Phase transitions in an expanding universe: stochastic gravitational waves in standard and non-standard histories

Abstract

We undertake a careful analysis of stochastic gravitational wave production from cosmological phase transitions in an expanding universe, studying both a standard radiation as well as a matter dominated history. We analyze in detail the dynamics of the phase transition, including the false vacuum fraction, bubble lifetime distribution, bubble number density, mean bubble separation, etc., for an expanding universe. We also study the full set of differential equations governing the evolution of plasma and the scalar field during the phase transition and generalize results obtained in Minkowski spacetime. In particular, we generalize the sound shell model to the expanding universe and determine the velocity field power spectrum. This ultimately provides an accurate calculation of the gravitational wave spectrum seen today for the dominant source of sound waves. For the amplitude of the gravitational wave spectrum visible today, we find a suppression factor arising from the finite lifetime of the sound waves and compare with the commonly used result in the literature, which corresponds to the asymptotic value of our suppression factor. We point out that the asymptotic value is only applicable for a very long lifetime of the sound waves, which is highly unlikely due to the onset of shocks, turbulence and other damping processes. We also point out that features of the gravitational wave spectral form may hold the tantalizing possibility of distinguishing between different expansion histories using phase transitions.

Theoretical uncertainties for cosmological first-order phase transitions

“…where Ω max,min are the maximum and minimum peak of the SGWB spectrum due to sound waves (sw) [1,97,150]. These are predicted by varying the thermodynamic parameters across the theoretical uncertainty band and utilising the following,…”

“…We estimate the outstanding factor in eq. (3.2), the timescale on which acoustic waves are active, from [150,151]…”

Phase transitions in the early universe

“…Strong phase transitions are a result of a sufficient amount of supercooling, i.e., the phase transition proceeds well below the critical temperature, where the two minima are degenerate, and hence take more time to complete. A more careful calculation of gravitational wave production in an expanding universe [98] shows that a better approximation for the effective source lifetime is (8.19) which assumes that the RMS fluid velocity stays constant until τ nl , when it immediately vanishes. Hence we can estimate 20) which goes as (H n R * ) 2 K 3/2 for fluid flow lifetimes much less than the Hubble time.…”

“…We can now present a simple model for the gravitational wave power spectrum from firstorder phase transitions, which captures the parametric understanding of the power outlined in this section, gives a simple functional form based on numerical simulations [12,71], and includes an explicit attenuation factor due to the decay of the flow [98]. The contribution to the fractional density in gravitational waves in a logarithmic frequency interval d ln f from a phase transition with mean bubble centre separation to Hubble length ratio H n R * generating a kinetic energy fraction K is In this very simple model the power at small s is over-estimated, and at large s under-estimated, compared to the sound shell model, and has significant deviations around the peak for wall speeds near the speed of sound.…”

Probing electroweak phase transition with multi-TeV muon colliders and gravitational waves

“…Taking v b = 0.6 as a benchmark, we are now able to calculate Ω GW (f ) for each FOEWPT data point. 2 The suppression factor coming from the short duration of the sound wave period has been taken into account [102,103].…”