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

Abstract

A model for the acoustic production of gravitational waves at a first-order phase transition is presented. The source of gravitational radiation is the sound waves generated by the explosive growth of bubbles of the stable phase. The model assumes that the sound waves are linear and that their power spectrum is determined by the characteristic form of the sound shell around the expanding bubble. The predicted power spectrum has two length scales, the average bubble separation and the sound shell width when the bubbles collide. The peak of the power spectrum is at wave numbers set by the sound shell width. For a higher wave number k, the power spectrum decreases to k −3 . At wave numbers below the inverse bubble separation, the power spectrum goes to k 5 . For bubble wall speeds near the speed of sound where these two length scales are distinguished, there is an intermediate k 1 power law. The detailed dependence of the power spectrum on the wall speed and the other parameters of the phase transition raises the possibility of their constraint or measurement at a future space-based gravitational wave observatory such as LISA. DOI: 10.1103/PhysRevLett.120.071301 Interest in gravitational waves from a first-order electroweak phase transition in the early Universe [1-3] has greatly increased following the European Space Agency's approval of a space-based gravitational wave observatory [4] and the detection of gravitational waves from a merging black hole binary [5]. At the same time, it has been realized that early work on gravitational waves from a thermal phase transition [6,7] greatly underestimated their energy density [8]. The first three-dimensional hydrodynamic simulations [8,9] revealed that the dominant source of gravitational waves was acoustic production from sound waves generated by the explosive growth of bubbles of the stable phase. 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].The simulations in Refs. [8,9] revealed a power spectrum peaked at a wavelength around the average bubble separation R Ã , with a power law k −p at wave number k ≫ R −1 Ã . Where the power law was clear, the index was somewhere in the range −3 ≲ p ≲ −4. There was also evidence for some structure in the peak: where the bubble wall speed v w was closer to the speed of sound, the peak was broader. Understanding the gravitational wave power spectrum is of great importance for LISA's detection prospects [13], so it is vital to have a better physical understanding of the numerical simulations.In this Letter, I outline a model for the acoustic gravitational wave power spectrum based on the observation that the shel...

Gravitational waves from first order cosmological phase transitions in the Sound Shell Model

“…The power laws observed in [12] are confirmed, and the shape around the peak of the power spectra reproduced for a range of wall speeds. We correct an error made in [13], leading to revised predictions for the power laws at low wavenumber.…”

“…In a previous work [13], one of us outlined a model for the acoustic gravitational wave power spectrum, called the Sound Shell Model. It is based on the observation that the sound waves set up by the compression shells around the expanding bubble of the stable phase continue to propagate after the phase boundaries driving them have disappeared.…”

“…In [13], it was shown that the model makes clear predictions for the general shape of the velocity and gravitational wave power spectra: they are double broken power laws, with the breaks in slope at wave-numbers k dictated by the mean bubble separation R * , and the sound shell thickness ∆R * = R * ∆ w , where ∆ w = |v w − c s |/v w for wall speeds which are not much less than c. For large k∆R * , the gravitational wave power law index is −3. For wall speeds near the speed of sound c s , the sound shell is thin, and there is a characteristic k 1 power law in the range R −1 * k ∆R −1 * .…”

Detecting gravitational waves from cosmological phase transitions with LISA: an update

“…The shocks decay by generating entropy, on the same timescale, setting the lifetime of the acoustic phase. The autocorrelation time in acoustic production is set by the sound-crossing time of the flow length scale τ c ∼ L f /c s ∼ R * /c s [54]. Applying these timescales, one finds for long-lasting acoustic production (c.f.…”

“…The ansatz below, based on the simulations in [19,20] and the analytic understanding in [54,62], is most reliable when the sound shell surrounding the bubble wall is large. This means, in turn, that the wall speed must be far from the Chapman-Jouguet speed, in which case the fluid just reaches sonic velocity in the frame of the shock.…”