Slow-roll violation in production of primordial black hole

Transcription

1 Slow-roll violation in production of primordial black hole Hayato Motohashi Center for Gravitational Physics Yukawa Institute for Theoretical Physics HM & Wayne Hu, PRD 92, (2015), [arxiv: ] PRD 96, (2017), [arxiv: ] PRD 96, (2017), [arxiv: ] nd workshop on gravity and cosmology by young researchers

2 This talk Sasaki, Suyama, Tanaka, Yokoyama, Kocsis, Suyama, Tanaka, Yokoyama, Inflation References Carr, Kohri, Sendouda, Yokoyama, Carr, Kuhnel, Sandstad, Sasaki, Suyama, Tanaka, Yokoyama, DM GW source

3 - High density region collapses at horizon reentry if 0 > 0 D Zel dovich, Novikov (1966) Hawking (1971) Carr (1975) - PBH fraction at formation! # $%& # '(' = 2,-. / 0 10 = erfc, : 7 8, - ; < = - > >? 8 > - 0 M holds at the horizon reenty! 6 : O P ; < Q - P - > Musco, Miller, Harada, Yoo, Kohri, Young, Byrnes, Sasaki, >R > P, MD = S D =

4 - High density region collapses at horizon reentry if 0 0E/ E > 0 D PBH fraction at formation! # $%& # '(' = 2,-. / 0 10 = erfc, : Gaussian / 0 ; < 6:7 8-0 Q holds at the horizon reenty! 6 : S T ; < U - T - > RD = > Zel dovich, Novikov (1966) Spherical collapse of closed universe 7 8, - ; < Hawking (1971) Carr (1975) = - > >? 8 > >? 8 > A B 0 D : Rare event Nearly scale inv. Δ U 6 for a few efolds around the reenty Musco, Miller, Harada, Yoo, Kohri, Young, Byrnes, Sasaki, >V > T, QD = W D =

5 Large peak in Δ " # Δ " # Δ # " 1 PBH A large peak of Δ " # can source PBH. Leading-order slow-roll CMB $ % 1 Δ # " 1# Δ # " 10 IJ End of inf. 8B # $ Approximation $ % $ ' is often used in the literature Δ # 3 <E " 24B #, $ ' <? = 3 3 Is $ % $ ' valid? $ ' 1 = 1 + $ % 2(3 $ % ) <ln$ % <? # $ % 1/1 # $ ' 3 4 /3 # /2 Naively, a large ( )* +, (- seems necessary for PBH.

6 Monochromatic mass function - PBH mass = Horizon mass! =! # = $%& '# ( = ) *+# 10*$ / / 1 10 M N? )O! # *! Horizon reentry - PBH fraction at formation 6 & 89: = 89:>? & ;<; = A B B 2 ( RD: P * = Q%+ ' % H 'I R S $ 2 = / E ( Ω / U P * I 4 E$ 1 10 EF / / 1 2 C = 89: G H I.)* K K )/*

7 CD E= CMB pivot scale Large mass PBH Small mass PBH C 1 C FGHI C + C - efolds between CMB and PBH scales Δ" ln & '()*+ ),- &. + ),- = ln 18 = => ln??. = > ln A A &

8 No go for slow-roll HM, Hu, ! # $ % & ) * ' ( & ( + +, & + 10 */ # 8 8 7/# ; < = 1.3 Δ@ 18 7 # ln 8 8 Given (Ω GHI, K) Δ ' # (N GHI ) and Δ@ Δ ' # N OPH 10 */ SR: Δ # ' R *7 S Δ ln R S Δ@ Δ ln R S

9 No go for slow-roll For (Ω #$%, ') = (Ω *+, ',-. ) HM, Hu, ',-. 10 <1D ' : Smallest PBH mass that does not evaporate by matterradiation equality barring merging and accretion Lower bound ü Δ 0 1 (2 #$% ) : from Δ ;+$ 10 <= ü 2 #$% exits the horizon Δ> 42 after 2 ;+$ = 0.05Mpc <D Slow-roll violation Δ ln G H Δ> > 0.38 for any single-field canonical inflation 10 : L M, N M, O ln N M O ln 2 PBH ',-. Δ> 42

10 No go for slow-roll HM, Hu, (Ω #$, & '() ), -). /,0 Ω 678, & = (10 ;< Ω #$, & '() ), -). /,0 (10 ;< Ω #$, 30& ), -). /,0 > 0.38 > 0.37 > 0.99 PBH = LIGO event scenario Sasaki, Suyama, Tanaka, Yokoyama, D PBH 30& E F, G F, H ln G F H ln K ΔA 17

11 Case study 1: Inflection model Garcia-Bellido, Morales, Ezquiaga, Garcia-Bellido, Morales, SR-V approx.!"!# &' & Δ * & ) * / 0

12 Case study 1: Inflection model HM, Hu, Exact SR-V approx.!"!# &' & Slow-roll violation

13 Case study 1: Inflection model HM, Hu, See also Germani, Prokopec, Exact SR-V approx. Δ # & " $ 24) # * +

14 Case study 1: Inflection model HM, Hu, ( ), + ),, ln + ), ln / Impossible to suppress 10 $% See also Ballesteros, Taoso, for different inflection potential for 10 $% suppression of & '.

