spectrum for 51 < ℓ ⩽ 96 to a fiducial value, and set it to zero for ℓ>96. The BR estimator is only used up to ℓ = 32 for cosmological analysis, so we marginalize over the 32 < ℓ ⩽ 51 spectrum. The likelihood is therefore p(L|d) = ∫p(L, H|d)dH, where L and H refer to the low, ℓ ⩽ 32, and higher, 32 < ℓ ⩽ 51, parts of the power spectrum. Examination of the BR estimator shows it to have a smooth distribution. Tests of the results to various input choices, including the choice of resolution, are given in Appendix A. We note that by using the low-ℓ likelihood only up to ℓ ⩽ 32, this breaks the likelihood into a low- and high-ℓ part, which introduces a small error by ignoring the correlation between these two parts of the spectrum. However, this error is small, and it is unfeasible, in a realistic sampling time, to use the BR estimator over the entire ℓ range probed by WMAP.

In Figure 1 we show slices through the C_ℓ distribution obtained from the BR estimator, compared to the pixel likelihood code. The estimated spectra agree well. Some small discrepancies are due to the pixel code using N_side = 16, compared to the higher resolution N_side = 32 used for the Gibbs code.

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Our observations of the acoustic peaks in the TT and TE spectrum imply that most of the ions and electrons in the universe combined to make neutral hydrogen and helium at z ≃ 1100. Observations of quasar spectra show diminishing Gunn–Peterson troughs at z < 5.8 (Fan et al. 2000, 2001) implying that the universe was nearly fully ionized by z = 5.7. How did the universe make the transition from being nearly fully neutral to fully ionized? The astrophysics of reionization has been a very active area of research in the past decade. Several recent reviews (Barkana & Loeb 2006; Fan et al. 2006; Furlanetto et al. 2006; Meiksin 2007) summarize the current observations and theoretical models. Here, we highlight a few of the important issues and discuss some of the implications of the WMAP measurements of optical depth.

What objects reionized the universe? While high-redshift galaxies are usually considered the most likely source of reionization, active galactic nuclei (AGNs) may also have played an important role. As galaxy surveys push toward ever higher redshift, it is unclear whether the known population of star-forming galaxies at z ∼ 6 could have ionized the universe (see, e.g. Bunker et al. 2007). The EE signal clearly seen in the WMAP five-year data (2008, Section 2) implies an optical depth, τ ≃ 0.09. This large optical depth suggests that higher redshift galaxies, perhaps the low-luminosity sources appearing in z > 7 surveys (Stark et al. 2007), played an important role in reionization. While the known population of AGNs cannot be a significant source of reionization (Bolton & Haehnelt 2007; Srbinovsky & Wyithe 2007), an early generation of supermassive black holes could have played a role in reionization (Ricotti & Ostriker 2004; Ricotti et al. 2007). This early reionization would also have an impact on the CMB.

Most of our observational constraints probe the end of the epoch of reionization. Observations of z > 6 quasars (Becker 2001; Djorgovski et al. 2001; Fan et al. 2006; Willott et al. 2007) find that the Lyα optical depth rises rapidly. Measurements of the afterglow spectrum of a gamma-ray burst at z = 6.3 (Totani et al. 2006) suggest that universe was mostly ionized at z = 6.3. Lyα emitter surveys (Taniguchi et al. 2005; Malhotra & Rhoads 2006; Kashikawa et al. 2006; Iye et al. 2006; Ota et al. 2008) imply a significant ionized fraction at z = 6.5. The interpretation that there is a sudden change in the properties of the intergalactic medium (IGM) remains a subject of active debate (Becker et al. 2007; Wyithe et al. 2008).

The WMAP data place new constraints on the reionization history of the universe. The WMAP data best constrains the optical depth due to reionization at moderate redshift (z < 25) and only indirectly constrains the redshift of reionization. If reionization is sudden, then the WMAP data imply that z_reion = 11.0 ± 1.4, as shown in Figure 5, and now excludes z_reion = 6 at more than 99.9% CL. The combination of the WMAP data implying that the universe was mostly reionized at z ∼ 11 and the measurements of rapidly rising optical depth at z ∼ 6–6.5 suggest that reionization was an extended process rather than a sudden transition. Many early studies of reionization envisioned a rapid transition from a neutral to a fully ionized universe occurring as ionized bubbles percolate and overlap. As Figure 5 shows, the WMAP data suggest a more gradual process with reionization beginning perhaps as early as z ∼ 20 and strongly favoring z > 6. This suggests that the universe underwent an extended period of partial reionization. The limits were found by modifying the ionization history in CAMB to include two steps in the ionization fraction at late times (z < 30): the first at z_r with ionization fraction x_e, the second at z = 7 with x_e = 1. Several studies (Cen 2003; Chiu et al. 2003; Wyithe & Loeb 2003; Haiman & Holder 2003; Yoshida et al. 2004; Choudhury & Ferrara 2006; Iliev et al. 2007; Wyithe et al. 2008) suggest that feedback produces a prolonged or perhaps even, multiepoch reionization history.

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While the current WMAP data constrain the optical depth of the universe, the EE data does not yet provide a detailed constraint on the reionization history. With more data from WMAP and upcoming data from Planck, the EE spectrum will begin to place stronger constraints on the details of reionization (Kaplinghat et al. 2003; Holder et al. 2003; Mortonson & Hu 2008). These measurements will be supplemented by measurements of the Ostriker–Vishniac effect by high-resolution CMB experiments which is sensitive to ∫n²_edt (Ostriker & Vishniac 1986; Jaffe & Kamionkowski 1998; Gruzinov & Hu 1998), and discussed in, e.g., Zhang et al. (2004).

As the E-mode signal is probed with higher accuracy, it becomes increasingly important to test how much the constraint on τ, z_reion, and the other cosmological parameters, depend on details of the Galactic foreground removal. Tests were done by Page et al. (2007) to show that τ was insensitive to a set of variations in the dust template used to clean the maps. In Figure 6 we show the effect on ΛCDM parameters of changing the number of bands used in the template-cleaning method: discarding the Ka band in the "QV" combination, or adding the W band in the "KaQVW" combination. We find that τ (and therefore z_reion) is sensitive to the maps, but the dispersion is consistent with noise. As expected, the error bars are broadened for the QV-combined data, and the mean value is τ = 0.080 ± 0.020. When the W band is included, the mean value is τ = 0.100 ± 0.015. We choose not to use the W-band map in our main analysis, however, because there appears to be excess power in the cleaned map at ℓ = 7. This indicates a potential systematic error, and is discussed further by Hinshaw et al. (2009). The other cosmological parameters are only mildly sensitive to the number of bands used. This highlights the fact that τ is no longer as strongly correlated with other parameters, as in earlier WMAP data (Spergel et al. 2003, 2007), notably with the spectral index of primordial fluctuations, n_s (Figure 3).

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We also test the parameters obtained using the Gibbs-cleaned maps described by Dunkley et al. (2008) and in Section 2.1.1, which use the K, Ka, Q, and V band maps. Their mean distributions are also shown in Figure 6, and have mean optical depth less than 1σ higher than the KaQV template-cleaned maps, obtained using an independent method. The other cosmological parameters are changed by less than 0.3σ compared to the template-cleaned results. This consistency gives us confidence that the parameter constraints are little affected by foreground uncertainty. The difference in central values from the two methods indicates an error due to foreground removal uncertainty on τ, in addition to the statistical error, of ∼0.01.

