  Production of gravitational waves in the early universe is discussed in a
cosmologically consistent analysis within a first order phase transition
involving a hidden sector feebly coupled with the visible sector. Each sector
resides in its own heat bath leading to a potential dependent on two
temperatures, and on two fields: one a standard model Higgs and the other a
scalar arising from a hidden sector $U(1)$ gauge theory. A synchronous
evolution of the hidden and visible sector temperatures is carried out from the
reheat temperature down to the electroweak scale.The hydrodynamics of two-field
phase transitions, one for the visible and the other for the hidden is
discussed, which leads to separate tunneling temperatures, and different sound
speeds for the two sectors. Gravitational waves emerging from the two sectors
are computed and their imprint on the measured gravitational wave power
spectrum vs frequency is analyzed in terms of bubble nucleation signature,
i.e., detonation, deflagration, and hybrid. It is shown that the two-field
model predicts gravitational waves accessible at several proposed gravitational
wave detectors: LISA, DECIGO, BBO, Taiji and their discovery would probe
specific regions of the hidden sector parameter space and may also shed light
on the nature of bubble nucleation in the early universe. The analysis
presented here indicates that the cosmologically preferred models are those
where the tunneling in the visible sector precedes the tunneling in the hidden
sector and the sound speed $c_s$ lies below its maximum, i.e.,
$c^2_s<\frac{1}{3}$. It is of interest to investigate if these features are
universal and applicable to a wider class of cosmologically consistent models.
