  We study the impact of gravitational waves originating from a first order
phase transition on structure formation. To do so, we perform a second order
perturbation analysis in the $1+3$ covariant framework and derive a wave
equation in which second order, adiabatic density perturbations of the
photon-baryon fluid are sourced by the gravitational wave energy density during
radiation domination and on sub-horizon scales. The scale on which such waves
affect the energy density perturbation spectrum is found to be proportional to
the horizon size at the time of the phase transition times its inverse
duration. Consequently, structure of the size of galaxies and bigger can only
be affected in this way by relatively late phase transitions at $\ge
10^{6}\,\text{s}$. Using cosmic variance as a bound we derive limits on the
strength $\alpha$ and the relative duration $(\beta/H_*)^{-1}$ of phase
transitions as functions of the time of their occurrence which results in a new
exclusion region for the energy density in gravitational waves today. We find
that the cosmic variance bound forbids only relative long lasting phase
transitions, e.g. $\beta/H_*\lesssim 6.8$ for $t_*\approx
5\times10^{11}\,\text{s}$, which exhibit a substantial amount of supercooling
$\alpha>20$ to affect the matter power spectrum.

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