The likelihood code used for cosmological analysis has a number of variable components that have been fixed using our best estimates. Here we consider the effect of these choices on the five-year ΛCDM parameters. The first two are the treatment of the residual point sources, and the treatment of the beam error, both discussed by Nolta et al. (2009). The multifrequency data are used to estimate a residual point source amplitude of A_ps = 0.011 ± 0.001 μK²sr, which scales the expected contribution to the cross-power spectra of sources below our detection threshold. It is defined by Hinshaw et al. (2007) and Nolta et al. (2009), and is marginalized over in the likelihood code. The estimate comes from QVW data, whereas the VW data give 0.007 ± 0.003 μK²sr, both using the KQ85 mask described by Hinshaw et al. (2009). The right panels in Figure 6 shows the effect on a subset of parameters of lowering A_ps to the VW value, which leads to a slightly higher n_s, Ω_ch², and σ₈, all within 0.4σ of the fiducial values, and consistent with more of the observed high-ℓ signal being due to CMB rather than unresolved point sources. We also use A_ps = 0.011 μK²sr with no point source error, and find a negligible effect on parameters (<0.1σ). The beam window function error is quantified by 10 modes, and in the standard treatment we marginalize over them, following the prescription by Hinshaw et al. (2007). We find that removing the beam error also has a negligible effect on parameters. This is discussed further by Nolta et al. (2009), who consider alternative treatments of the beam and point source errors.

The next issue is the treatment of a possible contribution from SZ fluctuations. We account for the SZ effect in the same way as in the three-year analysis, marginalizing over the amplitude of the contribution parameterized by the Komatsu–Seljak model (Komatsu & Seljak 2002). The parameter A_SZ is unconstrained by the WMAP data, but is not strongly degenerate with any other parameters. In Figure 6 we show the effect on parameters of setting the SZ contribution to zero. Similar to the effect of changing the point source contribution, the parameters depending on the third peak are slightly affected, with a <0.25σ increase in n_s, Ω_ch², σ₈, and a similar decrease in the baryon density.

Another choice is the area of sky used for cosmological interpretation, or how much we mask out to account for Galactic contamination. Gold et al. (2009) discuss the new masks used for the five-year analysis, with the KQ85 mask used as standard. We test the effect of using the more conservative KQ80 mask, and find a more noticeable shift. The quantity Ω_bh² is increased by 0.5σ, and n_s, Ω_ch², and σ₈ all decreased by ∼0.4σ. This raised concerns that the KQ85 mask contains residual foreground contamination, but as discussed by Nolta et al. (2009), this shift is found to be consistent with the effects of noise, tested with simulations. We also confirm that the effect on parameters is even less for ΛCDM models using WMAP with external data, and that the choice of mask has only a small effect on the tensor amplitude, raising the 95% confidence level by ∼5%.

Finally, we test the effect on parameters of varying aspects of the low-ℓ TT treatment. These are discussed in Appendix B, and in summary we find the same parameter results for the pixel-based likelihood code compared to the Gibbs code, when both use ℓ ⩽ 32. Changing the mask at low-ℓ to KQ80, or using the Gibbs code up to ℓ ⩽ 51, instead of ℓ ⩽ 32, has a negligible effect on parameters.

A number of recent CMB experiments have probed smaller angular scales than WMAP can reach and are therefore more sensitive to the higher order acoustic oscillations and the details of recombination (e.g., Hu & Sugiyama 1994, 1996; Hu & White 1997). Since the three-year WMAP analysis, there have been new temperature results from the Arcminute Cosmology Bolometer Array Receiver (ACBAR), both in 2007 (Kuo et al. 2007) and in 2008 (Reichardt et al. 2008). They have measured the angular power spectrum at 145 GHz to 5' resolution, over ∼ 600 deg². Their results are consistent with the model predicted by the WMAP five-year data, shown in Figure 7, although ACBAR is calibrated using WMAP, so the data are not completely independent.

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Figure 7 also shows data from the BOOMERANG, Cosmic Background Imager (CBI), and Very Small Array (VSA) experiments, which agree well with WMAP. There have also been new observations of the CMB polarization from two ground-based experiments, QUaD, operating at 100 and 150 GHz (Ade et al. 2008), and CAPMAP, at 40 and 100 GHz (Bischoff et al. 2008). Their measurements of the EE power spectrum are shown by Nolta et al. (2009), together with detections already made since 2005 (Leitch et al. 2005; Sievers et al. 2007; Barkats et al. 2005; Montroy et al. 2006), and are all consistent with the ΛCDM model parameters.

In our combined analysis in Komatsu et al. (2009) we use two different data combinations. For the first we combine four datasets. This includes the 2007 ACBAR data (Kuo et al. 2007), using 10 bandpowers in the range 900 < ℓ < 2000. The values and errors were obtained from the ACBAR Web site. We also include the three external CMB datasets used in Spergel et al. (2003): the CBI (Mason et al. 2003; Sievers et al. 2003; Pearson et al. 2003; Readhead et al. 2004), the VSA (Dickinson et al. 2004), and BOOMERANG (Ruhl et al. 2003; Montroy et al. 2006; Piacentini et al. 2006). As in the three-year release we only use bandpowers that do not overlap with the signal-dominated WMAP data, due to nontrivial cross-correlations, so we use seven bandpowers for CBI (in the range 948 < ℓ < 1739), five for VSA (894 < ℓ < 1407), and seven for BOOMERANG (924 < ℓ < 1370), using the log-normal form of the likelihood. Constraints are also found by combining WMAP with the 2008 ACBAR data, using 16 bandpowers in the range 900 < ℓ < 2000. In this case the other CMB experiments are not included. We do not use additional polarization results for parameter constraints as they do not yet improve limits beyond WMAP alone.

The acoustic peak in the galaxy correlation function is a prediction of the adiabatic cosmological model (Peebles & Yu 1970; Sunyaev & Zel'dovich 1970; Bond & Efstathiou 1984; Hu & Sugiyama 1996). It was first detected using the Sloan Digital Sky Survey (SDSS) luminous red galaxy (LRG) survey, using the brightest class of galaxies at mean redshift z = 0.35 by Eisenstein et al. (2005). The peak was detected at 100 h⁻¹ Mpc separation, providing a standard ruler to measure the ratio of distances to z = 0.35 and the CMB at z = 1089, and the absolute distance to z = 0.35. More recently, Percival et al. (2007b) have obtained a stronger detection from over half a million SDSS main galaxies and LRGs in the DR5 sample. They detect BAO with over 99% confidence. A combined analysis was then undertaken of SDSS and Two Degree Field Galaxy Redshift Survey (2dFGRS) by Percival et al. (2007a). They find evidence for BAO in three catalogs: at mean redshift z = 0.2 in the SDSS DR5 main galaxies plus the 2dFGRS galaxies, at z = 0.35 in the SDSS LRGs, and in the combined catalog. Their data are shown in Figure 8, together with the WMAP best-fit model. The BAO are shown by dividing the observed and model power spectra by P(k)_smooth, a smooth cubic spline fit described by Percival et al. (2007a). The observed power spectra are model-dependent, but were calculated using Ω_m = 0.25 and h = 0.72, which agrees with our maximum-likelihood model.

