A comment about the cosmology on a bubble wall Mehrdad Mirbabayi International Centre for Theoretical Physics Trieste Italy

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A comment about the cosmology on a bubble wall

Mehrdad Mirbabayi

International Centre for Theoretical Physics, Trieste, Italy

Abstract: The interface between a big bubble of true AdSDvacuum expanding inside a

false AdSDvacuum is a model of an inﬂating D−1 dimensional universe. It looks like an

interesting setup to study fundamentals of inﬂation. A recent computation shows that the

prediction of this model for the wavefunction of the universe disagrees with that of Hartle

and Hawking. We show that this discrepancy is because the eﬀective D−1 dimensional

description of the model is spin-0 (Nordstr¨om) gravity rather than spin-2 (Einstein) gravity.

1 Introduction and Conclusion

The Coleman-De Luccia (CDL) instanton is maximally symmetric [1]. In DEuclidean

dimensions, it is O(D) symmetric. We are particularly interested in the “thin wall” limit, in

which the two vacua are separated by a D−1-sphere. In Lorentzian signature, this becomes

aD−1-hyperboloid, which is the same as the dSD−1spacetime.

This hyperbolic wall could be interpolating two AdSDvacua, with inverse curvature

lengths k−> k+>0. In that case, if the wall tension is close to a critical value σc, the dS

curvature lengths `can be very large, k+`1. An interesting idea is to consider observers

(and matter ﬁelds) who are localized on such a bubble wall, and view this setup as a UV

completion for a D−1 dimensional de Sitter cosmology [2, 3, 4]. In D= 5, these authors

identify the eﬀective lower dimensional gravitational constant with

κ4=2k+k−

k−−k+

κ5,(1.1)

where κ5≡8πG5is the bulk gravitational constant. The eﬀective 4dFriedmann equation is

(in the limit k−−k+k+)

˙a2+ 1

a2=1

`2≈κ4

33k−−k+

κ5−σ.(1.2)

The critical tension is σc= 3(k−−k+)/κ5, above which the false vacuum cannot decay. At

least not via the CDL instanton.

arXiv:2210.14276v1 [hep-th] 25 Oct 2022

When decay is allowed, the CDL instanton looks as a higher dimension embedding of the

Hartle-Hawking (HH) no-boundary instanton [5]. The bubble-wall is a complex manifold

that can be thought of as the expanding phase of dS4glued to a 4dhemisphere at the

moment of reﬂection symmetry. Nevertheless, the regularized CDL instanton was found in

[4] to give for the leading exponent

|Ψ|2∼e−8π2`2/κ4,(1.3)

which has the opposite sign with respect to the HH exponent.

This discrepancy might appear insigniﬁcant given that Hartle and Hawking’s result is

a saddle point approximation to a path integral, which is anyway deﬁned up to an overall

normalization factor. However, the prediction is unambiguous if we are dealing instead of

a cosmological constant with an inﬂaton potential. In that case, HH wavefunction prefers

larger `, and as a result a shorter period of inﬂation to follow after the birth of the universe.

Let us therefore suppose that the two AdS5vacua belong to a richer multiﬁeld landscape

in which the wall tension can continuously and slowly vary. Then, there can be various

bubble sizes and the wavefunction (1.3) prefers inﬂation to begin with smaller curvature

length. So the disagreement with HH is real. The subsequent classical evolution is even

more puzzling since the inﬂaton potential, which we naturally expect to replace σ−σc,

appears with the wrong sign in the Friedmann equation (1.2): As the inﬂaton rolls down,

the expansion rate increases.

On the other hand, both of these signs are the correct ones in a 4dtheory with scalar-

mediated gravity rather than Einstein gravity. Let us deﬁne a metric gµν via

ds2=e2χ(−dτ2+dΩ2),(1.4)

where dΩ2is the metric of a 3-sphere. Consider the following action

S[χ, ψ] = −1

2κ4Zd4x√−gR +Sm[ψ, gµν ],(1.5)

where ψstands for all matter ﬁelds (conﬁned to 4d), and their action Smis deﬁned on the

manifold given by (1.4). The minus sign before the ﬁrst term ensures that high frequency

ﬂuctuations of χhave the correct sign kinetic term:

S=−3

κ4Zd4x√−ggµν ∂µχ∂νχ+··· (1.6)

The nonlinear equation of motion for χis the same as the trace of Einstein equation, but

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AcommentaboutthecosmologyonabubblewallMehrdadMirbabayiInternationalCentreforTheoreticalPhysics,Trieste,ItalyAbstract:TheinterfacebetweenabigbubbleoftrueAdSDvacuumexpandinginsideafalseAdSDvacuumisamodelofaninatingD1dimensionaluniverse.Itlookslikeaninterestingsetuptostudyfundamentalsofination.Arecentc...

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A comment about the cosmology on a bubble wall Mehrdad Mirbabayi International Centre for Theoretical Physics Trieste Italy

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