Stochastic Gravitational Wave Production After Inflation

TL;DR

This paper investigates gravitational waves produced during preheating after inflation, focusing on non-thermal, resonant dynamics that generate large inhomogeneities. Using lattice simulations (LatticeEasy) for λφ^4 and m^2 φ^2 inflation, it computes the resultant GW spectra from the evolving stress-energy tensor in the linearized gravity regime, yielding a sizable stochastic background with peak amplitude Ω_{gw} h^2 ≈ 10^{-10} and peak frequencies in the 10^6–10^9 Hz range. The findings imply that preheating can imprint a detectable high-frequency GW signal, whose peak location reddens as inflation scales decrease, and offer guidance for future high-frequency detector development. The work lays groundwork for extending analyses to broader inflationary classes and for improving the gravitational-wave calculation by solving the full perturbed Einstein equations.

Abstract

In many models of inflation, the period of accelerated expansion ends with preheating, a highly non-thermal phase of evolution during which the inflaton pumps energy into a specific set of momentum modes of field(s) to which it is coupled. This necessarily induces large, transient density inhomogeneities which can source a significant spectrum of gravitational waves. In this paper, we consider the generic properties of gravitational waves produced during preheating, perform detailed calculations of the spectrum for several specific inflationary models, and identify problems that require further study. In particular, we argue that if these gravitational waves exist they will necessarily fall within the frequency range that is feasible for direct detection experiments -- from laboratory through to solar system scales. We extract the gravitational wave spectrum from numerical simulations of preheating after $λφ^4$ and $m_φ^2 φ^2$ inflation, and find that they lead to a gravitational wave amplitude of around $Ω_{gw}h^2\sim 10^{-10}$. This is considerably higher than the amplitude of the primordial gravitational waves produced during inflation. However, the typical wavelength of these gravitational waves is considerably shorter than LIGO scales, although in extreme cases they may be visible at scales accessible to the proposed BBO mission. We survey possible experimental approaches to detecting any gravitational wave background generated during preheating.

Figures (6)

• Figure 1: The gravitational spectrum for the $\lambda \phi^4$ model with $\lambda=10^{-14}$ and $g^2/\lambda=1.2$ (dash line) and $120$ (full line) respectively . As expected, it is peaked around $10^7\sim 10^8$ Hz and spans about 2 decades. The horizon size at the time of preheating imposes the low frequency cut-off, while the high frequency cut-off is due to the fact that high momentum $\chi$ particles are energetically too expensive to be created. Notice that the power is roughly inversely proportional to $g^2$.
• Figure 2: The variance of the $\phi$ (dash) and $\chi$ (full) fields for a $\lambda \phi^4$ inflation model with $g^2/\lambda=120$. The time coordinate is the conformal time $\tau$ in units of $2.5\times 10^{-12} \textrm{GeV}^{-1}$ while the variances are in $m_{\hbox{\tiny Pl}}^2$ units. The associated hubble parameter during the rapid rise while in major resonance phase is $H\approx 10^{11}$ GeV, thus this phase lasts less than a Hubble time.
• Figure 3: The variance of the $\phi$ (dash) and $\chi$ (full) fields for $m_{\phi}^2\phi^2$ inflation model with $q=2.5\times 10^{5}$. The time coordinate is the cosmic time $t$ in units of $m_{\phi}^{-1}=8.33\times 10^{-14} \textrm{GeV}^{-1}$ while the variances are in $m_{\hbox{\tiny Pl}}^2$ units. The associated hubble parameter during the rapid rise while in major resonance phase is $H\approx 10^{12}$ GeV, thus this phase lasts much less than a Hubble time.
• Figure 4: The gravitational spectrum for a $m_{\phi}^2\phi^2$ inflation model, with parameters $g^2=2.5 \times 10^{-7}$ and $m_{\phi}=10^{-6}m_{\hbox{\tiny Pl}} \approx 10^{13}$ GeV. The resonance parameter here $q=2.5 \times 10^{5}$. The slight rise at high frequencies after the sharp drop is a numerical artifact.
• Figure 5: The gravitational spectrum for a $m_{\phi}^2\phi^2$ inflation model, with parameters $g^2=2.5 \times 10^{-7}$ and $m_{\phi}\approx10^{-7}m_{\hbox{\tiny Pl}} =10^{12}$ GeV. The resonance parameter here $q=2.5 \times 10^{5}$. Although this lower mass model is ruled out by the CMB, it serves as a useful prototype to illustrate that the peak will be reddened for a lower scale inflation.