Growth of matter fluctuations in $f(R,T)$ Gravity

Cosmological observations favor a spatially flat, isotropic and homogeneous universe with majority of energy-density in the form of some unknown quantities termed dark energy and dark matter observations . Regardless of the immeasurable successes in both theoretical and observational aspects of cosmology from the past few decades, the nature of dark energy and dark matter remain unknown and linger on to be the most outstanding problem in theoretical physics. Alternative scenarios have been widely investigated in literature to suffice the late-time acceleration by either modifying the matter-energy content in the so-called scalar field dark energy models which cohere to general relativity, endorsing however the existence of new fields filling the cosmos or by changing the gravitational forces felt by cosmic objects growft (see growft3 for a review).
Modified theories of gravity furnish an alternative mathematical treatment by proposing the current acceleration to be due to gravitational effects rather than due to presence of exotic matter-energy sources growft ; growft3 . In such extended theories of gravity, the Ricci Scalar $R$ in the action is replaced by either a well-motivated function of $R$ or by other curvature invariants such as Torsion scalar $\mathcal{T}$ , Gauss-Bonnet scalar $G$, non-metricity $Q$ and trace of energy momentum tensor $T$, which results in modified Friedmann equations with supplementary degrees of freedom which upon tuning fits the observations elegantly.
In this spirit, Harko et al harko introduced the $f(R,T)$ gravity which represents a straightforward generalization of $f(R)$ gravity (see extended for a review on modified gravity theories). In this theory, the action contain a combined function of $R$ and $T$. $f(R,T)$ gravity has been successful to addressing major cosmological enigmas such as dark matter in22 dark energy in21 , massive pulsars in23 , super-Chandrasekhar white dwarfs in25 , gravitational waves in36 , wormholes in26 , baryogenesis baryo , bouncing cosmology bounce ; bounce2 , viscous cosmology arora , redshift drift drift , inflationary cosmology inflation big bang nucleosynthesis bang and varying speed of light scenarios physical .
Since both modified theories of gravity and scalar field dark energy models fit the current observations, one may be interested in distinguishing them. This can be achieved at the perturbation level (see growft15 for a recent analysis). In particular, the idea of employing the growth index $\gamma$ growft16 to distinguish modified gravity from the $\Lambda$CDM is well known. One may find numerous studies aimed at deriving the expression of growth index for various cosmological models including DGP growft21 ; growft23to25 , scalar field dark energy growft21 ; growft17to20 ; growft22 , Finsler-Randers growft26 , $f(R)$ gravity ft27 and $f(\mathcal{T})$ gravity growft .
Although $f(R,T)$ gravity models have been richly employed to investigate the scalar density perturbations frt/per , as far as I know, no analysis concerning the growth rate of clustering have been attempted. In this paper, I shall fill in that gap. The manuscript is organized as follows: In Section II, I present an overview of $f(R,T)$ gravity and solve the field equations for $f(R,T)=R+\lambda T$ assuming a dust universe ($\omega=0$). In Section III I derive the expressions for $\delta(z)$ and $f(z)$ and show their behavior graphically and also their dependency on the model parameter $\lambda$. In Section IV I present an analytical expression for the growth index which is redshift dependent and the expression reduces to $3/5$ for $\lambda=0$, which is the growth index for a dust universe. Finally in Section V I summarize the results and conclude the work.

The action in $f(R,T)$ is defined as harko

$$\mathcal{S}=\frac{1}{2\kappa^{2}}\int\sqrt{-g}\left[f(R,T)+\mathcal{L}_{m}\right]d^{4}x,$$

(1)

where $\kappa^{2}=\frac{8\pi G_{N}}{c^{4}}$. Here $c$ denotes the speed of light and $G_{N}$ the gravitational constant. Additionally, $\mathcal{L}_{m}=-p$ represents matter Lagrangian density with $p$ being the cosmological pressure. In this work, I shall work with natural units and therefore I set $\kappa^{2}=\frac{8\pi G_{N}}{c^{4}}=1$.

