A Smooth Exit from Eternal Inflation?
S. W. Hawking1 and Thomas Hertog2
1
2

DAMTP, CMS, Wilberforce Road, CB3 0WA Cambridge, UK

Institute for Theoretical Physics, University of Leuven, 3001 Leuven, Belgium

Abstract

arXiv:1707.07702v2 [hep-th] 4 Mar 2018

The usual theory of inflation breaks down in eternal inflation. We derive a dual description of
eternal inflation in terms of a deformed CFT located at the threshold of eternal inflation. The
partition function gives the amplitude of different geometries of the threshold surface in the HartleHawking state. Its local and global behavior in dual toy models shows that the amplitude is low for
surfaces which are not nearly conformal to the round three-sphere and essentially zero for surfaces
with negative curvature. Based on this we conjecture that the exit from eternal inflation does not
produce an infinite fractal-like multiverse, but is finite and reasonably smooth.

1

 I.

INTRODUCTION

Eternal inflation [1] is a near de Sitter regime deep into the phase of inflation in which the
quantum fluctuations in the energy density of the inflaton are large. In the usual account of
eternal inflation the quantum diffusion dynamics of the fluctuations is modeled as stochastic
effects around a classical slow roll background. Since the stochastic effects dominate the
classical slow roll it is argued eternal inflation produces universes that are globally highly
irregular, with exceedingly large or infinite constant density surfaces [2–5].
However this account is questionable, because the dynamics of eternal inflation wipes out
the separation into classical backgrounds and quantum fluctuations that is assumed. We
therefore put forward a different model of eternal inflation based on gauge-gravity duality.
A reliable theory of eternal inflation is important to sharpen the predictions of slow roll
inflation. This is because the dynamics of eternal inflation specifies the prior over the zero
modes, or slow roll backgrounds, in the theory which in turn determines its predictions for
the precise spectral properties of CMB fluctuations on observable scales.
We work in the no-boundary quantum state [6]. This gives the ground state and is
heavily biased towards universes with a low amount of inflation [7]. However we do not
observe the entire universe. Instead our observations are limited to a small patch mostly
along part of our past light cone. Probabilities for local observations in the no-boundary
state are weighted by the volume of a surface Σf of constant measured density, to account
for the different possible locations of our past light cone [8]. This transforms the probability
distribution for the amount of inflation and leads to the prediction that our universe emerged
from a regime of eternal inflation [8, 9]. Thus we must understand eternal inflation in order
to understand the observational implications of the no-boundary wave function.
The standard saddle point approximation of the no-boundary wave function breaks down
in eternal inflation. However, gauge-gravity duality or dS/CFT [10–12] enables an alternative form of the wave function on a surface Σf in the large three-volume limit in which
the wave function is specified in terms of the partition function of certain deformations of a
Euclidean CFT defined on Σf . Euclidean AdS/CFT generalized to complex relevant deformations implies an approximate realisation of this [13–18]. This follows from the observation
[17] that all no-boundary saddle points in low energy gravity theories with a positive scalar
potential V admit a geometric representation in which their weighting is fully specified by

2

 FIG. 1: Two representations in the complex time-plane of the same no-boundary saddle point
associated with an inflationary universe. The saddle point action includes an integral over time τ
from the no-boundary origin or South Pole (SP) to its endpoint υ on Σf . Different contours for
this give different geometric representations of the saddle point, each giving the same amplitude
for the final real configuration (hij (~x), φ(~x)) on Σf . The interior saddle point geometry along the
nearly vertical contour going upwards from the SP consists of a regular, Euclidean, locally AdS
domain wall with a complex scalar profile. Its regularized action specifies the tree-level probability
in the no-boundary state of the associated inflationary, asymptotically de Sitter history. Euclidean
AdS/CFT relates this to the partition function of a dual field theory yielding (1.1).

an interior, locally AdS, domain wall region governed by an effective negative scalar potential −V . We illustrate this in Fig. 1. Quantum cosmology thus lends support to the view
that Euclidean AdS/CFT and dS/CFT are two real domains of a single complexified theory
[10, 13, 19–21]. In the large three-volume limit this has led to the following proposal for a
holographic form of the semiclassical no-boundary wave function [17],
−1
ΨN B [hij , φ] = ZQF
T [h̃ij , α̃] exp(iSst [hij , φ]/~) .

