Magnetic field generation during inflation Chiara Caprini IPhT - - PowerPoint PPT Presentation

Magnetic field generation during inflation

Lorenzo Sorbo and CC arXiv:1407.2809 Chiara Caprini IPhT CEA-Saclay

Cosmological magnetic fields

(z < 4)

• magnetic fields of microGauss amplitude observed in galaxies, clusters

and high redshift objects

Cosmological magnetic fields

(z < 4)

• correlated on scales of the order of the object size : difficult to explain
• magnetic fields of microGauss amplitude observed in galaxies, clusters

and high redshift objects

Cosmological magnetic fields

• magnetic fields of microGauss amplitude observed in galaxies, clusters

and high redshift objects

• lower bound on magnetic field amplitude in the intergalactic medium

from observation of blazars with gamma ray telescopes

(z < 4)

Vovk et al 1112.2534

BMpc > 6 · 10−18G (Bλ ∝ λ−1/2)

• correlated on scales of the order of the object size : difficult to explain

Cosmological magnetic fields

• magnetic fields of microGauss amplitude observed in galaxies, clusters

and high redshift objects

• lower bound on magnetic field amplitude in the intergalactic medium

from observation of blazars with gamma ray telescopes

(z < 4)

Vovk et al 1112.2534

BMpc > 6 · 10−18G (Bλ ∝ λ−1/2)

• correlated on scales of the order of the object size : difficult to explain
• the origin is not understood: after recombination (related to structure

formation) or primordial?

Primordial magnetic fields

• a primordial field permeates the universe: it could explain
• observations in all structures and at high redshift
• the lower bound in the intergalactic medium

• a primordial field permeates the universe: it could explain
• observations in all structures and at high redshift
• the lower bound in the intergalactic medium
• many proposed generation mechanisms but no preferred one

Primordial magnetic fields

• a primordial field permeates the universe: it could explain
• observations in all structures and at high redshift
• the lower bound in the intergalactic medium
• many proposed generation mechanisms but no preferred one

CAUSAL :

• phase transitions, MHD turbulence,

charge separation + vorticity...

• problem: small correlation length and blue spectrum

too small seeds on cosmologically relevant scales

Primordial magnetic fields

• a primordial field permeates the universe: it could explain
• observations in all structures and at high redshift
• the lower bound in the intergalactic medium
• many proposed generation mechanisms but no preferred one

NON CAUSAL :

• INFLATION

Primordial magnetic fields

• generation at all scales, spectrum can

be red

• a primordial field permeates the universe: it could explain
• observations in all structures and at high redshift
• the lower bound in the intergalactic medium
• many proposed generation mechanisms but no preferred one

NON CAUSAL :

• INFLATION

need to break conformal invariance

• therwise no amplification of vacuum fluctuations

L = −1 4FµνF µν

Primordial magnetic fields

• generation at all scales, spectrum can

be red

S = Z d4x√−g ✓ −f 2(φ) 4 FµνF µν ◆

Turner and Widrow 1988 Ratra 1992

Simple model for MF generation

• test field that does not change the background evolution

Fµν

S = Z d4x√−g ✓ −f 2(φ) 4 FµνF µν ◆

• test field that does not change the background evolution

Fµν

Turner and Widrow 1988 Ratra 1992

Simple model for MF generation

• a model for the function:

Martin and Yokoyama 0711.4307 Demozzi et al 0907.1030

f(φ) → f(τ) = a(τ)n

S = Z d4x√−g ✓ −f 2(φ) 4 FµνF µν ◆

• test field that does not change the background evolution

Fµν

Turner and Widrow 1988 Ratra 1992

Simple model for MF generation

• a model for the function:

f(φ) → f(τ) = a(τ)n

• equation of motion for the gauge field

¨ Aσ + ✓ k2 − n(n + 1) τ 2 ◆ Aσ = 0

S = Z d4x√−g ✓ −f 2(φ) 4 FµνF µν ◆

• test field that does not change the background evolution

Fµν

Turner and Widrow 1988 Ratra 1992

Simple model for MF generation

• a model for the function:

f(φ) → f(τ) = a(τ)n

• equation of motion for the gauge field

¨ Aσ + ✓ k2 − n(n + 1) τ 2 ◆ Aσ = 0

amplification at large scales

kτ ⌧ 1

same equation for both helicities parameter n controls the EM field spectrum

S = Z d4x√−g ✓ −f 2(φ) 4 FµνF µν ◆

• test field that does not change the background evolution

Fµν

Turner and Widrow 1988 Ratra 1992

Simple model for MF generation

• a model for the function:

f(φ) → f(τ) = a(τ)n

• equation of motion for the gauge field

¨ Aσ + ✓ k2 − n(n + 1) τ 2 ◆ Aσ = 0

• generates EM field : after reheating, conductivity in the universe is

very large, E-field dissipates away and B-field stays

• MF power spectrum

Simple model for MF generation

dρB d ln k

• end

= H4 ✓ k H ◆f(n)

• MF power spectrum

Simple model for MF generation

• interesting regime: scale invariant spectrum

high values of B-field at large scales for high scale inflation!

