The pattern of growth in viable f(R) cosmologies

TL;DR

The paper addresses how viable $f(R)$ gravity theories modify the growth of cosmic structures beyond the standard $ ext{ΛCDM}$ background evolution. It analyzes linear perturbations in the Jordan frame, confirms equivalence with the Einstein-frame description, and interprets the effects as an effective dark fluid with shear. A key finding is a characteristic scale-dependent growth pattern set by the scalaron Compton wavelength $\lambda_C$, with a suppression of deviations on scales larger than $\lambda_C$ and an enhanced growth on smaller scales due to a modified gravity strength $G_{\rm eff}$. The results indicate that weak lensing surveys could detect these differences even when the background remains indistinguishable from $ ext{ΛCDM}$, making $f(R)$ a testable modification to gravity, subject to tight local constraints on $|f_R^0|$.

Abstract

We study the evolution of linear perturbations in metric f(R) models of gravity and identify a potentially observable characteristic scale-dependent pattern in the behavior of cosmological structures. While at the background level viable f(R) models must closely mimic LCDM, the differences in their prediction for the growth of large scale structures can be sufficiently large to be seen with future weak lensing surveys. While working in the Jordan frame, we perform an analytical study of the growth of structures in the Einstein frame, demonstrating the equivalence of the dynamics in the two frames. We also provide a physical interpretation of the results in terms of the dynamics of an effective dark energy fluid with a non-zero shear. We find that the growth of structure in f(R) is enhanced, but that there are no small scale instabilities associated with the additional attractive "fifth force". We then briefly consider some recently proposed observational tests of modified gravity and their utility for detecting the f(R) pattern of structure growth.

Figures (6)

• Figure 1: $f(R)$ (top panel) and $f_R$ (bottom panel) vs $R$ obtained using the designer approach for several given expansion histories. The three solid black lines correspond to $w_{\rm{eff}}=-1$ with $A=0$ (f=-2$\Lambda$), $A<0$ ($-f/2\Lambda >1$, $f_R>0$) and $A>0$ ($-f/2\Lambda <1$, $f_R<0$). The red short-dashed line corresponds to $w_{\rm{eff}}=-1.01$ with $A=0$. The blue dot-dashed line is for $w_{\rm{eff}}=-0.99$ with $A=0$, while the blue dot line corresponds to $w_{\rm{eff}}=-0.99$ but with $A$ adjusted to make $|f_R|=10^{-6}$ today. The blue dot on the lower panel corresponds to $10\times f_R$ which makes it easier to see. Finally, the green long dash denotes a model with $A=0$ and a time varying $w_{\rm{eff}}(a)$ which crosses from $-0.99$ at early epochs to $-1.01$ for $z<1$.
• Figure 2: These plots show the numerical solutions for the transfer function $\Phi_+/\Phi_+^{initial}$, the ratio of the potentials $\Phi/\Psi$ and the slip $\chi=f_{RR}\delta R$ as functions of the scale factor $a$ for different scales for an $f(R)$ model with $w_{\rm{eff}}=-1$ and $f_R^0=-10^{-4}$. The dotted line corresponds to the scale $k=0.01$h/Mpc, the short-dashed line to $k=0.1$h/Mpc and the long-dashed line to $k=0.5$h/Mpc. Finally, the solid line shows the scale-independent behavior in the $\Lambda$CDM case. The right panel shows the tiny oscillations in $\chi$ at early times, shown for $k=0.1$h/Mpc.
• Figure 3: The evolution of the growth factor for the CDM $[\Delta_m(k,a)/a]/[\Delta_m(k,a_i)/a_i]$ as a function of redshift $z$ and scale $k$. The left panel corresponds to an $f(R)$ model with $w_{\rm{eff}}=-1$ and $f_R^0=-10^{-4}$. In the right panel we show the corresponding $\Lambda$CDM pattern for comparison. One can see the scale-dependent behavior of the growth factor in $f(R)$ as opposed to the scale-independence of the $\Lambda$CDM case. The dashed line crossing diagonally on the left plot corresponds to the Compton transition wavelength given by $Q=1$ (\ref{['Q']}). Deviations from $\Lambda$CDM become important on scales below $\lambda_C$. For modes with $\lambda<\lambda_C$ during matter domination, there is an enhancement in the growth due to the "fifth force" introduced by the modifications to GR. Eventually, the universe starts accelerating and the growth slows down. However, in comparison to the $\Lambda$CDM case, such slowing is delayed in a scale dependent way.
• Figure 4: The function probed by the cross-correlation of large scale structure with ISW, $\Delta_m\cdot d\Phi_+/dz$, as a function of scale $k$ and redshift $z$. The left panel corresponds to an $f(R)$ model with $w_{\rm{eff}}=-1$ and $f_R^0=-10^{-4}$ and shows a characteristic scale-dependent pattern. The right panel corresponds to $\Lambda$CDM. The dashed line crossing through the left panel corresponds to $Q=1$ (\ref{['Q']}), i.e. it corresponds to the Compton wavelength of the scalaron $\lambda_C$ (\ref{['lambda_C']}). For scales $\lambda<\lambda_C$, during matter domination, one can clearly notice the effect of the "fifth force" which suppresses the cross correlation and can actually make the correlation negative. Therefore, a negative cross correlation signal at early redshifts (corresponding to matter era), is a signature of $f(R)$. The acceleration of the background will eventually contrast the "fifth force" and lead to a positive cross-correlation.
• Figure 5: The parameters $\bar{\omega}=(\Psi-\Phi)/\Phi$ (left) and $\eta=\Phi/\Psi$ (right) as functions of scale $k$ and redshift $z$ for an $f(R)$ model with $w_{\rm{eff}}=-1$ and $f_R^0=-10^{-4}$. The long-dashed line corresponds to the Compton transition scale $Q=1$ (\ref{['Q']}). One can notice the time- and scale-dependent pattern. The parameter $\bar{\omega}$ evolves from $\bar{\omega}\simeq0$ on scales above $\lambda_C$ (\ref{['lambda_C']}), to $\bar{\omega}\simeq1$ on scales $\lambda\ll\lambda_C$. Analogously, $\eta$ evolves from $\eta\simeq1/2$ for $\lambda\gg\lambda_C$ to $\eta\simeq1$ for scales below $\lambda_C$.