The origin, evolution and signatures of primordial magnetic fields.

The universe is magnetized on all scales probed so far. On the largest scales, galaxies and galaxy clusters host magnetic fields at the micro Gauss level coherent on scales up to ten kpc. Recent observational evidence suggests that even the intergalactic medium in voids could host a weak ∼ 10(-16) Gauss magnetic field, coherent on Mpc scales. An intriguing possibility is that these observed magnetic fields are a relic from the early universe, albeit one which has been subsequently amplified and maintained by a dynamo in collapsed objects. We review here the origin, evolution and signatures of primordial magnetic fields. After a brief summary of magnetohydrodynamics in the expanding universe, we turn to magnetic field generation during inflation and phase transitions. We trace the linear and nonlinear evolution of the generated primordial fields through the radiation era, including viscous effects. Sensitive observational signatures of primordial magnetic fields on the cosmic microwave background, including current constraints from Planck, are discussed. After recombination, primordial magnetic fields could strongly influence structure formation, especially on dwarf galaxy scales. The resulting signatures on reionization, the redshifted 21 cm line, weak lensing and the Lyman-α forest are outlined. Constraints from radio and γ-ray astronomy are summarized. Astrophysical batteries and the role of dynamos in reshaping the primordial field are briefly considered. The review ends with some final thoughts on primordial magnetic fields.

Large-scale dynamo growth rates from numerical simulations and implications for mean-field theories.

Understanding large-scale magnetic field growth in turbulent plasmas in the magnetohydrodynamic limit is a goal of magnetic dynamo theory. In particular, assessing how well large-scale helical field growth and saturation in simulations match those predicted by existing theories is important for progress. Using numerical simulations of isotropically forced turbulence without large-scale shear with its implications, we focus on several additional aspects of this comparison: (1) Leading mean-field dynamo theories which break the field into large and small scales predict that large-scale helical field growth rates are determined by the difference between kinetic helicity and current helicity with no dependence on the nonhelical energy in small-scale magnetic fields. Our simulations show that the growth rate of the large-scale field from fully helical forcing is indeed unaffected by the presence or absence of small-scale magnetic fields amplified in a precursor nonhelical dynamo. However, because the precursor nonhelical dynamo in our simulations produced fields that were strongly subequipartition with respect to the kinetic energy, we cannot yet rule out the potential influence of stronger nonhelical small-scale fields. (2) We have identified two features in our simulations which cannot be explained by the most minimalist versions of two-scale mean-field theory: (i) fully helical small-scale forcing produces significant nonhelical large-scale magnetic energy and (ii) the saturation of the large-scale field growth is time delayed with respect to what minimalist theory predicts. We comment on desirable generalizations to the theory in this context and future desired work.

Cosmic microwave background trispectrum and primordial magnetic field limits.

Primordial magnetic fields will generate non-gaussian signals in the cosmic microwave background (CMB) as magnetic stresses and the temperature anisotropy they induce depend quadratically on the magnetic field. We compute a new measure of magnetic non-gaussianity, the CMB trispectrum, on large angular scales, sourced via the Sachs-Wolfe effect. The trispectra induced by magnetic energy density and by magnetic scalar anisotropic stress are found to have typical magnitudes of approximately a few times 10(-29) and 10(-19), respectively. Observational limits on CMB non-gaussianity from WMAP data allow us to conservatively set upper limits of a nG, and plausibly sub-nG, on the present value of the primordial cosmic magnetic field. This represents the tightest limit so far on the strength of primordial magnetic fields, on Mpc scales, and is better than limits from the CMB bispectrum and all modes in the CMB power spectrum. Thus, the CMB trispectrum is a new and more sensitive probe of primordial magnetic fields on large scales.

Mean-field dynamo action in renovating shearing flows.

We study mean-field dynamo action in renovating flows with finite and nonzero correlation time (τ) in the presence of shear. Previous results obtained when shear was absent are generalized to the case with shear. The question of whether the mean magnetic field can grow in the presence of shear and nonhelical turbulence, as seen in numerical simulations, is examined. We show in a general manner that, if the motions are strictly nonhelical, then such mean-field dynamo action is not possible. This result is not limited to low (fluid or magnetic) Reynolds numbers nor does it use any closure approximation; it only assumes that the flow renovates itself after each time interval τ. Specifying to a particular form of the renovating flow with helicity, we recover the standard dispersion relation of the α(2)Ω dynamo, in the small τ or large wavelength limit. Thus mean fields grow even in the presence of rapidly growing fluctuations, surprisingly, in a manner predicted by the standard quasilinear closure, even though such a closure is not strictly justified. Our work also suggests the possibility of obtaining mean-field dynamo growth in the presence of helicity fluctuations, although having a coherent helicity will be more efficient.

Cosmic microwave background bispectrum from primordial magnetic fields on large angular scales.

