Gravitational Waves from Phase Transitions

Gravitational Waves from Phase Transitions

(an analytical study)

CC, Ruth Durrer, Thomas Konstandin, Geraldine Servant

Phys.Rev. D77 (2008) + Phys.Rev. D79 (2009)

Outline

motivation

sources of GW from phase transition: expected frequency and amplitude

analytical model of the sources

analytical model of the spectra

Why primordial sources?

Small perturbations in FRW metric:

ds

= a

(η)(dη

− (δ

+ 2h

)dx

dx

) G

= 8πG T

¨h

(k, η) + 2

η ˙h

(k, η) + k

h

(k, η) = 8πGa

(η)Π

(k, η) Π

(k, η)

Once emitted, propagate without interaction

Direct probe of physical processes in the early universe (gravitons) Primordial source: stochastic background of GWs (example: inflation)

(h

= h

= 0)

Source:

Ω_G = ! ˙h_ij ˙h^ij"

Gρ_c = !

dk k

dΩ_G(k) d log(k) GW energy density:

Characteristic frequency of GWs produced at time

η

H

= a

˙a

! !

!

= η

k

≥ H

LISA detection at low frequency:

at about 1 mHz

Frequency window from a cosmological source: k_eq ! 10⁻¹⁵Hz

10

Hz ≤ k ≤ 1Hz

k_inf ! 1GHz

Characteristic frequency

100 GeV (EW phase transition):

{

Ω

∼ 10

k

≥ 10

Hz

10^!4 10^!3 10^!2 10^!1 1 10 100f !Hz"

10^!18 10^!16 10^!14 10^!12 10^!10 10^!8

"_GW h²

LISA

BBO Corr

E_I#3 10¹⁶GeV

E_I#5 10¹⁵ GeV

LIGO III

GW from phase transitions

Collision of bubbles walls Turbulent motions

Magnetic fields

duration of the phase transition

speed of the bubbles walls FIRST ORDER:

v

≤ 1

R ! v

β

β

! 0.01 H

size of the bubbles at collision

f ! β ! 10

β H

T

100 GeV mHz

Energy density in bubble walls to radiation energy density:

(phase transition in a thermal bath)

v

Ω

Ω

= 4 3

v

1 − v

T

= (ρ + p)γ

v

v

4

3 ρ

Bubbles:

Turbulence:

Magnetic fields:

T

= (ρ + p)v

v

3

Ω

Ω

= 2

3! v

"

T

= 1

8π B

B

^Ω

∗B

Ω

= !

8πρ

Scaling of the GW amplitude

Scaling of the GW amplitude

δG

= 8πGT

β

h ∼ 8πG T

T ∼ ρ^rad Ω^∗_kin Ω^∗_rad

˙h ∼ 8πG T β

characteristic time of evolution tensor perturbation

energy density:

10

10

¹⁰

Ω

∼ Ω

! H

β

"

! Ω

Ω

"

ρ

∼ ˙h

8πG

Analytic study of the GW signal

GW power spectrum:

h

(k, η) = !

ηin

dτ G(τ, η)Π

(k, τ)

Wave equation:

!Π

(k, τ

)Π

(q, τ

)" = δ(k − q)Π(k, τ

, τ

)

Anisotropic stress power spectrum:

Energy momentum tensor:

Wick

theorem Power spectrum of the source

dΩ

d ln k = k

| ˙h|

Gρ

! ˙h

(k, η)˙h

(q, η)" = δ(k − q)|˙h|

(k, η)

Π

= P

T

4 point correlation function

T

(k, τ) = w(τ ) 1 − v

(τ)

!

d

p v

(k − p, τ)v

(p, τ)

Bubble walls power spectrum

Hydrodynamics of bubble growth at late times:

symmetric phase broken phase

v1 v2

combustion front

(conservation of energy and momentum) DETONATIONS:

v

> c

, v

= c

v

< c

, v

< c

DEFLAGRATIONS:

symmetric phase at rest (Steinhardt 82)

broken phase at rest

shock wave in the symmetric phase

ρ₁ = aT₁⁴ + ρ_vac p₁ = aT₁⁴/3 − ρ^vac ρ₂ = aT₂⁴

p₂ = aT₂⁴/3

Bubble walls power spectrum

Two point correlation function

Velocity profile of a spherical bubble:

x and y in the same bubble the velocity is not zero

p: probability that there is a centre in the intersection region

|x − y| ≤ 2R(t)

!v

(x, t)v

(y, t)"

!v

(x, t)v

(y, t)" = v

R

p V

!

