  We investigate the curvature perturbations induced by the randomness of the
quantum tunneling process during cosmological first-order phase transitions
(PTs) and for the first time ultilize curvature perturbations to constrain the
PT parameters. We find that the observations of the cosmic microwave background
spectrum distortion and the ultracompact minihalo abundance can give strict
constraints on the PTs below 100GeV, especially for the low-scale PTs and the
weak PTs. The current constraint on the PT parameters is largely extended by
the results in this work.
