Misao Sasaki YITP, Kyoto University 9 June, 009 ICG, Portsmouth
contents 1. Inflation and curvature perturbations δn formalism. Origin of non-gaussianity subhorizon or superhorizon scales 3. Non-Gaussianity from superhorizon scales two typical models: curvaton & multi-brid inflation 4. Multi-brid inflation 5. Summary
1. Inflation and curvature perturbations single-field inflation, no other degree of freedom 8π G 1 = φ ( φ), φ 3 φ ( φ) =0 3 + + + t = t ( φ ) slow-roll evolution H V H V V(φ) log L H const. Linde 8,... 3 φ H -1 a L=H - 1 ~ const. inflation hot (bigbang) universe log a t=t end
N : Number of e-folds e of inflation 4 number of e-folds e counted backward in time from the end of inflation ~ log(redshift) at ( end ) at () = exp[ Nt ( t )] end tend 1+ z( φ) N = N( φ) = Hdt = ln t( φ ) 1+ z( φend) log L N=N(φ) L=H -1 ~ t L=H - 1 ~ const t=t(φ) inflation t=t end log a(t)
Cosmological curvature perturbations - standard single-field slow-roll case - Inflaton fluctuations (vacuum fluctuations=gaussian) 1 k φ k = ϕ, ~ iwkt k ϕk e ; wk = H w a k 5 Oscillation freezes out at k/a < H (~ classical Gaussian fluctuations on superhorizon scales) ϕ k H k H H δφ 3 k = k a π k/ a~ H ~ ; Curvature perturbations R c = H δφ φ conserved on superhorizon scales ( 3) ( 3) R ~ Δ R c 3a
6 δn N formalism spectrum of R c : P H ( k) = curvature perturbation = e-folding number perturbation (δn) t end φ end H N( φ) = Hdt = dφ t( φ) φ φ N H δ N( φ) = δφ = δφ = c φ φ R H N P ( k ) = δφk πφ = φ k / a= H ; R general slow-roll inflation R πφ k/ a= H ~ scale-invariant k/ a= H k/ a= H δ N A δφ N = δφ A φ nonlinear generalization Lyth, Malik & MS ( 04) A k Starobinsky ( 85) H = π k/ a= H MS & Stewart ( 96)
Theory vs Observation 7 Prediction of the standard single-field slow-roll inflation: almost scale-invariant Gaussian random fluctuations perfectly consistent with CMB experiments
However, nature may not be so simple... 8 Tensor (gravitational wave) perturbations? tensor perturbations induce B-mode polarization of CMB tensor-scalar ratio: r<0. (95%CL) WMAP+BAO+SN ( 08) cf. chaotic inflation: r~0.15 Non-Gaussianity? ( - )gravitational pot.: minus Komatsu & Spergel ( 01) Φ =Φ + f Φ + g Φ + -9< f NL <111 (95%CL) (f NL =51±30 at 1σ) 3 Gauss NL Gauss NL Gauss WMAP ( 08) What would the presence of non-gaussianity mean?
9. Origin of Non-Gaussianity three typical origins: Self-interaction of inflaton field quantum physics, subhorizon scale during inflation Multi-component field classical physics, nonlinear coupling to gravity superhorizon scale during and after inflation Nonlinearity of gravity classical general relativistic effect, subhorizon scale after inflation
10 Origin of non-gaussianity and cosmic scales log L k: comoving wavenumber L = a k classical/local NL effect classical NL gravity quantum NL effect L= H 1 inflation t=t end log a(t) hot Friedmann
Origin of NG (1): Self-interaction of inflaton field 11 Non-Gaussianity from subhorizon scales (QFT effect) interaction is very small for potential-type self-couplings ex. chaotic inflation V 1 m φ = free field! V = λφ λ~10 4 15 Maldacena ( 03) (gravitational interaction is Planck-suppressed) non-canonical kinetic term strong self-interaction large non-gaussianity
example : DBI inflation Silverstein & Tong (004) 1 kinetic term: 1 K ~ f ( ) 1- f( ) g μν μ ν φ φ φ φ f γ 1 1 ~ (Lorenz factor) -1 perturbation expansion: K = K + δ K + δ K + δ K + 0 1 3 γ -1 = 0 γ 3 γ 5 δφ~ δφ + γ δφ + 0 0 large non-gaussianity for large γ Seery & Lidsey ( 05),..., Langlois et al. ( 08), Arroja et al. ( 09)
bispectrum (3-pt function) of curvature perturbation from DBI inflation R ( p ) R ( p ) R ( p ) c 1 c c 3 ( P P ) ~ δ ( pj) fnl( p1, p, p3) R( p1) R( p) + cyclic j f NL ~ γ Alishahiha et al. ( 04) 13 f NL for equilateral configuration is large p ~ p ~ p 1 3 f NL f equil NL p p 1 p 3 ( p + p + p = ) 1 3 0 equil WMAP 5yr constraint: 151< < 53 ( 95% CL) f NL
Origin of NG ():( Nonlinearity of gravity 14 ex. post-newton metric in harmonic coordinates ( 1 ) ( 1 ) ds = + Ψ Ψ + dt + + Ψ + Ψ + dr + Newton potential NL terms may be important after the perturbation scale re-enters the Hubble horizon Effect on CMB bispectrum seems small (but non-negligible?) f NL ~ O( 1~ 10) Bartolo et al. (007)
Origin of NG (3): Superhorizon scales 15 Even if δφ is Gaussian, δt μν may be non-gaussian due to its nonlinear dependence on δφ This effect is small for a single-field slow-roll model ( linear approximation is extremely good) But it may become large for multi-field models Lyth & Rodriguez ( 05),... Non-Gaussianity in this case is local: Salopek & Bond ( 90),... f ( p, p, p ) f = NL 1 3 local NL const.
