Implications of string theory
for cosmology
Yasuhiro Sekino(KEK)
at KMI, Nagoya University
May 19, 20111

String theory• Theory of quantum gravity: Important at Planck scale– Should also have implications for physics at lower scales.
e.g., Why is our universe flat?
– There was inflation in the early universe.
– Quantum fluctuations during inflation is observed as
temperature fluctuations of CMB.
• Difficult to make precise predictions for cosmology– String theory is formulated in a limited class ofbackground; Have to rely on low energy effective theory.
2Topics of this talk
• Bubble nucleation: Spectrum of fluctuations,
holographic duality
w/ B. Freivogel, L. Susskind, C.‐P. Yeh, hep‐th/0606204;
w/ L. Susskind, arXiv:0908.3844 [hep‐th]• Bubble collisions, phases of eternal inflation
w/ R. Bousso, B. Freivogel, S. Shenker, L. Susskind, I.‐S. Yang,
C.‐P. Yeh, arXiv:0807.1947 [hep‐th] ;w/ S. Shenker, L. Susskind, arXiv:1003.1347[hep‐th]• CMB fluctuations from quantum effects of many fields:
w/ Y. Habara, H. Kawai, M. Ninomiya, arXiv:1103.0299 [hep‐th]3

“String landscape”
String theory has meta‐stable de Sitter vacua (V>0)(not only stable vacua with V=0)
V(Φ)• To understand physics, consider one
false vacuum and one true vacuum
V(ΦF)>0, V(ΦT)=0
ΦF(true vacuum: our universe)• If we ignore gravity, first order phase transition:ΦΦT– Nucleation of bubbles of true vacuum (Callan, Coleman, …)– The whole space eventually turns into true vacuum.

• What happens in a theory with gravity?

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False vacuum: de Sitter space• de Sitter space: hyperboloid in

Hubble parameter (expansion rate)• Causal structure: Two points separated
by > H‐1 (horizon) are causally disconnected.
• Inflation in false vacuum leads to a completely
different picture than the case without gravity.

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Bubble of true vacuum
Described by Coleman‐De Luccia instanton(Euclidean “bounce” solution).
Euclidean geometry:• Interpolates between true and false vacua• Deformed S4 (Euclidean de Sitter is S4 )• Rotationally symmetric: SO(4)• Nucleation rate:

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Lorentzian geometry: given by analytic continuation• Flat space and de Sitter patched
across a domain wall.
• SO(3,1) symmetric.
• Bubble wall is constantly accelerated.
• Open FRW universe (shaded region)
inside the bubble. Spatial slice: H3• Beginning of FRW universe: non‐singular(equivalent to Rindler horizon)

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Eternal inflation• A single bubble does not cover the whole
space. (Fills only the horizon volume).
• Many bubbles will form in the de Sitter region
with the rate
(per unit physical 4‐volume).
(In fact, bubble collisions are inevitable.)• But if, bubble nucleation cannot catch up
the expansion of space, and false vacuum exists
forever (“Eternal Inflation”; Guth, Linde, Vilenkin, …).
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Spectrum of fluctuations andholographic duality
w/ B. Freivogel, L. Susskind, C.‐P. Yeh,
“A Holographic framework for eternal inflation,”
hep‐th/0606204;
w/ L. Susskind,“Census taking in the hut: FRW/CFT duality,”
arXiv:0908.3844 [hep‐th](perturbative calculation ignoring bubble collisions)9Correlation functions in the bubble• Effect of the ancestor vacuum enters through theinitial condition.
• Vacuum is chosen by the Euclidean prescription:
We need the whole global space.
• Euclidean CDL geometry:• Analytic continuation to FRW:

Calculation of the correlator• e.o.m. for a minimally coupled scalar:• Calculation of the correlator is essentially a1‐dimensional scattering problem:• Correlator can be written in termsof the reflection coefficient.
• Bound state exists when mass issmall compared toCorrelator inside the bubble• Final result:(R: Geodesic distance on)• 1st term: “Flat space piece” (Minkowski correlator
written in hyperbolic slicing: Initial condition usuallyassumed for inflation)• 2nd term: Effect of the ancestor vacuum.

Classification by “scaling dimensions”• The ancestor piece can be expressed as(Ω: angular separation on)

Spectrum of the Dimensions: Δ = 2,3,4,…, and additional
lower values (massless case: Δ=0, 1+α (0<α<1)).
• Term with lowest dimension is most important– Decays most slowly after bubble nucleation.
– Amplitude:– Terms withgives large effect (in the IR)

Holographic duality
Conjecture (FSSY ‘06):
The open FRW created by bubble nucleationis dual to a conformal field theory on(at the boundary of the 3‐hyperboloid )• SO(3,1): conformal sym in 2D (as in AdS/CFT).
• The dual has 2 less dimensions than the bulk.
– The dual theory contains gravity (Liouville field).
(It is a 2D gravity coupled to matter with c>25).
– Liouville field plays the role of time.
• (Different from dS/CFT correspondence.)

