An emergent universe from a loop

Closed, singularity-free, inflationary cosmological models have recently been studied in the context of general relativity. Despite their appeal, these so called emergent models suffer from a number of limitations. These include the fact that they rely on an initial Einstein static state to describe the past-eternal phase of the universe. Given the instability of such a state within the context of general relativity, this amounts to a very severe fine tuning. Also in order to be able to study the dynamics of the universe within the context of general relativity, they set the initial conditions for the universe in the classical phase. Here we study the existence and stability of such models in the context of Loop Quantum Cosmology and show that both these limitations can be partially remedied, once semiclassical effects are taken into account. An important consequence of these effects is to give rise to a static solution (not present in GR), which dynamically is a center equilibrium point and located in the more natural semiclassical regime. This allows the construction of emergent models in which the universe oscillates indefinitely about such an initial static state. We construct an explicit emergent model of this type, in which a nonsingular past-eternal oscillating universe enters a phase where the symmetry of the oscillations is broken, leading to an emergent inflationary epoch, while satisfying all observational and semiclassical constraints. We also discuss emergent models in which the universe possesses both early- and late-time accelerating phases.

Figure 1

The top panel shows the plots for the function $A$ against $a$ for $j=100$ (solid line), $j=1000$ (dashed line) and $j=3000$ (dot-dashed line). The bottom panel shows plots of the function $B$ against $a$ for $V=0$ (solid line), $V>0$ (dot-dashed line) and $V<0$ (dashed line). Note the dramatic change in the shape of the function as the sign of $V$ is altered.

Figure 2

The top panel shows the plots of functions $A$ (solid line) and $B$ (dashed line) against $a$ for $V=0$. The vertical dotted line denotes the position of $a=a_{*}$. The bottom panel is the corresponding phase portrait demonstrating the center equilibrium point that occurs in this case. The plot is compactified using $x=\arctan (H)$ and $y=\arctan (\ln a)$, in order to present the entire phase space. The vertical dotted lines in the bottom panel demarcate the region in which the Hubble parameter is less than the Planck scale. The horizontal dotted line marks $a_i$, where the quantum regime begins. All axes are in Planck units, and $j=100$.

Figure 3

The top panel shows the plots of functions $A$ (solid line) and $B$ (dashed line) against $a$ for $V=0.005$. For this value of $V$, the equilibrium points are close to coalescing. The vertical dotted line denotes the position of $a=a_{*}$. The bottom panel is the corresponding phase portrait demonstrating the center equilibrium point and saddle point which occurs in this case with the dashed line indicating the separatrix. The shaded area is the unphysical region where ${\dot{\varphi }}^2$ would be negative. The plot is compactified as in Fig. 2. The dotted lines in the bottom panel are as described in Fig. 2. All axes are in Planck units, and $j=100$.

Figure 4

The top panel shows the plots of functions $A$ (solid line) and $B$ (dashed line) against $a$ for $V=-0.013$. The vertical dotted line denotes the position of $a=a_{*}$. The bottom panel is the corresponding phase portrait demonstrating the center equilibrium point that occurs in this case. The plot is compactified as in Fig. 2. The dotted lines in the bottom panel are as described in Fig. 2. All axes are in Planck units and $j=100$.

Figure 5

The figure depicts two possible forms of emergent potentials that allow for conventional reheating. The solid line is the form of the potential motivated by the inclusion of a $R^2$ term in the Einstein-Hilbert action, and the dotted line the form motivated by the inclusion of higher-order terms.

Figure 6

Time evolution of the scale factor (top panel) and scalar field (bottom panel) with the field initially on the asymptotically flat region of the potential (29) with $\alpha ={10}^{-12}$ and $\beta =0.1$. The field increases in value from initial conditions close to the LS ( center) solution defined by Eq. (14).

Figure 7

Plots illustrating the magnification of Fig. 6 around the region where emergence commences, the oscillations cease and inflation begins.

Figure 8

Plots illustrating the generic form of the potential that leads to early- and late-time accelerating phases. The potential exhibits a decaying tail as $\varphi \to -\infty$. As the field moves up this tail and increases in value, the universe can oscillate about the LS point. The inflationary regime rises (possibly) towards the Planck scale for large values of $\varphi$. As the field turns round, it can drive a phase of inflation and, if the potential exhibits a sufficiently steep middle section around $\varphi \approx 0$, reheating may proceed through gravitational particle production. Consequently, the field may survive until the present epoch, where it can act as the source of dark energy by slowly rolling along the tail towards $\varphi \to -\infty$.