Inflationary Trajectories Pascal M. Vaudrevange 19.09.2007
Scanning Inflaton Goals: Reconstruction of Primordial Power Spectra Reconstruction of Inflaton Potential
Inflation driven by a scalar field - I FLRW metric: ds 2 = dt 2 a(t) 2 (dx 2 + dy 2 + dz 2 ) Friedmann Equation 3H 2 = Eom for a scalar field 1 M 2 p ( ) 1 2 φ 2 + V φ + 3H φ + V,φ = 0 Inflation 0 ɛ = Ḣ H 2 1 ending when ɛ = 1 Number of efolds N, dn = Hdt = 1 ɛ 1 d ln k
Inflation driven by a scalar field - II Power Spectrum of Scalar/ Tensor Perturbations P s = k 3 2π 2 R kr k = 1 H 2 ( 8π 2 Mp 2 = A s ɛ k=ah P t = 2 H 2 ( ) k nt π 2 = A t k=ah M 2 p standard parametrization k pivot amplitude A s scalar/ tensor spectral index n s, n t running of the spectral index n run = dn s /d ln k tensor scalar ratio r = P t /P s k k pivot ) ns 1,
Chebyshev expansion trajectory expansion of f {P s, P t, ln P s, ln P t, ɛ, n t, ln(ɛ), ln( n t ),... } f (x) = c j T j (x) = f (x j )w j (x) T j (cos(x)) = cos(jx) power spectra: solid line: P s, dotted line: P t coefficients: V = λφ 4, V = 1 2 m2 φ 2
MCMC Reconstruction of Primordial Power Spectra CMB+ LSS data sets P s, P t to order 5 ln P s, ln P t to order 5 r = 0.228 +0.037 0.055 r < 0.075 using standard parametrization: r < 0.36
MCMC Reconstruction of Primordial Power Spectra P s, P t to order 5 ln P s, ln P t to order 5
Reconstructing Simulated Data Sets uniform prior on ln P s, ln P t uniform prior on P s, P t Parameter Initial Planck CMBPol Planck CMBPol n s 0.96 0.9595 +0.0035 0.0035 0.9591 +0.0024 0.0024 0.9548 +0.0051 0.0052 0.957 +0.0023 0.0023 n run 0.0551 0.055 +0.0066 0.0065 ln[a s] 19.9578 19.958 +0.0016 0.0017 0.056 +0.0034 0.0034 19.9577 +0.0011 0.0011 r 0.1 < 0.077(95%CL) 0.0852 +0.0081 0.008 n s 0.96 0.9595 +0.0036 0.0036 n run 0.0551 0.0558 +0.0065 0.0067 ln[a s] 19.9578 19.958 +0.0017 0.0017 0.959 +0.0024 0.0023 0.0561 +0.0034 0.0033 19.9577 +0.0012 0.0011 r 0.01 < 0.0088(95%CL) 0.0072 +0.0014 0.0015 0.0511 +0.0094 0.0095 19.9582 +0.0028 0.0026 0.0578 +0.0028 0.0029 19.9572 +0.0011 0.001 < 0.13(95%CL) 0.0925 +0.0047 0.0047 0.9549 +0.0043 0.0044 0.0517 +0.0084 0.0086 19.9583 +0.0023 0.0023 0.957 +0.0022 0.0022 0.0578 +0.0028 0.0028 19.9572 +0.0011 0.001 < 0.042(95%CL) 0.00868 +0.00092 0.00092 Simulated data sets with Planck s and CMBPol s errors keeping other parameters fixed
Influence of Priors uniform prior at A, B non-uniform at C!
