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A Holographic Framework for Eternal Inflation Yasuhiro Sekino

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A Holographic Framework for Eternal Inflation Yasuhiro Sekino On the Universe Created through Quantum Tunnelling Yasuhiro Sekino (Okayama Institute for Quantum Physics) Collaboration with Ben Freivogel (UC Berkeley), Leonard Susskind (Stanford), Chen-Pin Yeh (葉振斌; Stanford), hep-th/0606204 + work in progress. Motivation for this work: • “String Landscape” – String theory seems to have a large # of vacua with positive cosmological constant. – Creation of universe through tunneling (bubble nucleation) is quite common. Two important Problems: • Construct a non-perturbative framework – Presence of the Landscape has not been proven. – In fact, it is not known what a positive c.c. vacuum means in string theory (beyond low-energy level). • Find observational consequence of an universe created by tunnelling – What is the signature in the CMB ? – Can we have information on the “ancestor” vacuum? In this talk, • We study fluctuations a universe created by tunnelling. – Our analysis is based on the semi-classical gravity and QFT in curved space-time. • Main result: there is a peculiar long-range correlation. – This will play crucial roles in the above two problems. Plan of the talk: • Coleman-De Luccia instanton (decay of de Sitter space) • Correlation functions • Holographic dual description Decay of de Sitter space • We consider simplest case: single scalar, two minima (spacetime: 3+1 D) U(ΦF) >0 (de Sitter vacuum), U(ΦT)=0 (zero c.c. in the true vacuum) • Coleman-De Luccia (CDL) instanton (’82): – Euclidean classical solution – Topologically, a 4-sphere. Interpolates two cc’s. Preserves SO(4) out of SO(5). In the thin-wall limit Lorentzian geometry • Bubble of true vacuum is nucleated and then expands. (future half of the diagram is physical) Penrose diagram • A piece of flat space patched with a piece of de Sitter space (green curve: domain wall): I Σ IV III V II Open universe inside the bubble • Region I: open (k=-1) FRW universe – Constant time slice (blue lines): 3-hyperboloid Σ I – Symmetry: SO(3,1) IV – No singularity at the beginning of the FRW universe (red line) III • Here, we ignore V II – Non-zero final cosmological const. – Slow-roll inflation in region I. • Eternal inflation: – Infinite # of bubbles will form. – They will collide and form clusters. (We consider one bubble in this talk) Calculation of the correlation function • 2-pt functions of the linearized fluctuations (massless scalar, and TT mode of gravitons) • Study correlators in the FRW region. (We need the whole geometry to define the state). • Obtain correlator in Euclidean space, and analytically continue it to Lorentzian (Hartle-Hawking prescription). • We find that the massless correlator does not decay in the limit of infinite spatial separation. (Subtleties of masslessness in compact space, hyperbolic geometry) Coordinate system I Σ IV III V II Correlator in Euclidean space 1 X Subtlety of the massless Green’s function • Massless Green’s fn on S3 doesn’t really exist: We can’t solve (We cannot have a source in compact space.) • We define massless Green’s fn as a limit (k Æ i) of massive Green’s fn (discarding an infinite constant). • Physical quantity has a smooth massless limit. – Constant shift of massless field is a “gauge symmetry”: We have to take derivs of correlator w.r.t. the two points. For the thin wall example, The Euclidean correlator: k The integrand has poles at k=2i, 3i, … : single poles, from the normalization of k=i: double pole Lorentzian correlator • Analytic continuation to Region I: k “Non-normalizable mode” • The correlator does not decay at infinite separation • Gauge invariant correlator • Graviton case: 2D scalar curvature (along the sphere) – Boundary geometry is dynamical • NN mode for scalar: found by M. Sasaki and T. Tanaka; For graviton: argued to be absent (Sasaki et al, Turok et al). Interpretation of non-normalizable mode • We cannot throw away the NN mode. If we do so, becomes singular at (beginning of the FRW). – Without NN mode, (1-loop expectation value) diverges, but with NN mode, it doesn’t. • Presence of the NN mode is not strange from the viewpoint of de Sitter space. – This is of the same origin as the well-known super- horizon fluctuations in de Sitter. (Fluctuations at the domain wall, in the late time limit.) From the viewpoint of observer in FRW • To study physics in the open universe, we should fix the boundary condition at spatial infinity. – The correlation function gives probability distribution for boundary conditions (classical deformations of the background). – We find it interesting that probabilities for different universes can be obtained in a simple well-defined way. • Excitation of NN mode will introduce a novel kind of anisotropy in the CMB. (work in progress) Holographic dual theory Our proposal: I Σ Open universe with zero c.c. created by IV the CDL instanton is dual to CFT on S2 which contains gravity. III (The S2 is at the “boundary” Σ of H3.) V • Symmetry: SO(3,1)=2D conformal group II • The Dual has 2 less dims than the bulk. – Time is represented by a dynamical field (Liouville field of the 2D gravity). The dual theory contains gravity • This is natural since the boundary geometry is dynamical. • How to describe time-dependent physics? Remember the Wheeler-DeWitt theory: – Wave function is time-independent (due to diffeo inv). – In the large volume limit, the scale factor is treated 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). What the bulk correlator tells us: • One bulk field corresponds to a tower of CFT operators: (a) k • From single poles at k=2i, 3i, … we get dim ∆=3,4,… operators: (b) k • From a double pole at k=i we get dim ∆=0 and 2 operators. – Dim 2 piece for graviton is transverse- traceless on S2 . Can be interpreted as the CFT energy-momentum tensor. Conclusions • In an open universe created by the CDL instanton, – Boundary condition for massless field (especially, graviton) at the spatial infinity is dynamical. – Correlation fn gives the probability distribution for classical deformations of the universe. – This is a source of anisotropy of the universe. • Holographic dual description: – CFT on S2 (at spatial infinity) which contains gravity – Evidence: CFT energy-momentum tensor exists. – Matter sector of the CFT not identified. .
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