Helical magnetic fields in the early universe • What is magnetic helicity • The myth of catastrophic quenching • Magn helicity in decaying turbulence • The chiral magnetic effect • Gravitational waves from the resulting turbulence Axel Brandenburg (Nordita, Stockholm) 1
elix Curling tendril Climbing plant Biophysics: Helix Chirality 2
Helicity & topology: math. underpinnings Helical structures on the Sun (X-ray emission) Coronal mass ejections 3
Helical structures on the Sun (X-ray emission) Coronal mass ejections 4
Moffatt coined the term in hydro/MHD Coronal mass ejections 5
Magnetic helicity measures linkage of flux 1 A B H d V S 1 V 2 B A H A d B d S 1 H 2 L S 1 1 1 2 Therefore the unit is A d S 1 2 Maxwell squared S 2 6
What produces helicity? north equator south Cyclones: Down: faster No preferred helicity, Up: slower but + and – possible ω u if result of instability W W g g u u ω ω 0 0
Catastrophic quenching d A B 2 J B dt J B J B j b 2 2 J B k B , j b k b 1 f J B J B j b 2 2 J B k B , j b k b d 1 f 1 2 2 2 k B 2 k B 2 k b 1 1 f dt molecular value!! k 2 f 2 2 2 k ( t t ) B b 1 e 1 s k 1 8
Early Universe Energy momentum tensor Conformal time, rescaled equations 9
Kolmogorov turbulence forward cascade nonlinearity 2 1 1 cos kx cos 2 kx 2 2 k 2 k constant flux e cm 2 /s 3 cm 3 /s 2 1 u 2 E k dk E ( k ) 2 e a b E k C k K a =2/3, b = 5/3 a e e cm : 3 2 1 s : 2 3 a k 10
Magnetic helicity: inverse cascade helical vs Initial slope nonhelical E ~ k 4 Christensson et al. (2001, PRE 64, 056405) 11
Helical decay law: Biskamp & Müller (1999) EL H const e 3 3 / 2 U / L E / L e d E d / t e 3 / 2 5 / 2 d E / d t E / L E / H t 2 / 3 E 12
Forced – decaying Helical – nonhelical 13
Self-similar turbulent decay instantaneous scaling exponents 14
Collapsed spectra and pq diagrams Sect. 2.2 Assessement of the turbulent convective dynamo view 15
Chiral magnetic effect: caused by chirality of fermions electrons left-handed positrons: right-handed But also spontaneous Electrons have handedness spin flip unless 2 k T m c B e 6x10 9 K 16
Chiral magnetic effect In the presence of B -field chiral electrons produce a current J=…+ m B with a em is the fine Leads to a dynamo effect structure constant quantum effect m is a pseudoscalar m 2 k k
Total chirality conserved Uncurled induction equation & chemical potential (spin flipping) Coupling between electromagnetic field and chem potential Conservation equation Therefore
Total system of equations Momentum & continuity equations Together with:
Characteristic velocities Two velocities from chiral magnetic effect Different orderings
Dimensional arguments • [ E ( k , t ) ] = [ r ] cm 3 s -2 = g s -2 • E ( k , t ) = C m r m a b • cm: 3 = -a + 2 b • s : - 2 = -b • b = 2, a = 2 b - 3 = 4 - 3 = 1 • E ( k , t ) = C m rm 2 • to determine C m from simulations 21
The spectrum from chiral effect Spectrum build-up from high wavenumbers E M (k,t) k -2 k m/2 Governed solely by chiral chemical potential
Eventual saturation • [ E ( k , t ) ] = g s -2 • E ( k , t ) = C l r a m b l c • g: 1 = a – c • cm: 0 = – b – c • s : – 2 = 2 c • c = – 1, b = – c = 1, a = 1 + c = 0 • E ( k , t ) = C l m/l • to determine C l from simulations 23
The final spectrum
Inverse cascade! Growth at one wavenumber Then: saturation caused by initial chemical potential
Early universe: use conservation law Conseration equation Maximally helical:
Inserting actual numbers =(0.2e-18G) 2 Mpc G Magnetic helicity Inverse length scale
Science 328, 73 – 75 (2010) • Chiral magnetic effect alone may be too weak to explain B-field • But the magnetic stress could still explain gravitational waves 28
Cosmological GWs 29
Correspondence of spectra • If spectral slope of B is -5/3, then • Spectral slope of B 2 is -5/3-2 = -11/3 • But for slope 4, we don’t get 4 -2 = 2, but 0. 30
Comparison with LISA sensitivity limits • Frequency ~3 mHz 10% – Slope corresponds to 1% turbulence spectrum – magnetic energy 10% Större skallängt or 1% or radiation • Observable w/ LISA – arm length 2-5 milion km – Duration 2 or 5 yr 31
Quadratic scaling • Acoustic driving strongest GW field • Quadratic scaling 32
GWs from chiral magnetic effect 33
GW energy & strain spectra • Magnetic helicity causes circular polarized GWs • Can reach 100%, and inverse cascade apparent 34
Correspond to + and x modes Seto (2006) 35
Circular polarization in space & time • Both plus and cross polarization together • Combine the two as a function of space & time • Get circular polarization 36
Conclusions • Magnetic helicity nearly perfectly conserved – Catastrophic quenching in periodic boxes • Inverse cascade in decaying turbulence – Important in early Universe • Can be initiated by chiral magnetic effect (CME) – But may not yet explain lower observational limits • CME also drives gravitational waves – Currently somewhat too weak 37
Inverse cascading Conseration equation
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