Small Field Polynomial Inflation: Reheating, Radiative Stability and Lower Bound

We revisit the renormalizable polynomial inflection point model of inflation, focusing on the small field scenario which can be treated fully analytically. In particular, the running of the spectral index is predicted to be $\alpha = -1.43 \times 10^{-3} +5.56 \times 10^{-5} \left(N_{\rm CMB}-65 \right)$, which might be tested in future. We also analyze reheating through perturbative inflaton decays to either fermionic or bosonic final states via a trilinear coupling. The lower bound on the reheating temperature from successful Big Bang nucleosynthesis gives lower bounds for these couplings; on the other hand radiative stability of the inflaton potential leads to upper bounds. In combination this leads to a lower bound on the location $\phi_0$ of the near inflection point, $\phi_0 > 3 \cdot 10^{-5}$ in Planckian units. The Hubble parameter during inflation can be as low as $H_{\rm inf} \sim 1$ MeV, or as high as $\sim 10^{10}$ GeV. Similarly, the reheating temperature can lie between its lower bound of $\sim 4$ MeV and about $4 \cdot 10^8 \ (10^{11})$ GeV for fermionic (bosonic) inflaton decays. We finally speculate on the "prehistory" of the universe in this scenario, which might have included an epoch of eternal inflation.