Stochas(c Infla(on and Primordial Black Holes

Transcription

1 Stochas(c Infla(on and Primordial Black Holes Vincent Vennin IHP Trimester, «Analy(cal Methods» Paris, 17th September 2018

2 Outline Quantum State of Cosmological Perturba(ons The Stochas(c-δN Infla(on Formalism Primordial Black Holes Based on: VV and A. Starobinsky, (EPC) H. Assadullahi, H. Firouzjahi, M. Noorbala, VV, D. Wands, (JCAP) VV, H. Assadullahi, H. Firouzjahi, M. Noorbala, D. Wands, (PRL) C. Pa^son, VV, H. Assadullah, D. Wands, (JCAP) September 2018 IHP Trimester 0/15

3 Cosmological Perturbations in In5lation Infla(on is a high energy phase of accelerated expansion in the early Universe ds2 = dt2 + a2 (t)dx2 with a > 0 Quantum vacuum fluctua(ons are stretched to cosmological scales k wavelength Par(cle Produc(on à a/a Hubble radius (me CMB à Minkowski vacuum Structure Forma(on Quantum fluctua(ons sourcing the background September /15

4 Cosmological Perturbations in In5lation One scalar degree of freedom: v / (curvature perturba(on) / T/T (CMB T fluctua(on) i = O k2r 3+ ki Wigner func(on Evolution Equation with ki = W (v k,p k )= W (v, p, t) {W 1 coshr k 1X e 2in' k ( n=0 1) n tanh n r k n k,n k i Two-mode squeezed state (Gaussian state) dx 2 2 (v k x 2 )e ip kx (v k + x 2 ) (v, p, t),h(v, p, t)} Poisson Bracket For quadra8c Hamiltonians 2/15

5 Quantum State of Cosmological W (v, p, t) {W (v, p, t),h(v, p, t)} Poisson Bracket September 2018 IHP Trimester 3/15

6 Cosmological Perturbations in In5lation One scalar degree of freedom: v / (curvature perturba(on) / T/T (CMB T fluctua(on) i = O k2r 3+ ki Wigner func(on Evolution Equation with ki = W (v k,p k )= W (v, p, t) {W 1 coshr k 1X e 2in' k ( n=0 1) n tanh n r k n k,n k i Two-mode squeezed state (Gaussian state) dx 2 2 (v k x 2 )e ip kx Quantum Mean Value and Stochastic Average E Z DÔ (ˆv, ˆp) ' W (v, p) O (v, p)dvdp quant Super-Hubble limit (v k + x 2 ) (v, p, t),h(v, p, t)} Poisson Bracket For quadra8c Hamiltonians Example: hvpi! squeezed e N + i 2 ~ 4/15

7 Stochas(c Formalism Starobinsky, 1986 The physical scales probed in the CMB are super-hubble at the end of infla(on ˆ (x) = ˆcg + Z dk (2 ) 3/2 k ah k (t) e ikx â k +h.c. Upshot: Derive a (stochas(c and classical) effec6ve theory for the coarse-grained part of the field, integra(ng out the small wavelength modes. September 2018 IHP Trimester 5/15

8 with At leading order in slow roll: ˆ 1 = Z Stochas(c Formalism (2 ) ˆcg + apple V 0 ˆcg 3H 2 = ˆ 1 k ah Starobinsky, 1986 k (t) e ikx â k +h.c. Modes smaller than the coarsegraining scale are constantly escaping the Hubble radius and source the coarse-grained sector. ( ah) 1 k 1 super-h sub-h N Large Squeezing Approxima(on: ˆ 1! 1 quantum operator stochas6c variable September 2018 IHP Trimester 6/15

9 Primordial Black Holes from Infla(on Primordial density perturba(ons when modes re-enter the Hubble radius ager infla(on k=ah Rare fluctua(ons exceeding cri(cal value collapse to form black holes Mass frac(on (M) < 10 8 P ( ) > c 1 (M) = Z 1 c P ( )d 7/15

10 The Stochas(c-δN Formalism t f t in uniform density slice δρ(x)=0 ~x = const. (t, x) =N (t, x) N 0 (t) N spa(ally flat slice Ψ(x)=0 number of e-fold is a stochas(c variable N ( ) coarse grained = N hn i Moments obey an itera(ve rela(on (Vennin and Starobinsky 2015) hn n i 00 v 0 v 2 hn n i 0 n = N n 1 vm 2 Pl September 2018 IHP Trimester 8/15

11 Full PDF required for PBHs! Define characteris(c func(on (includes all the moments) N (t, )= e itn ( ) Z = e itn ( ) P (N, )dn Obeys par(al differen(al 2 2 v + it vm 2 N (t, )=0 Pl Inverse Fourier transform gives full probability distribu(on P (N, )= 1 Z 1 e itn N (t, )dt 2 1 9/15

12 Example: v( )=v 0 apple1+ 0 p v v 2 v 00 v 02 not the naïve criterion! full result slow roll classical limit classical limit stochastic limit 10 0 P well/ 0 well / 0 10/15

13 Example: v( )=v 0 apple1+ 0 p Classical Regime v P (v0n ) v =0.02 v =0.03 v =0.04 v =0.05 v =0.06 NLO NNLO well v 0 N Is the Gaussian approxima(on sufficient? 11/15

14 Example: v( )=v 0 apple1+ 0 p 4.0 Stochas(c Regime 1 v P (N /µ 2 ) µ 2 = 2 well v 0 M 2 Pl ( end)/ well well N /µ /15

15 Example: v( )=v 0 apple1+ 0 p Stochas(c Regime v mass fraction Observationally excluded well µ 2 = µ = end + well = end well 2 well v 0 M 2 Pl 13/15

16 Example: v( )=v 0 apple1+ 0 p Observationally excluded = end + well = end well µ P (N /µ 2 ) Stochas(c Regime µ 2 = 2 well v 0 M 2 Pl N /µ /15 ( end)/ well

17 Conclusions Stochas6c-δN needed to calculate primordial density perturba(ons beyond perturba(ve approach In the classical regime, Gaussian approxima6on may fail! Primordial Black Hole bounds require N<1 in quantum diffusion regime Extension to mul6-field? Transient slow-roll viola6on (inflec(on point models)? September 2018 IHP Trimester 15/15