Can the Universe create itself?

A Universe with CTCs can be stable against vacuum polarization.

“We show that a Universe with CTCs can be stable against vacuum polarization.”

A Universe with CTCs can be classically stable and self-consistent if and only if the potentials in this Universe are retarded.

“And, it can be classically stable and self-consistent if and only if the potentials in this Universe are retarded -which gives a natural explanation of the arrow of time in our universe.”

Self-consistent vacuums exist for a multiply connected de Sitter space in which the renormalized energy-momentum tensor does not diverge as one approaches the Cauchy horizon and solves Einstein's equations.

“We illustrate that such modelswith CTCs -are not necessarily inconsistent by demonstrating self-consistent vacuums for Misner space and a multiply connected de Sitter space in which the renormalized energy-momentum tensor does not diverge as one approaches the Cauchy horizon and solves Einstein's equations.”

Self-consistent vacuums exist for Misner space in which the renormalized energy-momentum tensor does not diverge as one approaches the Cauchy horizon and solves Einstein's equations.

“We illustrate that such modelswith CTCs -are not necessarily inconsistent by demonstrating self-consistent vacuums for Misner space and a multiply connected de Sitter space in which the renormalized energy-momentum tensor does not diverge as one approaches the Cauchy horizon and solves Einstein's equations.”

For a multiply connected de Sitter space there also exists a self-consistent vacuum state for a conformally coupled scalar field.

“In this paper, we find that for a multiply connected de Sitter space there also exists a self-consistent vacuum state for a conformally coupled scalar field (see section IX).”

The conformal relations given by Eq. (79) and Eq. (80) define a conformal map between static de Sitter space and Rindler space.

“The conformal relations given by Eq. ( 79) and Eq. ( 80) define a conformal map between the static de Sitter space and the Rindler space.”

In region P of Misner space, the only self-consistent potential is the advanced potential.

“Similarly, in region P the only self-consistent potential is the advanced potential (see Fig. 6a).”

For any finite boost parameter b, the energy-momentum tensor of the adapted Minkowski vacuum in Misner space diverges as one approaches the Cauchy horizon (ξ → 0).

“Obviously for any finite b, T ν µ M,ren diverges as one approaches the Cauchy horizon (ξ → 0).”

• cites [14]Not matched

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A particle detector in Vilenkin's tunneling universe has the same response function as an inertial particle detector in an eternal de Sitter space.

“We find that the integration of the second term is zero and therefore, the result is the same as that for an inertial particle detector in an eternal de Sitter space [100,153].”

• cites [100]Not matched

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• cites [153]Not matched

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In the time-nonorientable de Sitter space, the contribution of G_CM(-X,X') to the energy-momentum tensor is zero.

“Inserting Eq. ( 57) and Eq. ( 58) into Eq.( 45), we find that the contribution of G CM (-X, X ′ ) to the energy-momentum tensor is zero.”

For the time-nonorientable de Sitter space, the area of the event horizon is one half that of an eternal de Sitter space.

“For this time-nonorientable de Sitter space, the area of the event horizon is one half that of an eternal de Sitter space.”

A particle detector moving in Misner space with the adapted Rindler vacuum along the specified geodesic cannot be excited and detects nothing.

“Therefore such a detector cannot be excited and so it detects nothing [30].”

• cites [30]Not matched

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In region R of Misner space, a part-retarded-and-part-advanced potential is not self-consistent.

“Therefore in region R both the retarded and the advanced potential are self-consistent, but the part-retarded-and-part-advanced potential is not self-consistent.”

Inserting Eq. (27) into Eq. (88) and using Eqs. (79-83), one obtains the energy-momentum tensor of a conformally coupled scalar field in the adapted conformal Minkowski vacuum in dS/B, given by Eq. (93).

“Inserting Eq. ( 27) into Eq.( 88), and using Eqs.(79-83), we obtain the energy-momentum tensor of a conformally coupled scalar field in the adapted conformal Minkowski vacuum in our multiply connected de Sitter space dS/B.”

The renormalized energy-momentum tensor in the time-nonorientable de Sitter space is the same as that in an eternal de Sitter space, given by Eq. (49).

“Therefore the renormalized energy-momentum tensor is the same as that in an eternal de Sitter space, which is given by Eq. ( 49).”

dS/B is not a manifold at the two-sphere defined by W = V = 0 and X^2 + Y^2 + Z^2 = r0^2.

“Similar to the case of Misner space, dS/B is not a manifold at the two-sphere defined by W = V = 0 and X 2 + Y 2 + Z 2 = r 2 0 .”

The semiclassical Einstein equations are satisfied if and only if Λ - 3/r0^2 + 1/(120π r0^4) = 0.

“we find that the semiclassical Einstein's equations are satisfied if and only if Λ - 3 r 2 0 + 1 120πr 4 0 = 0. (91 )”

The energy-momentum tensor of a conformally coupled scalar field in the adapted Minkowski vacuum in Vilenkin's tunneling universe is T_ab^ren = -1/(960π2 r_04) g_ab, the same as for an eternal de Sitter space.

“Therefore, the energy-momentum tensor of a conformally coupled scalar field in the adapted Minkowski vacuum in Vilenkin's tunneling universe is T ab ren = - 1 960π 2 r 4 0 g ab ,(49) which is the same as that for an eternal de Sitter space.”

For a conformally coupled scalar field in the conformal Minkowski vacuum in a simply connected de Sitter space, T_ab^ren = -1/(960π^2 r0^4) g_ab.

“Since the renormalized energy-momentum tensor for Minkowski space in the Minkowski vacuum is zero, we have T b a ren = 0, and thus for a conformally coupled scalar field in the conformal Minkowski vacuum in a simply connected de Sitter space dS T ab ren = - 1 960π 2 r 4 0 g ab ,(89)”

A particle detector moving in the multiply connected Minkowski spacetime with the adapted Minkowski vacuum perceives no particles, even though the renormalized energy-momentum tensor has the form of radiation.

“Such a particle detector perceives no particles, though the renormalized energy-momentum tensor of the field has the form of radiation.”

The results obtained by doing quantum field theory in the Euclidean section and analytically extending to the Lorentzian section are the same as those obtained with the method of images.

“The results are the same as that obtained with the method of images.”

The response functions for detectors on worldlines with constant ξ, y, and z in Misner space with the adapted Rindler vacuum are zero.

“We [30] have also calculated the response functions for detectors on worldlines with constant ξ, y, and z and worldlines with constant ξ, y, and z -both are zero.”

• cites [30]Not matched

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The static coordinates defined by Eq. (63) only cover region R of de Sitter space.

“It is obvious that the static coordinates defined by Eq. ( 63) only cover region R.”

With the above conformal transformation, Misner space is naturally transformed to the multiply connected de Sitter space dS/B with β = br0.

“With the above conformal transformation, Misner space is naturally transformed to the multiply connected de Sitter space dS/B with β = br 0 . (83 )”

In region F of Misner space, the only self-consistent potential is the retarded potential.

“The only self-consistent potential in region F is the retarded potential.”

As L → 1 (approaching the Cauchy horizon), the n ≠ 0 contribution tends to zero; thus as the Cauchy horizon is approached from region F, the detector comoving in the steady-state coordinate system perceives pure Hawking radiation with Gibbons-Hawking temperature.

“As L → 1 (approaching the Cauchy horizon), we also have d dT n =0 F n → 0. Thus as the Cauchy horizon is approached from the side of region F , the particle detector comoving in the steady-state coordinate system perceives pure Hawking radiation with Gibbons-Hawking temperature.”

The multiply connected de Sitter space dS/B is constructed by identifying points (V,W,X,Y,Z) with (V cosh nb + W sinh nb, W cosh nb + V sinh nb, X,Y,Z) on de Sitter space.

“Similar to Misner space, our multiply connected de Sitter space is constructed by identifying points (V, W, X, Y, Z) with (V cosh nb + W sinh nb, W cosh nb + V sinh nb, X, Y, Z) on de Sitter space dS.”

If Λ = 0, the solutions to Eq. (91) are r0 = (360π)^(-1/2) and r0 = ∞.

“If Λ = 0, the solutions to Eq. ( 91) are r 0 = (360π) -1/2 and r 0 = ∞ [89].”

• cites [89]Not matched

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At the intersection of two light cones in multiply connected de Sitter space, the maximum value of T_ab T^ab is at the points with θ = 0 or θ = π.

“At the intersection of two light cones in multiply connected de Sitter space, it is easy to show that the maximum value of T ab T ab is at the points with θ = 0 or θ = π on the intersection.”

A particle detector moving along a geodesic with χ, θ, φ = constants in the time-nonorientable de Sitter space has response function dF/dT = r_0/(2π) ∆E/(e^{πr_0∆E}-1), representing a thermal spectrum with temperature twice the Gibbons-Hawking temperature.

