Menagerie of Euclidean constructions for 3D holographic cosmologies

This paper generalizes the Antonini-Sasieta-Swingle construction of 3D holographic cosmologies by introducing a gluing procedure that adds arbitrary AdS tubes to heavy-particle wormhole solutions, enabling the creation of homogeneous and isotropic models while identifying conditions under which these cosmological saddles dominate the Euclidean path integral over alternative configurations.

The Gist

Imagine the universe as a giant, invisible stage where the laws of physics play out a cosmic drama. For a long time, physicists have been trying to understand how to describe a "closed universe"—a universe that expands from a Big Bang and eventually collapses back into a Big Crunch, with no edges or outside.

In recent years, a specific recipe (called the AS2 construction) was proposed to create a "cosmic snapshot" of such a universe using the tools of quantum mechanics. The idea was that if you arrange certain quantum ingredients just right, they would naturally form a picture of this closed universe.

However, in this new paper, the authors (Mark Van Raamsdonk and Alejandro Vilar L´opez) act like a team of cosmic architects and detectives. They take that original recipe, build a massive "menagerie" (a zoo) of new, more complex versions of it, and then run a strict quality control test. Their conclusion? The original recipe usually fails the test. It doesn't actually produce the closed universe as the main event; instead, it produces something else entirely.

Here is a breakdown of their findings using simple analogies:

1. Building the Cosmic Zoo (The Construction)

The authors start with a basic blueprint for a closed universe. Think of this blueprint as a Euclidean wormhole.

• The Analogy: Imagine a tunnel connecting two separate rooms (the past and future of our universe). In the middle of this tunnel, there is a "baby universe" that pops into existence and then disappears.
• The Innovation: The original recipe treated the matter inside this universe like a smooth, uniform shell of dust. The authors, however, built a much more detailed version. They replaced the smooth shell with individual heavy particles (like distinct bricks or marbles).
• The Expansion: They realized they could glue extra tubes (AdS cylinders) onto this wormhole. Imagine taking a single tunnel and attaching many extra hallways to it. Each new hallway connects to a different "room" (a copy of our universe). This allows them to create a huge variety of complex scenarios, including ones that look almost perfectly smooth and uniform, just like our real universe.

2. The "Dominance" Test (The Necessary Condition)

The big question is: When we look at the quantum "snapshot" created by this recipe, does the closed universe actually show up as the main picture? Or is it just a faint, rare background noise?

The authors set up a litmus test to see if the closed universe is the "dominant" reality.

• The Analogy: Imagine you have a photo of a party. If the photo is "dominated" by the party, you should see a huge crowd of people interacting. If you slice the photo in a different direction (vertically instead of horizontally), you should still see a massive, connected crowd.
• The Rule: For the closed universe to be the main event, the quantum state must have a huge amount of "entanglement" (a deep, invisible connection) between the past and future sides of the wormhole. It's like saying, "For this universe to exist, the past and future must be so tightly linked that they are practically one big, entangled system."

3. The Verdict: The Original Recipe Fails

When the authors applied this test to the original AS2 recipe, they found a problem.

• The Problem: The original recipe creates a state where the past and future are only weakly connected. It's like trying to hold a massive, heavy balloon (the closed universe) with a single, thin thread. The thread snaps.
• The Result: Because the connection isn't strong enough, the universe doesn't form as the main picture. Instead, the "dominant" picture is a boring, empty universe where the particles just pair up with their neighbors and don't create a baby universe at all.
• The Metaphor: It's like trying to build a castle out of sand. The original recipe tried to build a huge castle, but the sand was too loose. The "castle" (the closed universe) collapsed, and what you actually got was just a pile of sand (a non-cosmological state).

4. How to Fix It (Making the Universe Win)

The authors ask: "Can we tweak the recipe so the closed universe actually wins?"

