Semi-Analytic Calculation of Gravitational Wave Spectrum
induced from Primordial Curvature Perturbations

寺田 隆広(KEK, JSPS fellow)[Kohri, Terada, 1804.08577 [gr-qc], PRD 97 (2018) 123532]

1

GW Astrophysics↓
GW Cosmology[Image credit: LIGO/T. Pyle]

We can probe
the very early Universe
or High Energy Physics.

!3[Image credit: NASA/WMAP Science Team]Beginning of a Trend?
• Ananda, Clarkson, Wands, gr-qc/0612013• Baumann, Steinhardt, Takahashi, Ichiki, hep-th/0703290• Assadullahi, Wands, 0901.0989• Inomata, Kawasaki, Mukaida, Tada, Yanagida, 1611.06130• Espinosa, Racco, Riotto, 1804.07732• Kohri, Terada, 1804.08577• Cai, Pi, Sasaki, 1810.11000• Bartolo, De Luca, Franciolini, Peloso, Riotto, 1810.12218• Bartolo, De Luca, Franciolini, Peloso, Racco, Riotto, 1810.12224• Byrnes, Cole, Patil, 1811.11158• Inomata, Nakama, 1812.00674• For other source of stochastic gravitational waves, see e.g.
Kuroyanagi, Chiba, Takahashi, 1807.00786.

What is the secondary GW?
[Ananda, Clarkson, Wands, gr-qc/0612013][Baumann, Steinhardt, Takahashi, Ichiki, hep-th/0703290][Inomata, Kawasaki, Mukaida, Tada, Yanagida, 1611.06130]!5

What is the secondary GW?
• Inﬂation produces[Ananda, Clarkson, Wands, gr-qc/0612013][Baumann, Steinhardt, Takahashi, Ichiki, hep-th/0703290][Inomata, Kawasaki, Mukaida, Tada, Yanagida, 1611.06130]

scalar and tensor (1st-order) perturbations
✓
◆
1
2
2
2
2
i
j
ds = a (1 + 2 )d⌘ + a (1 2 ) ij + hij dx dx
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• Inﬂation produces[Ananda, Clarkson, Wands, gr-qc/0612013][Baumann, Steinhardt, Takahashi, Ichiki, hep-th/0703290][Inomata, Kawasaki, Mukaida, Tada, Yanagida, 1611.06130]

scalar and tensor (1st-order) perturbations
✓
◆
1
2
2
2
2
i
j
ds = a (1 + 2 )d⌘ + a (1 2 ) ij + hij dx dx
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q

hk

00
0
2
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Z

✓
d3 q
eij (k)qi qj 2
3/2
(2⇡)

q

4
H 1
k q+
3(1 + w)
!5

0
q+

q

H 1

k q
0
k q+

k q

◆

Enhanced scalar perturbations
Initial condition
Enhanced primordial
curvature perturbations
are allowed at small scales.
[Inomata, Nakama, 1812.00674]

Dynamics

1.0

MD

Φ(x)

0.8

In a radiation-dominated (RD) era,
the source decays.
In a matter-dominated (MD) era,
the source does not decay.

0.6
0.4

RD

0.2
0.0
0

5

10

15
x

20

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25

30
!6

Relation to PBH scenario
produce

Primordial
Black Holes(PBHs)
explain

Dark Matter

Enhancedcurvature
perturbations(or MD era)predict/constrain[Saito, Yokoyama, 0812.4339]

constrain

Binary BH merger rateestimated from
LIGO/Virgo Observations!7

produceSecondarygravitational waves(GWs)

Relation to PBH scenario
produce

Primordial
Black Holes(PBHs)
explain

Dark Matter

Enhancedcurvature
perturbations(or MD era)predict/constrain[Saito, Yokoyama, 0812.4339]

constrain

Binary BH merger rateestimated from
LIGO/Virgo Observations!7

produceSecondarygravitational waves(GWs)

Relation to PBH scenario
produce

Primordial
Black Holes(PBHs)
explain

Dark Matter

Enhancedcurvature
perturbations(or MD era)predict/constrain[Saito, Yokoyama, 0812.4339]

constrain

Binary BH merger rateestimated from
LIGO/Virgo Observations!7

produceSecondarygravitational waves
(GWs)
PBH

Relation to PBH scenario
produce

Primordial
Black Holes(PBHs)
explain

Dark Matter

Enhancedcurvature
perturbations(or MD era)predict/constrain[Saito, Yokoyama, 0812.4339]

constrain

produceSecondarygravitational waves
(GWs)
PBH

Binary BH merger rateestimated from
LIGO/Virgo Observations!7

Secondary, stochastic GWs

[Espinosa, Racco, Riotto, 1804.07732]

sensitive

Collider exp.
(top mass,
Higgs mass)

produceproduce

Primordial
Black Holes(PBHs)
explain

Dark Matter

Higgs instability
probed during inflation
Enhancedcurvature
perturbations(or MD era)predict/constrain[Saito, Yokoyama, 0812.4339]

constrain

produceSecondarygravitational waves
(GWs)
PBH

Binary BH merger rateestimated from
LIGO/Virgo Observations!8

Secondary, stochastic GWs

Motivation (What we do)
The secondary GW power spectrum is obtainedby multiple integral of an oscillatory function.

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dtf (k, k 0 , t)

!9◆2P⇣ (k)P⇣ (k 0 )Motivation (What we do)
The secondary GW power spectrum is obtainedby multiple integral of an oscillatory function.

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!9◆2P⇣ (k)P⇣ (k 0 )primordial quantity

Motivation (What we do)
The secondary GW power spectrum is obtainedby multiple integral of an oscillatory function.
“universal part”

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dtf (k, k 0 , t)

◆2P⇣ (k)P⇣ (k 0 )describing time evolution primordial quantity!9Motivation (What we do)
The secondary GW power spectrum is obtainedby multiple integral of an oscillatory function.
“universal part”

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Z

dk

Z

dk 0

✓Z

dtf (k, k 0 , t)

◆2P⇣ (k)P⇣ (k 0 )describing time evolution primordial quantity[Orlofsky, Pierce, Wells, 1612.05279]

ΩGW /Aζ 2

10
1
0.100
0.010
0.001

!9

0.01

0.05 0.10
k/k*

0.50

1

Details (1/3)[Ananda, Clarkson, Wands, gr-qc/0612013] [Baumann, Steinhardt, Takahashi, Ichiki, hep-th/0703290][Inomata, Kawasaki, Mukaida, Tada, Yanagida, 1611.06130]✓ 2◆
Z 1 Z 1+v
2
2 2 2
4v
(1 + vu )Ph (⌘, k) = 4
dv
duI 2 (v, u, x)P⇣ (kv)P⇣ (ku)
4vu
0
|1 v|
<latexit sha1_base64="I7dXSD5EGWgtJL81iPiQeqcHOxE=">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</latexit>

where x ⌘ k⌘<latexit sha1_base64="q8HKV8n3XM04RwWMhXSndq+H32s=">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</latexit>u = |kv = k̃/kk̃|/k<latexit sha1_base64="r2J+wJ4tUeu5oBN75fGhr6q1wDo=">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</latexit><latexit sha1_base64="w5Y88L9tCRt+qjRDlbje1ZfMEEE=">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</latexit>!10Details (1/3)[Ananda, Clarkson, Wands, gr-qc/0612013] [Baumann, Steinhardt, Takahashi, Ichiki, hep-th/0703290][Inomata, Kawasaki, Mukaida, Tada, Yanagida, 1611.06130]✓ 2◆
Z 1 Z 1+v
2
2 2 2
4v
(1 + vu )Ph (⌘, k) = 4
dv
duI 2 (v, u, x)P⇣ (kv)P⇣ (ku)
4vu
0
|1 v|
<latexit sha1_base64="I7dXSD5EGWgtJL81iPiQeqcHOxE=">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</latexit>

where x ⌘ k⌘<latexit sha1_base64="q8HKV8n3XM04RwWMhXSndq+H32s=">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</latexit>u = |kv = k̃/kk̃|/k<latexit sha1_base64="r2J+wJ4tUeu5oBN75fGhr6q1wDo=">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</latexit><latexit sha1_base64="w5Y88L9tCRt+qjRDlbje1ZfMEEE=">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</latexit>

We want to analytically calculate this function.
Deﬁnitions: I(v, u, x) =Z x0<latexit sha1_base64="rlZW+MjM/lwOegUSkV737qoKoIE=">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</latexit>

!10

a(¯
⌘)
dx̄
kGk (⌘, ⌘¯)f (v, u, x̄)a(⌘)

Details (1/3)[Ananda, Clarkson, Wands, gr-qc/0612013] [Baumann, Steinhardt, Takahashi, Ichiki, hep-th/0703290][Inomata, Kawasaki, Mukaida, Tada, Yanagida, 1611.06130]✓ 2◆
Z 1 Z 1+v
2
2 2 2
4v
(1 + vu )Ph (⌘, k) = 4
dv
duI 2 (v, u, x)P⇣ (kv)P⇣ (ku)
4vu
0
|1 v|
<latexit sha1_base64="I7dXSD5EGWgtJL81iPiQeqcHOxE=">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</latexit>

where x ⌘ k⌘<latexit sha1_base64="q8HKV8n3XM04RwWMhXSndq+H32s=">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</latexit>u = |kv = k̃/kk̃|/k<latexit sha1_base64="r2J+wJ4tUeu5oBN75fGhr6q1wDo=">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</latexit><latexit sha1_base64="w5Y88L9tCRt+qjRDlbje1ZfMEEE=">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</latexit>

We want to analytically calculate this function.
Deﬁnitions: I(v, u, x) =Z x0<latexit sha1_base64="rlZW+MjM/lwOegUSkV737qoKoIE=">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</latexit>

a(¯
⌘)
dx̄
kGk (⌘, ⌘¯)f (v, u, x̄)a(⌘)

6(w + 1)6(1 + 3w)(w + 1)
f (v, u, x̄) =(v x̄) (ux̄) +(x̄@⌘¯ (v x̄) (ux̄) + x̄@⌘¯ (ux̄) (v x̄))
3w + 5(3w + 5)23(1 + 3w)2 (1 + w) 2+x̄ @⌘¯ (v x̄)@⌘¯ (ux̄)(3w + 5)2<latexit sha1_base64="63yUuwpOKUYLzj0rLd8aVU5hqfk=">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</latexit>

We have used H = aH = 2/((1 + 3w)⌘)<latexit sha1_base64="J0qgtfyBZYqFE9oiisqoFZD+XOI=">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</latexit>!10x̄ ⌘ k ⌘¯<latexit sha1_base64="cZwwjVpiaWKFPomeSQFDa+EYIVs=">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</latexit>w ⌘ P/⇢<latexit sha1_base64="F0lrOvknhQEI4oElcswZ2g7vOyI=">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</latexit>

Details (2/3)
For deﬁniteness, consider w = 1/3 (RD era).

I(v, u, x) =<latexit sha1_base64="rlZW+MjM/lwOegUSkV737qoKoIE=">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</latexit>

Z x
0

a(¯
⌘)
dx̄
kGk (⌘, ⌘¯)f (v, u, x̄)
a(⌘)

!11

Details (2/3)
For deﬁniteness, consider w = 1/3 (RD era).