One bulk field corresponds to a tower of operators(roughly speaking, KK reduction in the time direction)• Operators which scale like:
“RG‐invariant” operators (defined at the UV scale)• Operators which scale like:
“RG‐covariant” operators (defined at reference scale,such as effective action, energy momentum tensor)
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Graviton correlator• There is a non‐normalizable mode (Δ=0).
– Boundary (2D) curvature correlator is finite:

Gravity is not decoupled at the boundary (because of
the compactness of the Euclidean geometry).
• Dimension 2 piece of graviton is transverse (conserved)‐
traceless in 2D.
– Identified as energy‐momentum tensor of 2D CFT– Evidence for the existence of local 2D theory.

Bubble collisions andphases of eternal inflation
w/ R. Bousso, B. Freivogel, S. Shenker,
L. Susskind, I.‐S. Yang, C.‐P. Yeh,
“Future foam: Non‐trivial topology from bubble collisionsin eternal inflation”
arXiv:0807.1947 [hep‐th] ;w/ S. Shenker, L. Susskind,“Topological Phases of eternal inflation,”
arXiv:1003.1347[hep‐th]17Outline• 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 global de Sitter.
(First proposed by Winitzki, ’01)
• The cosmology of the true vacuum region is qualitatively
different in each phase.

18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: radiusrarer: volume of nucleation sites
19Model for eternal inflation• Mandelbrot model (Fractal percolation)– Start from a white cell.
(White: inflating, Black: non‐inflating
Cell: One horizon volume)– Divide the cell into cellswith half its linear size.
(The space grows by a factor of 2.
Time step:)

Picture of the 2D version– Paint each cell in black with probability P.
(P ~= nucleation rate per horizon volume: constant)– Subdivide the surviving (white) cells, and paint cellsin black w/ probability P. Repeat this infinite times.
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Mandelbrot model defines a fractal• If P > 1 ‐ (1/2)3 =7/8, the whole space turns black, since(the rate of turning black) > (the rate of branching).
(No eternal inflation)• If P < 7/8, white region is a fractal. Non‐zero fractal
dimension dF (rate of growth of the cells):

Physical volume of de Sitter region grows.
(Eternal inflation)• Fractals in eternal inflation: noted by Vilenkin, Winitzki, 21…

Three phases of eternal inflation
From the result on the 3D Mandelbrot model[Chayes et al, Probability Theory and Related Fields 90 (1991) 291]

In order of increasing P (or Γ), there are(white = inflating, black = 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.
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Courtesy of Vitaly Vanchurin
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Geometry of the true vacuum region• Mandelbrot model: the picture of the de Sitter side.
(de Sitter region outside the light cone of thenucleation site is not affected by the bubble.)• To find the spacetime in the non‐inflating region
inside (the cluster of) bubbles, we need to understandthe dynamics of bubble collisions.
• In the following, we study this using the intuition
gained from simple examples of bubble collisions.
24Black island phase (isolated cluster of bubbles)
Small deformations of open FRW universe.
• Basic fact: A collision of two bubbles (of thesame vacuum) does not destroy the bubble[c.f. Bousso, Freivogel, Yang, ‘07]

– Spatial geometry approaches smooth H3 at late time.
– Residual symmetry SO(2,1): spatial slice has H2 factor– Negative curvature makes the space expand.

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Collision of two bubbles• De Sitter space: hyperboloid in• One bubble: plane at• Second bubble: plane at
Residual sym: SO(2,1)• Parametrization of de Sitter w/ manifest

factor :• Parametrization of flat space:• Profile of domain wall in (t, z) space (flatdS

is attached)

flat
dS

dS

• Energy on the domain wall decays at late time

• The spatial geometry approach smooth H3

Tubular phase (tube‐like structure of bubbles)
In the late time limit: spatial slice is a negatively curved
space whose boundary has infinite genus.
Late time geometry:– Negatively curved space with non‐trivial boundary topology:
H3 modded out by discrete elements of isometry– Boundary genus = # of elements

Genus 1 case (universe:bulk of a torus)28• Simpler example: true vacuum with toroidal boundary[Bousso, Freivogel, YS, Shenker, Susskind, Yang, Yeh, ‘08]

– Ring‐like initial configuration of bubbles(with the hole larger than horizon size)– Solve a sequence of junction conditions

2

3
1

2
0

3
1

2
0

3
1

2
0

3
1

2

– Approaches flat spacetime at late time.
(Negatively curved spatial slice with toroidal boundary)29

White island phase (isolated inflating region)
An observer in the black region is “surrounded” by the white
region (contrary to the intuition from Mandelbrot model).
• Simple case: two white islands (with S2 symmetry)[Kodama et al ’82, BFSSSYY ’08]

