Chuo University 6 December, 011 Open Inflation in the String Landscape Misao Sasaki (YITP, Kyoto University) D. Yamauchi, A. Linde, A. Naruko, T. Tanaka & MS, PRD 84, 043513 (011) [arxiv:1105.674 [hep-th]] K. Sugimura, D. Yamauchi & MS, arxiv:1110.4773 [gr-qc]
1. Brief overview of Inflation
Observed CMB anisotropy Map isotropic component T CMB =.73 K COBE-DMR (1990) WMAP (003~) dipole (motion of of solar system) multipole components ( δ T/ T ) = v = CMB l ( δt/ T CMB ) 1-3 10 371 km/s -5 l 0 10 Large Scale Structure WMAP 7 years data ( δt/ T CMB ) l 700 10-5
Horizon problem Why the detection of δt/t at θ >10º was so important? Because in the standard Friedmann universe, the size of causal volume (horizon size) grows like ~ ct.
Origin of horizon problem Expansion of the Universe Hubble horizon size ch -1 ~ct a(t 1 ) a(t ) a(t) t 1/ for hot bigbang universe a(t 3 ) Horizon grows faster than the cosmic expansion in the standard Friedmann (Bigbang) Universe
Horizon problem in Big Bang Universe time we are here t=1.4x10 10 yr Last Scattering Surface (t=4x10 5 yr) size of causal region t=0
There are ~10 4 causally independent patches on LSS Last Scattering Surface (t=3x10 5 yr) ~1º horizon size Now (t~10 10 yr) t=0
Horizon problem Why the detection of δt/t at θ >10º was so important? Because in the standard Friedmann universe, the size of causal volume (horizon size) grows like ~ ct. The angle sustaining the horizon size at LSS is ~ 1º. Thus, any causal, physical process cannot produce correlation on scales θ >1º. But (δt/t) θ>10º 0 means there exists non-zero correlation.
Inflationary Universe Universe dominated by a scalar (inflaton) field For sufficiently flat potential: 8πG 1 ( φ) ɺ φ ( φ) 3 Hɺ 3ɺ φ = 1 H V( φ) H V V V(φ) H is almost constant ~ exponential expansion = inflation φ slowly rolls down the potential: slow-roll (chaotic) inflation Inflation ends when φ starts damped oscillation. φ decays into thermal energy (radiation) Birth of Hot Bigbang Universe φ Linde (1983)
Hubble horizon during inflation a(t)~e Ht ; H~const. A small region of the universe c H -1 Universe expands exponentially, while the Hubble horizon size remains almost constant. An initially tiny region can become much larger than the entire observable universe solves the horizon problem.
Length Scales of the Inflationary Universe log L L=c H -1 L a(t) Size of the observable universe Inflationary Universe Bigbang Universe log a(t)
Flatness of the Universe small universe expands by a factor >10 30 Size of our observable universe looks perfectly flat Birth of a gigantic universe Flatness can be explained only by Inflation
Seeds of cosmological perturbations Zero-point (vacuum) fluctuations of φ : δφ = δφ k k 3 H ɺ k k ω( t) k 0 ; () t () ik i x k t e π c δφɺɺ + δφ + δφ = ω = a() t λ( t) physical wavelength λ(t) a(t) harmonic oscillator with friction term and time-dependent ω δφ k δφk const. frozen when λ > c H -1 (on superhorizon scales) gravitational wave modes also satisfy the same eq.
Generation of curvature perturbations curvature perturbation R gravitational potential Ψ δφ is frozen on flat (R=0) 3-surface (t=const. hypersurface) Inflation ends/damped osc starts on φ =const. 3-surface. t T = const., R 0 end of inflation hot bigbang universe x i R 0 δφ = 0 R = 0 δφ 0
CMB anisotropy from curvature perturbation Photons climbing up from grav potential well are redshifted. E obs Ψ For Planck distribution, d T T obs ( n ) 1 = Ψ( x emit) T Temit 0 xemit = nd; n = line of sight c=1 units E emit In an expanding universe, this is modified to There is also the standard Doppler effect: T ( n ) = n i v ( xemit ) T T 1 ( n ) = Ψ( x emit) T 3 Sachs-Wolfe effect
T 1 ( n ) = Ψ( x LSS) n i v ( x LSS) + ( minor corrections) T 3 Ψ v Observer Last Scattering Surface (LSS)
CMB anisotropy spectrum WMAP 7 year data (010)
Amplitude of curvature perturbation: R = H ɺ πφ k/ a= H Mukhanov (1985), Sasaki (1986) 1 18 M pl ~. 4 10 GeV: Planck mass 8πG 3 4π k n 1 ( ) S V V P k = Ak ; n 1 3 3 R S = M pl ( π) V V Power spectrum index: n S R 5 1 4 16 COBE~ 10 V / ( φ)~ 10 GeV, WMAP 1 = 0. 049 ± 0. 017 ns 1~ 0. 04 for a typical model Tensor (gravitational wave) spectrum: 3 4π k n 1 ( ) ( ) T φ P k ; 3 3 P T k = Ak n T = ɺ R = ( π) V 8 P( k) to be observed by PLANCK! T Liddle-Lyth (199)
Summary of inflationary cosmology inflation (accelerated expansion) is a mechanism to solve horizon and flatness problems. slow-roll inflation can explain the observed structure of the universe. but need to identify the inflaton in unified theory, perhaps in string theory. any hint from observation/experiments?