V

d

x

(x − x

)

(y − x

)

x y

x0

v

(x, t) =

! (v

/R) (x − x

)

for r

< |x − x

| < R

0 otherwise

Bubble walls power spectrum

Power spectrum: Fourier transform of 2 p. correlation function

P (t)

probability that a point is in the broken phase at time t

P (t) ! 1 − exp(−Γ(t))

correlation function with

compact support

|x − y| ≤ 2R(t)

analytic power spectrum

small scale part

!v

(k, t)v

(q, t)" = δ(k − q)v

R(t)

P (t)[ A(kR)δ

+ B(kR)ˆ k

ˆk

]

A(k → 0) ∝ k

B(k → 0) ∝ k

B(k ! π/R)

A(k ! π/R) ∼ ^{∝ k}

Turbulence and magnetic field power spectra

Divergence-free vector fields

Correlated over the scale peak at

large scale part of the spectra

Kolmogorov (turbulence) Iroshnikov Kraichnan (magnetic field)

Small scale part of the spectra:

2R k ∼ π/R

k

k

, k

P

(k ! π/R) ∝ k

!v

(k)v

(q)" = δ(k − q) (δ

− ˆk

ˆk

)P

(kR)

Large scale part of the GW spectrum: k^3

k → 0

dΩ

d ln k = k

| ˙h|

Gρ

Generic CAUSAL power spectrum:

The anisotropic stress power spectrum is the CONVOLUTION:

White noise for the anisotropic stress

→ k

n > −3/2 m < −3/2

P (k) ∝ v

R

! (Rk)

for Rk < π (Rk)

for Rk > π

Π(k) ∝ !

0

dq q

P ( |k − q|)P (q) → v

R

" 1

2n + 3 −

1

2m + 3

#

.

Small scale part of the GW spectrum?

1) totally incoherent:

!Π

(k, τ)Π

(q, ζ)" = δ(k − q)Π(k, τ, τ) δ(τ − ζ) β

2) totally coherent: correct one according to SIMULATIONS

3) top hat in wavenumbers:

!Π^ij

(k, τ)Π

(q, ζ)" = δ(k − q)[Π(k, τ)Θ(kζ − kτ)Θ(x

+ Π(k, ζ)Θ(kτ − kζ)Θ(x

|ζ − τ| < x

k

!Π

(k, τ)Π

(q, ζ)" = δ(k − q) !

Π(k, τ) !

Π(k, ζ)

!Π

(k, τ)Π

(q, ζ)" = δ(k − q)Π(k, τ, ζ)

Time dependence of anisotropic stress power spectrum

Totally coherent

time Fourier transform, differentiability of time dependence

10^!4 0.001 0.01 0.1 1 10 100

10^!16 10^!14 10^!12 10^!10 10^!8

K"

spectrum for a continuous source

C

k

k

Peak at characteristic timescale

slope due to

continuity feature at characteristic lengthscale

slope due to continuity and source

causal

k ! β

k ! π/R

k

k

not in

simulations !

t L!t"

t_!"1#Β t_!

v#Β$

C

→ k

Conclusions

GW from bubbles observable for high fluid velocity or long lasting phase transition

GW production at EW symmetry breaking interesting for LISA Analytical model of the sources: bubbles, turbulence and

magnetic fields

Large scale part of the spectrum rises as k^3 (causality)

Peak frequency correspond to the typical time/length scale of the source

Small scale part of the spectrum very model dependent