Varying the action (1) with respect to the metric $g_{\mu\nu}$ yields the following field equation

$$\kappa^{2}T_{\mu\nu}-f^{1}_{,T}(R,T)(T_{\mu\nu}+\Xi_{\mu\nu})=\Pi_{\mu\nu}f^{1}_{,R}(R,T)+f^{1}_{,R}(R,T)R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}f(R,T)$$

(2)

where $T_{\mu\nu}$ represents the stress-energy-momentum tensor. Note that for a perfect fluid, $T_{\mu\nu}$ can be written as

$$T_{\mu\nu}=-pg_{\mu\nu}+(\rho+p)u_{\mu}u_{\nu},$$

(3)

where $\rho$ represents energy density, $u_{\mu}$ the four-velocity and

$$\Pi_{\mu\nu}=g_{\mu\nu}\square-\nabla_{\mu}\nabla_{\nu}.$$

(4)

$$\Xi_{ij}\equiv g^{\mu\nu}\frac{\delta T_{\mu\nu}}{\delta g^{ij}},$$

(5)

where I used the notation $f^{i}_{,X}\equiv\frac{d^{i}f}{dX^{i}}$ .

I shall now derive an analytic expression for the matter density fluctuations $\delta(z)$ and growth rate $f(z)$ in the framework of $f(R,T)$ gravity. The second order differential equation describing the evolution of $\delta(z)$ at the sub-horizon scales reads

$$\ddot{\delta}+2\nu H\dot{\delta}-4\pi G_{N}\mu\rho\delta=0.$$

(14)

In extended theories of gravity, the quantity $\mu=G_{eff}/G_{N}$ depends on the scale factor, while for the dark energy models adhered to general relativity, $G_{eff}$ equals Newton’s gravitational constant and therefore $\mu=1$ growft . $\nu$ is the damping coefficient which is a function of redshift and spatial Fourier frequency growft41 . Readers are encouraged to see growft21 ; growft27 ; growft38to40 ; growft41 for further clarifications.
Now from Eq. 37 in Ref. frt/per , the expression of $G_{eff}$ for $f(R,T)=R+f(T)$ gravity can be written as

$$G_{eff}/G_{N}=1-f^{1}_{,T}-k^{2}f^{1}_{,T}/(a^{2}(1-f^{1}_{,T})\rho).$$

(15)

Since I work with natural units, then by setting $G_{N}=1$, the expression of $G_{eff}$ becomes

$$G_{eff}=1-f^{1}_{,T}-k^{2}f^{1}_{,T}/(a^{2}(1-f^{1}_{,T})\rho).$$

(16)

Finally, the parameter $\mu$ for $f(T)=\lambda T$ can be written as

$$\mu=G_{eff}/G_{N}=1-\lambda-k^{2}\lambda/(a^{2}(1-\lambda)\rho).$$

(17)

It may be noted that for $\lambda=0$, $\mu=1$ since $G_{eff}=G_{N}$ as it should be.
In most cases, Eq. 14 does not have an analytic solution and can only be approached numerically. Fortunately, an analytic solution does exists for the $f(R,T)$ gravity model employed in this work. Upon substituting the expression of $H(z)$ (Eq. 12) and $\rho(z)$ (Eq. 13) in Eq. 14, the expression of matter perturbation $\delta$ reads