“Inserting this into Eq.( 17) we get dF dT = r 0 2π ∆E e πr0∆E -1 ,(60) which represents a thermal spectrum with a temperature equal to twice the Gibbons-Hawking temperature.”

In Vilenkin's tunneling universe, the boundary condition ∂φ/∂τ|Σ = 0 does not produce any renormalized energy-momentum tensor.

“Substituting Eqs.(42)(43)(44) and Eq. ( 47) into Eq.( 45), we find that lim X ′ →X D ab ′ G (1) CM (X -, X ′ ) = 0, which shows that the boundary condition (40) does not produce any renormalized energy-momentum tensor; but the action of”

In Misner space, T_ab T^ab is zero everywhere except at the intersection of two light cones.

“From the above discussion of T ab T ab in Misner space, we know that T ab T ab is zero everywhere except at the intersection of two light cones.”

The static coordinates can be extended to region F by the complex transformation given in Eq. (69).

“However, similar to the Rindler coordinates, these static coordinates can be extended to region F by the complex transformation”

For the adapted Rindler vacuum in Misner space, unless b = 2π, the energy-momentum tensor blows up as one approaches the Cauchy horizon.

“We [30] have found that unless b = 2π, T ν µ R,ren blows up as one approaches the Cauchy horizon (ξ → 0) (as also does T µν R,ren T µν R,ren ).”

• cites [30]Not matched

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In region F of Misner space, any part-retarded-and-part-advanced potential is not self-consistent.

“It is easy to see that any part-retarded-and-part-advanced potential is also not self-consistent in F.”

Region R, which is multiply connected, has a finite four-volume V_I = (4/3)π b r0^4, where b = β/r0 and β is the de Sitter boost parameter.

“Region R (for its fundamental cell see Fig. 4) which is multiply connected has a finite four-volume V I = 4 3 πbr 4 0 (here b = β/r 0 , β is the de Sitter boost parameter).”

The total probability of forming a bubble in V_II is P_II = (4/3)π b ε (cosh^3(τ0/r0) - 1).

“The total probability of forming a bubble in V II is P II = 4 3 πbǫ(cosh 3 τ0 r0 -1).”

Taking L to be the Hubble radius at the recombination epoch (∼10^23 cm), the maximum metric fluctuation is (δg μν)max ∼ 10^-76.

“If we take L to be the Hubble radius at the recombination epoch (∼ 10 23 cm), we have (δg µν ) max ∼ 10 -76 .”

With retarded potentials in region R, T_ab T^ab does not diverge as the Cauchy horizon separating R from P is approached from the R side, indicating no instability.

“But, as we have shown, with retarded potentials in R, T ab T ab does not diverge as the Cauchy horizon separating R from P is approached from the R side, indicating no instability.”

The condition τ > τ0 is also sufficient for a bubble formed in F not to collide with itself.

“In fact this is also a sufficient condition, which can be shown by the conformal mapping between region F in the multiply connected de Sitter space and region F-O in Misner space defined by Eqs.(81)(82)(83).”

The 0 < τ < τ0 part of the fundamental cell in F has a finite four-volume V_II = V_I (cosh^3(τ0/r0) - 1).

“The 0 < τ < τ 0 part of the fundamental cell in F has a finite four-volume V II = V I (cosh 3 τ0 r0 -1).”

The models considered so far have all obeyed the weak energy condition.

“The models considered so far have all obeyed the weak energy condition, and these models have all been multiply connected; in other words, they have a genus of 1, like a donut, since one of the later baby universes is connected with the original one.”

These two future light cones are identified creating a periodic boundary condition for the causally connected region F + R.

“These two future light cones are identified creating a periodic boundary condition for the causally connected region F + R.”

T_ab^ren remains formally ill-defined on the Cauchy horizon itself (ξ=0 in Eq. (32) with b=2π), a set of measure zero.

“It's true that T ab ren remains formally ill-defined on the Cauchy horizon itself [ξ = 0 in Eq. ( 32) with b = 2π], a set of measure zero.”

Our multiply connected de Sitter space is divided into four regions R, L, F, and P, which are separated by Cauchy horizons CH.

“Our multiply connected de Sitter space is divided into four regions R, L, F, and P, which are separated by Cauchy horizons CH.”

In F the retarded potential is self-consistent.

“This diagram shows that in F the retarded potential is self-consistent.”

When analyzing classical fields in this model, the only self-consistent solution without a blow up as the Cauchy horizon is approached occurs when there is a pure retarded potential in the causally connected region of the model.

“When analyzing classical fields in this model, the only self-consistent solution without a blow up as the Cauchy horizon is approached occurs when there is a pure retarded potential in the causally connected region of the model.”

Likewise the advanced potential in region P would not destroy the Cauchy horizon.

“Likewise the advanced potential in region P would not destroy the Cauchy horizon.”

If the boost b = 2π, there exists a Rindler-type vacuum for Misner space in which the renormalized energy-momentum tensor does not blow up as the Cauchy horizon is approached but produces a cosmological constant throughout spacetime that self-consistently solves Einstein's equations for this geometry.

“If the boost b = 2π, we show that we can find a Rindler-type vacuum where the renormalized energy-momentum tensor does not blow up as the Cauchy horizon is approached but rather produces a cosmological constant throughout the spacetime which self-consistently solves Einstein's equations for this geometry.”

• cites [30]Not matched

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The models considered so far have all been multiply connected with genus 1.

“The models considered so far have all obeyed the weak energy condition, and these models have all been multiply connected; in other words, they have a genus of 1, like a donut, since one of the later baby universes is connected with the original one.”

A treatment in the Euclidean section shows that if b=2π, T_ab^ren=0 everywhere, including at ξ=0.

“In fact, a treatment in the Euclidean section shows this is the case, for in the Euclidean section, if b = 2π, T ab ren = 0 everywhere, including at ξ = 0.”

This tiny oscillating universe has a small but non-zero probability to tunnel through the barrier out to become a de Sitter-type inflating universe.

“This tiny oscillating universe has a small but non-zero probability to tunnel through the barrier out to become a de Sitter-type inflating universe”

The solution of the Euclidean Einstein's equations representing the tunneling regime (open circle) in Fig. 2 is a solution to the Euclidean Einstein's equations with a positive cosmological constant and a conformally coupled scalar field in its ground state.

“The solution of the Euclidean Einstein's equations representing the tunneling regime (open circle) in Fig. 2.This is a solution to the Euclidean Einstein's equations with a positive cosmological constant and a conformally coupled scalar field in its ground state.”

A retarded potential in R and an advanced potential in L (or vice versa) are self-consistent.

“This diagram shows that a retarded potential in R and an advanced potential in L (or vice versa) are self-consistent.”

The collision of an electromagnetic wave with its images cannot destroy the Cauchy horizon, since the proper time from the collision (event p) to the origin is always bigger than the proper time from E to the origin.

“The “collision” of an electromagnetic wave with its images cannot destroy the Cauchy horizon, since the proper time from the “collision” (event p) to the origin is always bigger than the proper time from E to the origin.”

The adapted Rindler vacuum for Misner space has T_ab^ren = 0 throughout the space if b = 2π.

“But as we have shown [30], the adapted Rindler vacuum for Misner space has T ab ren = 0 throughout the space if b = 2π”

• cites [30]Not matched

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The simplest solutions we have obtained so far are not geodesically complete to the past.

“Perhaps the most obvious problem with the model we have proposed is that the simplest solutions we have obtained so far are not geodesically complete to the past.”

The region of CTCs has a finite four-volume equal to 4πbr^4_0/3.

“The region of CTCs has a finite four-volume equal to 4πbr 4 0 /3”

The advanced potential in F is not self-consistent, since the collision of an electromagnetic wave with its images will destroy the Cauchy horizon.

“This diagram shows that the advanced potential in F is not self-consistent, since the collision of an electromagnetic wave with its images will destroy the Cauchy horizon.”

The left dark disk is a tiny radiation-dominated closed oscillating universe, which oscillates between Big Bangs and Big Crunches.

“The left dark disk is a tiny radiation-dominated closed oscillating universe, which oscillates between Big Bangs and Big Crunches.”

This is a Euclidean space bouncing between the state with maximum radius a2 and the state with (non-zero) minimum radius a1.

“This is a Euclidean space bouncing between the state with maximum radius a2 and the state with (non-zero) minimum radius a1.”

If the boost b = 2π, we show that we can find a Rindler-type vacuum where the renormalized energy-momentum tensor does not blow up.

“If the boost b = 2π, we show that we can find a Rindler-type vacuum where the renormalized energy-momentum tensor does not”

• AssumptionRests on an assumption: The early universe contained a region of CTCs.

A part-advanced-and-part-retarded potential in R (or L) is also not self-consistent, as T_ab would also diverge as the Cauchy horizon is approached.

“This diagram shows that a part-advanced-and-part-retarded potential in R (or L) is also not self-consistent, as T ab T ab would also diverge as the Cauchy horizon is approached.”

