EARLY COSMOLOGY AND

FUNDAMENTAL GENERAL RELATIVITY

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Contents

1 Topics 737

2 Participants 7382.1 ICRANet participants . . . . . . . . . . . . . . . . . . . . . . 7382.2 Past collaborations . . . . . . . . . . . . . . . . . . . . . . . . 7382.3 Ongoing Collaborations . . . . . . . . . . . . . . . . . . . . . . 7382.4 Graduate Students . . . . . . . . . . . . . . . . . . . . . . . . 738

3 Brief description of Early Cosmology 7393.1 Birth and Development of the Generic Cosmological Solution . 7393.2 Classical Mixmaster . . . . . . . . . . . . . . . . . . . . . . . . 739

3.2.1 Chaos covariance of the Mixmaster model . . . . . . . 7393.2.2 Chaos covariance of the generic cosmological solution . 7403.2.3 Inhomogeneous inflationary models . . . . . . . . . . . 7403.2.4 The Role of a Vector Field . . . . . . . . . . . . . . . . 740

3.3 Dissipative Cosmology . . . . . . . . . . . . . . . . . . . . . . 7413.4 Extended Theories of Gravity . . . . . . . . . . . . . . . . . . 7413.5 Interaction of neutrinos and primordial GW . . . . . . . . . . 7423.6 Coupling between Spin and Gravitational Waves . . . . . . . . 743

4 Brief Description of Fundamental General Relativity 7444.1 Perturbation Theory in Macroscopic Gravity . . . . . . . . . . 7444.2 Schouten’s Classification . . . . . . . . . . . . . . . . . . . . . 7454.3 Inhomogeneous spaces and Entropy . . . . . . . . . . . . . . . 7454.4 Polarization in GR . . . . . . . . . . . . . . . . . . . . . . . . 7464.5 Averaging Problem in Cosmology and Gravity . . . . . . . . . 7474.6 Astrophysical Topics . . . . . . . . . . . . . . . . . . . . . . . 747

5 Selected Publications 7485.1 Early Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . 7485.2 Fundamental General Relativity . . . . . . . . . . . . . . . . . 754

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6 APPENDICES 760

A Early Cosmology 761A.1 Birth and Development of the Generic Cosmological Solution . 762A.2 Classical Mixmaster . . . . . . . . . . . . . . . . . . . . . . . . 765

A.2.1 Chaos covariance of the Mixmaster model . . . . . . . 765A.2.2 Chaos covariance of the generic cosmological solution . 766A.2.3 Inhomogeneous inflationary models . . . . . . . . . . . 767A.2.4 The Role of a Vector Field . . . . . . . . . . . . . . . . 768

A.3 Dissipative Cosmology . . . . . . . . . . . . . . . . . . . . . . 770A.4 Extended Theories of Gravity . . . . . . . . . . . . . . . . . . 774A.5 Interaction of neutrinos and primordial GW . . . . . . . . . . 775A.6 Coupling between Spin and Gravitational Waves . . . . . . . . 779

B Fundamental General Relativity 781B.1 Perturbation Theory in Macroscopic Gravity . . . . . . . . . . 781B.2 Schouten’s Classification . . . . . . . . . . . . . . . . . . . . . 782B.3 Inhomogeneous Spaces and Entropy . . . . . . . . . . . . . . . 783B.4 Polarization in GR . . . . . . . . . . . . . . . . . . . . . . . . 784B.5 Averaging Problem in Cosmology and Gravity . . . . . . . . . 786B.6 Astrophysical Topics . . . . . . . . . . . . . . . . . . . . . . . 788

Bibliography 789

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1 Topics

Early Cosmology

– Birth and Development of theGeneric Cosmological Solution

– Classical Mixmaster

– Dissipative Cosmology

– Extended Theories of Gravity

– The interaction between relic neutrinosand primordial gravitational waves

– On the coupling between Spin andCosmological Gravitational Waves

Fundamental General Relativity

– Perturbation Theory in Macroscopic Gravity:On the Definition of Background

– On Schouten’s Classification of the non-Riemannian Geometrieswith an Asymmetric Metric

– Gravitational Polarization in General Relativity:Solution to Szekeres’ Model of Gravitational Quadrupole

– Averaging Problem in Cosmology and Macroscopic Gravity

– Astrophysical Topics

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2 Participants

2.1 ICRANet participants

- Vladimir Belinski

- Riccardo Benini

- Giovanni Montani

2.2 Past collaborations

- Nicola Nescatelli

2.3 Ongoing Collaborations

- Massimiliano Lattanzi (Oxford, UK)