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The scale of the BAO is analyzed to estimate the geometrical distance measure at z = 0.2 and z = 0.35,

$$\begin{equation} D_V(z)=\big[(1+z)^2D_A^2cz\big/H(z)\big]^{1/3}, \end{equation} \tag{ 9 }$$

where D_A is the angular diameter distance and H(z) is the Hubble parameter. They find r_s/D_V(0.2) = 0.1980 ± 0.0058 and r_s/D_V(0.35) = 0.1094 ± 0.0033. Here, r_s is the comoving sound horizon scale at recombination. Our ΛCDM model, using the WMAP data alone, gives r_s/D_V(0.2) = 0.1946 ± 0.0079 and r_s/D_V(0.35) = 0.1165 ± 0.0042, showing the consistency between the CMB measurement at z = 1089 and the late-time galaxy clustering. However, while the z = 0.2 measures agree to within 1σ, the z = 0.35 measurements have mean values almost 2σ apart. The BAO constraints are tighter than the WMAP predictions, which shows that they can improve upon the WMAP parameter determinations, in particular on Ω_Λ and Ω_mh².

In Komatsu et al. (2009) the combined bounds from both surveys are used to constrain models as described by Percival et al. (2007a), adding a likelihood term given by −2ln L = X^TC⁻¹X, with

$$\begin{equation} X^T= [r_s/D_V(0.2)-0.1980, r_s/D_V(0.35)-0.1094]\\ \end{equation} \tag{ 10 }$$

and C¹¹ = 35, 059, C¹² = −24, 031, C²² = 108, 300, including the correlation between the two measurements. Komatsu et al. (2009) also consider constraints using the SDSS LRG limits derived by Eisenstein et al. (2005), using the combination

$$\begin{equation} A(z)=D_V(z)\sqrt{\Omega _mH_0^2}\big/cz \end{equation} \tag{ 11 }$$

for z = 0.35 and computing a Gaussian likelihood

$$\begin{equation} {-}2\ln L=(A-0.469(n_s/0.98)^{-0.35})^2/0.017^2. \end{equation} \tag{ 12 }$$

We can compare the predicted fluctuations from the CMB to the shape of galaxy power spectra, in addition to the scale of acoustic oscillations (e.g., Eisenstein & Hu 1998). The SDSS galaxy power spectrum from DR3 (Tegmark et al. 2004) and the 2dFGRS spectrum (Cole et al. 2005) were shown to be in good agreement with the WMAP three-year data, and used to place tighter constraints on cosmological models (Spergel et al. 2007), but there was some tension between the preferred values of the matter density (Ω_m = 0.236 ± 0.020 for WMAP combined with 2dFGRS and 0.265 ± 0.030 with SDSS). Recent studies used photometric redshifts to estimate the galaxy power spectrum of LRGs in the range 0.2 < z < 0.6 from the SDSS fourth data release (DR4), finding Ω_m = 0.30 ± 0.03 (for h = 70, Padmanabhan et al. 2007) and Ω_mh = 0.195 ± 0.023 for h = 0.75 (Blake et al. 2007).

More precise measurements of the LRG power spectrum were obtained from redshift measurements: Tegmark et al. (2006) used LRGs from SDSS DR4 in the range 0.01 h Mpc⁻¹ < k < 0.2 h Mpc⁻¹ combined with the three-year WMAP data to place strong constraints on cosmological models. However, there is a disagreement between the matter density predicted using different minimum scales, if the nonlinear modeling used by Tegmark et al. (2006) is adopted. Using the three-year WMAP data combined with the LRG spectrum we find Ω_m = 0.228 ± 0.019, using scales with k < 0.1 h Mpc⁻¹, and Ω_m = 0.248 ± 0.018 for k < 0.2 h Mpc⁻¹. These constraints are obtained for the six-parameter ΛCDM model, following the nonlinear prescription described by Tegmark et al. (2006). The predicted galaxy power spectrum P_g(k) is constructed from the "dewiggled" linear matter power spectrum P_m(k) using P_g(k) = b²P_m(k)(1 + Qk²)/(1 + 1.4k), and the parameters b and Q are marginalized over. The dewiggled spectrum is a weighted sum of the matter power spectrum at z = 0 and a smooth power spectrum (Tegmark et al. 2006). Without this dewiggling, we find Ω_m = 0.246 ± 0.018 for k < 0.2 h Mpc⁻¹, so its effect is small. We also explored an alternative form for the bias, motivated by third-order perturbation theory analysis, with P_g(k) = b²P^nl_m(k) + N (see, e.g. Seljak 2001; Schulz & White 2006). Here, P^nl_m is the nonlinear matter power spectrum using the Halofit model (Smith et al. 2003) and N is a constant accounting for nonlinear evolution and scale-dependent bias. Marginalizing over b and N we still find a discrepancy with scale, with Ω_m = 0.230 ± 0.021 using scales with k < 0.1 h Mpc⁻¹, and Ω_m = 0.249 ± 0.018 for k < 0.2 h Mpc⁻¹. Constraints using this bias model are also considered in Hamann et al. (2008a). These findings are consistent with results obtained from the DR5 main galaxy and LRG sample (Percival et al. 2007c), who argue that this shows evidence for scale-dependent bias on large scales, which could explain the observed differences in the early SDSS and 2dFGRS results. While this will likely be resolved with future modeling and observations, we choose not to use the galaxy power spectra to place joint constraints on the models reported by Komatsu et al. (2009).

In the last decade, Type Ia supernovae have become an important cosmological probe, and have provided the first direct evidence for the acceleration of the universe by measuring the luminosity distance as a function of redshift. The observed dimness of high-redshift supernovae (z ∼ 0.5) was first measured by Riess et al. (1998), Schmidt et al. (1998), and Perlmutter et al. (1999), confirmed with more recent measurements including those of Nobili et al. (2005), Krisciunas et al. (2005), Clocchiatti et al. (2006), and Astier et al. (2006), and extended to higher redshift by Riess et al. (2004) who found evidence for the earlier deceleration of the universe. The sample of high-redshift supernovae has grown by over 80 since the three-year WMAP analysis. Recent HST measurements of 21 new high-redshift supernova by Riess et al. (2007) include 13 at z > 1, allowing the measurement of the Hubble expansion H(z) at distinct epochs and strengthening the evidence for a period of deceleration followed by acceleration. The ESSENCE Supernova Survey has also recently reported results from 102 supernovae discovered from 2002 to 2005 using the 4-m Blanco Telescope at the Cerro Tololo Inter-American Observatory (Miknaitis et al. 2007), of which 60 are used for cosmological analysis (Wood-Vasey et al. 2007). A combined cosmological analysis was performed for a subset of the complete supernova dataset by Davis et al. (2007) using the MCLS2k2 light curve fitter (Jha et al. 2007). More recently, a broader "Union" compilation of the currently observed SNIa, including a new nearby sample, was analyzed by Kowalski et al. (2008) using the SALT1 light curve fitter (Guy et al. 2005).

