  We study dynamic heterogeneities in the out-of-equilibrium coarsening
dynamics of the spherical ferromagnet after a quench from infinite temperature
to its critical point. A standard way of probing such heterogeneities is by
monitoring the fluctuations of correlation and susceptibility, coarse-grained
over mesoscopic regions. We discuss how to define fluctuating coarse-grained
correlations (C) and susceptibilities (Chi) in models where no quenched
disorder is present. Our focus for the spherical model is on coarse-graining
over the whole volume of $N$ spins, which requires accounting for N^{-1/2}
non-Gaussian fluctuations of the spin. The latter are treated as a perturbation
about the leading order Gaussian statistics. We obtain exact results for these
quantities, which enable us to characterise the joint distribution of C and Chi
fluctuations. We find that this distribution is qualitatively different, even
for equilibrium above criticality, from the spin-glass scenario where C and Chi
fluctuations are linked in a manner akin to the fluctuation-dissipation
relation between the average C and Chi. Our results show that coarsening at
criticality is clearly heterogeneous for d>4 and suggest that, as in other
glassy systems, there is a well-defined timescale on which fluctuations across
thermal histories are largest. Surprisingly, however, neither this timescale
nor the amplitude of the heterogeneities increase with the age of the system,
as would be expected from the growing correlation length. For d<4, the strength
of the fluctuations varies on a timescale proportional to the age of the
system; the corresponding amplitude also grows with age, but does not scale
with the correlation volume as might have been expected naively.