$$\delta(z)=2^{\frac{16\lambda^{2}+3R+12\lambda+2}{32\lambda^{2}+24\lambda+4}}3^{-\frac{R}{16\lambda^{2}+12\lambda+2}-\frac{\lambda}{\lambda+1}-\frac{2}{3(\lambda+1)}}\left(\left(\left(\frac{1}{z+1}\right)^{\frac{3\lambda}{3\lambda+2}+\frac{3}{3\lambda+2}}\right)^{\frac{2}{3(\lambda+1)}}\right)^{\frac{16\lambda^{2}+R+12\lambda+2}{32\lambda^{2}+24\lambda+4}}\left(\left(\frac{1}{z+1}\right)^{\frac{3\lambda}{3\lambda+2}+\frac{3}{3\lambda+2}}\right)^{\frac{3\lambda+1}{6\lambda+6}}\\ \times\left((\lambda+1)\left(\frac{1}{z+1}\right)^{\frac{3\lambda}{3\lambda+2}+\frac{3}{3\lambda+2}}\right)^{-\frac{3\lambda+2}{3\lambda+3}}\\ \times\left(\frac{\lambda(\lambda+1)^{2}\left(\left(\frac{1}{z+1}\right)^{\frac{3\lambda}{3\lambda+2}+\frac{3}{3\lambda+2}}\right)^{\frac{2}{3(\lambda+1)}}}{\lambda-1}\right)^{-\frac{R}{32\lambda^{2}+24\lambda+4}}\left(K_{\frac{R}{16\lambda^{2}+12\lambda+2}}\left(3S\right)\times I_{\frac{R}{16\lambda^{2}+12\lambda+2}}\left(3S\right)\right)$$

(18)

where

$$R=\sqrt{-1728\lambda^{6}-2304\lambda^{5}+1276\lambda^{4}+3576\lambda^{3}+2185\lambda^{2}+546\lambda+49}$$

(19)

and

$$S=\sqrt{\frac{\left(\left(\frac{1}{z+1}\right)^{\frac{3\lambda}{3\lambda+2}+\frac{3}{3\lambda+2}}\right)^{\frac{2}{3(\lambda+1)}}\lambda(\lambda+1)^{2}}{\lambda-1}}$$

(20)

and $I$ and $K$ represent the modified Bessel functions of the first and second kind respectively. The growth rate of clustering $f$ introduced in growft16 is defined as

$$f(a)=\frac{d\text{ln}\delta}{d\text{ln}a}=\Omega_{m}^{\gamma}(a)$$

(21)

which can be re-written in terms of redshift $z$ as

$$f(z)=\frac{-(1+z)}{\delta}\frac{d\delta}{dz}=\Omega_{m}^{\gamma}(z).$$

(22)

Where $\Omega_{m}=\frac{8\pi G\rho_{m}}{3H_{0}^{2}}$. Substituting Eq. 18 in Eq. 22, the expression of growth rate becomes

