Holographic black hole cosmologies

TL;DR

The paper constructs a microscopic holographic model of big-bang/big-crunch cosmologies sourced by a lattice of black holes, realized via a Euclidean path integral that entangles multiple CFT copies on abutting boundary geometries connected by tubes. In three dimensions, the authors present explicit cosmological saddles and analyze their dominance relative to other bulk saddles by computing genus-two and replica contributions to the action, finding a critical horizon size where the cosmology becomes the dominant saddle, with finite and infinite lattice limits giving $R^{crit}_1$ and $R^{crit}_{\infty}$ values that translate to $r_{crit} \approx 1.12\, obreakspace\ell_{AdS}$ and $d_{crit} \approx 1.2\, obreakspace\ell_{AdS}$. The mixed-state construction, its purifications, and the entanglement- wedge/island perspectives provide a framework for understanding how cosmology can be encoded holographically and suggest generalizations to higher dimensions and alternative purifications. Overall, the work connects a concrete Euclidean CFT path integral to a Lorentzian cosmology with a lattice of black holes, offering a controlled setting to study holographic big-bang dynamics and the conditions under which cosmology dominates the quantum gravitational path integral.

Abstract

We describe and study a holographic construction of big-bang / big-crunch cosmological spacetimes where the matter consists of a lattice of black holes. The cosmological spacetime is dual to an entangled state of a collection of holographic CFTs associated with the second asymptotic regions of the black holes. For a cosmology with spatial slice geometry $Σ$, this state is constructed via a Euclidean path integral for the CFT on a geometry obtained by connecting two copies of $Σ$ by a lattice of tubes. In three-dimensional gravity, we describe the cosmological solutions and the associated Euclidean saddles explicitly. For the case of (globally) flat cosmology, we determine when the Euclidean solution associated with the cosmology provides the dominant saddle compared to other natural candidates that preserve the symmetries of the boundary space. We find that the cosmological saddle dominates when the black holes are sufficiently large and close together. Our cosmology has a mixed state version where the physics behind the black hole horizons is unspecified and the Euclidean construction involves a pair of CFTs with an ensemble of operator insertions correlated between the two CFTs. Various purifications (adding second asymptotic regions for the black holes) correspond to various ways to promote this ensemble to an interaction by adding auxiliary degrees of freedom that couple the two CFTs in the Euclidean picture. These auxiliary degrees of freedom provide a Hilbert space for the cosmology in the Lorentzian picture.

Figures (17)

• Figure 1: Left: a pair of Euclidean CFTs with an ensemble of operator insertions. Each pair of insertions is summed over all operators with a Boltzmann-type weight. The dual geometry has a spatial slice (green) with a collection of thermal mixed-state black holes. Center: the sum of operator insertions can be obtained by integrating out CFT degrees of freedom on a collection of tubes connecting the two CFTs. The dual geometry in this case has a second asymptotic region for each black hole. Right: The path integral with tubes can be sliced to define a pure entangled state of a collection of non-interacting CFTs (red) on spatial spheres. This state is dual to the Lorentzian cosmology.
• Figure 2: The CFT path integral on a cylinder produces the thermofield double state of the a pair of CFTs on the boundary spheres. The same state is constructed by the Euclidean path integral on a pair of disks with a correlated insertions of operators.
• Figure 3: Left: Euclidean path integral constructing an entangled state of CFTs (associated with the red boundaries). Right: Spatial slice of the dual spherical black hole cosmology; the funnels represent the second asymptotic regions for the black holes in the cosmology.
• Figure 4: Gluing together patches of hyperbolic disk
• Figure 5: a) The hyperbolic funnel geometry of the $\tau = 0$ slice. b) a simpler representation of this, with the asymptotic boundary represented as a finite circle. c) The spacetime geometry (\ref{['Euc']}) corresponding to this spatial patch. d) The asymptotic boundary of this spacetime geometry.