Towards Multi-field Inflation with a Random Potential

Transcription

1 Towards Multi-field Inflation with a Random Potential Jiajun Xu LEPP, Cornell Univeristy Based on H. Tye, JX, Y. Zhang, arxiv: and work in progress 1

2 Outline Motivation from string theory A scenario with random potential Primordial perturbations Power spectrum, (Non-Gaussianity) Anything observable? Conclusions 2

3 Inflation is an elegant explanation to: The flatness problem The horizon problem... The primordial seeds of structure formation 3

4 Inflation with a Single Field one scalar field + flat potential V (φ) H 2 = sufficient expansion + nearly scale invariant density perturbations V 3M 2 pl δρ ρ H2 φ φ 4

5 A Multi-field Perspective Many (~100) scalar fields from string theory (moduli, dilaton...) + complicated potential Some moduli are stabilized, some participate in inflation with total number D 1 A convoluted inflaton path (random walk) 5

6 An Example of Random Potential α i = M 4 i e Si inst V (ρ i, φ i ) = V 0 (ρ j )+α i cos φ i βij + β ij cos + U(ρ i, φ i ) +... ( φi f i ( φi f i φ j f j are axion fields serving as inflatons V 0 (ρ i ) is the moduli potential, contains vacuum energy for inflation, relatively flat U(ρ i, φ i ) couples moduli and axions, introduces randomness ) ) periodic and regular impurities & randomness 6

7 Inflation with a Random Potential S = d 4 x g 3M 2 plh 2 = 1 2 φ I 2 + V ( R 2 1 ) 2 gµν µ φ I ν φ I V ( φ) Ḣ = 1 2M 2 pl φ2 I ɛ Ḣ H 2 = φ2 I 2M 2 pl H2 σ 2 φ2 I V ( φ) provides the energy for inflation σ 2 <H 2 M 2, so that pl V ( φ) ɛ < 1 The inflaton scatters while drifting down the potential, with drift velocity v, the refraction index χ σ2 v 2 1 7

8 The Fokker-Planck Description P [ t = v( φ) ] [ P + I J D IJ ( φ) ] P The simplest case: D IJ = λδ IJ P ( φ,t) = (4πλ t) D 2 exp ( v( φ,t)=const φ vt 2 4λ t ) P ( φ,t) is the field space probability density No density perturbations so far, this is the homogeneous background, no matter how convoluted the path is. 8

9 Primordial Perturbations ds 2 = (1 2Φ)dt 2 + a(t) 2 (1 + 2Φ)dx i dx i ζ = Φ H δρ ρ, Q I = δφ I φ I H Φ ζ = Ḣ φ Q (k ah) δφ Q φ δt E = Q φ δn = H Q φ T a 1 a t 1/2 δt T Hδt E = δn Horizon Crossing End of Inflatio Radiation Domination MD / Last Scattering t 9

10 Adiabatic and Entropic Modes φ 1 Q s Q 1 perturbations One adiabatic mode tangent to the inflaton δρ 0 path, leads to. Q σ Q 2 Q σ Background path φ 2 (D-1) Entropic modes Q s orthogonal to the inflaton path, do not perturb the δρ =0 energy density. [C. Gordon, D. Wands, B. A. Bassett, R. Maartens, astro-ph/ ] ζ = Ḣ σ Q σ, Q σ Q I e I σ Entropic perturbations only exist for multi-field inflation. 10

11 Q s Q σ A shorter inner path gives extra time delay. In single field inflation, after horizon ζ =0 crossing. A bending path convert into time Q s delay, leading to superhorizon evolution of. ζ ζ = 2Ḣ σ ėi σ Q I end of inflation 11

12 Power Spectrum For single field inflation, one evaluates at the time of horizon crossing. ensures the result does not change by the end of inflation. For multi-field inflation, generically, ζ 2 ζ 2 t In principle, one can integrate t ζ =0 ζ 0 ζ 2 However, the inflaton path is a random walk, so is a random variable! θ ζ = 2Ḣ σ e I σq I te e I σ = dei σ dt = dθ dt 12

13 Random Jumps in ζ The inflaton path θ ζ The random jumps in ζ ζ = 2Ḣ σ θ Q κ N e N e N e =0 From t to t E, there are N random jumps, e / N e giving ζ 2H σ Θ Q κ Θ 2 θ 2 N e N e 13

14 P ζ (k) [ H 4 4π 2 σ 2 + ] H4 4π 2 σ 2 ϑn e k=ah ϑn e 4(D 1)Θ 2 N e N e The first term comes from the adiabatic mode. The second term is the contribution from the entropic modes. The first term is wrong, it should exhibit fluctuations due to the randomness in σ 2. The entropic contributions dominate, P ζ (k) H4 4π 2 σ 2 ϑn e H2 8π 2 M 2 pl ɛϑn e n s 1 d ln P ζ d ln k 2ɛ η 1 N e 14

15 Fluctuations in the Power Spectrum 3~4 efolds, 200~300 multiples/efold multiples/bin 15

16 On small angular scales, PLANCK has much better systematics than WMAP, and cosmic variance is negligible. There may be a better chance to see the fluctuations. Also check both TT and TE spectrums. 16

17 Small Non-Gaussianity Generically, order of magnitude f NL ζ3 ζ 2 2 ζ Ne N e Θ f NL D 3 Θ 3 ( ) 3/2 N e N e ( ) 2 D 2 Θ 2 N e N e 1 DΘ N e N e 17

18 Conclusions Multi-field Inflation with a random potential is a natural scenario motivated by the string landscape The random walk nature of the inflaton allows the entropic modes to feed into the adiabatic mode in a random way. Our analysis is only the first step towards fully quantifying such an effect. The final power spectrum has a dominant contribution from entropic modes, while the adiabatic mode gives fluctuations, might be observable by PLANCK. Non-Gaussianity is generically small. 18