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- 1. Introduction
Imprints of Cosmic Phase Transition on Gravitational Waves (GWs) - - PowerPoint PPT Presentation
Imprints of Cosmic Phase Transition on Gravitational Waves (GWs) - - PowerPoint PPT Presentation
Imprints of Cosmic Phase Transition on Gravitational Waves (GWs) Takeo Moroi (Tokyo) Refs: Jinno, TM and Nakayama, arXiv:1112.0084 [hep-ph] 1. Introduction Early universe: environment with high-energy particles High temperature high energy
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Early universe: environment with high-energy particles High temperature ∼ high energy How “deep” can we probe with various objects?
- Last scattering of photon: ∼ 1 eV
- Last scattering of neutrino: ∼ 1 MeV
- “Last scattering” of GWs: inflation
The history of our universe is imprinted in GWs
⇒ We may be able to extract information about high energy
physics which cannot be probed by colliders
⇒ GW spectrum may be precisely measured in (far) future
by, for e.g., DECIGO
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Today’s subject: phase transition (or SSB)
- QCD phase transition
- EW symmetry breaking
- Peccei-Quinn symmetry
- GUT
- · · ·
In models with SSB, cosmic phase transition may occur
⇒ Is there any effect on observables? ⇒ If yes, what can we learn?
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The spectrum of GWs is affected by cosmic phase transition
- 1. Primordial GWs are produced during inflation (via quan-
tum fluctuation)
- 2. Spectrum of GWs is deformed during the cosmic phase
transition Outline
- 1. Introduction
- 2. Gravitational Waves: Production and Evolution
- 3. Phase Transition
- 4. Effects of Cosmic Phase Transition on GWs
- 5. Summary
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- 2. GWs: Production and Evolution
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Story:
- 1. Primordial GWs are produced during inflation (via quan-
tum fluctuation)
- 2. Evolution of the amplitudes of GWs depends how the
universe expands
- 3. Spectrum of GWs is deformed during the cosmic phase
transition Gravitational wave:
- Fluctuation of the metric (propagating mode)
- Its evolution is governed by the Einstein equation
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Metric: ds2 = −dt2 + a2(t)(δij + 2hij)dxidxj Physical mode: transverse and traceless (hi
i = hij ,j = 0)
Fourier amplitude (using comoving wave-number
k) hij(t, x) = 1 MPl
∑ λ=+,× ∫
d3 k (2π)3˜ h(λ)
- k (t)ǫ(λ)
ij ei k x
MPl ≃ 2.4 × 1018 GeV: Reduced Planck scale ǫ(λ)
ij : polarization tensor (transverse & traceless)
˜ h(λ)
- k (t): Canonically normalized
L(flat) = 1 2M2
Pl ∫
d3xR + · · · ≃
∫
d3 k (2π)3
∑ λ 1
2 ˙ ˜ h
(λ)
- k ˙
˜ h
(λ) − k + · · ·
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Gravitational wave in de Sitter background: a ∝ eHinft
L =
∫
d3 k (2π)3a3 ∑
λ 1
2 ˙ ˜ h
(λ)
- k ˙
˜ h
(λ) − k − 1
2
(k
a
)2
˜ h(λ)
- k ˜
h(λ)
− k
⇒ ˜ h(λ)
- k
behaves as massless scalar field Quantum fluctuation generated during inflation
∆2
h ≡ 1
V
k3
2π2
×
1 M 2
Pl ∑ λ
- |˜
h(λ)
- k |2
- inflation ≃
1 M 2
Pl ∑ λ (Hinf
2π
)2
The primordial GW amplitude is proportional to Hinf
⇒ The effects of GWs become observable when the energy
scale of the inflation is high
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The tensor-to-scalar ratio
r ≡ ∆2
h
∆2
R
(= 16ǫ) R: curvature perturbation (∆2
R ≃ 2.42 × 10−9)
ǫ: Slow-roll parameter
- WMAP 7 years
⇒ r < 0.24
- PLANCK / Future CMB interferometric observations
⇒ r as small as 0.1 − 0.01 will be detected
- Future experiments to detect GWs (DECIGO, · · ·)
⇒ GW spectrum will be observed if r > ∼ ∼ 10−3
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GW evolution after inflation
¨ ˜ h
(λ)
- k
+ 3H ˙ ˜ h
(λ)
- k
+ k2 a2(t) ˜ h(λ)
- k
= 0
with
H = ˙ a a
Before the horizon-in: k ≪ aH
˜ h
k ∼ const.
