Amplification of magnetic fields from turbulence in the early Universe

Amplification of magnetic fields from turbulence in the early Universe Jacques M. Wagstaff 1 arxiv:1304.4723 Robi Banerjee 1, Dominik Schleicher 2, Günter Sigl 3 1 Hamburg University 2 Institut für Astrophysik, Georg-August-Universität Göttingen 3 II Institut für Theoretische Physik, Universität Hamburg Rencontres de Moriond, La Thuile, Aosta valley, Italy, March 28th, 2014 Wagstaff, Banerjee, Schleicher and Sigl (Universities of Somewhere and Elsewhere) Moriond 2014 1 / 24

Magnetic fields in the Universe Observational evidence: Stars, solar system, Milky Way (e.g.taylor et al. 2009) Galaxies - low and high redshift - B = O(10)µG (Hammond et al. 2012) Clusters and superclusters of galaxies - µg (Kim et al. 1990) Voids in the Large Scale Structure - 10 7 ng (Neronov&Vovk 2010) Wagstaff, Banerjee, Schleicher and Sigl (Universities of Somewhere and Elsewhere) Moriond 2014 2 / 24

Extragalactic magnetic fields (Neronov and Vovk 2010) Theoretically: Inflation (Turner&Widrow 1988) (10 25 10 1 )ng Phase transitions (Sigl et al. 1997) 10 20 ng (EW) 10 11 ng (QCD) Battery Mechanisms (Harrison 1970) 10 20 ng Amplification required! Wagstaff, Banerjee, Schleicher and Sigl (Universities of Somewhere and Elsewhere) Moriond 2014 3 / 24

The dynamo mechanism Converts turbulent kinetic energy into magnetic energy Large-scale dynamo - galaxy differential rotation Small-scale dynamo - gravitational collapse Wagstaff, Banerjee, Schleicher and Sigl (Universities of Somewhere and Elsewhere) Moriond 2014 4 / 24

The small-scale dynamo mechanism Amplification of magnetic fields from turbulence Merge Stretch Twist B 2 exp (2Γt) Depends on environment: Prandtl: P m R m /R e = ν/η Turbulence: v(l) l ϑ 1/3 ϑ 1/2 Kolmogorov ϑ Burgers (incompressible) (highly compressible) Fold Wagstaff, Banerjee, Schleicher and Sigl (Universities of Somewhere and Elsewhere) Moriond 2014 5 / 24

The small-scale dynamo mechanism Amplification of magnetic fields from turbulence Merge Stretch Twist B 2 exp (2Γt) Grows on eddy-turnover time scale τ eddy = l/v Growth rate Γ depends on the kinetic Reynolds number R e = τ diss /τ eddy = vl/ν Fold Wagstaff, Banerjee, Schleicher and Sigl (Universities of Somewhere and Elsewhere) Moriond 2014 7 / 24

Simulations of the small-scale dynamo (Schekochihin et al. 2004) B 2 v 2 Wagstaff, Banerjee, Schleicher and Sigl (Universities of Somewhere and Elsewhere) Moriond 2014 8 / 24

The small-scale dynamo - analytic model The Kazantsev model (Kazantsev 1967, Subramanian 1997, Brandenburg&Subramanian 2005) Decompose: Velocity spectrum: B = B + δb v = v + δv δv i (r 1, t 1 )δv j (r 2, t 2 ) = T ij (r)δ(t 1 t 2 ) Assumptions: Gaussian Homogeneous and Isotropic in space Instantaneously correlated in time δb i (r 1, t)δb j (r 2, t) = M ij (r, t) Induction equation Stochastic differential equation Wagstaff, Banerjee, Schleicher and Sigl (Universities of Somewhere and Elsewhere) Moriond 2014 9 / 24