. String theory landscape Lerche, Lust & Schellekens ( 87), Bousso & Pochinski ( 00), Susskind, Douglas, KKLT ( 03),... There are ~ 10 500 vacua in string theory vacuum energy ρ v may be positive or negative typical energy scale ~ Μ P 4 some of them have ρ v <<Μ P 4 0 which?
1 Is there any way to know what kind of landscape we live in? Or at least to know what kind of neighborhood we live in?
de Sitter (ds) space ds space: ρ v >0, O(4,1) symmetry M = (8 πg) : Planck mass 1 P ds = dt + a ( t ) dω : a = H cosh Ht, H = ρ / 3 M 1 3 ( S ) v P 3-sphere 3 Ht a e for t ( sin ) dω = d χ + sin χ dθ + θ dφ ( S ) Volume ~ a e 3 3Ht
Anti-de Sitter space AdS space: ρ v <0, O(3,) symmetry ds = dt + a ( t ) dω : a = H cos Ht, H = ρ / 3 M 1 3 ( H ) v P hyperbolic space collapses within t~ 1/H 3 ( H ) ( sin ) dω = dχ + sinh χ dθ + θdφ
A universe jumps around in the landscape by quantum tunneling 4 it can go up to a vacuum with larger ρ v ( ds space ~ thermal state with T =H/π ) if it tunnels to a vacuum with negative ρ v, it collapses within t ~ M P / ρ v 1/. so we may focus on vacua with positive ρ v : ds vacua ρ v 0 Sato, MS, Kodama & Maeda ( 81)
Quantum Tunneling = motion through a classically forbidden region = described by motion with imaginary time 5 E V(x) 1 dx 1 dx dx( τ ) + V = E + V = E = ( V E) dt dτ dτ τ = it tunneling probability: ψ ( x ) exp[ S]; out xout xout dx τ out dx S = ( V E) dx = dx dτ x = 0 x0 dτ dτ τ out 1 dx = V dτ const. + + dτ Euclidean Lorentzian 1 dx = + V dτ + const. dτ x 0 x out x Euclidean bounce action = instanton
6 3. Anthropic landscape Not all of ds vacua are habitable. anthropic landscape Susskind ( 03) A universe jumps around in the landscape and settles down to a final vacuum with ρ v,f ~ M P H 0 ~(10-3 ev) 4. ρ v,f must not be larger than this value in order to account for the formation of stars and galaxies. Just before it has arrived the final vacuum (=present universe), it must have gone through an era of (slow-roll) inflation and reheating, to create matter and radiation. ρ vac ρ matter ~ T 4 : birth of Hot Bigbang Universe
Most plausible state of the universe before inflation is a ds vacuum with ρ v ~ M P4. ds = O(4,1) O(5) ~ S 4 7 false vacuum decay via O(4) symmetric (CDL) instanton O(4) O(3,1) inside bubble is an open universe Coleman & De Luccia ( 80) bubble wall false vacuum τ + x = R t + x = R
8 τ + x = creation of open universe R bubble wall open (hyperbolic) space t x = con st. τ + = x con st. analytic continuation t + x = R
Natural outcome would be a universe with Ω 0 <<1. 9 empty universe: no matter, no life Anthropic principle suggests that # of e-folds of inflation inside the bubble (N=H t) should be ~ 50 60 : just enough to make the universe habitable. Garriga, Tanaka & Vilenkin ( 98), Freivogel et al. ( 04) Observational data excluded open universe with Ω 0 <1. Nevertheless, the universe may be slightly open: 1 Ω = 10 ~10 0 3 may be confirmed by Planck+BAO Colombo et al. ( 09)
30 What if 1-Ω 0 is actually confirmed to be non-zero:~10 - -10-3? revisit open inflation! see if we can say anything about Landscape
4. Open inflation in the landscape constraints from scalar-type perturbations Simplest polynomial potential = Hawking-Moss model m ν 3 λ 4 φ 4 potential: V = φ φ + φ 3 4 tunneling to a potential maximum ~ stochastic inflation Hawking & Moss ( 8) Starobinsky ( 84) 31 HM transition slow-roll inflation V < H φ too large fluctuations of φ unless # of e-folds >> 60 Linde ( 95)
