Primordial fluctuations without scalar fields

We revisit the question of whether fluctuations in hydrodynamical, adiabatical matter could explain the observed structures in our Universe. We consider matter with variable equation of state $w=p_0/{\epsilon }_0$ and a concomitant (under the adiabatic assumption) density dependent speed of sound, $c_s$. We find a limited range of possibilities for a setup when modes start inside the Hubble radius, then leaving it and freezing out. For expanding universes, power-law $w({\epsilon }_0)$ models are ruled out (except when $c_s^2\propto w\ll 1$, requiring post-stretching the seeded fluctuations); but sharper profiles in $c_s$ do solve the horizon problem. Among these, a phase transition in $c_s$ is notable for leading to scale-invariant fluctuations if the initial conditions are thermal. For contracting universes all power-law $w({\epsilon }_0)$ solve the horizon problem, but only one leads to scale-invariance: $w\propto {\epsilon }_0^2$ and $c_s\propto {\epsilon }_0$. This model bypasses a number of problems with single scalar field cyclic models (for which $w$ is large but constant).