15 Improve approximation Large SR violation! "# $ % > 0.38!& SR-V :, -, / Particularly bad Standard SR : Δ 2 3 Optimized SR : Δ $ % Not good $ % Works well - Minimize truncation error by optimization :; = = > - Apply for Horndeski, GLPV, subclass of DHOST Unitary gauge C 3 C A3 = DE F A, - G A 3 C K3 = DE F K 4G K 3 I 3 G A 3 : 3 D 3 3 M N, G K 3 : 3 D 3 I3 M 3 N, HM, Hu, , Kobayashi et al, Gleyzes et al, , Kase et al, Langlois et al,

16 Unitary gauge vs comoving gauge For canonical inflation!" $ # =!& ' $ #!) unitary gauge (!) = 0) = comoving gauge (!& ' $ # = 0)

17 Unitary gauge vs comoving gauge For noncanonical inflation!" $ # =!& ' $ #!) unitary gauge (!) = 0) = comoving gauge (!" $ # = 0), -./.0 =, !9 -./.0 =!9 123 : :; 5 6 < 7 8 For theories with 2nd-order EOM for scalar perturbation Einstein eq constraint eq!9 123, 123 Δ = Γ, 123, -./.0, 123 = : A2 B CDE :F, -./.0, 123 if G 7 8 Γ 0 (canonical case: Γ = 0), 123 const. HM, Hu, Δ J!9 123, 123

18 Optimized slow-roll approximation 1. Write down the formal solution of Mukhanov-Sasaki equation by using Green function (Generalized SR) 2. First order iteration * ln Δ $ +, % = ( ), -., / ln, -, = $: $: ; 6 <=7 $: : > / ln, = 2 + $ 6 ln ln Window function $: $: 5 > BC > D E F E G H I) $ 6 J I K 6 L MI I 6 N MI 5 > $C > D O F O B K 6 L PI I 6 N PI HM, Hu, , Stewart, astro-ph/ , %1 Sound horizon Source function - Function of 2, 4 5 etc - Information of model 4 5 2/2 $ J I + ln 4 5 /+T/2 4 5 L UI + ln V U /+T N UI + ln W U /+T

19 Optimized slow-roll approximation 1. Write down the formal solution of Mukhanov-Sasaki equation by using Green function (Generalized SR) 2. First order iteration *, %= ln Δ $ +, % = ( ), -., / ln, 3. Taylor expand /(ln,) around the evaluation point ln, 2 * ln Δ $ % = / ln, ln, 2 / 5 (ln, 2 ) 4. Truncate at 9 and optimize ln, 2 so that 8 5:7 ln, 2 = 0 HM, Hu, , Sound horizon

20 Optimized slow-roll approximation ln Δ $ % = ' ln ( ) + + Standard SR /,-. 0, ln ( ) ', (ln ( ) ) ln Δ $ = ' 0 : correction = 4(0. (0)' 0 ) HM, Hu, , ln ( ) = 0 : horizon exit 0. (0) /Δ: 0.35 for Δ: 3 ( %A ' ln ( = 2 ln D + $ E ln D F ln ln G H IJ H K L M L N O.P E Q G $ E R. S E T U. Ė V U. G H $J H K W M W I E Q G S E T X. Ė V X.

21 Optimized slow-roll approximation / HM, Hu, , ln Δ $ % = ' ln ( ) + + 0, ln ( ) ', (ln ( ) ) Standard SR,-. ln Δ $ = ' 0 : correction = 4(0. (0)' 0 ) ( %D ln ( ) = 0 : horizon exit 0. (0) /Δ: 0.35 for Δ: 3 Optimized SR ln Δ $ = ' ln (. : correction = 4(0 $ (ln (. )' ln (. ) ln ( ) = ln ( with 0. ln (. = 0 ~1 efold before horizon exit 0 $ ln ( for Δ: 3 1/Δ: $

22 HM, Hu, Case study 2: Running mass model Drees, Erfani, ! =! # + % & '& ln * * &! # 1 +. / ln * + 23 ln * &

23 HM, Hu, Case study 2: Running mass model Slow-roll violation OSR still works well

24 HM, Hu, Case study 3: Slow roll step model Parametrize ln # $ directly ln # $ = & ' + & ) * &, 1 + tanh * * 1 2

25 HM, Hu, Case study 3: Slow roll step model Parametrize ln # $ directly ln # $ = & ' + & ) * &, 1 + tanh * * 1 2 For Δ* < 10, all approximations do not work.

26 Summary PBH production requires slow-roll violation! "# $ % > 0.38!& Previous slow-roll analyses need reconsideration - Inflection model: No sufficient peak in Δ. / - Running mass model: Shift PBH mass scale Improved approximation: Optimized slow roll - Slow-roll step model: OSR remains a good description for models with 10 1 amplification of Δ. / in Δ2 > 10. Applies for Horndeski, GLPV, subclass of DHOST. Unitary gauge = comoving gauge if 3 4#5 const.