Treh ' 1016GeV B ' 10−11Gauss dρB d ln k

• end

= H4 ✓ k H ◆f(n) BMpc ' 10−5 ✓Treh Mpl ◆2 Gauss

• MF power spectrum

Simple model for MF generation

HOWEVER THERE ARE CONSTRAINTS

• interesting regime: scale invariant spectrum

dρB d ln k

• end

= H4 ✓ k H ◆f(n)

• MF power spectrum

Simple model for MF generation

• avoid strong coupling of the theory :

−f 2 4 FµνF µν → −1 4FµνF µν + i ¯ ψγµ ✓ ∂µ + iAµ f ◆ ψ f ≥ 1 → n < 0

Demozzi et al 0907.1030

HOWEVER THERE ARE CONSTRAINTS

• interesting regime: scale invariant spectrum

dρB d ln k

• end

= H4 ✓ k H ◆f(n)

• MF power spectrum

Simple model for MF generation

• avoid back-reaction of the EM field

energy density on the background:

n > −2

HOWEVER THERE ARE CONSTRAINTS

• avoid strong coupling of the theory :

f ≥ 1 → n < 0 ρEM ≤ ρinf →

Martin and Yokoyama 0711.4307

• interesting regime: scale invariant spectrum

dρB d ln k

• end

= H4 ✓ k H ◆f(n)

• MF power spectrum

Simple model for MF generation

HOWEVER THERE ARE CONSTRAINTS

• avoid strong coupling of the theory :

f ≥ 1 → n < 0 Treh ' 1016GeV

• avoid back-reaction of the EM field

energy density on the background:

n > −2 ρEM ≤ ρinf →

the spectrum is blue

BMpc < 10−32Gauss

• interesting regime: scale invariant spectrum

dρB d ln k

• end

= H4 ✓ k H ◆f(n)

• 1. lower the scale of inflation
• 2. reduce the duration of EM field production

Simple model for MF generation

two methods to reduce back-reaction:

• 1. lower the scale of inflation
• 2. reduce the duration of EM field production

Simple model for MF generation

Ferreira et al 1305.7151

inflation at 10 MeV + magnetogenesis active only when O(Mpc) scales exit the horizon = MF produced can fulfil lower bound in the IGM

two methods to reduce back-reaction:

Simple model for MF generation

Ferreira et al 1305.7151

• 1. lower the scale of inflation
• 2. reduce the duration of EM field production

two methods to reduce back-reaction:

inflation at 10 MeV + magnetogenesis active only when O(Mpc) scales exit the horizon = MF produced can fulfil lower bound in the IGM problem: too low-scale inflation for BICEP2

Problems and possible solutions

in the previous model the spectrum can vary (parameter n) but the amplitude is fixed decouple amplitude from spectrum: try to increase the EM field amplitude keeping a MF spectrum as red as possible very low scale inflation is required to enhance the MF amplitude and make the spectrum redder: problem with BICEP2 the gauge field sources the tensor perturbations

Add helicity to magnetogenesis

L = f 2(τ) ✓ −1 4Fµν F µν − ξ 8nFµν ˜ F µν ◆

Add helicity to magnetogenesis

• 1. avoid strong coupling
• 2. avoid strong back-reaction by the EM
• 3. get the most red spectrum possible

f(τ) = a(τ)n L = f 2(τ) ✓ −1 4Fµν F µν − ξ 8nFµν ˜ F µν ◆ −2 < n < 0

add axial coupling :

• 1. controls the EM field amplitude
• 2. the MF generated is helical so it

evolves by inverse cascade

Add helicity to magnetogenesis

f(τ) = a(τ)n L = f 2(τ) ✓ −1 4Fµν F µν − ξ 8nFµν ˜ F µν ◆ ξ

• 1. avoid strong coupling
• 2. avoid strong back-reaction by the EM
• 3. get the most red spectrum possible

the evolution through inverse cascade amplifies the magnetic field at large scales : brings energy there where we need it!

