Transdimensional tunneling in the multiverse

Transitions between dS4×S2 vacua with different flux quantum numbers n proceed through nucleation of magnetically charged branes.

“Transitions between dS 4 × S 2 vacua with different flux quantum numbers n were discussed in Ref. [7], where it was shown that such transitions proceed through nucleation of magnetically charged branes.”

• cites [7] J.J. Blanco-Pillado, 2009, J. Cosmol. Astropart. Phys. sourceSupported

The theory has black brane solutions with the necessary properties.

“The theory does have black brane solutions with the necessary properties [14,15].”

• cites [14] G.W. Gibbons, 1995, Class. Quant. Grav. sourceSupported
• cites [15] sourcePartly supported

  Section 5 (General Charged Branes; page unavailable) gives metrics for charged p-branes and describes the solutions as boost-symmetric, supporting the existence of brane solutions. But the claim does not specify the “necessary properties”; the paper also says its solutions suffer from singularities and leaves their stability unresolved. Our copy therefore cannot establish that they have all the properties the citing authors require. (LLM, conf=0.78)

  Experimental — accuracy not yet measured

Transitions from dS6 to AdS2×S4 can be mediated by nucleation of black holes in dS6.

“As discussed in Ref. [8], transitions from dS 6 to AdS 2 ×S 4 can be mediated by nucleation of black holes in dS 6 .”

• cites [8] S.M. Carroll, 2009, J. High Energy Phys. sourcePartly supported

  Section III says that a positive-cosmological-constant theory admits AdSₚ₊₂ × Sᵩ compactification solutions, and “Populating a landscape of vacua” describes nucleation-driven transitions from higher-dimensional de Sitter space toward negative-energy vacua. But that section says the nucleated region relaxes to a big crunch when its vacuum energy is negative. The cited passages do not establish the more specific claim that black-hole nucleation in dS₆ mediates a transition to AdS₂ × S⁴. (LLM, conf=0.82)

  Experimental — accuracy not yet measured

Nariai-type instantons are similar to Hawking-Moss instantons.

“Nariai-type instantons are similar to Hawking-Moss instantons [27].”

• cites [27] sourceSupported

The 4d effective theory description used by Carroll, Johnson and Randall is equivalent to the authors' description.

“This description was used by Carroll, Johnson and Randall (CJR) [8] and is equivalent to ours, but the geometry of the instanton and the resulting spacetime structure are not evident in this approach.”

• cites [8] S.M. Carroll, 2009, J. High Energy Phys. sourceSupported

Using results of [15] and [14], it can be shown that (46) is the only solution which is both non-singular and boost-invariant.

“Using the results of Refs.[15] and [14], it can be shown that ( 46) is the only solution which is both non-singular and boost-invariant.”

• cites [15] sourcePartly supported

  Section 5, “General Charged Branes” (page unavailable), identifies boost-symmetric solutions, but does not establish that any particular solution is uniquely both boost-invariant and non-singular. The Discussion (page unavailable) says the exact solutions generically have naked singularities at the brane core and proposes that a defect core might smooth them out. The cited paper therefore supports the boost-symmetry aspect, not the claimed non-singularity or uniqueness of equation (46). (LLM, conf=0.82)

  Experimental — accuracy not yet measured
• cites [14] G.W. Gibbons, 1995, Class. Quant. Grav. sourcePartly supported

  Section 2 says that, for odd p, the maximal analytic extension of the higher-dimensional solution is completely nonsingular. But our copy does not identify the citing paper’s equation (46) or establish that any solution is the only one that is both nonsingular and boost-invariant. The Introduction discusses reduction along a boost, not that uniqueness claim. (LLM, conf=0.76)

  Experimental — accuracy not yet measured

CJR's numerical calculation of the instanton action as a function of magnetic charge Qm agrees with the authors' approximate formula (58).

“CJR performed a numerical calculation of the instanton action as a function of the magnetic charge Q m ; the result is shown in their Fig. 17.We have verified that their calculation agrees with our approximate formula (58).”

• cites [8] S.M. Carroll, 2009, J. High Energy Phys. sourceSupported

Einstein-Maxwell theory with Λ6=0 has magnetically charged, asymptotically flat black brane solutions.

