1. Vector modes, vorticity and magnetic fields Kobayashi, RM, Shiromizu, Takahashi: PRD, astro-ph/0701596 Caprini, Crittenden, Hollenstein, RM: PRD, arXiv:0712.1667 Lu, Ananda, Clarkson, RM: JCAP, arXiv:0812.1349 Roy Maartens Institute of Cosmology and Gravitation University of Portsmouth April 2009 Kobayashi, RM, Shiromizu, Takahashi: PRD, astro-ph/0701596 Caprini, Crittenden, Hollenstein, RM: PRD, arXiv:0712.1667 Lu, Ananda, Clarkson, RM: JCAP, arXiv:0812.1349 Roy Maartens Institute of Cosmology and Gravitation University of Portsmouth April 2009

2. Generating B and ω at O(1) Maxwell in Minkowski spacetime 4-current Faraday tensor

3. Maxwell’s equations 2 General spacetime –The fundamental object is the Faraday tensor –Electric and magnetic fields depend on the observer –Charge and current densities depend on the observer

4. Maxwell’s equations 3 Expanding FRW universe background: at O(1):

5. Then Maxwell’s equations in an expanding FRW universe become, at O(1) where is the comoving derivative is the conformal time derivative are the comoving (lab) fields

6. In the cosmic plasma, up to O(1): Then are both transverse (‘pure’ vector) The metric at O(1), with only vector perturbations, is (in Poisson gauge) To generate B from zero at O(1), we need a current – and the current must be transverse, i.e. with nonzero curl We need a nonzero difference between vorticities

7. Take the curl of the curl B equation and use the curl E equation to get the wave equation: Thus, at O(1) This is the basis of Harrison’s mechanism (1970) Electrons are tightly coupled to photons, but not so tightly to protons. Radiation fluid (γ+e) spins down at a different rate to the proton fluid.

8. At O(1), the vorticity is given in terms of velocity and metric perturbations by This follows from the covariant definition of vorticity:

9. At O(1), according to Harrison, we need In particular, the proton and photon velocities must be different – whereas the electron velocity can be the same as the photon velocity and we can in principle still get a current that sources B The velocities evolve according to the momentum balance equations:

10. The force density terms are a sum of scattering terms and electromagnetic force densities: Lorentz force = O(2) The are Thomson and Coulomb rates. Then we get the momentum balance equations:

11. The velocities, and hence the vorticities, are driven together by Thomson/ Coulomb scattering: where

12. Caprini, Crittenden, Hollenstein, RM: PRD, arXiv:0712.1667

13. How to get vorticity? Harrison’s mechanism: * electrons bound to photons (Thomson) * protons can have different velocity * if there is primordial vorticity – then a rotational current arises at O(1) * this generates a B-field The key issue = how to get primordial vorticity at O(1)? NOT from inflation Defects (eg strings) actively generate vector perturbations at O(1) This should source vorticity in the plasma?

14. BUT - the total plasma vorticity is conserved: momentum balance is –and B is sourced by the initial total vorticity:

15. Thus there is a closed loop, independent of the defects The defects cannot induce plasma vorticity at O(1) Defects induce metric vector perturbations – but the plasma velocities respond to conserve total plasma vorticity:

16. Ways out of the closed loop O(1) anisotropic stress in the plasma But anisotropic stress is suppressed at high T - by collisions At O(2): defects can directly generate plasma motions via turbulence, as they move through the plasma (Battefeld et al, arXiv:0708.2901)

17. Alternative: O(2) vectors from density perturbations Density perturbations generate at O(2) a cosmological background of vector modes and O(1)

18. Then a a Lu, Ananda, Clarkson, RM, JCAP, 0812.1349

19. In principle, the O(2) vector background affects –ISW –Lensing –redshift-space distortions But – no vorticity is generated in the dust matter at O(2): (This is basically the Kelvin-Helmholtz theorem.) In fact, at all nonlinear orders, no vorticity can be generated in a perfect fluid:

20. 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

21. 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.

22. To generate vorticity, we need: Nonadiabaticity (with ) O(2) – also generates B (Biermann battery) Anisotropic stress O(1) – but need vector stress Momentum exchange with another fluid O(2) – also generates B (Kobayashi, RM, Shiromizu, Takahashi, PRD, astro-ph/0701596)