P s, P t order 5 monotonic P t non-monotonic P t non-monotonic ln P t uniform prior on P T dashed red : in between nodal points solid black: at nodal points
ln P s, ln P t order 5, uniform in P s, P t Parameter Value Ω b h 2 0.02202 +0.00079 0.0008 Ω c h 2 0.1177 +0.0058 0.0061 θ 1.0422 +0.0031 0.0031 τ 0.107 +0.034 0.033 n s 0.936 +0.036 0.035 n run 0.034 +0.03 0.029 ln[a s ] 19.925 +0.072 r 0.072 < 0.41(95%CL)
ɛ trajectories standard ln ɛ5 ln ɛ5 uniform ɛ
Reconstruction of Simulated Data Sets ɛ order 2 ɛ order 5 keeping other parameters fixed
Acceleration trajectories order 5 ɛ ln( n t ) dashed red line : at nodal points dotted green line : at nodal points with rejection solid black line : in between nodal points
ln( n t ) to order 5, uniform prior on ɛ Parameter Value Ω b h 2 0.02234 +0.00085 0.00081 Ω c h 2 0.1263 +0.0068 0.0067 θ 1.045 +0.0033 0.0033 τ 0.1 +0.03 0.03 H 1 1.82 +0.14 0.14 n s 0.885 +0.04 0.04 n t 0.049 +0.013 0.012 n run 0.121 +0.032 0.031 log[a s ] 19.899 +0.069 0.071 r 0.39 +0.099 0.1
MCMC reconstruction of the inflaton potential ln( n t ) to order 5 with uniform prior on ɛ d ln H = ɛ d ln k 1 ɛ, dφ 2ɛ d ln k = ( 1 ɛ V (ln k) = 3H(ln k) 2 1 1 ) 3 ɛ(ln k)
Degeneracy of the potential reconstruction dh d ln k = H 3 (k) H 2 (k) 8π 2 M 2 p P s (k) H 1 from P t = 2 π 2 H 2 M 2 p
Roulette Inflation K M 2 P ( 3 = 2 ln V s + ξg 2 s 2e 3φ 2 V (T, T ) = e K/M2 P Ŵ = g 3 2 s M 3 P ) ( ) +..., V s = 1 9 τ 3/2 2 1 τ 3/2 2. ( W0 4π + ) A i e a i T i ( K i jd i Ŵ D j Ŵ 3 Ŵ Ŵ MP 2 ) + D-terms. SUGRA approximation to large volume IIB-compactification by Conlon, Quevedo, 2005 Kähler moduli stabilized by non-perturbative effects
Resulting potential V (V, T 2,..., T n ) = 12W0 2ξ n (4V ξ)(2v + ξ) 2 + 12e 2a i τ i ξa 2 i (4V ξ)(2v + ξ) 2 + 16(a ia i ) 2 τi e 2ai τi 3αλ 2 (2V + ξ) i=2 + 32e 2a i τ i ai A 2 i τ i(1 + a i τ i ) (4V ξ)(2v + ξ) n + i,j=2 i<j + 8W 0A i e a i τ i cos(a i θ i ) (4V ξ)(2v + ξ) A i A j cos(a i θ i a j θ j ) (4V ξ)(2v + ξ) 2 e (a i τ i +a j τ j ) [ 32(2V + ξ)(a i τ i + a j τ j +2a i a j τ i τ j ) + 24ξ ] T i = τ i + iθ i ( 3ξ (2V + ξ) + 4a iτ i V (τ, θ) 8(a 2A 2 ) 2 τe 2a 2τ 4W 0a 2 A 2 τe a2τ cos (a 2 θ) 3αλ 2 V m Vm 2 + V, )
Roulette Inflation Racetrack scenario Kähler modulus inflation
axionic direction initial conditions large number of efolds bifurcation points isocurvature perturbations? N = 40... 50 (from N(k) = 62 ln k 6.96 10 5 Mpc 1 +, with = ln 1016 GeV V 1/4 + 1 4 ln V k k V 1 end 3 ln V 1/4 end ρ 1/4 reh )
extremely low P t template P s A S (k/k pivot ) ns 1 with pivot point N = 45 dash-dot line: n s = 0.95,n run = 0 dotted line: n s = 0.95, n run = 0.055
Stochastic regime shaded region: quantum kicks classical motion self reproduction distribution of initial values entering from stochastic region raining down from other holes settling to their minima
Trans-Planckian effects in the Milne Universe FRW metric ds 2 = a(η) 2 ( dη 2 dr 2 f (r) 2 dω 2) Physical scales grow like k a Probing Trans-Planckian Regime? Consistency? Milne Universe f (r) = sinh r, a(η) = e η Scalar field L = φ,µ φ,µ (m 2 R 6 )φ φ Choice of vacuum Adiabatic Minkowsky Vacuum Conformal Vacuum
Calculate 0 T µν 0 Minkowsky vacuum: 0 T µν 0 = 0 conformal vacuum: thermal bath Challenge to Trans-Planckian Challenge: Milne universe is inflating T µν either zero or thermal bath No contributions to Tµν from smallest scales No Trans-Planckian effect!?
Conclusions Scanning Inflation generalized parametrization of inflationary history many degrees of freedom pandorra s box simulated data sets (implicit) priors matter adjusting priors future experiments can beat priors for r not too small Roulette Inflation axionic direction lots of efolds initial conditions low tensors stochastic regime with self reproduction Trans-Planckian effects in the Milne Universe Minkowski space reparametrized as FLRW-type No trans-planckian effects for Minkowski observer Milne observer