$$f(z)=\frac{\begin{multlined}\begin{multlined}3\lambda(\lambda+1)^{2}I_{\frac{R}{16\lambda^{2}+12\lambda+2}-1}\left(3S\right)Y+3\lambda^{3}I_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3S\right)Y+6\lambda^{2}I_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y\\ \begin{multlined}+3\lambda I_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3S\right)Y-3\lambda^{3}K_{\frac{R}{16\lambda^{2}+12\lambda+2}-1}\left(3S\right)Y-6\lambda^{2}K_{\frac{R}{16\lambda^{2}+12\lambda+2}-1}\left(3S\right)Y\\ \begin{multlined}-3\lambda K_{\frac{R}{16\lambda^{2}+12\lambda+2}-1}\left(3\sqrt{\frac{\left(\left(\frac{1}{z+1}\right)^{\frac{3(\lambda+1)}{3\lambda+2}}\right)^{\frac{2}{3(\lambda+1)}}\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y\\ -3\lambda^{3}K_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y-6\lambda^{2}K_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y\end{multlined}-3\lambda K_{\frac{R}{16\lambda^{2}+12\lambda+2}-1}\left(3\sqrt{\frac{\left(\left(\frac{1}{z+1}\right)^{\frac{3(\lambda+1)}{3\lambda+2}}\right)^{\frac{2}{3(\lambda+1)}}\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y\\ -3\lambda^{3}K_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y-6\lambda^{2}K_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y\end{multlined}+3\lambda I_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3S\right)Y-3\lambda^{3}K_{\frac{R}{16\lambda^{2}+12\lambda+2}-1}\left(3S\right)Y-6\lambda^{2}K_{\frac{R}{16\lambda^{2}+12\lambda+2}-1}\left(3S\right)Y\\ \begin{multlined}-3\lambda K_{\frac{R}{16\lambda^{2}+12\lambda+2}-1}\left(3\sqrt{\frac{\left(\left(\frac{1}{z+1}\right)^{\frac{3(\lambda+1)}{3\lambda+2}}\right)^{\frac{2}{3(\lambda+1)}}\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y\\ -3\lambda^{3}K_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y-6\lambda^{2}K_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y\end{multlined}-3\lambda K_{\frac{R}{16\lambda^{2}+12\lambda+2}-1}\left(3\sqrt{\frac{\left(\left(\frac{1}{z+1}\right)^{\frac{3(\lambda+1)}{3\lambda+2}}\right)^{\frac{2}{3(\lambda+1)}}\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y\\ -3\lambda^{3}K_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y-6\lambda^{2}K_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y\end{multlined}3\lambda(\lambda+1)^{2}I_{\frac{R}{16\lambda^{2}+12\lambda+2}-1}\left(3S\right)Y+3\lambda^{3}I_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3S\right)Y+6\lambda^{2}I_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y\\ \begin{multlined}+3\lambda I_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3S\right)Y-3\lambda^{3}K_{\frac{R}{16\lambda^{2}+12\lambda+2}-1}\left(3S\right)Y-6\lambda^{2}K_{\frac{R}{16\lambda^{2}+12\lambda+2}-1}\left(3S\right)Y\\ \begin{multlined}-3\lambda K_{\frac{R}{16\lambda^{2}+12\lambda+2}-1}\left(3\sqrt{\frac{\left(\left(\frac{1}{z+1}\right)^{\frac{3(\lambda+1)}{3\lambda+2}}\right)^{\frac{2}{3(\lambda+1)}}\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y\\ -3\lambda^{3}K_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y-6\lambda^{2}K_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y\end{multlined}-3\lambda K_{\frac{R}{16\lambda^{2}+12\lambda+2}-1}\left(3\sqrt{\frac{\left(\left(\frac{1}{z+1}\right)^{\frac{3(\lambda+1)}{3\lambda+2}}\right)^{\frac{2}{3(\lambda+1)}}\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y\\ -3\lambda^{3}K_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y-6\lambda^{2}K_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y\end{multlined}+3\lambda I_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3S\right)Y-3\lambda^{3}K_{\frac{R}{16\lambda^{2}+12\lambda+2}-1}\left(3S\right)Y-6\lambda^{2}K_{\frac{R}{16\lambda^{2}+12\lambda+2}-1}\left(3S\right)Y\\ \begin{multlined}-3\lambda K_{\frac{R}{16\lambda^{2}+12\lambda+2}-1}\left(3\sqrt{\frac{\left(\left(\frac{1}{z+1}\right)^{\frac{3(\lambda+1)}{3\lambda+2}}\right)^{\frac{2}{3(\lambda+1)}}\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y\\ -3\lambda^{3}K_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y-6\lambda^{2}K_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y\end{multlined}-3\lambda