Imprints of a Second Order Electroweak Phase Transition on the Stochastic Gravitational Wave Background

TL;DR

The paper investigates whether a second-order (or crossover) electroweak phase transition occurring at $T \,\sim\ 150\,\mathrm{GeV}$ can leave detectable imprints on the stochastic gravitational wave background in a scenario where the Higgs field acts as the inflaton via non-minimal coupling. It develops the Higgs inflation framework in both Jordan and Einstein frames, estimates the reheating temperature $T_{re}$ (with $N_{re}$) and demonstrates the transition is very weakly first order, effectively signaling a slow-roll toward a new minimum that deforms the radiation-dominated equation of state. This EoS deformation imparts a characteristic modulation to the primordial GW spectrum for modes entering the horizon during the transition, parameterized by the total EoS $w$ and the scale $k_s$, and can yield a signal detectable by LiteBIRD for certain $w$ and reheating temps $T_R$. The work connects inflationary dynamics, reheating, and EWPT physics to provide a unique observational handle on the early Universe via gravitational waves.

Abstract

In this work we shall study the impact of a second order electroweak phase transition occurring at $\sim 150\,$GeV on the energy spectrum of the stochastic gravitational background. Specifically, we assume that the non-minimally coupled Higgs field controls the inflationary era, we find the reheating temperature for the Higgs inflationary model and we demonstrate that the Higgs effective potential exhibits a very weak first order phase transition. This weak first order phase transition is an indication that the electroweak phase transition may not actually proceed as a first order phase transition, but it will proceed as a crossover or second order phase transition. This second order phase transition proceeds with the Higgs field slow-rolling its potential toward to its new minimum. This slow-rolling may deform the radiation domination total equation of state, and the aim of this work is to pinpoint the observational imprints of this total equation of state deformation on the energy spectrum of the primordial gravitational waves, that affects modes that enter the horizon at temperatures $T\sim 150\,$GeV or lower.

Figures (5)

• Figure 1: The reheating temperature $T_{re}$ (GeV) versus the scalar spectral index for the Higgs inflation for various values of the scalar spectral index (depending on the total duration of the inflationary era which we took to be $50-60$$e$-foldings) and for various values of the reheating background EoS parameter $w$.
• Figure 2: The reheating temperature $T_{re}$ (GeV) versus the scalar spectral index for the Higgs inflation for various values of the scalar spectral index (depending on the total duration of the inflationary era which we took to be $50-60$$e$-foldings) and for various values of the reheating background EoS parameter $w$.
• Figure 3: The effective potential of the SM for various temperatures near the critical temperature $T_c\sim 147.693\,$GeV. The phase transition is a very weak first order transition, so weak that it qualifies for a crossover or second order phase transition.
• Figure 4: The $h^2$-scaled gravitational wave energy spectrum for the Higgs inflation model with a Higgs second order phase transition,. The deformed background EoS has the value $w=0.25$ and we considered $T_R=500\,$GeV, $T_R=10^7\,$GeV, and $T_R=10^{12}\,$GeV.
• Figure 5: The $h^2$-scaled gravitational wave energy spectrum for the Higgs inflation model with a Higgs second order phase transition,. The deformed background EoS has the value $w=0.15$ and we considered $T_R=500\,$GeV, $T_R=10^7\,$GeV, and $T_R=10^{12}\,$GeV.