Profile likelihoods in cosmology: When, why, and how illustrated with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>CDM</mml:mi></mml:math>, massive neutrinos, and dark energy

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 $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ model indicate that the profile likelihood method gives correct coverage, $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ 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 ${w}_{0}\mathrm{CDM}$ 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.