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We study the non-equilibrium dynamics of the spherical ferromagnet quenchedto its critical temperature, as a function of the magnetization of the initialstate. The two limits of unmagnetized and fully magnetized initial conditionscan be understood as corresponding to times that are respectively much shorterand much longer than a magnetization timescale, as in a recent fieldtheoretical analysis of the $n$-vector model. We calculate exactly thecrossover functions interpolating between these two limits, for themagnetization correlator and response and the resulting fluctuation-dissipationratio (FDR). For $d>4$ our results match those obtained recently from aGaussian field theory. For $d<4$, non-Gaussian fuctuations arising from thespherical constraint need to be accounted for. We extend our framework from thefully magnetized case to achieve this, providing an exact solution for therelevant integral kernel. The resulting crossover behaviour is very rich, withthe asymptotic FDR $X^\infty$ depending non-monotonically on the scaled age ofthe system. This is traced back to non-monotonicities of the two-timecorrelator, themselves the consequence of large magnetization fluctuations onthe crossover timescale. We correct a trivial error in our earlier calculationfor fully magnetized initial states; the corrected FDR is {\em consistent} withrenormalization group expansions to first order in $4-d$ for the longitudinalfluctuations of the O(n) model in the limit $n\to\infty$.

Fluctuation-dissipation relations in critical coarsening: crossover from unmagnetized to magnetized initial states