The curvature perturbations and induced gravitational waves induced by the first-order phase transition during reheating

TL;DR

This work presents a novel mechanism in which a scalar field $\chi$ undergoing a first-order PT modulates the inflaton decay rate $\Gamma$ during reheating, producing superhorizon curvature perturbations without requiring significant vacuum energy release. The stochastic bubble nucleation introduces spatially varying reheating times, leading to curvature perturbations $\delta_H$ that seed second-order scalar-induced GWs with amplitudes governed by the PT parameter $\beta/H_*$ and the ratio $\alpha=\beta/\Gamma$. The authors compute the resulting GW spectrum using a Green-function approach, predicting a distinctive double-peak signature: a large-scale peak set by $k_{\text{cut}}$ and a small-scale peak amplified during the matter-to-radiation transition, with a peak amplitude $\Omega_{\text{GW,peak}}\approx 2.6\times10^{-3}\delta_H^4$. With peak frequencies tied to the end of reheating, the scenario offers a detectable GW signal for space-based detectors like LISA, TianQin, and Taiji, providing a new window into PT dynamics in the early Universe.

Abstract

We propose a novel mechanism where a first-order phase transition modulates the decay rate of a massive field. This modulation, even if the scalar field has negligible energy density, subsequently generates an observable stochastic gravitational-wave background. The stochastic nature of bubble nucleation leads to the asynchrony of phase transitions, generating superhorizon-scale density perturbations through spatial variations in the decay rate $Γ$. These perturbations subsequently source second-order gravitational waves with peak amplitudes governed by the phase transition parameter \(β/H_*\) and decay rate $Γ$. We apply this mechanism in the reheating scanario where the decay rate of inflaton are modulated by the scalar field that undergoes a first-order phase transition. Numerical calculations reveal that the gravitational wave energy spectrum typically reaches \(Ω_{\text{GW}} \sim 10^{-10}\), demonstrating prospects for detection by space-based interferometers like LISA, TianQin and Taiji. This work establishes a new approach to probe phase transition processes in the early Universe without requiring significant vacuum energy release.

Figures (3)

• Figure 1: This figure shows the evolution of the energy densities of radiation $\rho_r$, inflaton in false vacuum $\rho_{m_1}$, and inflaton in true vacuum $\rho_{m_2}$ over time when $\beta/H_* = 100$ and $\alpha = 4$ ($\alpha=1/4$) for left (right) panel. The solid lines describe the evolution of the background energy density, while the dashed lines describe the evolution of the energy density in the delayed decay regions.
• Figure 2: This figure shows the numerical results of $\delta_H$ with differen parameters. The upper (lower) panel describes the change of density perturbations with $\frac{\beta/H_*}{\alpha+1}$ ($\Gamma/H_*$) when a fixed decay rate $\Gamma$ (PT parameter $\beta$). The orange and cyan dashed line in the upper panel represent the constraints of S4 and PBB on density perturbations, respectively.
• Figure 3: This figure we plot the energy spectrum of second-order scalar-induced GWs. The left panel shows the component of the GW energy spectrum independent of density perturbations $\delta_H$, denoted as $\Omega_{\text{GW,0}}/\delta^4_H$, where the black dashed line represents the peak of this component. The right panel presents the GW energy spectrum under the parameter choices of $\beta/H_* = 47$, $\alpha = 4/3$, $k_{\text{max}}/k_{\text{cut}} = 100$, and $k_{\text{cut}} = 0.001\ \text{Hz}$, where the left curve describes the GWs induced by density perturbations $\delta_H$, and the right curve describes the second-order GWs amplified by the rapid transition from the early matter-dominated era to the radiation-dominated era. The cyan and orange dashed lines represent the constraints of PBB and S4 on GWs, respectively.