After the horizon-in: k ≫ aH
−3H ˙ ˜ h
2
- k = ˙
˜ h
k
¨ ˜ h
k +
k2 a2(t) ˙ ˜ h
k˜
h
k = d
dt
1
2 ˙ ˜ h
2
- k + 1
2 k2 a2˜ h2
- k
+ H k2
a2˜ h2
- k
⇒ d dt
k2
a2˜ h2
- k
- sc
≃ −4 ˙ a a
k2
a2˜ h2
- k
- sc
⇔
˙
˜ h
2
- k
- sc
≃
k2
a2˜ h2
- k
- sc
⇒
˜
h2
- k
- sc ∼ a−2
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Amplitude of GWs
⇒ k > ∼ aH: ˜ h
k ∼ const.
⇒ k < ∼ aH: ˜ h2
- kosc ∼ a−2
Scale Inflation RD Scale factor a Horizon Physical Wavelength
- |˜
h(λ)
- k |2
- sc ≃
- |˜
h(λ)
- k |2
- inflation ×
a(t)
a|k=aH
−2
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Energy density: ρGW(t) ≡
∫
d ln k ρGW(t; k) ρGW(t; k) = 1 V
k3
2π2
× ∑ λ 1
2 ˙ ˜ h
(λ)
- k ˙
˜ h
(λ) − k + 1
2
(k
a
)2
˜ h(λ)
- k ˜
h(λ)
− k
≃ 1 V
k3
2π2
× (k
a
)2 ∑ λ
- |˜
h(λ)
- k |2
- sc
≃
( Hinf
2πMPl
)2 a(t)
a|k=aH
−4
M 2
PlH2 k=aH
For modes which enter the horizon at the RD epoch:
ρGW(t; k) ≃
( Hinf
2πMPl
)2
ρrad(t)
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Present GW spectrum:
Ω(tot)
GW = ρ(tot) GW (tNOW)
ρcrit ≡
∫
d ln k ΩGW(k)
In the case without phase transition (i.e., standard case):
Ω(SM)
GW (k) ≃ 1.7 × 10−15r0.1γ : kEW ≪ k ≪ kRH.
r0.1: the tensor-to-scalar ratio in units of 0.1 γ =
g∗(Tin(k))
g∗0
g∗s0 g∗s(Tin(k))
4/3 ( k
k0
)nt
Ω(SM)
GW (k) is insensitive to k
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In future, GW spectrum may be measured
⇒ BBO / DECIGO
Expected sensitivity
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- 3. Phase Transition
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The spectrum of GWs is affected by phase transitions
⇔ There may exist significant entropy production at the time
- f phase transition
Model: two real scalar fields φ and χ
V (φ) = g 24(φ2 − v2
φ)2 + h
2χ2φ2 φ: scalar field responsible for symmetry breaking χ: degrees of freedom in thermal bath
“Thermal mass” is generated for φ in the thermal bath
⇒ Cosmic phase transition occurs
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Potential of φ surrounded by the thermal bath (at φ ∼ 0)
VT(φ) = g 24(φ2 − v2
φ)2 + h
24T 2φ2 + · · · ≡ V0 + h 24(T 2 − T 2
c )φ2 + · · ·
Critical temperature: temperature for V ′′
T (φ = 0) = 0
Tc =
- 2g
h vφ
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Approximately, the phase transition occurs when V ′′
T (φ = 0) = 0
⇔ Tunneling rate is suppressed when g ≪ 1
Expectation value of φ:
φ =
0 : T > Tc vφ: T < Tc
Entropy is produced due to the phase transition
- Temperature just before the phase transition: Tc
- Temperature just after the phase transition: TPT > Tc
⇒ ρrad(TPT) = ρrad(Tc) + V0
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Expansion rate at the phase transition:
HPT ≡
- ρrad(Tc) + V0
3M 2
Pl
The mode which enters the horizon at the phase transition:
kPT ≡ a(tPT)HPT ⇒ k < kPT: out-of-horizon at t = tPT
Present frequency: fPT = kPT/2πaNOW
fPT ≃ 2.7 Hz ×
(
TPT 108 GeV
)
⇒ [fPT]g/h2≪1 ≃ 0.50 Hz ×
g1/4vφ
108 GeV