16 In the rest of this talk we focus on this case, ie non-gaussianity generated on superhorizon scales
17 3.. Non-Gaussianity from superhorizon scales - two typical models for f NL curvaton model Linde & Mukhanov (1996), Lyth & Wands (001), Moroi & Takahashi (001),... multi-brid inflation model MS (008), Naruko & MS (008) both may give large f NL local local NL - but in the case of curvaton scenario tensor-scalar ratio r will be very small.
Curvaton model 18 Inflation is dominated by inflaton = φ Curv. perts. are dominated by curvaton = χ 1 Vtot = V( φ) + m χ χ mχ H 1 during inflation: V( φ) m χ χ 8 φ dominates curvature perturbations during inflation πgv ( φ) 3 H H δφ~, δχ ~ V'( φ) δφ ( χ + δχ) χ π π m χ linear approx. is valid for δφ δχ contribution is small including NL corrections
After inflation, φ thermalizes, χ undergoes damped oscillation. 19 ρ = ρ a, ρ a 4 3 φ γ χ Assume χ dominates final amp of curvature perturbations: R c q δχ δχ qδχ 1 qδχ + + + 4 q χ χ χ q χ ρ q 1 f NL ~ 1/q χ q ρ + ρ fraction of χ at decay time χ γ t= t decay Large NG is possible if q<<1 But tensor mode is strongly suppressed: PT( k) PT ( k) PT( k) r = r=, PRφ( k) PRχ( k) P φ( k) P χ ( k) P φ( k) R R R Lyth & Rodriguez ( 05) Malik & Lyth ( 06) MS, Valiviita & Wands ( 06)
4. Multi-brid inflation 0 hybrid inflation: inflation ends by a sudden destabilization of vacuum multi-brid inflation = multi-field hybrid inflation: 1 μν Lφ = g μφa νφa V( φ) Slow-roll eom 1 = 3H V A, φ 3H A φ N as a time: A dn ( 8πG= M = 1) Planck = V = Hdt dφ A 1 V 1 logv = = dn 3V φ 3 φ A A V N=0 inflation φ
simple -brid example: the case of radial inflationary orbits on (φ 1,φ ) φ φ = φcosθ 1 φ = φsinθ θ = const. φ = φ( N, θ) N( φ, θ) = N( φ, φ ) 1 1 θ φ δ N = N( φ + δφ, φ + δφ ) 1 1 N( φ, φ ) 1 N=0 φ = φ ( θ) f f N=const. φ 1
Three types of δn ( ) indicates field perturbations φ φ 1 originally adiabatic (standard scenario) entropy adiabatic end of inflation or phase transition
condition to end the inflation φ 1, φ inflaton fields (-brid inflation) χ waterfall field 3 1 ( ) λ σ V = V0( χφ ; ) U( φ); V0 = g1φ1 + gφ χ + χ 4 λ during inflation g V g φ + g φ > σ 1 1 4 σ ( = ) = 4λ 0 χ 0 inflation ends when φ + g φ = σ 1 1 V 0 A g φ A A A g φ > σ A A χ < σ
Simple analytically soluble model exponential potential: V = V exp[ mφ + m φ ] parametrize the end of inflationφ δn to nd order in δφ : 0 1 1 σ = c osγ, φ = δφ1g1cosγ + δφgsin γ g1 g ( mδφ1 m1δφ) δ N = + mg cosγ + mg sin γ σ ( mg cosγ + mgsin γ) 1 1 1 1 MS ( 08) 1f, f, g1 g σ sinγ 3 4 Spectrum of curvature perturbation P R g cos γ + g sin γ H ( k) = ( mgcosγ + sin γ) π 1 1 1 mg k= Ha spectral index: tensor/scalar: n = 1 ( m + m ) s 1 P ( k ) ( cos sin ) 8 mg T γ + m g γ r = = P ( k) g cos g sin γ R 1 1 1 γ +
