http://arxiv.org/abs/2110.05121

We investigate the joint density-velocity evolution in $f(R)$ gravity using smooth, compensated, spherical top-hats as a proxy for the non-linear regime. Using the Hu-Sawicki model as a working example, we solve the coupled continuity, Euler and Einstein equations using an iterative hybrid Lagrangian-Eulerian scheme. The novel aspect of this scheme is that the metric potentials are solved for analytically in the Eulerian frame. The evolution is assumed to follow GR at very early epochs and switches to $f(R)$ at a pre-determined epoch $a_{switch}$. Choosing $a_{switch}$ too early is computationally expensive because of high frequency oscillations; choosing it too late potentially destroys consistency with $\Lambda$CDM. To make an informed choice of $a_{switch}$ we perform an eigenvalue analysis of the background model which gives a ballpark estimate of the magnitude of oscillations. There are two length scales in the problem: the width of the top-hat and the comoving Compton wavelength of the scalar field associated with the modification. The evolution is determined by their ratio $Q$. We vary the top-hat width to consider three regimes: $Q>>1$ (strong), $Q \sim 1$ (intermediate) and $Q<<1$ (weak). Two values of the smoothing parameter are considered to capture the profile-dependence of the results. When $Q>>1$, the evolution is scale-independent, the density-velocity divergence relation (DVDR) is unique and we give a fitting formula for the same. When $Q<<1$, the evolution is very close to GR, except for a formation of a spike near the top-hat edge, a feature which has been noted in earlier literature in the context of the chameleon mechanism. We are able to qualitatively explain this feature in terms of the analytic solution for the metric potential. When $Q \sim 1$, the evolution is highly profile-dependent and no unique DVDR exists.

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S. Nadkarni-Ghosh and S. Chowdhury
Tue, 12 Oct 21
33/73

Comments: 21 pages + appendix, 12 figures, comments welcome