- Alexander Kirillov (Nizhnii Novgorod, Ru)

- Roustam Zalaletdinov (Tashkent, Uz)

- Irene Milillo (Roma 2, IT and Portsmouth, UK)

- Giovanni Imponente (Centro Fermi, Roma)

- Nakia Carlevaro (Florence, IT)

2.4 Graduate Students

- Orchidea Maria Lecian

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3 Brief description of Early Cos-mology

3.1 Birth and Development of the Generic

Cosmological Solution

In section “Birth and Development of the Generic Cosmological Solution”we propose a historical review about the generic cosmological solution, fromits birth at the end of the 60’s, up to the most advanced and recent de-velopments. The review follows a chronological order discussing the mostimportant papers by Vladimir A. Belinski et al, ending with three papersby Giovanni Montani which is the leader of the group of ICRANet peopleworking now on this research line.

3.2 Classical Mixmaster

In the section “Classical Mixmaster” the most important results achieved onthe classical dynamics of homogeneous model of the type IX of the Bianchiclassification are reviewed together with its generalization to the more im-portant topic of the generic cosmological solution. The people involved inthis line of research are Riccardo Benini, Giovanni Imponente and GiovanniMontani

3.2.1 Chaos covariance of the Mixmaster model

In “Chaos covariance of the Mixmaster model” we face the study of the subtlequestion concerning the covariance chaoticity of the Bianchi type VIII and IXmodel. We introduce the Arnowitt-Deser-Misner formalism for General Rel-ativity, and adopt Misner-Chitre like variables. This way, the time evolutionis that of a ball on a billiard characterized by a constant negative curvature.

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The statistical properties (Kirillov and Montani, 2002) are described usingthe ensemble representation of the dynamics, while the problem of a correctdefinition of the Lyapunov exponent in such a relativistic system is resolvedadopting a generic time-variable (Imponente and Montani, 2001).

3.2.2 Chaos covariance of the generic cosmological so-lution

In “Chaos covariance of the generic cosmological solution” the question ofcovariance is extended to the more general frame of the generic cosmolog-ical solution (Benini and Montani, 2004). The problem is reformulated interms of the Hamilton approach to General Relativity, and Misner-Chitrelike variables are adopted. The problem of the dependence of the chaos onthe choice of the gauge is solved with a quite general change of coordinateson the space-time manifold, allowing us to solve the super Hamiltonian con-straint and the super-momentum one without fixing the forms of the lapsefunction and of the shift-vector. The analysis developed for the homogeneousMixmaster model is then extended to this more generic case.

3.2.3 Inhomogeneous inflationary models

In “Inhomogeneous inflationary models” we consider the inflationary scenarioas the possible way to interpolate the rich and variegate Kasner dynamics ofthe Very Early Universe (Imponente and Montani, 2004), in order to reachthe present state observable FLRW Universe, via a bridge solution. Hencewe show how it is possible to have a quasi-isotropic solution of the Einsteinequations in presence of the ultrarelativistic matter and a real self-interactingscalar field.

3.2.4 The Role of a Vector Field

In “The Role of a Vector Field” we study the effects of an Abelian vectorfield on the dynamics of a generic (n + 1)-dimensional homogeneous modelin the BKL scheme; the chaos is restored for any number of dimensions,and a BKL-like map, exhibiting a peculiar dependence on the dimensionnumber, is worked out (R Benini and Montani, 2005). Within the same spiritof the Mixmaster analysis, an unstable n-dimensional Kasner-like evolutionarises, nevertheless the potential term inhibits the solution to last up to thesingularity and induces the BKL-like transition to another epoch. There aretwo most interesting features of the resulting dynamics: the map exhibits

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a dimensional-dependence, and it reduces to the standard BKL one for thefour-dimensional case.

3.3 Dissipative Cosmology

In section “Dissipative Cosmology” we analyze the dynamics of the gravita-tional collapses (both in the Newtonian approach and in the pure relativisticlimit) including dissipative effects. The physical interest in dealing with dissi-pative dynamics is related to the thermodynamical properties of the system:both the analyzed regimes are characterized by a thermal history which cannot be regarded as settled down into the equilibrium. At sufficiently hightemperatures, micro-physical processes are no longer able to restore the ther-modynamical equilibrium and stages where the expansion and collapse in-duce non-equilibrium phenomena are generated. The average effect of havingsuch kind of micro-physics results into dissipative processes appropriately de-scribed by the presence of bulk viscosity, phenomenologically described by apower-law of the energy density (Carlevaro and Montani, 2007), (Carlevaroand Montani, 2005). With respect to dissipative dynamics, we also study theearly singularity proposed in the scheme of matter creation. The attentionis focused on those scenarios for which it is expected that the Universe hasbeen created as a vacuum fluctuation, thus the study of the particle cre-ation should be added for a complete analysis of its dynamics (Montani andNescatelli, 2007).

