The starting point for our construction is a special case described in [12], where ME is a time-reflection-symmetric Euclidean wormhole solution and M is a closed universe big-bang–big-crunch cosmology.

The spatial geometry Σ of the slices fixed by time-reflection symmetry is common to M and ME. It is a compact space obtained by gluing together geodesic polygons, as illustrated in Fig. 2.

When gluing together polygons, we can smoothly glue any two sides with equal geodesic length; the sides may be from two different polygons or two different sides of the same polygon, however we assume that all the polygons are oriented and that the gluings respect orientation. We consider gluing a collection of polygons such that the final surface is compact and without boundaries. The resulting space may include conical defects at the vertices. These correspond to the locations of massive particles. For a conical deficit δ, we require a particle of mass 8πGm=δ(2.1)in order that Einstein’s equations are satisfied. We restrict to the situation with no conical excesses so that all particles have positive mass.⁴

See Ref. [12] for a description of the most general such compact space.

Given the geometry Σ with metric dΣ2, the Euclidean wormhole geometry ME can be described most simply via the metric ds2=du2+cosh2(u/ℓ)dΣ2.(2.2)This has two conformal boundaries, each with the same conformal geometry as Σ, at u=±∞.

The associated cosmology M is described by the analytic continuation ds2=-dt2+cos2(t/ℓ)dΣ2.(2.3)These geometries are locally AdS3 away from the conical singularities. The geometry near any one of the conical defects can be represented as in Fig. 3; the time evolution of each polygon adjacent to the vertex can be represented as a geodesic prism in Friedmann-Robertson-Walker (FRW) coordinates with the chosen vertex at the center of the hyperbolic disk.

There is a nice relation between the total amount of matter in the cosmology, the 2D spatial volume and the genus given by [12] Mtot=14G(V2πℓ2+χ).(2.4)

Starting from one of the wormhole solutions that we have constructed in the previous section, we can construct a very large family of additional solutions whose Lorentzian continuation includes the same cosmology by a gluing procedure that we now describe. These new solutions generalize the AS2 construction in [19].

We give two equivalent descriptions of the construction. First, starting with the representation in Fig. 4, we can obtain new solutions by truncating the top and bottom of one or more of the cylinders in a reflection-symmetric way and gluing them together. For each AdS, we have a continuous parameter that controls the amount of the cylinder that we keep. We could also choose to glue together the top and bottom cylinders after some permutation, though only certain choices here will respect the time-reflection symmetry. Together with the parameters associated with embedding each polygon in hyperbolic space, we have all together a large number of parameters characterizing the Euclidean solution when the number of polygons is large.

For each of these glued solutions, the slice left invariant by time-reflection symmetry now includes n copies of H2 (the gluing surfaces) when we have truncated and glued n of the cylinders. In the Lorentzian continuation, we have n copies of Lorentzian AdS together with a single closed universe cosmology.⁶

See the discussion for a generalization where we have arbitrary numbers of closed universes.

An alternative way to understand the AS2-type solutions is to note that the wormhole geometry for a given Σ has a pair of conformal boundaries related by time-reflection symmetry. Consider any extremal surface that ends on one of these conformal boundaries and avoids any of the conical singularities (particles). Consider also the image of this surface under the time-reflection symmetry. We can excise the part of the geometry between each of these surfaces and the conformal boundaries and then identify the two surfaces (or glue in a tube of AdS3 that connects them), as shown in Fig. 6 (left). In this way, we can add an arbitrary number of additional AdS3 “tubes” to the geometry while preserving the Z2 symmetry.

We can perform a similar construction but include one or more of the particles in the glued tube region, as shown in Fig. 6 (right). In this case, we have an additional asymptotically AdS geometry in the Lorentzian solution that includes the particles that intersect the gluing surface.

The gluing picture suggests a generalization to arbitrary dimensions. Starting from any Euclidean wormhole solution that gives rise to a cosmology under analytic continuation, the asymptotic regions τ→±∞ become arbitrarily close to pure AdS (since all contributions in the Euclidean Friedmann equation dilute faster than the negative cosmological constant as the scale factor grows). Thus, we can perform a similar gluing procedure, removing regions bounded by geodesic surfaces at either end of the wormhole and then identifying the solution across the two surfaces.

