CMB constraints on dark matter-proton scattering: investigating prior-volume effects using profile likelihoods

Abstract

We present profile-likelihood constraints on velocity-independent dark matter-proton scattering, including cases in which only a fraction of dark matter has such non-gravitational interactions. Frequentist profile-likelihood techniques provide prior-independent constraints, circumventing prior-volume effects that we show arise in Bayesian constraints on this model. In the limit where the scattering cross section or the fraction of interacting dark matter approaches zero, the other interacting dark matter model parameters become unconstrained, causing the posterior distribution to favor that region of parameter space. Using Planck 2018 cosmic microwave background anisotropy data, we find a clear impact of prior-volume effects on the posteriors used to place constraints on dark matter scattering. Compared to the frequentist analysis, the Bayesian method consistently overestimates the constraints on the cross section. Given the potentially biased upper limits on models subject to prior-volume effects, such as this one, we recommend supplementing Bayesian constraints with frequentist statistics to better assess the impact of priors.

Figures (7)

• Figure 1: Profile likelihoods of the dark matter-proton scattering cross section $\sigma_0$ for different fixed dark matter particle masses $m_\chi$, as indicated by the color bar, for a fixed interaction fraction $f_\chi=1$. Points indicate computations of the profile likelihood, while the lines show the corresponding Gaussian fit. The dashed horizontal line shows the $\Delta\chi^2=6.18$ cutoff value for a 95.45% C.L., used to determine the upper limits on the cross section.
• Figure 2: Left: Profile-likelihood constraints on the cross section as a function of mass for the interaction fractions $f_\chi$ indicated in the legend. Shaded regions show the area of parameter space that is disfavored by the 95.45% confidence interval. Right: Same as the left panel, but showing Bayesian constraints from our MCMC analyses. The edges of the excluded shaded regions correspond to the 95.45% credible interval contours of the 2D marginal posteriors. For each MCMC analysis, we adjust the priors to accommodate larger cross sections for small interaction fractions as outlined in \ref{['tab:f_sigma_priors']}.
• Figure 3: The ratio of the profile-likelihood upper limits on the interaction cross section $\sigma_{\rm ProfLkl}$ to the MCMC limits $\sigma_{\rm MCMC}$ for fixed interaction fractions $f_\chi$ as indicated by different colors. The MCMC posteriors are more constraining than the profile likelihoods across the full mass range for every choice of fraction.
• Figure 4: Upper: Bayesian constraints obtained for three different priors on the cross section $\sigma_0$ as indicated, all for fixed interaction fraction $f_\chi=1$. The shaded regions indicate the parameter space excluded from the 95.45% credible region. Lower: The same plot as Fig. \ref{['fig:FreqBayesRatio']} but for different priors on the cross section as indicated, all for fixed interaction fraction $f_\chi=1$. The Bayesian posteriors are clearly influenced by the choice of prior: as the prior on $\log_{10}(\sigma_0/\mathrm{cm^2})$ widens, the Bayesian limits become more constraining since the posterior is pulled towards the larger prior volume that is indistinguishable from $\Lambda$CDM.
• Figure 5: Marginal posteriors for the three dark matter-proton scattering parameters for four combinations of priors on the cross section $\sigma_0$ and interaction fraction $f_\chi$ indicated by different colors. Here, shaded regions mark the favored $95.45\%$ credible regions of the parameter space. Expanding the range in either prior to include smaller values causes the posteriors to shift towards the $\Lambda$CDM limit, resulting in tighter constraints on that parameter. Marginalizing over a broader prior volume in the $\Lambda$CDM limit for the cross section results in looser constraints on the fraction, and vice versa. The colored points (with corresponding vertical lines in the one-dimensional plots) show the global maximum likelihood estimates calculated within the corresponding parameter ranges for each prior choice, also recorded in \ref{['tab:f_sigma_priors']}. We find all the MLEs in a region of parameter space indistinguishable from $\Lambda$CDM with values consistent with the MLE for $\Lambda$CDM.