(1.1)

Here the sources (h̃ij , α̃) are conformally related to the argument (hij , φ) of the wave function,
Sst are the usual surface terms, and ZQF T in this form of dS/CFT are partition functions of
deformations of Euclidean AdS/CFT duals. The boundary metric h̃ij stands for background
and fluctuations.
The holographic form (1.1) has led to a fruitful and promising application of holographic
techniques to early universe cosmology (see e.g. [14, 22–26]). No field theories have been
identified that correspond to top-down models of realistic cosmologies where inflation transitions to a decelerating phase. However it turns out that many of the known AdS/CFT
duals are ideally suited to study eternal inflation from a holographic viewpoint. This is
3

 2
and
because supergravity theories in AdS4 typically contain scalars of mass m2 = −2lAdS

with a negative exponential potential for large φ. This gives rise to eternal inflation in the
dS domain of the theory governed effectively by −V . In fact the Breitenlohner-Freedman
bound in AdS corresponds precisely to the condition for slow roll eternal inflation in dS.
In this paper we consider toy-model cosmologies of this kind in which a single bulk scalar
drives slow roll eternal inflation. Conventional wisdom based on semiclassical gravity asserts
that surfaces of constant scalar field become highly irregular on the largest scales. We study
this from a holographic perspective with the dual defined on a global constant density surface
Σf in the far future. The holographic form (1.1) of the wave function replaces the bulk scalar
by a source α̃ that turns on a low dimension scalar operator in the dual. The dependence of
the partition function on the geometry hij of the future conformal boundary in the presence
of a constant source α̃ 6= 0 then specifies a holographic measure on the global structure
of constant density surfaces in eternal inflation in the large volume regime. In this dual
framework, the separate bulk background and fluctuation degrees of freedom in traditional
approaches to eternal inflation are replaced by field theory degrees of freedom on a future
‘end-of-the-world’ brane. As such the field theory automatically includes the backreaction
effect of the fluctuations on the classical background1 .
The F–theorem and its extensions imply that the holographic amplitude of the round S 3
is a local maximum of the distribution, in contrast with expectations based on semiclassical
gravity. Here we compute the global holographic probability distribution for small and large
anisotropic deformations (squashings) of the future boundary in dual vector models. We
find that the amplitude of surfaces with conformal structures far from the round one is
exponentially small. Finally we put forward a general argument that indicates that the
amplitude is zero for all highly deformed conformal boundaries with a negative Yamabe
invariant. This raises doubt about the widespread idea that eternal inflation produces
a highly irregular universe with a mosaic structure of bubble like patches separated by
inflationary domains.
1

In a complete model of cosmology Σf should be thought of as a surface at the threshold of eternal
inflation. However it is beyond the current state of holographic cosmology to study the transition from
eternal inflation to classical cosmology.

4

 II.

A HOLOGRAPHIC MEASURE ON ETERNAL INFLATION

As an example of a bulk gravity model that leads to eternal inflation in the dS domain
of the theory we consider the well known consistent truncation of M-theory on AdS4 × S 7
down to gravity coupled to a single scalar φ with potential
√
V (φ) = −2 − cosh( 2φ) ,

(2.1)

2
in units where Λ = −3 and hence lAdS
= 1. The scalar has mass m2 = −2. Therefore in the

large three-volume regime it behaves as
φ(~x, r) = α(~x)e−r + β(~x)e−2r + · · ·
where r is the overall radial coordinate in Euclidean AdS, with scale factor er .

(2.2)
The

Fefferman-Graham expansion implies that in terms of the variable r the asymptotically
(Lorentzian) dS domain of the theory is to be found along the vertical line τ = r + iπ/2 in
the complex τ -plane [17]. This is illustrated in Fig. 1 where r changes from real to imaginary
values along the horizontal branch of the AdS contour from xA to xT P . This also means
that in the dS domain the potential (2.1) in the original Euclidean action of the AdS theory
acts as a positive effective potential
√
Ṽ (φ) = −V = 2 + cosh( 2φ) .

