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.