Primordial Black holes from Inflation and

Gravitational Waves

Misao Sasaki

Kavli IPMU, University of Tokyo

TG2019 SFU, Vancouver

24 Jan. 2019

Inflation & PBH formation

2

3

cosmic spacetime diagram

a/k0

aend a0

Rad. dom. 1/Hr ~a2

aeq

Matt. dom. 1/Hm ~a3/2

Inflation Hinf ~ 10-5MPl

???

log L

N= log a

1/H0

current Hubble radius

constant

comoving

scale

ç???

“reheating”

0~ -10~ -60~ -120

aini

( = lnHinf

H0 ) ~ -30

a(1GeV)

4

curvature perturbation from inflation• inflaton (~massless) vacuum fluctuations (=Gaussian)

2 2 1 2

, ~ ;kiw tk k k

k

kk e w Haw

φ ϕ ϕ −= =!

≫

rapid expansion renders oscillations frozen at k/a < H (fluctuations become “classical” on superhorizon scales)

22

3

22 / ~

~ ;k kk a H

H k HHak

ϕ δφπ

⎛ ⎞⇒ = ⎜ ⎟

⎝ ⎠≪

• curvature perturbation on comoving slices

··· conserved on superhorizon scale for single-field slow-roll models

··· almost scale-invariant

ℛc = −H·ϕ

δϕ

(almost scale-invariant if is also slowly varying)·ϕ

ϵ = −·H

H2≪ 1for

5

observational constraint on inflationPlanck 2015 results XX

power-law piece-wise continuous (9 segments)𝒫ℛ = A(k /k*)ns−1

ns ≈ 0.968 ⋯ almost scale-invariant

6

Planck constraint

There are some constraints on small scales, but quite weak.

Bringmann et al., arXiv:1110.2482

constraints on small scales are from BHs

observational constraint on inflation

disappears if CDM = WIMP

7

No resolution to say anything precise about higher k.

?

Bayesian reconstruction of the primordial power spectrum

with l < 2300. (Planck 2015)

8

N= log a

a/k*

a* aend a0aini are-entry aeq

a/k0

A peak in the primordial curvature perturbation, which leaves horizon and gets frozen at a*.

Rad.dom. 1/Hr ~a2

log L

1/H0

PBH formation

ç???

k* = Ha*

9

N= log a

a/k*

a* aend a0aini are-entry aeq

a/k0

The peak re-enters horizon during radiation era.If the amplitude > O(0.1), PBH will form.

Rad.dom. 1/Hr ~a2

log L

1/H0

ç???

PBH formation

k* = Ha*

10

a/k*

a* aend a0aini are-entry aeq

a/k0

Rad.dom. 1/Hr ~a2

PBH mass:

Inverse relation:

log L

1/H0

N= log a

1N

a/k*

a* aend a0aini are-entry aeq

a/k0

Rad.dom.1/Hr ~a2

Nosi

PBH mass:

Inverse relation:

The formation of Primordial Black

Holes

log L

1/H0

N= log a

1 1

22 58 0 875810 10 .

PBH HN NM

M M M e MH

J PlPl Pl

1N

1N

11 6

144 42 10

. lnNM§ ·

¨ ¸© ¹

PBH

g

a/k*

a* aend a0aini are-entry aeq

a/k0

Rad.dom.1/Hr ~a2

Nosi

PBH mass:

Inverse relation:

The formation of Primordial Black

Holes

log L

1/H0

N= log a

1 1

22 58 0 875810 10 .

PBH HN NM

M M M e MH

J PlPl Pl

1N

1N

11 6

144 42 10

. lnNM§ ·

¨ ¸© ¹

PBH

g

ç???

PBH mass scale does NOT depend on the reheating

physics

PBH formation

Primordial Black Holes

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What are Primordial BHs?

PBH = BH formed before recombination epoch (ie at z >>1000)

Hubble size region with δρ/ρ=O(1) collapses to form BH

Such a large perturbation may be produced by inflation

PBHs may dominate Dark Matter.

