Slow-roll violation in production of primordial black hole Hayato - PowerPoint PPT Presentation

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, 043501 (2015), [arXiv:1503.04810] PRD 96, 023502 (2017), [arXiv:1704.01128] PRD 96, 063503 (2017), [arXiv:1706.06784] 2018.03.03-04 2nd workshop on gravity and cosmology by young researchers

Sasaki, Suyama, Tanaka, Yokoyama, 1603.08338 Kocsis, Suyama, Tanaka, Yokoyama, 1709.09007 This talk Inflation DM GW source References Carr, Kohri, Sendouda, Yokoyama, 0912.5297 Carr, Kuhnel, Sandstad, 1607.06077 Sasaki, Suyama, Tanaka, Yokoyama, 1801.05235

Zel’dovich, Novikov (1966) Hawking (1971) - High density region collapses at horizon reentry Carr (1975) if 0 ≡ 0@/ ̅ @ > 0 D ∼ 0.3 − 0.7 - PBH fraction at formation > < =- . / 0 10 = erfc # $%& , - 6 7 8 > ! ≡ # '(' = 2 ∫ 67 8 ≈ , - ; >?8 , - : - 0 ∝ M holds at the horizon reenty Musco, Miller, 1201.2379 Harada, Yoo, Kohri, 1309.4201 > Young, Byrnes, Sasaki, 1405.7023 < P- > , O P 6 S >RP ⟹ ! ≈ Q - ; M D = 6 6 0 D = 0.95 − 2.2 :

Zel’dovich, Novikov (1966) Hawking (1971) - High density region collapses at horizon reentry Carr (1975) if 0 ≡ 0E/ ̅ E > 0 D ∼ 0.3 − 0.7 Spherical collapse of closed universe - PBH fraction at formation > < =- . / 0 10 = erfc # $%& , - 6 7 8 > ! ≡ # '(' = 2 ∫ 67 8 ≈ , - ; >?8 , - : < => A B ≪ 0 D : Rare event @ > Gaussian / 0 = 6:7 8 ; >?8 6 for a few Nearly scale inv. Δ U efolds around the reenty - 0 ∝ Q holds at the horizon reenty Musco, Miller, 1201.2379 Harada, Yoo, Kohri, 1309.4201 RD > Young, Byrnes, Sasaki, 1405.7023 < T- > , S T 6 W >VT ⟹ ! ≈ U - ; Q D = 6 6 0 D = 0.95 − 2.2 :

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

Monochromatic mass function - PBH mass = Horizon mass Horizon reentry 2 3 4 # * ! ⊙ $%& ) / ∗ *+# ≃ 10 *$ ! = ! # = '# ( = / ∗1 N ? ≃ 10 M )O ! ⊙ % H RD: P * = Q%+ 'I R ∗ S $ - PBH fraction at formation ' E 2 @( ( Ω U P I / ∗ & 89: = 89: > ? * 4 E$ = 6 ≡ & ;<; = @2 / ∗1 B∗ ( > ? @C = A B∗1 2 )/* = 89: G H / ∗ K C ≈ 10 EF / ∗1 I.)* K ⊙

CD E= CMB pivot scale Large mass PBH Small mass PBH C C 1 C FGHI C + - efolds between CMB and PBH scales & '()* + ),- & 0 + Δ" ≡ ln = ln 1.13456 78 & . + ),- ≈ 18 − = => ln ? ∗ ? ∗. − = > ln A A ⊙

HM, Hu, 1706.06784 No go for slow-roll + * &( 7/# + ≈ 10 */ 0 123 4 + % & # 8 +,& ! ≈ ' ( ) ; < = 1.3 $ 5.7# 8 ⊙ 7 8 Δ@ ≈ 18 − # ln 8 ⊙ Given (Ω GHI , K) # (N GHI ) and Δ@ ⟹ Δ ' Δ ln R S Δ@ # ∝ R S *7 ⟹ Δ ln R S SR: Δ ' # N OPH ≈ 10 */ Δ '

