Non-linear density–velocity dynamics in <i>f</i>(<i>R</i>) gravity from spherical collapse

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