Carr (1975), ….

Carr & Lidsey (1991), …

Ivanov, Naselsky & Novikov (1994), ...

conventionally during radiation-dominated era

12

Supermassive BHs may originate from PBHs.(M ≳ 106M⊙)….

Bean & Magueijo (2002), ...

13

Curvature perturbation to PBH

If , it collapses to form BH

Spins of PBHs are expected to be very small

Hamiltonian constraint (Friedmann eq.)

( )3 2 1 ( ) ~~ cR H δρ ρ⇔ /Young, Byrnes & MS ‘14

3 2( ) ~R H3 0( )R !

2 36 16( )( , ) ( , ) ( , )H t x R t x G t xπ ρ+ = + ⋅ ⋅ ⋅

gradient expansion/separate universe approach

R(3) ≈ −4a2

∇2ℛc ≈8πG

3δρc

δρc

ρ≈ ℛc

k2

a2= H2at

examples

14

hybrid-type inflation

CR grows near the saddle point non-Gauss may become large

non-minimal curvaton

2 2

12

12

( )

( )

L f g

h m

µνµ νφ χ χ

φ χ

= − ∂ ∂

−

Garcia-Bellido, Linde & Wands ‘96, … Domenech & MS ‘16

Abolhasani, Firouzjahi & MS ’11,..Pattison et al. 1707.00537

LIGO BHs = PBHs?

15

MS, Suyama, Tanaka & Yokoyama ‘16

fraction of PBH in dark matter

LIGO ev

ent

rate

[G

pc-3yr

-1]

3-body interaction leads to formation of

BH binaries

2 2

120 20PBHkM M M

T

−

−

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠⊙ ⊙" "

100MeVkpc

tightest constraint at

PBH merger rate

10~M M⊙(cf. Ali-Haïmoud et al., 1709.06576)

ff =

testing PBH hypothesis

16

Nakamura et al. PTEP 2016 (2016) 093E01

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testing PBH hypothesis 2Kocsis, Suyama, Tanaka, Yokoyama, arXiv:1709.09007

BBH Merger Rate at time t:

1 2 1 2 1 2i n t r , ,( ) ( ) ( ) :t tP m m t g m g m m m m mα∝ = +

36 2237 21

α< <• PBH binary scenario

• Dynamical formation in dense stellar systems 4α ≈

O'Leary et al (2016)

mass function

intrinsic probability

clearly distinguishable!

PBH constraints: recent update

1015 1020 1025 1030 1035

MPBH [g]

105

104

103

102

101

f=

PB

H/

DM

BH

Eva

por

atio

n

FemtoKepler

CMBEROS/MACHO

HSC M31 constraint (95% limit)

1015 1010 105 100MPBH [M]

Figure 5 The red-color shaded region show the 95% C.L. upper bound on the PBH mass fraction to DMin the halo regions of MW and M31, derived from our microlensing search of M31 stars based on the“one-night” HSC/Subaru data. To derive this constraint, we took into account the effect of finite sourcesize assuming that all source stars in M31 have a solar radius (see text for details). The effect weakensthe upper bounds at M < 107M. Our constraint can be compared with other observational constraintsas shown by the gray shaded regions: extragalactic -rays from PBH evaporation [32], femtolensing of-ray burst (“Femto”) [33], microlensing search of stars from the satellite 2-years Kepler data (“Kepler”)[18], MACHO/EROS/OGLE microlensing of stars (“EROS/MACHO”) [15], and the accretion effects onthe CMB observables (“CMB”) [34], updated from the earlier estimate [35].

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Niikura et al. ‘1701.02151v2

micro-lensing

λopt > RSch(MPBH)wave effect:

MPBH ≈ 1020 − 1023g Tre-entry ~ 102 -103 TeV

if LIGO BHs=PBHsMS, Suyama, Tanaka, Yokoyama ‘16

window only at

for ~5 yrs obs

PBHs = CDM?

