Parameter constraints for flat cosmologies from CMB and 2dFGRS power spectra

TL;DR

This work presents a joint likelihood analysis of flat cosmologies using a compressed CMB power spectrum and the 2dFGRS power spectrum, deriving tight constraints on $h$ and $\Omega_m$ (e.g., scalar-only $h=0.665\pm0.047$, $\Omega_m=0.313\pm0.055$) and related densities $\Omega_b h^2$ and $\Omega_c h^2$. It introduces and exploits the horizon-angle degeneracy, showing that CMB peak locations tightly constrain the combination $\Omega_m h^{3.4}$ (or $\Omega_m h^{3}$ when peak heights are included), and that including 2dFGRS breaks degeneracies to yield precise parameter estimates. Allowing tensor modes broadens the allowed region but maintains consistency with the data, while extending the analysis to a variable equation of state for dark energy yields $w<-0.52$ (95% CL) when combined with an external $h$ prior. The study provides a robust CDM+flatness framework, makes concrete predictions for the CMB power spectrum (e.g., the first peak near $\ell\approx 222$ for scalar models), and offers public data resources for further testing with upcoming experiments like MAP.

Abstract

We constrain flat cosmological models with a joint likelihood analysis of a new compilation of data from the cosmic microwave background (CMB) and from the 2dF Galaxy Redshift Survey (2dFGRS). Fitting the CMB alone yields a known degeneracy between the Hubble constant h and the matter density Omega_m, which arises mainly from preserving the location of the peaks in the angular power spectrum. This `horizon-angle degeneracy' is considered in some detail and shown to follow a simple relation Omega_m h^{3.4} = constant. Adding the 2dFGRS power spectrum constrains Omega_m h and breaks the degeneracy. If tensor anisotropies are assumed to be negligible, we obtain values for the Hubble constant h=0.665 +/- 0.047, the matter density Omega_m=0.313 +/- 0.055, and the physical CDM and baryon densities Omega_c h^2 = 0.115 +/- 0.009, Omega_b h^2 = 0.022 +/- 0.002 (standard rms errors). Including a possible tensor component causes very little change to these figures; we set a upper limit to the tensor-to-scalar ratio of r<0.7 at 95% confidence. We then show how these data can be used to constrain the equation of state of the vacuum, and find w<-0.52 at 95% confidence. The preferred cosmological model is thus very well specified, and we discuss the precision with which future CMB data can be predicted, given the model assumptions. The 2dFGRS power-spectrum data and covariance matrix, and the CMB data compilation used here, are available from http://www.roe.ac.uk/~wjp/

Figures (7)

• Figure 1: Top panel: the compilation of recent CMB data used in our analysis (see text for details). The solid line shows the result of a maximum-likelihood fit to these data allowing for calibration and beam uncertainty errors in addition to intrinsic errors. Each observed data set has been shifted by the appropriate best-fit calibration and beam correction. Bottom panel: the solid line again shows our maximum-likelihood fit to the CMB power spectrum now showing the nodes (the points at which the amplitude of the power spectrum was estimated) with approximate errors calculated from the diagonal elements of the covariance matrix (solid squares). These data are compared with the compilation of Wang et al. (2002) (stars) and the result of convolving our best fit power with the window function of Wang et al. (crosses). In order to show the important features in the CMB angular power spectrum plots we present in this paper we have chosen to scale the x-axis by $(\log\ell)^{2.5}$.
• Figure 2: Two parameter likelihood surfaces for scalar only models. Contours correspond to changes in the likelihood from the maximum of $2\Delta\ln{\cal L}=2.3, 6.0, 9.2$. Dashed contours are calculated by only fitting to the CMB data, solid contours by jointly fitting the CMB and 2dFGRS data. Dotted lines show the extent of the grid used to calculate the likelihoods.
• Figure 3: As Fig. \ref{['fig:like_sca']}, but now considering a wider class of models that possibly include a tensor component.
• Figure 4: Likelihood contours for $\Omega_m$ against $h$ for scalar only models, plotted as in Fig. \ref{['fig:like_sca']}. Variables were changed from $\Omega_bh^2$ and $\Omega_ch^2$ to $\Omega_m$ and $\Omega_b/\Omega_m$, and a uniform prior was assumed for $\Omega_b/\Omega_m$ covering the same region as the original grid. The extent of the grid is shown by the dotted lines. The dot-dash line follows the locus of models through the likelihood maximum with constant $\Omega_mh^{3.4}$. The solid line is a fit to the likelihood valley and shows the locus of models with constant $\Omega_mh^{3.0}$ (see text for details).
• Figure 5: The top panel shows three scalar only model CMB angular power spectra with the same apparent horizon angle, compared to the data of Table \ref{['tab:calib']}. Although these models have approximately the same value of $\Omega_m h^{3.4}$, they are distinguishable by peak heights. Such additional constraints alter the degeneracy observed in Fig. \ref{['fig:ommvsh']} slightly from $\Omega_m h^{3.4}$ to $\Omega_m h^{3.0}$. Three scalar only models that lie in the likelihood ridge with $\Omega_m h^{3.0}$ are compared with the data in the bottom panel. For all of the models shown, parameters other than $\Omega_m$ and $h$ have been adjusted to their maximum-likelihood positions.