Riccardo Benini

Classical and Quantum Features of the Mixmaster Singularity

This review paper is devoted to analyzing the main properties of the cosmological singularity associated with the homogeneous and inhomogeneous Mixmaster model. After the introduction of the main tools required to treat the cosmological issue, we review in detail the main results obtained over the last forty years on the Mixmaster topic. We first assess the classical picture of the homogeneous chaotic cosmologies and, after a presentation of the canonical method for the quantization, we develop the quantum Mixmaster behavior. Finally, we extend both the classical and the quantum features to the fully inhomogeneous case. Our survey analyzes the fundamental framework of the Mixmaster picture and completes it by accounting for recent and peculiar outstanding results.

Frame independence of the inhomogeneous mixmaster chaos via Misner-Chitre-like variables

We outline the covariant nature, with respect to the choice of a reference frame, of the chaos characterizing the generic cosmological solution near the initial singularity, i.e., the so-called inhomogeneous mixmaster model. Our analysis is based on a gauge independent Arnowitt-Deser-Misner reduction of the dynamics to the physical degrees of freedom. The resulting picture shows how the inhomogeneous mixmaster model is isomorphic point by point in space to a billiard on a Lobachevsky plane. Indeed, the existence of an asymptotic (energylike) constant of the motion allows one to construct the Jacobi metric associated with the geodesic flow and to calculate a nonzero Lyapunov exponent in each space point. The chaos covariance emerges from the independence of our scheme with respect to the form of the lapse function and the shift vector; the origin of this result relies on the dynamical decoupling of the space points which takes place near the singularity, due to the asymptotic approach of the potential term to infinite walls. At the ground of the obtained dynamical scheme is the choice of Misner-Chitre-like variables which allows one to fix the billiard potential walls.

Inhomogeneous quantum Mixmaster: from classical towards quantum mechanics

Starting from the Hamiltonian formulation for the inhomogeneous Mixmaster dynamics, we approach its quantum features through the link of the quasi-classical limit. We fix the proper operator-ordering which ensures that the WKB continuity equation overlaps the Liouville theorem as restricted to the configuration space. We describe the full quantum dynamics of the model in some detail, providing a characterization of the (discrete) spectrum with analytic expressions for the limit of high occupation number. One of the main achievements of our analysis relies on the description of the ground state morphology, showing how it is characterized by a non-vanishing zero-point energy associated with the universe anisotropy degrees of freedom.

Oscillatory regime in the multidimensional homogeneous cosmological models induced by a vector field

We show that in multidimensional gravity, vector fields completely determine the structure and properties of singularity. It turns out that in the presence of a vector field the oscillatory regime exists in all spatial dimensions and for all homogeneous models. By analysing the Hamiltonian equations we derive the Poincare return map associated with the Kasner indexes and fix the rules according to which the Kasner vectors rotate. In correspondence to a four-dimensional spacetime, the oscillatory regime here constructed overlaps the usual Belinski–Khalatnikov–Liftshitz one.

Crystalline structure of accretion disks: features of a global model.

In this paper, we develop the analysis of a two-dimensional magnetohydrodynamical configuration for an axially symmetric and rotating plasma (embedded in a dipolelike magnetic field), modeling the structure of a thin accretion disk around a compact astrophysical object. Our study investigates the global profile of the disk plasma, in order to fix the conditions for the existence of a crystalline morphology and ring sequence, as outlined by the local analysis pursued in Coppi [Phys. Plasmas 12, 7302 (2005)] and Coppi and Rousseau [Astrophys. J. 641, 458 (2006)]. In the linear regime, when the electromagnetic back-reaction of the plasma is small enough, we show the existence of an oscillating radial behavior for the flux surface function, which very closely resembles the one outlined in the local model, apart from a radial modulation of the amplitude. In the opposite limit, corresponding to a dominant back-reaction in the magnetic structure over the field of central object, we can recognize the existence of a ringlike decomposition of the disk, according to the same modulation of the magnetic flux surface, and a smoother radial decay of the disk density, with respect to the linear case. In this extreme nonlinear regime, the global model seems to predict a configuration very close to that of the local analysis, but here the thermostatic pressure, crucial for the equilibrium setting, is also radially modulated. Among the conditions requested for the validity of such a global model, the confinement of the radial coordinate within a given value sensitive to the disk temperature and to the mass of the central objet, stands; however, this condition corresponds to dealing with a thin disk configuration.

Stability of a self-gravitating homogeneous resistive plasma

Abstract In this paper, we analyze the stability of a homogeneous self-gravitating plasma, having a non-zero resistivity. This study provides a generalization of the Jeans paradigm for determining the critical scale above which gravitational collapse is allowed. We start by discussing the stability of an ideal self-gravitating plasma embedded in a constant magnetic field. We outline the existence of an anisotropic feature of the gravitational collapse. In fact, while in the plane orthogonal to the magnetic field the Jeans length is enhanced by the contribution of the magnetic pressure, outside this plane perturbations are governed by the usual Jeans criterion. The anisotropic collapse of a density contrast is sketched in detail, suggesting that the linear evolution provides anisotropic initial conditions for the non-linear stage, where this effect could be strongly enforced. The same problem is then faced in the presence of non-zero resistivity and the conditions for the gravitational collapse are correspondingly extended. The relevant feature emerging in this resistive scenario is the cancelation of the collapse anisotropy in weakly conducting plasmas. In this case, the instability of a self-gravitating resistive plasma is characterized by the standard isotropic Jeans length in any directions. The limit of very small resistivity coefficient is finally addressed, elucidating how reminiscence of the collapse anisotropy can be found in the different values of the perturbation frequency inside and outside the plane orthogonal to the magnetic field.

