Spherical collapse of dark energy with an arbitrary sound speed

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We consider a generic type of dark energy fluid, characterised by a constant equation of state parameter w and sound speed cₛ, and investigate the impact of dark energy clustering on cosmic structure formation using the spherical collapse model. Along the way, we also discuss in detail the evolution of dark energy perturbations in the linear regime. We find that the introduction of a finite sound speed into the picture necessarily induces a scale-dependence in the dark energy clustering, which in turn affects the dynamics of the spherical collapse in a scale-dependent way. As with other, more conventional fluids, we can define a Jeans scale for the dark energy clustering, and hence a Jeans mass Mⱼ for the dark matter which feels the effect of dark energy clustering via gravitational interactions. For bound objects (halos) with masses M >> Mⱼ, the effect of dark energy clustering is maximal. For those with M << Mⱼ, the dark energy component is effectively homogeneous, and its role in the formation of these structures is reduced to its effects on the Hubble expansion rate. To compute quantitatively the virial density and the linearly extrapolated threshold density, we use a quasi-linear approach which is expected to be valid up to around the Jeans mass. We find an interesting dependence of these quantities on the halo mass M, given some w and cₛ. The dependence is the strongest for masses lying in the vicinity of M Mⱼ. Observing this M-dependence will be a tell-tale sign that dark energy is dynamic, and a great leap towards pinning down its clustering properties.