Cosmological parameters from CMB and other data: a Monte-Carlo approach

TL;DR

The paper addresses estimating cosmological parameters from CMB and complementary data in a high-dimensional space using fast Markov Chain Monte Carlo methods. It develops a scalable Bayesian framework with importance sampling to incorporate new data and to explore flat and non-flat models, including parameters for neutrino mass, dark energy equation of state w, and tensor amplitudes. The authors derive robust constraints (e.g., m_nu < 0.3 eV, w < -0.75, r_10 < 0.7) and show near-flat geometry with a cosmological-constant-like dark energy, while finding no strong evidence for tensor modes or significant neutrino mass; they also demonstrate the method's ability to compare models and assess data consistency. Their approach provides a practical, extensible tool for updating cosmological inferences as new data arrive, with publicly available COSMOMC software. The work reinforces a concordance cosmology and offers a blueprint for high-dimensional parameter estimation in cosmology.

Abstract

We present a fast Markov Chain Monte-Carlo exploration of cosmological parameter space. We perform a joint analysis of results from recent CMB experiments and provide parameter constraints, including sigma_8, from the CMB independent of other data. We next combine data from the CMB, HST Key Project, 2dF galaxy redshift survey, supernovae Ia and big-bang nucleosynthesis. The Monte Carlo method allows the rapid investigation of a large number of parameters, and we present results from 6 and 9 parameter analyses of flat models, and an 11 parameter analysis of non-flat models. Our results include constraints on the neutrino mass (m_nu < 0.3eV), equation of state of the dark energy, and the tensor amplitude, as well as demonstrating the effect of additional parameters on the base parameter constraints. In a series of appendices we describe the many uses of importance sampling, including computing results from new data and accuracy correction of results generated from an approximate method. We also discuss the different ways of converting parameter samples to parameter constraints, the effect of the prior, assess the goodness of fit and consistency, and describe the use of analytic marginalization over normalization parameters.

Figures (7)

• Figure 1: The CMB temperature anisotropy band-power data used in this paper. The line shows the model with the parameters at their mean values, given all data after marginalizing in 6 dimensions (i.e. first column of Table. \ref{['inflationtable']}).
• Figure 2: Left: 2000 samples from the posterior distribution of the parameters plotted by their $\Omega_{\mathrm{m}}$ and $\Omega_\Lambda$ values. Points are colored according to the value of $h$ of each sample, and the solid line shows the flat universe parameters. We assume the base parameter set with broad top-hat priors. Right: bottom layer (green): supernova constraints; next layer up (red): CMB data alone; next (blue): CMB data plus HST Key Project prior; top layer (yellow): all data combined (see text). 68 and 95 per cent confidence limits are shown.
• Figure 3: Posterior constraints for 9-parameter flat models using all data. The top nine plots show the constraints on the base MCMC parameters, the remaining plots show various derived parameter constraints. Red lines include CMB, HST, SnIA and BBN constraints, black lines also include the 2dF data. The solid lines show the fully marginalized posterior, the dotted lines show the relative mean likelihood of the samples. The curves are generated from the MCMC samples using a Gaussian smoothing kernel $1/20$th the width of each plot.
• Figure 4: All-data posterior constraints for flat inflationary models using. The contours show the $68\%$ and $95\%$ confidence limits from the marginalized distribution. The shading shows the mean likelihood of the samples, and helps to demonstrate where the marginalized probability is enhanced by a larger parameter space rather than by a better fit to the data (e.g. low $n_s$ values fit the data better).
• Figure 5: Posterior constraints for $11$-parameter non-flat models (black lines) using all data, compared with $6$ (red) and $9$ (green) parameter models. Dotted lines show the mean likelihood of the samples for the $11$-parameter model. Some sampling noise is apparent due to the relatively small number of samples used.