General analytic methods for solving coupled transport equations: From cosmology to beyond

We propose a general method to analytically solve transport equations during a phase transition without making approximations based on the assumption that any transport coefficient is large. Using a cosmic phase transition in the minimal supersymmetric standard model as a pedagogical example, we derive the solutions to a set of 3 transport equations derived under the assumption of supergauge equilibrium and the diffusion approximation. The result is then rederived efficiently using a technique we present involving a parametrized ansatz which turns the process of deriving a solution into an almost elementary problem. We then show how both the derivation and the parametrized ansatz technique can be generalized to solve an arbitrary number of transport equations. Finally we derive a perturbative series that relaxes the usual approximation that inactivates vacuum-expectation-value dependent relaxation and $CP$-violating source terms at the bubble wall and through the symmetric phase. Our analytical methods are able to reproduce a numerical calculation in the literature.

Figure 1

The density $n_L$ in the symmetric phase near the bubble wall as a function of $z$. The exact solution with first-order corrections to the ultrathin wall approximation is given by the magenta line and the solution without such corrections is given in blue. The parameters used to calculate $n_L(Z)$ are $L_W=250/T$ (top left), $L_W=100/T$ (top right), $L_W=25/T$ (bottom left) and $L_W=10/T$ (bottom right).

Figure 2

Reproducing Winslow and Tulin’s purely numerical calculation. Here the blue, magenta and golden lines correspond to the densities $n_{Q_3}$, $n_{u_3}$ and $n_H$ respectively.