Formation of Primordial Black Holes in Double Inflation

Masahiro Kawasaki(ICRR and Kavli-IPMU, University of Tokyo)

Based on MK Mukaida Yanagida, arXiv:1605.04974 MK Kusenko Tada Yanagida arXiv:1606.07631 Inomate MK Tada in preparation

1. Introduction

• Primordial Black Holes (PBHs) have attracted interest for many years

• PBHs can be formed by gravitational collapse of over-density region with Hubble radius in the early universe

• PBHs with mass < 1015 g would have evaporated by now through Hawking radiation

• PBHs with larger masses can exist in the present universe but various constraints on their abundance are imposed

• PBHs can give a significant contribution to dark matter if MBH = 1021-24 g

• Large density fluctuations δ with O(0.1) are required for PBH formation but δ ~ O(10-5 ) on CMB scale

2

Hawking (1971)

Hawking (1974)

• We consider double inflation (preinflation+new inflation)

Preinflation (no specific model is required) accounts for perturbations on large scale observed by Planck

New inflation ( after preinflation) with e-fold Nnew < 50 produces large curvature perturbations on small scales

• This double inflation is realized in supergravity framework

3

P(k)

k

preinflation new inflation

kCMB kPBH

Today’s Talk

1. Introduction

2. PBH formation in radiation dominated universe

3. New inflation model and PBH formation

4. New inflation in supergravity

5. Production of gravitational waves

6. Conclusion 　

4

2. PBH formation in radiation dominated universe• Region with Hubble radius collapses if its over-density is

higher than δc (≈0.3)

• PBH mass

• PBH abundance is estimated by Press-Schechter formalism

• PBH mass fraction β=ρPBH(M)/ρ

• Present PBH fraction to DM

5

MPBH ' 3.6M

k

106Mpc1

2

' 4.5M

T

0.1GeV

2

(M) =

Z

c

d1p

22(k)

exp

2

22(k)

[ fraction per log M ]

fPBH,tot =

Zd logMfPBH(M)

P(k)P(k) O(102)

for PBH formation

2(k) : variance of the comoving density perturbation coarse-grained on k1

2(k) =

Zd log k0 W 2

(k0/k)16

81

(k0/k)4P(k0)

fPBH(M) =PBH(M)

DM' 1.3 108 (M)

MPBH

M

1/2

3. New Inflation after Preinflation• Potential for new inflation

Hubble

• Slow-roll parameter

• Curvature perturbation

PBH formation

6

'

V (')

=1

2

V 0

V

2

=1

2

c

v2 ' 2ng

'n1

v2+ · · ·

2

=V 00

V= 2n(n 1)g

'n2

v2+ · · ·

V (') =v2 g'n

2 cv2' 1

2v4'2

n = 3, 4, · · · Mp = 1

P1/2 O(0.1) for c v4

P1/2 =

Hinf

2

1p2

' 1

2p3

v4

c+ v2'+ 2ng'n1 1

2p3

v4

c(' . c

v2)

v4

Hinf 'v2p3

1 1 ) inflation

New inflation after preinflation• Initial value for the inflaton ?

Interaction with the inflaton of preinflation Hubble induced mass term

Just before new inflation

• Shape of power spectrum of curvature perturbationsdepends on κ > 0 or κ < 0spectral index

7

'

V H2'2

V () +V H2'2 cv2'+ · · · ) ' cv2

H2

'ini c

v2

H v2

P1/2 1

2p3

v4

c O(0.1) for c v4

ns 1 ' 2 6 ' 3c2

v4 2 4n(n 1)g

'n2

v2

(e.g. V m22'2)

• κ < 0

power spectrum has a peak when

• κ ~0

power spectrum is almost flat around

• κ > 0

power spectrum takes the largest value at

8

V

'V 00 0

'

P

V 00 0

' 'ini

V

'

'

Pred-tilted ns 1 < 0

) ns 1 = 0

'

P

flat

PBHs with very wide range of mass are produced

PBHs with wide range of mass are produced

c2/v4 1

O(0.1)' 'ini

PBHs with narrow range of mass are produced

V

'

|| . c2/v4 v4 1

Power spectrum of curvature perturbations• Estimate power spectrum using slow-roll approximation

• Another peak at k~106 Mpc-1

Hubble induced term cannot neglected at the beginning of new inflation, which flattens the potential

However, it should be examined more carefully 9

0.01 100.00 106 1010 1014 1018 1022

10-7

10-5

0.001

0.100

k[Mpc-1]

ζ

dashed

v = 104

= 0.18

c = 1.56 1016

g = 4.26 103

solid

v = 104

= 0.44

c = 5.20 1016

g = 5.04 103

μ distortion of CMB due to Silk damping

κ < 0

reenter the horizon before new inflation

MK Kusenko Tada Yanagida (2016)

flat peak

Abundance of produced PBHs

• Total PBH fraction to DM can be 1

• Another peak at GW150914 recently detected by LIGO/Virgo collaboration?

