Cosmic acceleration from matter–curvature coupling

Abstract

We consider $f\left( {R,T} \right)$ modified theory of gravity in which, in general, the gravitational Lagrangian is given by an arbitrary function of the Ricci scalar and the trace of the energy–momentum tensor. We indicate that in this type of the theory, the coupling energy–momentum tensor is not conserved. However, we mainly focus on a particular model that matter is minimally coupled to the geometry in the metric formalism and wherein, its coupling energy–momentum tensor is also conserved. We obtain the corresponding Raychaudhuri dynamical equation that presents the evolution of the kinematic quantities. Then for the chosen model, we derive the behavior of the deceleration parameter, and show that the coupling term can lead to an acceleration phase after the matter dominated phase. On the other hand, the curvature of the universe corresponds with the deviation from parallelism in the geodesic motion. Thus, we also scrutinize the motion of the free test particles on their geodesics, and derive the geodesic deviation equation in this modified theory to study the accelerating universe within the spatially flat FLRW background. Actually, this equation gives the relative accelerations of adjacent particles as a measurable physical quantity, and provides an elegant tool to investigate the timelike and the null structures of spacetime geometries. Then, through the null deviation vector, we find the observer area–distance as a function of the redshift for the chosen model, and compare the results with the corresponding results obtained in the literature.

Appendix

From the definition of the interaction/coupling energy–momentum tensor, Eq. (7), one has

$$\begin{aligned} \kappa {\nabla ^\mu }T_{\mu \nu }^{[{{\mathrm{int}}} ]} = {\nabla ^\mu }\left[ {{f_T}T_{\mu \nu }^{[m]} + \frac{1}{2}\left( {f - R{f_R}} \right) {g_{\mu \nu }} + \left( {{\nabla _\mu }{\nabla _\nu } - {g_{\mu \nu }}} \Box \right) {f_R}} \right] , \end{aligned}$$

(75)

wherein using the definition of Einstein tensor, it reads

$$\begin{aligned} \kappa {\nabla ^\mu }T_{\mu \nu }^{[{{\mathrm{int}}} ]} = {\nabla ^\mu }\left[ {{f_T}T_{\mu \nu }^{[m]} + \frac{f}{2}{g_{\mu \nu }} + {f_R}{G_{\mu \nu }} - {f_R}{R_{\mu \nu }} + \left( {{\nabla _\mu }{\nabla _\nu } - {g_{\mu \nu }}} \Box \right) {f_R}} \right] .\nonumber \\ \end{aligned}$$

(76)

As for any scalar and vector fields, one has ${\nabla _\mu }{\nabla _\nu }\varphi ={\nabla _\nu }{\nabla _\mu }\varphi$ and

$$\begin{aligned} {\nabla _\mu }{\nabla _\nu }{A_\alpha } - {\nabla _\nu }{\nabla _\mu }{A_\alpha } = {R_{\alpha \beta \mu \nu }}{A^\beta }, \end{aligned}$$

(77)

thus, one gets

$$\begin{aligned} {\nabla ^\mu }{\nabla _\mu }{\nabla _\nu }{f_R} = {\nabla ^\mu }{\nabla _\nu }{\nabla _\mu }{f_R} = {\nabla _\nu }{\nabla ^\mu }{\nabla _\mu }{f_R} + R_{\mu \alpha }{}^{\mu }{}_{\nu }{\nabla ^\alpha }{f_R} = {\nabla _\nu }\square {f_R} + {R_{\mu \nu }}{\nabla ^\mu }{f_R}. \end{aligned}$$

(78)

Also, by

$$\begin{aligned} {\nabla _\nu }f\left( {R,T} \right) = {f_R}{\nabla _\nu }R + {f_T}{\nabla _\nu }T \end{aligned}$$

(79)

and

$$\begin{aligned} {\nabla ^\mu }{G_{\mu \nu }} = 0\quad \Rightarrow \quad {\nabla ^\mu }{R_{\mu \nu }} = \frac{1}{2}{\nabla _\nu }R, \end{aligned}$$

(80)

one attains

$$\begin{aligned} \frac{{{g_{\mu \nu }}}}{2}\left( {{\nabla ^\mu }f} \right) - {f_R}\left( {{\nabla ^\mu }{R_{\mu \nu }}} \right) = \frac{1}{2}{f_T}{\nabla _\nu }T. \end{aligned}$$

(81)

Knowing ${\nabla _\mu }{g_{\alpha \beta }} = 0$, and substituting relations (78) and (81) into relation (76), leads to

$$\begin{aligned} \kappa {\nabla ^\mu }T_{\mu \nu }^{[{{\mathrm{int}}} ]} = T_{\mu \nu }^{[m]}{\nabla ^\mu }{f_T} + {G_{\mu \nu }}{\nabla ^\mu }{f_R} + \frac{1}{2}{f_T}{\nabla _\nu }T \end{aligned}$$

(82)

wherein, with relation (8), it poses some restrictions on the choice of the function of f(R, T) on T as

$$\begin{aligned} T_{\mu \nu }^{[m]}{\nabla ^\mu }{f_T} + \frac{1}{2}{f_T}{\nabla _\nu }T=0. \end{aligned}$$

(83)

Note that, such restrictions not only do not contradict with the diffeomorphism invariance of the action (1), but they also arise out of it. That is, the price for a priori inclusion of the trace of the energy–momentum tensor [that itself will emerge from the variation of the Lagrangian of the matter, definition (2)] is to have these restrictions as self-consistency.

Now, writing the result for the index $\nu =0$ of the comoving FLRW metrics, it yields

$$\begin{aligned} -T_{00}^{[m]}{\dot{f}_T} + \frac{1}{2}{f_T}{\dot{T}^{[m]}} = 0, \end{aligned}$$

(84)

and by substituting $T_{00}^{[m]} = {\rho ^{[m]}}$ and ${\dot{T}} = - {\dot{\rho } ^{[m]}} = 3H{\rho ^{[m]}}$, one achieves the constraint Eq. (10).