Two- (multi-)field model: quasi-open inflation a heavy field σ undergoes false vacuum decay another light field φ starts rolling after fv decay V mφ φ σ = σ σ + φ (, ) V ( ) Linde, Linde & Mezhlumian ( 95) ~ perhaps naturally/easily realized in the landscape 3 σ φ N = κφ ~ < 60 If inflation is short, 4 too large perturbations from supercurvature mode of φ δσ sc (3) p = psc K ; K + p + K Yplm(, r Ω ) = H H π π ~ F R H F : Hubble at false vacuum H R : Hubble after fv decay MS & Tanaka ( 96) 0
creation of open universe & supercurvature mode 33 wavelength > curvature radius supercurvature mode open universe ds vacuum bubble wall
Two-field model : a slightly more complicated two-field model V Sugimura, Yamauchi & MS ( 11) m β σ φ φ ( more parameters) φ ( φ σ ) = Vσ ( σ ) + φ + (, 0 ) 34 tunneling from σ=σ fv to σ=0 makes φ heavy at false vacuum kills the supercurvature mode after tunneling, φ becomes light and starts slow-rolling δφ : non-gaussianity due to interaction? (need study)
To summarize: 35 The models of the tunneling in the landscape with the simplest potentials such as V m ν 3 λ 4 = φ φ + φ or 3 4 V mφ φ σ = σ σ + φ (, ) V ( ) are ruled out by observations, assuming that inflation after the tunneling is short, N ~ 60. NB. a slightly more complicated two-field model may work. The same models are just fine if N >> 60 (if Ω 0 =1) This means that we are testing the models of the landscape in combination with the probability measures, which may or may not predict that the last stage of inflation is short.
How about more general single field models? if ρ fv ~ M P4, the universe will most likely tunnel to a point where the energy scale is still very high unless potential is fine-tuned. Linde, MS & Tanaka ( 99) 36 H F rapid-roll stage will follow right after tunneling. perhaps no strong effect on scalar-type pert s: FV decay R H ~ πφɺ rapid roll slow roll inflation C suppressed by 1/φ ɺ at rapid-roll phase need detailed analysis future issue φ F
37 but tensor perturbations may not be suppressed at all. h TT ~ H M P? Memory of H F (Hubble rate in the false vacuum) may remain in the perturbation on the curvature scale If H F ~M P, we would see a huge tensor perturbation!? tensor perturbations and their effect on CMB
5. Single field open inflation - evolution after tunneling - 38 curvature dominant phase Right after tunneling, H is dominated by curvature: V ( ) a t, ɺ φ φ t 4 H F aɺ κ = ρ + a 3 1 a rapid-roll phase kinetic energy grows until 1 ɺ H* φ V at t ε* 1 V ε κ V φ F : slow-roll parameter
rapid-roll phase for ɺ φ 1 * V at t 1 V ε κ V H for ε* 1, V dominates (curvature dominance ends) at t H H ε ε ε *~ > 1, ρ starts to decay at - continued * ( / ) 1 1 * * * no rapid-roll phase. slow-roll inflation starts at H F t~ > φ F 1 H * ( ) t~ > H / ε < H ε 1 1/ a 1 1 * * * aɺ κ 1 1 = ɺ φ + V + a 3 a dln ρ 3ɺ φ = dln a ɺ φ / + V ( φ ) rapid-roll continues until tracking ɺ φ V is realized during rapid-roll phase ( ) 39
exponential potential model V ( κε φ ) exp ( ) ( ) * R exp κε* ( φ φ* ) V / V = const. ε = const. V = H H + H R ε * = 0.1 added to realize slow-roll inflation ( H ) * H R 40 Log 10 [r/h * ] ε * = 10 4 ε * = 10 curvature term potential kinetic term ε * = 0.5 ε * = 1 Log 10 [a(t) H * ]