The metastable de Sitter phase decays along a hyperboloid H3 near the bottom of the figure to form a single open bubble universe with a hot Big Bang phase and an epoch of recombination.

“The metastable de Sitter phase is shown in grey.It decays along a hyperboloid H 3 near the bottom of the figure to form a single open bubble universe with a hot Big Bang phase and an epoch of recombination which is also shown.”

The de Sitter bubble expands forever.

“The de Sitter bubble expands forever.”

After CH, there are no CTCs.

“After CH, there are no CTCs.”

• Also stated in: § FIG. 3. (a) · plain

  “After CH there are no CTCs.”

In this model the closed null curves generating the Cauchy horizon (CH) pass through a hot Big Bang region.

“In this model the closed null curves generating the Cauchy horizon (CH) pass through a hot Big Bang region”

This reaches a point of maximum expansion at which point it tunnels to a doorknob-shaped configuration.

“This reaches a point of maximum expansion at which point it tunnels to a doorknob-shaped configuration.”

• AssumptionRests on an assumption: A super-civilization can create an expanding bubble of de Sitter metastable vacuum.

Even a short run of inflation, of 67 e-folding times or more, is sufficient to explain why the universe is as large as it is observed to be.

“Thus, even a short run of inflation, of 67 e-folding times or more, is sufficient to explain why the universe is as large as it is observed to be.”

The inflationary state preceding the Big Bang could not have been infinite in duration; it must have had a beginning also.

“as Borde and Vilenkin have shown, the inflationary state preceding the Big Bang could not have been infinite in duration -it must have had a beginning also.”

• cites Borde and VilenkinNot matched

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Hawking and Penrose proved singularity theorems showing that, with reasonable constraints on the energy-momentum tensor, if Einstein's equations are correct and the expansion of the universe is as observed today, there is no way to avoid an initial singularity.

“Hawking and Penrose proved a number of singularity theorems [48,49,40] showing that, with some reasonable constraints on the energy-momentum tensor, if Einstein's equations are correct and the expansion of the universe is as observed today, there is no way to avoid an initial singularity in the model; that is, initial singularities would form even in models that were not exactly uniform.”

• cites [48]Not matched

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Inflation also gives a natural explanation for why the observed radius of curvature of the universe is so large.

“Inflation also gives a natural explanation for why the observed radius of curvature of the universe is so large (a ≥ cH -1 0 ≃ 3000h -1 Mpc; here H 0 = 100h km s -1 Mpc -1 is the Hubble constant).”

If one of those later inflating pieces simply turns out to be the inflating piece that one started out with, then the Universe can be its own mother.

“If one of those later inflating pieces simply turns out to be the inflating piece that one started out with, then the Universe can be its own mother.”

All three Friedmann models (k = 0, k = 1, and k = -1) that were expanding at the present epoch had a beginning in the finite past.

“But now there was a problem: all three Friedmann models (k = 0, k = 1, and k = -1) that were expanding at the present epoch had a beginning in the finite past (see e.g.[39,40]).”

• cites [39]Not matched

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• cites [40]Not matched

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Li and Gott found that the self-consistent vacuum in Misner space is the adapted Rindler vacuum.

“Li and Gott [30] have found that it is the “adapted” Rindler vacuum.”

• cites [30]Not matched

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A little piece of inflationary state expands to produce a large volume of inflationary state, little pieces of which resemble the starting piece.

“A little piece of inflationary state expands to produce a large volume of inflationary state, little pieces of which resemble the starting piece.”

The inflationary power spectrum with CDM has been amazingly successful in explaining the qualitative features of observed galaxy clustering.

“The inflationary power spectrum with CDM has been amazingly successful in explaining the qualitative features of observed galaxy clustering (cf.[73][74][75][76][77][78][79][80][81][82]).”

• cites [73]Not matched

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The laws of physics may allow the Universe to be its own mother.

“Interestingly, the laws of physics may allow the Universe to be its own mother.”

• AssumptionThe authors state this as an assumption: The early universe contains a region of CTCs.
• AssumptionThe authors state this as an assumption: The probability of a particular multiple connection forming is not exactly zero.
• AssumptionRests on an assumption: An infinite number of baby universes are created.

The observed Zeldovich-Peebles-Yu-Harrison fluctuation spectrum with index n = 1 has been naturally predicted as the result of random quantum fluctuations.

“Another success of inflation is that the observed Zeldovich-Peebles-Yu-Harrison fluctuation spectrum with index n = 1 [66][67][68] has been naturally predicted as the result of random quantum fluctuations [69][70][71][72].”

• cites [66]Not matched

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• cites [67]Not matched

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• cites [69]Not matched

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• cites [72]Not matched

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Guth's proposal of inflation offered an explanation of why the initial conditions in the Big Bang should be approximately, but not exactly uniform.

“Guth's proposal of inflation [54] offered an explanation of why the initial conditions in the Big Bang should be approximately, but not exactly uniform.”

• cites [54]Not matched

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Inflation is well suited to reproducing conditions as they were earlier.

“Interestingly, inflation is well suited to this.”

General relativity allows for the possibility of closed timelike curves (CTCs).

“A remarkable property of general relativity is that it allows solutions that have closed timelike curves (CTCs) [2][3][4][5][6][7][8] (for review see [9,10]).”

• cites [2]Not matched

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There is the possibility of forming baby universes at late times where new pieces of inflating states are formed.

“Also there is the possibility of forming baby universes at late times where new pieces of inflating states are formed.”

• cites [32]Not matched

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In the Friedmann models the universe began in a singularly dense state at a finite time in the past.

“In the Friedmann models the universe began in a singularly dense state at a finite time in the past.”

Linde and Mezhlmian showed there are reasonable potentials which could produce bubble universes with different values of Ω.

“Linde and Mezhlmian [107,108] have shown how there are reasonable potentials which could produce such bubble universes with different values of Ω.”

• cites [107]Not matched

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Borde and Vilenkin have proved that if the Universe were infinitely old (i.e., if the de Sitter space were complete) then the bubbles would percolate immediately and inflation would never get started.

“Borde and Vilenkin have proved that if the Universe were infinitely old (i.e., if the de Sitter space were complete) then the bubbles would percolate immediately and inflation would never get started (see [126,127] and references cited therein).”

• cites [126]Not matched

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• cites [127]Not matched

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Light from events within the future light cone of E' never reaches events inside the future light cone of E.

“Light from events within the future light cone of E ′ never reaches events inside the future light cone of E.”

Turner and Fukugita, Futamase, and Kasai showed that a flat ΩΛ=1 model produces about 10 times as many gravitational lenses as a flat model with Ω=1.

“Turner [119] and Fukugita, Futamase, and Kasai [120] showed that a flat Ω Λ = 1 model produces about 10 times as many gravitational lenses as a flat model with Ω = 1”

• cites [119]Not matched

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• cites [120]Not matched

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In order that there be a greater than 5% chance that no bubble should have collided with our bubble by now, so as to be visible in our past light cone, ε < 0.01 for Ω = 0.4, Λ = 0, h = 0.63 today.

“In order that there be a greater than 5% chance that no bubble should have collided with our bubble by now, so as to be visible in our past light cone, ǫ < 0.01 for Ω = 0.4, Λ = 0, h = 0.63 today [88], but this is no problem since we expect tunneling probabilities through a barrier to be exponentially small.”

• cites [88]Not matched

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In a standard chaotic inflationary potential V(φ), one could build in a bump so that one would randomly walk to the top via quantum fluctuations and then roll down until lodged behind the bump in a metastable local minimum.

“In a standard chaotic inflationary potential V (φ) [109], one could simply build in a bump, so that one would randomly walk to the top of the curve via quantum fluctuations and then roll down till one lodged behind the bump in a metastable local minimum.”

• cites [109]Not matched

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Gott has shown how an open inflationary model might be produced.

“Gott [89] has shown how an open inflationary model might be produced.”

• cites [89]Not matched

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The amount of large scale power seen in the observations suggests an inflationary CDM power spectrum with 0.2 < Ωh < 0.3.

“The amount of large scale power seen in the observations suggests an inflationary CDM power spectrum with 0.2 < Ωh < 0.3 [83][84][85][86][87][88].”

• cites [83]Not matched

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• cites [84]Not matched

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• cites [85]Not matched

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• cites [86]Not matched

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• cites [87]Not matched

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• cites [88]Not matched

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Since the number of bubbles forming increases exponentially with time without limit, our universe is expected to form at a finite but arbitrarily large time after the beginning of the inflationary state.

“Since the number of bubbles forming increases exponentially with time without limit, our universe is expected to form at a finite but arbitrarily large time after the beginning of the inflationary state.”

Borde and Vilenkin have proved that in the inflationary scenario the universe must have a beginning.