I(v, u, x) =Z x0<latexit sha1_base64="rlZW+MjM/lwOegUSkV737qoKoIE=">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</latexit>

a(¯
⌘)
dx̄
kGk (⌘, ⌘¯)f (v, u, x̄)
a(⌘)

a(¯
⌘ )/a(⌘) = x̄/x<latexit sha1_base64="xOnzq7NiSdmMO6HVn3IZQC3+miA=">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</latexit>!11

Details (2/3)
For deﬁniteness, consider w = 1/3 (RD era).

I(v, u, x) =Z x0<latexit sha1_base64="rlZW+MjM/lwOegUSkV737qoKoIE=">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</latexit>

a(¯
⌘)
dx̄
kGk (⌘, ⌘¯)f (v, u, x̄)
a(⌘)

a(¯
⌘ )/a(⌘) = x̄/x<latexit sha1_base64="xOnzq7NiSdmMO6HVn3IZQC3+miA=">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</latexit>kGk (⌘, ⌘¯) = sin(x<latexit sha1_base64="EMiTCP8a7iOVvP+6w4fPrj1zN3I=">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</latexit>

!11

x̄)

Details (2/3)
For deﬁniteness, consider w = 1/3 (RD era).

I(v, u, x) =Z x0<latexit sha1_base64="rlZW+MjM/lwOegUSkV737qoKoIE=">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</latexit>

a(¯
⌘)
dx̄
kGk (⌘, ⌘¯)f (v, u, x̄)
a(⌘)

a(¯
⌘ )/a(⌘) = x̄/x9(x) = 2x<latexit sha1_base64="xOnzq7NiSdmMO6HVn3IZQC3+miA=">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</latexit>kGk (⌘, ⌘¯) = sin(xx̄)

psin(x/ 3)
p
x/ 3

p

cos(x/ 3)!

<latexit sha1_base64="NkMRE8i3lwOmpQRFC58+fKD11+g=">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</latexit><latexit sha1_base64="EMiTCP8a7iOVvP+6w4fPrj1zN3I=">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</latexit>

12fRD (v, u, x) = 3 3 6
u v x<latexit sha1_base64="1T7f+jxFByyO9U6lydw6WYErkVQ=">AAAENXichVHNbtNAEJ7Y/JRQaAoXJC4rqlaO0kZrN5TyJ1XAgWObkrZS3USOu0mtOrZjry0Xyy/AC3DgBFIPiAfgwo0LL8ChB24ICXHgUCQuHBj/lLapEsZa7+zMfN/Mp2k7puFxSg8Kgnju/IWLY5eKl8evXJ0oTV5b82zf1VlDt03b3WhrHjMNizW4wU224bhM67VNtt7efZzk1wPmeoZtPeN7DtvqaV3L6Bi6xjHUmizsd1qRylnIo/qTOJaCWX82LJOHM0TtuJoeyUoc+c15EuAJmwsxUU3W4RKRF4lPAgwpRNVtL6/2SRhHqtd3eTQfx1kmTwWnUxXpdo3MkQUi+UhRCZpKOeGqkOSJL2SukTJRPcMawn2cGaBOTXWN7g6vEtWyLb/XZi5R1SKK6vd9bTsTUcVuyhGM+CGRsr4KmbtbHqFqROuThEGYaftHOFTKyV6DWjId5VZpilZpauSsI+fOFOS2bJc+gArbYIMOPvSAgQUcfRM08PDbBBkoOBjbgghjLnpGmmcQQxGxPlYxrNAwuov/Lr4286iF74TTS9E6djHxuIgkME0/07f0kH6i7+h3+mcoV5RyJLPs4d3OsMxpTby4sfr7v6ge3hx2jlEjZ+bQgcV0VgNnd9JIokLP8MHzl4er9+rT0Qx9Q3/g/K/pAf2ICqzgl76/wuqvRsxzNImL2Q6eGFclDy7mrLOmVGValVdqU0uP8qWNwU24BRJu5g4swVNYhgbowrigCPeFB+J78Yv4VfyWlQqFHHMdTpn48y+0ZCVQ</latexit>

✓

ux
vx
ux
vx
18uvx2 cos p cos p + (54 6(u2 + v 2 )x2 + u2 v 2 x4 ) sin p sin p
3
3
3
3
◆
p
p
ux
vx
ux
vx
2 2
+2 3ux(v x9) cos p sin p + 2 3vx(u2 x2 9) sin p cos p
3
3
3
3

!11

Details (3/3)
For deﬁniteness, consider w = 1/3 (RD era).
✓
✓
3
4
ux
vx
2
2
3
2
p
p
IRD (v, u, x) = 3 3
uv(u
+
v
3)x
sin
x
6uvx
cos
cos
4u v x
x3
3
3
◆
p
p
ux
vx
ux
vx
ux
vx
+6 3ux cos p sin p + 6 3vx sin p cos p3(6 + (u2 + v 2 3)x2 ) sin p sin p
3
3
3
3
3
3
✓
✓ ✓✓
◆ ◆
✓✓
◆ ◆
v u
v u
p
+ (u2 + v 2 3)2 sin x Ci
1
x + Ci
1+ p
x
3
3
✓
◆
✓✓
◆ ◆
◆
2
v+u
v+u
3 (u + v)
p
Ci 1
x
Ci
1+ p
x + log
3 (u v)2
3
3
✓
✓✓
◆ ◆
✓✓
◆ ◆
v u
v u
Z x
p
+ cos x
Si
1
x
Si
1+ p
x
sin x̄
3
3
Si(x)
=
dx̄
✓✓
◆ ◆
✓✓
◆ ◆◆◆
x̄
0
v+u
v+u
Z 1
p
+Si
1
x + Si1+ px
cos x̄
3
3
Ci(x) =dx̄x̄x
<latexit sha1_base64="roX3tmMYDHPFmKCk0AwZ11+GP2w=">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</latexit><latexit sha1_base64="CK4pHlggtlOG25GpKJxhNSwI3KA=">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</latexit><latexit sha1_base64="ScpaayjF1nylSYHRTPCTlNdPWMo=">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</latexit>

See also [Espinosa, Racco, Riotto, 1804.07732]!12

Details (3/3)
For deﬁniteness, consider w = 1/3 (RD era).
✓
✓
3
4
ux
vx
2
2
3
2
p
p
IRD (v, u, x) = 3 3
uv(u
+
v
3)x
sin
x
6uvx
cos
cos
4u v x
x3
3
3
◆
p
p
ux
vx
ux
vx
ux
vx
+6 3ux cos p sin p + 6 3vx sin p cos p3(6 + (u2 + v 2 3)x2 ) sin p sin p
3
3
3
3
3
3
✓
✓ ✓✓
◆ ◆
✓✓
◆ ◆
v u
v u
p
+ (u2 + v 2 3)2 sin x Ci
1
x + Ci
1+ p
x
3
3
✓
◆
✓✓
◆ ◆
◆
2
v+u
v+u
3 (u + v)
p
Ci 1
x
Ci
1+ p
x + log
3 (u v)2
3
3
✓
✓✓
◆ ◆
✓✓
◆ ◆
v u
v u
Z x
p
+ cos x
Si
1
x
Si
1+ p
x
sin x̄
3
3
Si(x)
=
dx̄
✓✓
◆ ◆
✓✓
◆ ◆◆◆
x̄
0
v+u
v+u
Z 1
p
+Si
1
x + Si1+ px
cos x̄
3
3
Ci(x) =dx̄x̄x
<latexit sha1_base64="roX3tmMYDHPFmKCk0AwZ11+GP2w=">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</latexit><latexit sha1_base64="CK4pHlggtlOG25GpKJxhNSwI3KA=">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</latexit>

Oscillation average in the late-time limit
2 (v, u, x ! 1) =
IRD

1
2

✓

2

2

3(u + v
3)
4u3 v 3 x

◆2✓<latexit sha1_base64="ScpaayjF1nylSYHRTPCTlNdPWMo=">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</latexit>

4uv + (u2 + v 2+⇡ 2 (u2 + v 2<latexit sha1_base64="8UqSsxN5NPjpzi/cd7uYYL7/gNo=">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</latexit>

3) log3)2 ⇥(v + u3
32(u + v)(u v)2p ⌘3)◆2

See also [Espinosa, Racco, Riotto, 1804.07732]!12

General casesln ⌦GW (k)<latexit sha1_base64="yKkbGmbKs8xxnGSDlhdX81cyBVc=">AAACfnicSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQtIxuTkKcT456amJ8ZXx5SkVpRUu4fX1mpka8YLKBvoGYCBAibDEMpQZoCCgHyB5QwxDCkM+QzJDKUMuQypDHkMJUB2DkMiQzEQRjMYMhgwFADFYhmqgWJFQFYmWD6VoZaBC6i3FKgqFagiESiaDSTTgbxoqGgekA8ysxisOxloSw4QFwF1KjCoGlw1WGnw2eCEwWqDlwZ/cJpVDTYD5JZKIJ0E0ZtaEM/fJRH8naCuXCBdwpCB0IXXzSUMaQwWYLdmAt1eABYB+SIZor+savrnYKsg1Wo1g0UGr4HuX2hw0+Aw0Ad5ZV+SlwamBs1m4AJGgCF6cGMywoz0DA30DANNlB2coFHBwSDNoMSgAQxvcwYHBg+GAIZQoL31DEsZ1jGsZ2JgUmPSZdKHKGVihOoRZkABTBYAJz6TBw==</latexit>

RD
Consider (for simplicity) a scale-invariant spectrumP⇣ (k) = const.
<latexit sha1_base64="zjUBhK5r7iWmSENcbGUeqI5Q7M4=">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</latexit>

ln k<latexit sha1_base64="eiuXEAmEw6yyN7awEDboOEiO3Os=">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</latexit>!13General casesln ⌦GW (k)<latexit sha1_base64="yKkbGmbKs8xxnGSDlhdX81cyBVc=">AAACfnicSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQtIxuTkKcT456amJ8ZXx5SkVpRUu4fX1mpka8YLKBvoGYCBAibDEMpQZoCCgHyB5QwxDCkM+QzJDKUMuQypDHkMJUB2DkMiQzEQRjMYMhgwFADFYhmqgWJFQFYmWD6VoZaBC6i3FKgqFagiESiaDSTTgbxoqGgekA8ysxisOxloSw4QFwF1KjCoGlw1WGnw2eCEwWqDlwZ/cJpVDTYD5JZKIJ0E0ZtaEM/fJRH8naCuXCBdwpCB0IXXzSUMaQwWYLdmAt1eABYB+SIZor+savrnYKsg1Wo1g0UGr4HuX2hw0+Aw0Ad5ZV+SlwamBs1m4AJGgCF6cGMywoz0DA30DANNlB2coFHBwSDNoMSgAQxvcwYHBg+GAIZQoL31DEsZ1jGsZ2JgUmPSZdKHKGVihOoRZkABTBYAJz6TBw==</latexit>

