Gravitational waves from phase transitions

Chiara Caprini IPhT - CEA Saclay

CC, R. Durrer and G. Servant, arXiv: 0909.0622CC, R. Durrer, T. Konstandin and G. Servant, arXiv: 0901.1661CC, R. Durrer and G. Servant, arXiv: 0711.2593

Once emitted, propagate without interaction: direct probe of physical processes in the early universe

First order phase transitions are sources of GW

Primordial sources: stochastic background of GW

Temperature of the phase transition : characteristic frequency

Strength of the phase transition : amplitude

Signal potentially interesting for LISA, PTA, advanced LIGO

Analytical evaluation of the GW signal in terms of free parameters

Gravitational waves

Stochastic background of GW

Gµν = 8πG Tµν

tensor anisotropic stress• Source:

• energy density of GW

ds2 = a2(t)(dt2 − (δij + 2hij)dxidxj)

ds2 = a2(t)(dt2 − (δij + 2hij)dxidxj)

hij +2thij + k2hij = 8πGa2Πij

Πij(k, t)

ΩGW =hij hij

Gρc=

dk

k

dΩGW

d log k

Small perturbations in FRW metric: (hii = hi

j|j = 0)

hij(k, η)h∗ij(q, η) = δ(k− q)|h|2(k, η) dΩGW

d log k=

k3|h|2

Gρc

GW characteristic frequency

(G. Hobbs, arXiv:0802.1309)

k∗ ≥ H∗

• characteristic frequency of GW produced at time t∗

k100GeV 10−5

Hz

k100MeV 10−8

Hz

• Example: phase transitions

H∗ = a∗/a∗ = t−1∗

• LISA: low frequency

10−4

Hz ≤ k ≤ 1Hz

ΩG ∼ 10−12

GW from phase transitions : frequency

• Collision of bubbles walls • Turbulent motions in the primordial plasma• Magnetic fields

duration of the PT

FIRST ORDER

R vbβ−1 size at collision

k∗ β , R−1k∗ 10

−2 β

H∗

T∗100 GeV

mHz

speed of the wallvb ≤ 1

β−1 0.01H−1

∗

δGij = 8πGTij β2h ∼ 8πG T

T ∼ ρradΩ∗

kin

Ω∗rad

h ∼ 8πG T

β

characteristic time of evolution tensor perturbation

energy density:

ΩG ∼ Ωrad

H∗β

2 Ω∗

kin

Ω∗rad

2

ρG ∼h2

8πG

GW from phase transitions : amplitude

duration of the source with respect

to Hubble time

energy density of the source with respect to

radiation energy density

GW from phase transitions : amplitude

example: turbulence Tij = (ρ + p)vivjΩ∗

T

Ω∗rad

=23v2

Jouguet detonation

vb = 0.87

α =ρvac

ρrad=

13v2 =

13

10−5

10−4 10−2

ΩG ∼ Ωrad

H∗β

2 Ω∗

kin

Ω∗rad

2

To determine the GW signal :GW power spectrum

Anisotropic stresspower spectrum

Πij(k, t1)Π∗ij(q, t2) = δ(k− q)Π(k, t1, t2)

k structure at equal time

0.01 0.1 1 10 100106

105

104

0.001

0.01

0.1

K

flat: spatially uncorrelated,

causality

slope depending on source power

spectrum :

Kolmogorov turbulence

characteristic length scale : bubble size

k−11/3

dΩGW

d log k∝ k3

tfin

tin

dt1t1

tfin

tin

dt2t2

cos[k(t1 − t2)] Π(k, t1, t2)

Time correlation of the anisotropic stress

|t1 − t2| <1k

Π(k, t1, t2) = Π(k, t1)Θ[t1 − t2]Θ[1− k(t1 − t2)] + t1 ↔ t2

Π(k, t1, t2) =

Π(k, t1)

Π(k, t2)Completely coherent

BUBBLES : • different collision events are uncorrelated• single collision event is coherent

MHD TURBULENCE :

• motions decorrelate with eddy turnover time• decorrelation time depends on eddy size

Top hat decorrelation correlated for

This affects the peak and the high frequency slope of the GW spectrum

General form of the GW power spectrum

k3

low frequency tail : causality of the source

107 105 0.001 0.1 10 1000 1051030

1026

1022

1018

1014

1010

K

peak position :

coherent source

decorrelating source

R vb/β

k∗ R−1

k∗ β high frequency tail : depends on both power spectrum and time correlation

GW spectrum from bubbles Analytic result, arXiv:0901.1661

k3

k∗ β

k−1

coherent

high frequency slope: coherent and thin wall approximation

peak Ω∗kin

Ω∗rad

0.2

0.001 0.1 10 1000 1051020

1018

1016

1014

1012

1010

kΒ

h2d

GW

dlogk

GW spectrum from bubbles Simulations by Huber and Konstandin, arXiv: 0806.1828

k3

k∗ β k−1

coherent coherent and thin wall approximation

peak Ω∗kin

Ω∗rad

0.03

GW spectrum from MHD turbulence

k∗ R−1peakdecorrelating source k−5/3

high frequency slope:

Kolmogorov

k−3/2 Iroshnikov Kraichnan

Ω∗MHD

Ω∗rad

0.2

Analytic result, arXiv: 0909.0622

0.001 0.1 10 1000 1051018

1016

1014

1012

1010

k R

h2d

GW

dlogk

k3

Total GW spectrum

Ω∗kin

Ω∗rad

Ω∗MHD

Ω∗rad

0.2

LISA AGIS

BBO

104 0.001 0.01 0.1 110141013101210111010109108

f Hz

h2d

GW

dlogk

β

H∗ 100T∗ 100 GeV

LISA AGISAdv. LIGO Corr

BBO Corr

BBO

104 0.01 1 10010181016101410121010108

f Hz

GWh2

T∗ = 5 · 106 GeV β/H∗ = 50Ω∗

MHD

Ω∗rad

0.2

GW from first order phase transitions

due to bubble collisions and subsequent MHD turbulence

characteristic frequency : duration of the PT for bubble collision size of the bubbles for MHD turbulence

low frequency tail: k^3 due to causality high frequency tail: depends on the time and space correlation of the source

EWPT characteristic frequency : about mHz

if strong enough, detectable by LISA

Summary

GW power spectrum from magnetic fields

104 0.001 0.01 0.1 1 10 1001016

1014

1012

1010

108

K

105 104 0.001 0.01 0.1 1 10109

107

105

0.001

0.1

K

107 105 0.001 0.1 101018

1016

1014

1012

1010

108

K

causal n=2, EWPT, non helical causal n=2, EWPT, maximally helical

inflationary n=-1.8, maximally helical