We can conclude that the Universe cannot be created like an isotropicsystem and only after a certain time it becomes close to our usual concep-tion of isotropy. In this respect, this analysis encourages the idea of an earlyUniverse as characterized by a certain degree of anisotropy and inhomogene-ity.

The people involved in this line of research are Giovanni Montani, NakiaCarlevaro and Nicola Nescatelli (past collaborator).

3.4 Extended Theories of Gravity

In section ”Extended Theories of Gravity”, we analyze the dynamical impli-cations of an exponential Lagrangian density for the gravitational field, asreferred to an isotropic FRW Universe (Lecian and Montani, 2007). Then, wediscuss the features of the generalized deSitter phase, predicted by the newFriedmann equation. The existence of a consistent deSitter solution arisesonly if the ratio between the vacuum energy density and that associated with

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the fundamental length of the theory acquires a tantalizing negative char-acter. This choice allows us to explain the present universe dark energy asa relic of the vacuum-energy cancellation due to the cosmological constantintrinsically contained in our scheme. The corresponding scalar-tensor de-scription of the model is addressed too, and the behavior of the scalar fieldis analyzed for both negative and positive values of the cosmological term.In the first case, the Friedmann equation is studied both in vacuum and inpresence of external matter, while, in the second case, the quantum regimeis approached in the framework of repulsive properties of the gravitationalinteraction, as described in recent issues in Loop Quantum Cosmology. Inparticular, in the vacuum case, we find a pure non-Einsteinian effect, ac-cording to which a negative cosmological constant provides an acceleratingdeSitter dynamics, in the region where the series expansion of the exponen-tial term does not hold.The people involved in this research line are Orchidea Maria Lecian andGiovanni Montani.

3.5 The interaction between relic neutrinos

and primordial gravitational waves

In section “The interaction between relic neutrinos and primordial gravita-tional waves” we study the generalization to other regions of the frequencydomain of the anisotropic stress of free streaming relic neutrinos that actsas an effective viscosity, absorbing gravitational waves in the extremely lowfrequency region, thus resulting in a damping of the B-modes of CMBR(Lattanzi and Montani, 2005). In particular, we have considered GWs thatenter the horizon before the electroweak phase transition (EWPT). This cor-responds to an observable frequency today ν0 & 10−5 Hz, i.e., to all wavespossibly detectable by interferometers.In order to study this issue, one has to solve the Boltzmann equation for thephase space density f of cosmological neutrinos. It is found that the inten-sity of GWs is reduced to ∼ 90% of its value in vacuum (see Fig A.1), itsexact value depending only on one physical parameter, namely the densityfraction of neutrinos. Neither the wave frequency nor the detail of neutrinointeraction affect the value of the absorbed intensity, resulting in an universalbehaviour in the frequency range considered.The importance of our results relies in the fact that the damping affects GWsin the frequency range where the LISA space interferometer and future, sec-ond generation ground-based interferometers can possibly detect a signal of

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cosmological origin.The people involved in this research line are Massimiliano Lattanzi and Gio-vanni Montani.

3.6 On the coupling between Spin and Cos-

mological Gravitational Waves

In section “On the coupling between Spin and Cosmological GravitationalWaves” we study the influence of Spin on the dynamic concerning thermalequilibrium of primordial universe (Lattanzi and Montani, 2007). In a homo-geneous and isotropic flat universe described by the FRW metric, we considera fluid of spinning particles. The equations of motion for such particles inthe frame of general relativity were derived by Papapetrou in 1951. In ourwork we consider the case of absence of precession, so that the generalizedmomentum is equal to the standard momentum.Considering the dynamic of thermal equilibrium for this case, due to thesymmetry proprieties of the metric tensor, the Boltzmann equation for theevolution of the distribution function of the spinning particles, remains inal-tered by the presence of the Spin. So we add a small tensorial perturbationhij in the metric looking for a coupling between Spin and cosmological gravi-tational waves. The resulting Boltzmann equation gives a first order variationof the distribution function that is proportional to the product between theSpin and the time derivative of hij. Even if the Spin alters some componentsof the anisotropic stress tensor, the final result is that these componentsare those that don’t couple with the evolution of hij and that there is notcoupling between Spin and cosmological gravitational waves.

