In a cosmological first-order phase transition, bubbles of the stable phase nucleate and expand in the supercooled metastable phase. In many cases, the growth of bubbles reaches a stationary state, with bubble walls propagating as detonations or deflagrations. However, these hydrodynamical solutions may be unstable under corrugation of the interface. Such instability may drastically alter some of the cosmological consequences of the phase transition. Here, we study the hydrodynamical stability of deflagration fronts. We improve upon previous studies by making a more careful and detailed analysis. In particular, we take into account the fact that the equation of motion for the phase interface depends separately on the temperature and fluid velocity on each side of the wall. Fluid variables on each side of the wall are similar for weakly first-order phase transitions, but differ significantly for stronger phase transitions. As a consequence, we find that, for large enough supercooling, any subsonic wall velocity becomes unstable. Moreover, as the velocity approaches the speed of sound, perturbations become unstable on all wavelengths. For smaller supercooling and small wall velocities, our results agree with those of previous works. Essentially, perturbations on large wavelengths are unstable, unless the wall velocity is higher than a critical value. We also find a previously unobserved range of marginally unstable wavelengths. We analyze the dynamical relevance of the instabilities, and we estimate the characteristic time and length scales associated to their growth. We discuss the implications for the electroweak phase transition and its cosmological consequences.Comment: 45 pages, 13 figures. v2: Minor corrections, references added. v3: Typos corrected, minor modifications and references added (version accepted in PRD

Although inflation is a natural candidate to generate the lengths of coherence of magnetic fields needed to explain current observations, it needs to break conformal invariance of electromagnetism to obtain significant magnetic amplitudes. Of the simplest realizations are the kinetically-coupled theories $f^2(\phi)F_{\mu\nu}F^{\mu\nu}$ (or $IFF$ theories). However, these are known to suffer from electric fields backreaction or the strong coupling problem. In this work we shall confirm that such class of theories are problematic to support magnetogenesis during inflationary cosmology. On the contrary, we show that a bouncing cosmology with a contracting phase dominated by an equation of state with $p>-\rho/3$ can support magnetogenesis, evading the {backreaction/strong-coupling problem}. Finally, we study safe magnetogenesis in a particular bouncing model with an ekpyrotic-like contracting phase. In this case we found that $f^2(\phi)F^2$-instabilities might arise during the final kinetic-driven expanding phase for steep ekpyrotic potentials.Comment: 33 pages, 7 figures, v2: references added, typos corrected, v3 Final version published in NP