Closed universes in two dimensional gravity

Report 1: Equations (3.4) and (3.5) are results from gravitational path integral computations, and from a
dual boundary perspective should be properly interpreted as some sort of ensemble average. We have
added a footnote clarifying this. The ensemble average interpretation is discussed later in the paper.
Before equations (3.6) and (3.7), we added one sentence explaining the summation convention.
We have expanded the first few paragraphs of the discussion to address how the conclusion relates
to previous work on closed universes. We have also explained what additional assumptions are needed
for the argument to apply to more general theories in higher dimensions.

Report 2: We have added some clarifying comments near equations 2.6 and 3.17.

Report 3: We have expanded the discussion of prior work on AdS cosmologies, including the sug-
gested references, in the introduction. We have also clarified how these works differ in aim from the
present work. The discussion has also been expanded to discuss previous works on closed universes.
We added one sentence explaining the connection to Marolf-Maxfield model in the introduction when
we summarize the non-perturbative physics. We also added references to earlier works and explained
the motivations of this paper at the end of the introduction. We have also added a reference for tensor
networks.
We have expanded the first few paragraphs of the discussion to address how the conclusion re-
lates to previous work on closed universes. The key challenge is to understand how to describe the
experience of an observer in a closed universe with a one-dimensional Hilbert space, when all inner
products/observables that are naively defined receive large corrections from wormholes. A large part
of the discussion section is devoted to this problem, which we have tried to emphasize.
We have added a sentence clarifying the H appearing in tr[exp(−βH)Oi] in section 2. Section 2
contains a more detailed explanation of the conceptual setup useful for readers not familiar with JT
gravity. There is an additional argument regarding the dimensionality of the Hilbert space given in
the introduction and in section 2 in the JT context that consists of computing overlaps between closed
universe states that is independent of the object tr[exp(−βH)Oi].
We added some discussions about why we didn’t study both perturbative and non-perturbative
aspects in JT gravity+matter at the beginning of section 3. Calculations are more explicit in the
topological model, whereas JT with matter has additional complications.