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International Journal of Modern Physics DVol. 17, Nos. 3 & 4 (2008) 551–555c© World Scientific Publishing Company

CHILD UNIVERSE UV REGULARIZATION?∗

E. I. GUENDELMAN

Physics Department,Ben-Gurion University of the Negev,

Beer-Sheva 84105, Israelguendel@bgu.ac.il

Received 29 November 2007Communicated by D. V. Ahluwalia-Khalilova

It is argued that high energy density excitations, responsible for UV divergences inquantum field theories, including quantum gravity, are likely to be the source of childuniverses which carry them out of the original space–time. This decoupling preventsthe high UV excitations from having any influence on physical amplitudes. Child uni-verse production could therefore be responsible for UV regularization in quantum fieldtheories which take into account gravitational effects. Finally, we discuss child universeproduction in the last stages of black hole evaporation, the prediction of the absenceof trans-Planckian primordial perturbations, the connection with the minimum lengthhypothesis, and in particular the connection with the maximal curvature hypothesis.

Keywords: Cosmology; quantum gravity; child universes; regularization of ultravioletdivergences.

PACS Number(s): 04.60.-m, 04.70.-s, 11.27.+d

1. Introduction

Quantum field theory, and quantum gravity in particular, suffer from UV diver-gences. While some quantum field theories are of the renormalizable type, quantumgravity is not and the UV divergences cannot be hidden in a finite number of“counterterms.” Perturbative renormalizability does not appear to be available forquantum gravity.

In an apparently unrelated development, the “child universe” solutions havebeen studied.a,b These child universes are regions of space that evolve in such a waythat they disconnect from the ambient space–time. Inflationary bubbles of a falsevacuum correspond to this definition.1,2 In this case an exponentially expanding

∗This essay received an “honorable mention” in the 2007 Essay Competition of the GravityResearch Foundation.aSee for example, Ref. 1.bFor a review and more references, see Ref. 2.

551

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552 E. I. Guendelman

inflationary bubble arises from an ambient space–time with zero pressure which thefalse vacuum cannot displace. The inflationary bubbles disconnect from the ambientspace, generating a child universe.

Here we want to explore the possibility that high energy density excitations,associated with the dangerous UV sector of quantum field theory, are the sourceof child universes, which will carry the UV excitations out of the original ambientspace–time. Child universe production could be therefore responsible for UV soft-ening in quantum field theory that takes into account gravitational effects. It alsoimplies the existence of a maximum energy density and curvature.

We will now show, using a simple model, that very high UV excitations have anappreciable tendency to disconnect from the ambient space–time.

2. The Superhigh UV Bubble

We describe now the model which we will use to describe a high UV excitationwhich will be associated with the production of a child universe. This model for ahigh UV excitation consists of a bubble with very high surface tension and a veryhigh value of bulk energy density inside the bubble.

The entire space–time region consists of two regions and a boundary: (1) RegionI — de Sitter space; (2) Region II — Schwarzschild space and the domain wallboundary separating regions I and II.

Region I: The line element is given by

ds2 = −(1− χ2r2)dt2 + (1− χ2r2)−1dr2 + r2dΩ2, (1)

where χ is the Hubble constant, which is given by

χ2 =83πGρ0, (2)

ρ0 being the vacuum energy density of the child universe and G = 1m2

P, where

mP = 1019 GeV.Region II: The Schwarzschild line element is given by

ds2 = −(

1− 2GM

r

)dt2 +

(1− 2GM

r

)−1

dr2 + r2dΩ2. (3)

The Einstein field equations,

Rµν − 12gµνR = 8πGTµν , (4)

are satisfied in regions I and II and determine also the domain wall evolution,1 usingthe methods developed by Israel.3 Using Gaussian normal coordinates, which assignto any point in space three coordinates on the bubble, consider then a geodesicnormal to the bubble which reaches any given point after a distance η (the sign of

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Child Universe UV Regularization? 553

η depends upon which side of the bubble the point is found on). Then the energy–momentum tensor Tµν is given by

Tµν(x) = −ρ0gµν (η < 0) for the child universe, Region I, (negative pressure),

= 0 (η > 0) for the Schwarzschild region, Region II,

= −σhµνδ(η) for the domain wall boundary. (5)

where σ is the surface tension and hµν is the metric tensor of the wall, i.e. hµν =gµν − nµnν , with nµ being the normal to the wall.

The equation of motion of the wall gives1

Mcr =1

2Gχ

γ3z6m

(1− 1

4γ2

) 12

3√

3(z6m − 1)

32

, (6)

where Mcr is the mass at (or above) which there is classically a bubble that expandsto infinity into a disconnected space, the child universe. In the above equation

γ =8πGσ√

χ2 + 16π2G2σ2,

z3m =

12

√8 +

(1− 1

2γ2

)2

− 12

(1− 1

2γ2

),

(7)

where z3 = χ2+r3

2GM and χ2+ = χ2 + κ2, κ = 4πGσ. rm is the location of the maximum

of the potential barrier that prevents bubbles with mass less than Mcr from turninginto child universes.

