Mobility in the Cosmic Landscape Henry Tye Cornell University - - PowerPoint PPT Presentation

Mobility in the Cosmic Landscape

Henry Tye Cornell University

String Pheno ‘08, U Penn., 05/28/08

hep-th/0611148 ArXiv:0708.4374 [hep-th] with Qing-Guo Huang, ArXiv:0803.0663 [hep-th]

• A very brief review of the cosmological

constant problem in the string landscape.

• Renormalization group analysis of the

mobility in the landscape.

• Some intuitive understanding.
• Estimate the exponentially small critical CC.
• There is no eternal inflation in this scenario.
• What is the inflationary scenario compatible

with this landscape picture ? Signatures ?

Outline :

Flux compactification in Type IIB string theory

where all moduli of the 6-dim. “Calabi-Yau” manifold are stabilized

Giddings, Kachru, Polchinski, Kachru, Kallosh, Linde, Trivedi and many others, 2001....

KKLT vacua

• There are many meta-stable manifolds/

vacua, 10500 or more, probably infinite, with a positive cosmological constant.

• There are solutions with zero as well as

negative cosmological constants.

How our universe ends up with such a small CC ?

• String theory has many meta-stable vacua,

10500 or more. This is the cosmic landscape.

• They span a wide range of CC.
• Some of them have very small CCs.
• It is easy to convince ourselves that one of

the string vacuum sites in the landscape describes our universe.

• In the cosmic landscape, why such a small

CC vacuum site is selected ? So the CC problem becomes a selection problem.

Bousso and Polchinski

a cartoon :

Landscape : like a random potential in multi-dimensions

The vastness of cosmic landscape

• At a typical meta-stable site, count the

number of parameters or the number of light scalar fields. This gives the number of moduli, or directions in the field space.

• The number of light scalars can be dozens
• r hundreds (even thousands).
• This number at any neighborhood in the

landscape may be taken as the dimension d

• f the landscape around that neighborhood.
• The landscape potential is not periodic. It is

very complicated.

What are the conditions under which a low CC universe will emerge naturally ?

MOBILE trapped exponentially long lifetimes High CC sites Low CC sites Sharp transition NO eternal inflation !

The wavefunction of the universe

moduli

• pen string modes

may be crudely approximated by that of a D3-brane

• r

a stack of D3-branes cosmic scale factor

Strategy :

• Treat the landscape as a d-dimensional

random potential

• Use the scaling theory developed for

random potential (disordered medium)

• justify the key points of the above scenario
• calculate some of the properties of the

landscape, e.g., the critical CC

• argue why no eternal inflation
• why we should end up with an exponentially

small C.C.

Anderson localization transition

• random potential/disorder medium
• insulation-superconductivity transition
• quantum mesoscopic systems
• conductivity-insulation in disordered

systems

• percolation
• strongly interacting electronic systems
• doped systems, alloys, . . . . . . . . .

Some references :

• P. W. Anderson, Absence of Diffusion in Certain Random Lattices,
• Phys. Rev. Lett. 109, 1492 (1958).
• E. Abrahams, P. W. Anderson, D. C. Licciardello and T. V.

Ramakrishnan, Scaling Theory of Localization : Absence of Quantum Diffusion in Two Dimensions, Phys. Rev. Lett. 42, 673 (1979).

• B. Shapiro, Renormalization-Group Transformation for Anderson

Transition, Phys. Rev. Lett. 48, 823 (1982).

• P. A. Lee and T. V. Ramakrishnan, Disordered electronic

systems, Rev. Mod. Phys. 57, 287 (1985).

• M. V. Sadovskii, Superconductivity and Localization,'' (World

Scientific, 2000).

• Les Houches 1994, Mesoscopic Quantum Physics.

Define a dimensionless conductance g in a d-dim. hypercubic region of size L

(g(L) ∼ (L/a)(d−2)) The wavefunction is free

gd(L) = σLd−2

in d = 3, g = σ(Area)/L ∼ σL.) roscopic measure of the disorder. We Conducting/mobile (metalic) with finite conductivity

Conductance = Mobileness Conductivity = Mobility conductivity

g(a) ∼ |ψ(a)| ∼ e−a/ξ

|ψ(r)| ∼ exp(−|r − r0|/ξ)

How does g scale ? Given g at scale a, what is g at scale L as L becomes large ?

gd(L) ∼ e−L/ξ

g(a) ∼ |ψ(a)| ∼ e−a/ξ

gd(L) = σLd−2

Insulating, localized, trapped, eternal inflation Conducting, mobile zero conductivity

βd(gd(L)) = d ln gd(L) d ln L

gd(L) ∼ e−L/ξ

gd(L) = σLd−2

lim

g→0 βd(g) → ln g

gc lim

g→∞ βd(g) → d − 2

O

βd(g) ≈ 1 ν ln g gc ≈ 1 ν g − gc gc

ro: βd(gc) = 0,

so this zero of βd(g) corresponds to an unstable fixed point

• r g(a) > gc,

alue.

