Statistics of Primordial Magnetic Fields

1
Statistics of Primordial Magnetic Fields

• Iain Brown
• ICG, Portsmouth

2
Motivation

• Non-Linearity
• Standard model generally linear and Gaussian
• All information contained within power spectra
• Non-linearity non-Gaussianity 3-point
  correlations
• Scalar/vector/tensor decomposition 3-point
  correlations
• Observable correlations between E and B modes?

3
Motivation

• Magnetic fields?
• Galactic fields, ?G, lckpc
• Cluster fields, nG-?G, lcMpc
• Inter-cluster fields, nG? ?G?
• Grasso and Rubenstein, astro-ph/0009061,
  Giovannini, astro-ph/0312614
• Cosmological or Astrophysical?
• Early-time / continuous / recombination /
  late-time
• e.g. Berezhiani and Dolgov, astro-ph/0305595
  Widrow, astro-ph/0207240 Mataresse et. al.,
  astro-ph/0410687

4
Motivation

• Observable consequences?
• Impact on plasma physics ? impact on cosmological
  perturbations ? impact on microwave background
• Impact on structure formation
• Statistical imprint
• Stress-energy inherently non-linear
• Non-Gaussianities?
• Correlations between scalar/vector/tensor modes?
• Correlations between E and B modes?

5
Motivation

• Aim
• Construct simulated primordial magnetic fields
• Generate statistics of the stress-energy tensor
• 1-Point probability distribution function,
  skewness, kurtosis
• 2-Point power spectra
• 3-Point bispectra
• Consider generic functions all
  rotationally-invariant forms
• While model-specific, techniques and results
  could be relevant elsewhere defect models?

6
Basic Physics

• Assume
• Robertson-Walker cosmology
• Fields enter at half order, stresses at first
• ? conductivity ? E0
• Impact of magnetic field from stress-energy alone
• Large-scale tangled fields ?
• cut-off scale kc
• damping purely with universal expansion
• Gaussian statistics on the field test case
• Mack et. al. astro-ph/0105504, Lewis,
  astro-ph/0406096

7
Basic Physics

• Symmetrised scalar/vector/tensor decomposition

• Constraints on the power spectrum
• Causal field ngt2
• Nucleosynthesis bounds n?-3
• Singularity at n-3

8
Simulations

• Grid in Fourier space of sidelength ldim
• Automatic ultraviolet cutoff at kldim
• Nyquist wavenumber, kcldim/2
• Adjust to remove leakage of modes kcldim/4
• Infrared cutoff inavoidable
• Solenoidal fields ?
• Average n simulations and estimate errors

9
Simulations

• Gaussian magnetic field and ?2 isotropic pressure

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1-Point Statistics

• Scalar probability distribution functions

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1-Point Statistics

• Skewness and kurtosis
• P.D.F. moments
• Skewness and kurtosis
• Measures of deviation from Gaussian
• Skewness tilt, kurtosis curvature
• 20 realisations
• Rotation invariance lttgt and lttSgt

12
1-Point Statistics

• ?2 with p degrees of freedom,
  ,?1?1.63, ?24
• ? ?1?1.63?0.01, ?23.99?0.05 (insensitive to n)
• ?s
• ?1?-0.24?0.03, ?21.10?0.01 (n0)
• ?1?-0.38?0.01, ?20.86?0.02 (n-2.5)

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2-Point Statistics

• Stresses
• Calculate
• Invariant measures lt??gt, lt?s?sgt, lt?av?avgt,
  lt?abt?abtgt, lt??sgt
• Relevant angles

14
2-Point Statistics

• Find

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2-Point Statistics
Approximations (ngt-3/2) in excellent agreement
16
3-Point Statistics

• Bispectra three wavevectors k, p, q
• Statistical isotropy ? triangular formations
• Geometric dependance
• Colinear (degenerate)
• r1, ?0
• Equilateral
• r1, ?120
• Invariant lt???gt, lt???sgt, lt??s?sgt, lttststsgt,
  lt??va?vagt, lt?s?va?vagt, lt??Tab?Tabgt, lt?s?Tab?Tabgt,
  lt?av?abT?bvgt, lt?abT?bcT?acTgt

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3-Point Statistics

• Relevant angles
• Analogous calculation to 2-point case
• Apply combinations of projection operators
• to recover separate bispectra

18
3-Point Statistics

• Analytic formulae not very instructive
• Fully general scalar and vector correlations
  calculated
• Tensor correlations left for future work
• Symmetries of projection and full correlation
  operators remove full tensor correlation in
  colinear case
• Approximations (kltltkc, ngt-1)

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3-Point Statistics

• Approximations (kltltkc, ngt-1)
• Calculated generically colinear case

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3-Point Statistics

• Realisations
• Heavily compromised by sparse mode-selection
• Average 1,500 runs at ldim192
• Present colinear results for simplicity
• n0, scalar n0, vector

21
3-Point Statistics

• n-2.5

22
Conclusions and Prospects (1)

• Non-linear effects modify the statistics of the
  CMB
• Presented an informative test case
• Statistics will help constrain a primordial
  magnetic field
• Robust and general code generated
• 1-Point
• Non-Gaussian (?2) pressure, n-dependent
  anisotropic pressure
• 2-Point
• Uncorrelated lt??sgt signal, ?0.7 for n-2.5, ?0
  for n0
• 3-Point
• Strong non-Gaussianities in all signals at
  large-scales
• Calculations of cross-correlations
• Excellent agreement between theory and simulations

23
Conclusions and Prospects (2)

• Wanted
• CMB!
• Faster realisations of non-degenerate bispectra
• Analytic tensor modes
• Non-Gaussian magnetic fields - ?2 field?
• Techniques and analysis not necessarily magnetic
  defect models?