The persons involved in this line of research are Massimiliano Lattanzi,Irene Milillo and Giovanni Montani.

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4 Brief Description of Funda-mental General Relativity

4.1 Perturbation Theory in Macroscopic Grav-

ity: On the Definition of Background

In section “Perturbation Theory in Macroscopic Gravity: On the Definitionof Background” the notion of background metric adopted in the perturbationtheory in general relativity is analysed and a new definition of backgroundis proposed. An existence theorem for a metric tensor which serves as thebackground metric for a specific scale has been proven (Montani, 1995). Itcan be shown that the average value of a tensor field remains invariant underaction of the averaging operator introduced in (Kirillov and Montani, 1997).Such an averaging procedure on a space-time manifold provides a naturalcriterium for a definition of background metric.A background metric that is invariant with respect to the class of averagingscan be introduced, and the following theorem considering the existence ofsuch a metric tensor for a specific scale is proven:Theorem.Given an averaging space-time procedure with an idempotent av-eraging kernel of the class of bounded and continuous functions on a space-time manifold M, there always exists a continuous and bounded backgroundmetric gαβ(x) for a characteristic scale d = VΣ where Σ is a compact 4-regionof M.

The people involved in this line of research are Roustam Zalaletdinov andGiovanni Montani.

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4.2 On Schouten’s Classification of the non-

Riemannian Geometries with an Asym-

metric Metric

In section ”On Schouten’s Classification of the non-Riemannian Geometrieswith an Asymmetric Metric”, after reviewing the Schouten classification ofnon-Riemannian geometries with an asymmetric metric tensor, we find theinverse of the “structure matrix”, which links the generalized connection withall the metric objects, in the linear approximation (Casanova et al., 1999).By adopting this approach for affine-connection geometries with an asymmet-ric metric, the structure and variety of such geometries can be investigatedin a fully-geometrical formalism without adopting any variational principle.The definition of autoparallel trajectories at different approximation ordershas been established. Because of the first-order approximation, the asym-metricity object and the antisymmetric part of the non-metricity tensor donot contribute to the determination of the autoparallel trajectory. In thiscase, the role of torsion and of the antisymmetric part of the metric tensorhas to be investigated according to the approximation order. As a physicalfield, if considered at zeroth order, torsion influences the dynamics by notallowing for a flat Minkowskian metric: in this as, the antisymmetric partof the metric tensor contributes to the determination of the solution only atfirst order. Contrastingly, if we require that torsion be of order 1, we find outthat the antisymmetric part of the metric tensor contribute only at secondorder(Casanova et al., 2007). The persons involved in this research line areSabrina Casanova, Orchidea Maria Lecian, Giovanni Montani, Remo Ruffiniand Roustam Zalaledtinov.

4.3 Approximate Symmetries, Inhomogeneous

Spaces and Gravitational Entropy

In section “Approximate Symmetries, Inhomogeneous Spaces and Gravita-tional Entropy” we treat the problem of finding an appropriate geomet-rical/physical index for measuring a degree of inhomogeneity for a givenspace-time manifold. Interrelations with the problem of understanding thegravitational/ informational entropy are also pointed out. We propose an ap-proach based on the notion of approximate symmetry (Zalaletdinov, 2000):with this respect a definition of a Killing-like symmetry is given and we pro-vide a classification theorem for all possible averaged space-times acquiring

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such symmetries upon averaging out a space-time with a homothetic Killingsymmetry.

The main idea of the Killing-like symmetry is to consider the most gen-eral form of deviation from the Killing equations. The expression for such adeviation covers the cases of semi-Killing, almost-Killing and almost symme-tries with additional equations. Also covered are standard generalizations ofKilling symmetry such as conformal and homothetic Killing vectors. The al-gebraic classification of the deviation gives an invariant way to introduce a setof scalar indexes measuring the degree of inhomogeneity of the space-timecompared with that isometries, or even weaker symmetry (e.g., conformalKilling).

The person involved in this line of research is Roustdam Zalaletdinov.

4.4 Gravitational Polarization in General Rel-

ativity: Solution to Szekeres’ Model of

Gravitational Quadrupole

In section “Gravitational Polarization in General Relativity: Solution toSzekeres’ Model of Gravitational Quadrupole”, we analyze a model for thestatic weak-field macroscopic medium. In this respect, the equation for themacroscopic gravitational potential is derived: such an equation is found tobe a biharmonic equation which is a non-trivial generalization of the Poissonequation of Newtonian gravity (Montani et al., 2000).

