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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
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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)
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• 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
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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
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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
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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
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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)
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• 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
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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
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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
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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
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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)
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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)
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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
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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)
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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)
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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
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Backup
18
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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