“Thus Borde and Vilenkin have proved that in the inflationary scenario the universe must have a beginning.”

However, this point is still the earliest point in Euclidean time since it is at the center of the hemisphere specified by the Euclidean boundary at V=0.

“However, this point is still the earliest point in Euclidean time since it is at the center of the hemisphere specified by the Euclidean boundary at V = 0.”

Thus, the universe should have a microwave background that is isotropic, except for small quantum fluctuations.

“Thus, the universe should have a microwave background that is isotropic, except for small quantum fluctuations.”

With ΩB h^2=0.0125, the Ω=1, ΩΛ=0 model power spectrum reaches its peak value at l=200.

“With Ω B h 2 = 0.0125, the Ω = 1, Ω Λ = 0 model power spectrum reaches its peak value at l = 200”

• cites [122]Not matched

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There would be a fine tuning of parameters needed to produce the observed results.

“So there would be what is called a “fine tuning of parameters” needed to produce the observed results.”

This model has an event horizon, which is the future light cone of an event E' antipodal to E.

“This model has an event horizon, which is the future light cone of an event E ′ (W = -r 0 , X = 0, Y = 0, Z = 0, V = 0) which is antipodal to E.”

Inflation is thus eternal to the future.

“Inflation is thus eternal to the future [128][129][130][131].”

• cites [128]Not matched

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• cites [129]Not matched

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If there were 670 e-folds or 670 million e-folds, then Ω currently would be only slightly less than 1.

“If there were 670 e-folds or 670 million e-folds, then Ω currently would be only slightly less than 1.”

Models with low Ω but Ω+ΩΛ=1 are also acceptable.

“Models with low Ω but Ω + Ω Λ = 1 are also acceptable.”

If Ω is too small, no galaxies will form.

“since if Ω is too small, no galaxies will form [110].”

• cites [110]Not matched

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These things have forced even enthusiasts of k=0 models to move to models with Ω<1 and a cosmological constant so that Ω+ΩΛ=1 and k=0.

“These things have forced even enthusiasts of k = 0 models to move to models with Ω < 1 and a cosmological constant so that Ω + Ω Λ = 1 and k = 0 [118].”

• cites [118]Not matched

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These calculations allow predictions of fluctuations in the microwave background.

“This allows predictions of fluctuations in the microwave background.”

An Ω=0.3, ΩΛ=0.7 model reaches its peak value also at l=200.

“an Ω = 0.3, Ω Λ = 0.7 model reaches its peak value also at l = 200 [122]”

• cites [122]Not matched

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Ages of globular cluster stars have a 2σ lower limit of about 11.6 billion years.

“Ages of globular cluster stars have a 2σ lower limit of about 11.6 billion years [115]”

• cites [115]Not matched

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One has to produce a particular amount of inflation, approximately 67 e-folds.

“Because one has to produce a particular amount of inflation, approximately 67 e-folds.”

Ratra and Peebles showed how to calculate quantum fluctuations in the H^3 hyperbolic geometry with a(t)=r0 sinh(t/r0) during the inflationary epoch inside the bubble in the single-bubble model.

“Ratra and Peebles [102,103] have shown how to calculate quantum fluctuations in the H 3 hyperbolic geometry with a(t) = r 0 sinh(t/r 0 ) during the inflationary epoch inside the bubble in the single bubble model.”

• cites [102]Not matched

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• cites [103]Not matched

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• AssumptionThe authors state this as an assumption: For the rest of the paper, the authors assume single-bubble open inflationary models for our Big Bang universe.

With a myriad of open universes being created, anthropic arguments come into play.

“With a myriad of open universes being created, anthropic arguments [62] come into play”

• cites [62]Not matched

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From inside one of the bubbles, our view could be isotropic.

“But, from inside one of the bubbles, our view could be isotropic [89].”

• cites [89]Not matched

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Extreme-Λ dominated models are ruled out by producing too many gravitational lenses.

“Thus, extreme-Λ dominated models are ruled out by producing too many gravitational lenses.”

So we are surrounded by an event horizon.

“So we are surrounded by an event horizon.”

The bubble formation rate per unit four volume r0^4 is expected to be exponentially small so the bubbles do not percolate.

“The bubble formation rate per unit four volume r 4 0 is thus expected to be exponentially small so the bubbles do not percolate.”

• AssumptionThe authors state this as an assumption: The authors assume an initial metastable vacuum which decays by forming bubbles through barrier penetration.
• AssumptionThe authors state this as an assumption: The bubble formation rate per unit four volume r0^4 is ε≪1.

A variety of observational samples and methods have suggested 0.2<Ωh<0.3.

“A variety of observational samples and methods have suggested this: counts in cells, angular covariance function on the sky, power spectrum analysis of 3D samples, and finally topological analysis, all showing that 0.2 < Ωh < 0.3 [82][83][84][85][86][87][88].”

• cites [82]Not matched

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• cites [83]Not matched

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• cites [84]Not matched

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Coleman and de Luccia showed that if a metastable symmetric vacuum with positive cosmological constant Λ were to decay by tunneling directly through a barrier to reach the current vacuum with a zero cosmological constant, then it would do this by forming a bubble of low-density vacuum of radius σ around an event E.

“Coleman and de Luccia [93] showed that if a metastable symmetric vacuum (with the Higgs field φ = 0), with positive cosmological constant Λ were to decay by tunneling directly through a barrier to reach the current vacuum with a zero cosmological constant (where the Higgs field φ = φ 0 ), then it would do this by forming a bubble of low-density vacuum of radius σ around an event E.”

• cites [93]Not matched

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An Ω=0.4, ΩΛ=0 model reaches its peak value at l=350.

“while an Ω = 0.4, Ω Λ = 0 model reaches its peak value at l = 350 [123].”

• cites [123]Not matched

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Studies of large scale structure have shown that with the inflationary CDM power spectrum, the standard Ω=1, h=0.5 model simply does not have enough power at large scales.

“Also, studies of large scale structure have shown that with the inflationary CDM power spectrum, the standard Ω = 1, h = 0.5 model simply does not have enough power at large scales.”

In order that inflation proceeds and the bubbles do not percolate, it is required that the probability of forming a bubble in de Sitter space per four volume r0^4 is ε < ε_cr where 5.8×10^-9 < ε_cr < 0.24.

“In order that inflation proceeds and the bubbles do not percolate, it is required that the probability of forming a bubble in de Sitter space per four volume r 4 0 is ǫ < ǫ cr where 5.8 × 10 -9 < ǫ cr < 0.24 [95].”

• cites [95]Not matched

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This implies that there must be 'something' instead of 'nothing' before tunneling.

“This implies that there must be “something” instead of “nothing” before tunneling.”

Li has shown that in this spacetime the energy-momentum tensor is finite everywhere.

“Li [23] has shown that in this spacetime the energymomentum tensor is finite everywhere.”

• cites [23]Not matched

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Cassidy and Hawking's results cannot be extended to the Cauchy horizon.

“In other words, Cassidy and Hawking's results [143] cannot be extended to the Cauchy horizon.”

• cites [143]Not matched

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The divergence of the effective action may imply that the effective action is not a good tool as the polarized hypersurfaces are approached.

“The divergence of the effective action may imply that the effective action is not a good tool as the polarized hypersurfaces are approached.”

Without appeal to quantum gravity, counter-examples to the chronology protection conjecture also exist.

“Without appeal to quantum gravity, counter-examples to the chronology protection conjecture also exist.”

Vilenkin obtained his tunneling universe by choosing a so-called 'tunneling boundary condition' for the Wheeler-DeWitt equation.

“Vilenkin obtained his tunneling universe by choosing a so-called “tunneling boundary condition” for the Wheeler-DeWitt equation [135,136].”

• cites [135]Not matched

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Visser has generalized Thorne's result to the Roman-ring case.

“Recently Visser [29] has generalized this result to the Roman-ring case.”

• cites [29]Not matched

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If quantum gravity effects are considered (e.g. in quantum cosmology), they can be separated by a complex geometric region (as a quantum barrier) instead of a Cauchy horizon.

“But if quantum gravity effects are considered (e.g. in quantum cosmology), they can be separated by a complex geometric region (as a quantum barrier) instead of a Cauchy horizon [22].”

• cites [22]Not matched

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In the path integral approach to quantum cosmology, complex geometries are required in order to make the path integral convergent and to overcome the difficulty that in general situations a Euclidean space cannot be directly joined to a Lorentzian space.

“(In the path integral approach to quantum cosmology, complex geometries are required in order to make the path integral convergent and to overcome the difficulty that in general situations a Euclidean space cannot be directly joined to a Lorentzian space [141]).”

• cites [141]Not matched

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Boulware and Tanaka and Hiscock have found that for sufficiently massive fields in Gott space and Grant space respectively, the energy-momentum tensor remains regular on the Cauchy horizon.

