  The timescape cosmology represents a potentially viable alternative to the
standard homogeneous cosmology, without the need for dark energy. Although
average cosmic evolution in the timescape scenario only differs substantially
from that of Friedmann-Lemaitre model at relatively late epochs when the
contribution from the energy density of radiation is negligible, a full
solution of the Buchert equations to incorporate radiation is necessary to
smoothly match parameters to the epoch of photon decoupling and to obtain
constraints from cosmic microwave background data. Here we extend the
matter-dominated solution found in earlier work to include radiation, providing
series solutions at early times and an efficient numerical integration strategy
for generating the complete solution. The numerical solution is used to
directly calculate the scale of the sound horizon at decoupling, and at the
baryon drag epoch. The constraints on these scales from the Planck satellite
data yield bounds on the timescape cosmological parameters, which are found to
also agree with the best-fit values from a recent analysis of SDSS-II supernova
data, while avoiding the problem of a primordial lithium-7 abundance anomaly.