HM, Hu, 1706.06784 No go for slow-roll ' ,-. ≈ 10 <1D ' ⊙ : Smallest PBH mass that does not evaporate by matter- For (Ω #$% , ') = (Ω *+ , ' ,-. ) radiation equality barring merging and accretion ⟹ Lower bound 1 2 ;+$ ≈ 10 <= 1 (2 #$% ) ≈ 0.02 ⟹ ×10 : from Δ 0 ü Δ 0 ü 2 #$% exits the horizon Δ> ≈ 42 after 2 ;+$ = 0.05Mpc <D PBH ' ,-. Slow-roll violation ×10 : Δ ln G H > 0.38 Δ> for any single-field canonical inflation Δ> ≈ 42 L M , N M , O ln N M O ln 2

HM, Hu, 1706.06784 No go for slow-roll , -) . / (Ω #$ , & '() ) ⇒ > 0.38 ,0 , -) . / (10 ;< Ω #$ , & '() ) Ω 678 , & = ⇒ > 0.37 ,0 , -) . / (10 ;< Ω #$ , 30& ⊙ ) ⇒ > 0.99 ,0 PBH 30& ⊙ PBH = LIGO event scenario ×10 D Sasaki, Suyama, Tanaka, Yokoyama, 1603.08338 ΔA ≈ 17 E F , G F , H ln G F H ln K

Case study 1: Inflection model Garcia-Bellido, Morales, 1702.03901 Ezquiaga, Garcia-Bellido, Morales, 1705.04861 SR-V approx. !# ≈ − & ' !" & & * + ≈ Δ ) 24. * / 0

HM, Hu, 1706.06784 Case study 1: Inflection model SR-V approx. !# ≈ − & ' !" & Exact Slow-roll violation

HM, Hu, 1706.06784 Case study 1: Inflection model See also Germani, Prokopec, 1706.04226 SR-V approx. & # $ ≈ Δ " 24) # * + Exact

HM, Hu, 1706.06784 Case study 1: Inflection model ( ) , + ) , , ln + ) , ln / Impossible to suppress ×10 $% See also Ballesteros, Taoso, 1709.05565 for different inflection potential for ×10 $% suppression of & ' .

̇ ̇ HM, Hu, 1503.04810, 1704.01128 Improve approximation ! "# $ % > 0.38 Large SR violation !& § SR-V : , - ≈ , / ⟹ Particularly bad - 4 3 ≈ ? § Standard SR : Δ 2 ⟹ Not good 56 4 7 8 9 8 $ % @AB> - 4 3 ≈ ? § Optimized SR : Δ 2 ⟹ Works well 56 4 7 8 9 8 $ % @ABQ R - Minimize truncation error by optimization :; = = > - Apply for Horndeski, GLPV, subclass of DHOST Unitary gauge C 3 C A3 = D E F A , - 3 : 3 I 3 − G A D 3 I 3 3 Kobayashi et al, 1105.5723 G A Gleyzes et al, 1304.4840, 1411.3712 3 : 3 C K3 = D E F K − G K 3 3 M N,× D 3 M N,× Kase et al, 1409.1984 3 4G K Langlois et al, 1703.03797

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

HM, Hu, 1704.01128 Unitary gauge vs comoving gauge $ ∝ !) $ = !& '# For noncanonical inflation !" # $ = 0 ) ⟹ unitary gauge ( !) = 0 ) = comoving gauge ( !" # 5 , -./.0 = , 123 − Δ ≡ J!9 123 − ̇ , 123 67 8 : 5 !9 -./.0 = !9 123 − 6 < 7 8 :; For theories with 2nd-order EOM for scalar perturbation Einstein eq ⊃ constraint eq !9 123 ∝ ̇ , 123 ⟹ Δ = Γ ̇ , 123 : A2 B CDE G , -./.0 − , 123 = :F 7 8 Γ ≈ 0 (canonical case: Γ = 0 ) , -./.0 ≈ , 123 if , 123 ≈ const.

