Dynamics of scalar perturbations in $f(R,T)$ gravity

In the context of $f(R,T)$ theories of gravity, we study the evolution of scalar cosmological perturbations in the metric formalism. According to restrictions on the background evolution, a specific model within these theories is assumed in order to guarantee the standard continuity equation. Using a completely general procedure, we find the complete set of differential equations for the matter density perturbations. In the case of sub-Hubble modes, the density contrast evolution reduces to a second-order equation. We show that for well-motivated $f(R,T)$ Lagrangians the quasistatic approximation yields to very different results from the ones derived in the frame of the concordance $\Lambda \mathrm{CDM}$ model constraining severely the viability of such theories.

Figure 1

${\delta }_k$ evolution for $f_A(R,T)$ model according to the quasistatic evolution given by (40) and $\Lambda \mathrm{CDM}$ given by (39). The depicted modes are $k=50$, 100, 200, 500 and 1000 (in $H_0$ units). The plotted redshift ranged from $z=1000$ to $z=0$ (today). The value of ${\Omega }_m^0$ was fixed to 0.27 for illustrative purposes. It is seen how whereas the $\Lambda \mathrm{CDM}$ is $k$ independent, the $f_A(R,T)$ model evolutions diverge for all the studied modes and leave the linear region at redshifts $z\approx 100$. For larger $k$ modes (deep sub-Hubble modes) the divergence happens at larger redshift (earlier in the cosmological evolution).

Figure 2

${\delta }_k$ evolution for $f_B(R,T)$ model according to the quasistatic evolution given by (37) and $\Lambda \mathrm{CDM}$ given by (39). Here we have assumed a value $c_1=-{10}^{-3}$. As previously, the dependence on $k$ leads to a strong growth of the matter perturbations for large values of $k$, whereas the behavior is similar to the $\Lambda \mathrm{CDM}$ model for the modes $k<200H_0$.

Figure 3

${\delta }_k$ evolution for $f_B(R,T)$ model according to the quasistatic evolution given by (37) and $\Lambda \mathrm{CDM}$ given by (39). Here we have assumed a positive value for the free parameter $c_1={10}^{-3}$, which leads to an oscillating behavior of the matter perturbations, which turns out stronger as $k$ is larger, and whose oscillations are observed for large small redshifts. The model mimics the $\Lambda \mathrm{CDM}$ model only those modes small enough $k<50H_0$.