The small-scale dynamo - Analytic model The Kazantsev model (Kazantsev 1967, Subramanian 1997, Brandenburg&Subramanian 2005) Kazantsev equation: Schrödinger equation-like equation: Ψ t { Tij = ĤΨ where (r) = U(r) Ψ(t) = M ij (t) Find solution for fluctuating field: B 2 exp(2γt) General types of turbulence: v(l) l ϑ and P m = ν/η 1 Γ = (163 304ϑ) 60 R (1 ϑ)/(1+ϑ) e /τ eddy (Schober et al. 2012) Velocity spectrum modelled so that: (Schober et al. 2012) if ϑ = 1/3 Kolmogorov turbulence (incompressible) if ϑ = 1/2 Burgers turbulence (highly compressible) Wagstaff, Banerjee, Schleicher and Sigl (Universities of Somewhere and Elsewhere) Moriond 2014 10 / 24

The small-scale dynamo Applications The SSD mechanism requires turbulence Turbulence driven by: gravitational collapse, accretion and Supernovae explosions First stars (e.g. Schleicher et al. 2010) First galaxies (e.g. Schober et al. 2012) Clusters of Galaxies (Subramanian et al. 2012) What about before structure formation? Turbulence from primordial density perturbations Turbulence injected during first-order phase transitions Wagstaff, Banerjee, Schleicher and Sigl (Universities of Somewhere and Elsewhere) Moriond 2014 11 / 24

Turbulence from primordial density perturbations Cosmological perturbations (Kodama&Sasaki 1984) The primordial density perturbations generate velocity fluctuations Fluid 3-velocity perturbations: v(k, η) = ik 2H 2 [Φ (k, η) + HΦ(k, η)] v(k, η) = i 3 3 ˆk [ 2 sin(kη/ 3) 2j 1 (kη/ ] 3) Φ 0 (k). At linear order in Φ, turbulent velocity is purely irrotational v P v (k) = 27 4 [ sin(kη/ 3) 2j 1 (kη/ 3)] 2 PΦ0 (k) Wagstaff, Banerjee, Schleicher and Sigl (Universities of Somewhere and Elsewhere) Moriond 2014 12 / 24

Velocity fluctuations in the radiation era Spectrum of velocity perturbations at T 0.2 GeV (neutrino era) 10 9 Spectrum v k 10 11 10 13 10 15 10 5 10 3 10 1 10 1 c v P v 5 10 5 c Wagstaff, Banerjee, Schleicher and Sigl (Universities of Somewhere and Elsewhere) Moriond 2014 13 / 24

Turbulence in the radiation era Transition to turbulence Injection of kinetic energy = Large enough R e 1 = Turbulence R e (l c ) 8 v(l c)l c l ν,γ mfp,c 1, τ eddy = al c v(l c ) With the expansion R e decreases and τ eddy increases. Turbulence due to neutrinos ν, R e 1 Viscous regime, R e < 1 Neutrinos decoupling at T 2.6 MeV Turbulence due to photons γ, R e 1 e ± annihilation from m e T 20 kev Viscous regime, R e < 1 Wagstaff, Banerjee, Schleicher and Sigl (Universities of Somewhere and Elsewhere) Moriond 2014 14 / 24

The small-scale dynamo in the radiation era The excited scales The photon era Coherence length: Comoving scale lc pc 10 4 10 1 10 2 10 5 10 8 Time l H Ν l D L c Γ l D 10 2 10 3 10 4 10 5 10 6 Temperature T ev l c < l H Many turnover times: τ eddy = al c /v < H 1 Silk damping: ld 2 = lc γ a dt Wagstaff, Banerjee, Schleicher and Sigl (Universities of Somewhere and Elsewhere) Moriond 2014 15 / 24

The small-scale dynamo in the radiation era The excited scales The neutrino era Comoving scale lc pc 10 1 10 2 10 5 10 8 10 11 Time l H L c L c Ν l D 10 3 10 2 10 1 10 0 10 1 Temperature T GeV Coherence length: l c < l H Many turnover times: τ eddy = al c /v < H 1 Damping scale: ld 2 = lc ν a dt Wagstaff, Banerjee, Schleicher and Sigl (Universities of Somewhere and Elsewhere) Moriond 2014 16 / 24

