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Gravitational waves from the early Universe
Sachiko Kuroyanagi (Nagoya University)
26 Aug 2017
Summer Institute 2017
Part 2
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GWs from inflation
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InflationAccelerated expansion in the early Universe
Solves horizon/flatness/monopole problem
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log(a) →
H-1
radiation → matter-dominated
today
H-1∝a3/2 or a2
aλaλ: physical scale
H-1: Hubble horizon(region of causality = the distance light can travel )
Horizon problem
aλpast
now
horizon grows faster than physical scale
no causal relation
H-1
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log(
Scal
e) →
log(a) →
H-1
radiation → matter-dominated
today
H-1∝a3/2 or a2
Inflation phase
aλexponential growth of the scale factor
H-1~const
inside the causal region↑
a exp(Ht)→
Horizon problem
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ϕ
VInflation is driven by a scalar field slowly rolling down in its potential
slow-‐roll
Friedmann equation
Exponential expansion
Standard scenario
a exp(Ht)
H = const.
What drives inflation?
→ new particle? modification of gravity?
energy density of a scalar field Equation of Motion
slow-roll approximation
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ϕ
V
Oscillation
scalar field
radiation :decay rate
→ matter-dominated like Universe for φ2 potential(radiation-dominated for φ4 potential)
ReheatingStandard scenarioThe scalar field decays into standard model particles
Equation of Motion
Friedmann equation
Reheating temperature
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GWs from inflation
GW Power ∝ H2inf ∝ Vinf
wide range
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Generation mechanism
InflationH-1~const
a/k
H-1
log(a) →
log(
Scal
e) →
wavelength of GW
Hubble horizon
scale factor of the Universe
Today
quantum fluctuation in space-time
→ becomes GWs
stretched over the horizon
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InflationH-1~const
log(a) →
log(
Scal
e) →
Direct detection
CMB
long wavelength
short wavelength
a/k
H-1
wavelength of GW
Generation mechanism
→ GWs are produced for all wavelength
Hubble horizon
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hij + 3Hhij 1a22hij = 16Gij
・H>k/a
・H<k/a
Let us neglect the matter contribution and perform Fourier transform
expansion of the Universe effect from matter
Equation for GWs in the expanding Universe
oscillation
→ behavior is determined by the balance between H (Hubble) and k (wavenumber = f/2π)
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MD
RDInflationReheatingH-1~const
H-1∝a3/2
H-1∝a2
H-1∝a3/2
log(a) →
log(
Scal
e) → in the conventional cosmological scenario
Hubble expansion history after inflation
a/k
H-1
Hubble horizon size
wavelength of GW
ϕ
Vϕ
Vdriven by a scalar field
slow-roll
decay of the scalar field
oscillations
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MD
RDInflationReheatingH-1~const
H-1∝a3/2
H-1∝a2
H-1∝a3/2
log(a) →
log(
Scal
e) → in the conventional cosmological scenario
Hubble expansion history after inflation
a/k
H-1
Hubble horizon size
wavelength of GW
2nd term
3rd term
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MD
RDInflationReheatingH-1~const
H-1∝a3/2
H-1∝a2
H-1∝a3/2
outside the horizon
log(a) →
log(
Scal
e) →
inside the horizon
a/k
H-1
Hubble horizon size
wavelength of GW
Evolution of GWs
decay
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RDInflationReheatingH-1~const
H-1∝a3/2
H-1∝a2