In Figure 9, we confirm that the recently observed supernovae are consistent with the ΛCDM model, which predicts the luminosity distance μ_th as a function of redshift and is compared to the Union combined dataset (Kowalski et al. 2008). In the cosmological analysis described by Komatsu et al. (2009), the Union data is used, consisting of 307 supernovae that pass various selection cuts. These include supernovae observed using the HST (Riess et al. 2007), from the ESSENCE survey (Miknaitis et al. 2007), and the Supernova Legacy Survey (SNLS; Astier et al. 2006). For each supernova the luminosity distance predicted from theory is compared to the observed value. This is derived from measurements of the apparent magnitude m and the inferred absolute magnitude M, to estimate a luminosity distance μ_obs = 5log[d_L(z)/Mpc] + 25. The likelihood is given by

$$\begin{equation} -2\ln L=\sum _i[\mu _{{\rm obs},i}(z_i)-\mu _{th,i}(z_i,M_0)]^2\big/\sigma ^2_{{\rm obs},i} \end{equation} \tag{ 13 }$$

summed over all supernovae, where a single absolute magnitude is marginalized over (Lewis & Bridle 2002), and σ_obs is the observational error accounting for extinction, intrinsic redshift dispersion, K-correction, and light curve stretch factors.

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The WMAP estimated value of the Hubble constant, H₀ = 71.9^+2.6_−2.7 km s⁻¹ Mpc⁻¹, assuming a flat geometry, is consistent with the HST Key Project measurement of 72 ± 8 km s⁻¹ Mpc⁻¹ (Freedman et al. 2001). It also agrees within 1σ with measurements from gravitationally lensed systems (Koopmans et al. 2003), SZ and X-ray observations (Bonamente et al. 2006), Cepheid distances to nearby galaxies (Riess et al. 2005), the distance to the Maser-host galaxy NGC4258 as a calibrator for the Cepheid distance scale (Macri et al. 2006), and a new measure of the Tully–Fisher zero point (Masters et al. 2006), the latter two giving H₀ = 74 ± 7 km s⁻¹ Mpc⁻¹. Lower measures are favored by a compilation of the Cepheid distance measurements for 10 galaxies using the HST by Sandage et al. (2006; H₀ = 62 ± 6 km s⁻¹ Mpc⁻¹), and new measurements of an eclipsing binary in M33 which reduce the Key Project measurement to H₀ = 61 km s⁻¹ Mpc⁻¹ (Bonanos et al. 2006). Higher measures are found using revised parallaxes for Cepheids (van Leeuwen et al. 2007), raising the Key Project value to 76 ± 8 km s⁻¹ Mpc⁻¹. In Komatsu et al. (2009) the Hubble Constant measurements are included for a limited set of parameter constraints, using the Freedman et al. (2001) measurement as a Gaussian prior on H₀.

Weak gravitational lensing is produced by the distortion of galaxy images by the mass distribution along the line of sight (see Refregier 2003 for a review). There have been significant advances in its measurement in recent years, and in the understanding of systematic effects (e.g., Massey et al. 2007), and intrinsic alignment effects (Hirata et al. 2007), making it a valuable cosmological probe complementary to the CMB. Early results by the RCS (Hoekstra et al. 2002), VIRMOS-DESCART (Van Waerbeke et al. 2005), and the Canada–France–Hawaii Telescope Legacy Survey (CFHTLS; Hoekstra et al. 2006) lensing surveys favored higher amplitudes of mass fluctuations than preferred by WMAP. However, new measurements of the two-point correlation functions from the third year CFHTLS Wide survey (Fu et al. 2008) favor a lower amplitude consistent with the WMAP measurements, as shown in Table 3. This is due to an improved estimate of the galaxy redshift distribution from CFHTLS-Deep (Ilbert et al. 2006), compared to that obtained from photometric redshifts from the small Hubble Deep Field, which were dominated by systematic errors. Their measured signal agrees with results from the 100 Square Degree Survey (Benjamin et al. 2007), a compilation of data from the earlier CFHTLS-Wide, RCS, and VIRMOS-DESCAT surveys, together with the GABoDS survey (Hetterscheidt et al. 2007), with average source redshift z ∼ 0.8. Both these analyses rely on a two-dimensional measurement of the shear field. Cosmic shear has also been measured in two and three dimensions by the HST COSMOS survey (Massey et al. 2007), using redshift information to providing an improved measure of the mass fluctuation. Their measures are somewhat higher than the WMAP value, as shown in Table 3, although not inconsistent. Weak lensing is also produced by the distortion of the CMB by the intervening mass distribution (Zaldarriaga & Seljak 1999; Hu & Okamoto 2002), and can be probed by measuring the correlation of the lensed CMB with tracers of large-scale structure. Two recent analyses have found evidence for the cross-correlated signal (Smith et al. 2007; Hirata et al. 2008), both consistent with the five-year WMAP ΛCDM model. They find a 3.4σ detection of the correlation between the three-year WMAP data and NRAO VLA Sky Survey (NVSS) radio sources (Smith et al. 2007), and a correlation at the 2.1σ level of significance between WMAP and data from NVSS, and from SDSS LRGs and quasars (Hirata et al. 2008).

Correlation between large-scale CMB temperature fluctuations and large-scale structure is expected in the ΛCDM model due to the change in gravitational potential as a function of time, and so provides a test for dark energy (Boughn et al. 1998). Evidence of a correlation was found in the first-year WMAP data (e.g., Boughn & Crittenden 2004; Nolta et al. 2004). Two recent analyses combine recent large-scale structure data (Two Micron All Sky Survey, SDSS LRGs, SDSS quasars, and NVSS radio sources) with the WMAP three-year data, finding a 3.7σ (Ho et al. 2008) and 4σ (Giannantonio et al. 2008) detection of ISW at the expected level. Other recent studies using individual datasets find a correlation at the level expected with the SDSS DR4 galaxies (Cabré et al. 2006), at high redshift with SDSS qusars (Giannantonio et al. 2006), and with the NVSS radio galaxies (Pietrobon et al. 2006; McEwen et al. 2007).

The Lyα forest seen in quasar spectra probes the underlying matter distribution on small scales (Rauch 1998). However, the relationship between absorption line structure and mass fluctuations must be fully understood to be used in a cosmological analysis. The power spectrum of the Lyα forest has been used to constrain the shape and amplitude of the primordial power spectrum (Viel et al. 2004; McDonald et al. 2005; Seljak et al. 2005; Desjacques & Nusser 2005), and recent results combine the three-year WMAP data with the power spectrum obtained from the LUQAS sample of VLT-UVES spectra (Viel et al. 2006) and SDSS QSO spectra (Seljak et al. 2006). Both groups found results suggesting a higher value for σ₈ than consistent with WMAP. However, measurements by Kim et al. (2007) of the probability distribution of the Lyα flux have been compared to simulations with different cosmological parameters and thermal histories (Bolton et al. 2008). They imply that the temperature–density relation for the IGM may be close to isothermal or inverted, which would result in a smaller amplitude for the power spectrum than previously inferred, more in line with the five-year WMAP value of σ₈ = 0.796 ± 0.036. Simulations in larger boxes by Tytler et al. (2007) fail to match the distribution of flux in observed spectra, providing further evidence of disagreement between simulation and observation. Given these uncertainties, the Lyα forest data are not used for the main results presented by Komatsu et al. (2009). However, constraints on the running of the spectral index are discussed, using data described by Seljak et al. (2006). With more data and further analyses, the Lyα forest measurements can potentially place powerful constraints on the neutrino mass and a running spectral index.