“It is well known that Einstein-Maxwell theory with Λ 6 = 0 has magnetically charged, asymptotically flat black brane solutions [14,15].”

• cites [14] G.W. Gibbons, 1995, Class. Quant. Grav. sourceSupported
• cites [15] sourcePartly supported

  Section 4 (page unavailable) imposes asymptotically Minkowskian boundary conditions, and Section 5 derives magnetically charged p-brane solutions with unrestricted dilaton coupling. But Section 6 says the paper’s solutions have a singularity at r+ rather than inside an event horizon, so they do not establish the claimed black-brane solutions. The cited passages also do not state the Λ6=0 condition explicitly. (LLM, conf=0.84)

  Experimental — accuracy not yet measured

Nariai-type instantons describe nucleation of regions of size comparable to the horizon, filled with the unstable vacuum at the maximum of V(ψ).

“As discussed in [8], they describe nucleation of regions of size comparable to the horizon, filled with the unstable vacuum at the maximum of V (ψ).”

• cites [8] S.M. Carroll, 2009, J. High Energy Phys. sourcePartly supported

  The section ‘Summary of the cosmologically interesting solutions’ says the Nariai solution has an event horizon where the radion field is at its potential maximum. It does not establish the claimed nucleation of horizon-sized regions filled with an unstable vacuum at the maximum of V(ψ); the field identified in its Nariai discussion is the radion, φ. No page number is available in our copy. (LLM, conf=0.83)

  Experimental — accuracy not yet measured

At late times t→∞, the 6d metric in the bubble interior approaches that of dS6, with a(t)≈H6^-1 sinh(H6 t) and r(t)≈H6^-1 cosh(H6 t).

“At late times, t → ∞, the metric approaches that of dS 6 [6], with a(t) ≈ H -1 6 sinh(H 6 t) and r(t) ≈ H -1 6 cosh(H 6 t).”

• cites [6] sourcePartly supported

  Section 4.2 (eq. 4.16; page unavailable) says that, in the special case of a positive higher-dimensional cosmological constant, the asymptotic metric locally resembles de Sitter space. The later discussion says it asymptotes to (d+4)-dimensional de Sitter space, which includes six dimensions when d=2. Our copy does not give the claimed sinh and cosh formulas for a(t) and r(t). (LLM, conf=0.86)

  Experimental — accuracy not yet measured

The instanton describes nucleation of a pair of oppositely charged black holes with an initial separation of 2H6^-1.

“In our case, the instanton describes nucleation of a pair of oppositely charged black holes with an initial separation of 2H -1 6 .”

The corresponding instanton action can be estimated as S_inst ≈ -16π^3/(3H6^4) + 2πM/H6.

“The corresponding instanton action can be estimated as S inst ≈ - 16π 3 3H 4 6 + 2πM H 6 ,(34)”

An alternative tunneling channel is the charged Nariai instanton, which describes nucleation of maximally large black holes having r+ = rc.

“An alternative tunneling channel is given by the so-called charged Nariai instanton, which describes nucleation of maximally large black holes having r + = r c [24,25].”

• cites [24] sourceNot checked — no full text of the source
• cites [25] M.K. ParikhNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The corresponding instanton is the Euclideanized 6d Reissner-Nordstrom-de Sitter solution.

“The corresponding instanton is the Euclideanized 6d Reissner-Nordstromde Sitter (RNdS) solution [16],”

• cites [16] sourceNot checked — no full text of the source

The higher-energy solutions in both q=2 and q=4 sectors are unstable.

“In both cases, the higher-energy solutions are unstable; see the discussion in Sec.III”

As Q is increased at fixed H6, the black hole horizon radius r+ grows until it coincides, at Q≈0.125H6^-3 ≡ Qc, with the cosmological horizon rc.

“As Q is increased at a fixed H 6 , the black hole horizon radius r + grows until it coincides, at Q ≈ 0.125H -3 6 ≡ Qc , with the cosmological horizon r c .”

Even a simple toy model based on a 6-dimensional Einstein-Maxwell theory exhibits vacua with 0, 2, and 4 compact dimensions.

“Even a simple toy model based on a 6-dimensional Einstein-Maxwell theory exhibits vacua with 0, 2, and 4 compact dimensions.”