We can describe a class of solutions that matches precisely with the original AS2 construction as follows. Take any two regular hyperbolic n-gons for any n≥3 and glue sides cyclically, moving clockwise around one n-gon and counterclockwise around the other n-gon. The AS2 construction based on this gluing is depicted in Fig. 7; we will see that it corresponds to a fully backreacted unsmeared version of the original construction.

The polygon can be decomposed into 2n right triangles of the type shown in Fig. 8. Using the hyperbolic law of cosines and momentarily setting ℓ=1, we have that coshR=cosh LcoshA2.(2.7)The hyperbolic sine law gives sinhR=sinhLsin(α/2)=sinh(A/2)sin(π/n).(2.8)Squaring the cosine relation and using the sine law to eliminate sinhL and sinh(A/2), we get 1=sinh2Rsin2α2sin2πn+sin2α2+sin2πn.(2.9)The conical deficit is δ=2π-2α, so, using (2.4), the mass of each particle is m=π-α4πG,(2.10)and the total mass is M=n(π-α)4πG.(2.11)Taking a limit where the number of particles goes to infinity keeping the total mass M fixed, each polygon becomes a disk of radius R in hyperbolic space, with coshR=2GM.(2.12)The circumference of the disk is the limiting value of C=nA, which gives C=2πsinhR=2π(2GM)2-1.(2.13)

We now discuss a variety of other interesting examples that generalize AS2.

Let us apply this criterion to the original AS2 construction. The two relevant slicings are depicted in Fig. 14. Here, we have infinitesimally adjusted the locations of the operator insertions so that half are on one side of the vertical green slicing and half are on the other side. Our criterion says that if the horizontal red slicing produces a state with a baby-universe cosmology as the dominant contribution, the vertical green slicing should produce a state of two CFTs dual to a two-sided black hole and, therefore, have order c entanglement. We can write this state explicitly as |Ψ⟩=OL⊗OR|ΨTFD(β)⟩,(3.1)where we start with the thermofield double (TFD) state at large β and then act with the same operator in each CFT. The operator O is defined by a path integral on an infinitesimal cylinder with a ring of operator insertions. This is not unitary, so the new state can have different entanglement from the original state, but, in order to end up with order c entanglement, starting with order 1 entanglement would seem to require extreme fine-tuning that essentially reverses the Euclidean evolution defining the thermofield double state. To check this, we performed a simulation with a pair of harmonic oscillators originally in the thermofield double state at some large β. Working in an E<E0 truncation of each Hilbert space, we chose Gaussian random Hermitian operators O (the norm of these does not affect the entanglement in the final normalized state, so we work at some fixed norm) and constructed the state (3.1) for each choice of O. This construction typically produced final states with energy of the order of E0 but entropy of a similar order of magnitude to the original entropy, very small for large β. An example of such simulations can be found in Fig. 15.

Our conclusion is that the original AS2 construction seems very unlikely to produce the cosmology as a dominant part of the wave function, since the construction did not involve any particular fine-tuning such that the ring operators O mimic backward Euclidean evolution. In the CFT situation, we expect that the state (3.1) will have energy of the order of c but entropy of the order of one, inconsistent with having a two-sided black hole as the dominant part of the dual gravitational state. From the bulk viewpoint, this argument indicates that there must be another saddle that dominates the computation of the norm of the state (3.1) and which does not include a cosmology. In the next section, we will discuss examples of such alternative saddles for the AS2 construction in three-dimensional gravity, but we emphasize that the previous argument is fully general, even if it does not tell us anything about what the alternative saddles are.

Next, we ask whether the baby universe can ever dominate in any of our more general constructions. To satisfy our necessary condition, we would like a construction where in the dual slicing both the energy and the entropy are of the order of c.

Consider the situation where we have a sphere of total volume (i.e., 2D area) V triangulated by a very large number 2n of hyperbolic triangles. We also have order 2n particles, so according to (2.4), the mass of each particle is of the order of V/(nGℓ2). Now, consider the construction where we have a separate AdS cylinder for each triangle and we connect the top and bottom of each of these. Assume there is some spatial reflection symmetry so that there is a natural vertical slicing along the reflection-invariant slice such that n of the cylinders coupling the two CFTs are on each side of this slice.

