Inconsistency of an inflationary sector coupled only to Einstein gravity

From a model-building perspective, the inflationary sector might very well have no direct couplings to other species, apart from inevitable gravitational interactions. Within the context of General Relativity, a thermal universe can still emerge after inflation if: $i)$ some radiation sector is excited towards the end of inflation, and $ii)$ the post-inflationary equation of state becomes sufficiently stiff $w \geq w_{\rm RD}\gtrsim 0.57$, with $w_{\rm RD}$ a threshold depending on the inflationary scale $H_*$ and the initial radiation-to-inflaton energy ratio $\Delta_*$. Furthermore, a stiff period in the expansion history enhances significantly the inflationary gravitational wave (GW) background, making this signal (potentially) observable by aLIGO, LISA and other experiments. The very same enhancement leads however to an inconsistency of the scenario: the energy of the GWs becomes too large compared to the rest of the radiation sector, violating standard BBN and CMB bounds on GW backgrounds. Except for very special scenarios where the initial radiation sector comprises hundreds of fields with couplings tuned to specific values, our result applies independently of $w$, $H_*$ and $\Delta_*$. This suggests that in order to reheat the universe, the inflationary sector should be coupled directly to other particle species. Alternatively the inflationary sector could be implemented in modified gravity theories.