  We study the asymptotic behaviour of solutions to the linear wave equation on
cosmological spacetimes with Big Bang singularities and show that appropriately
rescaled waves converge against a blow-up profile. Our class of spacetimes
includes Friedman-Lema\^itre-Robertson-Walker (FLRW) spacetimes with negative
sectional curvature that solve the Einstein equations in presence of a perfect
irrotational fluid with $p=(\gamma-1)\rho$. As such, these results are closely
related to the still open problem of past nonlinear stability of such FLRW
spacetimes within the Einstein scalar field equations. In contrast to earlier
works, our results hold for spatial metrics of arbitrary geometry, hence
indicating that the matter blow-up in the aforementioned problem is not
dependent on spatial geometry. Additionally, we use the energy estimates
derived in the proof in order to formulate open conditions on the initial data
that ensure a non-trivial blow-up profile, for initial data sufficiently close
to the Big Bang singularity and with less harsh assumptions for $\gamma<2$.
