early universe What is magnetic helicity The myth of catastrophic - PowerPoint PPT Presentation

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