“Boulware [21] and Tanaka and Hiscock [25] have found that for sufficiently massive fields in Gott space [8,142] and Grant space [18] respectively, the energy-momentum tensor remains regular on the Cauchy horizon.”

• cites [21]Not matched

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• cites [25]Not matched

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His 'tunneling from nothing' boundary condition demands that when the universe is big (a^2 Λ/3 > 1) there is only an outgoing wave in the superspace.

“His “tunneling from nothing” boundary condition demands that when the universe is big (a 2 Λ/3 > 1 where Λ is the cosmological constant and a is the scale factor of the universe) there is only an outgoing wave in the superspace [135,136].”

• cites [135]Not matched

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• cites [136]Not matched

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In ordinary quantum mechanics, a particle bounded in a well surrounded by a barrier has a finite probability to tunnel through the barrier to the outside if the height of the barrier is finite.

“In ordinary quantum mechanics, a particle bounded in a well surrounded by a barrier has a finite probability to tunnel through the barrier to the outside if the height of the barrier is finite (as in the α-decay of radioactive nuclei [132][133][134]).”

• cites [132]Not matched

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• cites [134]Not matched

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Cassidy and Hawking have admitted that back-reaction does not enforce chronology protection.

“In a recent paper, Cassidy and Hawking [143] have admitted that “back-reaction does not enforce chronology protection”.”

• cites [143]Not matched

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Quantum field theory is well-defined in a Euclidean space because there are no CTCs in a Euclidean space.

“Quantum field theory is well-defined in a Euclidean space because there are no CTCs in a Euclidean space [147].”

• cites [147]Not matched

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The usual quantum field theory cannot be directly extended to a spacetime with CTCs.

“Basically speaking, their theorems imply that the usual quantum field theory cannot be directly extended to a spacetime with CTCs [144].”

• cites [144]Not matched

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In such a case, if we trace the history of the Universe backward, inevitably we will enter a region of CTCs.

“In such a case, if we trace the history of the Universe backward, inevitably we will enter a region of CTCs.”

Krasnikov has found some two-dimensional spacetimes with CTCs for which the energy-momentum tensor of vacuum polarization is bounded on the Cauchy horizon.

“Krasnikov [27] has found some two-dimensional spacetimes with CTCs for which the energy-momentum tensor of vacuum polarization is bounded on the Cauchy horizon.”

• cites [27]Not matched

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Hawking and Penrose's singularity theorems do not apply if the Universe has had CTCs.

“It is interesting to note that Hawking and Penrose's singularity theorems do not apply if the Universe has had CTCs.”

In classical general relativity, a region with CTCs and a region without closed causal curves must be separated by a Cauchy horizon (compactly generated or non-compactly generated) which usually contains closed null geodesics if it is compactly generated.

“In classical general relativity, a region with CTCs and a region without closed causal curves must be separated by a Cauchy horizon (compactly generated or non-compactly generated) which usually contains closed null geodesics if it is compactly generated [20].”

• cites [20]Not matched

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These arguments imply that in the case of a spacetime with CTCs where the energy-momentum tensor of vacuum polarization diverges at the Cauchy horizon, quantum gravity effects should become important near the Cauchy horizon.

“But in any case, these arguments imply that in the case of a spacetime with CTCs where the energy-momentum tensor of vacuum polarization diverges at the Cauchy horizon, quantum gravity effects should become important near the Cauchy horizon.”

So the beginning point in the Vilenkin model is indeed like the south pole of the Earth.

“So the beginning point in the Vilenkin model is indeed like the south pole of the Earth [13].”

• cites [13]Not matched

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Kay, Radzikowski, and Wald have proved two theorems demonstrating that some fundamental quantities such as Hadamard functions and energy-momentum tensors must be ill-defined on a compactly generated Cauchy horizon in a spacetime with CTCs, as one extends usual quantum field theory in a global hyperbolic spacetime to an acausal spacetime with a compactly generated Cauchy horizon.

“Recently, Kay, Radzikowski, and Wald [144] have proved two theorems which demonstrate that some fundamental quantities such as Hadamard functions and energy-momentum tensors must be ill-defined on a compactly generated Cauchy horizon in a spacetime with CTCs, as one extends the usual quantum field theory in a global hyperbolic spacetime to an acausal spacetime with a compactly generated Cauchy”

• cites [144]Not matched

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By introducing a spherical reflecting boundary between two mouths of a wormhole, Li has shown that with some boundary conditions for geodesics (e.g. the reflection boundary condition) closed null geodesics may be removed from the Cauchy horizon separating the region with CTCs and the region without closed causal curves.

“By introducing a spherical reflecting boundary between two mouths of a wormhole, Li [23] has shown that with some boundary conditions for geodesics (e.g. the reflection boundary condition) closed null geodesics [usually the “archcriminal” for the divergence of the energy-momentum tensor as the Cauchy horizon is approached (see e.g.[16])] may be removed from the Cauchy horizon separating the region”

• cites [23]Not matched

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Following Li, Low has given another example of spacetime with CTCs but without closed causal geodesics.

“Following Li [23], Low [24] has given another example of spacetime with CTCs but without closed causal geodesics.”

• cites [23]Not matched

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It has been shown that if a compact Lorentzian spacetime undergoes topology changes, there must be CTCs in this spacetime.

“And, it has been shown that, if a compact Lorentzian spacetime undergoes topology changes, there must be CTCs in this spacetime [140,19,20].”

• cites [140]Not matched

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The chronology protection conjecture does not provide any restriction on spacetimes with CTCs but no Cauchy horizons since there is no indication that this type of spacetime is unstable against vacuum polarization.

“It should be mentioned that the chronology protection conjecture does not provide any restriction on spacetimes with CTCs but no Cauchy horizons since there is no any indication that this type of spacetime is unstable against vacuum polarization.”

This is essentially different from tunneling in ordinary quantum mechanics since in ordinary quantum mechanics tunneling always takes place from one classically allowed region to another classically allowed region where the current and the probability are conserved.

“Thus this is essentially different from tunneling in ordinary quantum mechanics since in ordinary quantum mechanics tunneling always takes place from one classically allowed region to another classically allowed region where the current and the probability are conserved.”

Therefore CTCs may play an important role in the creation of the Universe.

“Therefore CTCs may play an important role in the creation of the Universe.”

Cramer and Kay have shown that Kay, Radzikowski, and Wald's theorems also apply to Misner space where the Cauchy horizon is not compactly generated, in the sense that the energy-momentum tensor must be ill-defined on the Cauchy horizon itself.

“Cramer and Kay [150,151] have shown that Kay, Radzikowski, and Wald's theorems [144] also apply to Misner space (for Sushkov's automorphic field case [28] and Krasnikov's two-dimensional case [27], respectively) where the Cauchy horizon is not compactly generated, in the sense that the energy-momentum tensor must be ill-defined on the Cauchy horizon itself.”

• cites [150]Not matched

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• cites [151]Not matched

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• cites [144]Not matched

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Ordinary quantum field theory must be significantly changed or some new approach must be introduced when one tries to do quantum field theory on a spacetime with CTCs.

“The ordinary quantum field theory must be significantly changed or some new approach must be introduced when one tries to do quantum field theory on a spacetime with CTCs.”

Creation from 'nothing' is supposed to take place from a classically forbidden (Euclidean) region to a classically allowed (Lorentzian) region, so the conservation of current is obviously violated.

“But creation from “nothing” is supposed to take place from a classically forbidden (Euclidean) region to a classically allowed (Lorentzian) region, so the conservation of current is obviously violated.”

Vilenkin's tunneling universe was based on an analogy between quantum creation of universes and tunneling in ordinary quantum mechanics.

“Vilenkin's tunneling universe was based on an analogy between quantum creation of universes and tunneling in ordinary quantum mechanics [11].”

• cites [11]Not matched

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From the arguments in the last section, the Universe does not seem to be created from nothing.

“From the arguments in the last section, we find that the Universe does not seem to be created from nothing.”

CTCs do not mean that the energy-momentum tensor must diverge.

“Thus CTCs do not mean that the energy-momentum tensor must diverge.”

The universe with metric (72) is a type of Kantowski-Sachs universe.

“The universe with metric (72) is a type of Kantowski-Sachs universe [164].”

• cites [164]Not matched

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Otherwise the collision of light cones from R and light cones from L will destroy the Cauchy horizon.

“Otherwise the collision of light cones from R and light cones from L will destroy the Cauchy horizon.”

If T_ab T^ab is zero in some region (not a single point) in Misner space, the corresponding T_ab T^ab is zero in the corresponding region in multiply connected de Sitter space.

“(3) if T ab T ab is zero in some region (not a single point) in Misner space, the corresponding T ab T ab is also zero in the corresponding region in the multiply connected de Sitter space.”

Usually there is no well-defined quantum field theory in a spacetime with CTCs.

