Thermal relics as hot, warm and cold dark matter in power-law $f(R)$ gravity

Abstract

We investigate the thermal relics as hot, warm and cold dark matter in $\mathcal{L}={\epsilon }^{2-2\beta }{R}^{\beta }+16\pi m_{\text{Pl}}^{-2}{\mathcal{L}}_m$ gravity, where $\epsilon$ is a constant balancing the dimension of the field equation, and $1<\beta <(4+\sqrt{6})/5$ for the positivity of energy density and temperature. If light neutrinos serve as hot/warm relics, the entropic number of statistical degrees of freedom $g_{\ast s}$ at freeze-out and thus the predicted fractional energy density ${\Omega }_{\psi }{h}^2$ are $\beta -$dependent, which relaxes the standard mass bound $\Sigma m_{\nu }$. For cold relics, by exactly solve the simplified Boltzmann equation in both relativistic and nonrelativistic regimes, we show that the Lee-Weinberg bound for the mass of heavy neutrinos can be considerably relaxed, and the 'WIMP miracle" for weakly interacting massive particles (WIMPs) gradually invalidates as $\beta$ deviates from $\beta =1^{+}$. The whole framework reduces to become that of GR in the limit $\beta \to 1^{+}$.