Acoustic gravitational waves beyond leading-order in bubble over Hubble radius

Abstract We calculate the gravitational wave power spectrum from sound waves in a cosmological first-order phase transition in the unexplored regime of large bubbles, by which we mean that the mean bubble spacing R * is a non-negligible fraction of the Hubble length ℋ * -1 , i.e. R * ℋ * ≲ 𝒪(1). Since the amplitude of the gravitational wave signal increases with R * ℋ * , this is also the loud signal regime. In this regime the effects of gravity, hitherto neglected, become relevant. We carry out the calculation in cosmological perturbation theory expanding in the parameter R * ℋ * , or bubble over Hubble radius. The leading-order term is the standard result for acoustic production of gravitational waves. At next-to-leading-order we find three novel contributions: two contributions arise from general relativistic corrections to the dynamics of both sound waves and gravitational waves. A third contribution comes from secondary gravitational waves induced by curvature perturbations. These contributions suppress the gravitational wave peak amplitude. The suppression factor relative to the leading-order contribution scales as ( R * ℋ * ) 2 , and also depends on other transition parameters, such as the sound speed c s , the duration of the acoustic source, and the peak wavenumber of the velocity field k p . We investigate the range 0.3 ≲ R * ℋ * ≲ 0.7 in a simplified model of the velocity field, finding that the suppression factor lies between 2% and 15% when R * ℋ * ≃ 0.5, but is independent of the root mean squared fluid velocity. We provide analytical approximations to the next-to-leading-order corrections, and a recipe to join them smoothly across different frequency regimes. Our work improves the precision of current estimates of the gravitational wave power spectrum in the relatively unexplored regime of phase transitions with large bubbles.