Changing the prior: absolute neutrino mass constraints in nonlocal
  gravity
  Prior change is discussed in observational constraints studies of nonlocally
modified gravity. In the latter, a model characterized by a modification of the
form $\sim m^2 R\Box^{-2}R$ to the Einstein-Hilbert action was compared against
the base $\Lambda$CDM one in a Bayesian way. It was found that the competing
modified gravity model is significantly disfavored (at $22 \,$:$\, 1$ in terms
of betting-odds) against $\Lambda$CDM given CMB+SNIa+BAO data, because of a
dominant tension appearing in the $H_0 \,$-$\, \Omega_M$ plan. We identify the
underlying mechanism generating such a tension and show that it is mostly
caused by the late-time, quite smooth, phantom nature of the effective dark
energy described by the nonlocal model. We find possible solutions for it to be
resolved and explore a given one that consists in extending the initial
baseline from one massive neutrino eigenstate to three degenerate ones, whose
absolute mass $\sum m_\nu \, / \, 3$ is allowed to take values within a
reasonable prior interval. As a net effect, the absolute neutrino mass is
inferred to be non-vanishing at $2 \sigma$ level, best-fitting at $\sum m_\nu
\approx 0.21 {\, \rm eV}$, and the Bayesian tension disappears rendering the
nonlocal gravity model statistically equivalent to $\Lambda$CDM, given recent
CMB+SNIa+BAO data. We also discuss constraints from growth rate measurements $f
\sigma_8$ whose fit is found to be improved by a larger massive neutrino
fraction as well. The $\nu$-extended nonlocal model also prefers a higher value
of $H_0$ than $\Lambda$CDM, therefore in better agreement with local
measurements.