RD

MD

RD

Enhancement in the MD era

Dilution in the MD era

ln k<latexit sha1_base64="eiuXEAmEw6yyN7awEDboOEiO3Os=">AAACaHichVG7SgNBFD1ZXzG+ohYqNsEQsQp3RVCsgjaWvmICUWR3ncQhm91ldxPQ4A/YWKpYKYiIn2HjD1j4CWoZwcbCm82CaFDvMDNnztxz58yM7pjS84meIkpHZ1d3T7Q31tc/MDgUHx7Z8uyqa4isYZu2m9c1T5jSEllf+qbIO67QKropcnp5ubmfqwnXk7a16R84YqeilSxZlIbmM5XdNq1EeTeepDQFkWgHagiSCGPVjt9gG3uwYaCKCgQs+IxNaPC4FaCC4DC3gzpzLiMZ7AscIcbaKmcJztCYLfNY4lUhZC1eN2t6gdrgU0zuLisTSNEj3VKDHuiOnunj11r1oEbTywHPeksrnN2h4/GN939VFZ597H+p/vTso4iFwKtk707ANG9htPS1w9PGxuJ6qj5NV/TK/i/pie75BlbtzbheE+sXiPEHqD+fux1szaZVSqtrc8nMUvgVUUxiCjP83vPIYAWryPK5Eic4w3nkRYkrY8pEK1WJhJpRfAtl6hMhxItp</latexit>

ln keqln kR<latexit sha1_base64="KIrIeiwKkp7OTUubuSVuAHF2qj8=">AAACdXichVFNLwNBGH66vutrcZGIpFHEqd4VCXESLo5VikSl2V2DTbe7m91pg03/gD/gwIWkRPwMF3/AwU8QRxIXB2+3mwiCdzIzzzzzPu88M2N4thVIoseE0tLa1t7R2ZXs7unt61cHBjcCt+KbIm+6tutvGXogbMsReWlJW2x5vtDLhi02jdJyY3+zKvzAcp11eeSJnbK+71h7lqlLpoqqWrCdVKkYFqQ4lGGuViuqacpQFKmfQItBGnFkXfUaBezChYkKyhBwIBnb0BFw24YGgsfcDkLmfEZWtC9QQ5K1Fc4SnKEzW+Jxn1fbMevwulEziNQmn2Jz91mZwgQ90A290D3d0hO9/1orjGo0vBzxbDS1wiv2nwyvvf2rKvMscfCp+tOzxB7mI68We/cipnELs6mvHp++rC3kJsJJuqRn9n9Bj3THN3Cqr2Z9VeTOkOQP0L4/90+wMZPRKKOtzqYXl+Kv6MQIxjDF7z2HRawgizyfW8U56rhKvCmjyrgy2UxVErFmCF9Cmf4A85+Qyg==</latexit><latexit sha1_base64="sY2VqnkIbK3HB5EKuT696ivpXhM=">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</latexit>!14General casesln ⌦GW (k)<latexit sha1_base64="yKkbGmbKs8xxnGSDlhdX81cyBVc=">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</latexit>

RD

MD

RD

Enhancement in the MD era

Dilution in the MD era

ln k<latexit sha1_base64="eiuXEAmEw6yyN7awEDboOEiO3Os=">AAACaHichVG7SgNBFD1ZXzG+ohYqNsEQsQp3RVCsgjaWvmICUWR3ncQhm91ldxPQ4A/YWKpYKYiIn2HjD1j4CWoZwcbCm82CaFDvMDNnztxz58yM7pjS84meIkpHZ1d3T7Q31tc/MDgUHx7Z8uyqa4isYZu2m9c1T5jSEllf+qbIO67QKropcnp5ubmfqwnXk7a16R84YqeilSxZlIbmM5XdNq1EeTeepDQFkWgHagiSCGPVjt9gG3uwYaCKCgQs+IxNaPC4FaCC4DC3gzpzLiMZ7AscIcbaKmcJztCYLfNY4lUhZC1eN2t6gdrgU0zuLisTSNEj3VKDHuiOnunj11r1oEbTywHPeksrnN2h4/GN939VFZ597H+p/vTso4iFwKtk707ANG9htPS1w9PGxuJ6qj5NV/TK/i/pie75BlbtzbheE+sXiPEHqD+fux1szaZVSqtrc8nMUvgVUUxiCjP83vPIYAWryPK5Eic4w3nkRYkrY8pEK1WJhJpRfAtl6hMhxItp</latexit>

ln keqln kR<latexit sha1_base64="KIrIeiwKkp7OTUubuSVuAHF2qj8=">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</latexit><latexit sha1_base64="sY2VqnkIbK3HB5EKuT696ivpXhM=">AAACdnichVHLSsNAFD2N7/qquhEEKZaqq3IjguJKdONSrVXBSkniqKFpEpNpfYT+gD/gQlwoaBE/w40/4MJPEJcKunDhbRoQFfUOM3PmzD13zszormX6kughpjQ1t7S2tXfEO7u6e3oTff2rvlP2DJEzHMvx1nXNF5Zpi5w0pSXWXU9oJd0Sa3pxvr6/VhGebzr2ijx0xWZJ27HNbdPQJFOFRF/espPFQpCX4kAGYq9aLSRSlKEwkj+BGoEUolh0EjXksQUHBsooQcCGZGxBg89tAyoILnObCJjzGJnhvkAVcdaWOUtwhsZskccdXm1ErM3rek0/VBt8isXdY2USabqna3qmO7qhR3r/tVYQ1qh7OeRZb2iFW+g9Hsy+/qsq8Syx+6n607PENqZDryZ7d0Omfgujoa8cnTxnZ5bTwShd0BP7P6cHuuUb2JUX43JJLJ8izh+gfn/un2B1IqNSRl2aTM3ORV/RjiGMYJzfewqzWMAicnzuPs5whVrsTRlW0spYI1WJRZoBfAmFPgAn/pFY</latexit>!15General cases
Schematic

Precise
10 4

RD → MD/k

1000
ΩGW /Aζ 2<latexit sha1_base64="UkuZRFCSjGEgJMyaSersu02Rk6A=">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</latexit>

100
10

/k

n(k)

10

100

6 . n(k) .

4<latexit sha1_base64="Fl6M45B6Az1Rod2uWI+Fv7RF4QI=">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</latexit>

MD → RD1
0.1 -410/k3<latexit sha1_base64="acPFGMdKO5/jsHPjV8gqN2xv/Fw=">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</latexit>

0.001 0.010 0.100

1k/kmax<latexit sha1_base64="/fsbfU/Aw1d5xCyP8/z+Ied/LBY=">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</latexit>

Scale-invariant primordial spectrum is assumed as a simple example.
!16

General cases
Schematic

Precise
10 4

RD → MD/k

1000
ΩGW /Aζ 2<latexit sha1_base64="UkuZRFCSjGEgJMyaSersu02Rk6A=">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</latexit>

100
10

/k

n(k)

10

100

6 . n(k) .

4<latexit sha1_base64="Fl6M45B6Az1Rod2uWI+Fv7RF4QI=">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</latexit>

MD → RD1
0.1 -410/k3<latexit sha1_base64="acPFGMdKO5/jsHPjV8gqN2xv/Fw=">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</latexit>

0.001 0.010 0.100

1k/kmax<latexit sha1_base64="/fsbfU/Aw1d5xCyP8/z+Ied/LBY=">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</latexit>

Scale-invariant primordial spectrum is assumed as a simple example.
!16

Summary!17

Summary• Secondary GW is induced by primordial curvature
perturbations. (important e.g. in PBH scenarios)!17Summary• Secondary GW is induced by primordial curvature
perturbations. (important e.g. in PBH scenarios)• We analytically calculated the universal part ofthe power spectrum of the secondary GW.

!17

Summary• Secondary GW is induced by primordial curvature
perturbations. (important e.g. in PBH scenarios)• We analytically calculated the universal part ofthe power spectrum of the secondary GW.

• Applicable to GWs induced in

RD, MD, and more general cases.

!17

Summary• Secondary GW is induced by primordial curvature
perturbations. (important e.g. in PBH scenarios)• We analytically calculated the universal part ofthe power spectrum of the secondary GW.

• Applicable to GWs induced in

RD, MD, and more general cases.

• Please use our results to write your papers!
!17Appendix:
More quantitative explanations• Basic things• Radiation-dominated (RD) era• Matter-dominated (MD) era• More general cases

Observable & Basic definitions!19

Observable & Basic definitions
Energy fraction

1 d⇢GW (⌘)
1⌦GW (⌘, k) ==⇢tot (⌘) d ln k
24<latexit sha1_base64="2S6BPCb0JvWiUX932awRcfCPvuY=">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</latexit>
sha1_base64="s97oGQvUazViTmQRw1rWya2knJo=">AAACf3ichVG7SgNBFD1ZXzHGJHaCTTAoVuGuCIoiCDaWakwixBB2N5O4ZF/sbgIx+AO2FhZWChbiL9jZ+AMW+QSxVLCx8O4mIhqMd5iZO+fOuXOGozqG7vlE3Yg0Mjo2PhGdjE3FY9OJZCpe8Oymq4m8Zhu2e6gqnjB0S+R93TfEoeMKxVQNUVQb20G92BKup9vWgd92RNlU6pZe0zXFZ2i3kspQlsJIDyZyP8mgH3bqHkeowoaGJkwIWPA5N6DA41GCDILDWBkdxlzO9LAucIoYc5t8S/ANhdEGr3U+lfqoxeegpxeyNX7F4OkyM40FeqJbeqVHuqNn+vizVyfsEWhp8672uMKpJM9mc+//skzefRx/s4Zq9lHDWqhVZ+1OiAS/0Hr81snFa259f6GzSNf0wvqvqEsP/AOr9abd7In9yyF6vpS4XK3xPGWn5N++DCaF5axMWXmPEMUc5rHEhqxiCzvYRZ4fqOIM51JCWpE2eo5Kkb61M/gR0uYnsZuQSg==</latexit>
sha1_base64="fmGuwgGCXqt0ybRNstzlTOj3d+s=">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</latexit>
sha1_base64="mPBGowM3642PHeiGO+zCprPx2b8=">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</latexit>

!19

✓

k
a(⌘)H(⌘)◆2

Ph (⌘, k)

Observable & Basic definitions
Energy fraction

1 d⇢GW (⌘)
1⌦GW (⌘, k) ==⇢tot (⌘) d ln k
24<latexit sha1_base64="2S6BPCb0JvWiUX932awRcfCPvuY=">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</latexit>
sha1_base64="s97oGQvUazViTmQRw1rWya2knJo=">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</latexit>
sha1_base64="fmGuwgGCXqt0ybRNstzlTOj3d+s=">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</latexit>
sha1_base64="mPBGowM3642PHeiGO+zCprPx2b8=">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</latexit>✓ka(⌘)H(⌘)◆2Power spectrum0hhk (⌘)hk0 (⌘)i =<latexit sha1_base64="brn3Eva1tFHRZthuMTxl5nBwdrk=">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</latexit>

2
2⇡
3
0
0
(k + k ) 3 Ph (⌘, k)k!19

Ph (⌘, k)

