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

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Helical magnetic fields in the early universe

Axel Brandenburg (Nordita, Stockholm)

• What is magnetic helicity
• The myth of catastrophic quenching
• Magn helicity in decaying turbulence
• The chiral magnetic effect
• Gravitational waves from the

resulting turbulence

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elix

Curling tendril  Climbing plant Biophysics: Helix Chirality

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Helical structures on the Sun (X-ray emission)

Coronal mass ejections

Helicity & topology: math. underpinnings

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Helical structures on the Sun (X-ray emission)

Coronal mass ejections

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Moffatt coined the term in hydro/MHD

Coronal mass ejections

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Magnetic helicity measures linkage of flux



 

V

V H d B A

1



2



2 1

2     H

 

  

1 1

d d

1 S L

H S B A 

2

d

      

S

S A

1

S

1

  A B   

Therefore the unit is Maxwell squared

What produces helicity?

Cyclones: Down: faster Up: slower

north south

g W g W

 u ω  u ω

equator

u ω    No preferred helicity, but + and – possible if result of instability

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Catastrophic quenching

, b b j B B J b j B J B J k k          

b j B J B J     

2 f 2 1 2 1 1

2 2 b B B k k dt d k     



 

) ( 2 1 2 2

s 2 1

1

t t k f

e k k

 

 



b B

molecular value!!

B J B A      2 dt d

, b b j B B J k k     

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Energy momentum tensor Conformal time, rescaled equations

Early Universe

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Kolmogorov turbulence

 forward cascade

nonlinearity  

2 1 2 1 2

2 cos cos   kx kx k k 2 

constant flux e cm2/s3 e e k E(k)  





2 2 1 u

dk k E

cm3/s2 

 

b a K

k C k E e 

1 2 3 : cm   a a 3 2 : s 

a=2/3, b= 5/3

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Magnetic helicity: inverse cascade

Initial slope E~k4

Christensson et al. (2001, PRE 64, 056405)

helical vs nonhelical

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Helical decay law: Biskamp & Müller (1999)

const   EL H L E L U / /

2 / 3 3

  e t E d / d   e

H E L E t E / / d / d

2 / 5 2 / 3

    e

3 / 2 

 t E

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Forced – decaying

Helical – nonhelical

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Self-similar turbulent decay

instantaneous scaling exponents

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• Sect. 2.2

Assessement of the turbulent convective dynamo view

Collapsed spectra and pq diagrams

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Chiral magnetic effect: caused by chirality of fermions

Electrons have handedness But also spontaneous spin flip unless

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c m T k

e B



6x109K electrons left-handed positrons: right-handed

Chiral magnetic effect

In the presence of B-field chiral electrons produce a current J=…+mB with Leads to a dynamo effect

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k k  m   

aem is the fine structure constant  quantum effect m is a pseudoscalar

Total chirality conserved

Uncurled induction equation & chemical potential Coupling between electromagnetic field and chem potential Conservation equation Therefore

(spin flipping)

Total system of equations

Momentum & continuity equations Together with:

Characteristic velocities

Different orderings Two velocities from chiral magnetic effect

Dimensional arguments

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• [E(k,t) ] = [r] cm3 s-2 = g s-2
• E(k,t) = Cm r mab
• cm: 3 = -a + 2b
• s: -2 = -b
•  b = 2, a = 2b - 3 = 4 - 3 = 1
• E(k,t) = Cm rm2
•  to determine Cm from simulations

The spectrum from chiral effect

Governed solely by chiral chemical potential Spectrum build-up from high wavenumbers k EM(k,t) m/2 k-2

Eventual saturation

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• [E(k,t) ] = g s-2
• E(k,t) = Cl ramblc
• g: 1 = a – c
• cm: 0 = – b – c
• s: – 2 = 2c
•  c = – 1, b = – c = 1, a = 1+ c = 0
• E(k,t) = Cl m/l
•  to determine Cl from simulations

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

Magnetic helicity Inverse length scale =(0.2e-18G)2 Mpc G

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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

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Cosmological GWs

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Correspondence of spectra

• If spectral slope of

B is -5/3, then

• Spectral slope of

B2 is -5/3-2 = -11/3

• But for slope 4, we

don’t get 4-2 = 2, but 0.

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Comparison with LISA sensitivity limits

• Frequency ~3 mHz

– Slope corresponds to turbulence spectrum – magnetic energy 10%

• r 1% or radiation
• Observable w/ LISA

– arm length 2-5 milion km – Duration 2 or 5 yr

Större skallängt

1% 10%

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Quadratic scaling

• Acoustic driving  strongest GW field
• Quadratic scaling

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GWs from chiral magnetic effect

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GW energy & strain spectra

• Magnetic helicity causes circular polarized GWs
• Can reach 100%, and inverse cascade apparent

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Correspond to + and x modes

Seto (2006)

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Circular polarization in space & time

• Both plus and cross polarization together
• Combine the two as a function of space & time
• Get circular polarization

Conclusions

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• 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

Inverse cascading

Conseration equation