WMAP measures the baryon abundance at decoupling, with Ω_bh² = 0.02273 ± 0.00062, giving a baryon-to-photon ratio of η₁₀(WMAP) = 6.225 ± 0.170. Element abundances of deuterium, helium, and lithium also depend on the baryon abundance in the first few minutes after the big bang. Steigman (2007) reviews the current status of BBN measurements. Deuterium measurements provide the strongest test, and are consistent with WMAP, giving η₁₀(D) = 6.0 ± 0.4 based on new measurements by O'Meara et al. (2006). The ³He abundance is more poorly constrained at η₁₀(³He) = 5.6^+2.2_−1.4 from the measure of y₃ = 1.1 ± 0.2 by Bania et al. (2002). The neutral lithium abundance, measured in low-metallicity stars, is two times smaller than the CMB prediction, η(Li), from measures of the logarithmic abundance, [Li]_P = 12 + log₁₀(Li/H), in the range [Li]_P ∼ 2.1–2.4 (Charbonnel & Primas 2005; Meléndez & Ramírez 2004; Asplund et al. 2005). These measurements could be a signature of new early universe physics, e.g., Coc et al. (2004), Richard et al. (2005), and Jedamzik (2004), with recent attempts to simultaneously fit both the ⁷Li and ⁶Li abundances by Bird et al. (2008), Jedamzik (2008a, 2008b), Cumberbatch et al. (2007), and Jittoh et al. (2008). The discrepancy could also be due to systematics, destruction of lithium in an earlier generation of stars, or uncertainties in the stellar temperature scale (Fields et al. 2005; Steigman 2006; Asplund et al. 2005). A possible solution has been proposed using observations of stars in the globular cluster NGC 6397 (Korn et al. 2006, 2007). They find evidence that as the stars age and evolve toward hotter surface temperatures, the surface abundance of lithium in their atmospheres drops due to atomic diffusion. They infer an initial lithium content of the stars [Li]_P = 2.54 ± 0.1, giving η₁₀(Li) = 5.4 ± 0.6 in good agreement with BBN predictions. However, further observations are needed to determine whether this scenario explains the observed uniform depletion of primordial ⁷Li as a function of metallicity. The measured abundance of ⁴He is also lower than predicted, with η₁₀(⁴He) = 2.7^+1.2_−0.9 from a measure of Y_P = 0.240 ± 0.006, incorporating data from Izotov & Thuan (2004), Olive & Skillman (2004), and Gruenwald et al. (2002) by Steigman (2007). However, observations by Peimbert et al. (2007) and Fukugita & Kawasaki (2006) predict higher values more consistent with WMAP.

The number of strongly lensed quasars has the potential to probe cosmology, as a dark-energy-dominated universe predicts a large number of gravitational lenses (Turner 1990; Fukugita et al. 1990). The CLASS radio band survey has a large statistical sample of radio lenses (Myers et al. 2003; Koopmans et al. 2003; York et al. 2005), yielding estimates for Ω_Λ ≃ 0.72–0.78 (Mitchell et al. 2005; Chae 2007). Oguri et al. (2008) have recently analyzed the large statistical lens sample from the Sloan Digital Sky Quasar Lens Search (Oguri et al. 2006). For w = −1, flat cosmology, they find Ω_Λ = 0.74^+0.11_−0.15(stat.)^+0.13_−0.06(syst.). These values are all consistent with our best-fit cosmology. The abundance of giant arcs also has the potential to probe the underlying cosmology. However, although recent surveys have detected larger numbers of giant arcs (Gladders et al. 2003; Sand et al. 2005; Hennawi et al. 2008) than argued to be consistent with ΛCDM (Li et al. 2006; Broadhurst & Barkana 2008), numerical simulations (Meneghetti et al. 2007; Hennawi et al. 2007; Hilbert et al. 2007; Neto et al. 2007) find that the lens cross-sections are sensitive to the mass distribution in clusters as well as to the baryon physics (Wambsganss et al. 2007; Hilbert et al. 2008). These effects must be resolved in order to test ΛCDM consistency.

Clusters are easily detected and probe the high-mass end of the mass distribution, so probe the amplitude of density fluctuations and of large-scale structure. Cluster observations at optical wavelengths provided some of the first evidence for a low-density universe with the current preferred cosmological parameters (see, e.g., Fan et al. 1997). Observers are now using a number of different techniques for identifying cluster samples: large optical samples, X-ray surveys, lensing surveys (see, e.g., Wittman et al. 2006), and SZ surveys. The ongoing challenge is to determine the selection function and the relationship between astronomical observables and mass. This has recently seen significant progress in the optical (Lin et al. 2006; Sheldon et al. 2007; Reyes et al. 2008; Rykoff et al. 2008) and the X-ray (Sheldon et al. 2001; Reiprich & Böhringer 2002; Kravtsov et al. 2006; Arnaud et al. 2007; Hoekstra 2007). With large new optical cluster samples (Bahcall et al. 2003; Hsieh et al. 2005; Miller et al. 2005; Koester et al. 2007) and X-ray samples from ROSAT, XMM-LSS, and Chandra (Pierre et al. 2006; Burenin et al. 2007; Vikhlinin et al. 2008), cosmological parameters can be further tested, and most recent results for σ₈ are converging on values close to the WMAP best-fit value of σ₈ = 0.796 ± 0.036. Constraints from the RCS survey (Gladders et al. 2007) give Ω_m = 0.30^+0.12_−0.11 and σ₈ = 0.70^+0.27_−0.1. Rozo et al. (2007) argue for σ₈ > 0.76 (95% CL) from SDSS BCG samples. Mantz et al. (2008) find Ω_m = 0.27^+0.06_−0.05 and σ₈ = 0.77^+0.07_−0.06 for a flat model based on the Jenkins et al. (2001) mass function and the Reiprich & Böhringer (2002) mass–luminosity calibration. With a 30% higher zero point, Rykoff et al. (2008) find that their data are best fit by σ₈ = 0.85 and Ω_m = 0.24.  Bergé et al. (2007) report σ₈ = 0.92^+0.26_−0.30 for Ω_m = 0.24 from their joint CFHTLS/XMM-LSS analysis. In Rines et al. (2007) redshift data from SDSS DR4 is used to measure virial masses for a large sample of X-ray clusters. For Ω_m = 0.3, they find σ₈ = 0.84 ± 0.03.

With deep large-scale structure surveys, cosmologists have now been able to measure β, the amplitude of redshift space distortions as a function of redshift. Combined with a measurement of the bias, b, this yields a determination of the growth rate of structure f ≡ dln G/dln a = βb, where G is the growth factor. For Einstein gravity theories we expect f ≃ Ω^γ with γ ≃ 6/11 (see Polarski & Gannouji 2008 for a more accurate fitting function). Analysis of redshift space distortions in the 2dFGRS (Peacock et al. 2001; Verde et al. 2002; Hawkins et al. 2003) find β = 0.47 ± 0.08 at z ≃ 0.1, consistent with the ΛCDM predictions for the best-fit WMAP parameters. The Tegmark et al. (2006) analysis of the SDSS LRG sample finds β = 0.31 ± 0.04 at z = 0.35. The Ross et al. (2007) analysis of the 2dF-SDSS LRG sample find β = 0.45 ± 0.05 at z = 0.55. The Guzzo et al. (2008) analysis of 10,000 galaxies in the VIMOS-VLT Deep Survey finds that β = 0.70 ± 0.26 at z = 0.8 and infer dln G/dln a = 0.91 ± 0.36, consistent with the more rapid growth due to matter domination at this epoch expected in a ΛCDM model. The da Ângela et al. (2008) analysis of a QSO sample finds β = 0.60^+0.14_−0.11 at z = 1.4 and use the clustering length to infer the bias. Extrapolating back to z = 0, they find a matter density of Ω_m = 0.25^+0.09_−0.07. A second approach is to use objects with well-determined distances, such as galaxies and supernovae, to look for deviations from the Hubble flow (Strauss & Willick 1995; Dekel 2000; Zaroubi et al. 2001; Riess et al. 1995; Haugbølle et al. 2007). The Park & Park (2006) analysis of the peculiar velocities of galaxies in the SFI sample find σ₈Ω^0.6_m = 0.56^+0.27_−0.21. Gordon et al. (2007) correlate peculiar velocities of nearby supernova and find σ₈ = 0.79 ± 0.22. These measurements provide an independent consistency check of the ΛCDM model (see Nesseris & Perivolaropoulos 2008 for a recent review).

