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

SLIDE 1

Slow-roll violation in production of primordial black hole

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

Hayato Motohashi

Center for Gravitational Physics Yukawa Institute for Theoretical Physics

SLIDE 2

Inflation

References Carr, Kohri, Sendouda, Yokoyama, 0912.5297 Carr, Kuhnel, Sandstad, 1607.06077 Sasaki, Suyama, Tanaka, Yokoyama, 1801.05235 Sasaki, Suyama, Tanaka, Yokoyama, 1603.08338 Kocsis, Suyama, Tanaka, Yokoyama, 1709.09007

This talk

DM GW source

SLIDE 3

PBH fraction at formation

! ≡

#$%& #'(' = 2 ∫ ,- . / 0 10 = erfc ,- 678 ≈ 6 : 78 ,- ; < =-

> >?8 >

High density region collapses at horizon reentry

if 0 ≡ 0@/ ̅ @ > 0D ∼ 0.3 − 0.7

Zel’dovich, Novikov (1966) Hawking (1971) Carr (1975)

0 ∝ M holds at the horizon reenty

⟹ ! ≈

6 : OP Q- ; < P-

> >RP > ,

MD =

S 6 6 0D = 0.95 − 2.2

Musco, Miller, 1201.2379 Harada, Yoo, Kohri, 1309.4201 Young, Byrnes, Sasaki, 1405.7023

SLIDE 4

PBH fraction at formation

! ≡

#$%& #'(' = 2 ∫ ,- . / 0 10 = erfc ,- 678 ≈ 6 : 78 ,- ; < =-

> >?8 >

Gaussian / 0 =

@ 6:78 ; < =>

>?8 >

AB ≪ 0D : Rare event

High density region collapses at horizon reentry

if 0 ≡ 0E/ ̅ E > 0D ∼ 0.3 − 0.7

Spherical collapse of closed universe

Zel’dovich, Novikov (1966) Hawking (1971) Carr (1975)

0 ∝ Q holds at the horizon reenty

⟹ ! ≈

6 : ST U- ; < T-

> >VT > ,

QD =

W 6 6 0D = 0.95 − 2.2

Musco, Miller, 1201.2379 Harada, Yoo, Kohri, 1309.4201 Young, Byrnes, Sasaki, 1405.7023

Nearly scale inv. ΔU

6 for a few

efolds around the reenty RD

SLIDE 5

Large peak in Δ"

# A large peak of Δ"

# can source PBH.

Leading-order slow-roll Approximation $% ≈ $' is often used in the literature Is $% ≈ $' valid? Naively, a large

( )* +, (-

seems necessary for PBH. $% ≡ − ̇ 1/1# $' ≡ 34/3 #/2 $' $% = 1 + 1 2(3 − $%) <ln$% <?

#

@ Δ"

#

Δ"

# ≈

1# 8B#$% Δ"

# ≈

3 24B#$' , <E <? = − 3′ 3

CMB PBH $% ∼ 1 End of inf. Δ"

# ∼ 10IJ

Δ"

# ∼ 1

SLIDE 6

PBH mass = Horizon mass

! = !# =

$%& '#( = ) *+# ≃ 10*$ /∗ /∗1

2 3 4#

* !⊙

Monochromatic mass function

PBH fraction at formation

6 ≡

&89: &;<; = =89:>?

@(

=A

B∗ B∗1 @2 (>? @C

≈ 10EF

/∗ /∗1

2 C

=89:GH I.)* K K⊙ )/*

≃ 10M

N? )O !⊙

RD: P* =

Q%+ ' %H 'I R∗S$

=

/∗ /∗1 E2

( ΩUPI

*4E$

Horizon reentry

SLIDE 7

efolds between CMB and PBH scales

Δ" ≡ ln

&'()*+),- &.+),-

= ln

&0+ 1.1345678

≈ 18 − =

=> ln ?∗ ?∗. − = > ln A A⊙

C CD E=

CMB pivot scale Large mass PBH Small mass PBH

C+ CFGHI C1

SLIDE 8

! ≈

# $ %& '( ) * &(

+ +,& + ≈ 10*/

01234+ 5.7# 8 8⊙ 7/#

;< = 1.3 Δ@ ≈ 18 −

7 # ln 8 8⊙

No go for slow-roll

Given (ΩGHI, K) ⟹ Δ'

