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.