!"!!#"$#"%&'&()*+,-.,+/012345!"#$%&'()*+(&,-./0123
5647894444:;<=>?@A@
BC4@DED4FGHI4FD4AJKLMNOI4PD4QRSMI4TD4@MUVWMXVI4YD4Z[\[]VXVI4^D4_D4Z[[`abcDbbdcaI4``b`Dbab`eI4MR4fLHfD

!"!!#"$#"%&'&()*+,-.,+/012345!"#$%&'()*+(&,-./0123
aD !"#$%&'()`D ghijk*/0123
lD +(&,mD nopqrst

!"4uv
Carr & Hawking ’71, ’74, ‘75

wxyz{|˜™ijk4 (1) ∼原始 BH𝒪

š4vSGG›H4œ•Ž…

} ~•€•‚} ƒ„@4…†rst‚} ‡=•ˆ4uv4*‰‚} Š‹ƒA4rsŒ•Ž•†‚} •4c4‘’‚} L“”*•–—‚!"4uv4*+(&,-./0123!"4uv564789

l C`ežŸˆ••¨Œ(©ª•
YT & Vennin ‘21

Escrivà, Kitajima, YT,Yokoyama, Yoo’21, ‘22Abe, Inui, YT, Yokoyama
in prep.

=–¡¢£ijk

原始 BH`¤nopqrst
3Since this is second-order in the perturbations, thesourced GWs are intrinsically non-Gaussian. The sourceis also local, depending only on spatial derivatives of theperturbations, so the resulting bispectrum will peak inmomentum-space configurations where the wavevectors
have similar amplitude (no squeezed component). We
define the projector in Fourier space using the chiral basis&¥2¦§%&~k) ⌦ eR`m (~k),
Teij `m (~k) = eLij (~k) ⌦ eL`m (~k) + eR
(ijL,R
where eij are the polarisation tensors.

(10)

In Eq. (9) thescalar perturbation (⌘, ~k) can be written in terms of theinitial gauge-invariant comoving curvature perturbationas [28]!"4uv4*+(&,-./0123~2~

Saito & Yokoyama ’08, '09
10-6

10-710-8

10-9

10-10

10-11

10-12

10-13

10-14

10-15

10-7
10-8
10-9
10-10
10-11

LISA

10-12
10-13
10-14
10-15

10-5

10-4

10-3

10-2

10-1

Bartolo+ ’18
FIG. 2: The power spectrum of GWs generated by PBHs!"4uv

564789m C`e

rt

Heavy-tailed Curvature Peurbation

δN4¬xV

End of in .

δϕ(t, x)

ϕ0(t)

V(ϕ)

decay

ϕ0(t)} ®¬¯°,-ϕ(τ, x) = ϕ0(τ) + δϕ(τ, x){gμν(τ, x) = a (τ)ημν + hμν(τ, x)

ϕρr : Big-Bang universe

2fl!"4uv4*+(&,-./0123

ghijk*/0123

564789« C`eδN4¬xH

δϕ(t, x)V−1

Separate U. assumption³´µ8¶{|

ϕ0(t)} δN4¬-4±/¯°²44Lyth, Malik, Sasaki ‘04ζ = δN = N(ϕ0 + δϕ) − N(ϕ0)δρ=−H ·ρ0

N = dt H∫

End of in .

V(ϕ)

decay

ϕ0(t)

ϕ

ρr : Big-Bang universe±·›VU¸J›MKH²!"4uv4*+(&,-./0123
flH
∼
2π

ghijk*/0123

564789d C`e

01©¹•} ·[L4JXV››4ζ ≪ 1V ()ζ ≃ Nϕδϕ4º4‹VSJJMVR¼½¾¿¡4N} ·[L4›VL»H4ζ ∼ 1
ζ = ζNL(δϕ)4º4‹VSJJMVR!"4uv4*+(&,-./0123

ghijk*/0123

564789e C`eAÀfD}UVM›

11−3ζ
ζ = − ln (1 − 3ζG) ⇔ ζG = (1 − e )3
33 5 2
Atal+ ’19
= ζG + × ζG + ⋯Ezquiaga+ ’19
5 2
USRBiagetti+’21fNL} AÀVKU4Á›ULV4YP
Vln P()····· −3Ht
ϕ + 3Hϕ = 0 ⇒ ϕ = ϕ0eÂuv4ζ ∼ 11δϕ
ζ = δN = − ln 1 − 3H ·3 (| ϕf | )USRfNL