Global slicing (S3 ) of de Sitter

Penrose diagram– An observer can see only one boundary; the other boundary isbehind the black hole horizon. [c.f. “non‐traversability of a30
wormhole”, “topological censorship”]• In the white island phase, a white region will split.
– Late time geometry for the three white island case:[Kodama et al ’82]
W
W

W

– Singularity and horizons will
form so that the boundaries
are causally disconnectedfrom each other.
31• From the Mandelbrot model: A single white island is oforder Hubble size (due to frequent bubble nucleation)• The boundary moves away from a given observer, but its
area remains finite. (Effectively a closed universe)• Black hole in the bulk.
• This universe will eventually collapse.
– Simpler model: Shells of bubbles constantly
colliding to a given bubble– Any given observer in the true vacuum
will end up at singularity.

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Summary of this partThree phases of eternal inflation and their cosmology:• Black island phase:
Small deformation of an open FRW• Tubular phase:
Negatively curved space with an infinite genus boundary• White island:
Observer sees one boundary and one or more black holehorizons (behind which there are other boundaries).
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CMB fluctuations fromquantum effects of many fields
w/ Y. Habara, H. Kawai, M. Ninomiya,“CMB Fluctuations and String Compactification Scales,”arXiv:1103.0299 [hep‐th](This is about inflation after our universe was created)34

Outline• We propose a new mechanism for generating
temperature anisotropy of CMB during inflation.
• An aspect of fundamental theory: presence of a large
number of fields (KK modes, string excitations).
• If the size L of extra dimension is large,
there are KK modes with.
• We consider quantum effect of these light fields.

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“Usual” mechanism:• Expansion is driven by potential energyof a scalar field (inflaton):– Classical value of inflaton: defines time
– Fluctuation of inflaton translates: difference in local
expansion (or curvature perturbation);
proportional to dT/T in CMB.
– Tensor mode: Amplitude is universally H/mpl• Observation:• “Scalar mode is large because of slow rolling.”
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A model considered in this work
• Free massive scalars in de Sitter background: classically at the bottom of potential
can be KK modes or string excitations.
(Later, comment on the time‐dependent Hubble)• This is different from “N‐flation” or “curvaton.”37Quantization in de Sitter space• Equation of motion for a free scalar:– At late times, time‐ and space‐ dependences factorize.
– Fields with mass
do not oscillate(because the “friction” overdumps the oscillation).

• Correlation function:
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Einstein equations (scalar mode)• LHS: Einstein tensor linearized in metric fluctuations•: gauge invariant variables. In longitudinal gauge,
39• Energy momentum tensor: quadratic in matter fields•: scalar part

• Solving for F,• At late times,• When g <<1 (which is important in later applications),(and F = Y )40• Correlation function of F: sum over the species

<ff>
F

F
<ff>

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Tensor mode
• Equation of motion:

RHS: transverse traceless part of•consists of: solution of the homogeneous eq.
(usual gravitational wave): depends on•(effect of matter) is negligible at late times.
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KK modes and string states• Assume D dims. are compactified on TD with period L(and the size of other internal space is string scale)• Mass of KK modes:• Number of states:• The formula for F becomes• The upper limit should be mc ~3H/2 in general relativity,but…
43• If ms (string scale) < H, the integration will be cut off
at mc ~ ms for the following reason:• The fields f are intermediate states in the loop.

• In string theory, Schwinger proper time
is cut off at string scale. Intermediate
state with E > ms do not contribute tophysical processes.
44

CMB fluctuations• CMB temperature fluctuations:• Perturbation is frozen outside the horizon ( q¥ 3o )• Angle q on the sky corresponds to the
distance d at the end of inflation:• We assume
45

Amplitude• Angular power spectrum

is defined by• Fromand expanding46

Preferred values of the parameters• From observations,• For the moment, assume the inequality is saturated.
• Then,gives a relation between ms and L ,or between ms and gs using47

L vs. ms

gs vs. ms• We prefer weak coupling (easier with small D), andL not too large (easier with large D).
• Typical values:• The number of fields:

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Time‐dependent Hubble• In our simple model, spectrum is tilted toward blue

Not favored by observation. But ns can be lowered bydecreasing H with time.
• Time‐dependence of H is needed at the end of inflation.
• By including inflaton in our analysis,(*: quantities at horizon crossing).
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Non‐gaussianities• Three‐point function is given by triangle
diagram summed over species.

F

<ff>

F

<ff>

F

<ff>

• If we define non‐linearity parameter fNL by
and evaluate it at generic separation, we get small fNL

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Summary• String theory suggests that our universe was createdby bubble nucleation.
• The presence of the ancestor vacuum affects the
spectrum of fluctuations.
• Proposed a holographic description in terms of CFT on
the boundary.
• There are phases of eternal inflation characterized by
the way bubbles percolate in the false vacuum.
• Found a mechanism for generating CMB temperaturefluctuations from quantum effect of many fields.
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