  We study the hydrodynamics of bubble expansion in cosmological first-order
phase transition in the Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW)
background with probe limit. Different from previous studies for fast
first-order phase transition in flat background, we find that, for slow
first-order phase transition in FLRW background with a given peculiar velocity
of the bubble wall, the efficiency factor of energy transfer into bulk motion
of thermal fluid is significantly reduced, thus decreasing the
previously-thought dominated contribution from sound wave to the stochastic
gravitational-wave background.

General Relativity and Quantum Cosmology;Cosmology and Nongalactic Astrophysics;High Energy Physics - Phenomenology