Gaussianghijk*/0123∝e

Cai+ ’18!"4uv4*+(&,-./0123−3ζ564789

c C`ethPeakeory for PBH•ˆÃÄÅÆÇÈµ¹É¥(¦x

Carr ’75 (Press & Schechter’74)

R

1p3
δR(x) = d y WR(x − y)δ(y) ≳=∫3 ( ρ)!"4uv4ÊÊ∞ρPBH=žŸˆ4º4 ρtot ∫1/3

12πσR2e−δR2/2σR24π 3•ˆË4º4MPBH ∼ MH R=H−1 = 3 ρR!"4uv4*+(&,-./0123+(&,-564789

R=H −1aa C`e

Carr ’75 (Press & Schechter’74)} ¡ÌÍÆ4Î1/3Ï‚} ÐÃÄÑÒÓÔÕ} δR4Ñ012ÖÄ‚?

} MPBH ∼ MH

R=H −1} +(&*×1•ØÑÙÚ¡totfPBHÅÆÇÈµ¹É¥(¦= ρPBH /ρDM•ˆÃÄζG = AζGδ(log k − log k*)
20101010k* = 1.56 × 1013 Mpc−1
Mk* = 1020 g

1-10

10Peak-20

10PS

10-30

0.001

0.005 0.010
0.050 0.100
AgG Kitajima, YT, Yokoyama, Yoo ‘21𝒫

2cf.
Ω∼AΩGWζ r)(!"4uv4*+(&,-./0123+(&,-564789

a` C`e•8!,••¨Œ(©ª•,-+(&,-¢£ijk*Û˜µÜÝÞ©ßàŒ(Øζ原始 BHÄá4‹Px Compaction Func.
!"4uv4*+(&,-./0123+(&,-x564789

al C`e+(&,Bardeen, Bond, Kaiser, Szalay ’86
Yoo, Harada, Garriga, Kohri ’18
Yoo, Gong, Yokoyama ’19
Yoo, Harada, Hirano, Kohri ‘20ãäåµæçè+(&¬̂ = ĝ (ψ(r); μ, k∙)g(r)g(x)4º4012â

dk sin kr
ψ(r) =∫ k kr

μg(k)`êëÃÃÄ−1k∙

ré012ÖÄìíîÄ˜™

npk(μ, k∙) dμdk∙

𝒫x!"4uv4*+(&,-./0123+(&,-564789

am C`eCompaction Func.

8 1 −5ζ/2 ζ/2δ=−eΔe9 a 2H 2Shibata & Sasaki ’99
Harada, Yoo, Nakama, Koga ‘1512=
δ × 4πR dR
V(R) ∫02?
2= [1 − (1 + rζ′) ] >3

R=H −1˜™ijk*ðñζ

R = ae r‡4vSGG›H4òž

Mb.g.

Φ𝒞MMS!"4uv4*+(&,-./0123

th+(&,-𝒞RR￼=G

MMS − Mb.g.

564789aï C`eCompaction Func.
Shibata & Sasaki ’99
Harada, Yoo, Nakama, Koga ‘15=G

MMS − Mb.g.

R12=δ × 4πR dR
V(R) ∫0R2?
2= [1 − (1 + rζ′) ] >3thR=H −1

˜™ijk*ðñ

‡4vSGG›H4òžCompaction Func.4*ðñÑóôõöå÷ø1¯ =V(R) ∫0( → μ > μth(k∙, ⋯))!"4uv4*+(&,-./0123

Atal, Cid, Escrivà, Garriga ’19
Escrivà, Germani, Sheth '19- tting for fNL < 0￼

Escrivà, YT, Yokoyama, Yoo, ‘22𝒞𝒞

22× 4πR > ¯ th ≃5- fNL > 0, exp.-tail, …

+(&,-𝒞R𝒞𝒞fi

8 1 −5ζ/2 ζ/2δ=−eΔe9 a 2H 2564789a« C`e•ˆPrimordial black hole formation in the radiative era

Choptuik ’93Niemeyer & Jedamzik ’94, ‘972ù()ú
MPBH ≃ (μ − μth(k∙, ⋯))0.36!"4uv4*+(&,-./0123MH

R=H −1d ln MPBH
≃ 0.36d ln(μ − μth)

Musco, Miller, Polnarev ’08
Figure 1. Scaling behaviour for MBH as function of (δ − δc )γ c
radiative perfect fluid. For MBH ! MH , the points are well-fitt+(&,-564789

ad C`eûü4±01©¹•²
Kitajima, YT, Yokoyama, Yoo ‘21ζG = AζGδ(log k − log k*)
tuned AgG

