1. A Holographic Framework for Eternal Inflation Yasuhiro Sekino (Okayama Institute for Quantum Physics) Collaboration with Ben Freivogel (UC Berkeley), Leonard Susskind and Chen-Pin Yeh (Stanford), PR D74, 086003 (2006), hep-th/0606204.

2. Landscape of string vacua According to the “Landscape” picture, –Vacuum energy is a function of “moduli” fields –Many de Sitter (positive c.c.) minima –De Sitter spaces are unstable Problems: –No rigorous framework (Landscape idea is not proven) –Unsolved problem: probability for each vacuum Purpose of this work: We propose a holographic dual description (of a universe created by the decay of de Sitter space)

3. Decay of de Sitter space Simplified Landscape: single scalar, two minima (spacetime: 3+1 D) U(  F ) >0 (de Sitter vacuum), U(  T )=0 (zero c.c. in the true vacuum) Coleman-De Luccia (CDL) instanton (’82): –Classical solution with Euclidean signature –Topologically, a 4-sphere. Interpolates two cc’s. SO(4) symmetric.

4. Lorentzian continuation of the CDL instanton Penrose diagram A piece of flat space patched with a piece of de Sitter space (green curve: domain wall): Future half of the diagram is physical. (“Bubble of flat space is nucleated, and expands”)

5. Region I: –Open (k=-1) FRW universe –Constant time slices (blue lines) are 3-hyperboloids. The whole geometry has SO(3,1): –In FRW, generates spatial translation. True picture: –Infinitely many bubbles will form in de Sitter region. (“eternal inflation”) –Bubble collision cannot be ignored in general. (comment at the end.)

6. In the rest of the talk: We propose a holographic dual description for this FRW universe created by the CDL instanton. What the dual theory should be Correlation functions on the CDL background Properties of the dual theory Summary and remaining questions

7. Our proposal: The dual theory is a CFT on S 2 which contains gravity. (The S 2 is at the “boundary” of H 3.) Dual theory has the same symmetry as bulk geometry –SO(3,1): the conformal group in 2D. (Note that hyperboloid = Euclidean AdS) Our dual theory has 2 less dimensions than the bulk. –Time is represented by a dynamical field (Liouville field of the 2D gravity) This is different from the “dS/CFT correspondence” (which states that the dual is a 3D CFT at future infinity)

8. “Dynamical” gravity at the boundary Peculiar property of the CDL (in contrast to AdS): –Even though the FRW universe has infinite volume, the Euclidean geometry is compact. Quantization in an infinite space is usually done with a fixed boundary condition. Here, this is not appropriate. –Correlation fn (continued from Euclidean space) does not decay at infinite spatial separation. In other words, the boundary condition (especially, the boundary geometry) is dynamical. –This suggests that the dual CFT contains gravity. (c.f. “dual of brane world”, “dS/dS correspondence”)

9. Notion of time in “closed” system Time cannot be regarded as a parameter in a closed universe, since there is no “clock.” General formalism: Wheeler-DeWitt theory –Wave function (fn of 3-geometry and matter) is time- independent, because of the diffeo. invariance. –Usual procedure: In the large volume limit, treat the scale factor semi-classically. It plays the role of time. The dual theory: “holographic Wheeler-DeWitt theory” –Describes 3-space holographically. –In the large volume limit, Liouville ~ time. –Liouville will be time-like (coupled to large # of matter).

10. Calculation of the correlation functions We calculate 2-point fn in FRW region –Bring the points to the boundary, and interpret the bulk correlator (in the late time limit) as the CFT correlator. –Want to find energy-momentum tensor of CFT (traceless, conserved, dim=2) from the graviton correlator. Method of calculation: Obtain correlator in Euclidean, and continue it to Lorentzian (Hartle-Hawking prescription). –We study massless scalar, as an illustration for graviton. We find “non-normalizable mode” (correlations at infinitely separated points) in graviton correlator. (For scalar, noted by Sasaki et al.)

11. Coordinate system

12. Correlator in Euclidean space

14. Subtlety of the massless Green’s function Massless Green’s fn on S 3 doesn’t really exist: We can’t solve (We cannot have a source in compact space.) Constant shift of massless field is a “gauge symmetry” –Need to take derivs to get a physical quantity We define massless Green’s fn as a limit of massive Green’s fn (discarding an infinite constant). –Physical quantity has a smooth massless limit.

15. For the thin wall example, The Euclidean correlator: The integrand has poles at k=2i, 3i, … (single poles) k=i (double pole)

16. Lorentzian correlator Analytic continuation to Region I: Contour of integration for the Lorentzian correlator:

17. A bulk field corresponds to an (infinite) tower of CFT operators A pole corresponds to an operator w/ definite dimension. (Poles in the upper half plane contribute in the late limit.) The terms from (a) (single poles at k=2i, 3i, …) These are of the form of CFT correlators (dim  =3,4,…)

18. “Non-normalizable mode” Contribution from the k=i double pole: Gauge inv correlator: remains finite near the boundary Graviton case: 2D scalar curvature (along the sphere) –Boundary geometry is fluctuating. This piece should correspond to Liouville d.o.f. in CFT.

19. Dimension 2 piece (energy momentum tensor) Sub-leading term from k=i double pole : Interpretation: degeneracy of dimensions. –If the bound state pole shifts slightly, we get dim 2 piece and dim (2+  ) piece with opposite signs. Graviton case: The piece which remains at dim 2 is transverse-traceless in S 2. –corresponds to energy-momentum tensor of the CFT

20. Properties of the CFT Central charge: from a scaling argument, c ~ area of de Sitter (before the decay) in Planck unit Late time limit: Matter will be decoupled from Liouville. i.e. matter central charge should approach constant –to reproduce the holographic entropy in the bulk, using UV/IR relation for cutoff. Bubble collision: represented by instanton in the CFT. –On the boundary, it is a localized region in a different vacuum.

21. Conclusions From the calculation of correlation functions, we found –A bulk field corresponds to infinite # of CFT operators. –Candidate for the energy-momentum tensor exists. Work in progress: –Thick wall case (extra operators in CFT) –Higher dim case (CFT w/ higher derivative gravity) –3 pt function, and operator algebra –Observational consequence of non-normalizable mode Open question –Realization in string theory?