In the ΛCDM model, primordial scalar fluctuations are adiabatic and Gaussian, and can be described by a power-law spectrum,

$$\begin{equation} \Delta _{\cal R}^2(k) \propto \left(\frac{k}{k_0}\right)^{n_s-1}, \end{equation} \tag{ 14 }$$

producing CMB angular power spectra consistent with the data. Limits can also be placed on the amplitude of tensor fluctuations, or gravitational waves, that could have been generated at very early times. They leave a distinctive large-scale signature in the polarized B-mode of the CMB (e.g., Basko & Polnarev 1980; Bond & Efstathiou 1984) that provides a clean way to distinguish them from scalar fluctuations. However, we have not yet reached sensitivities to strongly constrain this signal with the polarization data from WMAP. Instead we use the tensor contribution to the temperature fluctuations at large scales to constrain the tensor-to-scalar ratio r. We define $r=\Delta _h^2(k_0)\big/\Delta _{\cal R}^2(k_0)$, where Δ²_h is the amplitude of primordial gravitational waves (see Komatsu et al. 2009), and choose a pivot scale k₀ = 0.002 Mpc⁻¹.

The WMAP data now constrain r < 0.43(95%CL). This is an improvement over the three-year limit of r < 0.65 (95% CL), and comes from the more accurate measurement of the second and third acoustic peaks. The dependence of the tensor amplitude on the spectral index is shown in Figure 10, showing the n_s − r degeneracy (Spergel et al. 2007): a larger contribution from tensors at large scales can be offset by an increased spectral index, and an overall decrease in the amplitude of fluctuations, shown in Table 4. The degeneracy is partially broken with a better measure of the TT spectrum. There is a significant improvement in the limit on models whose scalar fluctuations can vary with scale, with a power spectrum with a "running" spectral index,

$$\begin{equation} n_s(k)=n_s(k_0)+ \frac{1}{2}\frac{dn_s}{d\ln k} \ln \left(\frac{k}{k_0}\right). \end{equation} \tag{ 15 }$$

The limit from WMAP is now r < 0.58(95%CL), about half the three-year value r < 1.1 (Spergel et al. 2007).

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What do these limits tell us about the early universe? For models that predict observable gravitational waves, it allows us to exclude more of the parameter space. The simplest inflationary models predict a nearly scale-invariant spectrum of gravitational waves (Grishchuk 1975; Starobinsky 1979). In a simple classical scenario where inflation is driven by the potential V(ϕ) of a slowly rolling scalar field, the predictions (Lyth & Riotto 1999) are

$$\begin{equation} r \simeq 4\alpha /N, \end{equation} \tag{ 16 }$$

$$\begin{equation} 1-n_s \simeq (\alpha +2)/2N, \end{equation} \tag{ 17 }$$

for V(ϕ) ∝ ϕ^α, where N is the number of e-folds of inflation between the time when the horizon scale modes left the horizon and the end of inflation. For N = 60, the λϕ⁴ model with r ≃ 0.27, n_s ≃ 0.95 is now excluded with more than 95% confidence. An m²ϕ² model with r ≃ 0.13, n_s ≃ 0.97 is still consistent with the data. Komatsu et al. (2009) discuss in some detail what these measurements, and constraints for combined datasets, imply for a large set of possible inflationary models and potentials.

With r = 0 also fitting the data well, models that do not predict an observable level of gravitational waves, including multi-field inflationary models (Polarski & Starobinsky 1995; Garcia-Bellido & Wands 1996), D-brane inflation (Baumann & McAllister 2007), and ekpyrotic or cyclic scenarios (Khoury et al. 2001; Boyle et al. 2004), are not excluded if one fits for both tensors and scalars.

The running of the spectral index has been the subject of some debate in light of WMAP observations, with the three-year data giving limits of dn_s/dln k = −0.055 ± 0.030, showing some preference for decreasing power on small scales (Spergel et al. 2007). Combined with high-ℓ CBI and VSA CMB data, a negative running was preferred at ∼2σ. A running index is not predicted by the simplest inflationary models (see, e.g., Kosowsky & Turner 1995), and the detection of a scale dependence would have interesting consequences for early universe models. Deviations from a power-law index, and their consequences, have been considered by a number of groups in light of three-year data, including Easther & Peiris (2006), Kinney et al. (2006), Shafieloo & Souradeep (2008), and Verde & Peiris (2008), using various parameterizations. In this analysis, and in Komatsu et al. (2009), we consider only a running index parameterized as in Equation (15).

We show in Figure 11 that with a better determination of the third acoustic peak, coupled with improved beam determination, the five-year WMAP data do not significantly prefer a scale-dependent index. The limit on the running is dn_s/dln k = −0.037 ± 0.028 for models with no tensor contribution. The running is anticorrelated with the tensor amplitude (Spergel et al. 2003, 2007), so the positive prior on the tensor amplitude leads to a more negative running preferred, dn_s/dln k = −0.050 ± 0.034, when a tensor contribution is marginalized over. Both limits are well within 2σ of zero, showing no evidence of departure from a power-law spectral index.

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The simplest classical single-field inflation models predict solely adiabatic fluctuations, but entropy (or isocurvature) fluctuations are also predicted in a wide range of scenarios, including axions (Seckel & Turner 1985; Linde 1985), multifield inflation (Polarski & Starobinsky 1994; Garcia-Bellido & Wands 1996; Linde & Mukhanov 1997), and decay of fields such as the curvaton (Lyth & Wands 2002; Moroi & Takahashi 2001; Bartolo & Liddle 2002; Lyth et al. 2003). They may be correlated with the adiabatic fluctuations to some degree, depending on the model. Most physical scenarios generate only CDM or baryon entropy fluctuations (Bond & Efstathiou 1987; Peebles 1987), with perturbation

$$\begin{equation} {\cal S}_c = \frac{\delta \rho _c}{\rho _c} - \frac{3\delta \rho _\gamma }{4\rho _\gamma }, \end{equation} \tag{ 18 }$$

for CDM with density ρ_c. Neutrino modes are also possible (Bucher et al. 2000). It has been known for some time that the CMB data cannot be fit by pure entropy fluctuations (Stompor et al. 1996; Langlois & Riazuelo 2000), but a contribution may be allowed. Several groups have placed limits on a variety of models using the WMAP one-year and three-year data (Peiris et al. 2003; Valiviita & Muhonen 2003; Bucher et al. 2004; Beltran et al. 2004; Dunkley et al. 2005; Kurki-Suonio et al. 2005; Lewis 2006; Bean et al. 2006; Trotta 2007; Keskitalo et al. 2007), finding no strong evidence for entropy fluctuations. Significant levels have been found to be consistent with the CMB data (Bucher et al. 2004; Moodley et al. 2004; Bean et al. 2006), but require correlated admixtures of CDM and neutrino isocurvature perturbations that are hard to motivate physically.