1020

1100.01

1fPBH (M)

totfPBH

1010-10Peak-20

1010-30

0.005 0.010
AgG10PS-61010-8

PS0.001-4

Peak0.050 0.100

10-100.010

0.100
M / M k*15𝒫

k* = 1.56 × 1013 Mpc−1
Mk* = 1020 g!"4uv4*+(&,-./0123+(&,-564789

ae C`elocal-type NGYoo, Gong, Yokoyama ‘19ζζ(x) = ℱNG (ζG(x))̂̂ζ(r) = ℱNG (ζG(r))ýþÿ!

32- ζ = ζG + fNLζG
51- ζ = − log (1 − 3ζG) : “exp-tail” in USR3ζG

3 5 2= ζG + × ζG + ⋯5 2USRfNL!"4uv4*+(&,-./0123+(&,-564789

ac C`efNL

Escrivà, YT, Yokoyama, Yoo ‘22ζG = AζGδ(log k − log k*)
tuned AgG
1810180.01

fPBH (M)

tot
fPBH

100.01-12

10fNL ρ -1

-22

100.001

0.005 0.010
Ag G

Peak

fNL ρ 0PS

fNL ρ 1-410fNL ρ -1

Peak

fNL ρ 0PS

fNL ρ 110-6
10-80.050 0.100-10100.0100.100
M / Mk*0.500 15𝒫k* = 1.56 × 1013 Mpc−1
Mk* = 1020 g!"4uv4*+(&,-./0123+(&,-564789`b C`eexp.-tailKitajima, YT, Yokoyama, Yoo ‘211ζ = − log (1 − 3ζG)
31018

1Peak

8

100.01

fPBH (M)tot
fPBH

tuned AgG

0.01
10-12

PeakPSfNL = 5 / 2-4

Gauss

1010-6PS-22

1010-8fNL = 5 / 2
Gauss

0.001

0.010

0.100

-10

10Ag G

0.001

0.010

0.100
M / Mk*1

k* = 1.56 × 1013 Mpc−1
Mk* = 1020 g!"4uv4*+(&,-./0123+(&,-564789`a C`eInduced GWs

norst••¨Œ(©ª•
YT & Vennin ‘21

Escrivà, Kitajima, YT,Yokoyama, Yoo’21, ‘22Abe, Inui, YT, Yokoyamain prep.

=–¡¢£ijk

原始 BH`¤nopqrst
3

Since this is second-order in the perturbations, thesourced GWs are intrinsically non-Gaussian. The sourceis also local, depending only on spatial derivatives of theperturbations, so the resulting bispectrum will peak inmomentum-space configurations where the wavevectors
have similar amplitude (no squeezed component). We
define the projector in Fourier space using the chiral basis&¥2¦§%&~k) ⌦ eR`m (~k),
Teij `m (~k) = eLij (~k) ⌦ eL`m (~k) + eR
(ijL,R
where eij are the polarisation tensors.

(10)

In Eq. (9) thescalar perturbation (⌘, ~k) can be written in terms of theinitial gauge-invariant comoving curvature perturbationas [28]!"4uv4*+(&,-./0123~

2~

Saito & Yokoyama ’08, '09
10-6

10-7

10-8

10-9

10-10

10-11

10-12

10-13

10-14

10-15

10-7
10-8
10-9
10-10
10-11

LISA

10-12
10-13
10-14
10-15

10-5

10-4

10-3

10-2

10-1

Bartolo+ ’18
FIG. 2: The power spectrum of GWs generated by PBHs

norst

564789`l C`enorst••¨Œ(©ª•
YT & Vennin ‘21Abe, Inui, YT, Yokoyamain prep.

=–¡¢£ijk`¤nopqrst
3Since this is second-order in the perturbations, thesourced GWs are intrinsically non-Gaussian. The sourceis also local, depending only on spatial derivatives of theperturbations, so the resulting bispectrum will peak inmomentum-space configurations where the wavevectors
have similar amplitude (no squeezed component). We
define the projector in Fourier space using the chiral basis~k) ⌦ eR`m (~k),
Teij `m (~k) = eLij (~k) ⌦ eL`m (~k) + eR
(ijL,R
where eij are the polarisation tensors.