(2.3)

This is a potential for which the eternal inflation condition   ≤ Ṽ holds, with   ≡ Ṽ,φ2 /Ṽ 2 ,
for field values in the neighbourhood of its minimum. This close connection between AdS
supergravity truncations and eternal inflation stems from the fact that the BreitenlohnerFreedman stability bound on the mass of scalars in AdS corresponds precisely to the condition for eternal inflation in the de Sitter domain of the theory.
The variance of the wave function of inhomogeneous fluctuation modes is of order ∼ Ṽ / 
evaluated at horizon crossing. Hence the fluctuation wave function is broadly distributed
[5]. This is the hallmark of eternal inflation in quantum cosmology. It implies the universe’s
evolution is governed by the quantum diffusion dynamics of the fluctuations and their backreaction on the geometry rather than the classical slow roll [2–5]. As a consequence it is
usually argued the total wave function spreads out and loses its semiclassical WKB form.
The argument (hij , φ) of the wave function evaluated at υ in Fig. 1 is real. This means
that in saddle points associated with inflationary universes, the scalar field must become
5

 real along the vertical dS line in the τ -plane. The expansion (2.2) shows this requires its
leading coefficient α to be imaginary, which in turn means that the scalar profile is complex
along the entire interior AdS domain wall part of the saddle points. But the bulk scalar
sources a deformation by an operator O of dimension one with coupling α in the dual ABJM
theory. Hence the holographic measure on the ensemble of eternally inflating universes in
this model is specified by its partition function on deformed three-spheres in the presence of
an imaginary mass deformation. We first recall the general behavior of partition functions
for small perturbations away from the round S 3 and then we turn to large deformations in
a specific toy model.

A.

Local measure: perturbations around S 3

Locally around the round sphere, the F–theorem and its extension to spin-2 deformations
provide a general argument that the round sphere is a local minimum of the partition
function. The F–theorem for three-dimensional CFTs [27, 28] states that the free energy of
a CFT on S 3 decreases along an RG flow triggered by a relevant deformation. A similar
result was recently proved for metric perturbations of the conformal S 3 background [29, 30].
The coupling of the energy-momentum tensor of the CFT to the curved background metric
triggers a spin-2 deformation. The fact that the free energy is a local maximum for the
round sphere is essentially equivalent to the positive definiteness of the stress tensor twopoint function. Applied to the holographic no-boundary wave function (1.1) these results
imply that the pure de Sitter history in the bulk is a local maximum of the holographic
probability distribution, in contrast with expectations based on semiclassical bulk gravity
in eternal inflation.

B.

Global measure: squashed three-spheres

We now turn to large deformations. Our bulk model is a consistent truncation of Mtheory compactified on AdS4 × S 7 . Therefore the dual is the ABJM SCFT, and we are
faced with the problem of evaluating the partition function of supersymmetry breaking
deformations of this theory. We do not attempt this here. Instead we focus on a simplified
model of this setup where we consider an O(N ) vector model. This is conjectured to be

6

 dual to higher-spin Vassiliev gravity in four dimensions [31]. Higher-spin theories are very
different from Einstein gravity. However, ample evidence indicates that the behavior of the
free energy of vector models qualitatively captures that of duals to Einstein gravity when
one restricts to scalar, vector or spin 2 deformations [32–34]. This includes a remarkable
qualitative agreement of the relation between the vev and the source for the particular scalar
potential (2.1) [36]. We therefore view these vector models as dual toy models of eternal
inflation and proceed to evaluate their partition functions.
Specifically we consider the O(N ) vector model on squashed deformations of the threesphere,
r2
ds = 0
4
2

