Nonlinear dynamics from the relativistic Boltzmann equation in the Friedmann-Lemaître-Robertson-Walker spacetime

The dissipative dynamics of an expanding massless gas with constant cross section in a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) universe is studied. The mathematical problem of solving the full nonlinear relativistic Boltzmann equation is recast into an infinite set of nonlinear ordinary differential equations for the moments of the one-particle distribution function. Momentum-space resolution is determined by the number of nonhydrodynamic modes included in the moment hierarchy, i.e., by the truncation order. We show that in the FLRW spacetime the nonhydrodynamic modes decouple completely from the hydrodynamic degrees of freedom. This results in the system flowing as an ideal fluid while at the same time producing entropy. The solutions to the nonlinear Boltzmann equation exhibit transient tails of the distribution function with nontrivial momentum dependence. The evolution of this tail is not correctly captured by the relaxation time approximation nor by the linearized Boltzmann equation. However, the latter probes additional high-momentum details unresolved by the relaxation time approximation. While the expansion of the FLRW spacetime is slow enough for the system to move towards (and not away from) local thermal equilibrium, it is not sufficiently slow for the system to actually ever reach complete local equilibrium. Equilibration is fastest in the relaxation time approximation, followed, in turn, by kinetic evolution with a linearized and a fully nonlinear Boltzmann collision term.

Figure 1

Evolution of the normalized moments $M_n$ (56) [panel (a)] and the Laguerre moments $c_n$ (59) [panel (b)] as a function of the dimensionless time variable $\tau$.

Figure 2

Ratio between the out-of-equilibrium solution (62) and the equilibrium distribution as a function of $k/T_0$.

Figure 3

Evolution of the moments $M_{10}$ (left column) and $M_{20}$ (right column) as a function of the dimensionless time $\tau$ according to the nonlinear evolution equation (28), its linearized version (42), and within the RTA (55) for the ES-IC (76) [panels (a)–(b)], 1M-IC (77) [panels (c)–(d)], and 2M-IC (78) [panels (e)–(f)]. See text for further details.

Figure 4

Evolution of the Laguerre moments as a function of $n$ according to the nonlinear Boltzmann equation (red circle), linear approximation (blue triangle) and RTA (green square) for fixed values of $\tau =\{0.5,4,8\}$ (left, middle and right column respectively). For the initial conditions of the distribution function we use the ES-IC (76) [panels (a),(b),(c)], 1M-IC (77) [panels (d),(e),(f)], and 2M-IC (78) [panels (g),(h),(i)].

Figure 5

Snapshots of the full nonlinear distribution function as a function of $k/T_0$ at different values of $\tau =\{0,2,4,8\}$ (with fugacity $\lambda =1$). For the initial conditions of the distribution function we use the ES-IC (76) [panel (a)], 1M-IC (77) [panel (b)], and 2M-IC (78) [panel (c)].

Figure 6

Snapshots of the ratio $F(\tau ,k/T_0)\equiv f_k(\tau )/f_k^{\mathrm{eq}}(\tau )$ as a function of $k/T_0$ for $\tau =\{1,8,15\}$ (left, middle and right column) according to the nonlinear Boltzmann equation (red line), linear approximation (blue dashed line) and RTA (green dotten line). For the initial conditions of the distribution function we use the ES-IC (76) [panels (a),(b),(c)], 1M-IC (77) [panels (d),(e),(f)], and 2M-IC (78) [panels (g),(h),(i)].

Figure 7

Evolution of the ratio $F(\tau ,k/T_0)\equiv f_k(\tau )/f_k^{\mathrm{eq}}(\tau )$ as a function of $\tau$, for fixed values of momentum $k/T_0=10$ (left column) and $k/T_0=20$ (right column), for the full nonlinear (red line), linear (dotted blue line) and RTA collision term (green dotten line). For the initial conditions we use the ES-IC (76) [panels (a),(b)], 1M-IC (77) [panels (c),(d)], 2M-IC (78) [panels (e),(f)].

Figure 8

Time evolution of the produced entropy as a fraction of its initial value, $\mathrm{\Delta }\mathcal{S}(\tau )$ as defined in Eq. (79), for initial conditions (a) ES-IC (76), (b) 1M-IC (77), and (c) 2M-IC (78).