Inflation in an exponential scalar model and finite-time singularity induced instability

We investigate how a Type IV future singularity can be included in the cosmological evolution of a well-known exponential model of inflation. In order to achieve this we use a two scalar field model, in the context of which the incorporation of the Type IV singularity can be consistently done. In the context of the exponential model we study, when a Type IV singularity is included in the evolution, an instability occurs in the slow-roll parameters, and in particular in the second slow-roll parameter. Particularly, if we abandon the slow-roll condition for both the scalars we shall use, then the most consistent description of the dynamics of the inflationary era is provided by the Hubble slow-roll parameters ${\epsilon }_H$ and ${\eta }_H$. Then, the second Hubble slow-roll parameter ${\eta }_H$, which measures the duration of the inflationary era, becomes singular at the point where the Type IV singularity is chosen to occur, while the Hubble slow-roll parameter ${\epsilon }_H$ is regular there. Therefore, this infinite singularity indicates that the occurrence of the finite-time singularity is responsible for the instability in the scalar field model we study. This sort of instability has its imprint on the dynamical system that can be constructed from the cosmological equations, with the dynamical system being unstable. Also the late-time evolution of the two scalar field system is studied, and in the context of the theoretical framework we use, late-time and early-time acceleration are described in a unified way. In addition, the instability due to the singularity mechanism we propose, is discussed in the context of other inflationary scalar potentials. Finally, we discuss the implications of such a singularity in the Hubble slow-roll parameters and we also critically discuss qualitatively, what implications could this effect have on the graceful exit problem of the exponential model.

Figure 1

The Hubble rate $H(t)=\frac{c_1}{c_2+c_{3}t}+c_4{(t_s-t)}^{\alpha }$ as a function of the cosmic time with $c_1={10}^{12}$, $c_2={10}^{-20}$, $c_3={10}^{10}$, $c_4={10}^{-40}$, $t_s={10}^{-35}\sec$, and $\alpha =4/3$.

Figure 2

The time dependence of the scalar field $\chi (t)$, with $c_1={10}^{12}$, $c_2={10}^{-20}$, $c_3={10}^{10}$, $c_4={10}^{-40}$, $t_s={10}^{-35}\sec$, $\alpha =4/3$ and for the initial conditions $\chi (0)\simeq {10}^{-25}$, ${\chi }^{\prime }(0)\simeq 10$.

Figure 3

The scalar field’s $\chi (t)$ evolution as a function of cosmic time $t$ with $c_1={10}^{12}$, $c_2={10}^{-20}$, $c_3={10}^{10}$, $c_4={10}^{-40}$, $t_s={10}^{-35}\sec$, $\alpha =4/3$ and for the initial conditions $\chi (0)\simeq {10}^{-25}$, ${\chi }^{\prime }(0)\simeq 10$.

Figure 4

The time dependence of the scalar field $\chi (t)$, with $c_1={10}^{12}$, $c_2={10}^{-20}$, $c_3={10}^{10}$, $c_4={10}^{-40}$, $t_s={10}^{-35}\sec$, $\alpha =4/3$ and for the initial conditions $\chi (0)\simeq {10}^{-25}$, ${\chi }^{\prime }(0)\simeq 10$, including the $\phi$-dependent term.

Figure 5

The time dependence of the scalar field potential $V(\phi (t))$, with $c_1={10}^{12}$, $c_2={10}^{-20}$, $c_3={10}^{10}$, $c_4={10}^{-40}$, $t_s={10}^{-35}\sec$,$\alpha =4/3$ and for the initial conditions $\phi (0)\simeq {10}^{20}$ (${\sec }^{-1}$), ${\phi }^{\prime }(0)\simeq {10}^{-2}$ (${\sec }^{-2}$), including the $\phi$-dependent term.

Figure 6

The time dependence of the scalar field $\phi (t)$, with $c_1={10}^{12}$, $c_2={10}^{-20}$, $c_3={10}^{10}$, $c_4={10}^{-40}$, $t_s={10}^{-35}\sec$, $\alpha =4/3$ and for the initial conditions $\phi (0)\simeq {10}^{20}$ (${\sec }^{-1}$), ${\phi }^{\prime }(0)\simeq 10$ (${\sec }^{-2}$).

Figure 7

The time dependence of the perturbation $\delta H(t)$, with $c_1={10}^{12}$, $c_2={10}^{-20}$, $c_3={10}^{10}$, $c_4={10}^{-40}$, $t_s={10}^{-35}\sec$, $\alpha =4/3$ and for the initial conditions $\delta H({10}^{-75})={10}^{-120}$ (${\sec }^{-1}$).