 
(σ1 )2 +

1
1
(σ2 )2 +
(σ3 )2
1+A
1+B

 
,

(2.4)

where r0 is an overall scale and σi , with i = 1, 2, 3, are the left-invariant one-forms of SU (2).
Note that the Ricci scalar R(A, B) < 0 for large squashings [33]. We further turn on a mass
deformation O with coupling α. This is a relevant deformation which in our dual O(N )
vector toy model induces a flow from the free to the critical O(N ) model. The coefficient α
is imaginary in the dS domain of the wave function as discussed above. Hence we are led to
evaluate the partition function, or free energy, of the critical O(N ) model as a function of
the squashing parameters A and B and an imaginary mass deformation α ≡ m̃2 . The key
question of interest is whether or not the resulting holographic measure (1.1) favors large
deformations as semiclassical gravity would lead one to believe.
The deformed critical O(N ) model is obtained from a double trace deformation f (φ ·
φ)2 /(2N ) of the free model with an additional source ρf m̃2 turned on for the single trace
operator O ≡ (φ · φ). By taking f → ∞ the theory flows from its unstable UV fixed point,
where the source has dimension one, to its critical fixed point with a source of dimension
two [31]. To see this we write the mass deformed free model partition function as
Z
R 3 √ 2
2
Zfree [m ] = Dφe−Ifree + d x gm O(x) ,
where Ifree is the action of the free O(N ) model
 
 
Z
1
1
3 √
a
µ a
Ifree =
d x g ∂µ φa ∂ φ + Rφa φ .
2
8

(2.5)

(2.6)

Here φa is an N -component field transforming as a vector under O(N ) rotations and R
is the Ricci scalar of the squashed boundary geometry. Introducing an auxiliary variable
7

 m̃2 =

m2
ρf

+

O
ρ

yields
Z

2

Zfree [m ] =

DφDm̃2 e−Ifree +N

R

√
1
d3 x g [ρf m̃2 O− f2 O2 − 2f
(m2 −ρf m̃2 )2 ]

R

√
d3 x g(m2 −ρf m̃2 )2

,

(2.7)

which can be written as
Z

2

Zfree [m ] =

N

Dm̃2 e− 2f

Zcrit [m̃2 ] ,

(2.8)

with
2

Z

Zcrit [m̃ ] =

Dφe−Ifree +N

R

√
d3 x g [ρf m̃2 O− f2 O2 ]

.

Inverting (2.8) gives Zcrit as a function of Zfree :
Z
 
R
√   4
N f ρ2 R 3 √
4
−ρm̃2 m2
N d3 x g m
2
d
x
g
m̃
2
2f
Zcrit [m̃ ] = e 2
Dm e
Zfree [m2 ] .

(2.9)

(2.10)

The value of ρ can be determined by comparing two point functions in the bulk with those
in the boundary theory [34]. For the O(N ) model this implies ρ = 1, which agrees with the
transformation from critical to free in [35].
We compute Zcrit for a single squashing A 6= 0 and m̃2 6= 0 by first calculating the
partition function of the free mass deformed O(N ) vector model on a squashed sphere and
then evaluate (2.10) in a large N saddle point approximation2 . Evaluating the Gaussian
integral in (2.5) amounts to computing the following determinant
"
#!
−∇2 + m2 + R8
N
− log Zfree = F =
log det
,
2
Λ2

(2.11)

where Λ is a cutoff that we use to regularize the UV divergences in this theory. The
eigenvalues of the operator in (2.11) can be found in closed analytic form [37],
λn,q = n2 + A(n − 1 − 2q)2 −

1
+ m2 ,
4(1 + A)

q = 0, 1, . . . , n − 1, n = 1, 2, . . .

(2.12)

To regularize the infinite sum in (2.11) we follow [33, 34] and use a heat-kernel type
regularization. Using a heat-kernel the sum over eigenvalues divides in a UV and an IR
part. The latter converges and can readily be done numerically. By contrast the former
contains all the divergences and should be treated with care. We regularize this numerically
2

The generalization to double squashings A, B 6= 0 yields qualitatively similar results but requires extensive
numerical work and is discussed in [36].

8

 by verifying how the sum over high energy modes changes when we vary the energy cutoff.
From a numerical fit we then deduce its non-divergent part which we add to the sum over
the low energy modes to give the total renormalized free energy. The resulting determinant
after heat-kernel regularization captures all modes with energies lower than the cutoff Λ.
The contribution of modes with eigenvalues above the cutoff is exponentially small. For
more details on this procedure we refer to [33, 36].
Imm2
0.6

Rem2
0.5

0.4

- 0.4

- 0.2

0.2

0.4


im2

0.2

- 0.4

- 0.2

- 0.5

0.2

0.4


im2

- 0.2
- 0.4
- 0.6

FIG. 2: The real and imaginary parts of the solutions m2 of the saddle point equation (2.13) are
shown for three different values of a single squashing, i.e. A = −0.8 (blue), A = 0 (red) and
A = 2.06 (green). For large im̃2 we have Re(m2 ) → −R/8.