In this case, we have a state of the form |Ψ⟩=(OL⊗OR)⊗n|ΨTFD(β)⟩,(3.2)where now O is an operator that maps n copies of the CFT Hilbert space to a single copy. Here the entropy is of the order of n, and again, we do not expect the operators to change the order of magnitude of the entropy. Thus, in order that the final state has order c entanglement between the two CFTs, we would need n to be of the order of c and all the particle masses to be of the order of 1. This takes us outside the regime of validity of our classical construction (see Ref. [12] for a discussion of the quantum description that replaces it), but for large but finite c, we can certainly make the CFT construction.

In summary, to have a chance to get the cosmology as a dominant saddle with tubes of length β long enough so that the auxiliary AdS spacetimes do not contain black holes, we need the number of such tubes to be of the order of c. This means that most of the mass is in light particles and essentially every particle has order one entanglement with an auxiliary system. Note that in this case the boundary defines a CFT observable on a genus-c surface, and it is unclear to what extent a semiclassical analysis can be applied to the computation of such objects.⁹

We thank Alex Belin for emphasizing this to us.

Alternatively, we can take sufficiently short tubes so that the auxiliary AdS spacetimes contain black holes; this gives the construction of [17].

The discussion above assumed a fully microscopic construction with no additional ensemble averaging. It is well known that, in some contexts, wormhole solutions compute certain CFT correlators averaged over an ensemble of theories [4,5,18,30–41]. For three-dimensional geometries with matter insertions, there is a similar picture where the action for wormhole solutions matches with calculations in which CFT correlators are averaged over an ensemble of operator production expansion data in the CFT [34]. Such additional averaging applied to an AS2 construction built upon a given wormhole could also pick out the cosmological saddle. Failing our necessary condition can be taken as a sign that such additional averaging is required. In this case, knowing the details of this averaging would likely be required to understand better how much information the AS2 state (which will be mixed due to the averaging) carries about the cosmology.

In cases where our necessary condition fails, we expect that the cosmological saddle cannot be dominant, so there should be some other saddle with lower action. In this other saddle, the asymptotic regions associated with the two CFTs in the alternative slicing should not be classically connected, so the wormhole should be absent in the Euclidean solution. In the next section, we discuss alternative saddles of this type that can be dominant. These alternative saddles can be understood as contributions to the CFT correlator where operators in a given ring contract among themselves.

We have presented two different bulk saddles for the AS2 CFT configuration: one including a closed cosmology in the bulk (which we will call the cosmological saddle) and one in which contractions between matter particles are done locally in a single spatial cut and no closed cosmology exists in the time-reflection-symmetric slice (the noncosmological saddle). We now want to compare their on-shell actions to see which one dominates in the limit of large c. In the general situation in which the particles produce a finite backreaction, obtaining the on-shell actions is nontrivial due to the conformal transformation needed in each case to make the boundary metric flat. However, we can look at a simplifying limit that takes us essentially to the original AS2 construction; in this limit, discussed already in (2.12), individual particles do not backreact, and the cosmology is sustained by a large number of them, forming a continuous fluid of pressureless dust. More concretely, the limit is [42] 1≪n≪c=3ℓ2G,M=nm=O(G-1)  fixed,(4.6)so that 1≪mℓ≪c.

