FRW/CFT duality: A holographic framework for eternal inflation

Yasuhiro Sekino (KEK)

Based on collaborations with:L. Susskind (Stanford), S. Shenker (Stanford), B. Freivogel (Amsterdam),

R. Bousso (UC Berkeley), I.-S. Yang (Columbia), C.-P. Yeh (NTU).

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Outline

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Part I: Phases of eternal inflation• R. Bousso, B. Freivogel, YS, S. Shenker, L. Susskind,

I.-S. Yang, C.-P. Yeh, PRD78 (2008) 063538;• YS, S. Shenker, L. Susskind, PRD81 (2010) 123515.

Part II: Holographic description (FRW/CFT duality)• B. Freivogel, L. Susskind, YS, C.-P. Yeh, PRD 74, 086003 (2006); • YS, L. Susskind, PRD80 (2009) 083531;

A simplified model for the landscape

• Gravity is coupled to a scalar field which has a false and a true vacuum

V(ΦF)>0, V(ΦT)=0

(True vacuum: our universe)

What happens when gravity is coupled to a theory with metastable vacuum?

• If we ignore gravity, first order phase transition:– Bubbles of true vacuum form, and the whole space

turns into true vacuum.

• When there is gravity, inflation in the false vacuum leads to qualitatively different picture.

Φ

V(Φ)

ΦF ΦT

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Motivation

• Construct a non-perturbative formulation – So far, studies of landscape relies on low-energy

effective theory. (We don’t know how to do string theory on de Sitter space.)

– We will need holographic formulation (like AdS/CFT, which defines string theory on AdS by super-Yang-Mills).

• Find observational consequences of landscape

With these goals in mind, we will study the physics in the (simplified) landscape. 4

Bubble of true vacuum

Described by Coleman-De Luccia instanton(Euclidean “bounce” solution)

• SO(4) symmetric solution, which interpolates between true and false vacua.

• Euclidean geometry: Deformed S4

(Euclidean de Sitter is S4 )• Nucleation rate:

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Φ

V(Φ)

ΦF ΦT

Spacetime with one bubble

Lorentzian geometry: • Given by analytic continuation• The whole geometry has SO(3,1)

Causal structure: right figure• Bubble wall is constantly accelerated.• Open FRW universe inside the bubble

(shaded region). Spatial slice (dotted lines):H3. Isometry: SO(3,1).

• Beginning of FRW universe: non-singular.(similar to Rindler horizon)

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• A single bubble does not cover the whole space. (Fills only the horizon volume.)

• Many bubbles form in de Sitter region with therate (constant per unit physical 4-volume).

• But if , bubble nucleation cannot catch up the expansion of space, and the false vacuum exists forever:“Eternal Inflation” [Guth, Linde, Vilenkin, …]

• There are many confusing issues in eternal inflation, such as the “measure problem.” 7

Eternal inflation

Part I: Phases of eternal inflation

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Outline of Part I

• There are three phases of eternal inflation, depending on the nucleation rate.

• Phases are characterized by the existence of percolating structures (lines, sheets) of bubbles in de Sitter space. (First proposed by Winitzki, ’01.)

• The geometry in the true vacuum region is qualitatively different depending on the phase.

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View from the future infinity

• Consider conformal future of de Sitter. (future infinity in comoving coordinates)

• A bubble: represented as a sphere cut out from de Sitter.

• “Scale invariant” distribution of bubbles Bubbles nucleated earlier:

appear larger: radius are rarer: volume of nucleation sites

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Model for eternal inflation

• Mandelbrot model (Fractal percolation)– Start from a white cell.

(White: inflating, black: bubble,Cell: One horizon volume)

– Divide the cell into cellswith half its linear size. (The space grows by a factor of 2.Time step: )

– Paint each cell in black with probability P.(P ~ = nucleation rate per horizon volume: constant)

– Subdivide the surviving (white) cells, and paint cells in black w/ probability P. Repeat this infinite times.

Picture of the 2D version

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Properties of Mandelbrot model

• If P > 1 - (1/2)3 =7/8, the whole space turns black, since(the rate of survival)(the rate of branching) = (1-P)23 < 1(No eternal inflation)

• If P < 7/8, white region is not an empty set (It is a fractal). Non-zero fractal dimension dF (rate of growth of # of cells):

Physical volume of de Sitter region (# of cells) grows. (Eternal inflation) [Fractals in eternal inflation: noted by Vilenkin, Winitzki, ..]

