Equation of state during (p)reheating with trilinear interactions

TL;DR

This work determines the full post-inflationary expansion history when the inflaton couples to a massless daughter field via a trilinear interaction, by marrying 2+1D lattice simulations of preheating with Boltzmann evolution to track the equation of state from the end of inflation to radiation domination. The authors reveal a three-stage preheating dynamics: an initial tachyonic resonance driving partial fragmentation and a transient rise in the equation of state, followed by a late-time return of the inflaton-dominated, matter-like regime, and then perturbative reheating completing RD. The study yields precise shifts in CMB predictions, lowering the tensor-to-scalar ratio by up to a few parts in $10^{-4}$ and nudging the spectral tilt by about $10^{-3}$, while also predicting a heavily suppressed and redshifted stochastic gravitational-wave background from preheating. Overall, the results demonstrate that the post-inflationary expansion history, including non-perturbative effects, is crucial for accurate inflationary predictions and gravitational-wave forecasts, and provide a framework applicable to broader inflationary potentials and couplings.

Abstract

We characterize the post-inflationary evolution of the equation of state of the universe from the end of inflation until the onset of radiation domination, when the inflaton is coupled to a daughter field through a trilinear interaction. We consider an inflaton potential that is quadratic near the minimum and flattens in the inflationary regime. By simulating the dynamics in 2+1-dimensional lattices, we have tracked the long-term evolution of the equation of state for about ten e-folds of expansion, for various coupling strengths. The trilinear interaction initially excites daughter field modes through a process of tachyonic resonance immediately after inflation and triggers a temporary deviation of the equation of state from $\bar{w} = 0$ to a maximum value $\bar{w} = \bar{w}_{\rm max} < 1/3$. However, at much later times, the inflaton homogeneous mode once again dominates the energy density, pushing the equation of state towards $\bar{w} = 0$ until the onset of perturbative reheating. By combining the lattice results with a Boltzmann approach, we characterize the entire post-inflationary expansion history, which allows to calculate precise predictions for the inflationary CMB observables. We also accurately compute the redshift of the stochastic gravitational wave background produced during preheating, and show that taking the temporary return of the equation of state towards $\bar{w} = 0$ into account can reduce the amplitude by many orders of magnitude relative to previous estimates.

Figures (12)

• Figure 1: Stability chart for the daughter field coupled to the inflaton through a trilinear interaction. We depict the real part of the Floquet index $\nu_k$ as a function of the resonance parameter $\tilde{q}^{(h)}$ and momentum $\tilde{\kappa}$. The white area correspond to the stable region where ${\mathfrak Re}[\nu_k]=0$. The dashed red line shows the maximum momentum $\tilde{\kappa}_{\rm max} = \sqrt{\tilde{q}^{(h)}}$ experiencing tachyonic resonance for a given $\tilde{q}^{(h)}$.
• Figure 2: Value of the daughter field's variance $\langle \chi^2 \rangle_{\rm end}$ when the tachyonic resonance ends, obtained by solving Eqs. \ref{['eq:eomvarphi']} and \ref{['eq:modechi']} for different choices of $q_*^{(h)}$ and $q_*^{(\lambda)}$. The white dashed line indicates the value of $q_{\rm min}^{(h)}$ separating the two regimes described in the main text. The white area depicts the model parameters for which the potential is unstable. The blue diamonds and red circles indicate the two sets of cases simulated in the lattice, see Sect. \ref{['sec:RehCMB']} for more details.
• Figure 3: Evolution of the fraction of energy density stored in the daughter field $|\varepsilon^\chi|$ (light colors) and its oscillation average $\overline{|\varepsilon^\chi|}$ (dark colors). We depict the absolute value of each quantity. The left panel depicts cases for fixed $q_*^{(h)}=20$ and different values of $q_*^{(\lambda)}$, while the right panel shows cases for different $q_*^{(h)}$ and the condition $q_*^{(\lambda)}=(q_*^{(h)})$. The dashed horizontal lines indicate the estimate of $|\varepsilon^\chi|$ at the end of the tachyonic resonance, obtained with Eq. \ref{['eq:varend']} (note that in the right panel all dashed horizontal lines overlap).
• Figure 4: Left panels: Evolution of the energy ratios $\varepsilon_{\alpha}$ for Case A (top) and Case B (bottom) as a function of time $u$ and number of e-folds, obtained from 2+1-dimensional lattice simulations. Note that for the interaction energy ratio we plot the absolute value $|\varepsilon_i|$, as it can become negative. The vertical lines in each panel delimit Stages I, II and III of the post-inflationary evolution, described in the bulk text. Right panels: Evolution of the instantaneous equation of state $w$ (blue) and its oscillation-average $\bar{w}$ (red) for the same cases.
• Figure 5: Evolution of the effective equation of state as a function of number of e-folds $N$ from the end of inflation until the system reaches a radiation dominated state, for different parameter choices in Set A (left) and Set B (right). The solid lines show the initial evolution obtained from the 2+1-dimensional lattice simulations, while the dashed lines show the later evolution obtained by solving the Boltzmann equations \ref{['eq:BE1']}-\ref{['eq:BE3']}. The horizontal dotted-dashed line indicates $\bar{w}_{\rm rd}=1/3$.