(10)

In Eq. (9) thescalar perturbation (⌘, ~k) can be written in terms of theinitial gauge-invariant comoving curvature perturbationas [28]!"4uv4*+(&,-./0123~

2~

Saito & Yokoyama ’08, '09
10-6

10-7

10-8

10-9

10-10

10-11

10-12

10-13

10-14

10-15

10-7
10-8
10-9
10-10
10-11

LISA

10-12
10-13
10-14
10-15

10-5

10-4

10-310-2

10-1

Bartolo+ ’18
FIG. 2: The power spectrum of GWs generated by PBHs

norst564789`m C`e"•¹#$$

Adshead+ ‘21□ hk = S(ζp, ζk−p) ⇒ hk = dt′ Gk(t, t′)Sk(ζ, ζ; t′)∫⇒ ⟨hkh′k⟩ = dt′dt′′G(t, t′)G(t, t′′)⟨Sk(ζ, ζ; t′)S′k(ζ, ζ; t′)⟩∫23
ζ = ζG + FNLζG + GNLζG + ⋯ζhζζh+ζhζζnorst￼￼￼￼￼￼￼

￼￼￼￼￼

ζG!"4uv4*+(&,-./0123ζGFNLζζGFNLhζζG￼ζG564789`ï C`e"•¹#$$

Abe, Inui, YT, Yokoyama in prep.

2AζG

3AζG3AζG

3AζG3AζG4AζG4AζG!"4uv4*+(&,-./01234AζG⋯

norst564789`« C`eGW spectrum
Abe, Inui, YT, Yokoyama in prep.

-91013 23
ζ = − log (1 − 3ζG) = ζG + ζG + 3ζG + ⋯32AζG = 5 × 10−3−310-12

LISA-15

DECIGOGW h21010-18

10-2110Preliminary

2AζGk* = 2 × 1012 Mpc−1
Mk* = 6.1 × 1021 g

3AζG4AζG-4 0.001

0.005

5. × 100.010

34h ∼ ζ ,ζ ,⋯cf. Yuan+ ’19, Zhou ‘21

f [Hz]!"4uv4*+(&,-./0123

norst564789`d C`eû} HÀf}UVM›4º4ζ ∼ 14Í01©¹•.Ñ%jµ¡} !"4uv4*+(&,ζ•ˆ*2ù()ú'44*/012Þ¾Æç(&& ÂY4)*/012Þ*+,=} norst& HÀf}UVM›4ÍÆ4ƒQYF4Í-þ.4±/HVNMHL}UVM›40.!‚4cf. Hooshangi+ ’21, Cai+ ’21²& Á„41¾/012Þ*23

45
σ11 2110222̂ =μg(r)ψ(r)+RΔψ(r)−kγ
ψ(r)+RΔψ(r)∙∙∙))]
3γ(1 − γ 2) σ2 (3[ 1 − γ2 (dk 2n2σn =k∫ knpk(μ, k∙) dμ dk∙ =g(k),

γ=

⟨k 2⟩
⟨k 4⟩

,R∙ =3⟨k 2⟩⟨k 4⟩

1 dk sin kr
ψ(r) = 2σ0 ∫ k krg(k)13(3)dxδ (x − xp)δ(μ − μ(xp))δ(k∙ − k∙(xp))∑VΩ ∫Ω
∇g(x )=0dμ dk∙

p2
σ22μk∙2μk∙

2×3μ=μkfP,dμdk∙
1∙3/2(2π)σ0σ1 ( σ2 ) ( σ0 σ2 )3/25
ξ + erf
2 ][5ξ +2 ]28 31 25 28 1 2
5 2+ ξ exp − ξ + − + ξ exp − ξ[ 8 ] ( 5 2 )[ 2 ]}
5π {( 54 )21(ξ−γν)P1(ν, ξ) =exp −ν2 +1 − γ 2 )][ 2(2π 1 − γ 21𝒫𝒫

1 2
1f(ξ) = ξ(ξ − 3) erf
2[2!"4uv4*+(&,-./0123564789

45!"4uv4*+(&,-./0123

56478945

1
ζ = − log (1 − 3ζG)
3tuned AgG
1PeakfNL = 5 / 2-4Gauss

10

0.6-6

10

0.4

10-8-10

10

Mmax

0.8
M(mμȆ 2) / Mk*

fPBH (M)

0.01

rPS

0.2

0.001

0.010

!"4uv4*+(&,-./0123

0.100
M / Mk*

1

0.0
0.310

0.315

0.320
μmȆ2

0.325

5647890.330