Generalizing the Bogoliubov vs Boltzmann approaches in gravitational production

Abstract

We investigate the spectral behavior of scalar fluctuations generated by gravity during inflation and the subsequent reheating phase. We consider a non-perturbative Bogoliubov treatment within the context of pure gravitational reheating. We compute both long and short-wavelength spectra, first for a massless scalar field, revealing that the spectral index in part of the infrared (IR) regime varies between $-6$ and $-3$, depending on the post-inflationary equation of state (EoS), $0\leq w_ϕ\leq1$. Furthermore, we study the mass-breaking effect of the IR spectrum by including finite mass, $m_χ$, of the daughter scalar field. We show that for $m_χ/H_{\rm e} \gtrsim 3/2$, where $H_{\rm e}$ is the Hubble parameter during inflation, the IR spectrum of scalar fluctuations experiences exponential mass suppression, while for smaller masses, $m_χ/H_{\rm e}<3/2$, the spectrum remains flat in the IR regime regardless of the post-inflationary EoS. For any general EoS, we also compute a specific IR scale, $k_m$, of fluctuations below which the IR spectrum will suffer from this finite mass effect. In the UV regime, oscillations of the inflaton background lead to interference terms that explain the high-frequency oscillations in the spectrum. Interestingly, we find that for any EoS, $1/9 \lesssim w_ϕ\lesssim 1$, the spectral behavior turns out to be independent of the EoS, with a spectral index $ -6$. We have compared this Bogoliubov treatment for the UV regime to perturbative computations with solutions to the Boltzmann equation and found an agreement between the two approaches for any EoS, $0 \lesssim w_ϕ\lesssim 1$. We also explore the relationship between the gravitational reheating temperature and the reheating EoS employing the non-perturbative analytic approach, finding that reheating can occur for $w_ϕ\gtrsim 0.6$.