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

We investigate the thermal relics as hot, warm and cold dark matter in $\mathscr{L}=\varepsilon^{2-2β}R^β+{16π}m_{\text{Pl}}^{-2}\mathscr{L}_m$ gravity, where $\varepsilon$ is a constant balancing the dimension of the field equation, and $1<β<(4+\sqrt{6})/5$ for the positivity of energy density and temperature. If light neutrinos serve as hot/warm relics, the entropic number of statistical degrees of freedom $g_{*s}$ at freeze-out and thus the predicted fractional energy density $Ω_ψh^2$ are $β-$dependent, which relaxes the standard mass bound $Σm_ν$. 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 $β$ deviates from $β=1^+$. The whole framework reduces to become that of GR in the limit $β\to 1^+$.