Compactification transitions dS6 → AdS2×S4 occur through the nucleation of electrically charged black hole pairs.

“We find that compactification transitions dS 6 → AdS 2 × S 4 occur through the nucleation of electrically charged black hole pairs, and transitions from dS 6 to dS 4 × S 2 and AdS 4 × S 2 occur through the nucleation of magnetically charged spherical black branes.”

Transitions from dS6 to dS4×S2 and AdS4×S2 occur through the nucleation of magnetically charged spherical black branes.

“transitions from dS 6 to dS 4 × S 2 and AdS 4 × S 2 occur through the nucleation of magnetically charged spherical black branes.”

Positive-energy vacua are obtained for n > n0 ≡ 2 Λ6^{1/2} e.

“Positive-energy vacua are obtained for n > n 0 ≡ 2 Λ 6 1/2 e. (11 )”

No vacuum solutions exist for n > n_max = 2n0/√3.

“No vacuum solutions exist for n > n max = 2n 0 / √ 3.”

For small values of Qm satisfying Qm ≪ H6^-1, the 4d cosmological constant is necessarily negative.

“We note that it follows from Eq. ( 10) that for small values of Q m satisfying (57) the 4d cosmological constant is necessarily negative.”

These endpoints are non-singular provided B'(0)=B'(ξm)=1 and r'(0)=r'(ξm)=0.

“These endpoints are non-singular, provided that B ′ (0) = B ′ (ξ m ) = 1 (55 ) and r ′ (0) = r ′ (ξ m ) = 0. (56 )”

The role of the bubble wall is played by a spherical magnetically charged black brane (the same kind as in the inverse transition).

“The role of the bubble wall is played by a spherical magnetically charged black brane (the same kind of brane as in the inverse transition).”

Transitions from dS6 to unstable dS2×S4 (or dS4×S2) vacua are mediated by Nariai-type instantons, similar to Hawking-Moss instantons.

“Transitions from dS 6 to unstable dS 2 × S 4 (or dS 4 × S 2 ) vacua are mediated by Nariaitype instantons, which are similar to Hawking-Moss instantons.”

The action of the Nariai-type instanton is greater than that of the black brane instanton for all Qm up to some critical value Q.

“Its action is greater than that for the black brane instanton discussed above for all values of Q m , up to some critical value Q depends non-trivially on Λ 6 .9”

Transitions from dS6 to dS4×S2 and AdS4×S2 proceed through nucleation of spherical, magnetically charged black 2-branes.

“Transitions from dS 6 to dS 4 × S 2 and AdS 4 × S 2 proceed through nucleation of spherical, magnetically charged black 2-branes.”

A bubble with ψ∼ψc nucleates in the inflating background of ψ=0 vacuum.

“A bubble with ψ ∼ ψ c nucleates in the inflating background of ψ = 0 vacuum.”

The surface r = r0 is a non-singular event horizon.

“The surface r = r 0 is a non-singular event horizon.”

In the black brane case, the region inside the black brane horizon is a new (A)dS4×S2 universe.

“In the latter case, the region inside the black brane horizon is a new (A)dS 4 × S 2 universe.”

The numerical solution for e^2=2, Λ6=10^-4, n=10 shows functions only deviate slightly from pure de Sitter counterparts.

“This value of n corresponds to a small magnetic charge so one can see from the figures that the functions only deviate slightly from their pure de Sitter counterparts.”

For K≪Kmax, the solutions of (B3) can be approximated as x1≈12^{-1/6} and x2≈K^{-1/2}.

“For K ≪ K max , the solutions of (B3) can be approximated as x 1 ≈ 12 -1/6 , x 2 ≈ K -1/2 .(B9)”

The tunneling rate is Γ∼exp(S0−Sinst), with S0=−24π^2 M4^4/Λ4.

“Γ ∼ exp(S 0 -S inst ), (70”

• Also stated in: § V. FROM dS 4 × S 2 TO dS 6 · plain

  “S 0 = -24π 2 M 4 4 /Λ 4 ,(71)”

For K>Kmax, there are no solutions.

“For K > K max , there are no solutions.”

The brane nucleation rate is Γ ∼ exp(-2π^2 H6^-3 T2).