Observable & Basic definitions
Energy fraction

1 d⇢GW (⌘)
1⌦GW (⌘, k) ==⇢tot (⌘) d ln k
24<latexit sha1_base64="2S6BPCb0JvWiUX932awRcfCPvuY=">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</latexit>
sha1_base64="s97oGQvUazViTmQRw1rWya2knJo=">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</latexit>
sha1_base64="fmGuwgGCXqt0ybRNstzlTOj3d+s=">AAADT3ichVHLahRBFL094yMvzehKcNMYIjMgQ/UQMAQCgguzc5I4mUA6ttWV6ummqx9U1wzGpn7AH3DhSsGF+Avu3AiuXQT8AXEZwY0Lbz8yUYOxmqq6de45t8/luqkIMkXIkdFoXrh46fLM7Nz8wpWri61rCztZMpaMD1giErnr0oyLIOYDFSjBd1PJaeQKPnTD+0V+OOEyC5L4kTpM+X5ER3HgBYwqhJzWJ/thxEfUyW3Fn6r8wVDrts0VvWOGnXXbk5Tlls5t6ScnFJWomtPRFaHCD/TvrGmhjj7Ni9gM9bo5Ldtb0aYtuKfaNRbqnFaqjVps2jIY+arzuGfaCTZS9JnbEVU+oyLva8ef2tVOa4l0SbnMs4FVB0tQr37Seg82HEACDMYQAYcYFMYCKGT47YEFBFLE9iFHTGIUlHkOGuZQO0YWRwZFNMRzhK+9Go3xXdTMSjXDvwjcEpUmLJPP5C05Jh/JO/KV/PxnrbysUXg5xNuttDx1Fp/f2P7xX1WEtwL/VHWuZwUerJZeA/SelkjRBav0k2cvjrfXtpbz2+Q1+Yb+X5Ej8gE7iCff2ZtNvvXyHD8nTiRmPdzFqKy/B3M22Ol1LdK1NgnMwE24BW0cyF24BxvQhwEwwzKGxhODNr40G83ZaqgNo57udfhjNed/Aec93o0=</latexit>
sha1_base64="mPBGowM3642PHeiGO+zCprPx2b8=">AAADWnichVHLahRBFL097SMPNWN0IbhpHCIzIEP1IChCIOjC7JwkTiaQjm11p3q66eoH3TVDYlE/4A+4cKXgQvwFd24E1y4C/oC4jODGhbcfmajBWE1V3Tr3nNvncp2UB7kg5EBr6GfOnjs/Mzs3f+HipYXm5cXNPBlnLhu4CU+yLYfmjAcxG4hAcLaVZoxGDmdDJ3xQ5IcTluVBEj8W+ynbiegoDrzApQIhu/nJehSxEbWlJdiekA+HSrUtJugtI+wsW15GXWkqaWV+ckQRiag5HVURKnxX/c6aFuqo4zyPjVAtG9OyvdvKsDjzRLvGQiVppVqtxYaVBSNfdJ70DCvBRoo+pRVR4buUy76y/aldZTdbpEvKZZwMzDpoQb36SfM9WLALCbgwhggYxCAw5kAhx28bTCCQIrYDErEMo6DMM1Awh9oxshgyKKIhniN8bddojO+iZl6qXfwLx52h0oAl8pm8JYfkI3lHvpKf/6wlyxqFl328nUrLUnvh+bWNH/9VRXgL8I9Vp3oW4MHd0muA3tMSKbpwK/3k2YvDjXvrS/ImeU2+of9X5IB8wA7iyXf3zRpbf3mKnyMnGWY93MWozL8HczLY7HVN0jXXSGvlfj20GbgON6CNk7kDK7AKfRiAq5naUHuq0cYXvaHP6vMVtaHVmivwx9Kv/gKUSt9Q</latexit>✓ka(⌘)H(⌘)◆2

Ph (⌘, k)Power spectrum0hhk (⌘)hk0 (⌘)i =<latexit sha1_base64="brn3Eva1tFHRZthuMTxl5nBwdrk=">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</latexit>

2
2⇡
3
0
0
(k + k ) 3 Ph (⌘, k)kGravitational field: Green’s function methoda(⌘)hk (⌘) = 4<latexit sha1_base64="7gYoqJ9Mho2LcUvcpXdlvOUeEdA=">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</latexit>Z ⌘d¯⌘ Gk (⌘, ⌘¯)a(¯⌘ )Sk (¯⌘)
G00k (⌘, ⌘¯) +<latexit sha1_base64="AY4axVtVZt/rr2y23WEMbM00vbY=">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</latexit>

!19

✓

k

2

00

a (⌘)
a(⌘)

◆

Gk (⌘, ⌘¯) = (⌘⌘¯)

Power spectrum & “universal part”[Ananda, Clarkson, Wands, gr-qc/0612013] [Baumann, Steinhardt, Takahashi, Ichiki, hep-th/0703290][Inomata, Kawasaki, Mukaida, Tada, Yanagida, 1611.06130]✓ 2◆
Z 1 Z 1+v
2
2 2 2
4v
(1 + vu )Ph (⌘, k) = 4
dv
duI 2 (v, u, x)P⇣ (kv)P⇣ (ku)
4vu
0
|1 v|
<latexit sha1_base64="I7dXSD5EGWgtJL81iPiQeqcHOxE=">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</latexit>

where x ⌘ k⌘<latexit sha1_base64="q8HKV8n3XM04RwWMhXSndq+H32s=">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</latexit>u = |kv = k̃/kk̃|/k<latexit sha1_base64="r2J+wJ4tUeu5oBN75fGhr6q1wDo=">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</latexit><latexit sha1_base64="w5Y88L9tCRt+qjRDlbje1ZfMEEE=">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</latexit>!20

Power spectrum & “universal part”[Ananda, Clarkson, Wands, gr-qc/0612013] [Baumann, Steinhardt, Takahashi, Ichiki, hep-th/0703290][Inomata, Kawasaki, Mukaida, Tada, Yanagida, 1611.06130]✓ 2◆
Z 1 Z 1+v
2
2 2 2
4v
(1 + vu )Ph (⌘, k) = 4
dv
duI 2 (v, u, x)P⇣ (kv)P⇣ (ku)
4vu
0
|1 v|
<latexit sha1_base64="I7dXSD5EGWgtJL81iPiQeqcHOxE=">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</latexit>

where x ⌘ k⌘<latexit sha1_base64="q8HKV8n3XM04RwWMhXSndq+H32s=">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</latexit>u = |kv = k̃/kk̃|/k<latexit sha1_base64="r2J+wJ4tUeu5oBN75fGhr6q1wDo=">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</latexit><latexit sha1_base64="w5Y88L9tCRt+qjRDlbje1ZfMEEE=">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</latexit>

We want to analytically calculate this function.
Deﬁnitions: I(v, u, x) =Z x0<latexit sha1_base64="rlZW+MjM/lwOegUSkV737qoKoIE=">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</latexit>

!20

a(¯
⌘)
dx̄
kGk (⌘, ⌘¯)f (v, u, x̄)a(⌘)

Power spectrum & “universal part”[Ananda, Clarkson, Wands, gr-qc/0612013] [Baumann, Steinhardt, Takahashi, Ichiki, hep-th/0703290][Inomata, Kawasaki, Mukaida, Tada, Yanagida, 1611.06130]✓ 2◆
Z 1 Z 1+v
2
2 2 2
4v
(1 + vu )Ph (⌘, k) = 4
dv
duI 2 (v, u, x)P⇣ (kv)P⇣ (ku)
4vu
0
|1 v|
<latexit sha1_base64="I7dXSD5EGWgtJL81iPiQeqcHOxE=">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</latexit>

where x ⌘ k⌘<latexit sha1_base64="q8HKV8n3XM04RwWMhXSndq+H32s=">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</latexit>u = |kv = k̃/kk̃|/k<latexit sha1_base64="r2J+wJ4tUeu5oBN75fGhr6q1wDo=">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</latexit><latexit sha1_base64="w5Y88L9tCRt+qjRDlbje1ZfMEEE=">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</latexit>

We want to analytically calculate this function.
Deﬁnitions: I(v, u, x) =Z x0<latexit sha1_base64="rlZW+MjM/lwOegUSkV737qoKoIE=">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</latexit>

a(¯
⌘)
dx̄
kGk (⌘, ⌘¯)f (v, u, x̄)a(⌘)

6(w + 1)6(1 + 3w)(w + 1)
f (v, u, x̄) =(v x̄) (ux̄) +(x̄@⌘¯ (v x̄) (ux̄) + x̄@⌘¯ (ux̄) (v x̄))
3w + 5(3w + 5)23(1 + 3w)2 (1 + w) 2+x̄ @⌘¯ (v x̄)@⌘¯ (ux̄)(3w + 5)2<latexit sha1_base64="63yUuwpOKUYLzj0rLd8aVU5hqfk=">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</latexit>

We have used H = aH = 2/((1 + 3w)⌘)<latexit sha1_base64="J0qgtfyBZYqFE9oiisqoFZD+XOI=">AAACrHicSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQvIxuQmlmQkJ+ZUe9TaJnrYGulraBhqG5drxqSWJGrGCygb6BmAgQImwxDKUGaAgoB8gT0MMQwpDPkMyQylDLkMqQx5DCVAdg5DIkMxEEYzGDIYMBQAxWIZqoFiRUBWJlg+laGWgQuotxSoKhWoIhEomg0k04G8aKhoHpAPMrMYrDsZaEsOEBcBdSowqBpcNVhp8NnghMFqg5cGf3CaVQ02A+SWSiCdBNGbWhDP3yUR/J2grlwgXcKQgdCF180lDGkMFmC3ZgLdXgAWAfkiGaK/rGr652CrINVqNYNFBq+B7l9ocNPgMNAHeWVfkpcGpgbNxuMemEuKgLJpQFwLjCpD9IjBZIQZ6Rka6BkGmig7OEEjjYNBmkGJQQMYM+YMDgweDAEMoUCbmhlWMmxi2MykxxTCFM0UC1HKxAjVI8yAApjSAGXPnL4=</latexit>!20x̄ ⌘ k ⌘¯<latexit sha1_base64="cZwwjVpiaWKFPomeSQFDa+EYIVs=">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</latexit>w ⌘ P/⇢<latexit sha1_base64="F0lrOvknhQEI4oElcswZ2g7vOyI=">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</latexit>

Analytic Calculation
For deﬁniteness, consider w = 1/3 (RD era).

I(v, u, x) =<latexit sha1_base64="rlZW+MjM/lwOegUSkV737qoKoIE=">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</latexit>

Z x
0

a(¯
⌘)
dx̄
kGk (⌘, ⌘¯)f (v, u, x̄)
a(⌘)

!21

Analytic Calculation
For deﬁniteness, consider w = 1/3 (RD era).

I(v, u, x) =Z x0<latexit sha1_base64="rlZW+MjM/lwOegUSkV737qoKoIE=">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</latexit>

a(¯
⌘)
dx̄
kGk (⌘, ⌘¯)f (v, u, x̄)
a(⌘)

a(¯
⌘ )/a(⌘) = x̄/x<latexit sha1_base64="xOnzq7NiSdmMO6HVn3IZQC3+miA=">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</latexit>!21

Analytic Calculation
For deﬁniteness, consider w = 1/3 (RD era).

I(v, u, x) =Z x0<latexit sha1_base64="rlZW+MjM/lwOegUSkV737qoKoIE=">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</latexit>

a(¯
⌘)
dx̄
kGk (⌘, ⌘¯)f (v, u, x̄)
a(⌘)

a(¯
⌘ )/a(⌘) = x̄/x<latexit sha1_base64="xOnzq7NiSdmMO6HVn3IZQC3+miA=">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</latexit>kGk (⌘, ⌘¯) = sin(x<latexit sha1_base64="EMiTCP8a7iOVvP+6w4fPrj1zN3I=">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</latexit>

!21

x̄)

Analytic Calculation
For deﬁniteness, consider w = 1/3 (RD era).

I(v, u, x) =Z x0<latexit sha1_base64="rlZW+MjM/lwOegUSkV737qoKoIE=">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</latexit>

a(¯
⌘)
dx̄
kGk (⌘, ⌘¯)f (v, u, x̄)
a(⌘)

a(¯
⌘ )/a(⌘) = x̄/x9(x) = 2x<latexit sha1_base64="xOnzq7NiSdmMO6HVn3IZQC3+miA=">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</latexit>kGk (⌘, ⌘¯) = sin(xx̄)

psin(x/ 3)
p
x/ 3

p

cos(x/ 3)!