In the case of the strong gravitational polarization the equation essentiallyholds inside a macroscopic matter source: the scheme is equivalent to asystem of the Poisson equation and the nonhomogeneous modified Helmholtzequations. The general solution to this system is obtained by using Green’sfunction method and it does not exist a limit to Newtonian gravity. Incase of the insignificant gravitational quadrupole polarization, the equationfor macroscopic gravitational potential becomes the Poisson equation withthe matter density renormalized by the factor including the value of thequadrupole gravitational polarization of the source.

The persons involved in this line of research are Giovanni Montani, RemoRuffini and Roustdam Zalaletdinov.

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4.5 Averaging Problem in Cosmology and Macro-

scopic Gravity

In section “Averaging Problem in Cosmology and Macroscopic Gravity”, wediscuss the averaging problem using the approach of macroscopic gravity.We start modifying the averaged Einstein equations of macroscopic gravity(i.e., on cosmological scales) by the gravitational correlation tensor terms.Such a correlation tensor satisfies an additional set of structure and fieldequations. Then we focus on the cosmological solutions for spatially homo-geneous and isotropic macroscopic space-times. As a result, we find that,for a flat geometry, the gravitational correlation tensor terms have the formof a spatial curvature term which can be either negative or positive. Thisscheme exhibits a very non-trivial phenomenon from the point of view of thegeneral-relativistic cosmology: the macroscopic (averaged) cosmological evo-lution in a flat Universe is governed by the dynamical evolution equations foreither a closed or an open Universe depending on the sign of the macroscopicenergy-density with a dark spatial curvature term (Montani et al., 2002).

From the observational point of view, such a cosmological model givesa new paradigm to reconsider the standard cosmological interpretation andtreatment of the observational data. Indeed, such model has the Riemanniangeometry of a flat homogeneous, isotropic space-time and all measurementsand data are to be considered and designed for this geometry. The dynamicalinterpretation of the obtained data should be considered and treated for thecosmological evolution of either a closed or an open spatially homogeneous,isotropic Riemannian space-time.

The persons involved in this line of research are Giovanni Montani, RemoRuffini and Roustdam Zalaletdinov.

4.6 Astrophysical Topics

In section “Astrophysical Topics” we propose a review of different astrophys-ical topics by a brief discussion of very important papers by V. A. Belinskiet al.

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5 Selected Publications

5.1 Early Cosmology

1. G. Montani; “On the general behaviour of the universe near the cosmo-logical singularity”; Classical and Quantum Gravity, 12, 2505 (1995).

In this paper we discuss dynamical features characterizing the oscil-latory regime near a spacelike singularity in a generic inhomogeneouscosmological model, the effect of which leads to a profound modificationof the asymptotic behaviour toward that singularity, and creates condi-tions under which the system can evolve into a qualitatively turbulentregime. The well known pointwise ‘chaotic’ behaviour of the evolutionof the gravitational field toward such a singularity is shown to lead toa similarly complicated spatial structure on the spacelike slices whichapproach it.

2. A.A Kirillov, G. Montani; “Description of statistical properties of themixmaster universe”; Phys. Rev. D, 56, 6225 (1997).

Stochastic properties of the homogeneous Bianchi type-VIII and -IX(the mixmaster) models near the cosmological singularity are more dis-tinctive in the Hamiltonian formalism in the Misner-Chitre parametriza-tion. We show how the simplest analysis of the dynamical evolutionleads, in a natural way, to the construction of a stationary invariantmeasure distribution which provides the complete statistical descrip-tion of the stochastic behavior of these systems. We also establish thedifference between the statistical description in the framework of theMisner-Chitre approach and that one based on the BKL (BelinskiKha-latnikovLifshitz) map by means of an explicit reduction of the invariantmeasure in the continuous case to the measure on the map. It turns outthat the invariant measure in the continuous case contains an explicitinformation about durations of Kasner eras, while the measure in thecase of the BKL map does not.

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3. G. Imponente, G. Montani; “Covariance of the mixmaster chaoticity”;Phys. Rev. D, 63, 103501 (2001).

We analyze the dynamics of the mixmaster universe on the basis of astandard Arnowitt-Deser-Misner Hamiltonian approach showing howits asymptotic evolution to the cosmological singularity is isomorphicto a billiard ball on the Lobachevsky plane. The key result of ourstudy consists in the temporary gauge invariance of the billiard ballrepresentation, once provided the use of very general Misner-Chitre-like variables.

