The late to early time behaviour of an expanding plasma: hydrodynamisation from exponential asymptotics

We use exponential asymptotics to match the late time temperature evolution of an expanding conformally invariant fluid to its early time behaviour. We show that the rich divergent transseries asymptotics at late times can be used to interpolate between the two regimes with exponential accuracy using the well-established methods of hyperasymptotics, Borel resummation and transasymptotics. This approach is generic and can be applied to any interpolation problem involving a local asymptotic transseries expansion as well as knowledge of the solution in a second region away from the expansion point. Moreover, we present global analytical properties of the solutions such as analytic approximations to the locations of the square-root branch points, exemplifying how the summed transseries contains within itself information about the observable in regions with different asymptotics.