“Usually there is no well-defined quantum field theory in a spacetime with CTCs.(Kay-Radzikowski-Wald's theorems [144] enforce this claim, though they do not apply directly to an acausal spacetime without a Cauchy horizon.)”

• cites [144]Not matched

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The back-reaction of vacuum polarization does not enforce the chronology protection conjecture.

“However, we can conclude that the back-reaction of vacuum polarization does not enforce the chronology protection conjecture, a point Hawking himself also admits [143].”

• cites [143]Not matched

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The first thermodynamic law of event horizons δM_c = T δA is preserved in the time-nonorientable de Sitter space.

“This together with T = 2T G-H tells us that the first thermodynamic law of event horizons δM c = T δA is preserved, where M c is the mass within the horizon, and A is the area of the horizon [100].”

• cites [100]Not matched

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The conical singularity condition may be regarded as a geometrical explanation of the self-consistent condition β = 2πr0.

“This may be regarded as a geometrical explanation of the self-consistent condition in (96).”

The cold-to-hot structure of dS/B gives rise to an arrow of increasing entropy, from a cold region to a hot region.

“This gives rise to an arrow of increasing entropy, from a cold region to a hot region (Fig. 5).”

In Misner space, Li and Gott found that the adapted Rindler vacuum is self-consistent if b = 2π.

“Therefore, in the case of Misner space, Li and Gott [30] have tried to find a vacuum which is also self-consistent and found that the adapted Rindler vacuum is such a vacuum if b = 2π.”

• cites [30]Not matched

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De Sitter space can be represented by a timelike hyperbolic hypersurface embedded in five-dimensional Minkowski space, with r0 = (3/Λ)^(1/2).

“De Sitter space can be represented by a timelike hyperbolic hypersurface W 2 + X 2 + Y 2 + Z 2 -V 2 = r 2 0 ,(61) embedded in a five-dimensional Minkowski space (V, W, X, Y, Z) with the metric ds 2 = -dV 2 + dW 2 + dX 2 + dY 2 + dZ 2 , (62 ) where r 0 = (3/Λ) 1/2 [40,90].”

• cites [40]Not matched

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• cites [90]Not matched

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When a covering space exists, quantum field theory in a spacetime with CTCs can be done in the covering space with the method of images.

“However, in the case where a covering space exists, we can do it in the covering space with the method of images.In fact in most cases where the energy-momentum tensor in spacetimes with CTCs has been calculated, this method has been used (for the theoretical basis for the method of images see Ref. [15] and references cited therein).”

• cites [15]Not matched

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In multiply connected de Sitter space, T_ab T^ab is zero everywhere except at the intersection of two light cones.

“Thus, in multiply connected de Sitters pace, T ab T ab is also zero everywhere except at the intersection of two light cones.”

A particle detector moving along such a geodesic in this time-nonorientable spacetime perceives thermal radiation with temperature T = 2T_G-H.

“Therefore a particle detector moving along such a geodesic in this time-nonorientable spacetime perceives thermal radiation with temperature T = 2T G-H .”

If T_ab T^ab diverges on the Cauchy horizon in Misner space, the corresponding T_ab T^ab also diverges on the Cauchy horizon in multiply connected de Sitter space.

“(1) if T ab T ab diverges on the Cauchy horizon in Misner space, the corresponding T ab T ab also diverges on the Cauchy horizon in our multiply connected de Sitter space;”

The principle of self-consistency naturally gives rise to an arrow of time in the multiply connected de Sitter space.

“In this subsection we argue that the principle of self-consistency [178,179] naturally gives rise to an arrow of time in our multiply connected de Sitter space.”

• cites [178]Not matched

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

Hawking has given a general proof that any future chronology horizon is classically unstable unless light rays are diverging when they propagate near the chronology horizon.

“Hawking [20] has given a general proof along the above lines that any future chronology horizon is classically unstable unless light rays are diverging when they propagate near the chronology horizon.”

• cites [20]Not matched

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

If T_ab T^ab is finite in some region (except at the Cauchy horizon) in Misner space, the corresponding T_ab T^ab is finite in the corresponding region (not at the Cauchy horizon) in multiply connected de Sitter space.

“(2) if T ab T ab is finite in some region (except at the Cauchy horizon) in Misner space, the corresponding T ab T ab is also finite in the corresponding region (not at the Cauchy horizon) in the multiply connected de Sitter space;”

This periodic boundary condition (the self-consistent condition) is sufficient to fix the solutions of the universe.

“This periodic boundary condition (the self-consistent condition) is sufficient to fix the solutions of the universe.”

From the point of view of an observer at the center of a V = Λ/8π bubble, just after the tunneling event, he would see himself in a de Sitter space that was initially static but starts to inflate.

“From the point of view of an observer sitting at the center of V = Λ/8π bubble, he would see himself, just after the tunneling event, as sitting in a de Sitter space that was initially static but which starts to inflate.”

Li [26] has estimated the fluctuation of the metric of the background spacetime caused by vacuum polarization with absorption, which is a small number in most cases.

“Li [26] has estimated the fluctuation of the metric of the background spacetime caused by vacuum polarization with absorption, which is a small number in most cases.”

• cites [26]Not matched

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

The infinitely expanding de Sitter space, which begins expanding at the tunneling event, is a perfect starting point for making an infinite number of bubble universes, because it has a finite beginning and then expands forever.

“This infinitely expanding de Sitter space, which begins expanding at the tunneling event, is a perfect starting point (just like Vilenkin's tunneling universe) for making an infinite number of bubble universes, as this de Sitter space has a finite beginning and then expands forever.”

An event in F + R and an event in P + L are always causally independent in physics: they can never physically influence each other though they may be mathematically connected by some causal curves.

“An event in F +R and an event in P +L are always causally independent in physics: they can never physically influence each other though they may be mathematically connected by some causal curves (null curves or timelike curves).”

If the absorption is caused by electron-positron pair production by a photon in a photon-electron collision, the maximum value of the metric fluctuation is (δg μν)max ∼ l_P^2/(r_e L), where l_P is the Planck length, r_e is the classical radius of electron, and L is the spatial distance between the identified points in the frame of rest relative to the absorber.

“If the absorption is caused by electron-positron pair production by a photon in a photon-electron collision, the maximum value of the metric fluctuation is (δg µν ) max ∼ l 2 P /(r e L), where l P is the Planck length, r e is the classical radius of electron, L is the spatial distance between the identified points in the frame of rest relative to the absorber [26].”

• cites [26]Not matched

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

In our multiply connected de Sitter space, for inflation to proceed, there should be the additional requirement that bubbles do not collide with images of themselves (producing percolation).

“In our multiply connected de Sitter space, for inflation to proceed, there should be the additional requirement that bubbles do not collide with images of themselves (producing percolation).”

Converging rays are turned into diverging rays during the tunneling event without violating the weak energy condition.

“Thus, converging rays are turned into diverging rays (as in the wormhole solution) during the tunneling event without violating the weak energy condition.”

If the potential in R is chosen to be retarded, then the potential in L must be advanced.

“If we choose the potential in R to be retarded, then the potential in L must be advanced.”

In multiply connected de Sitter space, region F and region P are causally independent in physics: an event in F can never influence an event in P, and vice versa.

“From the above discussion we find that in the multiply connected de Sitter space region F and region P are causally independent in physics: the self-consistent potential in F is the retarded potential, while the self-consistent potential in P is the advanced potential, thus an event in F can never influence an event in P, and vice versa.”

This model produces, inside the black hole, to the future, and behind the event horizon, an expanding de Sitter phase that has a beginning, like Vilenkin's tunneling universe.

“This model thus produces, inside the black hole, to the future, and behind the event horizon, an expanding de Sitter phase that has a beginning, just like Vilenkin's tunneling universe.”

• cites [55]Not matched

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

There are no particles crossing the Cauchy horizons separating P from R and F from L.

“Thus, there are no particles crossing the Cauchy horizons separating P from R and F form L.”

If the metastable vacuum is at the GUT density, which is many orders of magnitude lower than the Planck density, then the number of trials becomes truly formidable (P ∼ 10^-10^18).

“but if it is at the GUT density which turns out to be many orders of magnitude lower than the Planck density, then the number of trials becomes truly formidable (P ∼ 10 -10 18 ) [32].”

• cites [32]Not matched

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

It would seem difficult for the super-civilization that made the metastable state that later gave rise to our universe to have been able to manipulate the physical constants in our universe.

“Thus, it would seem difficult for the super-civilization that made the metastable state that later gave rise to our universe to have been able to manipulate the physical constants in our universe.”

For b = 2π, in order that there be less than a 5% chance that a bubble forms in V_I (and thus less than 0.05% chance in V_II), ε should be less than 2 × 10^-3.

“For the case of b = 2π, in order that there be less than a 5% chance that a bubble forms in V I (and thus less than 0.05% chance in V II ), ǫ should be less than 2 × 10 -3 .”