4. Kirillov, A. A. and Montani, G.; “Quasi-isotropization of the inhomo-geneous mixmaster universe induced by an inflationary process”; Phys.Rev. D, 66, 064010 (2002).

We derive a generic inhomogeneous “bridge” solution for a cosmologicalmodel in the presence of a real self-interacting scalar field. This solutionconnects a Kasner-like regime to an inflationary stage of evolution andtherefore provides a dynamical mechanism for the quasi-isotropizationof the universe. In the framework of a standard Arnowitt-Deser-MisnerHamiltonian formulation of the dynamics and by adopting Misner-Chitre-like variables, we integrate the Einstein-Hamilton-Jacobi equa-tion corresponding to a “generic” inhomogeneous cosmological modelwhose evolution is influenced by the coupling with a bosonic field, ex-pected to be responsible for a spontaneous symmetry breaking config-uration. The dependence of the detailed evolution of the universe onthe initial conditions is then appropriately characterized.

5. Imponente, Giovanni and Montani, Giovanni; “Inhomogeneous de Sit-ter solution with scalar field and perturbations spectrum”; Mod. Phys.Lett., A19, 1281 (2004).

We provide an inhomogeneous solution concerning the dynamics of areal self interacting scalar field minimally coupled to gravity in a regionof the configuration space where it performs a slow rolling on a plateauof its potential. During the inhomogeneous de Sitter phase the scalarfield dominant term is a function of the spatial coordinates only. Thissolution specialized nearby the FLRW model allows a classical originfor the inhomogeneous perturbations spectrum.

6. Riccardo Benini and Giovanni Montani; “Frame independence of the in-homogeneous mixmaster chaos via Misner-Chitre-like variables”; Phys-ical Review D, 70, 103527 (2004).

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We outline the covariant nature, with respect to the choice of a refer-ence frame, of the chaos characterizing the generic cosmological solutionnear the initial singularity, i.e., the so-called inhomogeneous mixmastermodel. 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 modelis isomorphic point by point in space to a billiard on a Lobachevskyplane. Indeed, the existence of an asymptotic (energylike) constant ofthe motion allows one to construct the Jacobi metric associated withthe geodesic flow and to calculate a nonzero Lyapunov exponent in eachspace point. The chaos covariance emerges from the independence ofour scheme with respect to the form of the lapse function and the shiftvector; the origin of this result relies on the dynamical decoupling of thespace points which takes place near the singularity, due to the asymp-totic approach of the potential term to infinite walls. At the groundof the obtained dynamical scheme is the choice of Misner-Chitre-likevariables which allows one to fix the billiard potential walls.

7. R Benini, A A Kirillov and Giovanni Montani; “Oscillatory regime inthe multidimensional homogeneous cosmological models induced by avector field”; Classical and Quantum Gravity, 22, 1483 (2005).

We show that in multidimensional gravity, vector fields completely de-termine the structure and properties of singularity. It turns out that inthe presence of a vector field the oscillatory regime exists in all spatialdimensions and for all homogeneous models. By analyzing the Hamil-tonian equations we derive the Poincare return map associated with theKasner indexes and fix the rules according to which the Kasner vectorsrotate. In correspondence to a four-dimensional spacetime, the oscilla-tory regime here constructed overlaps the usual Belinski-Khalatnikov-Liftshitz one.

8. Nakia Carlevaro and Giovanni Montani; “On the gravitational collapseof a gas cloud in the presence of bulk viscosity; Classical and QuantumGravity, 22, 4715 (2005).

We analyse the effects induced by the bulk (or second) viscosity on thedynamics associated with the extreme gravitational collapse. The aimof the work is to investigate whether the presence of viscous correctionsto the evolution of a collapsing gas cloud influences the top-down frag-mentation process. To this end, we generalize the approach presentedby Hunter (1962 Astrophys. J. 136 594) to include in the dynamics