10

1016

1021

1026

1031

1036

10-9

10-7

10-5

0.001

0.100

MPBH [g]

ΩPBH/ΩDM

EGγ

Femtolensing

Kepler

EROS/MACHO

FIRAS

WMAP3

White Dwarf

PBH/DM =

Zd logMfPBH(M) ' 1

PBH fraction to DM in each logarithmic mass bin

MK Kusenko Tada Yanagida (2016)

MPBH 30M

κ < 0

Abundance of produced PBHs

• For PBHs with wide range of mass are produced

• For PBHs with very narrow range of mass are produced

• For both cases PBH can be DM

11

v = 4 104

c = 0.6v4

g = 65 103

v = 4 104

c = 0.4v4

= 0.25g = 0.30 103

c2/v4 1

O(0.1)

κ ~ 0

MK Mukaida Yanagida (2016)

O(0.1)

c2/v4 1

κ > 0

KeplerHawking

Star Formation

FemtolensingMACHO/EROS

FIRASWMAP3

NS in GCs

ΩPBH/Ωc

KeplerHawking

Star Formation

FemtolensingMACHO/EROS

FIRASWMAP3

NS in GCs

ΩPBH/Ωc

Another peak at large PBH mass• The corresponding perturbations reenter the horizon before

new inflation

• We need careful numerical calculation including all 1st order perturbations of metrics and scalar fields

• Chaotic inflation + New inflation with

• Peak height crucially depends on the Hubble induced term

12

cH=0.583

-4

10

10-19

10-9

10-14

PB

HD

MΩ

/

Ω

1016

1021

1026

1031

1036

M [g]PBH

cH=1

-4

10

10-19

10-9

10-14

1016

1021

1026

1031

1036

M [g]PBH

PB

HD

MΩ

/

Ω

V =

4m22'2 H2'2

= 1 = 0.58

Inomate MK Tada (2016)

4. New Inflation in Supergravity• Potential of a scalar field in supergravity

• Superpotential W (assuming Z2nR R-symmetry)

• Kähler potential

• Inflaton potential

• Potential minimum

13

V () = eK(DW )K(DW ) 3|W |2

W () = v2 g

n+ 1n+1

K() = ||2 +

4||4 + · · ·

V (') = v4

2v4'2 g

2n/21v2n + · · ·

Inflaton ' =p2Re

'

V (')

hV (')i ' 3v4v2

g

2/n

canceled by SUSY contribution µ4SUSY

m3/2 ' µ2SUSY/

p3 ' v2

v2

g

1/n

DW =@W

@+

@K

@W

K =

@2K

@@

1

Izawa Yanagida (1997)

New Inflation in Supergravity• Constant term in superpotential

Linear term

• Hubble induced term Inflaton obtains the Hubble induced mass during and after the first inflation(=preinflation) through the term

• Just before new inflation starts, the inflaton has the initial value

• Potential

• PBH formation 14

W = W () + c

V ' eK(Vpre + · · · ) H2'2 O(1)

'

Vlin = 2p2v2'

'i c

v2 No initial value problem

V (') = v4 2p2cv2'

2v4'2 gp

2v23 + · · ·

c v4

SUSY breaking sector

• Dynamical SUSY breaking models

• If the origin of Z is destabilized due to a large Yukawa coupling (~ 4π )

• New inflation model (n=3)

15

µ2SUSY =

2

4

hZi '

4

WSUSY 3

(4)2 c µ3

SUSY

c v4c µ3

SUSY

consistent µSUSY v4/3

Izawa Yanagida (1996), Intriligator Thomas(1996)

Chacko Luty Ponton (1998)

WSUSY =2

4Z (Z: SUSY breaking field)

5. Production of gravitational waves• PBHs form from large curvature perturbations

• 2nd order perturbations ~ induce a source term of tensor perturbations

16

h00~k+ 2Hh0

~k+ k2h~k = S(~k, t)

MPBH 0.1 10 M 10－9 10－8 10－7Observed GW frequency [Hz]

10－1210－1110－1010－910－810－710－610－510－410－3

Ω gw(f)

BBN + CMB

Spectral estimationSMBH binary limitf = 1yr－1

• GWs can be probed future detectors

• Pulsar timing experiments already give a stringent constraint on PBHs with

fGW 2 109Hz(MPBH/M)1/2

GWh2 108(P/102)2

EPTA (2015)

O(~k ~k~k0) Moore Cole Berry (2015)

Saito Yokoyama (2009) Bugaev Kulimai (2010)

6. Conclusion

• PBHs can be formed in double inflation (=preinflation + new inflation)

• The new inflation potential has linear and quadratic terms which play an important role in determining the initial value and amplitude of curvature perturbations

• In this model PBHs with wide range of mass can be produced and account for DM of the universe

• The model is realized in framework of supergravity

• Large curvature perturbations required for PBH production also produce gravitational waves which will be proved by future detectors

17

Backup

18

Power spectrum of curvature perturbations with double peaks

19

κ=0.128 (ch=0.583)double peak

μ-distortion

0.1 1000.0 107 1011 1015 1019

10-9

10-6

0.001

1

k[Mpc-1 ]

ℛ