We first compute the Euclidean on-shell action of the cosmological saddle from I=-116πG∫M(R+2ℓ2)-18πG∫∂M(K-1ℓ)+∑i=1nmi∫WidLi.(4.7)The second term contributes only at asymptotic boundaries (where we set our boundary conditions to induce a flat metric), while the last term is the contribution from each point particle with worldline Wi and length Li. Away from the conical defects we have R=-6/ℓ2, so the first term gives a contribution proportional to the volume of the manifold. At the defects, R is singular; we can get the value of its integral by working in a local FRW patch ds2=du2+cosh2(u/ℓ)hijdyidyj,(4.8)with hijdyidyj the metric of an infinitesimal hyperbolic disk Dε with a conical defect δ at the origin. The Gauss-Bonnet theorem implies for the metric on a constant-u slice ∫Dεd2yhR[h]=2δ,(4.9)as ε→0, with R[h] the scalar curvature of the metric h. For a three-dimensional infinitesimal tube around the defect Tε, and up to terms that vanish as ε→0, ∫TεR=∫du∫Dεd2yhR[h]=2δ∫Wdu,(4.10)with the integral over the worldline W measuring the proper length of the particle trajectory. The singular curvature contribution from all the defects is then -116πG∑i∫Tε,iR=-∑iδi8πG∫WidLi.(4.11)Using that δi=8πGmi, this exactly cancels the final term in (4.7). We, thus, need to evaluate I=14πGℓ2Vol[M]-18πG∫∂M(K-1ℓ).(4.12)

In order to induce the flat toroidal metric on the boundary, it is convenient to work with the metric of the AdS cylinder in the form ds2=ℓ2cosh2(R/ℓ)dτ2+dR2+ℓ2sinh2(R/ℓ)dϕ2,(4.13)with the asymptotic boundary at a large R=R∞. In the limit (4.6), the boundary insertions in Fig. 7 happen all at the same Euclidean time τ, giving a spherically symmetric configuration. We can split the geometry to compute the on-shell action into three pieces: two long AdS tubes of size β (as measured by τ) and the region that arises from chewing and gluing the tubes and that includes the cosmological slice. Each of the AdS tubes contributes IAdS(β)=-ℓβ8G,(4.14)with the boundary term in (4.12) regularizing the result. To obtain the contribution from the piece that results from gluing the chewed parts of the tubes, note that each particle follows a radial geodesic given by tanh(R/ℓ)=4G2M2-12GMcosh(τ),(4.15)where we have set τ=0 at the turning point of the geodesics, shown in (2.12) to be cosh(R/ℓ)=2GM. The geodesics have finite length in Euclidean time, τ∈(-τ∞,τ∞) with τ∞=arccosh(2GM4G2M2-1).(4.16)Using the axial symmetry, the equation (4.15) defines the surface of revolution traced by the cloud of particles. We can compute the action of this piece from the volume as Ichewed=2×14πGℓ2∫02πdϕ∫-τ∞τ∞dτ∫0R(τ)dRℓ2 cosh(R/ℓ)sinh(R/ℓ)=2Mℓarc cosh(sinh(R∞/ℓ)4G2M2-1)-ℓGτ∞=2Mℓlog(eR∞/ℓ4G2M2-1)-ℓGτ∞,(4.17)where in the last step we have kept only nonvanishing terms as R∞→∞. The divergence we obtain is the standard one associated with n boundary primary insertions of mass m; we could renormalize it away, but we prefer to keep it to check that it also appears in the alternative saddle. Collecting the previous results, the on-shell action for the cosmological saddle is Icosm=2c3[-β4-arccosh(2GM4G2M2-1)+2GMlog(eR∞/ℓ4G2M2-1)],(4.18)with c=3ℓ/(2G).

This is to be compared with the on-shell action of the noncosmological saddle. In this case, the particles contract pairwise with each neighbor insertion, so that they do not create significant backreaction in the limit (4.6). We can treat them in a probe approximation, in which case the Euclidean action is that of an AdS tube of length 2β plus a factor mL for each particle, with L the geodesic length between the insertions.¹⁰

Here, we are thinking of a full AdS tube with no conical defects at all; the contribution from the particles is included via the mL factor. Alternatively, we could use the action (4.12), which makes no reference to the point particles, but then we would have to remove infinitesimal conical defects at the particle trajectories. Both perspectives are, of course, equivalent.

We now compare the on-shell actions. Expanding at n≫1 Icosm-Inoncosm=2c3[2GMlog(nπ4G2M2-1)-arccosh(2GM4G2M2-1)].(4.21)This difference is always large and positive in the regime we are working, given that our assumption mℓ≪c implies n≫GM. We, thus, conclude that the noncosmological saddle always dominates. In retrospect, this is the case because the length of a geodesic connecting the insertions is much shorter in the noncosmological saddle than in the cosmological one: Lcosm-Lnoncosm=2ℓlog(nπ4G2M2-1),(4.22)so that the cosmological saddle is suppressed by the matter propagators.

