Anisotropic Stress: The Shear Bound on a Primordial Field at Recombination and at the Bounce

A magnetic field is not an isotropic fluid: it presses along its lines and pulls across them, and a universe containing a coherent primordial field is a slightly anisotropic universe. That anisotropy is bounded by the observed isotropy of the microwave sky, and this paper converts the bound into field strengths. Requiring the magnetic energy at recombination to be below one part in 1e5 of the total, an isotropy level matched to the observed quadrupole scale, gives a present-equivalent field of 2.56e+01 nanogauss for the coherent component. The same style of accounting applied at the bounce, under a stated shear budget of one per cent of the turnaround density, permits 4.69e+33 tesla there, equivalent adiabatically to 3.24e+02 nanogauss today, so the bounce's own shear budget is not the binding constraint; recombination isotropy is, for the coherent component. The shear bound is weaker than the spectral microwave bound but applies to a uniform field that spectral analyses handle poorly, and it is the correct rail for bounce models whose contraction aligns the field. Computed from stated assumptions, not measured.