K_{\frac{R}{16\lambda^{2}+12\lambda+2}-1}\left(3\sqrt{\frac{\left(\left(\frac{1}{z+1}\right)^{\frac{3(\lambda+1)}{3\lambda+2}}\right)^{\frac{2}{3(\lambda+1)}}\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y\\ -3\lambda^{3}K_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y-6\lambda^{2}K_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y\\ {\begin{multlined}\begin{multlined}-3\lambda 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-3\lambda^{3}K_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y-6\lambda^{2}K_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y\end{multlined}-3\lambda K_{\frac{R}{16\lambda^{2}+12\lambda+2}-1}\left(3\sqrt{\frac{\left(\left(\frac{1}{z+1}\right)^{\frac{3(\lambda+1)}{3\lambda+2}}\right)^{\frac{2}{3(\lambda+1)}}\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y\\ -3\lambda^{3}K_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y-6\lambda^{2}K_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y\end{multlined}+3\lambda I_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3S\right)Y-3\lambda^{3}K_{\frac{R}{16\lambda^{2}+12\lambda+2}-1}\left(3S\right)Y-6\lambda^{2}K_{\frac{R}{16\lambda^{2}+12\lambda+2}-1}\left(3S\right)Y\\ \begin{multlined}-3\lambda K_{\frac{R}{16\lambda^{2}+12\lambda+2}-1}\left(3\sqrt{\frac{\left(\left(\frac{1}{z+1}\right)^{\frac{3(\lambda+1)}{3\lambda+2}}\right)^{\frac{2}{3(\lambda+1)}}\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y\\ -3\lambda^{3}K_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y-6\lambda^{2}K_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y\end{multlined}-3\lambda K_{\frac{R}{16\lambda^{2}+12\lambda+2}-1}\left(3\sqrt{\frac{\left(\left(\frac{1}{z+1}\right)^{\frac{3(\lambda+1)}{3\lambda+2}}\right)^{\frac{2}{3(\lambda+1)}}\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y\\ -3\lambda^{3}K_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y-6\lambda^{2}K_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}\right)Y\\ {\begin{multlined}\begin{multlined}-3\lambda K_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3S\right)Y-3\lambda^{2}\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}K_{\frac{R}{16\lambda^{2}+12\lambda+2}}\left(3S\right)\\ \left(3\lambda^{2}-2\lambda-1\right)I_{\frac{R}{16\lambda^{2}+12\lambda+2}}\left(3S\right)\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}\end{multlined}-3\lambda K_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3S\right)Y-3\lambda^{2}\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}K_{\frac{R}{16\lambda^{2}+12\lambda+2}}\left(3S\right)\\ \left(3\lambda^{2}-2\lambda-1\right)I_{\frac{R}{16\lambda^{2}+12\lambda+2}}\left(3S\right)\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}\\ +K_{\frac{R}{16\lambda^{2}+12\lambda+2}}\left(3S\right)\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}+2\lambda K_{\frac{R}{16\lambda^{2}+12\lambda+2}}\left(3S\right)S\end{multlined}\begin{multlined}-3\lambda K_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3S\right)Y-3\lambda^{2}\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}K_{\frac{R}{16\lambda^{2}+12\lambda+2}}\left(3S\right)\\ \left(3\lambda^{2}-2\lambda-1\right)I_{\frac{R}{16\lambda^{2}+12\lambda+2}}\left(3S\right)\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}\end{multlined}-3\lambda K_{\frac{R}{16\lambda^{2}+12\lambda+2}+1}\left(3S\right)Y-3\lambda^{2}\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}K_{\frac{R}{16\lambda^{2}+12\lambda+2}}\left(3S\right)\\ \left(3\lambda^{2}-2\lambda-1\right)I_{\frac{R}{16\lambda^{2}+12\lambda+2}}\left(3S\right)\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}\\ +K_{\frac{R}{16\lambda^{2}+12\lambda+2}}\left(3S\right)\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}+2\lambda K_{\frac{R}{16\lambda^{2}+12\lambda+2}}\left(3S\right)S}}{2(\lambda-1)S(3\lambda+2)\left(I_{\frac{R}{16\lambda^{2}+12\lambda+2}}\left(3S\right)+K_{\frac{R}{16\lambda^{2}+12\lambda+2}}\left(3\sqrt{\frac{Y\lambda(\lambda+1)^{2}}{\lambda-1}}\right)\right)}$$

