Analyze This! A Cosmological Constraint Package for CMBEASY

TL;DR

This paper presents AnalyzeThis, a parallel MCMC package added to Cmbeasy to efficiently constrain cosmological parameters by combining CMB, SNe Ia, and LSS data. It couples a Metropolis-based MCMC framework with an adaptive, multivariate Gaussian proposal that is tuned during early burn-in and then frozen to preserve the correct posterior sampling. The AnalyzeThis class implements likelihoods for WMAP, ACBAR/CBI/VSA, 2dFGRS, SDSS, and multiple SNe Ia datasets, enabling joint cosmological constraints in a flat $\Lambda$CDM setting while marginalizing nuisance parameters. The approach yields faster convergence and tighter parameter bounds, demonstrating the impact of data sets on the inferred $\Omega_b h^2$, $\Omega_m h^2$, $h$, $\tau$, and $n_s$, and providing a publicly available, user-friendly tool for cosmological inference.

Abstract

We introduce a Markov Chain Monte Carlo simulation and data analysis package that extends the CMBEASY software. We have taken special care in implementing an adaptive step algorithm for the Markov Chain Monte Carlo in order to improve convergence. Data analysis routines are provided which allow to test models of the Universe against measurements of the cosmic microwave background, supernovae Ia and large scale structure. We present constraints on cosmological parameters derived from these measurements for a $Λ$CDM cosmology and discuss the impact of the different observational data sets on the parameters. The package is publicly available as part of the CMBEASY software at www.cmbeasy.org.

Figures (6)

• Figure 1: The graphical user interface of Cmbeasy. It can be used to marginalize, visualize and print the one and two dimensional likelihoods from the MCMC chains. Shown is the marginalized likelihood in the $\Omega_m h^2 - \Omega_b h^2$ plane of a $\Lambda$CDM model.
• Figure 2: Illustrating the Metropolis algorithm for two parameters. Filled circles represent points belonging to the chain, empty circles are proposed but rejected points not belonging to the chain. In this example, the chain would be $[\theta_0, \theta_1, \theta_1,\theta_3,\theta_4, \dots]$.
• Figure 3: Illustrating the naive Gaussian sampler with fixed step sizes for two parameters. The (unknown) true likelihood surface is shown in black, the proposal distribution with arrows in red. This proposal distribution does not does not take into account the degeneracy among the parameters $\theta^{(1)}$ and $\theta^{(2)}$, leading to slow mixing.
• Figure 4: Convergence properties using different proposal distributions for a cosmological model with seven parameters. For illustration purposes we display the average $R$-statistic as a function of number of models computed. The univariate Gaussian proposal distribution using fixed variances but adaptive overall step size $\alpha$ (black dashed curve) shows convergence after about 2800 iterations. The multivariate Gaussian proposal distribution with covariance matrix estimated from the previous chain points and adaptive overall step size $\alpha$ as suggested in this paper (red solid curve) converges after 500 iterations.
• Figure 5: CMB data set for constraining cosmological parameters.