Here, and in Komatsu et al. (2009), we quantify the relative contributions to the angular power spectrum following Beltran et al. (2004) and Bean et al. (2006), with

$$\begin{equation} C_\ell = (1-\alpha)C_\ell ^{\cal R} + \alpha C_\ell ^{\cal S} +2\beta \sqrt{\alpha (1-\alpha)}C_\ell ^{X}, \end{equation} \tag{ 19 }$$

summing the spectra from curvature fluctuations $C_\ell ^{\cal R}$, entropy fluctuations $C_\ell ^{\cal S}$ with power spectrum

$$\begin{equation} \Delta _S^2(k)\delta ^3(\mathbf k- \mathbf k^{\prime }) = (k/2\pi)^3\langle {\cal S}_c(\mathbf k){\cal S}_c(\mathbf k^{\prime })\rangle, \end{equation} \tag{ 20 }$$

and a cross-correlation spectrum C^X_ℓ with power spectrum

$$\begin{equation} \Delta _{X}^2 (k)\delta ^3(\mathbf k- \mathbf k^{\prime })= (k/2\pi)^3\langle -{\cal R}(\mathbf k) {\cal S}_c(\mathbf k^{\prime })\rangle. \end{equation} \tag{ 21 }$$

This follows the definition of the curvature perturbation $\cal R$ by Komatsu et al. (2009), which gives large-scale temperature anisotropy $\Delta T/T=-{\cal R}/5$. An anticorrelated spectrum with β = −1 gives a positive C^X_ℓ on large scales.

Limits are found for α₋₁, corresponding to anticorrelated models with β = −1. This could correspond to a curvaton scenario in which inflation is driven by an inflaton field, but CDM perturbations are generated by the decay of a distinct curvaton field (see, e.g., Lyth & Wands 2002). In this case, we make the assumption that the spectral index of the anti-uncorrelated entropy fluctuations is equal to the adiabatic spectral index. We do not find evidence for curvaton entropy perturbations, finding a limit from the WMAP of α₋₁ < 0.011(95%CL), shown in Figure 12 and Table 5. This is half the three-year limit, and places strong lower limits on the possible density of the curvaton at its decay, in this scenario, compared to the total energy density (Komatsu et al. 2009). If the curvaton field dominated at its decay, the perturbations would be purely adiabatic. Where has the improvement come from? The pure entropy spectrum and the cross-correlated spectrum both add large-scale power, so a similar degeneracy is seen with the spectral index, and σ₈, as in the case where the tensor amplitude is varied. The entropy spectrum is also out of phase with the adiabatic spectrum, so the improved TE measurements combine with the third-peak TT spectrum to tighten the limits.

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We also place limits on α₀, corresponding to an uncorrelated model with β = 0. In this case the entropy spectral index is set to be scale invariant. Komatsu et al. (2009) describe how this corresponds to entropy perturbations created by axions, which would constitute some part of the dark matter budget. The limit is α₀ < 0.16(95%CL), 10 times higher than the anticorrelated amplitude, but still preferring pure adiabatic fluctuations. Without the large-scale power contribution from the anticorrelated spectrum, a much larger amplitude is permitted, but with the same degeneracy with the spectral index and σ₈. This has implications for the maximum deviation from adiabaticity of axion dark matter and photons. Komatsu et al. (2009) provide a discussion of the theoretical implications of these limits, and those for combined data, for both models considered.

Neutrinos are expected to play an important role in the dynamics of the early universe. For standard parameters, they contribute about 40% of the energy density of the universe during the radiation epoch and about 11% of the energy density of the universe at z ∼ 1100 (very close to the energy density in baryons). Because neutrinos contribute to the expansion of the universe and stream relativistically out of density fluctuations, they produce a significant imprint on the growth rate of structure and on the structure of the microwave background fluctuations (e.g., Ma & Bertschinger 1995). The amplitude of these effects depend upon N_eff, the number of effective neutrino species. By "effective neutrinos species," we are counting any particle that is relativistic at z ∼ 1000–3000, couples very weakly to the baryon–electron–photon fluid, and has very weak self-interactions. Because we know neutrinos exist, we associate "neutrinos" with "light relativistic particle," but note that in the strictest sense we limit only light relativistic species, as the cosmological constraints are sensitive to the existence of any light species produced during the big bang or any additional contribution to the energy density of the universe (e.g., primordial magnetic fields).

Measurements of the width of z provide very tight limits on the number of neutrino species: N_ν = 2.984 ± 0.008 (Particle Data Book), consistent with the three light neutrino species in the standard model. Because of nonthermal effects due to the partial heating of neutrinos during the e^± annihilations, and other small corrections, the effective number of species is 3.0395 (Dicus et al. 1982; Mangano et al. 2002). Most analyses of the number of neutrino species with three-year WMAP data (Spergel et al. 2007; Ichikawa et al. 2008; Mangano et al. 2007; Hamann et al. 2007; de Bernardis et al. 2008) relied on combining CMB measurements with probes of the growth rate of structure. Since one of the signatures of the number of neutrino species is a change in the growth rate of structure, there are degeneracies between the properties of the neutrinos and of the dark energy. Neutrinos, however, leave a distinctive signature directly on the CMB power spectrum (see Bashinsky & Seljak 2004 for detailed discussion): the neutrinos not only suppress the CMB peak heights, they also shift the acoustic peak positions. While the effects that depend on shifts in the epoch of matter/radiation equality are degenerate with changes in the matter density, the effects of neutrino free streaming are distinct. Changes in the baryon/matter ratio and the baryon/photon ratio also have their own imprints on the Silk damping scale and on the acoustic scale. With five years of data, we are now able to see evidence of the effects of the neutrinos on the CMB power spectrum.

Figure 13 shows the limits on the number density of neutrinos and the density in dark matter. The degeneracy valley, shown in the right panel, corresponds to a constant ratio of matter density to radiation density, or equivalently a measurement of the expansion factor at matter radiation equality:

$$\begin{eqnarray} 1 +z_{\rm eq} &=& a_{\rm eq}^{-1} = \frac{\rho _c + \rho _{b}}{\rho _{\gamma } + \rho _{\nu }}\nonumber \\ &\simeq & 40500 \ \frac{\Omega _c h^2 + \Omega _b h^2}{1 + 0.23 N_{\rm eff}}. \end{eqnarray} \tag{ 22 }$$

With only three years of data and a lack of precision on the third peak position and height, WMAP was not able to make a clear detection of neutrinos (or relativistic species); however, the data did provide a ∼2σ hint of the effects of neutrino anisotropic stresses (Melchiorri & Serra 2006). Figure 13 shows that with the five-year data alone we now constrain the number density of relativistic species: N_eff > 2.3(95%CL). By bounding N_eff < 10, and choosing a uniform prior on N_eff, this level of significance depends somewhat on the prior. We therefore test the significance of the constraint by comparing two ΛCDM models: one with N_eff = 0, and one with the standard N_eff = 3.04. We find that the data prefer N_eff = 3.04. The best-fit model has Δ(−2ln L) = 8.2 less than the N_eff = 0 best-fit model, corresponding to evidence for relativistic species at >99.5% confidence. The CMB power spectra corresponding to these two models, and their fractional difference, are shown in Figure 14. The model with no neutrinos has a lower matter density, Ω_mh², in order to keep z_eq fixed. The improvement in likelihood between the two models comes from both the low-ℓ and high-ℓ TT spectrum, with a small contribution from the TE spectrum. We also check that this evidence does not go away if we relax the assumption of a power-law spectral index, by testing a model with a variable running, dn_s/dln k ≠ 0.

