Invariant solutions and Noether symmetries in hybrid gravity

Symmetries play a crucial role in physics and, in particular, the Noether symmetries are a useful tool both to select models motivated at a fundamental level, and to find exact solutions for specific Lagrangians. In this work, we apply Noether point symmetries to metric-Palatini hybrid gravity in order to select the $f(\mathcal{R})$ functional form and to find analytical solutions for the field equations and for the related Wheeler-DeWitt (WDW) equation. It is important to stress that hybrid gravity implies two definitions of curvature scalar: $R$ for standard metric gravity and $\mathcal{R}$ for further degrees of freedom related to the Palatini formalism. We use conformal transformations in order to find out integrable $f(\mathcal{R})$ models. In this context, we explore two conformal transformations of the forms $d\tau =N(a)dt$ and $d\tau =N(\varphi )dt$. For the former, we found two cases of $f(\mathcal{R})$ functions where the field equations admit Noether symmetries. In the second case, the Lagrangian reduces to a Brans-Dicke-like theory with a general coupling function. For each case, it is possible to transform the field equations by using normal coordinates to simplify the dynamical system and to obtain exact solutions. Furthermore, we perform quantization and derive the WDW equation for the minisuperspace model. The Lie point symmetries for the WDW equation are determined and used to find invariant solutions. In particular, hybrid gravity introduces a further term in cosmic dynamics whose interpretation is related to the signature of an auxiliary scalar field. Solutions are compared with $\mathrm{\Lambda }\mathrm{CDM}$.

Figure 1

Comparison of the scale factor (84) with that of $\mathrm{\Lambda }\mathrm{CDM}$ cosmology $a_{\mathrm{\Lambda }}(t)$ and the radiation solution $a_r(t)=a_{0r}\sqrt{t}$ where $t_0$ is the present time, $a_{\mathrm{\Lambda }}(t_0)=1$. For the solution (84) of the hybrid gravity, we set $|a_1|>1$.