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of the (uniform and spherically symmetric) cloud the negative pres-sure contribution associated with the bulk viscosity phenomenology.Within the framework of a Newtonian approach (whose range of valid-ity is outlined), we extend to the viscous case either the Lagrangian orthe Eulerian motion of the system addressed in Hunter (1962 Astro-phys. J. 136 594) and we treat the asymptotic evolution. We show howthe adiabatic-like behaviour of the gas is deeply influenced by viscouscorrection when its collapse reaches the extreme regime toward the sin-gularity. In fact, for sufficiently large viscous contributions, densitycontrasts associated with a given scale of the fragmentation processacquire, asymptotically, a vanishing behaviour which prevents the for-mation of sub-structures. Since in the non-dissipative case density con-trasts diverge (except for the purely adiabatic behaviour in which theyremain constant), we can conclude that in the adiabatic-like collapsethe top-down mechanism of structure formation is suppressed as soonas enough strong viscous effects are taken into account. Such a featureis not present in the isothermal-like collapse because the sub-structureformation is yet present and outlines the same behaviour as in the non-viscous case. We emphasize that in the adiabatic-like collapse the bulkviscosity is also responsible for the appearance of a threshold scale (de-pendent on the polytropic index) beyond which perturbations beginto increase; this issue, absent in the non-viscous case, is equivalent todealing with a Jeans length. A discussion of the physical character thatthe choice n = 5/6 takes place in the present case is provided.

9. Carlevaro, N. and Montani, G.; “Bulk Viscosity Effects on the EarlyUniverse Stability”; Modern Physics Letters A, 20, 1729 (2005).

We present a discussion of the effects induced by the bulk viscosityon the very early Universe stability. The matter filling the cosmologi-cal (isotropic and homogeneous) background is described by a viscousfluid having an ultrarelativistic equation of state and whose viscositycoefficient is related to the energy density via a power-law of the formζ = ζ0ρ

ν . The analytic expression of the density contrast (obtained forν = 1/2) shows that, for small values of the constant ζ0, its behavioris not significantly different from the non-viscous one derived by Lif-shitz. But as soon as ζ0 overcomes a critical value, the growth of thedensity contrast is suppressed forward in time by the viscosity and thestability of the Universe is favored in the expanding picture. On theother hand, in such a regime, the asymptotic approach to the initialsingularity (taken at t = 0) is deeply modified by the apparency ofsignificant viscosity in the primordial thermal bath, i.e. the isotropic

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and homogeneous Universe admits an unstable collapsing picture. Inour model this feature also regards scalar perturbations while in thenon-viscous case it appears only for tensor modes.

10. Lattanzi, M. and Montani, G.; “On the Interaction Between Ther-malized Neutrinos and Cosmological Gravitational Waves above theElectroweak Unification Scale”; Modern Physics Letters A, 20, 2607(2005).

We investigate the interaction between the cosmological relic neutri-nos, and primordial gravitational waves entering the horizon before theelectroweak phase transition, corresponding to observable frequenciestoday ν0 > 10−5Hz. We give an analytic formula for the tracelesstransverse part of the anisotropic stress tensor, due to weakly inter-acting neutrinos, and derive an integro-differential equation describingthe propagation of cosmological gravitational waves at these conditions.We find that this leads to a decrease of the wave intensity in the fre-quency region accessible to the LISA space interferometer, that is at thepresent the most promising way to obtain a direct detection of a cos-mological gravitational wave. The absorbed intensity does not dependneither on the perturbation wavelength, nor on the details of neutrinointeractions, and is affected only by the neutrino fraction fν . The trans-mitted intensity amounts to 88% for the standard value fν = 0.40523.An approximate formula for non-standard values of fν is given.

11. Imponente, G. and Montani, G.; “Bianchi IX chaoticity: BKL map andcontinuous flow”; Physica A, 338, 282 (2004).

We analyze the Bianchi IX dynamics (Mixmaster) in view of its stochas-tic properties; in the present paper we address either the original ap-proach due to Belinski, Khalatnikov and Lifshitz (BKL) as well as aHamiltonian one relying on the ArnowittDeserMisner (ADM) reduc-tion. We compare these two frameworks and show how the BKL mapis related to the geodesic flow associated with the ADM dynamics. Inparticular, the link existing between the anisotropy parameters and theKasner indices is outlined.

12. Imponente, G. and Montani, G.; “Covariant Feature of the MixmasterModel Invariant Measure”; International Journal of Modern Physics D,11, 1321 (2002).

We provide a Hamiltonian analysis of the Mixmaster Universe dynamicsshowing the covariant nature of its chaotic behavior with respect to anychoice of time variable. Asymptotically to the cosmological singularity,

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we construct the appropriate invariant measure for the system (whichrelies on the appearance of an “energy-like” constant of motion) in sucha way that its existence is independent of fixing the time gauge, i.e. thecorresponding lapse function. The key point in our analysis consists ofintroducing generic Misner-Chitr-like variables containing an arbitraryfunction, whose specification allows us to set up the same statisticalscheme in any time gauge.