“The brane nucleation rate is given by Γ ∼ exp(-2π 2 H -3 6 T 2 ).(59)”

Inflating black brane solutions approach the Nariai limiting case, as seen in numerical solutions in Figs. 10 and 11.

“One can see how the inflating black brane solutions approach the Nariai limiting case by looking at the numerical solutions presented in Figs. 10 and 11.”

Compactification transitions from dS6 to AdS2×S4 occur through nucleation of pairs of electrically charged black holes.

“We found that compactification transitions from dS 6 to AdS 2 × S 4 occur through nucleation of pairs of electrically charged black holes.”

The model admits tunneling transitions between (A)dS4×S2 vacua having the same topology but different values of magnetic flux.

“Apart from the topology-changing transitions that we discussed here, our model admits tunneling transitions between (A)dS 4 × S 2 vacua having the same topology, but characterized by different values of the magnetic flux.”

In the newly formed compactified region, ψ starts at ψ+ and rolls to ψ=0.

“In the newly formed compactified region, ψ starts at ψ + and rolls to ψ = 0.”

The initial radius of a nucleating brane is set by the de Sitter horizon H6^{-1}.

“The initial radius of a nucleating brane is set by the de Sitter horizon, H -1 6 .”

The extreme 2-brane in 6d is described by metric (46) with r0 = √3 Qm/(8π).

“In particular, the extreme 2-brane in 6d is described by the metric, ds 2 = 1 - r 0 r 2/3 (-dt 2 + dx 2 + dy 2 ) + 1 - r 0 r -2 dr 2 + r 2 dΩ 2 2 , (46 ) with r 0 = √ 3Q m 8π . (47 )”

As Q approaches Qc from below, the two instanton actions approach one another and coincide at Q = Qc.

“As Q approaches Qc from below, the two instanton actions approach one another and coincide at Q = Qc .”

More generally, the metric behind the black brane horizon has the form (63).

“More generally, the metric behind the black brane horizon has the form”

The charge of these black holes is quantized in units of the elementary charge of the theory.

“The charge of these black holes is quantized in units of the elementary charge of the theory,”

The role of the bubble walls in the resulting bubbles is played by magnetically charged branes, which can be either solitonic branes or black branes.

“The role of the bubble walls in the resulting bubbles is played by magnetically charged branes, which can be either solitonic branes (if the model is augmented by adding the appropriate Higgs fields) or black branes of the type described here.”

The mass of these black holes is determined by the condition that the temperature at the black hole horizon (r=r+) is the same as that at the cosmological horizon (r=rc) in the corresponding Reissner-Nordstrom-de Sitter solution.

“and their mass is determined by the condition that the temperature at the black hole horizon (r = r + ) is the same as that at the cosmological horizon (r = r c ) in the corresponding Reissner-Nordstrom-de Sitter solution.”

In the limit r+≪rc, the black holes become nearly extremal.

“In the limit when r + ≪ r c the black holes become nearly extremal.”

For a pure dS6 space, B(ξ)=H6^-1 cos(H6ξ), r(ξ)=H6^-1 sin(H6ξ).

“For a pure dS 6 space, these functions are given by B(ξ) = H -1 6 cos(H 6 ξ), r(ξ) = H -1 6 sin(H 6 ξ),(51)”

In the presence of a brane, the Einstein equations for B(ξ) and r(ξ) with the ansatz (3) for the Maxwell field have the form (52).

“In the presence of a brane, the Einstein equations for B(ξ) and r(ξ) with the ansatz (3) for the Maxwell field have the form”

The instanton action in the limit Qm ≪ H6^-1 can be estimated as S_inst ≈ S_dS6 + A3 H6^-3 T2.

“The instanton action in the limit (57) can be estimated as S inst ≈ S dS 6 + A 3 H -3 6 T 2 ,(58)”

A Euclidean continuation of the metric (50) is given by ds^2 = B^2(ξ)dΩ3^2 + dξ^2 + r^2(ξ)dΩ2^2.

“This can be found by setting t = iθ -iπ/2, ds 2 = B 2 (ξ)dΩ 2 3 + dξ 2 + r 2 (ξ)dΩ 2 2 . (53 )”

The region between the brane horizon ξ=0 and the cosmological horizon ξ=ξm is described by metric (50).