2-D MHD CONFIGURATIONS FOR ACCRETION DISKS AROUND MAGNETIZED STARS

The description of an accretion disk around a compact and strongly magnetized star within the framework of two-dimensional local MHD profiles is presented. After a brief review of the crystalline structure the disk acquires because of the coupling of the radial and vertical equilibria (as shown by B. Coppi in 2005), we face the question of making account for a non-zero accretion rate. In particular, the puzzle of a self-consistent configuration, for which the electron force balance holds in the azimuthal direction, is addressed. In the limit of a linearized perturbation theory, the accretion characteristics, as coming out of the two-dimensional MHD, are discussed when resistivity is involved in the model. Our analysis yields an oscillating radial accretion. The possible implications of non-linear resistivity features, in view of the growth of plasma instabilities are eventually presented within the ring-morphology preservation. Results and perspectives are given. In particular, we stress that one of us (B.C.) is investigating an accretion scenario involving very low relative values of the plasma resistivity. 2-D MHD Configurations for Accretion Disks around Magnetized Stars Axisymmetric thin disk model Problem accretion disk configuration around a central object Features • static configuration • compact: M ∼ 1.5− 2M , R ≈ RS • strongly magnetized: B ∼ 1012− 1014G Thin disk condition: z0(r)/r 1 Newton potential: χ(r, z) = − GM √ r2 + z2 Dipole Field approximation: ψ0 = μr2 (r2 + z2)3/2 Axisymmetric Field: ~B = − r ∂zψ~er + I r ~eφ + 1 r ∂rψ~ez Corotation theorem ω = ω(ψ) z0 is the half depth of the disk ψ = ψ0 + ψ1 2-D MHD Configurations for Accretion Disks around Magnetized Stars

Ring sequence decomposition of an accretion disk: The viscoresistive approach

We analyze a two-dimensional viscoresistive magnetohydrodynamical (MHD) model for a thin accretion disk which reconciles the crystalline structure outlined in Coppi B., Phys. Plasmas, 12 (2005) 7302 and Coppi B. et al., Astrophys. J., 641 (2006) 458, with real microscopic and macroscopic features of astrophysical accreting systems. In particular, we consider small dissipative effects (viscosity and resistivity, characterized by a magnetic Prandtl number of order unity), poloidal matter fluxes and a toroidal component of the magnetic field. These new ingredients allow us to set up the full equilibrium profile including the azimuthal component of the momentum conservation equation and the electron force balance relation. These two additional equations, which were identically satisfied in the original model, permit us to deal with non-zero radial and vertical matter fluxes, and the solution we construct for the global equilibrium system provides a full description of the radial and vertical dependence within the plasma disk. The main issue of our analysis is outlining a modulation of the matter distribution in the disk which corresponds to the formation of a ring-like sequence, here associated with a corresponding radial oscillation of the matter flux.

LINEAR TWO-DIMENSIONAL MHD OF ACCRETION DISKS: CRYSTALLINE STRUCTURE AND NERNST COEFFICIENT

We analyse the two-dimensional MHD configurations characterising the steady state of the accretion disk on a highly magnetised neutron star. The model we describe has a local character and represents the extension of the crystalline structure outlined in Ref. 1, dealing with a local model too, when a specific accretion rate is taken into account. We limit our attention to the linearised MHD formulation of the electromagnetic back-reaction characterising the equilibrium, by fixing the structure of the radial, vertical and azimuthal profiles. Since we deal with toroidal currents only, the consistency of the model is ensured by the presence of a small collisional effect, phenomenologically described by a nonzero constant Nernst coefficient (thermal power of the plasma). Such an effect provides a proper balance of the electron force equation via nonzero temperature gradients, related directly to the radial and vertical velocity components. We show that the obtained profile has the typical oscillating feature of the crystalline structure, reconciled with the presence of viscosity, associated to the differential rotation of the disk, and with a net accretion rate. In fact, we provide a direct relation between the electromagnetic reaction of the disk and the (no longer zero) increasing of its mass per unit time. The radial accretion component of the velocity results to be few orders of magnitude below the equatorial sound velocity. Its oscillating-like character does not allow a real matter in-fall to the central object (an effect to be searched into nonlinear MHD corrections), but it accounts for the outgoing of steady fluxes, favourable to the ring-like morphology of the disk.

Viscoresistive MHD configurations of plasma in accretion disks

We present a discussion of two-dimensional magneto-hydrodynamics (MHD) configurations, concerning the equilibria of accretion disks of a strongly magnetized astrophysical object. We set up a viscoresistive scenario which generalizes previous two-dimensional analyses by reconciling the ideal MHD coupling of the vertical and the radial equilibria within the disk with the standard mechanism of the angular momentum transport, relying on dissipative properties of the plasma configuration. The linear features of the considered model are analytically developed and the non-linear configuration problem is addressed, by fixing the entire disk profile at the same order of approximation. Indeed, the azimuthal and electron force balance equations are no longer automatically satisfied when poloidal currents and matter fluxes are included in the problem. These additional components of the equilibrium configuration induce a different morphology of the magnetic flux surface, with respect to the ideal and simply rotating disk.