Inflation with a smooth constant-roll to constant-roll era transition

In this paper, we study canonical scalar field models, with a varying second slow-roll parameter, that allow transitions between constant-roll eras. In the models with two constant-roll eras, it is possible to avoid fine-tunings in the initial conditions of the scalar field. We mainly focus on the stability of the resulting solutions, and we also investigate if these solutions are attractors of the cosmological system. We shall calculate the resulting scalar potential and, by using a numerical approach, we examine the stability and attractor properties of the solutions. As we show, the first constant-roll era is dynamically unstable towards linear perturbations, and the cosmological system is driven by the attractor solution to the final constant-roll era. As we demonstrate, it is possible to have a nearly scale-invariant power spectrum of primordial curvature perturbations in some cases; however, this is strongly model dependent and depends on the rate of the final constant-roll era. Finally, we present, in brief, the essential features of a model that allows oscillations between constant-roll eras.

Figure 1

Phase space structure of the solution (18), with the following initial conditions: $\phi (0)=10M_p$ (red line), $\phi (0)=M_p$ (green line), $\phi (0)=0.1M_p$ (blue line).

Figure 2

Stability of linear perturbations for the solution (18).

Figure 3

Behavior of the perturbations $\theta (\phi )$ for the constant-roll era with $\frac{\ddot{\phi }}{\dot{\phi }H}=-1$.

Figure 4

Behavior of the perturbations $\theta (\phi )$ for the constant-roll era with $\frac{\ddot{\phi }}{\dot{\phi }H}=-1/2$.

Figure 5

The Hubble rate as a function of the cosmic time (left) and as a function of the scalar field $\phi /M_p$.

Figure 6

The phase space of the solution $H(\phi )=C_1{I}_{-\beta }(2\sqrt{\alpha }{e}^{-\frac{1}{2}(\lambda x)})$, for the initial conditions $\phi (0)=10M_p$ (red line), $\phi (0)=M_p$ (green line), $\phi (0)=0.1M_p$ (blue line).

Figure 7

$\phi$ dependence of the function $f(\phi )=-({\sin }^2(\lambda \phi )+\cos (\lambda \phi ))$.

Figure 8

$\phi$ dependence of the Hubble rate $H(\phi )=C_1{e}^{(-\cos (\lambda \phi ))}$, for various scalar field intervals.

Figure 9

$\phi$ dependence of the scalar potential (41).

Figure 10

Phase space structure of the solution (40), with the following initial conditions: $\phi (0)=10M_p$ (red line), $\phi (0)=M_p$ (green line), $\phi (0)=0.1M_p$ (blue line).

Figure 11

$\phi$ dependence of the linear perturbations $\theta (\phi )$ for the solution $H(\phi )=C_1{e}^{(-\cos (\lambda \phi ))}$.