Primordial magnetic fields and gravitational waves

Primordial magnetic fields and
gravitational waves
Chiara Caprini
IPhT - CEA Saclay
France
CC, Ruth Durrer and Elisa Fenu, in preparation
CC, Ruth Durrer and Geraldine Servant, in preparation
CC, Ruth Durrer, Thomas Konstandin and Geraldine Servant, arXiv:0901.1661
CC, Ruth Durrer and Tina Kahniashvili, astro-ph/0304556
CC and Ruth Durrer, astro-ph/0603476, 0305059, 0106244

OUTLINE
Gravitational waves are sourced by magnetic tensor
anisotropic stresses
Model of the primordial magnetic field spectrum
Model of the time evolution of the magnetic field in
the early radiation phase
Evaluation and spectral shape of the emitted
gravitational waves energy density
Constraints on the magnetic field amplitude using
gravitational waves

Why gravitational waves?
Small perturbations in FRW metric:
ds 2 = a 2 (η)(dη 2 − (δij + 2hij)dx i dx j ) Gµν = 8πG Tµν
¨hij(k, η) + 2
η ˙ hij(k, η) + k 2 hij(k, η) = 8πGa 2 (η)Πij(k, η)
Source:
Πij(k, η) tensor anisotropic stress
Once emitted, propagate without interaction
Direct probe of physical processes in the early universe
Primordial source: stochastic background of GWs
GW energy density:
ΩG = 〈˙ hij ˙ h ij 〉
Gρc
=
(hi i = ∂jhi j = 0)
dk
k
dΩG(k)
d log(k)

GW power spectrum:
To determine the GW signal :
dΩG
d ln k = k3 | ˙ h| 2
Gρc
Wave equation:
Anisotropic stress
power spectrum:
Πij(k) =
hij(k, η) =
η
ηin
projector that extracts the tensor component
Pij = δij − ˆ ki ˆ kj
〈 ˙ hij(k, η) ˙ h ∗ ij(q, η)〉 = δ(k − q)| ˙ h| 2 (k, η)
dτG(τ, η)Πij(k, τ)
〈Πij(k, τ1)Π ∗ ij(q, τ2)〉 = δ(k − q)Π(k, τ1, τ2)
P l i P m j − 1
lm
PijP Tlm(k)
2
energy momentum tensor
of the source

Magnetic field is a source of GW
MF breaks FRW symmetries and has a non zero anisotropic stress:
T B µν =
− B2
2 g00 0
0 − B2
2 gij − BiBj
first order perturbation in FRW: stochastic field,
statistically homogeneous, isotropic and gaussian
GW: energy momentum tensor power spectrum
〈Tij(k)T ∗
lm(q)〉 ∝
4 point correlation
function
d 3
p
Wick
theorem
〈Bi〉 = 0 〈B 2 〉 = 0
d 3 q〈Bi(p)Bj(|k − p|)B ∗ l (q)B ∗ m(|k − q|)〉
Power spectrum
of the field

Primordial magnetic field generation
Phase transitions:
correlation scale for MF :
Inflation:
generation is causal, operates sub-horizon
(ɛ 0.01 for first order PT )
generation at every scale
1oo GeV
horizon at T∗
1 MeV 0.32 eV
TURBULENT PHASE VISCOUS PHASE
Re = vL
ν
ν ℓνe
≫ 1
σ ∝ T
Jedamzik et al 1996, Ahonen and Enqvist 1996,
Banerjee and Jedamzik 2004...
L∗ = ɛη∗ ɛ ≤ 1 η∗
Re 1
ν ℓγe σ ∝ T 3/2
Subramanian and Barrow 1997,
Banerjee and Jedamzik 2004...

energy density
helicity density H =
Magnetic field power spectrum
〈Bi(k)B ∗ j (q)〉 = δ(k − q)[ iɛijm ˆ k m (δij − A(k)
ˆ ki ˆ kj)S(k) +
]
EB =
∞
0
∞
0
dk k 2 S(k)
S(k) ∝ 〈B + (k)B + (−k) + B − (k)B − (−k)〉
dk kA(k)
A(k) ∝ 〈B + (k)B + (−k) − B − (k)B − (−k)〉
k → 0
at large scales : S(k) ∝ k n
S(k) ≥ |A(k)|
Divergence free
parity even
parity odd
A(k) ∝ k m
m ≥ n