To evaluate the holographic measure we must substitute our result for Zfree [A, m2 ] in
(2.10) and compute the integral in a large N saddle point approximation. The factor outside
the path integral in (2.10) diverges in the large f limit. We cancel this by adding the
appropriate counterterms. The saddle point equation then becomes
  2
 
m
2π 2
∂ log Zfree [m2 ]
2
p
− m̃ = −
.
∂m2
(1 + A)(1 + B) f

(2.13)

We are interested in imaginary m̃2 as discussed above. This means we need Zfree [A, m2 ]
for complex deformations m2 . Numerically inverting (2.13) in the large f limit we find a
saddle point relation m2 (m̃2 ). This is shown in Fig. 2, where the real and imaginary parts
of m2 are plotted as a function of im̃2 for three different values of A.
Notice that Re(m2 ) ≥ −R(A)/8. This reflects the fact that the determinant (2.11), which
is a product over all eigenvalues of the operator −∇2 +m2 +R/8, vanishes when the operator
has a zero eigenvalue. Since the lowest eigenvalue of the Laplacian ∇2 is always zero, the
first eigenvalue λ1 of the operator in (2.11) is zero when R/8 + m2 = 0. In the region of
configuration space where the operator has one or more negative eigenvalues the Gaussian
9

 FIG. 3: The holographic probability distribution in a dual toy model of eternal inflation as a
function of the coupling of the mass deformation m̃2 that is dual to the bulk scalar, and the
squashing A of the future boundary that parameterizes the amount of asymptotic anisotropy. The
distribution is smooth and normalizable over the entire configuration space and suppresses strongly
anisotropic future boundaries.

integral (2.5) does not converge, and (2.11) does not apply. This in turn means that the
−1
holographic measure Zcrit
[A, m̃2 ] is zero on such boundary configurations, as we now see.

Inserting the relation m2 (m̃2 ) in (2.10) yields the partition function Zcrit [A, m̃2 ]. We
show the resulting two-dimensional holographic measure in Fig. 3. The distribution is well
behaved and normalizable with a global maximum at zero squashing and zero deformation
corresponding to the pure de Sitter history, in agreement with the F–theorem and its spin2 extensions. When the scalar is turned on the local maximum shifts slightly towards
positive values of A. However the total probability of highly deformed boundary geometries
is exponentially small as anticipated3 . We illustrate this in Fig. 4 where we plot two onedimensional slices of the distribution for two different values of m̃2 .
3

The distribution has an exponentially small tail in the region of configuration space where the Ricci scalar
R(A) is negative and Zfree diverges. We attribute this to our saddle point approximation of (2.10).

10

 1

1

Z 2
2.0

Z 2
0.12
0.10

1.5

0.08
1.0

0.06
0.04

0.5
0.02
0

5

10

15

20

25

30

A

0

10

20

30

40

A

FIG. 4: Two slices of the probability distribution for m̃2 = 0.0 (left) and m̃2 = 0.05i (right).
C.

Global measure: general metric deformations

It is beyond the current state-of-the-art to evaluate partition functions in vector models
for more general large metric deformations. However the above calculation suggests that
the amplitude of such deformations is highly suppressed in the holographic measure.
This is because the action of any dual CFT includes a conformal coupling term of the
form Rφ2 . For geometries that are close to the round sphere this is positive and prevents the
partition function from diverging [39]. On the other hand the same argument suggests that
the conformal coupling likely causes the partition function to diverge on boundary geometries
that are far from the round conformal structure. These include in particular geometries with
patches of negative curvature or, more accurately, a negative Yamabe invariant4 .