#(NGHI) and Δ@

Δ'

# NOPH ≈ 10*/

SR: Δ'

# ∝ RS *7 ⟹ Δ ln RS

Δ ln RS Δ@

HM, Hu, 1706.06784

SLIDE 9

No go for slow-roll

For (Ω#$%, ') = (Ω*+, ',-.) ü Δ0

1(2#$%) ≈ 0.02 ⟹ ×10: from Δ0 1 2;+$ ≈ 10<=

ü 2#$% exits the horizon Δ> ≈ 42 after 2;+$ = 0.05Mpc<D

Slow-roll violation Δ ln GH Δ> > 0.38 for any single-field canonical inflation

HM, Hu, 1706.06784

Δ> ≈ 42 ×10:

LM, NM, O ln NM O ln 2

PBH ',-.

',-. ≈ 10<1D'⊙: Smallest PBH mass

that does not evaporate by matter- radiation equality barring merging and accretion ⟹ Lower bound

SLIDE 10

No go for slow-roll

(Ω#$, &'()) ⇒

, -) ./ ,0

> 0.38 Ω678, & = (10;<Ω#$, &'()) ⇒

, -) ./ ,0

> 0.37 (10;<Ω#$, 30&⊙) ⇒

, -) ./ ,0

> 0.99 ΔA ≈ 17 ×10D

EF, GF, H ln GF H ln K

PBH 30&⊙

HM, Hu, 1706.06784 Sasaki, Suyama, Tanaka, Yokoyama, 1603.08338

PBH = LIGO event scenario

SLIDE 11

Case study 1: Inflection model

Garcia-Bellido, Morales, 1702.03901 Ezquiaga, Garcia-Bellido, Morales, 1705.04861

SR-V approx. !" !# ≈ − &' & Δ)

* + ≈

& 24.*/0

SLIDE 12

Case study 1: Inflection model

Slow-roll violation

HM, Hu, 1706.06784

Exact SR-V approx. !" !# ≈ − &' &

SLIDE 13

Case study 1: Inflection model

Exact SR-V approx. Δ"

# $ ≈

& 24)#*+

See also Germani, Prokopec, 1706.04226 HM, Hu, 1706.06784

SLIDE 14

Case study 1: Inflection model

Impossible to suppress ×10$% HM, Hu, 1706.06784 See also Ballesteros, Taoso, 1709.05565 for different inflection potential for ×10$% suppression of &'.

(), +), , ln +) , ln /

SLIDE 15

Improve approximation

Large SR violation

! "# $% !&

> 0.38 § SR-V : ,- ≈ ,/ ⟹ Particularly bad § Standard SR : Δ2

3 ≈

4

5647898$%

⟹ Not good § Optimized SR : Δ2

3 ≈

4

5647898$%

⟹ Works well

Minimize truncation error by optimization :; = =>
Apply for Horndeski, GLPV, subclass of DHOST

?

@AB>

Kobayashi et al, 1105.5723 Gleyzes et al, 1304.4840, 1411.3712 Kase et al, 1409.1984 Langlois et al, 1703.03797

CA3 = DEFA,- GA

3

̇ I3 − GA

3:3

D3 I3 CK3 = DEFK 4GK

3

̇ MN,×

3

− GK

3:3

D3 MN,×

3

?

@ABQR

HM, Hu, 1503.04810, 1704.01128

Unitary gauge C3

SLIDE 16

Unitary gauge vs comoving gauge

For canonical inflation !"#

$ = !&'# $ ∝ !)

⟹ unitary gauge (!) = 0) = comoving gauge (!&'#

$ = 0)

SLIDE 17

Unitary gauge vs comoving gauge

For noncanonical inflation !"#

$ = !&'# $ ∝ !)

