Dynamics of linear perturbations in the hybrid metric-Palatini gravity

In this work we focus on the evolution of the linear perturbations in the novel hybrid metric-Palatini theory achieved by adding a $f(\mathcal{R})$ function to the gravitational action. Working in the Jordan frame, we derive the full set of linearized evolution equations for the perturbed potentials and present them in the Newtonian and synchronous gauges. We also derive the Poisson equation, and perform the evolution of the lensing potential, ${\mathrm{\Phi }}_{+}$, for a model with a background evolution indistinguishable from $\mathrm{\Lambda }\mathrm{CDM}$. In order to do so, we introduce a designer approach that allows one to retrieve a family of functions $f(\mathcal{R})$ for which the effective equation of state is exactly $w_{\text{eff}}=-1$. We conclude, for this particular model, that the main deviations from standard general relativity and the cosmological constant model arise in the distant past, with an oscillatory signature in the ratio between the Newtonian potentials, $\mathrm{\Phi }$ and $\mathrm{\Psi }$.

Figure 1

Evolution of $f(\mathcal{R})$, $F(\mathcal{R})$ and $\mathcal{R}$, as a function of the scale factor, $a$, obtained using the designer approach for a model with $w_{\text{eff}}=-1$. We have set $C_1=-1.0\times 10^{-8}$ and $C_2=-10.5$.

Figure 2

The evolution of the lensing potential, ${\mathrm{\Phi }}_{+}$, $\chi$ and the ratio between the Newtonian potentials, $\mathrm{\Phi }$ and $\mathrm{\Psi }$, as a function of the scale factor, $a$, for the designer model with $w_{\text{eff}}=-1$ under analysis. Note that, for $\mathrm{\Lambda }\mathrm{CDM}$, $\chi =0$ and $\mathrm{\Phi }/\mathrm{\Psi }=1$ throughout the entire evolution. We plot this for three different $k(h/\mathrm{Mpc})$ modes.

Figure 3

The evolution of the effective scalar mass, $m_{\varphi }^2$, for the designer model $w_{\text{eff}}=-1$ considered in this analysis.