4

The Yamabe invariant Y (h̃) is a property of conformal classes. It is essentially the infimum of the total
scalar curvature in the conformal class of h̃, normalized with respect to the overall volume. It is defined
as
Y (h̃) ≡ inf ω I(ω 1/4 h̃)
(2.14)
where the infimum is taken over conformal transformations ω(x) and I(ω h̃) is the normalized average
scalar curvature of ω 1/4 h̃,
 p
R   2
2
ω
R(
h̃)
+
8(∂ω)
h̃ d3 x
M
1/4
I(ω h̃) =
(2.15)
 R
 1/3
p
6 h̃ d3 x
ω
M
There always exists a conformal transformation ω(x) such that the metric h̃0 = ω 1/4 h̃ has constant scalar
curvature [40]. The infimum defining Y is obtained for this metric h̃0 .

11

 The Yamabe invariant is negative in conformal classes with a metric of constant R < 0.
Since the lowest eigenvalue of the conformal Laplacian is negative on such backgrounds this
means that the partition function of a CFT does not converge, thereby strongly suppressing
the amplitude of such conformal classes in the measure (1.1). Our result for the holographic
measure specified by the partition function of the deformed O(N ) model on squashed spheres
provides a specific example of this, as discussed above.
Conformal classes with negative Y (h̃) include the highly irregular constant density surfaces from eternal inflation featuring in a semiclassical gravity analysis. This general argument therefore suggests their amplitude will be low in a holographic measure. We interpret
this as evidence against the idea that eternal inflation typically leads to a highly irregular
universe with a mosaic structure of bubble like patches separated by inflationary domains5 .

III.

DISCUSSION

We have used gauge-gravity duality to describe the quantum dynamics of eternal inflation
in the no-boundary state in terms of a dual field theory defined on a global constant density
surface in the large volume limit. Working with the semiclassical form (1.1) of dS/CFT the
field theories are Euclidean AdS/CFT duals deformed by a low dimension scalar operator
that is sourced by the bulk scalar driving eternal inflation.
The inverse of the partition function specifies the amplitude of different shapes of the
conformal boundary surface. This yields a holographic measure on the global structure of
eternally inflating universes. We have computed this explicitly in a toy model consisting of
a mass deformed interacting O(N) vector theory defined on squashed spheres. In this model
we find that the amplitude is low for geometries far from the round conformal structure.
These include surfaces with negative scalar curvature which, we have argued, are strongly
suppressed in a holographic measure in general. Based on this we conjecture that eternal
inflation produces universes that are relatively regular on the largest scales. This is radically different from the usual picture of eternal inflation arising from a semiclassical gravity
5

This resonates with [9] where we argued that probabilities for local observations in eternal inflation can
be obtained by coarse-graining over the large-scale fluctuations associated with eternal inflation, thereby
effectively restoring smoothness. Our holographic analysis suggests that the dual description implements
some of this coarse-graining automatically.

12

 treatment.
Our conjecture strengthens the intuition that holographic cosmology implies a significant
reduction of the multiverse to a much more limited set of possible universes. This has
important implications for anthropic reasoning. In a significantly constrained multiverse
discrete parameters are determined by the theory. Anthropic arguments apply only to a
subset of continuously varying parameters, such as the amount of slow roll inflation.
However, the setup we have considered does not allow us to describe the transition from
the quantum realm of eternal inflation to a universe in the semiclassical gravity domain. This
is because our duals are defined in the UV and live at future infinity. It therefore remains
an open question whether the conjectured smoothness of global constant density surfaces
impacts the eternity of eternal inflation6 . To answer this will require a significant extension
of holographic cosmology to more realistic cosmologies. It has been suggested that in such
models, inflation corresponds to an IR fixed point of the theory [22]. The detailed exit from
inflation will then be encoded in the coupling between the field theory degrees of freedom
and the bulk dynamics in a manner somewhat analogous to the holographic description of
vacuum decay in AdS [41].
Acknowledgments: We thank Dio Anninos, Nikolay Bobev, Frederik Denef, Jim Hartle, Kostas Skenderis and Yannick Vreys for stimulating discussions over many years, and
we thank the referees of PRL for useful correspondence. SWH thanks the Institute for
Theoretical Physics in Leuven for its hospitality. TH thanks Trinity College and the CTC
in Cambridge for their hospitality. This work is supported in part by the ERC grant no.
ERC-2013-CoG 616732 HoloQosmos.

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