Gravitational waves from bubble collisions: An analytic derivation

Abstract

We consider gravitational wave production by bubble collisions during a cosmological first-order phase transition. In the literature, such spectra have been estimated by simulating the bubble dynamics, under so-called thin-wall and envelope approximations in a flat background metric. However, we show that, within these assumptions, the gravitational wave spectrum can be estimated in an analytic way. Our estimation is based on the observation that the two-point correlator of the energy-momentum tensor T (x)T (y) can be expressed analytically under these assumptions. Though the final expressions for the spectrum contain a few integrations that cannot be calculated explicitly, we can easily estimate it numerically. As a result, it is found that the most of the contributions to the spectrum come from single-bubble contribution to the correlator, and in addition the fall-off of the spectrum at high frequencies is found to be proportional to f −1 . We also provide fitting formulae for the spectrum.

Sound Shell Model for Acoustic Gravitational Wave Production at a First-Order Phase Transition in the Early Universe

“…Similarly, for a high wave number, the bubble wall collision signal goes to k −1 [7,11,12] and eventually reemerges. The power law predictions of the sound shell model have recently been compared against results from numerical simulations [20], with some success, particularly for the intermediate and high k power laws.…”

“…In fact, it had been pointed out long before that sound waves were a source of gravitational waves [10], but subsequent work had not appreciated that the sound wave source persisted for long after the phase transition completed, hence boosting the signal by orders of magnitude. The original model of gravitational radiation from the colliding bubble walls may still be relevant for nearvacuum transitions [11], and a semianalytic approach has recently been developed [12].…”

Gravitational waves from a supercooled electroweak phase transition and their detection with pulsar timing arrays

“…Several studies provide more accurate expressions for the GW spectrum from bubble collisions, beyond the simple dimensional analysis. They usually rely on the envelope approximation and numerical simulations [4,5,37], although some analytical formula have also been suggested [36,67]. Using the results of [37], the spectrum today can be described as follows:…”

Probing the baryogenesis and dark matter relaxed in phase transition by gravitational waves and colliders

“…Thus, the peak frequency at current epoch from the three sources can be written as f i = f * i a * /a 0 . For the bubble collision, the corresponding f * co = 0.62β/(1.8 − 0.1v b + v 2 b ) [23] and the phase transition GWs spectrum can be written as [23][24][25][26] …”