Quantum nature of the big bang: An analytical and numerical investigation

Analytical and numerical methods are developed to analyze the quantum nature of the big bang in the setting of loop quantum cosmology. They enable one to explore the effects of quantum geometry both on the gravitational and matter sectors and significantly extend the known results on the resolution of the big bang singularity. Specifically, the following results are established for the homogeneous isotropic model with a massless scalar field: (i) the scalar field is shown to serve as an internal clock, thereby providing a detailed realization of the “emergent time” idea; (ii) the physical Hilbert space, Dirac observables, and semiclassical states are constructed rigorously; (iii) the Hamiltonian constraint is solved numerically to show that the big bang is replaced by a big bounce. Thanks to the nonperturbative, background independent methods, unlike in other approaches the quantum evolution is deterministic across the deep Planck regime. Our constructions also provide a conceptual framework and technical tools which can be used in more general models. In this sense, they provide foundations for analyzing physical issues associated with the Planck regime of loop quantum cosmology as a whole.

Figure 1

Classical phase space trajectories are plotted in the $\varphi$, $p\sim \mu$ plane. For $\mu \ge 0$, there is a branch which starts with a big bang (at $\mu =0$) and expands out and a branch which contracts into a big crunch (at $\mu =0$). Their mirror images appear in the $\mu \le 0$ half plane.

Figure 2

Crosses denote the values of an eigenfunction $e_{\omega }(\mu )$ of $\Theta$ for $\epsilon =0$ and $\omega =20$. The solid curve is the eigenfunction ${\underline{e}}_{\omega }(\mu )$ of the WDW $\underline{\Theta }$ to which $e_{\omega }(\mu )$ approaches at large positive $\mu$. As $\mu$ increases, the set of points on ${\mathcal{L}}_{\epsilon }$ becomes denser and fills the solid curve. For visual clarity only some of these points are shown for $\mu >100$.

Figure 3

The exponential growth of $|e_{-|k|}^{\pm }|(\mu )$ in the genuinely quantum region is shown for three different values of $\omega$, where $\omega$ is given by $\Theta e_{-|k|}^{\pm }={\omega }^2{e}_{-|k|}^{\pm }$.

Figure 4

The amplification factor ${\lambda }^{\pm }$ in the genuinely quantum region is shown as a function of the parameter $\epsilon$ labeling the lattice and $\omega$.

Figure 5

The function $a(\epsilon )$ of Eq. (5.6) is plotted by connecting numerically calculated data points.

Figure 6

The function $b(\epsilon )$ of Eq. (5.6) is plotted by connecting numerically calculated data points.

Figure 7

Plot of the wave function $\Psi (\mu ,\varphi )$ obtained by directly evaluating the right side of (4.15). Parameters are $p_{\varphi }=500$, $\Delta p_{\varphi }/p_{\varphi }=0.05$, and $\epsilon ={\mu }_o$.

Figure 8

Expectation values and dispersion of $\widehat{|\mu {|}_{\varphi }}$ for the wave function presented in Fig. 7 are compared with classical trajectories.

Figure 9

Error functions $|{\Psi }_{(M^{\prime })}-{\Psi }_{(M)}{|}_{\mathrm{I}}$ (upper curve) and $|{\Psi }_{(M^{\prime })}-{\Psi }_{(M)}{|}_{\mathrm{II}}$ (lower curve) are plotted as a function of time steps. Here ${\Psi }_{(M^{\prime })}$ refers to the final profile of wave function for simulation with $M^{\prime }$ time steps. ${\Psi }_{(M)}$ refers to the final profile for the finest evolution (approximately $2.88\times {10}^6$ time steps). In both cases, the evolution started at $\varphi =0$ and the final profile refers to $\varphi =-1.8$.

Figure 10

The absolute value of the wave function obtained by evolving an initial data of method II. For clarity of visualization, only the values of $|\Psi |$ greater than ${10}^{-4}$ are shown. Being a physical state, $\Psi$ is symmetric under $\mu \to -\mu$. In this simulation, the parameters were $\epsilon =2{\mu }_o$, $p_{\varphi }^{\star }={10}^4$, and $\Delta p_{\varphi }/p_{\varphi }^{\star }=7.5\times {10}^{-3}$.

Figure 11

The expectation values (and dispersions) of $\widehat{|\mu {|}_{\varphi }}$ are plotted for the wave function in Fig. 10 and compared with expanding and contracting classical trajectories.

Figure 12

Comparisons between the relative dispersions $\Delta \mu /\mu$ as functions of $\varphi$ for all three methods specifying initial data and the result of direct construction.

Figure 13

A zoom on the absolute value of the wave function near the bounce point. Initial data was specified using method I.

Figure 14

A zoom on the absolute value of the wave function near the bounce point. Initial data was specified using method II.

Figure 15

A zoom on the absolute value of the wave function near the bounce point. Initial data was specified using method III.

Figure 16

The uncertainty product $\Delta \varphi \Delta p_{\varphi }$ for three methods of specifying the initial data.

Figure 17

Expectation values (and dispersions) of $\widehat{|\mu {|}_{\varphi }}$ are plotted near the bounce point, together with classical and effective trajectories (fainter and darker dots, respectively). While the classical description fails in this region, the effective description provides an excellent approximation to the exact quantum evolution. In this plot, $p_{\varphi }=3000$ and $\epsilon =2{\mu }_o$.

Figure 18

The darker lattice shows the difference between final and initial relative spreads $\Delta \mu /\mu$ for states obtained by evolving initial data of method II. The upper, fainter lattice shows the heuristic bound $\delta \mu /\mu$ given by Eq. (a10).

Figure 19

The darker lattice shows the difference between final and initial relative spreads $\Delta \mu /\mu$ for states obtained by evolving initial data of method III. The upper, fainter lattice shows the heuristic bound $\delta \mu /\mu$ given by Eq. (a12).