Each mode experiences different evolution = different amplitude for different frequency
outside the horizon
log(a) →
log(
Scal
e) →
inside the horizon
Hubble expansion history after inflation
Direct detection
CMB
a/k
H-1
Hubble horizon size
wavelength of GW
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GW amplitude
InflationH-1~const
a/k
H-1
log(a) →
log(
Scal
e) →
Hubble horizon size
scale factor of the Universe
Today
quantum fluctuation in space-time
wavelength of GW
initial condition
|hk| =1
apkMpl
decay
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InflationH-1~const
a/k
H-1
log(a) →
log(
Scal
e) →
scale factor of the Universe
Today
quantum fluctuation in space-time
low energy scale inflation
wavelength of GW
GW amplitude
initial condition
decay
|hk| =1
apkMpl
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InflationH-1~const
a/k
H-1
log(a) →
log(
Scal
e) →
scale factor of the Universe
Today
quantum fluctuation in space-time
wavelength of GW
aout,1 aout,2<
Horizon out at k=aH
GW amplitude
initial condition
decay
|hk| =1
apkMpl
→ a-1∝H
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InflationH-1~const
a/k
H-1
log(a) →
log(
Scal
e) →
scale factor of the Universe
Today
quantum fluctuation in space-time
wavelength of GW
aout,high aout,low<hk,high hk,low>→GW Power ∝ H2inf
GW amplitude
low-scale inflationinitial condition
decay
|hk| =1
apkMpl
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Inflation
reheating
radiationdominant
matterdominant
a ∝exp(Ht)
a ∝t2/3
a∝t1/2
a∝t2/3
Outside the horizon
Inside the horizon
Amplitude of GW background
k-‐2
k-2
k0
k
ΩGW
scale invariant spectrum
small scale modes begin to enter the horizon and damp with ∝a-1
k-2
k0
k-2
the expansion decelerates = damping ∝ a-1 becomes smaller
PT∝k0
Spectral shape
scale invariant =
→ ΩGW∝k2
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spec
tral
am
plit
ude
frequency = k/2π
∝ k-2
∝ k0
∝ k-2
Spectral shape
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spec
tral
am
plit
ude
frequency = k/2π
Spectral shape
∝ k0 Power spectrum PT(k)
For slow-roll inflation
→GW Power ∝ H2inf ∝ Vinf
ϕ
Vslow-‐roll
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spec
tral
am
plit
ude
frequency = k/2π
ΩGW, today∝ k-2
reheating temperature TRH
low high
Effect of reheating
→ matter-dominated Universe
Reheating after inflation
for Φ2 potential, H ∝a-3/2ϕ
V
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spec
tral
am
plit
ude
frequency = k/2π
→ radiation-dominated Universe
ΩGW, today∝ k0
for Φ4 potential, H ∝a-2ϕ
VEffect of reheatingReheating after inflation
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spec
tral
am
plit
ude
frequency = k/2π
ΩGW, today∝ k1
reheating temperature TRH
low high
Effect of reheating
→ kination-dominated Universe
Reheating after inflation
kinatic energy > potential energy → H ∝a-6
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Inflation (m2Φ2 potential)matter
dominatedradiation
dominatedreheating
MD
RD
MD
tilt of the spectrum+
deviation from the slow-roll
ϕ
V(Φ)
Spectral shape from numerical calculation
S. Kuroyanagi, T. Chiba and N. Sugiyama, Phys. Rev. D 79, 103501 (2009)
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S. Kuroyanagi, T. Chiba and N. Sugiyama, Phys. Rev. D 79, 103501 (2009)
Spectral shape from numerical calculation
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primordial spectrum with tilt
Damping due to the changes in effective number of degrees of freedom g*
log( T [MeV] )
log(
g* )
As the temperature of the universe decreases, relativistic matter particles become non-relativistic.
