Remarks on 2D quantum cosmology

TL;DR

This work provides a precise, multi-faceted treatment of two-dimensional quantum gravity with a positive cosmological constant coupled to a large central charge CFT. It combines a thorough classical ADM analysis with an exact quantum treatment via timelike Liouville theory, deriving Wheeler-DeWitt wavefunctions and identifying Hartle-Hawking and Vilenkin states, including a quantum bouncing cosmology. It connects the wavefunction to disk timelike Liouville path integrals, clarifying contour choices and saddle geometries, and discusses the quantum information content of big bang states in relation to the de Sitter entropy. The study advances understanding of quantum cosmology in a tractable model, offering insights into wavefunction interpretations, norm choices, and potential extensions to higher-genus topologies and supersymmetric settings.

Abstract

We consider two-dimensional quantum gravity endowed with a positive cosmological constant and coupled to a conformal field theory of large and positive central charge. We study cosmological properties at the classical and quantum level. We provide a complete ADM analysis of the classical phase space, revealing a family of either bouncing or big bang/crunch type cosmologies. At the quantum level, we solve the Wheeler-DeWitt equation exactly. In the semiclassical limit, we link the Wheeler-DeWitt state space to the classical phase space. Wavefunctionals of the Hartle-Hawking and Vilenkin type are identified, and we uncover a quantum version of the bouncing spacetime. We retrieve the Hartle-Hawking wavefunction from the disk path integral of timelike Liouville theory. To do so, we must select a particular contour in the space of complexified fields. The quantum information content of the big bang cosmology is discussed, and contrasted with the de Sitter horizon entropy as computed by a gravitational path integral over the two-sphere.

Figures (5)

• Figure 1: Penrose diagram of global de Sitter space (square) with the shaded region indicating the planar patch (\ref{['planar']}). For dS$_2$, each point in the square corresponds to an $S^0$, and thus two-points in the physical spacetime. The two asymptotia are given by the horizontal boundaries of the Penrose diagram.
• Figure 2: Left: For $E_{\rm m} < \frac{c}{24\pi}$ we have bouncing cosmologies with two exponentially expanding regions in the far past and far future. The red slice at $t=0$ is the minimal waist of length square $\frac{2\pi}{\Lambda}(\frac{c}{24\pi} - E_{\rm m})$. Right: For $E_{\rm m} > \frac{c}{24\pi}$ we have solutions which end in the past or future in a Milne type singularity. Here we show the case of a past singularity.
• Figure 3: Plot of $\text{Re} \Psi(\tilde{\varphi}_0)$ vs. $\tilde{\varphi}_0$ in (\ref{['psi']}) with $\alpha_-=0$ and $\alpha_+=1$ overlaid with the asymptotic approximations (\ref{['largephi0']}) and (\ref{['smallphi0']}) for (left) $\tilde{\varepsilon}_{\rm m}^2 = 0.5$ and (right) $\tilde{\varepsilon}_{\rm m}^2 = -0.5$.
• Figure 4: A complexified saddle for the Hartle-Hawking no boundary wavefunction can be obtained by gluing the Euclidean dS$_2$ hemisphere solution to the future half of Lorentzian dS$_2$. The saddle caps off smoothly in the Euclidean regime.
• Figure 5: Schläfli integration contour. The integration contour avoids the branch cut along the real axis $\mathbb{R}^+$.