Physics in the Universe Created by Bubble Nucleation

Yasuhiro Sekino (Okayama Institute for Quantum Physics)

Collaboration with Ben Freivogel (UC Berkeley), Leonard Susskind and Chen-Pin Yeh (Stanford), hep-th/0606204 + work in progress.

Subject of this talk:

• Universe created by bubble nucleation

• Motivation: String Landscape

– String theory seems to havea large # of vacua with positive cosmological constant.

– Creation of universe throughtunneling (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 me

ans in string theory (beyond low-energy level).

• Find observational consequence of an universe created by bubble nucleation– What is the signature on the CMB ?– Can we have information on the “ancestor” vacuum?

In this talk,

• We study fluctuations in such a universe on the basis of semi-classical gravity and QFT in curved space-time.

• We find a peculiar long-range correlation of fluctuations.– 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 theory

Decay of de Sitter space

• For simplicity, we consider: 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): – Euclidean classical solution– Topologically, a 4-sphere.

Interpolates two cc’s. SO(4) symmetric. In the thin-wall limit

Lorentzian geometry

Penrose diagram• A piece of flat space patched

with a piece of de Sitter space (green curve: domain wall):

• Bubble of true vacuum is nucleated and then expands. (future half of the diagram is physical)

Open universe inside the bubble

• Region I: open (k=-1) FRW universe– Constant time slice (blue lines):

3-hyperboloid– Symmetry: SO(3,1) – No singularity at the beginning of th

e FRW universe (red line)

• To get the real universe, we need– Slow-roll inflation in region I.– Non-zero final cosmological const.

• 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 geometry, 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

Correlator in Euclidean space

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.)

• 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.

For the thin wall example,

The Euclidean correlator:

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:

“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

• If we throw away the NN mode, becomes singular at (beginning of the FRW universe).

– Without NN mode, diverges. – This suggests that the NN mode is part of the theory.

• From the viewpoint of the de Sitter, this is a well-known super-horizon fluctuations. (Fluctuations at the domain wall, in the late time limit.)

From the viewpoint of observer in FRW

• NN mode (boundary condition at spatial infinity) should be frozen.

– The correlation function

gives probability distribution for boundary conditions.

• Excitation of NN mode (deformation of the boundary condition) will introduce a novel kind of anisotropy in the CMB. (work in progress)

Holographic dual theory

Our proposal:

Open universe with zero c.c. created by the CDL instanton is dual to CFT on S2 which contains to gravity. (The S2 is at the “boundary” of H3.)

• Symmetry: SO(3,1)=2D conformal group

• 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.

• Time-dependence: from the relation of Liouville and other fields. Remember in 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:

• From single poles at k=2i, 3i, …we get dim =3,4,… operators:

• 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, gra

viton) at the spatial infinity is dynamical.– Correlation fn gives the probability distribution of the b

oundary condition.– 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.