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Generation of magnetic fields and vorticity in the photon-baryon plasma Roy Maartens Institute of Cosmology and Gravitation University of Portsmouth CMF Ascona June 2009 Roy Maartens Institute of Cosmology and Gravitation University of Portsmouth CMF Ascona June 2009
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Pre-recombination plasma Density perturbations generate magnetic fields and vorticity at O(2) – see Takahashi’s talk Coulomb interaction dominates Thomson interaction: neglect e-p velocity difference Neglect photon anisotropic stress:
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Schematic of the process: 1 2 3 4 To lowest nontrivial order of breaking tight coupling, E ~ ρv (3,4) Then curl E ~ ρ curl v + grad ρ x v (1,2)
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O(2) vectors from density perturbations Density perturbations generate at O(2) a cosmological background of vector modes Note: O(2) scalars do not source O(2) vectors, and are independent of O(2) vectors
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O(1) vector perturbation equations: Initial conditions = 0, so that O(2) vector perturbation equations: O(1) scalar velocity and density perturbations: where
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Then Power spectra today a a Lu, Ananda, Clarkson, RM, JCAP, 0812.1349
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In principle, the O(2) vector background affects the late universe via –ISW –Lensing –redshift-space distortions But – no vorticity is generated in the dust matter at O(2): This is basically the Kelvin-Helmholtz theorem: at all nonlinear orders, no vorticity can be generated in a perfect fluid:
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For a general fluid, kinematic quantities defined via Energy-momentum tensor (in the energy frame): Momentum conservation (nonlinear): Vorticity propagation (nonlinear): only source is curl acceleration
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Use the nonlinear identity NB: curl grad ≠0 and the vanishing of anisotropic stress, to get where For a perfect (adiabatic) fluid, p nad =0, and vorticity is conserved to all orders.
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To generate vorticity, and hence B, we need: Nonadiabatic fluid, i.e., with O(2) effect – Biermann battery Two or more perfect fluids with momentum exchange, eg, photons + baryons O(2) (Kobayashi, RM, Shiromizu, Takahashi, PRD, astro-ph/0701596)
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Anisotropic photon stress, with vector part (from O(1)XO(1) scalars): Momentum balance where and
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Expect anisotropic stress may be important for magnetogenesis on small scales O(2) dynamics of photon anisotropic stress, governed by the O(2) Boltzmann equation: (all quantities assumed projected and tracefree) Octopole and 16-pole are higher-order away from tight coupling
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