−2 < n < 0

¨ Aσ + ✓ k2 + 2 σ ξ k τ − n (n + 1) τ 2 ◆ Aσ = 0

• equation of motion in this case :

Add helicity to magnetogenesis

L = f 2(τ) ✓ −1 4Fµν F µν − ξ 8nFµν ˜ F µν ◆

¨ Aσ + ✓ k2 + 2 σ ξ k τ − n (n + 1) τ 2 ◆ Aσ = 0

exponential amplification

• f only one helicity mode

at horizon crossing: generation of helical MF

• equation of motion in this case :

Add helicity to magnetogenesis

L = f 2(τ) ✓ −1 4Fµν F µν − ξ 8nFµν ˜ F µν ◆

¨ Aσ + ✓ k2 + 2 σ ξ k τ − n (n + 1) τ 2 ◆ Aσ = 0

amplification at super-horizon scales gives n-dependent spectral index to the power spectrum

• equation of motion in this case :

Add helicity to magnetogenesis

exponential amplification

• f only one helicity mode

at horizon crossing: generation of helical MF

L = f 2(τ) ✓ −1 4Fµν F µν − ξ 8nFµν ˜ F µν ◆

¨ Aσ + ✓ k2 + 2 σ ξ k τ − n (n + 1) τ 2 ◆ Aσ = 0

• equation of motion in this case :

Add helicity to magnetogenesis

L = f 2(τ) ✓ −1 4Fµν F µν − ξ 8nFµν ˜ F µν ◆

• magnetic field power spectrum

dρB d ln k

• end

= H4 e2πξ ξ|2n+1| ✓ k H ◆5−|2n+1|

amplitude can be tuned with the parameter ξ

back-reaction and strong coupling constraints are satisfied : what else?

Add helicity to magnetogenesis

back-reaction and strong coupling constraints are satisfied : what else? vector field during inflation induces METRIC PERTURBATIONS that must be smaller than the observed ones

Add helicity to magnetogenesis

constraint from TENSOR modes

back-reaction and strong coupling constraints are satisfied : what else? vector field during inflation induces METRIC PERTURBATIONS that must be smaller than the observed ones

Add helicity to magnetogenesis

GWs are sourced by the tensor part of the energy momentum tensor of the EM field :

¨ hσ + 2 ˙ a a ˙ hσ + k2hσ = 2 M 2

Pl

Πσ

ij(k) T EM ij

(k)

constraint from TENSOR modes

Tensor mode spectrum from the gauge field

PT = F(n) H4 M 4

Pl

e4 π ξ ξ6

Tensor mode spectrum from the gauge field

assume that BICEP2 is correct, in order to determine the energy scale of inflation in this model

PT = F(n) H4 M 4

Pl

e4 π ξ ξ6 H MPl (n, ξ) = ✓ rPζ F(n) ξ6 e4πξ ◆1/4 r = 0.2 Pζ = 2.5 · 10−9

the Hubble rate is tunable: we can get low scale inflation with large value of the tensor to scalar ratio good for the MF amplitude!

• evolve in time accounting for helicity : non-trivial amplification at large

scales by inverse cascade

Do we get interesting MF amplitudes today at large scales?

• get the value of the magnetic field today at the correlation scale

as a function of BL(n, ξ)

• evolve in time accounting for helicity : non-trivial amplification at large

scales by inverse cascade

Do we get interesting MF amplitudes today at large scales?

• get the value of the magnetic field today at the correlation scale

as a function of BL(n, ξ)

• evolve in time accounting for helicity : non-trivial amplification at large

scales by inverse cascade

• impose that it satisfies the lower bound in the IGM

BL > 6 · 10−18 r 1Mpc L Gauss

Do we get interesting MF amplitudes today at large scales?

• get the value of the magnetic field today at the correlation scale

as a function of BL(n, ξ)

• evolve in time accounting for helicity : non-trivial amplification at large

scales by inverse cascade

• impose that it satisfies the lower bound in the IGM
• get the energy scale of inflation for which this is satisfied as a function
• f n

ξ

( is fixed by the required MF amplitude)

BL > 6 · 10−18 r 1Mpc L Gauss

Do we get interesting MF amplitudes today at large scales?

it is possible to generate large enough MF with not too low scale inflation

Results

• 1.5
• 1.0
• 0.5

0.0 105 106 107 108 109 1010 1011 n rinf

1ê4HGeVL

BL > 6 · 10−18 G p 1Mpc/L BL > 2.5 · 10−17 G p 1Mpc/L

BL > 10−16 G p 1Mpc/L

r = 10−4

BL > 2.5 · 10−17 G p 1Mpc/L

Conclusions

• magnetogenesis during inflation is difficult, but it has the advantage that the

field can be generated also at very large scales

• the simplest model is constrained by back-reaction, strong coupling, BICEP2

(if confirmed)

• we propose a not too complicated model that works :

avoid back reaction and strong coupling constraints, and explain r=0.2 with low scale inflation

• parity violation is fundamental : helical field, evolving through inverse

cascade which amplifies the large scales

• the MF spectrum and the MF amplitude are governed by two separate

parameters

• the MF generated by this model can seed the galactic dynamo if the spectrum

is red enough and possibly explain MF in clusters by ejection from galaxies

Treh > 105 GeV