6. Tensor perturbations 41 some technical details... action 1 1 µν L = g R g µ φ νφ V φ κ CDL instanton ( ) ( ) ds = dt + a ( t ) dr + sin r dω E E E E E ( η ) = a( η ) d + dr + sin r dω φ = φ( η E ) C E E E E < η E < r E bubble wall π = η E = + η E r E =
4 analytic continuation to Lorentzian space (through r E =π/) η C =- r =const. η =const. 8 r C =0 C R η C =+ C-region: ~ outside the bubble r π = i r, η = η C E C E ( η ) ds = a( η ) d dr + cosh r dω 8 C C C C C time Bubble wall R-region: inside the bubble π π, η η, r = r + i = i a = i a ds = a ( η ) dη + dr + sinh r dω R C R C R C ( ) R R R R R time Euclidean vacuum C-region R-region
tensor mode function Euclidean vacuum h = a ( η ) X p m ( η l ) ( r, Ω ) TT ij C p ( ηc ) ( ) C Y ij C d a d + + ( p + 1) X 0; 1 p = K = dη C a ηc dη C new variable w p : a X d ( aw ) p p dη C Y ij : regular at r c =0 d p p κ + U ( ) ; T ηc w = p w UT( ηc) = φ ( ηc) dη C κ = ac( tc) ɺ φ( tc) ρ + σ = 1 U T 43 σ e ip η C bubble wall ip η C ρ + e e ip η C η C
analytic continuation from C-region to open univ. (=R-region) π η η = η C R C i effect of wall w = ρ e + e w = e ρ e + e e p + ip C ip η C p pπ/ ip η R pπ/ ip R C R epoch of bubble nucleation:ηr there will be time evolution of w p in R-region: d p κ U 0 ; T( ηr) + p w = UT( ηr) = φ dη R d d a or + H + ( p + 1) X 0 ; p = H dη R dη R a ηr 1 d p ( TT X ) p = aw h a dη ( ηr ) ( ) final amplitude of X p depends both on the effect of wall & on the evolution after tunneling 44
Effect of tunneling/bubble wall on ( ) p P( p) X T 45 high freq continuum + low freq resonance p > 1 p~ 0 wall fluctuation mode s ~0.0 s ~0.1 s ~1 s ~0.7 s ~ s κ dη φ C ( η ) S 1 s~ ; S1 = dt ɺ Cφ M P V wall tension scale-invariant C
rapid-roll phase (ε * -)dependence of P T (p) 46 ε <<1: usual slow roll ε~1: small p modes remember H at false vacuum ε >>1: No memory of H at false vacuum
7. CMB anisotropy 47 ε <<1: the same as usual slow roll inflation ε~1: small l modes remember initial Hubble ε >>1: No memory of initial Hubble l scales as (1 Ω) l at small l, scale-invariant at large l 0 small l modes enhanced for ε ~1 *
CMB anisotropy due to wall fluctuation (W-)mode ( C) ( W) 1 C = C + PW Cɶ l l l ; PW = dp P( ) 0 T p s scale-invariant part MS, Tanaka & Yakushige ( 97) ɶ ( 1 ) ( W) C Ω l l 0 48 C κ H / s l= * 10 4 l= 10 7 C T s Κ H 10 10 10 13 10 16 10 19 Α Β 10 5 Α 10 5,Β 1 Α Β 10 4 Α Β 10 3 Α Β 10 s = 10-5 s = 10-4 s = 10-3 s = 10 - W-mode dominates l= 0. 0.5 1.0.0 ε * Ε
8. Summary 49 Open inflation has attracted renewed interest in the context of string theory landscape anthropic principle + landscape 1-Ω 0 ~ 10-10 -3 Landscape is already constrained by observations If inflation after tunneling is short (N ~ 60): simple polynomial potentials aφ bφ 3 + cφ 4 lead to HM-transition, and are ruled out simple -field models, naturally realized in string theory, are ruled out due to large scalar-type perturbations on curvature scale
Tensor perturbations may also constrain the landscape single-field models it seems difficult to implement models with short slow-roll inflation right after tunneling in the string landscape. if ε<<1, energy scale must have been already very low. there will be a rapid-roll phase after tunneling. 1 V ε = ~ > 1 κ V right after tunneling unless ε>>1, the memory of pre-tunneling stage persists in the IR part of the tensor spectrum large CMB anisotropy at small l (1 Ω0) l due to either wall fluctuation mode or evolution during rapid-roll phase We are already testing the landscape! 50