The small-scale dynamo in the radiation era The Reynolds number 10 19 The neutrino era Reynolds numbers 10 15 10 11 10 7 10 3 R m R e P m 10 1 10 1 10 0 10 1 Temperature T GeV Time P m 1 we can neglect dissipative effects due to finite conductivity. Wagstaff, Banerjee, Schleicher and Sigl (Universities of Somewhere and Elsewhere) Moriond 2014 17 / 24

Magnetic field amplification Estimate the growth rate from the Kazantsev model: B 2 exp(2γt) Turbulence with spectrum: v(l) l ϑ and P m 1 (Kolmogorov: ϑ = 1/3), (Burgers: ϑ = 1/2) Γ = (163 304ϑ) 60 R (1 ϑ)/(1+ϑ) e /τ eddy (Schober et al. 2012) Large amplification factor in very short time! N Γ(t)dt B 2 e 2N Wagstaff, Banerjee, Schleicher and Sigl (Universities of Somewhere and Elsewhere) Moriond 2014 18 / 24

Magnetic field saturation (Schekochihin et al. 2004, Federrath et al. 2011) Saturation given by equipartition of magnetic and kinetic energy B 2 4πε(ρ + p) v 2 Wagstaff, Banerjee, Schleicher and Sigl (Universities of Somewhere and Elsewhere) Moriond 2014 19 / 24

Magnetic field saturation (Schekochihin et al. 2004, Federrath et al. 2011) Saturation given by equipartition of magnetic and kinetic energy B 2 4πε(ρ + p) v 2 Saturation efficiency at low Mach numbers (Federrath et al. 2011) Irrotational v : ε 10 3 10 4 Rotational v : ε 1 Wagstaff, Banerjee, Schleicher and Sigl (Universities of Somewhere and Elsewhere) Moriond 2014 20 / 24

Results The final magnetic field strengths The SSD mechanism can amplify to saturation tiny seed fields B seed 0 (10 30 10 20 ) ng Turbulence generated by primordial density perturbations: B rms 0 10 6 ε 1 2 ng on scales up to λc 10 1 pc Too weak on too short scales to explain observed intergalactic magnetic fields Turbulence generated by first-order phase transitions: B rms 0 (10 6 10 3 )ε 1 2 ng on scales λc (10 1 10 2 ) pc Strong enough to explain observed intergalactic magnetic fields Wagstaff, Banerjee, Schleicher and Sigl (Universities of Somewhere and Elsewhere) Moriond 2014 21 / 24

Intergalactic magnetic fields (Neronov and Vovk 2010) Wagstaff, Banerjee, Schleicher and Sigl (Universities of Somewhere and Elsewhere) Moriond 2014 22 / 24

Summary The small-scale dynamo in the radiation era Before structure formation the SSD can be effective at amplifying magnetic seed fields. Kinetic energy generated by primordial density perturbations and first-order phase transitions Fully developed turbulence is expected on scales l D l c l e where R e 1, many eddy-interactions and no damping Identified epoch in RD in which conditions are good for SSD Kazantsev model gives large growth rate of seed field: Γ R 1/2 e For seed fields 10 30 B seed 0 /ng 10 20 magnetic energy saturates rapidly to: B 0 10 3 ε 1/2 ng on scales up to λ c 100 pc These strong extragalactic fields can explain Fermi observations Provide initial magnetic fields for structure formation Other observable signatures? Wagstaff, Banerjee, Schleicher and Sigl (Universities of Somewhere and Elsewhere) Moriond 2014 23 / 24

Subsequent evolution of magnetic fields (Durrer&Neronov 2013) MHD turbulence decay Wagstaff, Banerjee, Schleicher and Sigl (Universities of Somewhere and Elsewhere) Moriond 2014 24 / 24