Neutrino Mass Priors for Cosmology from Random Matrices

Cosmological measurements of structure are placing increasingly strong constraints on the sum of the neutrino masses, $\Sigma m_\nu$, through Bayesian inference. Because these constraints depend on the choice for the prior probability $\pi(\Sigma m_\nu)$, we argue that this prior should be motivated by fundamental physical principles rather than the ad hoc choices that are common in the literature. The first step in this direction is to specify the prior directly at the level of the neutrino mass matrix $M_\nu$, since this is the parameter appearing in the Lagrangian of the particle physics theory. Thus by specifying a probability distribution over $M_\nu$, and by including the known squared mass splittings, we predict a theoretical probability distribution over $\Sigma m_\nu$ that we interpret as a Bayesian prior probability $\pi(\Sigma m_\nu)$. We find that $\pi(\Sigma m_\nu)$ peaks close to the smallest $\Sigma m_\nu$ allowed by the measured mass splittings, roughly $0.06 \, {\rm eV}$ ($0.1 \, {\rm eV}$) for normal (inverted) ordering, due to the phenomenon of eigenvalue repulsion in random matrices. We consider three models for neutrino mass generation: Dirac, Majorana, and Majorana via the seesaw mechanism; differences in the predicted priors $\pi(\Sigma m_\nu)$ allow for the possibility of having indications about the physical origin of neutrino masses once sufficient experimental sensitivity is achieved. We present fitting functions for $\pi(\Sigma m_\nu)$, which provide a simple means for applying these priors to cosmological constraints on the neutrino masses or marginalizing over their impact on other cosmological parameters.