Profile Likelihoods in Cosmology: When, Why and How illustrated with $\Lambda$CDM, Massive Neutrinos and Dark Energy

TL;DR

This paper reviews the use of profile likelihoods in cosmology, assessing their validity for different models with Planck data, and compares frequentist and Bayesian methods, providing a new code for profile-likelihood analysis.

Contribution

It provides a pedagogical review of when profile likelihoods give valid constraints in cosmology and introduces a new code 'pinc' for such analyses.

Findings

Profile likelihoods give correct coverage for ΛCDM with Planck data.

Boundary effects influence the validity of profile likelihoods in certain models.

Care is needed when applying graphical profile likelihood methods in cosmological parameter estimation.

Abstract

Frequentist parameter inference using profile likelihoods has received increased attention in the cosmology literature recently since it can give important complementary information to Bayesian credible intervals. Here, we give a pedagogical review of frequentist parameter inference in cosmology and focus on when the graphical profile likelihood construction gives meaningful constraints, i.e. confidence intervals with correct coverage. This construction rests on the assumption of the asymptotic limit of a large data set such as in Wilks' theorem. We assess the validity of this assumption in the context of three cosmological models with Planck 2018 Plik_lite data: While our tests for the $\Lambda$CDM model indicate that the profile likelihood method gives correct coverage, $\Lambda$CDM with the sum of neutrino masses as a free parameter appears consistent with a Gaussian near a boundary motivating the use of the boundary-corrected or Feldman-Cousins graphical method; for $w_0$CDM with the equation of state of dark energy, $w_0$, as a free parameter, we find indication of a violation of the assumptions. Finally, we compare frequentist and Bayesian constraints of these models. Our results motivate care when using the graphical profile likelihood method in cosmology. Along with this paper, we publish our profile-likelihood code "pinc".