Evolution of coupled scalar perturbations through smooth reheating. II. Thermal fluctuation regime

TL;DR

The paper investigates how curvature perturbations evolve through a smooth reheating phase when a thermal plasma coexists with the inflaton, modeled as a two-component system with an inflaton-plasma coupling $\Upsilon(\varphi,T)$. It develops a gauge-invariant, linear perturbation framework including a fluctuation-dissipation noise term whose autocorrelator is fixed by matching quantum-statistical results in a local frame, enabling numerical computation of the curvature power spectrum without slow-roll assumptions. The study probes the model dependence of thermally modified spectra via freeze-out parameters and tests the feasibility of embedding warm inflation within the Standard Model, finding that while there is residual model dependence, SM warm inflation remains viable with modest parameter tuning. The work highlights the role of pre-horizon-exit acoustic oscillations and demonstrates how thermal fluctuations alter the evolution of perturbations across reheating, with implications for high-momentum probes and potential observables beyond the CMB.

Abstract

Curvature perturbations with short wavelengths exit the Hubble horizon when the universe may contain a thermal plasma in addition to an inflaton field that drives its expansion. We solve the corresponding fluctuation-dissipation dynamics at linear order, building upon a previously established set of gauge-invariant evolution equations. The properties of the noise autocorrelator are constrained via a matching of equilibrium correlators to quantum-statistical physics deep inside the Hubble horizon. The curvature power spectrum is determined numerically, without slow-roll approximations or assumptions about the equilibration of the inflaton field. As applications, we scrutinize two issues from recent literature: the model dependence of the thermally modified power spectrum as a function of freeze-out parameters, and the viability of embedding warm inflation within the Standard Model (we offer support for the latter proposal). The role of pre-horizon-exit acoustic oscillations is illustrated.

Figures (7)

• Figure 1: Left: keeping the inflaton potential (cf. eqs. (\ref{['V']}), (\ref{['params']})) and the initial conditions fixed (cf. eqs. (\ref{['init']}), (\ref{['T_ref']})), the contours indicate the time $t^{ }_1$ (in units of $t^{ }_{\hbox{\tiny\rm{ref}}}$ from eq. (\ref{['t_ref']})) at which the "pivot scale", $k^{ }_* / a^{ }_0 = 0.05\,\hbox{Mpc}^{-1}_{ }$, reaches the value $k^{ }_*/(aH) = 10^2_{ }$, as a function of the parameters $\kappa^{ }_m$ and $\kappa^{ }_{\hbox{\tiny\rm{$T$}}}$ determining the friction (cf. eq. (\ref{['Upsilon_ansatz']})). The perturbations are initialized at this moment, cf. eqs. (\ref{['initial_dot_R_k']}) and (\ref{['initial_R_k']}). Middle and right: the background solution from eqs. (\ref{['bg_varphi']})--(\ref{['bg_H']}) for $\kappa^{ }_m = 10^{13}_{ }$, $\kappa^{ }_{\hbox{\tiny\rm{$T$}}} = 10^{8}_{ }$ and $\kappa^{ }_m = 10^{16}_{ }$, $\kappa^{ }_{\hbox{\tiny\rm{$T$}}} = 10^{10}_{ }$, respectively. The first choice corresponds to a weak regime, the second to a strong regime of warm inflation, in the sense defined under point (i) in sec. \ref{['ss:scales']}. The curvature perturbations corresponding to these solutions are shown in fig. \ref{['fig:cases_curvature']}.
• Figure 2: The curvature perturbations corresponding to the background solutions shown in fig. \ref{['fig:cases_background']}. The evolution was started at $k^{ }_*/(aH) = 10^2_{ }$. The dashed orange curve is a solution of eq. (\ref{['matrix_formalism']}) but without noise, obtained by setting tol$=10^{-3}_{ }$ in eq. (\ref{['tolerance']}). In the dotted blue curve, the noise average has been included. The red squares show an example trajectory from the stochastic eq. (\ref{['ito_1']}) (for better visibility, only a subset of points are shown). For the parameters at left, the noise has no visible effect. For the parameters at right, the noiseless solution undergoes acoustic oscillations while decaying. The noise compensates for the damping by generating new fluctuations. Around the time of the horizon exit, the curvature perturbations $\mathcal{R}^{ }_v$ and $\mathcal{R}^{ }_{\hbox{\tiny\rm{$T$}}}$ drag $\mathcal{R}^{ }_\varphi$ to a large common value.
• Figure 3: The probability distributions of the curvature power spectra for the two benchmarks from fig. \ref{['fig:cases_curvature']}, from the weak (left) and strong (right) regime (the individual bins are broader for a broader distribution, so the areas do not look the same). The skewed distribution on the right resembles the one found in ref. ballesteros, where its shape and relation to $\log^{ }_{10}\langle\mathcal{P}^{ }_\mathcal{R}\rangle$ were also determined.
• Figure 4: The main physical characteristics of the background solution at the "freeze-out point", $k^{ }_*/(a H) \equiv 1$, as a function of the parameters $\kappa^{ }_m$ and $\kappa^{ }_{\hbox{\tiny\rm{$T$}}}$ parametrizing $\Upsilon$ (cf. eq. (\ref{['Upsilon_ansatz']})). Left: the temperature $T$, normalized to the Hubble rate $H$. Middle: like at left, but for the friction coefficient $\Upsilon$. The "strong" and "weak" regime, as well as the corner where the thermal plasma has no influence, refer to item (i) in sec. \ref{['ss:scales']}. Right: the inflaton derivative, $\dot{\bar{\varphi}}$, normalized to $H^2_{ }$. Even if the value does not vary much, the variation has a non-trivial shape, which correlates in an intriguing way with curvature perturbations (cf. the text).
• Figure 5: Left: $\langle\mathcal{P}^{ }_\mathcal{R}\rangle / \mathcal{P}^{\hbox{\scriptsize vac}}_\mathcal{R}$ outside of the Hubble horizon, $k^{ }_*/(a H) \equiv 10^{-5}_{ }$, as a function of $\kappa^{ }_m$ and $\kappa^{ }_{\hbox{\tiny\rm{$T$}}}$ parametrizing $\Upsilon$ (cf. eq. (\ref{['Upsilon_ansatz']})). Right: $\langle\mathcal{P}^{ }_\mathcal{R}\rangle / \mathcal{P}^{\hbox{\scriptsize vac}}_\mathcal{R}$ replotted versus the freeze-out parameter $\Upsilon^{ }_*/H^{ }_*$ from fig. \ref{['fig:scans_bg']}(middle). We also compare a specific model ("only $\kappa^{ }_{\hbox{\tiny\rm{$T$}}}$ varies", implying $\Upsilon\sim T^3_{ }$) with fit forms from the literature (cf. the text). The grey band shows that $\langle\mathcal{P}^{ }_\mathcal{R}\rangle$ is not a single-valued function of $Q^{ }_* = \Upsilon^{ }_*/(3 H^{ }_*)$, but displays a large spread as $Q^{ }_*$ increases.