• AssumptionRests on an assumption: The tunneling probability ε is exponentially small.

We can smoothly extend F + R to P + L so that the total multiply connected de Sitter space dS/B is geodesically complete but not a manifold at a two-sphere.

“On the other hand, we can smoothly extend F + R to P + L so that the total multiply connected de Sitter space dS/B is geodesically complete but at the price that it is not a manifold at a two-sphere (section IX A).”

These closed null geodesics need not be infinitely extendible in affine distance toward the past.

“These closed null geodesics need not be infinitely extendible in affine distance toward the past.”

Thus it would not be surprising to find region R and region F for epochs 0 < τ < τ0 = 0.086 r0 clear of bubble formation events (and clear of real particles), which is all we require.

“Thus it would not be surprising to find region R and region F for epochs 0 < τ < τ 0 = 0.086r 0 clear of bubble formation events (and clear of real particles), which is all we require.”

• AssumptionRests on an assumption: The tunneling probability ε is exponentially small.

A retarded photon traveling around one of those closed null geodesics will be red-shifted more and more on each cycle, thus not causing an instability.

“A retarded photon traveling around one of those closed null geodesics will be red-shifted more and more on each cycle, thus not causing an instability.”

In this case the Cauchy horizons separating F from R and P from L are classically stable, as indicated by our detailed study of T_ab T^ab as these Cauchy horizons are approached.

“In this case the Cauchy horizons separating F from R and P from L are classically stable, as indicated by our detailed study of T ab T ab as these Cauchy horizons are approached.”

All that super-civilizations might reasonably be able to do would be to drive the baby universe up into a particular metastable vacuum.

“All, they might reasonably be able to do would be to drive the baby universe up into a particular metastable vacuum [32].”

• cites [32]Not matched

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

Region F has an infinite four-volume and thus there should be an infinite number of bubble universes formed.

“Region F (its fundamental cell is shown in Fig. 4) has an infinite four-volume and thus there should be an infinite number of bubble universes formed [89,98].”

• cites [89]Not matched

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

• cites [98]Not matched

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

This model describes two physically disconnected but mathematically connected universes.

“This model describes two physically disconnected but mathematically connected universes.”

There is a Cauchy horizon separating the region of CTCs from the later region that does not contain them.

“There is a Cauchy horizon (CH, see Fig. 7) separating the region of CTCs from the later region that does not contain them.”

If a future chronology horizon is replaced by a past chronology horizon, Hawking's proof breaks down because a wave packet propagating toward the future near the past chronology horizon will suffer a red-shift instead of a blue-shift.

“However, in Hawking's proof [20], if we replace a future chronology horizon with a past chronology horizon, then the proof breaks down because, in such a case, a wave packet propagating toward the future near the past chronology horizon will suffer a red-shift instead of a blue-shift.”

• cites [20]Not matched

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

All bubbles formed after the epoch τ0 in F in the multiply connected de Sitter space will never collide with themselves.

“Therefore all bubbles formed after the epoch τ 0 in F in the multiply connected de Sitter space will never collide with themselves.”

The topological multi-connectivity in F + R gives rise to a periodic boundary condition, which is a kind of self-consistent boundary condition.

“the topological multi-connectivity in F + R has already given rise to a periodic boundary condition (which is a kind of self-consistent boundary condition).”

If the probability of forming a bubble per volume r0^4 in de Sitter space is ε, then the total probability of forming a bubble in V_I is P_I = (4/3)π b ε.

“If the probability of forming a bubble per volume r 4 0 in de Sitter space is ǫ, then the total probability of forming a bubble in V I is P I = 4 3 πbǫ.”

Centered on this observer's antipodal point in de Sitter space, he would see a bubble of ordinary V = 0 vacuum surrounding a black hole of mass M.

“Centered on this observer's antipodal point in de Sitter space, he would see a bubble of ordinary V = 0 vacuum surrounding a black hole of mass M .”

In region P, the potentials are advanced, so Hawking's instability does not arise as one approaches the Cauchy horizon separating it from R.

“In region P, the potentials are advanced, so Hawking's instability does not arise as one approaches the Cauchy horizon separating it from R.”

From the de Sitter observer's point of view, the black hole evaporates by Hawking radiation, eventually leaving an empty V = 0 bubble in an ever-expanding de Sitter space.

“From the point of view of the de Sitter observer, the black hole also evaporates by Hawking radiation, eventually leaving an empty V = 0 bubble in an ever-expanding de Sitter space.”

To reach the N-th image one has to pass through the hot optically thick thermal radiation of the hot Big Bang N times.

“Thus, to reach the N -th image one has to pass through the hot optically thick thermal radiation of the hot Big Bang N times.”

In order that inflation proceeds and the bubbles do not percolate, it is required that ε < ε_cr where 5.8 × 10^-9 < ε_cr < 0.24.

“In order that the inflation proceeds and the bubbles (which expand to the future -as expected with the retarded potential in region F ) do not percolate, it is required that ǫ < ǫ cr where 5.8 × 10 -9 < ǫ cr < 0.24 [95].”

• cites [95]Not matched

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

This will cause the contribution of the N-th image to the renormalized energy-momentum tensor to be exponentially damped by a factor e^{-Nτ} where τ ≡ nzσ_t ≫ 1.

“And this will cause the contribution of the N -th image to the renormalized energy-momentum tensor to be exponentially damped by a factor e -N τ where τ ≡ nzσ t ≫ 1 (where n is the number density of target particles, z is the thickness of hot material, σ t is the total cross-section).”

Therefore a past chronology horizon, according to Hawking's argument, is classically stable in a world with retarded potentials.

“Therefore a past chronology horizon, according to Hawking's argument, is classically stable in a world with retarded potentials.”

It seems easier to have a Cauchy horizon in the early universe.

“It thus seems easier to have a Cauchy horizon in the early universe.”

The tunneling model ignores the zero-point-energy.

“The problem with this model is that it ignores the”

• Also stated in: § XI. CONCLUSIONS · plain

  “zero-point-energy”

We may need no boundary condition because we have a periodic boundary condition instead.

“But we may need no boundary condition since we have a periodic boundary condition instead.”

If this zero-point-energy is considered, the quantum behavior of the universe is like a particle of unit mass with total energy 1/2 moving in the potential U(a).

“If this “zero-point-energy” is considered, the quantum behavior of the universe is like a particle of unit mass with total energy 1/2 moving in the potential U (a).”

Since these models are all ones in which there are an infinite number of baby universes created, this multiply connected outcome must occur unless the probability for a particular multiple connectivity to exist is exactly zero.

“Since these models are all ones in which there are an infinite number of baby universes created, this multiply connected outcome must occur unless the probability for a particular multiple connectivity to exist is exactly zero.”

• AssumptionThe authors state this as an assumption: One of the created baby universes is the original universe that one started with.

If one of the baby universes turns out to be the one we started with, then we get a model with an epoch of CTCs that is over by now, bounded toward the future by a Cauchy horizon.

“If one of the baby universes simply turns out to be the one we started with, then we get a model with an epoch of CTCs that is over by now, bounded toward the future by a Cauchy horizon.”

The divergence of the energy-momentum tensor as one approaches the Cauchy horizon does not necessarily occur, particularly when the Cauchy horizon crosses through a hot Big Bang phase where absorption occurs.

“We have argued that the divergence of the energy-momentum tensor as one approaches the Cauchy horizon does not necessarily occur, particularly when the Cauchy horizon crosses through a hot Big Bang phase where absorption occurs.”

• AssumptionRests on an assumption: The dense absorber can make the Cauchy horizon stable against vacuum polarization effects.

If in the recycling model one of the Λ1 bubbles formed inside an Λ2 bubble inside a Λ1 region is the Λ1 region that one started out with, then we would have a multiply connected model including a region of CTCs.

“Now take this recycling model where it turns out that one of the Λ 1 bubbles formed inside an Λ 2 bubble inside a Λ 1 region is, in fact, the Λ 1 region that one started out with.In this case, we would have a multiply connected model such as we are proposing which would include a region of CTCs (Fig. 8).”

• cites [36]Not matched

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

• AssumptionThe authors state this as an assumption: One of the created baby universes is the original universe that one started with.

This would make every black hole produce an infinite number of universes.

“This would make every black hole produce an infinite number of universes.”

If the zero-point-energy 1/2 were neglected (as Hartle and Hawking did), the left classically allowed region would shrink to a point.

“If the “zero-point-energy” 1/2 were neglected (as Hartle and Hawking did), the left classically allowed region would shrink to a point.”

Self-consistency (non-divergence of T_ab as CH is approached) requires retarded potentials in R and F.

“Self-consistency (non-divergence of T ab T ab as CH is approached) requires retarded potentials in R and F.”

Since any closed null geodesics generating the Cauchy horizon must circulate through the optically thick region of the hot Big Bang phase after inflation has stopped, the renormalized energy-momentum tensor should not blow up as the Cauchy horizon is approached.