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Three phases of eternal inflation

Phase structure of the 3D Mandelbrot model is known. [Chayes et al, Probability Theory and Related Fields 90 (1991) 291]

In order of increasing P (or Γ), there are (white=false vacuum(inflating), black=bubble(non-inflating))• Black island phase: Black regions form isolated clusters;

percolating white sheets.• Tubular phase: Both regions form tubular network;

percolating black and white lines.• White island phase: White regions are isolated;

percolating black sheets.13

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Courtesy of Vitaly Vanchurin

Geometry of the true vacuum region

• Mandelbrot model: the picture of the de Sitter side. (de Sitter region outside the light cone of the nucleation site is not affected by the bubble.)

• To find the spacetime structure of the non-inflating region inside (the cluster of) bubbles, we need to understand the dynamics of bubble collisions.

• Exact analysis is difficult, but the intuition gained from simple examples of bubble collisions is helpful.

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Black island phase (isolated cluster of bubbles)

Small deformations of open FRW universe.• Basic fact: Collision of two bubbles (of the

same vacuum) does not destroy the bubbles.Bubbles merge into one smooth bubble, whose spatial slice approach H3.

• This was shown by solving junction condition after the collision. [c.f. Bousso, Freivogel, Yang, ‘07]

• Even if many bubbles collide, local geometry near the collision will be similar to the two-bubble case, so the geometry will approach smooth H3.

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dS dS dS

flat flat

• Symmetry for the two-bubble collision: SO(2,1)

– De Sitter space: hyperboloid in

– Bubbles: planes at and at

– Two bubbles collide at spacelike H2

• Parametrization of de Sitter:

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• Parametrization of flat space:

• World volume of domain wall:

• Energy on the domain wall decays at late time.(Negative curvature of H2 dilutes the energy.)

• The spatial geometry approach smooth H3

• In black island phase, local geometry near a collision will be similar to the two-bubble case, and the space will be smoothed at late times.

dS dS dS

flat flat

Tubular phase (tube-like structure of bubbles)

Spatial slice at late time will be a negatively curved space which has a boundary with high genus.

• True vacuum with a boundary w/ non-trivial genus: torus case [Bousso, Freivogel, YS, Shenker, Susskind, Yang, Yeh, ‘08]

– Ring-like initial configuration of bubbles.– If the hole in the middle is larger than horizon

size, it cannot shrink.

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– Solve a sequence of junction conditions

– Spacetime becomes smooth at late time.

– Spatial slice: negatively curved space with toroidal boundary. Constructed by modding out H3 by its discrete isometry.

– Space is bulk of a doughnut. Infinite distance to get to the surface. Non-contractible circle with finite length in the bulk.

– Negatively curved space with arbitrary genus can be constructed similarly.

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0 0 01 1 1 1

2 2 23 3

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An observer in the true vacuum is “surrounded” by inflating region (contrary to the intuition from Mandelbrot model).

• Simple case: two white islands (with S2 symmetry) [Kodama et al ’82, BFSSSYY ’08]

– An observer can see only one boundary; the other boundary is behind the black hole horizon. [c.f. “non-traversability of a wormhole”]

White island phase (isolated inflating region)

Global slicing (S3 ) of de Sitter Penrose diagram

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• In the white island phase, a white region will split.• Also when there are more than two white islands,

singularity and horizon will form so that the boundaries are causally disconnected.

– Late time geometry for the three white island case: [Kodama et al ’82]

W

WW

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Summary of Part I

Three phases of eternal inflation and their geometry:• Black island phase:

Small deformation of an open FRW• Tubular phase:

Negatively curved space with a high genus boundary.(Bubble collision provides a natural mechanism for producing non-trivial topology.)

• White island phase: Observer sees only one boundary and one or more black hole horizons (behind which there are other boundaries).

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Part II: Holographic duality

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FRW/CFT duality [Freivogel, Sekino, Susskind, Yeh, ’06]

Proposal: Universe created by bubble nucleation is

described by a conformal field theory (CFT)on S2 at the boundary of H3.

