Gradient expansion, curvature perturbations, and magnetized plasmas

Abstract

The properties of magnetized plasmas are always investigated under the hypothesis that the relativistic inhomogeneities stemming from the fluid sources and from the geometry itself are sufficiently small to allow for a perturbative description prior to photon decoupling. The latter assumption is hereby relaxed and predecoupling plasmas are described within a suitable expansion where the inhomogeneities are treated to a given order in the spatial gradients. It is argued that the (general relativistic) gradient expansion shares the same features of the drift approximation, customarily employed in the description of cold plasmas, so that the two schemes are physically complementary in the large-scale limit and for the low-frequency branch of the spectrum of plasma modes. The two-fluid description, as well as the magnetohydrodynamical reduction, is derived and studied in the presence of the spatial gradients of the geometry. Various solutions of the coupled system of evolution equations in the anti-Newtonian regime and in the quasi-isotropic approximation are presented. The relation of this analysis to the socalled separate universe paradigm is outlined. The evolution of the magnetized curvature perturbations in the nonlinear regime is addressed for the magnetized adiabatic mode in the plasma frame. I. MOTIVATIONSThe analyses of the large-scale galaxy distribution [1,2], of the high-redshift type Ia supernovae [3,4], and of the CMB observables [5,6] seem to converge, these days, on a concordance model sometimes called the ÃCDM scenario, where à stands for the dark energy component and CDM accounts for the dark matter component. The ÃCDM scenario is just the compromise between the number of ascertainable parameters and the quality of the observational data. The quest for a concordance lore is also able to shed some light on the presence of large-scale magnetic fields in nearly all gravitationally bound systems we observe. Since we do see magnetic fields today over large distance scales, it seems natural to scrutinize their impact on the CMB observables. This is the motivation of a program aimed at bringing the unconventional study of magnetized CMB anisotropies to the same standard of the more conventional adiabatic 1 paradigm (see [7][8][9][10] and references therein). While different approaches to the problem are certainly available [11][12][13][14][15] (see [16] for a more complete list of earlier references), the path followed in [7,8] led to the calculation of the temperature and polarization anisotropies induced by the magnetized (adiabatic and entropic) initial conditions. The parameters of the magnetized background have been estimated (for the first time) in [9,10] by using the TT and TE correlations 2 measured by the WMAP Collaboration. The obtained results 3 show that large-scale (comoving) magnetic fields larger than 3.5 nG are excluded to 95% C.L. and for magnetic spectral indices n B ¼ 1:6 0:8 À0:1 . These determinations have been conducted in the context of the minimal mÃCDM, where m stands for magneti...

Anomalous magnetohydrodynamics

“…The comoving electric and magnetic fields in three-dimensional notation are E = g 00 √ −g e and B = g 00 √ −g b. Using the standard ADM decomposition the comoving fields are E = ( √ γ/N) e and B = ( √ γ/N) b and coincide with the ones discussed, for instance, in [29]. Following the definitions spelled out in this appendix and consistently followed in the paper we have that…”

Primordial vorticity and gradient expansion

“…By fully inhomogeneous plasma we mean the situation where not only the concentrations of charged and neutral species depend, in an arbitrary manner, upon the spatial coordinates but where the geometry as well as the electromagnetic fields are not homogeneous. It has been recently argued [14] that such a description can be rather effective for the analysis of a wide range of phenomena including the physics of pre-decoupling plasmas. In the present paper the results of Ref.…”

“…[14] shall be first extended and then applied to a concrete situation with the purpose of obtaining an explicit set of equations describing the evolution of the vorticities of the various species of the plasma. The proposal of [14] is built on the fully inhomogeneous description of the geometry in terms of the Adler-Misner-Deser (ADM) variables [15,16] which are customarily exploited for the implementation of the general relativistic gradient expansion [17,18,19,20,21,22]. The second key ingredient of Ref.…”

“…The second key ingredient of Ref. [14] is the fully inhomogeneous description of cold plasmas in flat space which is the starting points of the analysis of nonlinear effects in kinetic theory and in magnetohydrodynamics (see, e.g. [23,24,25]).…”

“…The logic of the gradient expansion [17,18,19,20,21,22] can be combined with the tenets of the drift approximation [23,24,25] in the context of the ADM decomposition [15,16]. It will be shown hereunder that the resulting formalism [14] provides a more general description of the angular momentum transfer between the various species of the plasma. Consider therefore the standard ADM decomposition where the shift vectors are set to zero but the lapse function kept arbitrary, i.e.…”