The nature of dark matter (DM) and how it might interact with the particles of the Standard Model (SM) is one of greatest mysteries currently facing particle physics, and addressing these issues should provide some understanding of how the observed relic abundance was produced. One widely considered production mechanism, a weakly interacting massive particle (WIMP) produced as a thermal relic, provides a target cross section for DM annihilation into SM particles by solving the Boltzmann equation. In this thermal freeze-out mechanism, dark matter is produced in thermal equilibrium with the SM in the early universe, and drops out of equilibrium to its observed abundance as the universe cools and expands. In this paper, we study the impact of a generalized thermodynamics, known as Tsallis statistics and governed by a parameter $q$, on the target DM annihilation cross section. We derive the phase space distributions of particles in this generalized statistical framework, and check their thermodynamic consistency, as well as analyzing the impact of this generalization on the collisional term of the Boltzmann equation. We consider the case of an initial value of $q_0>1$, with $q$ relaxing to 1 as the universe expands and cools, and solve the generalized Boltzmann numerically for several benchmark DM masses, finding the corresponding target annihilation cross sections as a function of $q_0$. We find that as $q$ departs from the standard thermodynamic case of $q=1$, the collisional term falls less slowly as a function of $x = m_chi/T$ than expected in the standard case. We also find that the target cross section falls sharply from $sigma v simeq 2.2-2.6times10^{-26} textrm{cm}^3/textrm{s}$ for $q_0=1$ to, for example, $sigma v simeq 3times 10^{-34} textrm{cm}^3/textrm{s}$ for $q_0=1.05$ for a 100 GeV WIMP.
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