Inflation in AdS/CFT

TLDR: In this paper, it was shown that large enough bubbles of the dS phase, including those able to inflate, are described by mixed states in the CFT, and that no scattering process can create an inflating region, even by quantum tunneling, since a pure state can never evolve into a mixed state under unitary evolution.

Abstract: We study the AdS/CFT correspondence as a probe of inflation. We assume the existence of a string landscape containing at least one stable AdS vacuum and a (nearby) metastable de Sitter state. Standard arguments imply that the bulk physics in the vicinity of the AdS minimum is described by a boundary CFT. We argue that large enough bubbles of the dS phase, including those able to inflate, are described by mixed states in the CFT. Inflating degrees of freedom are traced over and do not appear explicitly in the boundary description. They nevertheless leave a distinct imprint on the mixed state. Analytic continuation allows us, in principle, to recover a large amount of nonperturbatively defined information about the inflating regime. Our work also shows that no scattering process can create an inflating region, even by quantum tunneling, since a pure state can never evolve into a mixed state under unitary evolution.

Figures

Fig. 3: Sketches of Penrose diagrams for (a) de Sitter, (b) Schwarzschild-AdS, and (c) de Sitter/Schwarzschild-AdS domain wall spacetimes, with constant-r surfaces indicated. The dashed vertical lines correspond to the points r = 0 in de Sitter, the dashed diagonal lines are horizons, the horizontal squiggly lines are the singularities, and the bold lines indicate the boundaries. The thick dotted lines indicate a possible trajectory of a shell across which the two spacetimes (a) and (b) may be patched together to obtain (c).

Fig. 13: Initial slices on a de Sitter Penrose diagram which are guaranteed to contain a piece of de Sitter I in their evolution. (The vertical lines are the poles r = 0 of de Sitter, the dashed diagonal lines are the cosmological horizons r = rd. Horizontal slices represent 3-spheres, with their equator along the vertical midpoint of the diagram.)

Fig. 10: Allowed and disallowed types of behavior of the de Sitter/SchwarzschildAdS domain wall at small r. The dotted curves depict the shell’s trajectory (on (a) de Sitter and (b) Schwarzschild-AdS Penrose diagrams) and the arrows the correspond to the outward-pointing normal. The checks indicate allowed scenario while the crosses label disallowed scenario. (a) In de Sitter, the shell starting from or ending at the left origin is allowed, whereas starting/ending on the right origin is not allowed. (b) In Schwarzschild-AdS, the shell starting towards the right from the past singularity or moving towards the left before hitting the future singularity is allowed, whereas the opposite behavior is not allowed.

Fig. 7: Sketch of Penrose diagrams obtained by continuous deformations of the initial data and parameters, starting with pure AdS and ending with de Sitter (including the future and past boundary I±) in Schwarzschild-AdS. Note that (b) is possible only for small black holes r+ < rA, and in (f) the de Sitter I is generically joined to the Schwarzschild-AdS singularities by some metric which depends sensitively on the evolution (and hence drawn by a dotted line) but is unimportant for our discussion.

Fig. 16: The domain wall trajectory on de Sitter and Schwarzschild-AdS Penrose diagrams which is necessitated by the type of set-up specified above: and initial slice Σ with monotonically increasing r, and a guarantee of dS I+ under evolution.

Fig. 17: Initial slice Σ with monotonically increasing r on a combined Penrose diagram (obtained by taking the left part of dS and right part of S-AdS in Fig. 16). The shaded region is uncertain under a reasonable evolution.

