Spherical collapse in $f(R)$ gravity

We use one-dimensional numerical simulations to study spherical collapse in the $f(R)$ gravity models. We include the nonlinear self-coupling of the scalar field in the theory and use a relaxation scheme to follow the collapse. We find an unusual enhancement in density near the virial radius which may provide observable tests of gravity. We also use the estimated collapse time to calculate the critical overdensity ${\delta }_c$ used in calculating the mass function and bias of halos. We find that analytical approximations previously used in the literature do not capture the complexity of nonlinear spherical collapse.

Figure 1

Fractional residuals in the solution for $f_R$ at the final step of the relaxation process as a function of the time step.

Figure 2

Evolution of the virial term. Observe that there are two points where it crosses zero. We are interested in the one that happens after turnaround.

Figure 3

Smoothed density profiles with Gaussians with different dispersion.

Figure 4

Comparison of the density profiles at the epoch of virialization for $f(R)$ gravity (blue) and GR (red). In each case the starting profile (Gaussian smoothed top-hat) and mass ($1.5\times {10}^{14}{\mathrm{M}}_{\odot }$) are the same. Virialization is reached at different epochs.

Figure 5

Velocity ratio computed shell by shell and normalized by the physical position of the shells. Shell number 100 represents the edge of the top-hat part of the initial overdensity.

Figure 6

${\delta }_c$ as a function of field strength $f_{R0}$ for mass ($1.5\times {10}^{14}{\mathrm{M}}_{\odot }$). The results are color coded to represent the dispersion of the smoothing Gaussian.