5 δ N L 1g1cos gsin 1gsin g1cos δφ γ + δφ γ, S δφ γ δφ γ mgcosγ + mg sinγ mgcosγ mg sinγ 1 1 1 1 A B H AB δ L N S = 0 for δφ δφ = δ π isocurvature perturbation f 3 δ N = δ N + δ N + S L ( ) local f 5 NL L 5gg ( mgcosγ mgsin γ ) local 1 1 1 NL = 6σ ( g1 cos γ + gsin γ) mg 1 1cosγ + mgsinγ local NL ( σ ) f = O gm/ for m, m ~ O( m), g, g ~ O( g). 1 1 possibility of large non-gaussianity local f NL is always positive in this model isocurvature contributes at nd order
Just an example... input parameters: m 1/ 18 1= MPl = ( 8π G) =. 43 10 GeV ~ 0. 005, m ~ 0. 035 1 m cosγ m 1 g = g g 1 sinγ 6 outputs: n = 1 ( m + m )~ 0. 96 s 1 1 r 8m ~ 0. 04 ( ) R 3H = σ 4λ~ 15. 10 P ( k)~ 5. 10 4 9 9 σ ~ λ 10 1/ 4 indep. of waterfall χ f 5gm ~ m σ local NL 6 1 g 40 λ 1/ 4
WMAP 5yr Komatsu et al. 08 7 WMAP+BAO+SN WMAP present example tensor-scalar ratio r is not suppressed AND f NL local ~ 50 (positive and large)
Another example: O() SSB model 1 V V0exp = M ( φ1 + φ) Inflation ends at g1φ 1 + gφ = σ ; g1 g symmetry breakdown at the end of inflation φ Alabidi & Lyth 06 8 φ 1 1 cos + sin = φ β φ β g gcos β, g gsin β 1 σ g
1, f cos +, f sin = φ β φ β curvature perturbation spectrum: spectral index: n s σ g = 1+ M φ sin cos = σ δ, φ = σ δ g cosβ g sin β 1, f, f P R 4 8 H 1 σ ( k) = r π = r 6 r 1 cos β cos 8 sin β δ πλ 9 tensor/scalar: M N P ( k ) T σ 8 M e r = = P ( k) g R 1 cosβ cosδ local 5M cosβsinδ non-gaussianity: f NL = 1 6 1 cosβcosδ local r 10 f ~ 5 NL M ~ > 01. β for 00., 1 δ β Again, f local NL may be large and positive.
local f NL and r f NL 100 1 M = 0 30 r 0.5 0.4 0.3 0. 0.1 1 δ β 4 g 1 λ = 80 60 40 0 1 3 0.000 0.005 0.010 0.015 0.00 Β n s = P R 1 5 096. 1 7 ( k) = 5. 10 9 1 30 0.000 0.005 0.010 0.015 0.00 Β
It seems f NL >0 always in multi-brid inflation 31 A 1 A B δ N = N Aδφ + N ABδφ δφ + A N N N = N A =, N AB =, A A B φ φ φ A B 1 N N ABN δ N = Rc + R c + ( C N ) C N A B Any good reason for N N N > 0? AB
Perhaps yes! 3 inflaton trajectories y y x = σ f f For hyperbolic end of inflation condition f NL may become negative. x y + x = σ f f
5. Summary 3 types of non-gaussianity 1. subhorizon quantum origin. superhorizon classical (local) origin These are important 33 3. gravitational dynamics classical origin DBI inflation --- type 1. f can be large equil NL curvaton scenario, multi-brid inflation --- type. local f NL can be large. sign is important, too. but curvaton scenario predicts r 1 if f NL is large. multi-brid inflation can give r~0.1
34 Non-Gaussianity plays an important role in determining (constraining) models of inflation 4-pt function (trispectrum) may be detected in addition to 3-pt function (bispectrum) Arroja & Koyama ( 08), Huang ( 09), NG may be detected in the very near future PLANCK launched a month ago (on 14 May)!