<latexit sha1_base64="NkMRE8i3lwOmpQRFC58+fKD11+g=">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</latexit><latexit sha1_base64="EMiTCP8a7iOVvP+6w4fPrj1zN3I=">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</latexit>

12fRD (v, u, x) = 3 3 6
u v x<latexit sha1_base64="1T7f+jxFByyO9U6lydw6WYErkVQ=">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</latexit>

✓

ux
vx
ux
vx
18uvx2 cos p cos p + (54 6(u2 + v 2 )x2 + u2 v 2 x4 ) sin p sin p
3
3
3
3
◆
p
p
ux
vx
ux
vx
2 2
+2 3ux(v x9) cos p sin p + 2 3vx(u2 x2 9) sin p cos p
3
3
3
3

!21

Analytic Calculation
For deﬁniteness, consider w = 1/3 (RD era).
IRD (v, u, x) =

✓

✓

3
4
ux
vx
2
2
3
2
p
p
uv(u
+
v
3)x
sin
x
6uvx
cos
cos
4u3 v 3 x
x3
3
3
◆
p
p
ux
vx
ux
vx
ux
vx
+6 3ux cos p sin p + 6 3vx sin p cos p3(6 + (u2 + v 2 3)x2 ) sin p sin p
3
3
3
3
3
3
✓
✓ ✓✓
◆ ◆
✓✓
◆ ◆
v u
v u
p
+ (u2 + v 2 3)2 sin x Ci
1
x + Ci
1+ p
x
3
3
✓
◆
✓✓
◆ ◆
◆
v+u
v+u
3 (u + v)2
p
Ci 1
x
Ci
1+ p
x + log
3 (u v)2
3
3
✓
✓✓
◆ ◆
✓✓
◆ ◆
v u
v u
Z x
p
+ cos x
Si
1
x
Si
1+ p
x
sin x̄
3
3
Si(x)
=
dx̄
✓✓
◆ ◆
✓✓
◆ ◆◆◆
x̄
0
v+u
v+u
Z 1
p
+Si
1
x + Si1+ px
cos x̄
3
3
Ci(x) =dx̄x̄x
<latexit sha1_base64="roX3tmMYDHPFmKCk0AwZ11+GP2w=">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</latexit><latexit sha1_base64="CK4pHlggtlOG25GpKJxhNSwI3KA=">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</latexit><latexit sha1_base64="ScpaayjF1nylSYHRTPCTlNdPWMo=">AAAC2nichVG/axRBFP6yURPjj5xJI9gsHgmx8HgrgQQhEJJGsMkPLwlk47E7mY1D9naX2bnjLss2dmorWFgpWIiVvZ2N/4BIOi3FMoKNhW/3NogG4xtm5pvvvW/mG56fhCo1RIdD1vCZs+dGRs+PXbh46fJ47crERhp3tJBNEYex3vK9VIYqkk2jTCi3Ei29th/KTX9/uchvdqVOVRzdM/1E7rS9vUgFSniGqVbtrmtkz2TLKp/p3bAXpu2btqsic5+XwPRbPXuQ381d39NZL7fdQHsic0WcVkyeHYNWrU4NKsM+CZwK1FHFSlx7Bxe7iCHQQRsSEQzjEB5SHttwQEiY20HGnGakyrxEjjHWdrhKcoXH7D6ve3zartiIz8WdaakW/ErIU7PSxhR9pNd0RB/oDX2ln/+8KyvvKLz0efcHWpm0xh9fXf/xX1Wbd4MHv1WnejYIMF96Vew9KZniF2Kg7x48O1q/vTaVTdNL+sb+X9AhvecfRN3v4tWqXHt+ip9jJ5qzAc+iVc7fjTkJNm41HGo4q7P1xaWqaaO4huuY4c7MYRF3sIImv/QWn/AZXyzXemg9sp4MSq2hSjOJP8J6+gvVmrJ7</latexit>!22Analytic Calculation
For deﬁniteness, consider w = 1/3 (RD era).
IRD (v, u, x) =

✓

✓

3
4
ux
vx
2
2
3
2
p
p
uv(u
+
v
3)x
sin
x
6uvx
cos
cos
4u3 v 3 x
x3
3
3
◆
p
p
ux
vx
ux
vx
ux
vx
+6 3ux cos p sin p + 6 3vx sin p cos p3(6 + (u2 + v 2 3)x2 ) sin p sin p
3
3
3
3
3
3
✓
✓ ✓✓
◆ ◆
✓✓
◆ ◆
v u
v u
p
+ (u2 + v 2 3)2 sin x Ci
1
x + Ci
1+ p
x
3
3
✓
◆
✓✓
◆ ◆
◆
v+u
v+u
3 (u + v)2
p
Ci 1
x
Ci
1+ p
x + log
3 (u v)2
3
3
✓
✓✓
◆ ◆
✓✓
◆ ◆
v u
v u
Z x
p
+ cos x
Si
1
x
Si
1+ p
x
sin x̄
3
3
Si(x)
=
dx̄
✓✓
◆ ◆
✓✓
◆ ◆◆◆
x̄
0
v+u
v+u
Z 1
p
+Si
1
x + Si1+ px
cos x̄
3
3
Ci(x) =dx̄x̄x
<latexit sha1_base64="roX3tmMYDHPFmKCk0AwZ11+GP2w=">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</latexit><latexit sha1_base64="CK4pHlggtlOG25GpKJxhNSwI3KA=">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</latexit>

Oscillation average in the late-time limit
12
IRD (v, u, x ! 1) =
2

✓

2

2

3(u + v
3)
4u3 v 3 x

◆2✓<latexit sha1_base64="ScpaayjF1nylSYHRTPCTlNdPWMo=">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</latexit>

2

4uv + (u + v+⇡ 2 (u2 + v 2<latexit sha1_base64="8UqSsxN5NPjpzi/cd7uYYL7/gNo=">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</latexit>

!22

2

3
3) log33)2 ⇥(v + u2(u + v)(u v)2p ⌘3)◆2

AnalyticallyCalculable GW spectrum
Example 1 in RD: Monochromatic caseP⇣ (k) = A⇣ (log k/k⇤ )<latexit sha1_base64="JosQBY922HoZ4AkvkG3OrjMNMKQ=">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</latexit>⌦GW (⌘, k) =

3A2⇣
64
⇥

k̃ 2

4
4
2

⇡ (3k̃

2

!2

k̃
2

2

⇣

3k̃

2

p

2) ⇥(2 3⌘22k̃ ⌘ k/k⇤<latexit sha1_base64="FuDh2pTmwYJw7vI5SR5GgX1Aul4=">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</latexit>✓3k̃) + 4 + (3k̃<latexit sha1_base64="5gMC9Wh33EaHkg7fjuyCKFx1jrg=">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</latexit>

22) log 1

4
3k̃ 2

◆2 !

⇥(2

k̃)

ΩGW /Aζ 2

101<latexit sha1_base64="Npt8E00adJ5UaH64oQzqqd6Y2hk=">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</latexit>

0.100

p
!1 =k⇤ / 3|~k1 | =k⇤<latexit sha1_base64="+NAlBRCkLS9mtkwu7YhzcBIV6NA=">AAACoHichVHLSsNQED2Nr1ofbXUjuCkWRVyUiQiKIIhu3NmqVfFBSeJtDU2TkKQVbf0Bf8CFuFBQEX/BnRt/wEU/QVwquHHhNI2IijqXe2fumTlzz2VU29Bdj6geklpa29o7wp2Rru6e3mgs3rfqWmVHE1nNMixnXVVcYeimyHq6Z4h12xFKSTXEmlqcb+TXKsJxdctc8fZtsV1SCqae1zXFYygXi9e2KkJLFHNyLTEzwn4sF0tSinxL/AzkIEgisLQVu8UWdmBBQxklCJjwODagwOW1CRkEm7FtVBlzONL9vMAhIswtc5XgCoXRIp8Fvm0GqMn3Rk/XZ2v8isHbYWYCw/RA1/RM93RDj/T2a6+q36OhZZ+92uQKOxc9Glh+/ZdVYu9h95P1p2YPeUz5WnXWbvtI4xdak185OH5enl4aro7QOT2x/jOq0x3/wKy8aBcZsXTyh54PJQ5n87wPeVTy98H8DFbHUzKl5MxEcnYuGFoYgxjCKE9mErNYQBpZfmkPp7jElTQkLUiLUqZZKoUCTj++mLTxDrSGmFI=</latexit><latexit sha1_base64="Hz9sgilQqXkrd1y3bep7Zl8eE40=">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</latexit>p!2 =k⇤ / 3|~k2 | =k⇤<latexit sha1_base64="TgvX++W1ndyety7zUnhOxviinjY=">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</latexit>

0.010
0.001

0.01

0.05 0.10

0.50p!GW = 2k⇤ / 3p~~|k1 + k2 | = 2k⇤ / 3

1<latexit sha1_base64="ldb0Ada+FZUswHA8ZJJM5Jz2x0E=">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</latexit>

k/k*<latexit sha1_base64="ia7THHCgro8RE/ZGIYM2Ef30EIw=">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</latexit>!23

AnalyticallyCalculable GW spectrum
Example 2 in RD: Scale-invariant case<latexit sha1_base64="hzXv39wX+7jKGsnKvpvV951hdlg=">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</latexit>P⇣ (k) = A⇣

2
⌦GW (⌘, k) '0.8222A⇣<latexit sha1_base64="ihwIyGGpYqG/VnXxRTI7UyCey2Q=">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</latexit>

Example 3 in RD: Power-law caseP ⇣ = A⇣<latexit sha1_base64="DuQOCQ1fxRuSoZTN7QU10TH6ic8=">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</latexit>

2⌦GW (⌘, k) =Q(ns )A⇣<latexit sha1_base64="HeuHOypApW4CIC54Zr0zUPxkijA=">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</latexit>

✓

k
k⇤

✓

◆2(ns 1)

k
k⇤

◆ns 1

1000
100

Q(ns ) 10<latexit sha1_base64="q0ONZGwS4e83aa2Le4+3y5vnOV8=">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</latexit>

1
0.1

0.5

1.0

1.5
ns

!24

2.0

Enhancement in MD era[Assadullahi, Wands, 0901.0989] [Baumann, Steinhardt, Takahashi, Ichiki, hep-th/0703290]

Next, consider w = 0 (MD era).

I(v, u, x) =<latexit sha1_base64="rlZW+MjM/lwOegUSkV737qoKoIE=">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</latexit>

Z x
0

a(¯
⌘)
dx̄
kGk (⌘, ⌘¯)f (v, u, x̄)
a(⌘)

!25

Enhancement in MD era[Assadullahi, Wands, 0901.0989] [Baumann, Steinhardt, Takahashi, Ichiki, hep-th/0703290]

Next, consider w = 0 (MD era).

I(v, u, x) =Z x0<latexit sha1_base64="rlZW+MjM/lwOegUSkV737qoKoIE=">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</latexit>

a(¯
⌘)
dx̄
kGk (⌘, ⌘¯)f (v, u, x̄)
a(⌘)

a(¯
⌘ )/a(⌘) = x̄2 /x2<latexit sha1_base64="C5NMtoADaxOrL665GINuu8CJmnE=">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</latexit>!25

Enhancement in MD era[Assadullahi, Wands, 0901.0989] [Baumann, Steinhardt, Takahashi, Ichiki, hep-th/0703290]

Next, consider w = 0 (MD era).

