Nonminimal coupling and inflationary attractors

We show explicitly how the T model, E model, and Hilltop inflations are obtained from the general scalar-tensor theory of gravity with arbitrary conformal factors in the strong coupling limit. We argue that $\xi$ attractors can give any observables $n_s$ and $r$ by this method. The existence of attractors imposes a challenge to distinguish different models.

Figure 1

The numerical results of $n_s$ and $r$ for the scalar-tensor theory with the potential (36). We take $\omega (\varphi )=1$, $n=2$, $\alpha =1$, the power-law functions $f(\varphi )={\varphi }^k$ with $k=1/4$, $1/3$, $1/2$, 1, 2, and 3, and $N=60$. The coupling constant $\xi$ increases along the direction of the arrow in the plot. The E-model attractors (37) and (38) are reached if $\xi \gtrsim 100$.

Figure 2

The numerical results of $n_s$ and $r$ for the scalar-tensor theory with the potential (47). We take $\omega (\varphi )=1$, $n=2$, $\alpha =1$, the power-law functions $f(\varphi )={\varphi }^k$ with $k=1/4$, $1/3$, $1/2$, 1, 2, and 3, and $N=60$. The coupling constant $\xi$ increases along the direction of the arrow in the plot. The T-model attractors (48) and (49) are reached if $\xi \gtrsim 100$.

Figure 3

The dependence of (a) $n_s$ and (b) $r$ on $\xi$ for the scalar-tensor theory with the potential (55). We take $\omega (\varphi )=1$, $n=12$, $\mu =1/3$, the power-law functions $f(\varphi )={\varphi }^k$ with $k=1/3$, $1/2$, $2/3$, 1, 2, and 3, and $N=60$.

Figure 4

The numerical results of $(n_s,r)$ for the scalar-tensor theory with the potential (55). The coupling constant $\xi$ increases along the direction of the arrow in the plot.

Figure 5

The marginalized 68%, 95%, and 99.8% C.L. contours for $n_s$ and $r_{0.002}$ from Planck 2015 data and the attractors for T, E, and Hilltop models. The dashed lines are for the coupling function $f(\varphi )=\varphi$, and the solid lines represent $f(\varphi )={\varphi }^2$.