Future global stability of Maxwell-J\"uttner equilibria and vacuum for the massless Boltzmann equation on FLRW spacetimes

In this work we study the general relativistic massless Boltzmann equation on Friedmann-Lema\^itre-Robertson-Walker spacetimes with spatial topology $\mathbb{T}^3$ in the linear and decelerated expanding regimes, where the scale factor is $t^{\mathfrak{q}}$ with $\mathfrak{q}\in [0,1]$. The massless Boltzmann equation on these backgrounds admits non-stationary Maxwell-J\"uttner equilibria of the form $\exp(- |t^{2\mathfrak{q}}p|)$. For $0 \leq \mathfrak{q} \leq 1$, we prove future global-in-time existence and uniqueness of small perturbations of these equilibria in the case of hard ball interaction without symmetry assumptions. For $0\leq \mathfrak{q} < 1/3$, we prove that the perturbation -- measured in a suitable $L^2_p$ based energy norm -- decays at the superpolynomial time-decay rate of $t^{-3\mathfrak{q}}\exp(-t^{1-3\mathfrak{q}})$, whereas for $1/3< \mathfrak{q} \leq 1$ we obtain the polynomial time-decay rate of $t^{-3\mathfrak{q}}$. In the borderline case $\mathfrak{q}=1/3$, we show the time-decay of $t^{-3\mathfrak{q} -c}$ with a uniform constant $c>0$. Finally, for $\frac{1}{3}< \mathfrak{q}\leq 1$, we prove future global-in-time existence and uniqueness of small perturbations of the vacuum solution on $\mathbb{T}^3$.