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Freivogel, B. and Hubeny, V. E. and Maloney, A. and Myers, R. C. and Rangamani, M. and Shenker, S.
(2006) 'Ination in AdS/CFT.', Journal of high energy physics., 2006 (03). 007.
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arXiv:hep-th/0510046v4 2 Mar 2006
Inflation in AdS/CFT
Ben Freivogel
a,b
, Veronika E. Hubeny
b,c
, Alexander Maloney
a,d
,
Robert C. Myers
e,f
, Mukund Rangamani
b,c
, and Stephen Shenker
a
a
Department of Physics, Stanford University, Stanford, CA 94305, USA
b
Department of Physics & Theoretical Physics Group, LBNL, Berkeley, CA 94720, USA
c
Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, UK
d
Stanford Linear Accelerator Center, Menlo Park, CA 94025, USA
e
Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5, Canada
f
Department of Physics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
Abstract
We study the AdS/CFT correspondence as a probe o f inflation. We assume the
existence of a string landscape containing at least one stable AdS vacuum and a (nearby)
metastable de Sitter state. Standard arguments imply that the bulk physics in the vicinity
of the AdS minimum is described by a boundary CFT. We a rgue that large enough bubbles
of the dS phase, including those able to inflate, are described by mixed states in the
CFT. Inflating degrees of freedom are traced over and do not appear explicitly in the
boundary description. They nevertheless leave a distinct imprint on the mixed state. In
the supergravity approximation, analytic continuation connects AdS/CFT correlators to
dS/CFT correlators. This provides a framework for extracting further information as well.
Our work al so shows that no scattering process can create an inflating region, even by
quantum tunneling, since a pure state can never evolve into a mixed state under unitary
evolution.
October 2005
freivogel@berkeley.edu, veronika.hubeny@durham.ac.uk, maloney@slac.stanford.edu,
rmyers@perimeterinstitute.ca, mukund.rangamani@durham.ac.uk, sshenker@stanford.edu
SU-ITP-05-27 UCB-PTH-05/30 LBNL-58913 DCPT-05/45 SLAC-PUB-11505

Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. Inflation in asymptotically AdS spacetimes . . . . . . . . . . . . . . . . . . . . 4
2.1. Thin domain wall constructions . . . . . . . . . . . . . . . . . . . . . . . 5
2.2. Beyond the thin wall approximation . . . . . . . . . . . . . . . . . . . . . 13
2.3. A special parameter domain . . . . . . . . . . . . . . . . . . . . . . . . . 15
3. Properties of the boundary CFT . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1. The entropy puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2. Mixed states in asymptotically Schwarzschild-AdS geometrie s . . . . . . . . . . 18
3.3. Conditions for the appearance of mixed states . . . . . . . . . . . . . . . . . 21
4. Probes of inflation in AdS/CFT . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.1. Geodesics probes of domain wall spacetimes . . . . . . . . . . . . . . . . . . 27
4.2. From AdS/CFT to dS/CFT and beyond . . . . . . . . . . . . . . . . . . . 30
5. Can inflation begin by tunneling? . . . . . . . . . . . . . . . . . . . . . . . . 32
6. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Appendix A. Details of the thin wall geometries . . . . . . . . . . . . . . . . . . . 34
A.1. Effective potential and extrinsic curvatures . . . . . . . . . . . . . . . . . . 34
A.2. Thin wall trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Appendix B. False vacuum bubbles in scalar-gravity systems . . . . . . . . . . . . . 39
Appendix C. Construction allowing de Sitter I and r
d
< r
+
. . . . . . . . . . . . . . 43
Appendix D. Computation of dS-SAdS Propagators . . . . . . . . . . . . . . . . . 46
Appendix E. A pure state description of spacetimes with causall y disconnected regions? . . 47
Appendix F. Analyti city in Coleman-de Lucci a spacetimes . . . . . . . . . . . . . . . 50
1. Introducti on
Our current understanding of the cosmological evolution of the universe relies on the
existence of an early period of inflation
1
. Recent data suggest that the universe is now
undergoing another period of inflation. It is of central importance t o understand this
remarkable phenomenon as deeply as p ossible.
String theory, which currently is t he only viable candidate for a theory of quantum
gravity, has had partial success in describing inflationary physics. In recent years there has
been dramatic progress in constructing string vacua with stabilized moduli and positive
and negative cosmological constants [4,5,6,7,8,9,10,11]. These lead to de Sitter and Anti-
de Sitter (AdS) cosmologies, respectively. There are an enormous number of such vacua,
populating what is now called the “string landscape” [12]. A small piece of the la ndscape,
containing an AdS and a neighboring de Sitter vacuum, is sketched in Fig. 1. The richness
of the landscape allows us to view the parameters of such a potential as essentially free
parameters.
1
For reviews see [1,2,3] .
1