HM, Hu, 1503.04810, 1704.01128 Optimized slow-roll approximation Stewart, astro-ph/0110322 1. Write down the formal solution of Mukhanov-Sasaki equation by using Green function (Generalized SR) 2. First order iteration , ≡ %1 * +, ln Δ $ % = − ( , - . , / ln , Sound horizon ) Window function Source function - Function of 2, 4 5 etc - , = 6 789 $: − 6 <=7 $: − 6 789 $: $: ; : > - Information of model $: / ln , = −2 ln @ + $ 6 ln @ . 4 5 ≡ − ̇ 2/2 $ 5 > BC > D E F E G H − I) 6 4 5 − $ 6 J I − K 6 L MI − I ln 6 N MI J I ≡ + ln 4 5 /+T/2 − 4 5 ≈ 5 > L UI ≡ + ln V U /+T $C > D O F O − B 6 4 5 − K 6 L PI − I ln 6 N PI N UI ≡ + ln W U /+T

HM, Hu, 1503.04810, 1704.01128 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 , Sound horizon ) 3. Taylor expand /(ln ,) around the evaluation point ln , 2 * ln Δ $ % = / ln , 2 + 4 8 5 ln , 2 / 5 (ln , 2 ) 567 4. Truncate at 9 and optimize ln , 2 so that 8 5:7 ln , 2 = 0

HM, Hu, 1503.04810, 1704.01128 Optimized slow-roll approximation / ln Δ $ % = ' ln ( ) + + 0 , ln ( ) ' , (ln ( ) ) ,-. ( ≡ %A § Standard SR ln Δ $ = ' 0 : correction = 4(0 . (0)'′ 0 ) ln ( ) = 0 : horizon exit 1/Δ: 0 . (0) ≈ 1.06 ≃ 0.35 for Δ: ∼ 3 ' ln ( = −2 ln D + $ E ln D F G H IJ H K L M L N O − .P E Q G − $ E R . − S E T U. − . ln E V U. ≈ G H $J H K W M W − I E Q G − S E T X. − . ln E V X.

HM, Hu, 1503.04810, 1704.01128 Optimized slow-roll approximation / ln Δ $ % = ' ln ( ) + + 0 , ln ( ) ' , (ln ( ) ) ,-. ( ≡ %D § Standard SR ln Δ $ = ' 0 : correction = 4(0 . (0)'′ 0 ) ln ( ) = 0 : horizon exit 1/Δ: 0 . (0) ≈ 1.06 ≃ 0.35 for Δ: ∼ 3 § Optimized SR ln Δ $ = ' ln ( . : correction = 4(0 $ (ln ( . )'′′ ln ( . ) ln ( ) = ln ( . ≈ 1.06 with 0 . ln ( . = 0 1/Δ: $ 0 $ ln ( . ≈ −0.36 ~1 efold before horizon exit ≃ 0.04 for Δ: ∼ 3

̃ HM, Hu, 1706.06784 Case study 2: Running mass model Drees, Erfani,1102.2340 & ' & ln * * & % ! = ! # + . 3 ln * & ≈ ! # 1 + / −1 + 2 ln * + 2

HM, Hu, 1706.06784 Case study 2: Running mass model Slow-roll violation OSR still works well

HM, Hu, 1706.06784 Case study 3: Slow roll step model Parametrize ln # $ directly ln # $ = & ' + & ) * − & , 1 + tanh * − * 1 2

HM, Hu, 1706.06784 Case study 3: Slow roll step model Parametrize ln # $ directly ln # $ = & ' + & ) * − & , 1 + tanh * − * 1 2 For Δ* < 10 , all approximations do not work.

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 / in Δ2 > 10 . with 10 1 amplification of Δ . § Applies for Horndeski, GLPV, subclass of DHOST. Unitary gauge = comoving gauge if 3 4#5 ≈ const.