Constraints on just enough inflation preceded by a thermal era
  If the inflationary era is preceded by a radiation dominated era in which the
inflaton too was in thermal equilibrium at some very early time then the CMB
data places an upper bound on the comoving temperature of the (decoupled)
inflaton quanta. In addition, if one considers models of "just enough"
inflation, where the number of e-foldings of inflation is just enough to solve
the horizon and flatness problems, then we get a lower bound on the Hubble
parameter during inflation, $H_{\rm inf}$, which is in severe conflict with the
upper bound from tensor perturbations. Alternatively, imposing the upper bound
on $H_{\rm inf}$ implies that such scenarios are compatible with the data only
if the number of relativistic degrees of freedom in the thermal bath in the
pre-inflationary Universe is extremely large (greater than $10^9$ or
$10^{11})$. We are not aware of scenarios in which this can be satisfied.