(23)

where

$$Y=\left(\left(\frac{1}{z+1}\right)^{\frac{3(\lambda+1)}{3\lambda+2}}\right)^{\frac{2}{3(\lambda+1)}}$$

(24)

In Fig. 1 I show the evolution of Eq. 23 for different values of $\lambda$ and also for $\lambda=0$. the linear growth rate $f(z)$ is a extremely useful concept in physical cosmology which provides information about the evolution of density perturbations from a nearly smooth state to the highly clumpy state observed at the present epoch. From the figure it is clear that the linear growth rate is sensitive to the model parameter $\lambda$ and it is observed that as $\lambda$ increases, $f(z)$ assumes lower values at all redshift. This implies that the presence of the model parameter $\lambda$ enhances the effect of geometrical dark energy and acts against gravity and represses the growth of formation of structures. Note that for negative $\lambda$, the growth rate becomes imaginary and therefore not permissible.
In Fig. 2, I show the evolution of Eq. 18 for $0\leq\lambda\leq 1$ as a function of redshift. I find that, for $\lambda=0$, the density perturbations decrease faster with increasing redshift compared to the cases when $\lambda>0$. Also as $\lambda$ increases, $\delta(z)$ decreases slower. For $\lambda<0$, $\delta(z)$ increases with redshift which is unphysical. Hence, I find from the present analysis that negative $\lambda$ values do not yield satisfactory results.

In this section I shall derive an analytic expression for the growth index $\gamma$ for the $f(R,T)$ gravity model. The growth index is very useful in discriminating various dark energy models from modified gravity theories. It is not clear whether $\gamma$ is a constant or temporally evolving, owing to which, authors have studied the consequence of both a constant growth index and an one which is redshift dependent. However, in this work I shall focus on deriving an expression for a growth index which is redshift dependent. See growft ; growft21 ; growft23to25 ; growft17to20 ; growft22 ; growft26 ; ft27 for other studies related to redshift dependent growth index.
A general mathematical treatment for the growth index is proposed in growft41 , according to which the growth index $\gamma_{\infty}$ can be expressed as

$$\gamma_{\infty}=\frac{3(M_{0}+M_{1})-2(H_{1}+N_{1})}{2+2X_{1}+3M_{0}}.$$

(25)

Where the relevant quantities are defined as growft41

$$M_{0}=\mu\bigg{|}_{\omega=0},\hskip 10.84006ptM_{1}=\frac{d\mu}{d\omega}\bigg{|}_{\omega=0}$$

(26)

and

$$H_{1}=-\frac{X_{1}}{2}=\frac{d(d\text{ln}E/d\text{ln}a)}{d\omega}\bigg{|}_{\omega=0},\hskip 10.84006ptN_{1}=\frac{d\nu}{d\omega}\bigg{|}_{\omega=0},$$

(27)

where $E(z)=H(z)/H_{0}$ represents the normalized Hubble parameter. For modified gravity theories the quantity $\nu=1$, therefore $N_{1}=0$.
Note that in growft41 , all the cosmological functions are written in terms of the variable $\omega=\text{ln}\Omega_{m}(a)$, implying $\Omega_{m}(a)\rightarrow 1$ when $\omega\rightarrow 0$. Therefore, I can write growft41 ; growft

$$M_{1}=\frac{d\mu}{d\omega}\bigg{|}_{\omega=0}=\Omega_{m}\frac{d\mu}{d\Omega_{m}}\bigg{|}_{\Omega_{m}=1}.$$

(28)

From Eq. 17, I find for the $f(R,T)$ gravity model,

$$\mu=-\frac{3\lambda(\lambda+1)^{2}\left(8\lambda^{2}+6\lambda+1\right)\left(\frac{1}{z+1}\right)^{\frac{6\lambda}{3\lambda+2}+\frac{6}{3\lambda+2}}}{a^{2}(1-\lambda)(3\lambda+2)(2(3\lambda+1)+\lambda(8\lambda+3))}-\lambda+1.$$

(29)

Now, after some calculations I obtain $\frac{d\mu}{d\Omega_{m}}=0$ and therefore $M_{1}=0$.
Lastly, $H_{1}$ can be written as growft ; growft41

$$H_{1}=-\frac{X_{1}}{2}=\frac{d(d\text{ln}E/d\text{ln}a)}{d\omega}\bigg{|}_{\omega=0}=\Omega_{m}\frac{d(d\text{ln}E/d\text{ln}a)}{d\Omega_{m}}\bigg{|}_{\Omega_{m}=1}.$$