I(v, u, x) =Z x0<latexit sha1_base64="rlZW+MjM/lwOegUSkV737qoKoIE=">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</latexit>

a(¯
⌘)
dx̄
kGk (⌘, ⌘¯)f (v, u, x̄)
a(⌘)

a(¯
⌘ )/a(⌘) = x̄2 /x2<latexit sha1_base64="C5NMtoADaxOrL665GINuu8CJmnE=">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</latexit>1
kGk (⌘, ⌘¯) =((1 + xx̄) sin(x
xx̄

x̄)

<latexit sha1_base64="pZHudWZMMLxKThiwx5QJhfmdrM8=">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</latexit>

!25

(x

x̄) cos(xx̄))

Enhancement in MD era[Assadullahi, Wands, 0901.0989] [Baumann, Steinhardt, Takahashi, Ichiki, hep-th/0703290]
1.0

Next, consider w = 0 (MD era).

I(v, u, x) =0<latexit sha1_base64="rlZW+MjM/lwOegUSkV737qoKoIE=">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</latexit>

Φ(x)

Z x

0.8

a(¯
⌘)
dx̄
kGk (⌘, ⌘¯)f (v, u, x̄)
a(⌘)

0.6
0.4
0.2
0.0
0

5

10

15
x

(x) = 1a(¯⌘ )/a(⌘) = x̄2 /x2<latexit sha1_base64="1041Zd4Ja5ChMMCTBlBLDVVzrJg=">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</latexit><latexit sha1_base64="C5NMtoADaxOrL665GINuu8CJmnE=">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</latexit>1
kGk (⌘, ⌘¯) =((1 + xx̄) sin(x
xx̄

x̄)

<latexit sha1_base64="pZHudWZMMLxKThiwx5QJhfmdrM8=">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</latexit>

!25

(x

x̄) cos(xx̄))

fMD (v, u, x) =<latexit sha1_base64="X1FKt4r3/ZFHdSH859iMbRnmwlc=">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</latexit>

6
5

20

25

30

Enhancement in MD era[Assadullahi, Wands, 0901.0989] [Baumann, Steinhardt, Takahashi, Ichiki, hep-th/0703290]
1.0

Next, consider w = 0 (MD era).

I(v, u, x) =0<latexit sha1_base64="rlZW+MjM/lwOegUSkV737qoKoIE=">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</latexit>

Φ(x)

Z x

0.8

a(¯
⌘)
dx̄
kGk (⌘, ⌘¯)f (v, u, x̄)
a(⌘)

0.6
0.4
0.2
0.0
0

5

10

15

20

25

30

x

(x) = 1a(¯⌘ )/a(⌘) = x̄2 /x2<latexit sha1_base64="1041Zd4Ja5ChMMCTBlBLDVVzrJg=">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</latexit><latexit sha1_base64="C5NMtoADaxOrL665GINuu8CJmnE=">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</latexit>1
kGk (⌘, ⌘¯) =((1 + xx̄) sin(x
xx̄

x̄)

(x

x̄) cos(xx̄))

fMD (v, u, x) =<latexit sha1_base64="X1FKt4r3/ZFHdSH859iMbRnmwlc=">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</latexit>

65

Non-decaying source!

<latexit sha1_base64="pZHudWZMMLxKThiwx5QJhfmdrM8=">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</latexit>!25

Enhancement in MD era[Assadullahi, Wands, 0901.0989] [Baumann, Steinhardt, Takahashi, Ichiki, hep-th/0703290]
1.0

Next, consider w = 0 (MD era).

I(v, u, x) =0<latexit sha1_base64="rlZW+MjM/lwOegUSkV737qoKoIE=">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</latexit>

Φ(x)

Z x

0.8

a(¯
⌘)
dx̄
kGk (⌘, ⌘¯)f (v, u, x̄)
a(⌘)

0.6
0.4
0.2
0.0
0

5

10

15

20

25

30

x

(x) = 1a(¯⌘ )/a(⌘) = x̄2 /x2<latexit sha1_base64="1041Zd4Ja5ChMMCTBlBLDVVzrJg=">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</latexit><latexit sha1_base64="C5NMtoADaxOrL665GINuu8CJmnE=">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</latexit>1
kGk (⌘, ⌘¯) =((1 + xx̄) sin(x
xx̄

x̄)

(x

x̄))

<latexit sha1_base64="X1FKt4r3/ZFHdSH859iMbRnmwlc=">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</latexit>

65

Non-decaying source!

<latexit sha1_base64="pZHudWZMMLxKThiwx5QJhfmdrM8=">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</latexit><latexit sha1_base64="Y29DcK0TuTQkjK2MoXRyaCUw1lM=">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</latexit>x̄) cos(xfMD (v, u, x) =18
2
IMD (v, u, x ! 1) =25

!25

← Not diluted!

Enhancement in MD era[Assadullahi, Wands, 0901.0989] [Baumann, Steinhardt, Takahashi, Ichiki, hep-th/0703290]
1.0

Next, consider w = 0 (MD era).

I(v, u, x) =0<latexit sha1_base64="rlZW+MjM/lwOegUSkV737qoKoIE=">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</latexit>

Φ(x)

Z x

0.8

a(¯
⌘)
dx̄
kGk (⌘, ⌘¯)f (v, u, x̄)
a(⌘)

0.6
0.4
0.2
0.0
0

5

10

15

20

25

30

x

(x) = 1a(¯⌘ )/a(⌘) = x̄2 /x2<latexit sha1_base64="1041Zd4Ja5ChMMCTBlBLDVVzrJg=">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</latexit><latexit sha1_base64="C5NMtoADaxOrL665GINuu8CJmnE=">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</latexit>1
kGk (⌘, ⌘¯) =((1 + xx̄) sin(x
xx̄

x̄)

(x

x̄) cos(x18
2IMD (v, u, x ! 1) =Constant metric distortion<latexit sha1_base64="X1FKt4r3/ZFHdSH859iMbRnmwlc=">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</latexit>

65

Non-decaying source!

<latexit sha1_base64="pZHudWZMMLxKThiwx5QJhfmdrM8=">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</latexit><latexit sha1_base64="Y29DcK0TuTQkjK2MoXRyaCUw1lM=">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</latexit>x̄))fMD (v, u, x) =25

← Not diluted!

After MD

!25

Propagating GW

Calculable GW spectrum
Example 1 in MD: Monochromatic caseP⇣ (k) = A⇣ (log k/k⇤ )<latexit sha1_base64="JosQBY922HoZ4AkvkG3OrjMNMKQ=">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</latexit>

3
⌦GW =
25
<latexit sha1_base64="UNA3SEV3A587/iroR6raoY/MwcM=">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</latexit>

✓

k⇤
aH

◆2

1

✓

k
2k⇤

◆2 ! 2

A2⇣ ⇥(2k⇤k)

Example 2 in MD: Scale-invariant case with a cutoﬀ
P⇣ (k) = A⇣ ⇥(kmax⌘8⇣< 1792k̃ 1 2520 + 768k̃ + 105k̃ 2⌘4 ⇣⇥ ⇣: 1 2k̃ 1
105k̃ 2 + 72k̃ + 16 32k̃ 1

k)<latexit sha1_base64="wLdXhgg47zo5aQx9pCsybmCCSEY=">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</latexit><latexit sha1_base64="qbKNTUMTyIHfDf7ETuus/EM4pCk=">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</latexit>

k
aH

◆2

16k̃ 210 4⌘(0 < k  kmax )(kmax < k  2kmax )1000k̃ ⌘ k/k⇤

ΩGW /Aζ 2

A2⇣
⌦GW =
14000

✓

<latexit sha1_base64="FuDh2pTmwYJw7vI5SR5GgX1Aul4=">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</latexit>

100
10
1
0.1 -410!260.001 0.010 0.100
k/kmax110100General cases
The MD → RD transition
I(v, u, x) =

Z xR

dx̄

R

kGMD!RD(⌘, ⌘¯)fMD (v, u, x̄)
k

x
xR
Z x
⇣ x̄ ⌘kGRD¯)fMD!RD (v, u, x̄)+dx̄k (⌘, ⌘xxR0<latexit sha1_base64="M2A2bij7oXzqsdxy/FwCsL+4960=">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</latexit>⇣ x ⌘ ✓ x̄ ◆2

The RD → MD transition
I(v, u, x) =

Z xeq

eq

kGRD!MD(⌘, ⌘¯)fRD (v, u, x̄)
k

x
xeq
Z x
⇣ x̄ ⌘2+dx̄
kGMD¯)fRD!MD (v, u, x̄)k (⌘, ⌘xxeq
0<latexit sha1_base64="eVhuMW//hshof6xujMrfk4I9lTI=">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</latexit>dx̄

⇣ x ⌘2 ✓ x̄ ◆!27

General cases
The MD → RD transition
I(v, u, x) =

Z xR

dx̄

R

kGMD!RD(⌘, ⌘¯)fMD (v, u, x̄)
k

x
xR
Z x
⇣ x̄ ⌘kGRD¯)fMD!RD (v, u, x̄)+dx̄k (⌘, ⌘xxR0<latexit sha1_base64="M2A2bij7oXzqsdxy/FwCsL+4960=">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</latexit>⇣ x ⌘ ✓ x̄ ◆2

The RD → MD transition
I(v, u, x) =

Z xeq

eq

kGRD!MD(⌘, ⌘¯)fRD (v, u, x̄)
k

x
xeq
Z x
⇣ x̄ ⌘2+dx̄
kGMD¯)fRD!MD (v, u, x̄)k (⌘, ⌘xxeq
0<latexit sha1_base64="eVhuMW//hshof6xujMrfk4I9lTI=">AAAD6XicjVFLa9RQFD6T+Kjjo9O68bG5OHSYwWG4KYIiCEUL6kJop05baDohSW+mIc8mN0NqyMqdO1eKrlRExJ/hxj/goj9BXVZw48KTB0xnBqs3JPnuOd93znfu1XzbDDmlBxVBPHHy1OmZM9Wz585fmK3Nza+HXhTorKd7thdsamrIbNNlPW5ym236AVMdzWYbmnUvy28MWRCanvuY7/ts21EHrmmYusoxpMxVnj5sDttRO26ROw3ZdLlC+0msJDJnMU/YXpqmpMA7qaypQRLj3mYGbxLZCFR9gpunA3Owy1v9xXFiqU6nqhd0QixyX7FIv0x2l7PGXtn80XKaNmXG1TbJ62QwbRFDGbHTfI6yC5aTkep6buRoLCCyXG2Q6yQfcMJAPz5+wJHvo6ONm/0Pe5PDHDWr1Oq0Q/NFpoFUgjqUa8WrfQAZdsADHSJwgIELHLENKoT4bIEEFHyMbUOCsQCRmecZpFBFbYQshgwVoxZ+B7jbKqMu7rOaYa7WsYuNb4BKAgv0K/1ID+kX+ol+o7//WivJa2Re9vGvFVrmK7PPLq39+qfKwT+H3ZHqWM8cDLiVezXRu59Hsin0Qj988uJw7XZ3IWnQt/QH+n9DD+hnnMAd/tTfr7Lua6jiBUiTxz0N1hc7Eu1IqzfqS3fLq5iBq3ANmnjeN2EJHsAK9ECvfBdqwmXhimiJz8WX4quCKlRKzUUYW+K7PxLiG+o=</latexit>dx̄