⟹ unitary gauge (!) = 0) = comoving gauge (!"#

$ = 0)

,-./.0 = ,123 −

5 678

!9-./.0 = !9123 −

: :; 5 6<78

For theories with 2nd-order EOM for scalar perturbation Einstein eq ⊃ constraint eq !9123 ∝ ̇ ,123 ⟹ Δ = Γ ̇ ,123 ,-./.0 − ,123 =

: A2 BCDE :F G 78

Γ ≈ 0 (canonical case: Γ = 0) ,123 ≈ const.

HM, Hu, 1704.01128

Δ ≡ J!9123 − ̇ ,123 ,-./.0 ≈ ,123 if

SLIDE 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 ,

HM, Hu, 1503.04810, 1704.01128

Window function Source function , ≡ %1

Function of 2, 45 etc
Information of model

Stewart, astro-ph/0110322 Sound horizon

, = 6 789 $:

$:;

− 6 <=7 $:

:>

− 6 789 $:

$:

/ ln , = −2 ln @ + $

6 ln @ .

ln

5> BC>DEFEGH − I) 6 45 − $ 6 JI − K 6 LMI − I 6 NMI

ln

5> $C>DOFO − B 6 45 − K 6 LPI − I 6 NPI

≈ 45 ≡ − ̇ 2/2$ JI ≡ + ln 45 /+T/2 − 45 LUI ≡ + ln VU /+T NUI ≡ + ln WU /+T

SLIDE 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 ,2 + 4

567 *

85 ln ,2 / 5 (ln ,2)

4. Truncate at 9 and optimize ln ,2 so that

85:7 ln ,2 = 0 , ≡ %=

Sound horizon HM, Hu, 1503.04810, 1704.01128

SLIDE 20

Optimized slow-roll approximation

ln Δ$ % = ' ln () + +

,-. /

0, ln () ' , (ln ()) § Standard SR ln Δ$ = ' 0 : correction = 4(0.(0)'′ 0 ) ≃ 0.35 for Δ: ∼ 3

ln () = 0 : horizon exit 0.(0) ≈ 1.06 1/Δ:

( ≡ %A

HM, Hu, 1503.04810, 1704.01128 ' ln ( = −2 ln D + $

E ln D F

ln

GH IJHKLMLNO − .P E QG − $ E R. − S E TU. − . E VU.

ln

GH $JHKWMW − I E QG − S E TX. − . E VX.

≈

SLIDE 21

Optimized slow-roll approximation

ln Δ$ % = ' ln () + +

,-. /

0, ln () ' , (ln ()) § Standard SR ln Δ$ = ' 0 : correction = 4(0.(0)'′ 0 ) ≃ 0.35 for Δ: ∼ 3 § Optimized SR ln Δ$ = ' ln (. : correction = 4(0$(ln (.)'′′ ln (. ) ≃ 0.04 for Δ: ∼ 3

ln () = 0 : horizon exit 0.(0) ≈ 1.06 1/Δ: 0$ ln (. ≈ −0.36 ln () = ln (. ≈ 1.06 with 0. ln (. = 0 ~1 efold before horizon exit 1/Δ:$

( ≡ %D

HM, Hu, 1503.04810, 1704.01128

SLIDE 22

Case study 2: Running mass model

HM, Hu, 1706.06784 Drees, Erfani,1102.2340

! = !

# + % & '& ln * *&

≈ !

# 1 + ̃ . / −1 + 2 ln * + 2

3 ln * &

SLIDE 23

Case study 2: Running mass model

HM, Hu, 1706.06784

Slow-roll violation OSR still works well

SLIDE 24

Case study 3: Slow roll step model

HM, Hu, 1706.06784

Parametrize ln #$ directly ln #$ = &' + &)* − &, 1 + tanh * − *1 2

SLIDE 25

Case study 3: Slow roll step model

HM, Hu, 1706.06784

Parametrize ln #$ directly ln #$ = &' + &)* − &, 1 + tanh * − 1 2 For Δ < 10, all approximations do not work.

SLIDE 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 101 amplification of Δ.

/ in Δ2 > 10.