(30)

For the relevant $f(R,T)$ gravity model used in this work,

$$d\text{ln}E/d\text{ln}a=-\frac{3(\lambda+1)}{3\lambda+2}.$$

(31)

Since $d\text{ln}E/d\text{ln}a$ is a $constant$, therefore $\Omega_{m}\frac{d(d\text{ln}E/d\text{ln}a)}{d\Omega_{m}}\bigg{|}_{\Omega_{m}=1}=H_{1}=-\frac{X_{1}}{2}=0$. Hence I obtain,
{$M_{0},M_{1},N_{1},H_{1},X_{1}$} = {($-\frac{3\lambda(\lambda+1)^{2}\left(8\lambda^{2}+6\lambda+1\right)\left(\frac{1}{z+1}\right)^{\frac{6\lambda}{3\lambda+2}+\frac{6}{3\lambda+2}}}{a^{2}(1-\lambda)(3\lambda+2)(2(3\lambda+1)+\lambda(8\lambda+3))}-\lambda+1$), $0,0,0,0,0$}.
Upon substituting all the respective values of the coefficients in Eq. 25, the expression of growth index in $f(R,T)$ gravity reads

$$\gamma_{\infty}=\frac{3\left(\frac{3\lambda(\lambda+1)^{2}\left(8\lambda^{2}+6\lambda+1\right)\left(\frac{1}{z+1}\right)^{\frac{2}{3\lambda+2}+2}}{a^{2}(\lambda-1)(3\lambda+2)(\lambda(8\lambda+9)+2)}-\lambda+1\right)}{\frac{9\lambda(\lambda+1)^{2}(2\lambda+1)(4\lambda+1)\left(\frac{1}{z+1}\right)^{\frac{2}{3\lambda+2}+2}}{a^{2}(\lambda-1)(3\lambda+2)(\lambda(8\lambda+9)+2)}-3\lambda+5}.$$

(32)

For $\lambda=0$, the expression reduces to $\gamma_{\infty}=3/5$, which is the growth index for a dust universe.

I studied the growth of matter fluctuations in the framework of $f(R,T)$ modified gravity for the simplest functional choice $f(R,T)=R+\lambda T$, where $R$ denote the Ricci scalar, $T$ the trace of energy momentum tensor and $\lambda$ a constant. This is by far the most widely studied $f(R,T)$ gravity model in the literature harko ; in25 ; bounce ; in26 ; in36 . However, to the best of my knowledge, no studies related to matter density fluctuations have been attempted in $f(R,T)$ gravity thus far.
To carry out the analysis, I first solve the field equations assuming a dust universe ($\omega=0$) for the Hubble parameter $H(z)$ and employ it to solve the equation of matter density fluctuations and finally for the growth rate $f(z)$ and show their behavior with redshift for some values of $\lambda$ with observational constraints on $f(z)$. I found that positive $\lambda$ yields remarkable values of $f(z)$ which suits perfectly with observations, while negative $\lambda$ makes $f(z)$ imaginary and therefore should be refrained from using in future studies. I find that as $\lambda$ increases, $f(z)$ assumes lower values at each redshift.
Furthermore, I find that, for $\lambda=0$, the density perturbations decrease faster with increasing redshift compared to the cases when $\lambda>0$. Also as $\lambda$ increases, $\delta(z)$ decreases slower. For $\lambda<0$, $\delta(z)$ increases with redshift which is unphysical.
Additionally, following the prescription of growft41 , I also present in this work an analytical expression for the growth index which is redshift dependent and the expression reduces to $3/5$ for $\lambda=0$, which is the growth index for a dust universe.

I thank Tiberiu Harko for helpful suggestions. I thank the anonymous reviewer for useful criticisms and encouraging comments that helped me to improve the work significantly. I acknowledge DST, New-Delhi, Government of India for the provisional INSPIRE fellowship selection [DST/INSPIRE/03/2019/003141].