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Komatsu et al. (2009) combine WMAP data with other distance indicators (which constrain Ω_ch²) and finds a stronger limit on the number density of neutrino species: N_eff = 4.4 ± 1.5. These limits will continue to improve as CMB measurements of the higher peaks improve. The CMB constraints on the number of relativistic species at redshift ∼1000–3000 complement constraints from BBN and from particle accelerators. Measurements of the abundance of helium are sensitive to the expansion rate of the universe during its first few minutes (Steigman et al. 1977). The agreement between the best-fit value from BBN, N_eff = 3.24 ± 1.2 (95% confidence interval; Cyburt et al. 2005; Particle Data Book 2007), with the best-fit CMB value is another consistency check for standard cosmology.

Cosmological data places limits on the mass of neutrinos. Atmospheric and solar neutrino experiments show that neutrinos are massive (see Mohapatra et al. 2007), and measure the difference between the square of their masses, m²_νi − m²_νj. Cosmological measurements constrain the sum of the masses ∑m_ν due to their effect on the propagation of perturbations, on the clustering of matter, and on the expansion rate of the universe (Bond & Szalay 1983; Ma 1996; Hu et al. 1998). The mass has a large effect on the matter power spectrum, as massive neutrinos do not cluster as well as cold dark matter, leading to a suppression in power on small scales. Neutrinos also affect the CMB at earlier times: if the fraction of dark matter that is warm is raised, acoustic oscillations in the photon–baryon plasma are less strongly damped for modes that entered the horizon while the neutrinos were relativistic, raising the acoustic peak amplitudes. The radiationlike behavior at early times also changes the expansion rate, shifting the peak positions.

These effects are somewhat degenerate with other parameters, so CMB data alone cannot limit the mass as well as when combined with other data. With the three-year WMAP data alone the limits were ∑m_ν < 1.8 eV (Spergel et al. 2007), and <0.66 eV when combined with other data. Since the three-year WMAP analysis there have been many studies of the constraints, as discussed by Komatsu et al. (2009).

The five-year WMAP data now give an upper limit on the total mass to be ∑m_ν < 1.3 eV(95%CL), as shown in Table 6. We have checked that this upper limit is robust to the choice of cosmological models. The upper limit is raised by <10% when we include tensor fluctuations, a running spectral index, or a constant w ≠ −1 equation of state of dark energy, as shown in Figure 15. This dependence on additional parameters is consistent with earlier investigations by, e.g., Crotty et al. (2004) and Zunckel & Ferreira (2007). A larger neutrino mass raises the amplitude of the higher acoustic peaks, hence the observed degeneracy with σ₈ (Figure 15). Stronger constraints come from combining the CMB data with probes of the expansion rate and clustering of matter at later times: Komatsu et al. (2009) find ∑m_ν < 0.61(95%CL) for WMAP combined with additional data.

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In most cosmological analyses the primordial helium abundance is fixed to be Y_P = 0.24, motivated by observations discussed in Section 3.2.9. The effect of the abundance on the CMB spectrum is small, but provides an independent cross-check of the BBN results, and probes for any difference between the helium abundance during the first few minutes, and after 300,000 years. The abundance affects the CMB at small scales due to the recombination process. The number density of electrons before recombination depends on the helium fraction through n_e = n_b(1 − Y_P), where n_b is the baryon number density. Changing the electron number density changes the mean free path of Compton scattering, which affects the Silk damping scale. A larger Y_p increasingly damps the power on small scales, as shown by Trotta & Hansen (2004).

Constraints from the first-year WMAP data were presented by Trotta & Hansen (2004), Huey et al. (2004), and Ichikawa & Takahashi (2006), with 99% upper limits of Y_P < 0.65 inferred (Trotta & Hansen 2004). A subsequent analysis of the three-year data gave Y_P < 0.61 at 95% confidence, tightened to 0.25 ± 0.10 with small-scale CMB data (Ichikawa et al. 2008). We now find Y_P < 0.45 (95% CL) with the five-year WMAP data. Higher values allowed by the three-year data are disfavored with a better measure of the third acoustic peak height. With future small-scale CMB measurements, for example from the Planck satellite, constraints should significantly improve (Ichikawa et al. 2008; Hamann et al. 2008b).

In combination with other data, WMAP observations place strong constraints on the geometry of the universe (Spergel et al. 2007). The CMB measures with high accuracy the angular scale at which acoustic oscillations are imprinted at the last scattering surface, θ_* = 05952 ± 00017. However, this alone does not provide a good measure of the geometry, as there is a degeneracy with the expansion rate of the universe since last scattering. This is shown in Figure 16, indicating the degeneracy between the dark energy density Ω_Λ and the curvature Ω_k. With WMAP alone the curvature is weakly constrained, with marginalized limits Ω_k = −0.099^+0.085_−0.100, and Ω_Λ < 0.76(95%CL), assuming a Hubble prior of 20 < H₀ < 100 and Ω_Λ > 0. Without this prior on the positivity of Ω_Λ, limits on the curvature are weakened. The same degeneracy is seen, although slightly broadened, when the dark energy equation of state is allowed to vary. However, in both cases the Hubble constant decreases with increasingly negative curvature, taking values inconsistent with observation. This degeneracy can be used to constrain the curvature by combining observations (Jungman et al. 1996). In the three-year WMAP analysis, Spergel et al. (2007) showed that the degeneracy is truncated with the addition of only one piece of additional cosmological data (Type Ia supernovae, or the HST measurement of the Hubble constant, or galaxy power spectra), tightly constraining any deviations from flatness. Komatsu et al. (2009) draw similar conclusions with currently available data, and discuss the current limits on the spatial curvature from recent observations.

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The ΛCDM model requires a nonzero dark energy density Ω_Λ = 0.742 ± 0.030 to fit the data, which is assumed to be in the form of a cosmological constant. We do not have an explanation for this component of the universe. A natural explanation could be a vacuum energy density (Carroll et al. 1992), but if so, we are faced with the fine-tuning problem to explain its observed value, 120 orders of magnitude smaller than expected from field theory arguments. Alternative explanations include quintessence (Peebles & Ratra 1988; Wetterich 1988; Ferreira & Joyce 1998) or modifications to gravity (Deffayet et al. 2002). Testing the dark energy equation of state today, and as a function of cosmic time will help identify the possible explanation.

The CMB by itself cannot place strong limits on the equation of state w = p/ρ, but by measuring the acoustic peak positions and heights, and constraining Ω_mh² with the third peak, limits the range of models to a degeneracy between Ω_m and w, as shown in Figure 16. The dark energy in these models is allowed to cluster. With a prior on the Hubble constant H₀ < 100, WMAP alone places weak limits w = −1.06^+0.41_−0.42, with Ω_Λ = 0.73^+0.10_−0.11. If flatness is not assumed, the WMAP data cannot constrain w or Ω_Λ due to the geometric degeneracy, also shown in Figure 16. However, the situation is significantly improved when WMAP is combined with astronomical data measuring the expansion rate and clustering of matter at late times. Komatsu et al. (2009) discuss limits obtained from various data in combination with WMAP, and find w constrained to be −1 to within 6% for a flat universe and constant equation of state.