1. 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

2. 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:

3. 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)

4. 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

5. O(1) vector perturbation equations: Initial conditions = 0, so that O(2) vector perturbation equations: O(1) scalar velocity and density perturbations: where

6. Then Power spectra today a a Lu, Ananda, Clarkson, RM, JCAP, 0812.1349

7. 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:

8. 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

9. 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.

10. 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)

11. Anisotropic photon stress, with vector part (from O(1)XO(1) scalars): Momentum balance where and

12. 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