“The region between the brane horizon, ξ = 0, and the cosmological horizon, ξ = ξ m , is described by the metric (50).”

The Einstein-Maxwell theory in 6 dimensions admits a rich landscape of vacua.

“The Einstein-Maxwell theory in 6 dimensions admits a rich landscape of vacua”

At Kmax, H=0.

“and H = 0.”

As K is increased, the roots approach one another until they coincide at some critical value Kmax.

“As K is increased, the roots approach one another until they coincide at some critical value K max (corresponding to some flux quantum number n max ).”

The approximate action (34) is in good agreement with the exact action at small values of Q.

“As expected, it is in a good agreement with the exact action at small values of Q.”

As the bubble expands, the field rolls off to infinity in the bubble interior.

“As the bubble expands, the field rolls off to infinity in the bubble interior.”

Compact, non-singular instanton solutions require the range of ξ to be finite, 0 < ξ < ξm, with endpoints corresponding to zeros of B(ξ), B(0)=B(ξm)=0.

“We are interested in compact, non-singular instanton solutions, so the range of ξ has to be finite, 0 < ξ < ξ m , with the endpoints corresponding to zeros of B(ξ), B(0) = B(ξ m ) = 0. (54 )”

Metric (61) describes an expanding, open 4d FRW universe with 2 extra dimensions compactified on a sphere.

“The metric (61) describes an expanding, open 4d FRW universe with 2 extra dimensions compactified on a sphere.”

Metric (61) applies only in the vicinity of t=0, where the 4d universe is curvature-dominated.

“The metric (61) applies only in the vicinity of t = 0, where the 4d universe is curvaturedominated.”

The critical value Kmax is 3^{4/3}/4.

“K max = 3 4/3 4 .(B6)”

For sufficiently small K, eq. (B3) has two roots.

“For sufficiently small values of K, there are two roots.”

Topology change from dS6 to dS4×S2 and to AdS4×S2 can occur through the nucleation of spherical black 2-branes.

“We shall now argue that topology change from dS 6 to dS 4 × S 2 and to AdS 4 × S 2 can occur through the nucleation of spherical black 2-branes.”

Decompactification transitions dS4×S2→dS6 have a daughter vacuum of higher effective dimension than the parent vacuum.

“We also discussed decompactification transitions dS 4 × S 2 → dS 6 , in which the effective dimension of the daughter vacuum is higher than that of the parent vacuum.”

The magnetic charge is subject to the charge quantization condition Qm = 2πn/e.

“Then the magnetic charge is subject to the usual charge quantization condition, Q m = 2πn e ,(4)”

• AssumptionThe authors state this as an assumption: The model includes electrically charged particles with elementary charge e.

This agrees with the intuitive expectation Γ ∼ exp(-M/T_dS), where T_dS = H6/2π is the Gibbons-Hawking temperature of dS6.

“This agrees with the intuitive expectation that Γ ∼ exp(-M/T dS ), where T dS = H 6 /2π is the Gibbons-Hawking temperature of dS 6 .4”

Topology-changing transitions involve nucleation of charged black branes or black holes.

“As we shall see, such transitions also involve nucleation of charged black branes or black holes.”

The instanton can be pictured as a 0-brane whose worldline runs along a big circle on S6.

“Thus, the instanton can be pictured as a 0-brane whose worldline runs along a big circle on S 6 , as illustrated in Fig. 3.”

Flux vacua in the 6d Einstein-Maxwell theory are described by solutions with a (6-q)-dimensional maximally symmetric space of constant curvature and a static q-dimensional sphere of fixed radius.

“Flux vacua are described by solutions of this model with the spacetime metric given by a (6 -q)-dimensional maximally symmetric space of constant curvature and a static q-dimensional sphere of fixed radius, namely a metric of the form,”

In the limit r+≪rc, the instanton describes nucleation of nearly extremal black holes in dS6.

“Thus, the instanton in this limit describes the nucleation of nearly extremal black holes in dS 6 .”

For q=2, the only Maxwell field ansatz consistent with the symmetries of the metric is a monopole-like configuration on the extra-dimensional 2-sphere.