temperature decreases→ contribution to ρ and s decreases→ step-like changes in H→ step shape in ΩGW becomes non-relativistic when T~m
Effective number of degrees of freedom
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=0anisotropic stress term
Damping due to the neutrino anisotropic stress
Before neutrino decoupling (T>2MeV)Anisotropic stress is suppressed by the coupling with matter (e±)
Neutrino anisotropic stress
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Damping due to the neutrino anisotropic stress
initially =0
gives energy→ Damping only when H~k/a
Before neutrino decoupling (T>2MeV)Anisotropic stress is suppressed by the coupling with matter (e±)
After neutrino decoupling (T<2MeV)Neutrino anisotropic stress affects GWs as a viscosity when they enter the horizon
Neutrino anisotropic stress
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anisotropic stress term ∝ρν ~0
Damping due to the neutrino anisotropic stress
Before neutrino decoupling (T>2MeV)Anisotropic stress is suppressed by the coupling with matter (e±)
After the Universe becomes matter-dominatedThe energy density of radiation becomes negligible
After neutrino decoupling (T<2MeV)Neutrino anisotropic stress affects GWs as a viscosity when they enter the horizon
Neutrino anisotropic stress
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Before neutrino decoupling (T>2MeV)Anisotropic stress is suppressed by the coupling with matter (e±)
After the Universe becomes matter-dominatedThe energy density of radiation becomes negligible
anisotropic stress term
After neutrino decoupling (T<2MeV)Neutrino anisotropic stress affects GWs as a viscosity when they enter the horizon
① Neutrino decoupling
② Start of matter domination Damping due to the neutrino anisotropic stress
Neutrino anisotropic stress
→ ②
→ ①
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“Planck 2015 results. XX. Constraints on inflation”, Planck Collaboration, A&A, 594 (2016) A20
Constraint on inflation from Planck
Dependence on inflation models
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Dependence on inflation modelsKuroyanagi et al. PRD 79, 103501 (2009)Kuroyanagi et al. PRD 90, 063513 (2014)
-19
-18
-17
-16
-15
-14
-13
-12
-11
-15 -10 -5 0 5
TRH=109GeV
log 1
0 ΩG
W
log10 f[Hz]
DECIGOUpgraded DECIGOUltimate DECIGO
-19
-18
-17
-16
-15
-14
-13
-12
-11
-15 -10 -5 0 5
TRH=109GeV
log 1
0 ΩG
W
log10 f[Hz]
λφ4/4m2φ2/2Natural inflation: f=7Mpl
R2 inflation ← quantum limit
no MD↓
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GWs from cosmic strings
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Cosmological history
inflation now
Log(time)
cosmic phase transitions
Theories
We are here
reheatingcosmic string
cosmic superstring
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Phase transition in the Universe
Φ
potential VΦ1Φ2
Φ=0
vacuumΦ=Φ2
vacuumΦ=Φ1
height = energyhigher energy
low energy low energy
In 3-dimension space,it looks like a wall
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2- dimensional potential in 3-dimension space
becomes a string
→ Cosmic string
Φ=0high energy in a string
Phase transition in the Universe
Line Density:
Φ=σ
Z(@i)
2d
2x
Z
r
2d
2x
2
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2: Cosmic superstrings
Important notes
1: Phase transition
“The remarkable fact is that, although many possibilities remain, every one of them predicts the formation of topological or embedded cosmic strings at the end of inflation. So it seems that cosmic strings are almost unavoidable.”
Jeannerot et al., PRD 68 103514 (2003)
“In all acceptable spontaneous symmetry breaking schemes, cosmic string formation is unavoidable”
Kibble, Lecture at COSLAB 2004, arXiv:astro-ph/0410073
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Cosmic string network
loop
infinite string
infinite string becomes a loop by reconnection
strings emit gravitational waves especially from singular structures
kink cusp
loops lose energy and shrink by emitting GWs and eventually evaporate
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kinks on infinite strings
cusps on loops
GWs from cosmic strings
→ GW from cusps are dominant
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3 main parameters to characterize cosmic string
・Gμ:tension = line density
・α:initial loop size L~αH-1
・p:reconnection probabilityPhase transition origin: p=1Cosmic superstring: p<<1
Generation mechanism
Network evolution
What determines the GW amplitude?
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The energy density of cosmic strings
∝a-2 ~ (line density × length)/volume
energy density
a: scale factor
∝a-4
∝a-3
∝a-2 Cosmic strings
radiation
matter
X
a: scale factor of the Universe
Evolution of cosmic string network
∝a1 ∝a3
L∝a1 V∝a3
→ becomes dominant component of the Universe?
