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

Mobility in the Cosmic Landscape Henry Tye Cornell University hep-th/0611148 ArXiv:0708.4374 [hep-th] with Qing-Guo Huang, ArXiv:0803.0663 [hep-th] String Pheno ‘08, U Penn., 05/28/08

Outline : • 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 ?

Flux compactification in Type IIB string theory where all moduli of the 6-dim. “Calabi-Yau” manifold are stabilized • There are many meta-stable manifolds/ vacua, 10 500 or more, probably infinite, with a positive cosmological constant. • There are solutions with zero as well as negative cosmological constants. Giddings, Kachru, Polchinski, KKLT vacua Kachru, Kallosh, Linde, Trivedi and many others, 2001....

How our universe ends up with such a small CC ? • String theory has many meta-stable vacua, 10 500 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 or hundreds (even thousands). • This number at any neighborhood in the landscape may be taken as the dimension d of 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 ? High CC sites MOBILE NO eternal inflation ! Sharp transition Low CC sites trapped exponentially long lifetimes

The wavefunction of the universe moduli open string modes cosmic scale factor may be crudely approximated by that of a D3-brane or a stack of D3-branes

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 d ( L ) = σ L d − 2 conductivity in d = 3, g = σ (Area) /L ∼ σ L .) roscopic measure of the disorder. We ( g ( L ) ∼ ( L/a ) ( d − 2) ) The wavefunction is free Conducting/mobile (metalic) with finite conductivity Conductance = Mobileness Conductivity = Mobility

| ψ ( r ) | ∼ exp( − | r − r 0 | / ξ ) g ( a ) ∼ | ψ ( a ) | ∼ e − a/ ξ

How does g scale ? Given g at scale a, what is g at scale L as L becomes large ? g ( a ) ∼ | ψ ( a ) | ∼ e − a/ ξ g d ( L ) = σ L d − 2 g d ( L ) ∼ e − L/ ξ Insulating, localized, Conducting, mobile trapped, eternal inflation zero conductivity

β d ( g d ( L )) = d ln g d ( L ) d ln L g d ( L ) = σ L d − 2 g d ( L ) ∼ e − L/ ξ g → 0 β d ( g ) → ln g lim g →∞ β d ( g ) → d − 2 lim g c

O

ro: β d ( g c ) = 0, β d ( g ) ≈ 1 ν ln g ≈ 1 g − g c g c g c ν so this zero of β d ( g ) corresponds to an unstable fixed point or g ( a ) > g c , or g ( a ) < g c in alue. see that the g d ( L ) = σ L d − 2 g d ( L ) ∼ e − L/ ξ Insulating, localized, Conducting, mobile trapped, eternal inflation

What is the critical g c ? ∆ ln g c = ln g c ( d ) − ln g c ( d + 1) = k > 0 g c ( d ) ≃ e − ( d − 3) k g c (3) Shapiro : β d ( g ) = ( d − 1) − ( g + 1) ln(1 + 1 /g ) g c = e − ( d − 1) ν → 1 d=1 : Anderson, Thouless, Abraham, Fisher, Phys. Rev. B 22, 3519 (1980)

Condition for mobility g ( a ) ∼ | ψ ( a ) | ∼ e − a/ ξ g c = e − ( d − 1) d > a ξ + 1 ξ , Γ 0 ∼ | ψ ( a ) | 2 ∼ e − 2 a/ ξ . d ~ 100 (1.3) at scale , what happ Γ 0 > e − 2( d − 1) Mobile

The Quantum Landscape For the scenario to work : • 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.

Harder to trap in higher dimensions An 1-dim. attractive delta-function potential always has a bound state but not a 3-dim one. Spherical square well : V (r) = − V 0 r < R = 0 r > R k 0 R k 02 = 2mV 0 ψ (r) ∼ e − ar a 2 = 2m|E|

Resonance Tunneling in QM: Tunneling are exponentially small T A → B ∼ T B → C probabilities When the condition is right : T A → C ∼ 1 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 ? T A → B = T B → C = T 0 T A → C = T 0 / 2 T ( n ) � T 0 /n

In a d-dimensional hyper-cubic lattice Γ nr naive : ∼ 2 d Γ 0 t 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. Γ t ∼ n d Γ 0 actually : For large enough d (and maybe n ) n ∼ 1 /Hs the tunneling can be fast.

Tunneling is much faster at high CC. , , 1 2 − 1 1 = − 1 D’ 2 , , 11 − 1 1 10 = − 1 110 D Similar in Coleman de Luccia M 4 M 4 p p 3 8 [ VB ] Hawking-Moss : VD − Γ ( B → C ) ∼ e

Estimate of critical C.C. assume a random distribution : fraction of sites : d = s ( Λ c ) / ξ + 1 ξ ∼ s ( Λ s ) ∼ 1 ∼ Λ − 1 / 4 s m s Λ c ∼ d − d M 4 For flat distribution : s d > 60

The scenario Saltatory relaxation Abbott, Brown and Teitelboim Feng, March-Russell, Sethi and Wilczek

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. 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 ?

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.