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A Bayesian model selection analysis of WMAP3

David Parkinson, Pia Mukherjee, Andrew R Liddle

TL;DR

The paper tackles whether the scalar spectral index deviates from unity and whether slow-roll inflation models requiring a tensor component are supported by WMAP3 data, using a Bayesian model selection framework implemented in CosmoNest. It computes Bayesian evidences for competing models, including $n_S$ varying (with $r$ fixed) and the two-parameter $(n_S,r)$ inflationary plane, highlighting prior dependence and methodological advantages of nested sampling. The results show substantial evidence for $n_S\neq 1$ when combining WMAP3 with external data, while WMAP3 alone is inconclusive; in the $(n_S,r)$ plane, the evidence is highly prior-dependent and can even favor the Harrison–Zel'dovich case under certain priors. The work emphasizes the role of priors and systematics in model selection and provides CosmoNest as a public tool for rigorous Bayesian evidence computation in cosmology.

Abstract

We present a Bayesian model selection analysis of WMAP3 data using our code CosmoNest. We focus on the density perturbation spectral index $n_S$ and the tensor-to-scalar ratio $r$, which define the plane of slow-roll inflationary models. We find that while the Bayesian evidence supports the conclusion that $n_S \neq 1$, the data are not yet powerful enough to do so at a strong or decisive level. If tensors are assumed absent, the current odds are approximately 8 to 1 in favour of $n_S \neq 1$ under our assumptions, when WMAP3 data is used together with external data sets. WMAP3 data on its own is unable to distinguish between the two models. Further, inclusion of $r$ as a parameter weakens the conclusion against the Harrison-Zel'dovich case (n_S = 1, r=0), albeit in a prior-dependent way. In appendices we describe the CosmoNest code in detail, noting its ability to supply posterior samples as well as to accurately compute the Bayesian evidence. We make a first public release of CosmoNest, now available at http://www.cosmonest.org.

A Bayesian model selection analysis of WMAP3

TL;DR

The paper tackles whether the scalar spectral index deviates from unity and whether slow-roll inflation models requiring a tensor component are supported by WMAP3 data, using a Bayesian model selection framework implemented in CosmoNest. It computes Bayesian evidences for competing models, including varying (with fixed) and the two-parameter inflationary plane, highlighting prior dependence and methodological advantages of nested sampling. The results show substantial evidence for when combining WMAP3 with external data, while WMAP3 alone is inconclusive; in the plane, the evidence is highly prior-dependent and can even favor the Harrison–Zel'dovich case under certain priors. The work emphasizes the role of priors and systematics in model selection and provides CosmoNest as a public tool for rigorous Bayesian evidence computation in cosmology.

Abstract

We present a Bayesian model selection analysis of WMAP3 data using our code CosmoNest. We focus on the density perturbation spectral index and the tensor-to-scalar ratio , which define the plane of slow-roll inflationary models. We find that while the Bayesian evidence supports the conclusion that , the data are not yet powerful enough to do so at a strong or decisive level. If tensors are assumed absent, the current odds are approximately 8 to 1 in favour of under our assumptions, when WMAP3 data is used together with external data sets. WMAP3 data on its own is unable to distinguish between the two models. Further, inclusion of as a parameter weakens the conclusion against the Harrison-Zel'dovich case (n_S = 1, r=0), albeit in a prior-dependent way. In appendices we describe the CosmoNest code in detail, noting its ability to supply posterior samples as well as to accurately compute the Bayesian evidence. We make a first public release of CosmoNest, now available at http://www.cosmonest.org.

Paper Structure

This paper contains 13 sections, 5 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Bayesian model selection
  3. Application to WMAP3
  4. The spectral index
  5. The inflationary plane
  6. Systematic effects
  7. Conclusions
  8. The Nested Sampling Algorithm
  9. CosmoNest
  10. Evidence evaluation
  11. Posterior samples
  12. The information
  13. Public code release

Figures (4)

  • Figure 1: Marginalized likelihood of $n_{{\rm S}}$ for WMAP alone (dashed) and WMAP+all (solid), obtained using CosmoNest.
  • Figure 2: Schematic of the Nested Sampling algorithm.
  • Figure 3: The posterior weights $p_i$ assigned to each point in one of our HZ runs. The $x$-axis is the element number in the chain, and the vertical dashed line indicates where the live points start to be used. This transition could be shifted to the right by running the code for longer so as to generate a longer chain of discarded points.
  • Figure 4: Posterior samples from Nested Sampling (solid) compared to MCMC (dashed), for a $\Lambda$CDM HZ model using WMAP3 data only.