Local-gravity test of unified models of inflation and dark energy in $f(R)$ gravity

We consider $f(R)$ gravity theories which unify $R^n$ inflation and dark energy models. First, from the final Planck data of the cosmic microwave background, we obtain a condition, $1.977<n<2.003$. Next, under this constraint, we investigate local-gravity tests for three models. We find that the $R^n$ term can dominate over the dark energy term even at the Earth’s curvature scale, contrary to intuition; however, the $R^n$ term does not relax or tighten the constraints on the three models.

Figure 1

Potential in the Einstein frame for $R^2$ inflation model. We show parametric representations $V(R)$ and $\varphi (R)$ in the top and the bottom, respectively. In this model, the potential minimum is given by $V=0$ at $\varphi =0$. The potential as a function of the field value, $V(\varphi )$, is shown in Ref. [17].

Figure 2

Potential in the Einstein frame for Model 1 (red line) and power-law DE model (blue line). We show parametric representations $V(R)$ and $\varphi (R)$ in the top and the bottom, respectively. Dashed lines show $\varphi <0$. The potential is positive at the minimum.

Figure 3

Potential in the Einstein frame for Model 2 (red line) and Starobinsky DE model (blue line). We show parametric representations $V(R)$ and $\varphi (R)$ in the top and the bottom, respectively. Dashed lines show $\varphi <0$. The potential is positive at the minimum.

Figure 4

The allowed parameter range of $g$ and $b$ in $g$-AB model. This curve is obtained from the stability conditions (13).

Figure 5

Potential in the Einstein frame for Model 3 (red line) and AB model (blue line). We show parametric representations $V(R)$ and $\varphi (R)$ in the top and the bottom, respectively. Dashed lines show $\varphi <0$. The potential is positive at the minimum.

Figure 6

The relation between the Compton wavelength ${\lambda }_{\varphi }=m_{\varphi }^{-1}$ and the density $\rho$. Blue dashed line and red solid line correspond to the DE model and unified model (20), respectively. In this figure, we set $j=1$ and $n=2$.

Figure 7

Same as Fig. 6 except for $j=10$.

Figure 8

The relation between the Compton wavelength ${\lambda }_{\varphi }=m_{\varphi }^{-1}$ and the density $\rho$ for possible values of $n$ under the inflationary constraint (41). These lines correspond to $n=1.977$, 2, and 2.003 from bottom to top. In this figure, we fix ${\rho }_{\inf }$ in Eq. (75).

Figure 9

The relation between the Compton wavelength ${\lambda }_{\varphi }=m_{\varphi }^{-1}$ and the density $\rho$. Blue dashed line and red solid line correspond to the DE model and unified model (27), respectively. In this figure, we set $b=2$, $g=0.47$, and $n=2$.

Figure 10

Same as Fig. 9 except for $b=3$, $g=0.45$.

Figure 11

Same as Fig. 9 except for $b=9$, $g=0.3$.

Figure 12

Same as Fig. 9 except for $b=17$, $g=0.28$.