Some comments are in order regarding this computation. First, given two neighboring insertions in the noncosmological saddle, there are infinitely many topologically inequivalent geodesics joining them (wrapping the thermal circle an arbitrary number of times). We have chosen the shortest (and, thus, dominant) one, but, of course, in principle, one should sum over all possibilities, much like in a thermal two-point function. It is also important to note that, in the noncosmological saddle, the choice of pairing for the boundary insertions is not unique. This is most obviously exemplified by the two alternative ways to pair neighboring operators, but, in general, there are also contractions in which non-neighboring insertions are paired.¹¹

The discussion in this section is in the limit in which the particles behave as nonbackreacting probes. However, one could also try to build such saddles with a small number of particles with finite backreaction, via a slight generalization of the construction in Fig. 16.

The existence of the saddle we have described relies on the neighboring operators being able to pair up; there should be a nonvanishing two-point function if only these operators are inserted into the path integral. The local geometry near a pair of operators in the full saddle is the same as the local geometry in the saddle that computes the two-point function. However, even if this two-point function vanishes or we have an odd number of total operators, we expect that there would be similar saddles where collections of some small number q>2 of neighboring operators with a nonvanishing q-point function “contract” with each other, such that the local geometry near these operator insertions in the full saddle matches with the local geometry in the saddle computing the nonvanishing q-point function. For all such saddles, the full geometry should differ significantly from AdS3 only in the asymptotic region near the rings of operator insertions. In the absence of symmetry reasons, having vanishing q-point functions for all local groupings of operators would likely require some extreme fine-tuning in the choice of operators, if it is possible at all. This lines up with the general arguments in the previous section suggesting that the AS2 construction is not likely to have the cosmological saddle as the dominant one.

When the operator insertions creating the shell are charged under some symmetry we have to be more careful, since this can forbid the existence of a nonvanishing q-point function for all q≤n. If the symmetry is an exact boundary global symmetry, dual to some bulk gauge symmetry, the cosmological saddle cannot exist: Otherwise, in the Euclidean bulk we would have a closed spatial slice (the initial value surface of the cosmology) with nonvanishing gauge charge, violating Gauss’s law.

We can however imagine that the particles going through the cosmology are charged under some flavor symmetry of the bulk EFT, which can be broken explicitly at some high-energy scale or by quantum gravity effects. In this situation, it is tempting to conclude that the dominant saddle would be the one in which charged operators in the ket are connected with the oppositely charged operators in the bra through the closed universe slice. However, note that the general argument presented in the previous section still applies, regardless of whether or not the particles are charged: For sufficiently large β, we do not expect the operator insertions to generate the O(c) entanglement needed to make the cosmological saddle the dominant one. One could imagine a competing noncosmological configuration in which the Euclidean spacetime is a solid torus and, possibly after some local interactions, the charge is transported from bra to ket via the lightest possible particles (to minimize the e-mL suppression). Clarifying whether this or some other type of saddle dominates over the cosmological one is a nontrivial and interesting problem but, as long as the general argument of the previous section applies, an alternative dominant saddle is expected to exist.

It was suggested in [19] that a way to avoid noncosmological saddles where the matter insertions self-contract within the individual rings would be to choose operators that insert particles with some global charge. In this case, at least naively, the insertions on one ring give rise to particle worldlines that must end on the other ring.

In quantum gravity, it is expected that there are no exact global symmetries (see, e.g., [43]), so the charge carried by any such particles can be only approximately conserved (as for baryon charge in the Standard Model). Thus, there should be some nonvanishing amplitude where we insert some number of these particles on one side of the torus (the Euclidean past) and none on the other. Similarly, there could be an alternative saddle where the particles worldlines from the Euclidean past and future do not join up. The existence of these saddles require bulk interactions, and so the saddles are suppressed at large N. But they can still be dominant if this suppression is less than the suppression (due to additional action) that comes from adding a wormhole part of the geometry for the charged particles to propagate through. It would be interesting to understand through a more detailed calculation whether inserting a large number of particles carrying some approximately conserved charge can cause the cosmological saddle to dominate. This would be fascinating, as it could imply that any cosmology produced by this construction exhibits something like baryon asymmetry. If this is the case, it is also possible that the explanation for baryon asymmetry in our actual Universe could have a similar origin in quantum gravity.