We expect this representation of a high UV excitation to be relevant even fora purely gravitational excitation, which can be associated, after an appropriateaveraging procedure, with an effective energy–momentum, a procedure that getsmore and more accurate in the UV limit.

Let us now focus our attention on the limit where σ → ∞ (while ρ0 is fixed),which we use as our first model of a super UV excitation. We see that γ → 2 andMcr → 0. Alternatively, we could obtain another model of a super UV excitation byconsidering the energy density inside the bubble, ρ0 → ∞, while keeping σ fixed.This leads to Mcr → 0 as well. Finally, letting both σ → ∞ and ρ0 → ∞ whilekeeping their ratio fixed also leads to Mcr → 0. In all these limits we also get theradius of the critical bubble rm → 0.

In Ref. 1 the above expression for Mcr was explored for a case where energydensity scales (bulk and surface) were much smaller than the Planck scale, like theGUT scale. This gave a value for Mcr = 56 kg À mP . Here we take the alternativeview that the scale of the excitations is much higher than the Planck scale, givingnow an arbitrarily small critical mass. Defining the “scale of the excitation” throughρ0 ≡ M4

exc, the prefactor 12Gχ in Eq. (6) goes like ( mp

Mexc)2mP . We see that for

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554 E. I. Guendelman

trans-Planckian excitations, i.e. if Mexc À mP , we obtain a very big reduction forMcr. This is a kind of “seesaw mechanism,” since the higher the Mexc, the smallerthe Mcr.

This means that in these models for high UV excitations there is no barrier tothe high UV excitation being carried out to a disconnected space by the creationof a child universe. Notice also the interesting “UV–IR mixing” that takes placehere: although we go to very high UV limits in the sense that the energy densityin the bulk or the surface energy density is very high, the overall critical mass goesto zero.

We should note that these limits, where we take the surface tension or the energydensity to very big values, can be achieved as we go to early times (corresponding tothe time of the creation of the child universes) in models where these quantities aredynamical variables. In this context,4 when considering for example models withdynamical tension, one can show the existence of child universe production wherethe critical mass Mcr is indeed zero.

3. The Conjecture

This allows us to formulate the conjecture that the dangerous UV excitations thatare the source of the infinities and the nonrenormalizability of quantum gravity aretaken out of the original space by child universe production, i.e. the considerationof child universe production in the ultrahigh (trans-Planckian) sector of the theorycould result in a finite quantum gravity, since the superhigh UV modes, after sep-arating from the original space, will not be able to contribute anymore to physicalprocesses.

The hope is that in this way child universes could not only be of interest incosmology but could also become an element essential for the consistency of quan-tum gravity. One situation where all the elements required (high energy densities,as the temperature is very big) are necessary for obtaining a child universe appearsto be the late stages of black hole evaporation. If the ideas explained here are cor-rect, we should not get contributions to primordial density perturbations from thetrans-Planckian sector, since these perturbations would have disconnected from ourspace-time. Any attempt to measure distances smaller than the Planck length will,according to this, also be impossible since such a measurement will involve excit-ing a high UV excitation that will disconnect. This means that there must be aminimum length that we could measure, of the order of the Planck scale.

It appears there is a maximal energy density according to this, since now bubbleswith high energy density will be quickly disconnected, being replaced in the observ-able universe by regions of Schwarzschild space, which has zero energy density, i.e.a very big energy density must decay in the observable universe. The “maximalcurvature”5–8 hypothesis (here we focus on scalar curvature) is justified by thismaximal energy density result, if we use Eq. (4). An effective dynamics that takesinto account the effect of child universe production (i.e. integrates out this effect)

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Child Universe UV Regularization? 555

could resemble indeed that of Refs. 5–8. Notice that the maximal scalar curvaturehypothesis gives rise to very interesting dynamics (see for e.g. Ref. 9).

Acknowledgments

I thank S. Ansoldi for many discussions, encouragement and collaboration on fur-ther developments related to the subject matter of this paper (discussed in Ref. 4).I also thank G. Nayak for some discussions and D. Easson for discussions relatedto Refs. 5–9.

References

1. S. K. Blau, E. I. Guendelman and A. H. Guth, Phys. Rev. D 35 (1987) 1747.2. S. Ansoldi and E. I. Guendelman, gr-qc/0611034.3. W. Israel, Nuovo Cimento B 44, S10 (1966) 1 [Erratum, ibid B 48 (1967) 463; Nuovo

Cimento B 44 (1966) 1].4. S. Ansoldi and E. I. Guendelman, gr-qc/0706.1233.5. M. Markov, Pis’ma Zh. Eksp. Theor. Fiz. 36 (1982) 214.6. M. Markov, Pis’ma Zh. Eksp. Theor. Fiz. 46 (1987) 342.7. V. P. Frolov, M. A. Markov and V. F. Mukhanov, Phys. Rev. D 41 (1990) 383.8. V. F. Mukhanov and R. Brandenberger, Phys. Rev. Lett. 68 (1992) 1969.9. D. A. Easson, J. High Energy Phys. 0302 (2003) 037.