  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.