Primordial magnetic fields lead to non-Gaussian signals in the cosmic microwave background (CMB) even at the lowest order, as magnetic stresses and the temperature anisotropy they induce depend quadratically on the magnetic field. In contrast, CMB non-Gaussianity due to inflationary scalar perturbations arises only as a higher-order effect. We propose a novel probe of stochastic primordial magnetic fields that exploits the characteristic CMB non-Gaussianity that they induce. We compute the CMB bispectrum (b(l1l2l3)) induced by such fields on large angular scales. We find a typical value of l1(l1 + 1)l3(l3 + 1)b(l1l2l3) approximately 10(-22), for magnetic fields of strength B0 approximately 3 nG and with a nearly scale invariant magnetic spectrum. Observational limits on the bispectrum allow us to set upper limits on B0 approximately 35 nG.

Shear dynamo problem: Quasilinear kinematic theory.

Large-scale dynamo action due to turbulence in the presence of a linear shear flow is studied. Our treatment is quasilinear and kinematic but is nonperturbative in the shear strength. We derive the integrodifferential equation for the evolution of the mean magnetic field by systematic use of the shearing coordinate transformation and the Galilean invariance of the linear shear flow. For nonhelical turbulence the time evolution of the cross-shear components of the mean field does not depend on any other components excepting themselves. This is valid for any Galilean-invariant velocity field, independent of its dynamics. Hence the shear-current assisted dynamo is essentially absent, although large-scale nonhelical dynamo action is not ruled out.

Reconnecting flux-rope dynamo.

We develop a model of the fluctuation dynamo in which the magnetic field is confined to thin flux ropes advected by a multiscale model of turbulence. Magnetic dissipation occurs only via reconnection of the flux ropes. This model can be viewed as an implementation of the asymptotic limit R_{m}-->infinity for a continuous magnetic field, where magnetic dissipation is strongly localized to small regions of strong-field gradients. We investigate the kinetic-energy release into heat mediated by the dynamo action, both in our model and by solving the induction equation with the same flow. We find that a flux-rope dynamo is an order of magnitude more efficient at converting mechanical energy into heat. The probability density of the magnetic energy release in reconnections has a power-law form with the slope -3 , consistent with the solar corona heating by nanoflares.

Nonperturbative quasilinear approach to the shear dynamo problem.

We study large-scale dynamo action due to turbulence in the presence of a linear shear flow. Our treatment is quasilinear and equivalent to the standard "first-order smoothing approximation." However it is non perturbative in the shear strength. We first derive an integrodifferential equation for the evolution of the mean magnetic field, by systematic use of the shearing coordinate transformation and the Galilean invariance of the linear shear flow. We show that, for nonhelical turbulence, the time evolution of the cross-shear components of the mean field do not depend on any other components excepting themselves; this is valid for any Galilean-invariant velocity field, independent of its dynamics. Hence, to all orders in the shear parameter, there is no shear-current-type effect for non helical turbulence in a linear shear flow in quasilinear theory in the limit of zero resistivity. We then develop a systematic approximation of the integro-differential equation for the case when the mean magnetic field varies slowly compared to the turbulence correlation time. For nonhelical turbulence, the resulting partial differential equations can again be solved by making a shearing coordinate transformation in Fourier space. The resulting solutions are in the form of shearing waves, labeled by the wave number in the sheared coordinates. These shearing waves can grow at early and intermediate times but are expected to decay in the long time limit.

Nonlinear current helicity fluxes in turbulent dynamos and alpha quenching.

Large scale dynamos produce small scale current helicity as a waste product that quenches the large scale dynamo process (alpha effect). This quenching can be catastrophic (i.e., intensify with magnetic Reynolds number) unless one has fluxes of small scale magnetic (or current) helicity out of the system. We derive the form of helicity fluxes in turbulent dynamos, taking also into account the nonlinear effects of Lorentz forces due to fluctuating fields. We confirm the form of an earlier derived magnetic helicity flux term, and also show that it is not renormalized by the small scale magnetic field, just like turbulent diffusion. Additional nonlinear fluxes are identified, which are driven by the anisotropic and antisymmetric parts of the magnetic correlations. These could provide further ways for turbulent dynamos to transport out small scale magnetic helicity, so as to avoid catastrophic quenching.

Hyperdiffusion in nonlinear large- and small-scale turbulent dynamos.

The generation of large-scale magnetic fields is generically accompanied by the more rapid growth of small-scale fields. The growing Lorentz force due to these fields backreacts on the turbulence to saturate the mean-field and small-scale dynamos. For the mean-field dynamo, in a quasilinear treatment of this saturation, it is generally thought that, while the alpha effect gets renormalized and suppressed by nonlinear effects, the turbulent diffusion is left unchanged. We show here that this is not true and the effect of the Lorentz forces is also to generate additional nonlinear hyperdiffusion of the mean field. A combination of such nonlinear hyperdiffusion with diffusion at small scales also arises in a similar treatment of small-scale dynamos, and is crucial to understand its saturation.