Simulating first-order phase transition during inflation

TL;DR

The paper investigates ending inflation through a first-order phase transition by embedding a GUT-scale FoPT into Starobinsky inflation with an inflaton-controlled barrier that decays exponentially as the inflaton rolls. It constructs a two-field potential $U(\chi,\phi)=W(\chi)+V(\phi)$, where the barrier height scales as $a=e^{\beta\chi/M_{\rm Pl}}-1$ and vanishes at $a=0$, suppressing early bubble nucleation and enabling a rapid percolation near the end. Using 3D lattice simulations on an FLRW background, the authors demonstrate bubble nucleation, collisions, and end-of-inflation dynamics, and compute the gravitational-wave spectrum $\Omega_{\rm GW}(k)$ that shows distinctive high-frequency oscillations in agreement with analytical expectations. The work provides a robust GW signature for inflation-triggered FoPTs and motivates further studies of full inflation-era dynamics, multiple nucleation events, and related observational consequences such as particle production and primordial black holes.

Abstract

Ending the inflation by vacuum decay is considered infeasible due to the graceful exit problem. Even if considering an alternative field other than the inflaton to realize a first-order phase transition (FoPT) during inflation, it is usually challenging for concrete model building, as bubble nucleations might not be fast and dense enough to successfully end the inflation. In this work, we propose a FoPT at the grand-unification-theory (GUT) scale within the Starobinsky inflation. The key construction is an exponentially evolving potential barrier dynamically controlled by the rolling inflaton, so that almost no bubble is nucleated during the early inflationary era, but with massive bubble nucleations near the end of inflation. With lattice numerical simulations, we have successfully tested this GUT-FoPT during Starobinsky inflation, and the resulting gravitational-wave energy density spectrum reproduces previous analytical estimation with a distinctive oscillation feature at high frequencies.

Figures (6)

• Figure 1: A 3D visualization of the total potential.
• Figure 2: Comparisons of inflaton field value and Hubble parameter evolutions between the approximated slow-roll solution and the exact solution shifted to match $\epsilon_H(N=0)=1$.
• Figure 3: The number of bubbles per unit comoving Hubble volume (blue) and the corresponding inflaton $\chi$-field value (red) as functions of the e-folding number $N$, where the intersection point presents the necessary condition for bubble collisions within one e-folding number. The inset presents the fitted bubble profile nucleated at the time corresponding to the intersection point.
• Figure 4: Evolution of the cosmological parameters and field averages during the expansion. The panels illustrate the scale factor $a$ (top-left), the comoving Hubble parameter $\mathcal{H}$ (top-right), the equation-of-state parameter $w$ (bottom-left), and the mean values of the scalar fields $\chi$ and $\phi$ (bottom-right) as functions of time. At the initial time $\tau/R^*=0$, the $\chi$ field begins its evolution from a Gaussian random field with an initial mean value of 5.42, while the $\phi$ field is set to zero. As the system evolves to $\tau/R^*=1.1$, the mean value of the $\chi$ field decreases to 1.69, triggering a phase transition in the $\phi$ field that results in bubble nucleation. Subsequently, the mean value of the $\phi$ field increases, while the $\chi$ field oscillates around $\chi=0$ with a gradually diminishing amplitude. After $\tau/R^*=1.95$, the bubbles in the $\phi$ field have expanded to permeate the entire simulation volume.
• Figure 5: The 2d slices for bubbles dynamic of $\phi$ (top panels) and $\chi$(bottom panels) at different time.