• Option A: More Tubes: They suggest that if you add a massive number of extra tubes (of the order of the universe's complexity constant, $c$), you might force the connection to be strong enough. But this requires a level of complexity that pushes the math to its breaking point.
• Option B: The "Baryon" Idea: They mention a possibility where the particles carry a special "charge" (like a secret ID card). If the particles on the past side have a charge that must match the particles on the future side, they might be forced to connect through the wormhole, creating the universe. However, in quantum gravity, these charges usually leak away, so this is tricky.
• Option C: The "Polycosmos" (Many Universes): They propose that if you build a setup with many different closed universes and permute (shuffle) how they connect, the sheer number of possibilities might overwhelm the boring "no-universe" options. It's like flipping a coin a billion times; eventually, you might get a long string of heads, even if tails is more likely for a single flip.

5. The Black Hole Twist

The paper also looks at what happens when you heat up the system (turn up the temperature).

• The Shift: At low temperatures, the "closed universe" was the loser. But as you increase the temperature, the system transitions. The "boring" state turns into a black hole.
• The Surprise: In this hot, black-hole phase, the long wormhole does become the dominant picture. This is because the black hole naturally has the strong connections (entanglement) needed to pass the test. So, while the cold closed universe recipe fails, the hot black hole version works.

Summary

The authors built a massive library of complex 3D gravity models to test a popular theory about how closed universes are created in quantum mechanics. They found that the original, simple theory does not work because the quantum connections aren't strong enough to hold the universe together. The "real" outcome of that recipe is usually a universe without a baby universe.

However, they showed that with enough complexity (many tubes), specific charges, or by heating the system up to create black holes, it is possible to make the closed universe the dominant reality. They haven't found a simple, guaranteed way to make the original recipe work, but they have mapped out exactly why it fails and what it would take to fix it.

Technical Summary

Technical Summary: Menagerie of Euclidean constructions for 3D holographic cosmologies

Problem Statement
The paper addresses the interpretation of closed universe cosmologies within the framework of holography and the gravitational path integral. Specifically, it investigates the construction proposed by Antonini, Sasieta, and Swingle (AS2), which defines a state of two Conformal Field Theories (CFTs) on a sphere purported to be dual to a spacetime containing a disconnected closed universe (a "baby universe") alongside two asymptotically Anti-de Sitter (AdS) regions.

A central debate in the literature concerns whether the AS2 construction genuinely encodes the physics of a closed universe or if the cosmological interpretation is merely a subdominant saddle in the path integral. Previous arguments suggest that the standard AdS/CFT dictionary might interpret the AS2 state as a pair of AdS regions containing entangled low-energy particle gases, leading to a puzzling scenario where a single CFT state has two distinct gravitational duals. Furthermore, arguments regarding the breakdown of semiclassicality and the nature of the Hilbert space for closed universes have cast doubt on the dominance of the cosmological saddle. The authors aim to rigorously test the conditions under which a cosmological saddle can dominate the Euclidean path integral and to generalize the AS2 construction to explore these conditions more broadly.

Methodology
The authors employ a geometric construction in three-dimensional gravity with a negative cosmological constant ($\Lambda < 0$). Their methodology involves:

1. Generalized Euclidean Constructions: Building upon the work in [1], the authors construct a large class of exact solutions involving heavy matter particles (modeled as conical defects) in 3D gravity. They start with a parent Euclidean wormhole solution, which is a time-reflection symmetric geometry constructed by gluing hyperbolic polygons (representing spatial slices) to form a compact surface $\Sigma$.
2. Surgery and Gluing: They introduce a "surgery" procedure where arbitrary numbers of AdS cylinders (tubes) are glued to the parent wormhole. These tubes connect the past and future conformal boundaries. This procedure generalizes the AS2 setup, allowing for arbitrary spatial topologies (genus $g \ge 0$) and arbitrary distributions of matter particles, including configurations that are approximately homogeneous and isotropic.
3. Lorentzian Continuation: The Euclidean solutions are analytically continued to Lorentzian spacetimes. The authors demonstrate that these continuations yield spacetimes containing a closed universe cosmology (big-bang/big-crunch) entangled with one or more copies of asymptotically AdS spacetimes.
4. Dominance Criterion: The authors derive a necessary condition for the cosmological saddle to dominate the path integral. They argue that if the cosmological saddle is dominant, a specific orthogonal slicing of the Euclidean geometry (cutting through the length of the wormhole) must correspond to a state with order-$c$ entanglement entropy between the dual CFTs (where $c$ is the central charge). This is based on the Ryu-Takayanagi formula and the expectation that a dominant cosmological saddle implies a dual state dominated by a two-sided black hole in the alternative slicing.
5. Action Comparisons: To test dominance, the authors explicitly calculate and compare the on-shell Euclidean actions of the cosmological saddle against alternative "non-cosmological" saddles. These alternative saddles involve local contractions of matter particles (operator insertions) within the same CFT copy, avoiding the formation of a closed universe. They perform these calculations in both the limit of discrete particles and the thin-shell approximation.