Power spectrum at large scales and causality
〈Bi(x)Bj(x + r)〉 = Σ(r)(δij − ˆri ˆrj)+Γ(r)ˆriˆrj + Λ(r)ɛijℓˆrℓ
∂bij(r)
MF is divergence free: =0 (r
∂ri
2 Γ(r)) ′ =2rΣ(r)
MF generated by a CAUSAL process: (phase transition, charge separation...)
bij(r) =
S(k)
〈Bi(x)Bj(x + r)〉 = 0 for r > L L ≤ horizon
=
0
d 3 r e ik·r bii(r) =
∞
divergence free
S(k → 0) k 2
∞
dr (r 3 Γ(r)) ′
k 4
∞
+ dr r 4 bii(r)+O( )
0
0
dr
sin kr
kr (r3 Γ(r)) ′

Power spectrum at large scales and causality
〈Bi(x)Bj(x + r)〉 = Σ(r)(δij − ˆri ˆrj)+Γ(r)ˆriˆrj + Λ(r)ɛijℓˆrℓ
∂bij(r)
MF is divergence free: =0 (r
∂ri
2 Γ(r)) ′ =2rΣ(r)
MF generated by a CAUSAL process: (phase transition, charge separation...)
bij(r) =
S(k)
〈Bi(x)Bj(x + r)〉 = 0 for r > L L ≤ horizon
=
∞
divergence free
S(k → 0) k 2
dr (r 3 Γ(r)) ′
+
0
d 3 r e ik·r bii(r) =
∞
0
dr
sin kr
kr (r3 Γ(r)) ′
k 4
∞
dr r 4 bii(r)+O( )
0
(Loytsyanky)

Power spectrum at large scales and causality
A(k)
B 2 λ =
DIVERGENCE FREE + CAUSALITY IMPLY BLUE SPECTRA
= ɛijℓ ˆ k ℓ
dk k 2
d 3 r e ik·r bij(r)
e −k2 λ 2
∞
dr r 3 ∞
Λ(r)+ dr r 4 k k Λ(r)+O( )
3 k 5
0
m ≥ n
(k → 0)
Bλ = BL
MF generated by a NON-CAUSAL process: (inflation, pre big bang...)
n , m > −3
L
λ
0
n+3
2
S(k) n ≥ 2

0.15
0.10
0.05
0.00
Interpolating formula:
(turbulence, Von Karman 48)
blue
Power spectrum at all scales
Small scales: MHD spectrum: Kolmogorov, Iroshnikov Kraichnan....
causal spectrum n=2
peak at
k 2π/L
k 2
k −7/2
0 1 2 3 4 5 6
K = kL/2π
S(k) = EB L 3
1000
10
0.1
0.001
10 5
K n
(1 + K 2 ) (2n+7)/4
a-causal spectrum n>-3
red k n
feature at
k 2π/L
(horizon)
k −7/2
10
0.01 0.1 1 10 100
7
K = kL/2π

Time evolution: freely decaying turbulence
non-helical field: DIRECT TURBULENT CASCADE
EB(τ) =E ∗ B τ −2(n+3)/(n+5)
L(τ) =L∗τ 2/(n+5)
constant on
large scales
(n=2)
EBL 5 = E ∗ BL 5 ∗
E B
The evolution ends when
the entire Komogorov range
is dissipated:
0.1
0.01
0.001
10 4
10 5
λ(τfin) L(τfin)
τ =
η − ηin
τL
0.1 1 10 100 1000
K
τL L∗
vL
λ(τ)
dissipation scale
Re(λ) = vλ λ
1
ν
grows faster than L
MF with n=2 generated
at EWPT :
Tfin 180 MeV
eddy
turnover
time

Time evolution: freely decaying turbulence
helical field: INVERSE CASCADE
EB(τ) =E ∗ B τ −2/3
L(τ) =L ∗ τ 2/3
Magnetic energy is transferred
to large scales (n=2)
EBL = E ∗ BL∗
E B
0.1
0.01
0.001
10 4
Maximally helical MF
with n=2 generated at
EWPT :
10
0.01 0.1 1 10 100 1000
5
(Christensson et al 2002, Banerjee and Jedamzik 2004, Campanelli 2007...)
λ(τfin) L(τfin)
K
Tfin 22 MeV
MHD FIELDS SOURCE GW FOR SEVERAL HUBBLE TIMES

