Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation

TL;DR

The paper analyzes Wilkinson Microwave Anisotropy Probe (WMAP) 5-year data, in combination with distance measurements from Type Ia supernovae and baryon acoustic oscillations, to test the ΛCDM model and probe early-universe physics. It constrains the primordial power spectrum (tilt n_s and running), the amplitude of primordial gravitational waves (r), spatial curvature (Ω_k), dark energy (w and w(z)), non-Gaussianity (f_NL^local, f_NL^equil), parity-violating interactions (Δα), neutrino masses (∑m_ν) and the effective number of neutrino species (N_eff). Across all probes, the results show no convincing deviations from the standard model, while yielding tight, complementary constraints that sharpen the allowed parameter space for inflationary models and cosmic evolution. The work demonstrates the power of combining CMB with external distance measures to break degeneracies and to provide robust, model-tested limits on fundamental physics. The findings strengthen the standard cosmological paradigm and set the stage for future precision tests with larger data sets and complementary probes.

Abstract

(Abridged) The WMAP 5-year data strongly limit deviations from the minimal LCDM model. We constrain the physics of inflation via Gaussianity, adiabaticity, the power spectrum shape, gravitational waves, and spatial curvature. We also constrain the properties of dark energy, parity-violation, and neutrinos. We detect no convincing deviations from the minimal model. The parameters of the LCDM model, derived from WMAP combined with the distance measurements from the Type Ia supernovae (SN) and the Baryon Acoustic Oscillations (BAO), are: Omega_b=0.0456+-0.0015, Omega_c=0.228+-0.013, Omega_Lambda=0.726+-0.015, H_0=70.5+-1.3 km/s/Mpc, n_s=0.960+-0.013, tau=0.084+-0.016, and sigma_8=0.812+-0.026. With WMAP+BAO+SN, we find the tensor-to-scalar ratio r<0.22 (95% CL), and n_s>1 is disfavored regardless of r. We obtain tight, simultaneous limits on the (constant) equation of state of dark energy and curvature. We provide a set of "WMAP distance priors," to test a variety of dark energy models. We test a time-dependent w with a present value constrained as -0.33<1+w_0<0.21 (95% CL). Temperature and matter fluctuations obey the adiabatic relation to within 8.9% and 2.1% for the axion and curvaton-type dark matter, respectively. The TE and EB spectra constrain cosmic parity-violation. We find the limit on the total mass of neutrinos, sum(m_nu)<0.67 eV (95% CL), which is free from the uncertainty in the normalization of the large-scale structure data. The effective number of neutrino species is constrained as N_{eff} = 4.4+-1.5 (68%), consistent with the standard value of 3.04. Finally, limits on primordial non-Gaussianity are -9<f_{NL}^{local}<111 and -151<f_{NL}^{equil}<253 (95% CL) for the local and equilateral models, respectively.

Figures (21)