The solutions that we have focused on in this paper have a single cosmological wormhole and a single baby-universe cosmology in the Lorentzian picture. But it is straightforward to generalize this to solutions with multiple disconnected cosmologies. Start with some number n>1 of basic cosmological wormholes as constructed in Fig. 5. Each wormhole has a single connected boundary in the Euclidean past and a single connected boundary in the Euclidean future. Now we can perform gluing operations similar to the ones in Fig. 2. There, we connected the past and future boundaries of a single wormhole, but we can also connect the future boundaries of two different wormholes and the past boundaries of the same two wormholes in a way that preserves the time-reflection symmetry. More generally, we can connect any part of one boundary to any part of any other boundary, provided that we also add the gluing related by time-reflection symmetry. Through a combination of such gluings, we can produce many Euclidean solutions with a connected boundary whose Lorentzian continuations have multiple closed universes together with various copies of AdS. An example is shown in Fig. 18.

An interesting feature of the multiuniverse saddles constructed from the same underlying wormhole is that we have a whole family of saddles with the same boundary geometry obtained by permuting how the wormhole bottoms connect to the wormhole tops. For saddles with a large number of universes, this can provide an exponential (n!) enhancement to the contribution of the cosmological saddle to the path integral. An interesting possibility is that this (or, more generally, having a large number of cosmological saddles with the same boundary through some other mechanism) might cause the set of cosmological saddles to dominate over noncosmological saddles in the path integral overall even if noncosmological saddles have lower action. In this case, the corresponding Lorentzian state would describe with high probability an entangled polycosmos¹³

Polycosmos (n.): a collection of mutually disconnected but quantum entangled cosmological spacetimes.

The AS2 cosmologies supported by thin shells were introduced as the low-temperature phase of a family of states labeled by β [19], collectively known as partially entangled thermal states [44]. As we decrease β (increase the temperature), the dominant saddle transitions to one in which the time-reflection-symmetric slice is connected, forming a large Einstein-Rosen bridge that evolves in Lorentzian signature to a black hole with an expanded interior due to the presence of the matter shell. A family of such thin-shell black hole states were shown in [40,45] to account for the microscopic entropy of Schwarzschild-like black holes, with later works extending them to a variety of other contexts [46–53].

Given that in this work we have argued that the cosmological saddle is generically not the dominant one in the AS2 configuration, is the same statement true when we lower β and transition to the black hole phase? The first thing to notice is that the general argument in Sec. III ruling out the dominance of the cosmological saddle does not apply. Indeed, it is precisely as we lower β enough to make the black hole phase to dominate that the dual two-sided state develops O(c) entanglement, compatible with a connected slice in both the horizontal and vertical slicings in Fig. 13. Making more quantitative statements treating the particles forming the thin shell individually would require a more careful analysis, but we can use a toy model analogous to the one in the Appendix to estimate if the long wormhole saddle can dominate over the one in which the matter particles contract locally.

The long wormhole saddle is still penalized by the longer geodesics contracting between bra and ket, which give a factor ∼cGMlog(n/(GM)), where 1/n∼ε/ℓ characterizes the separation between individual insertions. But now the on-shell action of a three-dimensional black hole is negative and inversely proportional to its inverse temperature, and the long wormhole phase contains two black holes of inverse temperature proportional to β, while the locally contracted saddle has a single black hole of inverse temperature proportional to 2β. These two contributions do not cancel (contrary to what happens in the cosmological phase), so the long wormhole is favored by a factor ∼c/β. As an order of magnitude estimate, we then expect that the long wormhole phase can dominate when 1β≳GMlog(nGM).(5.1)For large n, this requires a small β, but the dependence is only logarithmic.¹⁴

Interestingly, if we make GM very large, we seem to require an accordingly small β to avoid the dominance of the locally connected saddle. It would be interesting to understand the implications of this fact for the heavy shell limit often taken in the black hole microstate counting works [40,45,46].