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Relevant equations to be solved (background):
- H2 =
( ˙
a a
)2
= ρrad + V0θ(tPT − t) 3M 2
Pl
- ˙
ρrad + 4Hρrad = V0δ(t − tPT) tPT: time of phase transition (i.e., T = Tc)
Effects of φ
- Deviation from the radiation-domination at t ∼ tPT
- Entropy production due to the phase transition
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Evolution of the universe (with h = 1):
H = 1 2t in RD V0 ρrad(Tc) ∼ O(1) × h2 g∗g g∗: Effective number of massless degrees of freedom
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- 4. Imprints of Phase Transition in GWs
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Behavior of GW amplitudes:
- k ≤ kPT: No effect of phase transition
- k ≥ kPT: Density is diluted due to the entropy production
Scale Inflation After inflation (RD, ...) Time H Phase Transition k < kPT k > kPT k = kPT
- 1
P h y s i c a l W a v e l e n g t h = a / k
⇒ ΩGW(k > ∼ kPT) is suppressed
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“Short wavelength (i.e., high frequency)” mode: k ≥ kPT
- 1. The amplitude is constant until the horizon-reentry
[ρGW(k)]k=aH ≃ ρGW(t = 0)
- 2. ρGW(k) ∝ a−4 once the mode enters the horizon
[ρGW(k)] (t) ≃ [ρGW(k)]k=aH
(ahorizon-in
aPT
)4 aPT
a(t)
4
≃ [ρGW(k)]k=aH
(Thorizon-in
Tc
)−4 TPT
T(t)
−4
≃
( Tc
TPT
)4
[ρGW(k)]no phase transition
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ΩGW(k > ∼ kPT) becomes suppressed R ≡ ΩGW(k) Ω(SM)
GW (k)
- k≫kPT
=
( Tc
TPT
)4
= ρrad(Tc) ρrad(Tc) + V0
Spectrum of GWs: result of numerical calculation
fPT = kPT 2πaNOW ≃ 2.7 Hz
(
TPT 108 GeV
)
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What can we learn from the GW spectrum?
- Position of the drop-off (∼ fPT)
⇒ “Reheating temperature” after the phase transition
- Magnitude of the drop-off (R)
⇒ Entropy production
- Slope of the drop-off (∼ dΩGW/d ln k)
⇒ Time scale of the reheating (instantaneous or ?)
GWs from white dwarf binaries are significant for small-f
⇒ It will be difficult to extract the signal of cosmic phase
transition in the GW spectrum if fPT <
∼ 0.1 Hz
[Farmer & Phinney]
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Detectability of the “drop-off” signal
- Drop-off of ΩGW should be bigger than the sensitivity
⇒ Lower bound on R
- ΩGW(k >
∼ kpt) should be observable ⇒ Upper bound on R
Comparison with the BBO-corr sensitivity:
- (r, fPT) = (0.1, 0.1 Hz)
0.005 < R < 0.98 ⇒ 1.5 × 10−6 < g < 0.014 (for h = 1)
- (r, fPT) = (0.1, 1 Hz)
0.17 < R < 0.83 ⇒ 6.0 × 10−5 < g < 0.0014 (for h = 1)
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- 5. Summary
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GW spectrum contains information about the early universe Example: Cosmic phase transition
⇒ If a cosmic phase transition occurred, its effect may
be imprinted in the spectrum of GWs Message: GWs are interesting because
- Various information about the early universe is imprinted
in GWs
- In a future, precise determination of the GW spectrum