B
Anisotropic stress power spectrum
For GW energy density need the trace of power spectrum at different times
1
0.01
10 4
10 6
10 8
Π(k, τ, τ) =
flat (causal n=2) or k inflationary
2n+3
non-helical
〈Πij(k, τ1)Π ∗ ij(q, τ2)〉 = δ(k − q)Π(k, τ1, τ2)
10
0.001 0.01 0.1 1 10 100 1000
10
K
d 3 p [
S(p)S(|k − p|) + A(p)A(k − p) ]
k −7/2
B
10
0.1
0.001
maximally
helical
S(k) = A(k)
10
0.001 0.01 0.1 1 10 100
5
K

dΩG
d ln k
Gravitational wave power spectrum strongly influence
G
= k
ρc
3
ηfin
ηin
dη1
a(η1)
ηfin
ηin
the GW spectrum
dη2
a(η2) cos(k(η1 − η2)) Π(k, η1, η2)
COMPLETELY COHERENT Π(k, η1, η2) = Π(k, η1) Π(k, η2)
dΩGW
d ln k
dΩG
d ln k
G
∝ k
ρc
3
(k→0)
ηfin
ηin
GW spectrum at large scales :
time Fourier transform
∝ k 3 ,k 2n+6 always blue on large scales
causal, flat inflationary
dη1
a(η1) cos(kη1) 2 Π(k, η1) + ...
k 2n+3

Gravitational wave power spectrum
GW spectrum at small scales : time Fourier transform, differentiability
of time dependence
k 2π
Peak at characteristic
timescale
causal
k 3
spectrum for a continuous source C , 0
τL
10 8
10 10
10 12
10 14
10 16
k −1
slope due to
continuity
4 10 0.001 0.01 0.1 1 10 100
K
C m → k −m
L∗ vLτL
k 2π
feature at characteristic
lengthscale
L∗
slope due to
continuity and
source
k −1 k −7/2

10 8
10 10
10 12
10 14
10 16
Gravitational wave power spectrum (preliminary)
4 10 0.001 0.01 0.1 1 10 100
K
dΩG
d ln k 0.1(Ω∗ B )2
I(k)
Ωrad
causal n=2, EWPT, non helical causal n=2, EWPT, maximally helical
inflationary n=-1.8,
maximally helical
0.1
0.001
10 5
10 7
105 10
4 10 0.001 0.01 0.1 1 10
9
K
10 8
10 10
10 12
10 14
10 16
107 10
5 10 0.001 0.1 10
18
K

Generation at 100 GeV, n=2 :
Bounds from Nucleosynthesis
Generation at inflation 10^14 GeV :
n = 0
n → −3
non-helical
helical
non-helical B0.1Mpc 10 −39 G
helical n = 0
n = −1.8
ΩG I∗
(Ω ∗ B )2
Ωrad
0.1 Ωrad
B0.1Mpc 10 −9 G
B0.1Mpc 10 −28 G
B0.1Mpc 10 −18 G
Generation at 100 MeV, n=2 : B0.1Mpc ≤ 10 −20 G
B0.1Mpc 10 −27 G
B0.1Mpc 10 −25 G
Ok without
dynamo
Ok with
dynamo
(preliminary helical, non-helical to be updated with the cascade)

Bounds at generation time
Nucleosynthesis bound directly on MF : account for dissipation
GW production stores energy at small scales
BUT :
Bound on the rms value :
Ω ∗ B 0.1 Ωrad
〈B 2 〉 10 −6 G
Bound at large scales : Bλ 10 −8 G
n =2
n+3
2 L
λ
Lew = 10 −4 pc λ = 1 kpc
B1kpc 10 −26 G
Crucial ingredients: blue spectrum and average at large scales
GW production does not help in this case!

CONCLUSIONS
Analytic formula for the MF spectrum interpolating large and small
scale behaviour : consider the entire spectrum, no joining at k=1/L
Using MHD decay laws, MF source of GW for several Hubble
times
Model of the anisotropic stress power spectrum at unequal time
strongly influence the GW spectrum (peak position and small
scales)
If coherent, peak at characteristic time, but the speed of turning on
and off strongly influence the small scale spectrum
Under these assumptions, primordial MF generated before
Nucleosynthesis are strongly constrained, even if helical

Time dependence of the anisotropic stress power
spectrum
1) totally incoherent:
〈Πij(k, τ)Π ∗ ij(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:
〈Πij(k, τ)Π ∗ ij(q, ζ)〉 = δ(k − q)Π(k, τ, ζ)
〈Πij(k, τ)Π ∗ ij(q, ζ)〉 = δ(k − q) Π(k, τ) Π(k, ζ)
|ζ − τ| < xc
k
〈Πij(k, τ)Π ∗ ij(q, ζ)〉 = δ(k − q)[Π(k, τ)Θ(kζ − kτ)Θ(xc − (kζ − kτ))
+ Π(k, ζ)Θ(kτ − kζ)Θ(xc − (kτ − kζ))]