Bounce inflation in $f(T)$ cosmology: A unified inflaton-quintessence field

We investigate a bounce inflation model with a graceful exit into the Friedmann-Robertson-Walker (FRW) decelerated Universe within $f(T)$-gravity framework, where $T$ is the torsion scalar in the teleparallelism. We study the cosmic thermal evolution, the model predicts a supercold universe during the precontraction phase, which is consistent with the requirements of the slow-roll models, while it performs a reheating period by the end of the contraction with a maximum temperature just below the grand unified theory (GUT) temperature. However, it matches the radiation temperature of the hot big bang at later stages. The equation-of-state due to the effective gravitational sector suggests that our Universe is self-accelerated by teleparallel gravity. We assume the matter component to be a canonical scalar field. We obtain the scalar field potential that is induced by the $f(T)$ theory. The power spectrum of the model is nearly scale invariant. In addition, we show that the model unifies inflaton and quintessence fields in a single model. Also, we revisited the primordial fluctuations in $f(T)$ bounce cosmology, to study the fluctuations that are produced at the precontraction phase.

Figure 1

($\dot{H}-H$) phase space diagram. The dot curve represents the zero acceleration boundary, it divided the phase space into two regions. The shaded region is the deceleration region, while the unshaded is the acceleration one. The labels (I)–(IV) give four possible behaviors.

Figure 2

The models: (a) For $\alpha >0$, we have an initial big bang singularity followed by an inflation period capable to evolve to FRW phase; (b) For $\alpha <0$, we have a bouncing behavior which avoids the trans-Planckian problems of inflationary models; (c) For $\alpha <0$, the velocity curve shows clearly a bouncing behavior. However, after the bouncing time $t_B\sim 4.14\times {10}^{-33}\mathrm{s}$, the model can also perform an early accelerated expansion period with a smooth transition (graceful exit) into a FRW model.

Figure 3

($\dot{H}-H$) phase-space diagram corresponds to Eq. (7): (a) In the $\dot{H}>0$, the bounce is at $H=0$, where the transition from $H<0$ to $H>0$ is at $\dot{H}>0$; (b) In the $\dot{H}<0$ and $H>0$ region, it shows an early accelerated cosmic expansion (inflation) as it appears in the (IV) region, then it is followed by a decelerated expansion phase after crossing the zero acceleration curve (FRW) as it appears in (III) region.

Figure 4

The equation-of-state parameter of the torsion is given by the solid line, while the effective (total) equation-of-state parameter is given by the dash-doted line.

Figure 5

The temperature evolution (47) shows a reheating after inflation, the maximum effective temperature by the end of reheating is just below the GUT temperature $\sim {\mathrm{\Theta }}_{\text{eff}}={\mathrm{\Theta }}_{\mathrm{rh},\max }\sim {10}^{26}\mathrm{K}$. Then the effective temperature evolves similar to the radiation temperature ${\mathrm{\Theta }}_{\mathrm{r}}$ of the hot big bang.

Figure 6

The equation-of-state parameter of the scalar field.

Figure 7

Matter equation of state that produces the observed power spectrum; (a) For $V_0=5.3\times 10^{78}$, (b) For $V_0=1.21\times 10^{-19}$.

Figure 8

The above plots represent the four ingredients (${\rho }_{\varphi }$, ${\rho }_{\varphi }\pm p_{\varphi }$, and ${\rho }_{\prime \mathrm{phi}}+3p_{\varphi }$), which are necessary to study the energy conditions (68); The three models are as follows: (a) For $V_0=5.3\times 10^{78}$, the SEC is generally violated except during $t\in (2.1\times {10}^{-33},{10}^{-32})\mathrm{s}$; (b) For $V_0=0$, the energy conditions (68) are fulfilled; (c) For $V_0=1.2\times {10}^{-19}$, only the SEC is violated during $t\in (0,3.4\times {10}^{-35}\mathrm{s}]\cup (10^{17}\mathrm{s},\infty )$. All the above plots are in agreement with Fig. 6, also a summary of the final results is given in Table 1. The numerical values $10^{79}$, which appear on the vertical axis represent the scale of the axis.

Figure 9

Schematic diagram to show the evolution of the hubble radius (100). Immediately after bounce, it shrinks allowing the relevant physical modes to exit the horizon, then it expands allowing these physical modes to reenter the horizon at later time.