⇣ x ⌘2 ✓ x̄ ◆!27

General cases
The MD → RD transition
I(v, u, x) =

Z xR

dx̄

R

kGMD!RD(⌘, ⌘¯)fMD (v, u, x̄)
k

x
xR
Z x
⇣ x̄ ⌘kGRD¯)fMD!RD (v, u, x̄)+dx̄k (⌘, ⌘xxR0<latexit sha1_base64="M2A2bij7oXzqsdxy/FwCsL+4960=">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</latexit>⇣ x ⌘ ✓ x̄ ◆2

The RD → MD transition
I(v, u, x) =

Z xeq

eq

kGRD!MD(⌘, ⌘¯)fRD (v, u, x̄)
k

x
xeq
Z x
⇣ x̄ ⌘2+dx̄
kGMD¯)fRD!MD (v, u, x̄)k (⌘, ⌘xxeq
0<latexit sha1_base64="eVhuMW//hshof6xujMrfk4I9lTI=">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</latexit>dx̄

⇣ x ⌘2 ✓ x̄ ◆!27

General cases
The MD → RD transition
I(v, u, x) =

Z xR

dx̄

R

kGMD!RD(⌘, ⌘¯)fMD (v, u, x̄)
k

x
xR
Z x
⇣ x̄ ⌘kGRD¯)fMD!RD (v, u, x̄)+dx̄k (⌘, ⌘xxR0<latexit sha1_base64="M2A2bij7oXzqsdxy/FwCsL+4960=">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</latexit>⇣ x ⌘ ✓ x̄ ◆2

The RD → MD transition
I(v, u, x) =

Z xeq

eq

kGRD!MD(⌘, ⌘¯)fRD (v, u, x̄)
k

x
xeq
Z x
⇣ x̄ ⌘2+dx̄
kGMD¯)fRD!MD (v, u, x̄)k (⌘, ⌘xxeq
0<latexit sha1_base64="eVhuMW//hshof6xujMrfk4I9lTI=">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</latexit>dx̄

⇣ x ⌘2 ✓ x̄ ◆!27

General cases
The MD → RD transition
I(v, u, x) =

Z xR

dx̄

R

kGMD!RD(⌘, ⌘¯)fMD (v, u, x̄)
k

x
xR
Z x
⇣ x̄ ⌘kGRD¯)fMD!RD (v, u, x̄)+dx̄k (⌘, ⌘xxR0<latexit sha1_base64="M2A2bij7oXzqsdxy/FwCsL+4960=">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</latexit>⇣ x ⌘ ✓ x̄ ◆2

The RD → MD transition
I(v, u, x) =

Z xeq

eq

kGRD!MD(⌘, ⌘¯)fRD (v, u, x̄)
k

x
xeq
Z x
⇣ x̄ ⌘2+dx̄
kGMD¯)fRD!MD (v, u, x̄)k (⌘, ⌘xxeq
0<latexit sha1_base64="eVhuMW//hshof6xujMrfk4I9lTI=">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</latexit>dx̄

⇣ x ⌘2 ✓ x̄ ◆!27

General cases
Schematic

Precise
10 4

RD → MD/k

1000
ΩGW /Aζ 2<latexit sha1_base64="UkuZRFCSjGEgJMyaSersu02Rk6A=">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</latexit>

100
10

/k

n(k)

10

100

6 . n(k) .

4<latexit sha1_base64="Fl6M45B6Az1Rod2uWI+Fv7RF4QI=">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</latexit>

MD → RD1
0.1 -410/k3<latexit sha1_base64="acPFGMdKO5/jsHPjV8gqN2xv/Fw=">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</latexit>

0.001 0.010 0.100

1k/kmax<latexit sha1_base64="/fsbfU/Aw1d5xCyP8/z+Ied/LBY=">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</latexit>

Scale-invariant primordial spectrum is assumed as a simple example.
!28

General cases
Schematic

Precise
10 4

RD → MD/k

1000
ΩGW /Aζ 2<latexit sha1_base64="UkuZRFCSjGEgJMyaSersu02Rk6A=">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</latexit>

100
10

/k

n(k)

10

100

6 . n(k) .

4<latexit sha1_base64="Fl6M45B6Az1Rod2uWI+Fv7RF4QI=">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</latexit>

MD → RD1
0.1 -410/k3<latexit sha1_base64="acPFGMdKO5/jsHPjV8gqN2xv/Fw=">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</latexit>

0.001 0.010 0.100

1k/kmax<latexit sha1_base64="/fsbfU/Aw1d5xCyP8/z+Ied/LBY=">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</latexit>

Scale-invariant primordial spectrum is assumed as a simple example.
!28

APPLICATION:PRIMORDIAL
BLACK HOLE

SCENARIOS[Kohri, Terada, 1802.06785]

Current PBH abundance

PBH fraction in CDM

1

d⇢PBH
fPBH (M ) =⇢CDM d ln M
<latexit sha1_base64="whx+nVPveP+p/yWASKQwt3702IA=">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</latexit>=<latexit sha1_base64="1q91lJoeKRf/082E9jYBCQjJazo=">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</latexit>✓g⇤ (T ) g⇤,s (Teq ) Tg⇤ (Teq ) g⇤,s (T ) Teq( (k(M )))

!30

◆

⌦m
T =Min[TM ,TR ] ⌦CDM

Current PBH abundance

PBH fraction in CDM

1

d⇢PBH
fPBH (M ) =⇢CDM d ln M

PBH formation probability<latexit sha1_base64="whx+nVPveP+p/yWASKQwt3702IA=">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</latexit>=<latexit sha1_base64="1q91lJoeKRf/082E9jYBCQjJazo=">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</latexit>✓g⇤ (T ) g⇤,s (Teq ) Tg⇤ (Teq ) g⇤,s (T ) Teq( (k(M )))

!30

◆

⌦m
T =Min[TM ,TR ] ⌦CDM

Current PBH abundance
Observational constraints
Note: These are not the latest constraints.

[Carr, Kühnel, Sandstad, 1607.06077]

PBH fraction in CDM

1

d⇢PBH
fPBH (M ) =⇢CDM d ln M

PBH formation probability<latexit sha1_base64="whx+nVPveP+p/yWASKQwt3702IA=">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</latexit>=<latexit sha1_base64="1q91lJoeKRf/082E9jYBCQjJazo=">AAADWHichVHNbtNAEB7HQNvw07RckLhYREU2qqI1QipCqlRRDlwi2tRpi+LIWrsbdxX/YW8iirG4Ai/AgRNICCEegwsvwKG8AeIYJC4cmNhOSxsBY3ln9pvvm5ndtSOPJ4KQQ6kinzl7bmZ2rnr+wsVL87WFxe0kHMQOazuhF8a7Nk2YxwPWFlx4bDeKGfVtj+3Y/fVxfmfI4oSHgSEOItb1qRvwHneoQMiqjVZNj/VEQ8mdqpi9mDqpa6U3MtXQsklkpaZgj0XKHmWZlv3BWk6mssf4EdPI0hMkxF3q+1QxbSaoaibc9anaV5uapilmzN19oRVOeWqlxqpSSJs8yDqG1VxWjqq1sFZ30sZ84DOXTlI+9jkFrd9rImjV6qRBclOmA70M6lDaRlh7DybsQQgODMAHBgEIjD2gkODXAR0IRIh1IUUsxojneQYZVFE7QBZDBkW0j6uLu06JBrgf10xytYNdPPxjVCqwRL6QD2REPpOP5Bv59ddaaV5jPMsBervQssiaf3ll6+d/VT56AfvHqn/OLKAHt/NZOc4e5cj4FE6hHz55Ndq601pKr5O35DvO/4Yckk94gmD4w3m3yVqvoYoPoJ++7ulg+2ZDJw1981Z97W75FLNwFa6Bive9AmtwHzagDY70UHomPZdeVL7KIM/IcwW1IpWay3DC5MXfOj3lrQ==</latexit>✓g⇤ (T ) g⇤,s (Teq ) Tg⇤ (Teq ) g⇤,s (T ) Teq( (k(M )))

!30

◆

⌦m
T =Min[TM ,TR ] ⌦CDM

PBH formation probability[Young, Byrnes, Sasaki, 1405.7023]
see also [Ando, Inomata, Kawasaki, 1802.06393]

Coarse-grained perturbations
2

(k) =

Z 1

d ln q w2

1<latexit sha1_base64="wDainmd0xaBah9xr/JuLZ0WMoRc=">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</latexit>⇣ q ⌘ 4(1 + w

2 ⇣ ⌘4)qeos
2T(q/k)P⇣ (q)2(5 + 3weos ) kk

2(1 + weos )2⇠P⇣ (k)
2(5 + 3weos )<latexit sha1_base64="KvXwm0OE6kVs7H+RI3QpRL31MEw=">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</latexit>

Formation probability in RD
Pressure prevents the formation → rare process( )=<latexit sha1_base64="IakpzbRSHg3rOAzcXaFhhP7YAUc=">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</latexit>

Z 1
c

1
p
exp
2⇡

✓

2 2

◆

[Press, Schechter, 1974]

0.4

0.3

d

0.2

0.1

-3

-2

-1

1

2

(exaggerated figure for illustration)!31

3

PBH formation probability[Young, Byrnes, Sasaki, 1405.7023]
see also [Ando, Inomata, Kawasaki, 1802.06393]

Coarse-grained perturbations
2

(k) =

Z 1

Window function

d ln q w2

1<latexit sha1_base64="wDainmd0xaBah9xr/JuLZ0WMoRc=">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</latexit>⇣ q ⌘ 4(1 + w

2 ⇣ ⌘4)qeos
2T(q/k)P⇣ (q)2(5 + 3weos ) kk

2(1 + weos )2⇠P⇣ (k)
2(5 + 3weos )<latexit sha1_base64="KvXwm0OE6kVs7H+RI3QpRL31MEw=">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</latexit>

Formation probability in RD
Pressure prevents the formation → rare process( )=<latexit sha1_base64="IakpzbRSHg3rOAzcXaFhhP7YAUc=">AAAC7XichVHLahRBFL3dPpKMj4yahSBK4xCZbIbqQVAEIZiNyzycJJBKmuqe25Mi/bK7ZshY1F7ci4uIoCBB/Aw3/oCLfIIIbiK4ceGd7gbRoN6mq06de86tW1V+FslCMXZk2adOnzk7NT3TOHf+wsXZ5qXL60U6zAPsBWmU5pu+KDCSCfaUVBFuZjmK2I9ww99bmuQ3RpgXMk0eqXGG27EYJDKUgVBEec2n3EclnDYv5CAWC859h8tEeZr3MVLC4wr3lQ6M2dHEh2psHB7mItCu0bx4nCvd5Zk0lZtyuJ85PMJQtWtdVceQrJTsdA3P5WBXLThV6b6pFI7XbLEOK8M5CdwatKCO5bR5CBz6kEIAQ4gBIQFFOAIBBX1b4AKDjLht0MTlhGSZRzDQIO+QVEgKQewejQNabdVsQutJzaJ0B7RLRH9OTgfm2Sf2jh2zj+w9+8x+/LWWLmtMehnT7FdezLzZZ1fXvv/XFdOsYPeX6589KwjhbtmrpN6zkpmcIqj8oycvjtfurc7rW+wN+0L9v2ZH7AOdIBl9C96u4OoBNOgB3D+v+yRY73Zc1nFXbrcWH9RPMQ3X4Ca06b7vwCI8hGXo0b5frTnrunXDTu3n9kv7VSW1rdpzBX4L+/AnxLq/Dw==</latexit>