Key Contributions and Results

• Generalization of AS2: The paper constructs a "menagerie" of solutions that significantly generalize the AS2 construction. Unlike the original AS2 model, which approximates matter as a uniform shell, this work treats matter as discrete particles creating conical defects. This allows for the construction of cosmologies with general spatial topologies and matter distributions, including those that are nearly homogeneous and isotropic.
• Necessary Condition for Dominance: The authors establish a necessary condition for the cosmological saddle to dominate: the dual CFT state must possess order-$c$ entanglement entropy. They argue that the original AS2 construction, which typically involves operator insertions that do not generate such high entanglement (especially at large $\beta$), fails to meet this condition.
• Subdominance of the Cosmological Saddle: Through explicit action calculations, the authors demonstrate that for the standard AS2 configuration (and its generalizations with light particles), the non-cosmological saddle (where particles contract locally) always has a lower action. The difference in action scales with the number of particles $n$ and the central charge $c$, specifically growing as $\sim c \cdot GM \log(n)$ or $\sim c \cdot GM \log(1/\epsilon)$ in the thin-shell limit. Consequently, the cosmological saddle is exponentially suppressed.
• Conditions for Potential Dominance: The paper identifies specific regimes where the cosmological saddle might dominate:
  • If the number of glued AdS tubes is of order $c$, creating a state with order-$c$ entanglement.
  • If the tubes are sufficiently short to contain black holes (as discussed in [17]).
  • If there is a vast number of distinct cosmological saddles (e.g., via multi-universe permutations) that collectively overwhelm non-cosmological saddles, even if individual cosmological saddles have higher action.
  • If the matter carries an approximate global charge (like baryon number), potentially suppressing local contractions, though the authors note that exact global symmetries are forbidden in quantum gravity.
• Black Hole Phase Transition: The authors briefly discuss the transition to the black hole phase (low $\beta$). They suggest that in this regime, the long-wormhole saddle (which includes the cosmology) may become dominant because the negative action of the black hole phase can outweigh the suppression from particle propagators, unlike in the cosmological phase.

Significance and Claims
The paper claims to provide a rigorous test for the validity of the AS2 construction as a description of closed universe cosmologies. By generalizing the construction and applying a necessary condition based on entanglement entropy, the authors argue that the original AS2 construction is likely subdominant. The cosmological interpretation is not the primary description of the CFT state; rather, the state is dominated by a geometry without a closed universe, where matter particles contract locally.

The significance of this work lies in clarifying the conditions under which holographic duals of closed universes can emerge. It suggests that simply inserting operators to create a shell is insufficient to generate a dominant cosmological saddle; specific entanglement structures or ensemble averaging are required. The authors conclude that while the cosmological spacetime may exist as a rare component of the wavefunction, extracting its physics requires mechanisms (such as projection or specific mixed-state prescriptions) that go beyond the standard saddle-point approximation of the pure state path integral. The work also opens avenues for understanding how "baryon asymmetry" or multi-universe configurations might alter the dominance of saddles, though these remain speculative within the current framework.