Theory and Numerics of Gravitational Waves from Preheating after Inflation

TL;DR

The work develops a comprehensive theory and numerical framework to compute the stochastic gravitational-wave background produced by post-inflation preheating in an expanding universe. By formulating GW generation through the transverse-traceless source $T_{ij}^{\rm TT}$, projecting it in Fourier space, and performing ensemble averages, the authors provide a robust method to obtain the present-day spectrum $d\rho_{\rm gw}/d\ln k$ from non-linear scalar-field dynamics. Key findings show that the dominant GW production occurs during the non-linear bubbly stage, with the peak frequency and amplitude primarily determined by the characteristic momentum $k_*$ amplified during preheating; analytical checks in the $\lambda\phi^4$ model confirm numerical results, while preliminary hybrid-inflation estimates suggest potential observability in some parameter regimes. The framework enables systematic predictions across inflationary scenarios and informs the prospects for direct GW detection as a probe of early-universe dynamics.

Abstract

Preheating after inflation involves large, time-dependent field inhomogeneities, which act as a classical source of gravitational radiation. The resulting spectrum might be probed by direct detection experiments if inflation occurs at a low enough energy scale. In this paper, we develop a theory and algorithm to calculate, analytically and numerically, the spectrum of energy density in gravitational waves produced from an inhomogeneous background of stochastic scalar fields in an expanding universe. We derive some generic analytical results for the emission of gravity waves by stochastic media of random fields, which can test the validity/accuracy of numerical calculations. We contrast our method with other numerical methods in the literature, and then we apply it to preheating after chaotic inflation. In this case, we are able to check analytically our numerical results, which differ significantly from previous works. We discuss how the gravity wave spectrum builds up with time and find that the amplitude and the frequency of its peak depend in a relatively simple way on the characteristic spatial scale amplified during preheating. We then estimate the peak frequency and amplitude of the spectrum produced in two models of preheating after hybrid inflation, which for some parameters may be relevant for gravity wave interferometric experiments.

Figures (10)

• Figure 1: Would be emission of a graviton $h_{ij}$ with momentum $\mathbf{k}$ from the annihilation of two scalar waves $\phi(\mathbf{p})$ and $\phi(\mathbf{k} - \mathbf{p})$ with momenta $\mathbf{p}$ and $\mathbf{k} - \mathbf{p}$. Helicity $2$ of the emitted graviton cannot match the helicity zero of the incoming scalar waves.
• Figure 2: Two-dimensional slice through a three-dimensional realization of the scalar field $\chi$ from a numerical simulation of preheating in the $\lambda \phi^4 +g^2\phi^2 \chi^2$ model of the next section. The horizontal axes correspond to spatial coordinates and the vertical axis corresponds to the field's values.
• Figure 3: Spectrum of energy density in gravity waves calculated along nine different directions in $\mathbf{k}$-space. The parameters for this run were the same as for Fig. \ref{['omegafq120']}, see sub-section \ref{['numerics']} for details.
• Figure 4: Spectrum of gravity wave energy density in physical variables today (\ref{['spectoday']}), accumulated up to the time $x_f = 240$, for the model (\ref{['lambdaphi4']}) with $q = 120$. The 2 spectra were obtained from simulations with different box sizes, and averaged over different directions in $\mathbf{k}$-space.
• Figure 5: The thick curve shows the total energy density in gravity waves (\ref{['rhogwtoday']}) accumulated up to the time $x_f$, as a function of $x_f$. The thin curve shows the evolution with time of the total particles number density, $n_{\mathrm{tot}} = n_{\chi} + n_{\phi}$, rescaled to fit on the same figure.