Non-linear evolution in $f(R)$ gravity: iterative modelling of the Chameleon mechanism

TL;DR

The paper analyzes non-linear growth in $f(R)$ gravity with Chameleon screening using an iterative scheme for the non-linear scalar field $\chi = \Phi - \Psi$. The method is demonstrated on smooth, compensated spherical top-hat perturbations, revealing density enhancements at the edges and a subtle inner-density feature, with the Chameleon effect strongest when the perturbation size is near the background Compton length ${\bar{x}_C}$. It shows that the density-velocity divergence relation becomes multi-valued in the Chameleon regime and that screening can suppress growth relative to GR. The approach is computationally efficient and readily extensible to 3D initial conditions using FFT-based solvers.

Abstract

We investigate the non-linear evolution of matter perturbations in $f(R)$ models with the Chameleon screening mechanism. The novel feature of our investigation is an iterative solution for the non-linear equation for the scalar field $χ= Φ- Ψ$, where $Φ$ and $Ψ$ are the potentials that characterise scalar perturbations of the metric. We demonstrate the scheme on spherical perturbations - smooth, compensated top-hats of varying length scales. We find that the effect of the Chameleon mechanism is seen most prominently on scales where the size of the top-hat is comparable to the Compton scale of the background. There is a density enhancement near the outer edge of the top-hat and the top-hat does not retain its shape. We explain this well-known observation in the context of the spatio-temporal evolution of the Compton scale. Additionally, we find a slight enhancement of the density near the origin, a feature not reported previously in the literature. On scales much smaller or much larger than the background Compton length, including the Chameleon screening has no appreciable effect on the perturbations. In the former, the growth is enhanced as compared to GR and is almost the same as GR in the latter. Finally, we examine the non-linear density velocity divergence (DVDR) relation and find that for evolution affected by Chameleon screening, the DVDR is no longer one-to-one even for a single profile. The relation between density and velocity depends on the location within the perturbation.

Figures (6)

• Figure 1: The spatial solutions for the dimensionless field $\chi$, its gradient $\nabla \chi$ accounting for the chameleon mechanism. Fractional $\chi$ is the difference between $\chi$ and $\chi^{(1)}$ divided by the maximum value of $\chi$. Here $\chi$ is the solution with the effects of the screening, and $\chi^{(1)}$ is the solution without the chameleon screening. Similarly, fractional $\Delta \chi$ is the difference between $\Delta \chi$ and $\Delta \chi^{(1)}$ divided by the minimum value of $\Delta \chi$.
• Figure 2: The Chameleon mechanism: spatio-temporal evolution of the density field and associated Compton scale. The initial profile (first panel), the evolved density (second panel), the scaled Compton wavelength (third panel) and the $Q$ factor (fourth panel) plotted for three different initial profiles. Profile A stays in the strong field regime. The Chameleon mechanism does not manifest in this regime. Profile C stays in the weak field limit and again does not exhibit the Chameleon screening. Only profile B shows a change in behaviour due to the Chameleon screening.
• Figure 3: Non-linear evolution of density and velocity upto $a=1$. The top panel plots the initial profiles at $a =0.001$. The second, third and fourth panels show the density contrast $\delta$, the spherically averaged density $\Delta$ and the infall velocity $v_{infall}$ respectively at $a=1$. The last panel shows the fractional Hubble parameter for each shell $\delta_v$, which is a measure of the velocity divergence at that point. For profile A, there is an enhanced growth as compared to GR, but no effect of the Chameleon mechanism on the evolution. For profile C, the length scale is much larger than the Compton scale and the results are in agreement with GR. In this regime, the extra force is sensitive to the gradient of the density. For a smoothly varying profile, the density variation is also on scales larger than the Compton scale, resulting in agreement with GR. For profile B, the Chameleon mechanism suppress growth significantly both compared to GR and compared to the case when Chameleon is not invoked.
• Figure 4: Evolution in the $\delta-\delta_v$ phasespace. As was seen in NC22, the evolution for profile A and profile C there is a unique density-velocity divergence relation at late times. However, for scales where the Chameleon mechanism is active, even for a single profile, the $\delta-\delta_v$ curve does not stay monotonic, but becomes a multi-valued function.
• Figure 5: Convergence test of the Taylor expansion. Comparison of successive terms in the Taylor expansion of $\chi$ at a=1 using 500 spatial points. $\mathcal{E}(N)$ has been calculated up to $\delta_0 =30$ and we observe convergence up to $\delta_0 =15$.