Z 1
c

1
p
exp
2⇡

✓

2 2

◆

[Press, Schechter, 1974]

0.4

0.3

d

0.2

0.1

-3

-2

-1

1

2

(exaggerated figure for illustration)!31

3

PBH formation probability[Young, Byrnes, Sasaki, 1405.7023]
see also [Ando, Inomata, Kawasaki, 1802.06393]

Coarse-grained perturbations
2

(k) =

Z 1

Window function

d ln q w2

1<latexit sha1_base64="wDainmd0xaBah9xr/JuLZ0WMoRc=">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</latexit>⇣ q ⌘ 4(1 + w

2 ⇣ ⌘4)qeos
2T(q/k)P⇣ (q)2(5 + 3weos ) kkTransfer function

2(1 + weos )2⇠P⇣ (k)
2(5 + 3weos )<latexit sha1_base64="KvXwm0OE6kVs7H+RI3QpRL31MEw=">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</latexit>

Formation probability in RD
Pressure prevents the formation → rare process( )=<latexit sha1_base64="IakpzbRSHg3rOAzcXaFhhP7YAUc=">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</latexit>

Z 1
c

1
p
exp
2⇡

✓

2 2

◆

[Press, Schechter, 1974]

0.4

0.3

d

0.2

0.1

-3

-2

-1

1

2

(exaggerated figure for illustration)!31

3

PBH formation probability[Young, Byrnes, Sasaki, 1405.7023]
see also [Ando, Inomata, Kawasaki, 1802.06393]

Coarse-grained perturbations
2

(k) =

Z 1

Window function

d ln q w2

1<latexit sha1_base64="wDainmd0xaBah9xr/JuLZ0WMoRc=">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</latexit>

Primordial curvature perturbations⇣ q ⌘ 4(1 + w

2 ⇣ ⌘4)qeos
2T(q/k)P⇣ (q)2(5 + 3weos ) kkTransfer function

2(1 + weos )2⇠P⇣ (k)
2(5 + 3weos )<latexit sha1_base64="KvXwm0OE6kVs7H+RI3QpRL31MEw=">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</latexit>

Formation probability in RD
Pressure prevents the formation → rare process( )=<latexit sha1_base64="IakpzbRSHg3rOAzcXaFhhP7YAUc=">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</latexit>

Z 1
c

1
p
exp
2⇡

✓

2 2

◆

[Press, Schechter, 1974]

0.4

0.3

d

0.2

0.1

-3

-2

-1

1

2

(exaggerated figure for illustration)!31

3

PBH formation probability
Formation probability in RD
Pressure prevents the formation → rare process( )=<latexit sha1_base64="IakpzbRSHg3rOAzcXaFhhP7YAUc=">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</latexit>

Z 1
c

1
p
exp
2⇡

✓

2 2

◆

[Press, Schechter, 1974]

0.4

0.3

d

0.2

0.1

-3

-2

-1

1

2

(exaggerated figure for illustration)!32

3

PBH formation probability
Formation probability in RD
Pressure prevents the formation → rare process( )=<latexit sha1_base64="IakpzbRSHg3rOAzcXaFhhP7YAUc=">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</latexit>

Z 1
c

1
p
exp
2⇡

✓

2 2

◆

0.4

0.3

d

0.2

0.1

[Press, Schechter, 1974]

-3

-2

-1

1

2

(exaggerated figure for illustration)

Formation probability in MD⇢/ a(t)⇢[Khlopov, Polnarev, 1980, 1985]

Perturbation grows in the MD era → low threshold
Anisotropy & angular momentum aﬀects the formation rate.

!32

3

PBH formation probability
Formation probability in RD
Pressure prevents the formation → rare process

( )=

Z 1
c

<latexit sha1_base64="IakpzbRSHg3rOAzcXaFhhP7YAUc=">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</latexit>

1
p
exp
2⇡

✓

2 2

◆

0.3

d

0.2

0.1

[Press, Schechter, 1974]-3⇢/ a(t)⇢[Khlopov, Polnarev, 1980, 1985]

Perturbation grows in the MD era → low threshold
Anisotropy & angular momentum aﬀects the formation rate.

( )=1.894 ⇥ 10 6 ⇥ fq I 6 2 exp
0.05556 5

⇣

-2

-1

1

2

(exaggerated figure for illustration)

Formation probability in MD

(

0.4

I 4/3

0.1474 2/3⌘<latexit sha1_base64="QLhTfMvinRZL9tqfGFHlHJlhh1U=">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</latexit>[Harada, Yoo, Kohri, Nakao, Jhingan, 1609.01588][Harada, Yoo, Kohri, Nakao, 1707.03595]!32( < 0.005)(

0.005)

3

PBH formation probability
Formation probability in RD
Pressure prevents the formation → rare process

( )=

Z 1
c

<latexit sha1_base64="IakpzbRSHg3rOAzcXaFhhP7YAUc=">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</latexit>

1
p
exp
2⇡

✓

2 2

◆

0.4

0.3

d

0.2

0.1

[Press, Schechter, 1974]

-3

-2

-1

1

2

(exaggerated figure for illustration)

Formation probability in MD⇢/ a(t)⇢[Khlopov, Polnarev, 1980, 1985]

Perturbation grows in the MD era → low threshold
Anisotropy & angular momentum aﬀects the formation rate.

( )=

(

1.894 ⇥ 10 6 ⇥ fq I 6 2 exp
0.05556 5

⇣

I 4/3

0.1474 2/3⌘( < 0.005)(

0.005)<latexit sha1_base64="QLhTfMvinRZL9tqfGFHlHJlhh1U=">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</latexit>[Harada, Yoo, Kohri, Nakao, Jhingan, 1609.01588][Harada, Yoo, Kohri, Nakao, 1707.03595]!32[Harada, Yoo, Kohri, Nakao, 1707.03595]

3

Inflation with running spectrum
PR (k) = As

✓

k
k⇤

◆ns 1+ ↵2s ln kk + 6s (ln kk )2 +···⇤⇤

Planck TT+lowP (TT,TE,EE+lowP)

at 68% CL at the pivot scale k*=0.05/Mpc.

[Planck 2015, “Constraints on Inflation”]

Interplay between PBH & GW
PBH mass spectrum

PBH-for-LIGO scenario
fPBH (M)

10 -2

Curvature perturbations
0.100

10 -8

0.001

0.0

0.5

1.0

1.5

2.0

2.5

3.0

10 -5

log10 (M/Msolar )10 -7Secondary GW spectrum

10 -9

0

5

10

15

10 -5

log10 (k/k* )ns =0.96↵s =0<latexit sha1_base64="X8MefuKmA6/V1ARmlyM5I8f4Crw=">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</latexit><latexit sha1_base64="P3YXBKCKQEea5HQxtg6gKUwCFtU=">AAACrnicSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQvIh8RXx5SkVpRUB9XW2ioYGsRZxugoQETcU8Nq4wWUDfQMwEABk2EIZSgzQEFAvsAehhiGFIZ8hmSGUoZchlSGPIYSIDuHIZGhGAijGQwZDBgKgGKxDNVAsSIgKxMsn8pQy8AF1FsKVJUKVJEIFM0GkulAXjRUNA/IB5lZDNadDLQlB4iLgDoVGFQNrhqsNPhscMJgtcFLgz84zaoGmwFySyWQToLoTS2I5++SCP5OUFcukC5hyEDowuvmEoY0BguwWzOBbi8Ai4B8kQzRX1Y1/XOwVZBqtZrBIoPXQPcvNLhpcBjog7yyL8lLA1ODZuNxD8wlRUDZNCAGRZUhesRgMsKM9AwN9AwDTZQdnKCRxsEgzaDEoAGMGXMGBwYPhgCGUKBNrQyrGbYwbGUyYApjimWKhyhlYoTqEWZAAUwZAA4Insg=</latexit>

ΩGW h2Pζ (k)

10 -5

TR = 109 GeV

s =0.026

10 -8
10 -11
10 -1410 -1310 -11

10 -9
f/Hz

!34

10 -7

Interplay between PBH & GW
PBH mass spectrum

PBH-DM scenario
1

fPBH (M)0.100

Curvature perturbations

0.010
0.001

0.100

10 -4
-17

-16

0

5

10

15

-13

10 -7

log10 (k/k* )ns =0.96↵s =0<latexit sha1_base64="jOkzgoBfq+KHZeHOcNYx+pNpi+s=">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</latexit>-14

Secondary GW spectrum

10 -7
10 -9

-15
log10 (M/Msolar )10 -5

ΩGW h2Pζ (k)

0.001

4<latexit sha1_base64="oO/6RkkNa7Rpdlr3Q5P7u2TuiQM=">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</latexit>

TR = 10 GeV

10 -10
10 -1310 -16s =0.0019485

10 -19
10 -6

0.01

10 -4
f/Hz

!35

1

Comparison with Espinosa et al.
I(v, u, x) =<latexit sha1_base64="rlZW+MjM/lwOegUSkV737qoKoIE=">AAAC+XichVFLaxRBEK6dqInxkU28CF4al8guLEuNBJRAIOhBveXhJoFMHHomPZt2Z2eGnt5lk2H+gH/Ag6cIgo+j19xy8ejFQ36CeIzoRdCa3hEfwVhDT3/9VX3VX1NeEspUIx5VrLEzZ8+NT5yfvHDx0uWp6vTMWhr3lS/afhzGasPjqQhlJNpa6lBsJErwnheKda97t8ivD4RKZRw91LuJ2OrxTiQD6XNNlFt9/KA+aPabw8YCc2SkXXw0ZI4WQ51t547HVTbMmRMo7mec1Q3hCM3zRp7xeoEaOeuye26XmVOT/VbCAmZ6l20abrWGLTTBTgK7BDUoYymuHoAD2xCDD33ogYAINOEQOKT0bYINCAlxW5ARpwhJkxeQwyRp+1QlqIIT26V/h06bJRvRueiZGrVPt4S0FCkZzOIHfIXH+A7f4Ef89s9emelReNml3RtpReJOPbm6+vW/qh7tGnZ+qU71rCGA28arJO+JYYpX+CP9YO/p8er8ymx2A5/jJ/K/j0d4SC+IBp/9F8ti5dkpfn46UZQNaOU0KvvvwZwEazdbNrbs5bna4p1yaBNwDa5DnSZzCxbhPixBm256D1/gewWszNq3XlqvR6VWpdRcgT/CevsDAMS6jw==</latexit>

Z x

a(¯
⌘)
dx̄
kGk (⌘, ⌘¯)f (v, u, x̄)
a(⌘)

x0min
<latexit sha1_base64="XgGJsailfBgd3CNe5RkeraSijwc=">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</latexit>

Espinosa et al. :xmin = k⌘min = 1<latexit sha1_base64="G1fadwfZ9Uivz/yL5DsA07GPNUI=">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</latexit>

GW production right after inflation

Kohri, Terada :xmin = k⌘min = 0<latexit sha1_base64="owkIj26fLgG5muJ1Rc+rcaPWU24=">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</latexit>

inflation → RD/MD → GW production[Espinosa, Racco, Riotto, 1804.07732][Kohri, Terada, 1804.08577]