• SO(3,1): 2D conformal symmetry

Guiding principle: Dual theory should be constructed using the d.o.f. which

can be seen by a single observer (“Census Taker”).(Different from “dS/CFT correspondence”.)

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Dual theory has two less dimensions than the bulk.

• The dual theory contains 2D gravity. – Liouville field (2D scale factor) plays

the role of time.• Late time: corresponds to the 2D theory

with finer cutoff.• One bulk field corresponds to a tower of CFT

operators.

We can see these from the study of bulk correlators.

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Calculation of correlation functions[See also Garriga-Montes-Sasaki-Tanaka, ‘98, ’99]

• Hartle-Hawking prescription:Analytic continuation from Euclidean

• Essentially a 1D scattering problem:

• Euclidean correlator: Written in terms of the reflection coefficient .

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Correlator in open FRW• Analytic continuation:

(R: Geodesic distance on )

• 1st term: “Flat space piece” (Minkowski correlator written in hyperbolic slicing)

• 2nd term (which depends on ): Effect of the false vacuum.

• Bound state: non-normalizable mode on H3

Decomposition in terms of scaling dimensions

• Correlation function can be written as discrete sum (over contributions from the poles)

(geodesic distance: )

• Each term has definite scaling dimension Δ. (Δ = 2,3,4,…; and 0, 1+α (0<α<1) for massless

fields)

• This is an expansion valid near the boundary R -> ∞. 29

Interpretation in the dual theory

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• One bulk field corresponds to a tower of CFT operators.

• “UV-IR relation” (by identifying T= L). – e-R: “reference” (length) scale– e-(T+R): UV cutoff (lattice) scale

• Late time corresponds to dual theory with finer cutoff. (At late time, Census Taker sees more and more stuff.)

• The prefactors of the correlator is interpreted as wave function renormalization.

Graviton correlator

• Graviton correlator remains finite at R -> ∞ (Graviton correlator has non-normalizable mode).– 2D Curvature correlator:

– Gravity is not decoupled at the boundary (unlike in AdS/CFT): Dual theory contains gravity.

• ∆=2 piece of graviton is transverse traceless along S2: – Identified as energy-momentum tensor of 2D CFT.

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Summary of Part II

• Holographic dual for universe created by bubble nucleation: 2D CFT on S2 at the boundary of H3

– Liouville plays the role of time.– One bulk field corresponds to a tower of CFT operators.– Evidence: dimension 2 energy-momentum tensor.– There is a well-defined way of calculating 3-point

functions. This will test OPE, and tell us central charge(which is of order de Sitter entropy).

• Implication of results of Part I: Need to sum over topology of 2D space, but need not to consider multiple boundaries.(Phase transitions: instability of 2D gravity?)

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Side remark: CMB spectrum

[See also Yamauchi et al, Phys.Rev. D84 (2011) 043513]

• Bubble nucleation predicts negative spatial curvature.– We are looking at distance at most 0.1 Rc

(Rc: curvature radius).

• Slow-roll inflation after tunneling:– Most likely,

(HI: Hubble in slow-roll inflation; HA: in false vacuum)– Just after the tunneling, there is a period of curvature

domination: 33

Evolution during curvature domination

• Tunneling gives an initial condition for slow-roll inflation.

– 1st term: the term that is usually considered.– 2nd term: effect of the ancestor vacuum.

• We want to find the 2nd term in the CMB fluctuations, assuming the duration of slow-roll inflation is minimal.

• The 1st term: order (agree with the standard estimate).Could the 2nd term be larger than ?

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Classification in terms of scaling dimensions

• Useful for the study of time dependence.

• Close the contour in the lower half plane,and write the correlator as a sum.– Pole at gives– The uppermost pole (smallest b): largest contribution– In particular, non-normalizable mode (b<0), if exists, is

larger than (fluctuations generated after tunneling)• The angular spectrum:

(Exponentially peaked at low l)

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Prospects for observation

• Curvature perturbation (scalar mode):– Small effect (at least in the single-field model)

• Tensor fluctuations (gravitons):– Effect on the B-mode polarization will be small.

• Extra fields (isocurvature perturbation):– There could be large effect if we have field with small

mass in the ancestor vacuum (e.g. axions), in principle.We don’t have clear example of large effect of the ancestor.(Maybe we have to think of the measure problem seriously

to find evidence for the landscape.)

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