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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 α = -1.43 × 10-3 +5.56 × 10-5(NCMB-65 ), 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 ϕ0 of the near inflection point, ϕ0 > 3 · 10-5 in Planckian units. The Hubble parameter during inflation can be as low as Hinf∼ 1 MeV, or as high as ∼ 1010 GeV. Similarly, the reheating temperature can lie between its lower bound of ∼ 4 MeV and about 4 · 108 (1011) 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.

The self coupling $$\lambda $$ λ of the Higgs boson in the Standard Model may show critical behavior, i.e. the Higgs potential may have a point at an energy scale $$\sim 10^{17-18}$$ ∼ 10 17 - 18 GeV where both the first and second derivatives (almost) vanish. In this case the Higgs boson can serve as inflaton even if its nonminimal coupling to the curvature scalar is only $${{\mathcal {O}}}(10)$$ O ( 10 ) , thereby alleviating concerns about the perturbative unitarity of the theory. We find that just before the Higgs as inflaton enters the flat region of the potential the usual slow-roll conditions are violated. This leads to “overshooting” behavior, which in turn strongly enhances scalar curvature perturbations because of the excitation of entropic (non-adiabatic) perturbations. For appropriate choice of the free parameters these large perturbations occur at length scales relevant for the formation of primordial black holes. Even if these perturbations are not quite large enough to trigger copious black hole formation, they source second order tensor perturbations, i.e. primordial gravitational waves; the corresponding energy density can be detected by the proposed space-based gravitational wave detectors DECIGO and BBO.

It is typically assumed that during reheating the inflaton decays with a constant decay width. However, this is not guaranteed and can have a strong impact on the dark matter (DM) genesis. In the context of the ultraviolet (UV) freeze-in mechanism, if the operators connecting the dark and visible sectors are of sufficiently high mass dimension, the bulk of the DM abundance is produced during and not after reheating. We study here the impact of a time-dependent decay width of the inflaton on the DM abundance, emphasizing the differences with respect to the cases where the decay is either instantaneous or constant. We also provide concrete examples for DM production via UV freeze-in, e.g., from 2-to-2 scatterings of standard model particles, or from inflaton scatterings or decays, elucidating how the time-dependence influences the DM yield.

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