No-boundary wave function, Wheeler-DeWitt equation, and path integral analysis of the bouncing quantum cosmology

Bouncing models are alternatives to inflationary cosmology that replace the initial big bang singularity by a “bouncing” phase. A deeper understanding of the initial conditions of the universe, in these scenarios, requires knowledge of quantum aspects of bouncing models. In this work, we propose two classes of bouncing models that can be studied with great analytical ease and hence, provide a test bed for investigating more profound problems in quantum cosmology of bouncing universes. Our model’s two key ingredients enable us to do straightforward analytical calculations: (i) a convenient parametrization of the minisuperspace of FRLW spacetimes and (ii) two distinct choices of the effective perfect fluids that source the background geometry of the bouncing universe. We study the quantum cosmology of these models using both the Wheeler-de Witt equations and the path integral approach. In particular, we found a bouncing model analogue of the no-boundary wave function and presented a Lorentzian path integral representation for the same. We also discuss the introduction of real scalar perturbations.

Figure 1

The red graph shows the first kind of effective potential $U_{\mathrm{eff}}^{(\mathrm{I})}(q)$ and the brown dashed graph shows the second kind of effective potential $U_{\mathrm{eff}}^{(\mathrm{II})}(q)$. The shaded portion, with $q>1$, denotes the classically allowed region, describing a bouncing universe.

Figure 2

The steepest descent/ascent curves in the complex $\mathcal{N}$-plane, that pass through the saddle points ${\overline{\mathcal{N}}}_{\pm }^{(\mathrm{I})}$, for the linear model are represented by the black curves. The contour-plots for the function $\mathrm{Re}[i2{\ell }_{\mathrm{p}}^2\{(n/6)-1{\}}^2{V}_3^{-1}{\tilde{\mathcal{S}}}_{\mathrm{cl}}^{(\mathrm{I})}]$ are also presented, using the color coding scheme given by the right inset, to better visualize the descent/ascent directions (the plots are for the parameter values $h_n=1$ and $q_1=3$). The dark red, horizontal line is the real line contour over which the integration of the lapse function needs to be performed to evaluate the no-boundary wave function. The real line contour can be smoothly deformed into the union of dashed red contours ${\mathcal{C}}_{+}^{(\mathrm{I})}$ and ${\mathcal{C}}_{-}^{(\mathrm{I})}$, thereby, enabling us to perform saddle point approximation. See text for more discussion.

Figure 3

The saddle point geometry that would be relevant to Euclidean path integral approach to matter bounce. The surface denotes the $x-\eta$ plane, with periodic identification assumed along the $x$ direction. The orange-colored portion denotes the Lorentzian part of the geometry, while blue denotes the Euclidean portion. There is a true singularity at the center of the Euclidean portion, unlike the Hawking-Hartle geometry which is smooth everywhere.

Figure 4

The steepest descent/ascent curves in the complex $\mathcal{N}$-plane for the quadratic model are represented by the black curves. The contour-plots for $\mathrm{Re}[i(9/8){\ell }_{\mathrm{p}}^2(n/6-1{)}^2{V}_3^{-1}{\tilde{\mathcal{S}}}_{\mathrm{cl}}^{(\mathrm{II})}]$ is also presented, with the color coding scheme given in the right inset, to better visualise the descent/ascent directions (the parameter values for the above plot correspond to: $h_n=1$ and $q_1=3$). The dashed purple lines are the branch cuts of the preexponential factor $\sqrt{\mathrm{sech}(3h_n\mathcal{N}/2)}$. The horizontal, green line is the real line contour. However, convergence of the $\mathcal{N}$-integral demands that we choose a slightly modified contour, namely, the red continuous curve ${\tilde{\mathcal{C}}}^{(\mathrm{II})}$, joining $-\infty e^{-i{0}^{+}}$ to $\infty e^{i{0}^{+}}$. This continuous red contour can be smoothly deformed into the dashed red curve ${\mathcal{C}}^{(\mathrm{II})}$, that pass through the saddle points, thereby enabling us to evaluate the saddle point approximation. See text for more discussion.

Figure 5

The typical form of the total effective potential $U_{\text{total}}=U_{\mathrm{eff}}-({\rho }_{\theta }/q^2)$ has been demonstrated, when a classical bouncing scenario is allowed. The two classical turning points are marked $q_{<}$ and $q_{>}$, respectively and the shaded portions with $q>q_{>}$ and $q<q_{<}$ denote the classically allowed regions. The (blue) region to the left of the plot ($0<q<q_{<}$) describes a big crunch scenario, wherein the classical domain the universe started from a zero size, reaches a maximum $q$ value, and then reduces back to zero. While the (red) region to the right ($q_{>}<q<\infty$) describes the bouncing scenario, where the universe reaches a minimum value of $q$ and then re-expands. Classically, a solution of the relevant Einstein’s equations describes either of these two scenarios. However, quantum mechanically, tunneling from one region to the other is allowed.

Figure 6

The red graph shows $\mathcal{T}({\sigma }_1)$ and the blue dashed graph shows $\mathcal{T}({\sigma }_2)$, with $V_3\sqrt{2M{\rho }_0}$ fixed to a value of 50 in both the cases. Note, however, that the WKB tunneling probability cannot be trusted for ${\sigma }_1$, ${\sigma }_2\approx 1$, since, in this regime WKB approximation fails.