1
φ
V
φ
Fig. 1: A typical scalar pot ential appearing in string theory, with de Sitter and
Anti-de Sitter minima.
These constructions do not answer many of the deep questions raised by inflation:
How can inflatio n begin? What measure should be used o n the multiverse of eternal
inflation? What is the holographic description of inflation? More g enerally, what degrees
of freedom are appropriate for a complete description of quantum gravity in this domain?
A substantial amount of work has been done on these topics [13,1 4 ,15,16,17,18, 19]. We
will not be able to answer these questions in this paper, but we will try to make some
progress by embedding inflation in our best understood and most powerful framework for
understanding quantum gravity, the AdS/CFT correspondence
2
[21,22,23 ,24].
Consider a stable supersymmetric A dS ground state, say the one indicated in Fig. 1.
The bulk correlators taken to the AdS boundary define a conformal field theory (CFT),
which encodes the bulk dynamics precisely. Given that small fluctuations around the
AdS minimum are captured by the CFT, it seems plausible that classical configurations
correspo nding to the excursions to the neighboring de Sitter minimum should be encoded in
the CFT somehow. For instance, correlat ors of the scalar field φ describing the horizontal
axis o f Fig. 1 should enable one to reconstruct the effective potential
3
. If we can construct
a region of space where φ is displaced from the AdS minimum to the dS minimum
4
, the
behavior o f such a bubble of false vacuum might probe some inflationary physics.
The fate of such bubbles has been investigated extensively while exploring the possi-
bility of “creating a universe in a laboratory” [25,26,27]. We will discuss these results in
more detail later, but for now we will just summarize the main points. Observed from the
boundary of AdS (where t he dual CFT is located), all such bubbles collapse into a black
2
The first work on this connection is [20].
3
Of course, a CFT that captures aspects of the landscape as a whole must be a com plicated
object indeed.
4
The finite energy excitations described by the AdS/CFT correspondence require that φ
approach the AdS minimum at the AdS boundary.
2

hole. If the bubble was large enough initially, an inflating region forms, but i t is b ehind
the black hole horizon. At first gla nce this seems discouraging. The standard AdS/CFT
observables are only sensitive to physics outside the horizon. But in recent years tools
have been developed, based largely on analyticity, to examine physics behind the horizon
in AdS/CFT [28,29,3 0,31]. We will discuss here how these tools enable us to observe the
inflating region.
A basic obstruction to making a universe in a laboratory classically was noted by
the authors of [26]. They argued using singularity theorems in general relativity that
an inflating region must classically always begin in a singularity. But AdS/CFT quite
comfortably describes geometries with both future and past singularities, like the eternal
Schwarzschild-AdS black hole [32,33]. So the observations of [26] should not prevent us
from studying inflation in AdS/CFT.
A more general worry about representing inflation in AdS/CFT is that the boundary
CFT must encode a very large number of degrees of freedom describing the inflating
region. Specifically, the authors of [34,35 ] have pointed out that in certain situat ions the
dS ent ropy of the inflating region is larger than the entropy of the Schwarzschild-AdS black
hole. Notions of black hole complementarity and holography suggest t hat this would be
hard to accommodate.
In our picture this puzzle is resolved in a simple way. We present arguments that the
geometries created by large bubbles of false vacuum must be represented as mixed states
in the boundary CFT. The large number of degrees of freedom in the region behind the
horizon are entangled with the degrees of freedom outside the horizon, as in the Hartle-
Hawking stat e representation of the eternal Schwarzschild-AdS bla ck hole
5
[36,32,33 ]. The
degrees of freedom behind the horizon are not explicitly represented; they are traced over.
They can be weakly entangled, though, so that even tracing over a large number of them
can yield a density matrix wi th entropy compatible wi th the black hole entropy.
We do not expect the degrees of freedom behind the horizon to be fully represented by
a CFT, and do not know how to calculate the full density matrix beyond the supergravity
approximation. If we do know the density matrix, a large amount of informati on about the
degrees of freedom that have been traced over can be extracted, again by using analyticity.
As an example, in the eternal Schwarzschild-AdS black hole, boundary operators on the
right hand boundary can be moved to the left hand boundary by continuing in complex
time. In our situation, boundary operators on the ri ght hand AdS boundary can be moved
by a suitable continuatio n in complex time to the de Sitt er boundary at future (or past)
infinity. The resulting correlators living on the boundary of de Sitter have t he form that
5
As we review later, the eternal Schwarzschild-AdS black hole is described as a pure entangled
state in the Hilb ert space of two copies of the boundary CFT, each living on a separate AdS
boundary. Traci ng over one of these Hilbert spaces leads to a thermal density matrix in the other.
3