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The energy density of cosmic strings
∝a-2 ~ (line density × length)/volume
energy density
a: scale factor
∝a-4
∝a-2 X
Cosmic strings
radiation
X
a: scale factor of the Universe
Evolution of cosmic string network
∝a1 ∝a3
L∝a1 V∝a3
→ follows the total energy density of the Universe
→ GWs
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horizon
Strings can communicate each other inside the horizon
reconnection
→ Loops disappear after continuous GW emission
Evolution of cosmic string networkScaling law The Universe always has O(1-10) strings per horizon
String network keeps producing loops
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i
GW power
Initial loop length =
(initial loop energy)(energy release rate per time)Lifetime of the loop =
Γ: numerical constant ~50-100
=
Loop length at time t
(energy of loop at time t =μl) =(initial energy of the loop =μαti)ー(enegry released to GWs =PΔt)
From the energy conservation law
ti: time when the loop formed
i
depends on Gμ and αEvolution of loops
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Gravitational waves from cosmic string loops
Gravitational waves coming from different directions overlap each other and form gravitational wave background
Search for bursts & stochastic background are both important
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Wide range GW background spectrum
horizon
Early Universe Late Universe
GW frequency ~ 1 / (loop size)
→ high frequency GWs → low frequency GWs
Scaling law The Universe always has O(1-10) strings per horizon
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10-18
10-16
10-14
10-12
10-10
10-8
10-6
10-18 10-16 10-14 10-12 10-10 10-8 10-6 10-4 10-2 100 102 104
Gμ=10-8, 10-10, 10-12, 10-14, 10-16
α=10-1, p=1
oldnewRadiation dominant
Matter dominant
ΩGW
frequency [Hz]
dependence on Gμ
large Gμ
small Gμ
GW power from cusps
h2∝(Gμ)2
life time of loops∝(Gμ)-1
Spectrum of the GWB
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loop size directly corresponds to the frequency of the GW
Gμ=10-8, p=1 α=10-1, 10-5, 10-9, 10-13, 10-17
10-14
10-13
10-12
10-11
10-10
10-9
10-8
10-7
10-6
10-18 10-16 10-14 10-12 10-10 10-8 10-6 10-4 10-2 100 102 104
ΩGW
small α
Spectrum of the GWBdependence
on α
frequency [Hz]
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ΩGW
Gμ=10-12, α=10-1
p=1,10-1, 10-2, 10-3
10-18
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
10-18 10-16 10-14 10-12 10-10 10-8 10-6 10-4 10-2 100 102 104
small p increases the number density of loopssmall p
large p
dependence on p
frequency [Hz]
Spectrum of the GWB
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Accessible cosmic string parameter space
G
α
点線:バースト実線:背景重力波
DECIGO
Adv-LIGO
eLISASKA
Kuroyanagi et al. PRD 87, 023522 (2013)
Loop size α
Line
den
sity
Gμ
reconnection probability p=1
Dotted: burstSolid: background
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Accessible cosmic string parameter space
Dotted: burstSolid: background
DECIGOAdv-LIGO
eLISASKA
Kuroyanagi et al. PRD 87, 023522 (2013)
Loop size α
Line
den
sity
Gμ
reconnection probability p=10-2
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Accessible cosmic string parameter space
Dotted: burstSolid: background
DECIGOAdv-LIGO
eLISASKA
Kuroyanagi et al. PRD 87, 023522 (2013)
Loop size α
Line
den
sity
Gμ
reconnection probability p=10-2
Gμ>10-12
Theoretical lower limit ofcosmic superstring theory
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Summary
→ It may provide information not only on inflation but also on the thermal history after inflation (eg. reheating)
・GWs from inflation are promising
(but the amplitude may be small)
・GWs from cosmic strings has relatively large
amplitude and testable by near-future experiments
GWs can become a powerful probe of the very early Universe
→ It will be useful constraint for the model building of the early Universe