Sub-Planckian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>ϕ</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> inflation in the Palatini formulation of gravity with an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> term

Abstract

The simplest model that can produce inflation is a massive noninteracting scalar particle with potential $V={m}^{2}{\ensuremath{\phi}}^{2}/2$. However, ${\ensuremath{\phi}}^{2}$ chaotic inflation is inconsistent with the observed upper bound on the tensor-to-scalar ratio, $r$. Recently it has been shown that, in the context of the Palatini formalism of gravity with an ${R}^{2}$ term, the ${\ensuremath{\phi}}^{2}$ potential can be consistent with the observed bound on $r$ while retaining the successful prediction for the scalar spectral index, ${n}_{s}$. Here we show that the Palatini ${\ensuremath{\phi}}^{2}{R}^{2}$ inflation model can also solve the super-Planckian inflaton problem of ${\ensuremath{\phi}}^{2}$ chaotic inflation, and that the model can be consistent with Planck scale-suppressed potential corrections, as may arise from a complete quantum gravity theory. If $\ensuremath{\alpha}\ensuremath{\gtrsim}{10}^{12}$, where $\ensuremath{\alpha}$ is the coefficient of the ${R}^{2}$ term, the inflaton in the Einstein frame, $\ensuremath{\sigma}$, remains sub-Planckian throughout inflation. In addition, if $\ensuremath{\alpha}\ensuremath{\gtrsim}{10}^{20}$, then the predictions of the model are unaffected by Planck-suppressed potential corrections in the case where there is a broken shift symmetry, and if $\ensuremath{\alpha}\ensuremath{\gtrsim}{10}^{32}$, then the predictions are unaffected by Planck-suppressed potential corrections in general. The value of $r$ is generally small, with $r\ensuremath{\lesssim}{10}^{\ensuremath{-}5}$ for $\ensuremath{\alpha}\ensuremath{\gtrsim}{10}^{12}$. We calculate the maximum possible reheating temperature, ${T}_{R\mathrm{max}}$, corresponding to instantaneous reheating, for the different regimes of $\ensuremath{\alpha}$. We find that for $\ensuremath{\alpha}\ensuremath{\approx}{10}^{32}$, ${T}_{R\mathrm{max}}$ is approximately ${10}^{10}\text{ }\text{ }\mathrm{GeV}$, with larger values of ${T}_{R\mathrm{max}}$ for smaller $\ensuremath{\alpha}$. For the case of instantaneous reheating, we show that ${n}_{s}$ is in agreement with the 2018 Planck results to within $1\text{\ensuremath{-}}\ensuremath{\sigma}$, with the exception of the $\ensuremath{\alpha}\ensuremath{\approx}{10}^{32}$ case, which is close to the $2\text{\ensuremath{-}}\ensuremath{\sigma}$ lower bound. Following inflation, the inflaton condensate is likely to rapidly fragment, which makes it possible for reheating to occur via the Higgs portal due to inflaton annihilations within oscillons. This typically results in delayed reheating, which is disfavored by the observed value of ${n}_{s}$. In contrast, reheating via inflaton decays to right-handed neutrinos can easily result in instantaneous reheating. We determine the scale of unitarity violation and show that, in general, unitarity is conserved during inflation, although the inflaton field is larger than the unitarity-violation scale. We conclude that the Palatini ${\ensuremath{\phi}}^{2}{R}^{2}$ inflation model provides a completely consistent model of inflation which can be sub-Planckian and consistent with Planck scale-suppressed potential corrections, can reheat successfully, and conserves unitarity during inflation.

Topics & Concepts