• Figure 1: Constraint on the primordial tilt, $n_s$ (§ \ref{['sec:PK_results_tilt']}). No running index or gravitational waves are included in the analysis. ( Left) One-dimensional marginalized constraint on $n_s$ from the WMAP-only analysis. ( Middle) Two-dimensional joint marginalized constraint (68% and 95% CL), showing a strong correlation between $n_s$ and $\Omega_bh^2$. ( Right) A mild correlation with $\tau$. None of these correlations are reduced significantly by including BAO or SN data, as these data sets are not sensitive to $\Omega_bh^2$ or $\tau$; however, the situation changes when the gravitational wave contribution is included (see Fig. \ref{['fig:tens']}).
• Figure 2: How the WMAP temperature and polarization data constrain the tensor-to-scalar ratio, $r$. ( Left) The contours show 68% and 95% CL. The gray region is derived from the low-$l$ polarization data (TE/EE/BB at $l\le 23$) only, the red region from the low-$l$ polarization plus the high-$l$ TE data at $l\le 450$, and the blue region from the low-$l$ polarization, the high-l TE, and the low-$l$ temperature data at $l\le 32$. ( Right) The gray curves show $(r,\tau)=(10,0.050)$, the red curves $(r,\tau)=(1.2,0.075)$, and the blue curves $(r,\tau)=(0.20,0.080)$, which are combinations of $r$ and $\tau$ that give the upper edge of the 68% CL contours shown on the left panel. The vertical lines indicate the maximum multipoles below which the data are used for each color. The data points with 68% CL errors are the WMAP 5-year measurements nolta/etal:prep. (Note that the BB power spectrum at $l\sim 130$ is consistent with zero within 95% CL.)
• Figure 3: Constraint on the tensor-to-scalar ratio, $r$, at $k=0.002~{\rm Mpc}^{-1}$ (§ \ref{['sec:GW_results']}). No running index is assumed. See Fig. \ref{['fig:run+tens']} for $r$ with the running index. In all panels we show the WMAP-only results in blue and WMAP+BAO+SN in red. ( Left) One-dimensional marginalized distribution of $r$, showing the WMAP-only limit, $r < 0.43\ \hbox{(95% CL)}$, and WMAP+BAO+SN, $r < 0.22\ \hbox{(95% CL)}$. ( Middle) Joint two-dimensional marginalized distribution (68% and 95% CL), showing a strong correlation between $n_s$ and $r$. ( Right) Correlation between $n_s$ and $\Omega_mh^2$. The BAO and SN data help to break this correlation which, in turn, reduces the correlation between $r$ and $n_s$, resulting in a factor of 2.2 better limit on $r$.
• Figure 4: Constraint on the tensor-to-scalar ratio, $r$, the tilt, $n_s$, and the running index, $dn_s/d\ln k$, when all of them are allowed to vary (§ \ref{['sec:GW_results']}). In all panels we show the WMAP-only results in blue and WMAP+BAO+SN in red. ( Left) Joint two-dimensional marginalized distribution of $n_s$ and $r$ at $k=0.002~{\rm Mpc}^{-1}$ (68% and 95% CL). ( Middle) $n_s$ and $dn_s/d\ln k$. ( Right) $dn_s/d\ln k$ and $r$. We find no evidence for the running index. While the inclusion of the running index weakens our constraint on $n_s$ and $r$, the data do not support any need for treating the running index as a free parameter: changes in $\chi^2$ between the power-law model and the running model are $\chi^2({\rm running})-\chi^2({\rm power-law})\simeq -1.8$ with and without the tensor modes for WMAP5+BAO+SN, and 1.2 for WMAP5.
• Figure 5: Constraint on three representative inflation models whose potential is positively curved, $V">0$ (§ \ref{['sec:summary_inflation']}). The contours show the 68% and 95% CL derived from WMAP+BAO+SN. ( Top) The monomial, chaotic-type potential, $V(\phi)\propto \phi^\alpha$linde:1983, with $\alpha=4$ (solid) and $\alpha=2$ (dashed) for single-field models, and $\alpha=2$ for multi-axion field models with $\beta=1/2$easther/mcallister:2006 (dotted). The symbols show the predictions from each of these models with the number of $e$-folds of inflation equal to 50 and 60. The $\lambda\phi^4$ potential is excluded convincingly, the $m^2\phi^2$ single-field model lies outside of (at the boundary of) the 68% region for $N=50$ (60), and the $m^2\phi^2$ multi-axion model with $N=50$ lies outside of the 95% region. ( Middle) The exponential potential, $V(\phi)\propto \exp[-(\phi/M_{pl})\sqrt{2/p}]$, which leads to a power-law inflation, $a(t)\propto t^p$abbott/wise:1984lucchin/matarrese:1985. All models but $p\sim 120$ are outside of the 68% region. The models with $p<60$ are excluded at more than 99% CL, and those with $p<70$ are outside of the 95% region. For multi-field models these limits can be translated into the number of fields as $p\rightarrow np_i$, where $p_i$ is the $p$-parameter of each field liddle/mazumdar/schunck:1998. The data favour $n\sim 120/p_i$ fields. ( Bottom) The hybrid-type potential, $V(\phi)=V_0+(1/2)m^2\phi^2=V_0(1+\tilde{\phi}^2)$, where $\tilde{\phi}\equiv m\phi/(2V_0)^{1/2}$linde:1994. The models with $\tilde{\phi}<2/3$ drive inflation by the vacuum energy term, $V_0$, and are disfavoured at more than 95% CL, while those with $\tilde{\phi}>1$ drive inflation by the quadratic term, and are similar to the chaotic type (the left panel with $\alpha=2$). The transition regime, $2/3<\tilde{\phi}<1$ are outside of the 68% region, but still within the 95% region.