• r g(a) < gc in

see that the

gd(L) ∼ e−L/ξ

Insulating, localized, trapped, eternal inflation

gd(L) = σLd−2

Conducting, mobile

What is the critical gc ?

∆ ln gc = ln gc(d) − ln gc(d + 1) = k > 0 gc(d) ≃ e−(d−3)kgc(3)

βd(g) = (d − 1) − (g + 1) ln(1 + 1/g)

gc = e−(d−1)

ν → 1

Shapiro :

d=1 : Anderson, Thouless, Abraham, Fisher, Phys. Rev. B 22, 3519 (1980)

Condition for mobility

Γ0 > e−2(d−1)

ξ, Γ0 ∼ |ψ(a)|2 ∼ e−2a/ξ. (1.3) at scale , what happ

g(a) ∼ |ψ(a)| ∼ e−a/ξ

d > a ξ + 1

gc = e−(d−1)

d ~ 100

Mobile

The Quantum Landscape

• Tunneling from a positive CC site to a negative CC

site is ignored (CDL crunch).

• Tunneling from a dS site to another dS site with a

larger CC is ignored.

• Improve on the above hand-waving argument.
• A sharp transition from fast tunneling to very

slow tunneling is necessary to avoid eternal inflation.

For the scenario to work :

Harder to trap in higher dimensions

An 1-dim. attractive delta-function potential always has a bound state but not a 3-dim one.

V (r) = −V0 r < R = 0 r > R k02 = 2mV0 k0R

Spherical square well :

a2 = 2m|E| ψ(r) ∼ e−ar

TA→C ∼ 1

TA→B ∼ TB→C Resonance Tunneling in QM:

are exponentially small Tunneling probabilities When the condition is right :

Resonance tunneling in QFT : Saffin, Padilla and Copeland

Tunneling from a typical meta-stable site below the Planck (or string) scale.

Why fast tunneling is possible ?

TA→C = T0/2 TA→B = TB→C = T0

T(n) T0/n

In a d-dimensional hyper-cubic lattice

Γnr

t

∼ 2d Γ0

Γt ∼ nd Γ0

naive :

actually :

The time for one e-fold of inflation is Hubble time 1/H ( So the lifetime of a typical site is longer than the Hubble scale, and eternal inflation seems unavoidable.

n ∼ 1/Hs

For large enough d (and maybe n) the tunneling can be fast.

Tunneling is much faster at high CC.

,

D’ D

, , , Hawking-Moss :

Γ(B → C) ∼ e

3 8[ M4 p VD − M4 p VB ]

1 2 − 1 1 = −1 2 1 11 − 1 10 = − 1 110

Similar in Coleman de Luccia

Estimate of critical C.C.

Λc ∼ d−dM4

s

For flat distribution : assume a random distribution : fraction of sites :

d > 60 d = s(Λc)/ξ + 1

ξ ∼ s(Λs) ∼ 1 ms ∼ Λ−1/4

s

The scenario

Saltatory relaxation

Abbott, Brown and Teitelboim Feng, March-Russell, Sethi and Wilczek

Open questions :

• Better knowledge of the shape and structure of the

string landscape, and the mobility of the universe.

• What is the starting wavefunction of the universe and

how does it collapse ?

• Why the cosmological constant is so very small ?
• What are the signatures of this inflationary scenario ?

Remarks

If the scenario is correct, we can appreciate string theory in this new light : it provides a vast landscape so that a very small CC vacuum is among its solutions, and the same vastness destabilizes all vacua except ones with very small CCs, thus allowing our universe to emerge, survive and grow.

Extended brane inflation

• Inflation takes place while the brane is moving in the

landscape (and in the bulk).

• It rolls, percolates (bouncing around), hops and

tunnels, happens at a rate of 1000 to 10,000 times per e-fold. So it may look like slow roll.

• Besides adiabatic perturbations, it also generates

entropic perturbations repeatedly.

• So this scenario will have large non-Gaussianity of

the squeezed type.

Freese, Spolyar Freese, Liu, Spolyar Davoudiasal, Sarangi, Shiu Sarangi, Shlaer, Shiu Podolsky, Majumder, Jokela Chialva, Danielsson Watson, Perry, Kane, Adams Huang . . . . .

Summary

• The universe is freely moving in the string

landscape when the vacuum energy density is above the critical value. Because of mobility, there is no eternal inflation.

• When the universe drops below the critical C.C.

value, it loses its mobility. Its lifetime there is exponentially long.

• The critical C.C. value is exponentially small

compared to the string/Planck scale.

• This scenario suggests an alternative inflationary

scenario that involves fast tunneling, scattering/ bouncing around, hopping as well as rolling.

A puzzle

• During nucleosynthesis, the radiation density

is around 1MeV4, which is ~1035 bigger than the CC. Since both radiation/matter and CC contributes to the stress tensor that couples to gravity, why the universe does not pick a larger CC ?

• If the universe was once as hot as 100 GeV
• r above, the vacuum energy density before

the EW phase transition VEW ~ (100 GeV)4.

• How does the dynamics know to pick a site

with VEW ~ (100 GeV)4 that will eventually end up with a CC ~ 10-11 eV4 ?