  We consider the asymptotics of flat, radiation-dominated isotropic universes
in four-dimensional theories with quadratic curvature corrections which may
arise when contributions related to the string parameter $\alpha'$ are switched
on. We show that all such universes are singular initially, and calculate all
early time asymptotics using the method of asymptotic splittings. We also
examine the late time asymptotic behaviour of these models and show that there
are no solutions which diverge as $e^{t^2}$, the known radiation solution of
general relativity is essentially the only late time asymptotic possibility in
these models.
