Anisotropic solution via Noether symmetry for $f(R)$ Palatini gravity

Abstract

In this work, we consider the Palatini $f(R)$ gravity theory with the matter dominated universe. Utilizing the dynamical equivalence between Palatini $f(R)$ gravity and Brans-Dicke like scalar tensor theory, we obtain a point-like Lagrangian for the locally rotational symmetric Bianchi I space time. We determine the physically interesting functional forms of $f(R)$ for this Lagrangian via Noether symmetry approach. We also explore some cosmological solutions of the field equations using these forms obtained by the existence of Noether symmetry. It is shown that these solutions are consistent with the accelerated behavior of the universe.

Appendix: Noether symmetries with gauge term

In the case of $B\neq0$ and $\tau\neq0$, Noether gauge symmetry equations given by (27)–(33) yield following two solutions,

$$\begin{aligned} &{\alpha=-\frac{c_{2}(m+4)B}{\rho_{m0}m(m+2)}-\frac {c_{1}}{2}B^{-\frac {2m^{2}-m+2}{2(m-1)}} \phi^{-\frac{3m}{4(m-1)}},} \end{aligned}$$

(81)

$$\begin{aligned} &{\beta=\frac{4 c_{2}\phi}{\rho_{m0}m}+c_{1}B^{-\frac {m(2m+1)}{2(m-1)}} \phi^{\frac{m-4}{4(m-1)}},} \end{aligned}$$

(82)

$$\begin{aligned} &{\tau=-\frac{c_{2}}{\rho_{m0}}t+c_{3},} \end{aligned}$$

(83)

$$\begin{aligned} &{G=c_{2} t+ c_{4},} \end{aligned}$$

(84)

with the potential (43) and

$$\begin{aligned} &{\alpha=\frac{(7m+8)c_{2}B}{\rho_{m0}(m+2)(m-4)}-\frac {c_{1}}{2}B^{-\frac {2m^{2}-m+2}{2(m-1)}}\phi^{-\frac{3m}{4(m-1)}},} \end{aligned}$$

(85)

$$\begin{aligned} &{\beta=-\frac{4 (2m+1)c_{2}\phi}{\rho_{m0}(m-4)}+c_{1}B^{-\frac {m-4}{2(m-1)}}\phi^{\frac{3m}{4(m-1)}},} \end{aligned}$$

(86)

$$\begin{aligned} &{\tau=-\frac{c_{2}}{\rho_{m0}}t+c_{3},} \end{aligned}$$

(87)

$$\begin{aligned} &{G=c_{2} t+ c_{4},} \end{aligned}$$

(88)

with the potential (46). In the above solutions, $c_{i}$ $(i=1,2,3,4)$ are a redefined integration constants. Considering the first of above solutions, we obtain three Noether symmetries of the Lagrangian (15) which are

$$\begin{aligned} &{\mathbf{X}_{1} = \frac{\partial}{\partial t},} \end{aligned}$$

(89)

$$\begin{aligned} &{\mathbf{X}_{2} = -B^{-\frac{m(2m+1)}{2(m-1)}}\phi^{\frac{m-4}{4(m-1)}} \biggl( \frac{B}{2\phi} \frac{\partial}{\partial B} - \frac{\partial }{\partial\phi} \biggr),} \end{aligned}$$

(90)

$$\begin{aligned} &{\mathbf{X}_{3} = -\frac{1}{\rho_{m0}} \biggl( t\frac{\partial}{\partial t} + \frac{(m+4)B}{m(m+2)} \frac{\partial}{\partial B} - \frac{4\phi }{m} \frac{\partial}{\partial\phi} \biggr) .} \end{aligned}$$

(91)

The first symmetry $\mathbf{X}_{1}$ (corresponds to translations in time) gives the energy conservation of the dynamical system in the form of (94) below, while the second and third symmetries $\mathbf{X}_{2}$ and $\mathbf{X}_{3}$ generate the constants of motions of the form (95) and (96) below, respectively. The non-commuting generators $\mathbf{X}_{i}$ satisfy the algebra given by

$$\begin{aligned} &{[\mathbf{X}_{1},\mathbf{X}_{3}] = - \frac{1}{\rho_{m0}} \mathbf{X}_{1},} \end{aligned}$$

(92)

$$\begin{aligned} &{[\mathbf{X}_{2},\mathbf{X}_{3}] = \frac{2m^{2}+3m-8}{2\rho_{m0}(m+2)(m-1)} \mathbf{X}_{2}.} \end{aligned}$$

(93)

The corresponding constants of motion associated with the above Noether symmetries are

$$\begin{aligned} &{I_{1} = 6a^{2} \dot{a}\dot{\phi} + 6a \phi \dot{a}^{2} + \frac{3a^{3}}{2\phi} \dot{\phi}^{2} - a^{3} U - 2\kappa\rho_{m0},} \end{aligned}$$

(94)

$$\begin{aligned} &{I_{2} = 3(2\ell+1) a \bigl(a^{2} \phi \bigr)^{-\frac{(\ell+1)}{(2\ell+1)}} \ \frac{d}{dt} \bigl(a^{2} \phi\bigr),} \end{aligned}$$

(95)

$$\begin{aligned} &{\begin{aligned}[b] I_{3} &= \frac{3(\ell+1)}{\ell-2} t I_{1} - \frac{3(4\ell+1)a }{\ell-2} \ \frac{d(a^{2} \phi)}{dt} \\ &\quad {}+ \frac{6(\ell+1) \kappa \rho_{m0}}{\ell-2}t. \end{aligned}} \end{aligned}$$

(96)