  The imprint of gravitational waves (GWs) on large-scale structures (LSS) is a
useful and promising way to detect or to constrain them. Tensor fossils have
been largely studied in the literature as an indirect way to detect primordial
GWs. In this paper we analyze a new effect induced by primordial GWs: a
correction to the density contrast of the underlying matter distribution of
LSS, as well as its radiation counterpart, induced by the energy density
fluctuation of the gravitational radiation. We perform our derivation of the
full analytical solution of the density contrast for waves entering the horizon
during radiation dominance. We account for two phases in the radiation era,
depending on the main contributor to the perturbed energy density of the
Universe. By comparing the density contrast of cold dark matter and radiation
-- sourced by linear gravitational waves only -- we conclude that the former
overcomes the latter at some time in the radiation era, a behaviour analogous
to their linear counterpart. Then we conclude by discussing the case of density
perturbations produced by GWs entering the Hubble radius during the matter era
as well as their evolution in the late dark-energy dominated phase.