“For q = 2, the only ansatz for the Maxwell field that is consistent with the symmetries of the metric is a monopole-like configuration on the extra-dimensional 2-sphere [10],”

• cites [10] sourceNot checked — no full text of the source

The multiverse is expected to include regions of different effective dimensionality.

“We therefore expect the multiverse to include regions of different (effective) dimensionality [5].”

• cites [5] sourceNot checked — no full text of the source

Instantons of this form are known to describe the production of particle-antiparticle pairs in de Sitter space.

“Instantons of this form are known to describe the production of particle-antiparticle pairs in de Sitter space [22].”

• cites [22] sourceNot checked — no full text of the source

The black holes are then driven apart by the de Sitter expansion.

“The black holes are then driven apart by the de Sitter expansion.”

Such instantons describe spontaneous nucleation of horizon-radius spherical branes in de Sitter space.

“As discussed in [22], such instantons describe spontaneous nucleation of horizon-radius spherical branes in de Sitter space.”

• cites [22] sourceNot checked — no full text of the source

Since the radius of S2 is stabilized by magnetic flux, the branes have to be magnetically charged.

“Since the radius of S 2 is stabilized by magnetic flux, the branes have to be magnetically charged.”

Observers in the newly formed compactified region see only a homogeneous FRW universe, approaching either dS or AdS space.

“Observers in this newly formed compactified region see only a homogeneous FRW universe, approaching either dS or AdS space.”

The Nariai instanton is the only relevant tunneling channel in the range Qc < Q < Qmax.

“The Nariai instanton is the only relevant tunneling channel in the range Qc < Q < Qmax , 7 where Qmax = Q 17), is the largest value of Q e above which no instanton solutions exist.”

The tunneling for dS4×S2→dS6 is described by the same CdL instanton as for the inverse transition dS6→dS4×S2.

“The tunneling is described by the same CdL instanton as for the inverse transition, dS 6 → dS 4 × S 2 , and the tunneling rate is”

Small perturbations cause ψ to roll away from the maximum.

“Small perturbations will cause ψ to roll away from the maximum.”

In the limit r+≪rc, the corresponding instanton can be pictured as a brane whose Euclidean worldsheet (a 3-sphere) is wrapped around a big circle of S6.

“Once again, in the limit when the black brane horizon radius r + is small compared to r c , the corresponding instanton can be pictured as a brane whose Euclidean worldsheet (which has the form of a 3-sphere) is wrapped around a”

Small perturbations cause fragmentation of these vacua into regions of AdS2×S4 (or AdS4×S2) and dS6.

“Small perturbations cause fragmentation of these vacua into regions of AdS 2 × S 4 (or AdS 4 × S 2 ) and dS 6 .”

In this limit, the instanton can be pictured as S6 (Euclideanized de Sitter) with a 0-brane worldline running along a big circle.

“The instanton in this limit can be pictured as S 6 (Euclideanized de Sitter) with a 0-brane worldline running along a big circle.”

In terms of the effective 4D theory, the decompactification transition from dS4×S2 to dS6 is the usual false vacuum decay.

“In terms of the effective 4D theory, this is the usual false vacuum decay.”

The same topological term factor e^{2α} would also appear in the nucleation rate of black branes discussed in Section IV.A.

“The same factor would also appear in the nucleation rate of black branes discussed in Section IV.A.”

This is a mixed slicing of de Sitter space, with open slices in 3 out of 5 spatial dimensions and closed spherical slices in the remaining two directions.

“This is a somewhat unfamiliar,”

These dS2×S4 solutions are unstable: small perturbations cause them to disintegrate into AdS2×S4 and dS6 regions.

“These solutions are known to be unstable: small perturbations cause them to disintegrate into AdS 2 × S 4 and dS 6 regions [25].”

• cites [25] M.K. ParikhNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Observers in the parent vacuum see nucleation and subsequent expansion of spherical bubbles, as in ordinary CdL vacuum decay.

“Observers in the parent vacuum see nucleation and subsequent expansion of spherical bubbles, as in ordinary CdL vacuum decay.”

As a result, the multiverse will be populated by vacua of all possible dimensionalities.

“As a result, the multiverse will be populated by vacua of all possible dimensionalities.”

• AssumptionRests on an assumption: Eternal inflation occurs.