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monocromatic PBH production

End of the 1st stage of inflation

ℒ = R +R2

6M2−

12

(∂χ)2 − V(χ)

χ

R2-inflation

Pi, Zhang, Huang & MS ‘18

f ≈ 0.73

sharp peak

sharp peak in P(k) spike in f

spike

f ≡ΩPBH

ΩCDM∝ exp [−

1𝒫(k) ]

μ2 ≈H2

2nd

H21st

21

μ = 8

Zhang, Hwang, Pi & MS ‘18

• non-negligible PBH formation means

• GWs are produced with amplitude:

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2nd order GWsSaito & Yokoyama ’09, Alabidi et al. ’13, …

𝒫S(k) ≳ 10−2.5

··hij + 3H ·hij − a−2Δhij = Sij

Sij =1a2

∂iℛc∂jℛc + ⋯ ∼k2

a2𝒫S(k)

hij ∼k2

a2H2𝒫S(k) ∼ 𝒫S(k) 𝒫GW(k) ∼ (𝒫S(k))2

2nd order GWs could dominate GW background at f>10-10 Hz (k>104 Mpc-1)

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352

Fig. 1. Top: The energy density of the induced GWs for the power spectrum for a peak width, ∆ =0.0, 1.0 × 10−3, 1.0 × 10−1, 1.0. Bottom: Energy density of scalar-induced GWs associated with PBHformation together with current pulsar constraint (thick solid line segment) and sensitivity of variousGW detectors (convex curves). Solid wedged lines indicate the energy density with the parameters(ΩPBHh2, MPBH) = (10−5, 102M⊙) (left), (10−1, 1020g) (right) for sufficiently small ∆ (thick lines) and∆ = 1.0 (thin lines).

Downloaded from https://academic.oup.com/ptp/article-abstract/126/2/351/1838316by gueston 14 July 2018

GWs test PBH scenarios!

Saito & Yokoyama,arXiv:0912.5317

(ΩPBHh2, MPBH) = (10−5,100M⊙)(ΩPBHh2, MPBH) = (10−1,1020g)

Non-Gaussianity Effect?

-14

-12

-10

-8

-6

-4

-2

ℛ(x) = ℛg(x)

RG Cai, S Pi & MS, arXiv:1810.11000

PBH constraints:If PBHs=CDM LISA must detect induced GWs!

MPBH = 1022 g

fGW= 3

×10-3 Hz

fGW= 3

×10-2 Hz

AR=10-2

AR=10-3

AR=10-4

- -

-

-

()

(

)

fGW= 3

×10-3 Hz

fGW= 3

×10-2 Hz

AR=10-2

AR=10-3

AR=10-4

- -

-

-

(||)

(

)

k3

FNL=0

FNL=10

FNL=20

FNL=50

-

-

-

/

Ω

+FNL [ℛ2g(x) − ⟨ℛ2

g(x)⟩]

log L

log a(t)

Size of the Observable Universe

L∝a(t)

Inflationary Universe Hot Bigbang Universe

probing inflation by GW astronomy

k=const.

L=H0-1~1028cm

~1011cm

Reheating stage

~1018cmLIGO BHs?

CMB B-modes

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PBH formation?

physics at N ~ 20 - 40physics at

N ~ 60

L=H-1

CDM BHs?

∗ Inflation has become the standard model of the Universe. further tests are needed to confirm inflation.

∗ Inflation can produce large curvature perturbation on small scales. PBHs are virtually the only probe on very small scales.

∗ LIGO BHs may be primordial. future GW detectors(+G lensing) will prove/disprove the scenario.

∗ CDM can be dominated by PBHs of M~1020g. secondary GWs may be detected by future GW detectors.

∗ Multi-frequency GW astronomy/astrophysics is arriving!

Summary

GWs will be an essential tool for proving/falsifying PBH scenarios

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