“Since any closed null geodesics generating the Cauchy horizon must circulate through the optically thick region of the hot Big Bang phase of the universe after the inflation has stopped, the renormalized energy-momentum tensor should not blow up as the Cauchy horizon is approached.”

• AssumptionRests on an assumption: The early universe contained a region of CTCs.

Depending on the coordinate system, a bubble of Λ2 forming inside a Λ1 universe could also be seen as a Λ1 bubble forming inside a Λ2 universe.

“They point out that depending on the coordinate system, a bubble of Λ 2 forming inside a Λ 1 universe could also be seen as a Λ 1 bubble forming inside of a Λ 2 universe.”

• cites [36]Not matched

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

In the Garriga-Vilenkin recycling model, there is a metastable vacuum with cosmological constant Λ1 and a true lowest energy vacuum with cosmological constant Λ2, with Λ1 at the GUT or Planck scale and Λ2 taken to be the present value of Λ.

“In this model there is a metastable vacuum with cosmological constant Λ 1 , and a true lowest energy vacuum with a cosmological constant Λ 2 .Λ 1 is at the GUT or Planck energy scale, while Λ 2 is taken to be the present value of Λ (as might be the case in a flat-Λ model).”

• cites [36]Not matched

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

Since our multiply connected versions already obey the weak energy condition, the associated simply connected versions would also obey it.

“Since our multiply connected versions already obey the weak energy condition, so would the associated simply connected versions.”

The low-entropy boundary condition could occur on the Cauchy horizon that ends the period of CTCs.

“This could occur on the Cauchy horizon that ends the period of CTCs.”

The region of CTCs should be in a pure vacuum state containing no real particles or Hawking radiation and no bubbles.

“and should be in a pure vacuum state containing no real particles or Hawking radiation and no bubbles.”

At late times in the de Sitter phase a particle detector would find the usual Hawking radiation just as in the usual vacuum for de Sitter space.

“At late times in the de Sitter phase a particle detector would find the usual Hawking radiation just as in the usual vacuum for de Sitter space.”

In our model, the multiply connected nature of the spacetime geometry forces an arrow of time and retarded potentials.

“Interestingly, in our model, the multiply connected nature of the spacetime geometry forces an arrow of time and retarded potentials.”

There are a number of models in which baby universes are created which can be converted into models in which the Universe creates itself, if one of those created baby universes turns out to be the original universe that one started with.

“Thus, there are a number of models in which baby universes are created which can be converted into models in which the Universe creates itself, if one of those created baby universes turns out to be the original universe that one started with.”

• AssumptionThe authors state this as an assumption: One of the created baby universes is the original universe that one started with.

A wormhole solution connecting two mouths must violate the weak energy condition.

“To do this, the wormhole solution must violate the weak energy condition [7].It must have some negative energy density material, for it is a diverging lens (converging light rays entering one wormhole mouth, diverge upon exiting the other mouth).”

• cites [7]Not matched

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

If one found oneself in an expanding de Sitter phase, it would be much more likely that it was the result of classical bounce from a contracting de Sitter phase, rather than the result of a contracting de Sitter phase that had tunneled to an oscillating phase and then back out to an expanding de Sitter phase.

“In this case, if one found oneself in an expanding de Sitter phase, it would be much more likely that it was the result of classical bounce from a contracting de Sitter phase, rather than the result of a contracting de Sitter phase that had tunneled to an oscillating phase and then back out to an expanding de Sitter phase.”

Gott space represents how a multiply connected spacetime with CTCs (Grant space) can be converted into a simply connected spacetime with CTCs by adding material that obeys the weak energy condition.

“Thus, Gott space represents how a multiply connected spacetime with CTCs (Grant space) can be converted into a simply connected spacetime with CTCs by adding to the solution material that obeys the weak energy condition.”

Given quantum mechanics, such multiple connectivities would not be absolutely forbidden, particularly in the Planck foam era.

“Given quantum mechanics, it would seem that such multiple connectivities would not be absolutely forbidden, particularly in the Planck foam era.”

The potential makes the system behave like a harmonic oscillator in the potential well near a=0.

“The potential makes the system behave like a harmonic oscillator in the potential well near a = 0.”

If one of these baby universes simply turns out to be the universe we started with, then a multiply connected model with early CTCs bounded by a Cauchy horizon is produced.

“If one of these baby universes simply turns out to be the universe we started with, then a multiply connected model with early CTCs bounded by a Cauchy horizon is produced.”

• AssumptionRests on an assumption: The early universe contained a region of CTCs.
• AssumptionRests on an assumption: One of the baby universes produced by inflation turns out to be the universe we started with.

It is the very presence of the initial region of CTCs that produces the strong causality that we observe later on.

“Thus, it is the very presence of the initial region of CTCs that produces the strong causality that we observe later on.”

If one of those bubble universes is the original one in which the black hole formed, then the solution is multiply connected with a region of CTCs.

“If one of those bubble universes simply turns out to be the original one in which the black hole formed, then the solution is multiply connected with a region of CTCs.”

• AssumptionThe authors state this as an assumption: One of the created baby universes is the original universe that one started with.

A wormhole connection between two widely separated mouths in an asymptotically flat spacetime increases the genus by one.

“Consider an asymptotically flat spacetime with two connected wormhole mouths that are widely separated.The existence of the wormhole connection increases the genus by one.Instead of a flat plane, it becomes a flat plane with a handle.”

When the zero-point-energy is considered, the initial state is not a point but a tiny oscillating (0 ≤ a ≤ a1) Big Bang universe that oscillates between Big Bangs and Big Crunches.

“Thus, when the”

• Also stated in: § XI. CONCLUSIONS · plain

  “zero-point-energy”

In the early universe, the negative divergence of entropy may turn out to be an advantage, since to produce the ordinary entropy arrow of time we observe today, we must necessarily have some kind of natural low-entropy boundary condition in the early universe.

“Yet, in the early universe this may turn out to be an advantage, since to produce the ordinary entropy arrow of time we observe in the universe today, we must necessarily have some kind of natural low-entropy boundary condition in the early universe [58,59].”

• cites [58]Not matched

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

• cites [59]Not matched

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

If Λ1 = Λ2, this model is just the multiply connected de Sitter space we have considered.

“(If Λ 1 = Λ 2 , this model is just the multiply connected de Sitter space we have considered.)”

Borde and Vilenkin have shown that if the initial inflationary state is metastable, then it must have had a finite beginning.

“Inflation has solved some of these problems, but Borde and Vilenkin have shown that if the initial inflationary state is metastable, then it must have had a finite beginning also.”

• cites Borde and VilenkinNot matched

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

The classical instability problem of a future Cauchy horizon in a spacetime with CTCs is solved in a world with retarded potentials for a Cauchy horizon that occurs to our past and which ends an epoch of CTCs.

“But this problem is solved in a world with retarded potentials for a Cauchy horizon that occurs to our past (a past chronology horizon) and which ends an epoch of CTCs.”

Our multiply connected solutions might in principle be made simply connected by adding extra positive mass density without violating the weak energy condition.

“Our solutions are already multiply connected, so they might in principle be made simply connected by the addition of some extra positive mass density, without violating the weak energy condition.”

Inflation is particularly useful in this regard, because starting with a tiny piece of inflating state, at later times a huge volume of inflating state is produced, little pieces of which look just like the one we started with.

“Inflation is particularly useful in this regard, for starting with a tiny piece of inflating state, at later times a huge volume of inflating state is produced, little pieces of which look just like the one we started with.”

In the Hartle-Hawking no-boundary proposal with Wheeler-DeWitt potential and E = 0, the Universe should start not as nothing but as an S3 universe of radius zero, a point.

“Unfortunately, tunneling is, as the name suggests, usually a process that involves tunneling from one classical state to another, thus, with the Wheeler-DeWitt potential and”

• Also stated in: § X. BABY UNIVERSE MODELS · plain

  “energy”

• Also stated in: § XI. CONCLUSIONS · plain

  “E = 0 that Hartle and Hawking adopted, the Universe, we argue, should really start not as nothing but as an S 3 universe of radius zeroa point.”

• cites Hartle and HawkingNot matched

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

The smoothness of the potential at a = 0 may indicate that any Big Bang and Big Crunch singularities are removed by quantum theory.

“The smoothness of the potential at a = 0 may indicate that any Big Bang and Big Crunch singularities are removed by quantum theory.”

A harmonic oscillator cannot sit at the bottom of the potential well because the uncertainty principle would not allow it.

“A harmonic oscillator cannot sit at the bottom of the potential well -the uncertainty principle would not allow it.”

The adapted Rindler vacuum for Misner space is a self-consistent vacuum for this spacetime because it solves Einstein's equations for this geometry.

“thus, this is a self-consistent vacuum for this spacetime since it solves Einstein's equations for this geometry.”