13. Imponente, Giovanni and Montani, Giovanni; “Covariant MixmasterDynamics”.

We provide a Hamiltonian analysis of the Mixmaster Universe dynam-ics on the base of a standard Arnowitt-Deser-Misner Hamiltonian ap-proach, showing the covariant nature of its chaotic behaviour with re-spect to the choice of any time variable, from the point of view either ofthe dynamical systems theory, either of the statistical mechanics one.

14. Imponente, G. and Montani, G.; “On the Quasi-Isotropic InflationarySolution”; International Journal of Modern Physics D, 12, 1845 (2003).

In this paper we find a solution for a quasi-isotropic inflationary Uni-verse which allows to introduce in the problem a certain degree of in-homogeneity. We consider a model which generalizes the (flat) FLRWone by introducing a first order inhomogeneous term, whose dynamicsis induced by an effective cosmological constant. The 3-metric tensoris constituted by a dominant term, corresponding to an isotropic-likecomponent, while the amplitude of the first order one is controlled by a”small” function. In a Universe filled with ultra relativistic matter anda real self-interacting scalar field, we discuss the resulting dynamics, upto first order, when the scalar field performs a slow roll on a plateau of asymmetry breaking configuration and induces an effective cosmologicalconstant. We show how the spatial distribution of the ultra relativisticmatter and of the scalar field admits an arbitrary form but neverthe-less, due to the required inflationary e-folding, it cannot play a seriousdynamical role in tracing the process of structures formation (via theHarrison-Zeldovic spectrum). As a consequence, this paper reinforcesthe idea that the inflationary scenario is incompatible with a classicalorigin of the large scale structures.

15. Carlevaro, Nakia and Montani, Giovanni; “Study of the Quasi-isotropicSolution near the Cosmological Singularity in Presence of Bulk-Viscosity”;International Journal of Modern Physics D.

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We analyze the dynamical behavior of a quasi-isotropic Universe in thepresence of a cosmological fluid endowed with bulk viscosity. We ex-press the viscosity coefficient as a power-law of the fluid energy density:ζ = ζ0ǫ

s. Then we fix s = 1/2 as the only case in which viscosity playsa significant role in the singularity physics but does not dominate theUniverse dynamics (as requested by its microscopic perturbative ori-gin). The parameter ζ0 is left free to define the intensity of the viscouseffects. Following the spirit of the work by E.M. Lifshitz and I.M. Kha-latnikov on the quasi-isotropic solution, we analyze both Einstein andhydrodynamic equations up to first and second order in time. As aresult, we get a power-law solution existing only in correspondence toa restricted domain of ζ0.

16. Giovanni Montani; “Influence of particle creation on flat and nega-tive curved FLRW universes”; Classical and Quantum Gravity, 18, 193(2001).

We present a dynamical analysis of (classical) spatially flat and nega-tive curved Friedmann-Lameıtre-Robertson-Walker (FLRW) universesevolving (by assumption) close to the thermodynamic equilibrium inthe presence of a particle creation process. This analysis is describedby means of a realiable phenomenological approach, based on the ap-plication to the comoving volume (i.e. spatial volume of unit comovingcoordinates) of the theory for open thermodynamic systems. In par-ticular we show how, since the particle creation phenomenon induces anegative pressure term, then the choice of a well-grounded ansatz forthe time variation of the particle number, leads to a deep modifica-tion of the very early standard FLRW dynamics. More precisely, forthe considered FLRW models, we find (in addition to the limiting caseof their standard behaviour) solutions corresponding to an early uni-verse characterized respectively by an ‘eternal’ inflationary-like birthand a spatial curvature dominated singularity. In both these cases theso-called horizon problem finds a natural solution.

5.2 Fundamental General Relativity

1. G Montani, R Ruffini and R Zalaletdinov; “The gravitational polar-ization in general relativity: solution to Szekeres’ model of quadrupolepolarization”; Classical and Quantum Gravity, 20, 4195 (2003).

A model for the static weak-field macroscopic medium is analysed andthe equation for the macroscopic gravitational potential is derived.

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This is a biharmonic equation which is a non-trivial generalization ofthe Poisson equation of Newtonian gravity. In the case of strong grav-itational quadrupole polarization, it essentially holds inside a macro-scopic matter source. Outside the source the gravitational potentialfades away exponentially. The equation is equivalent to a system ofthe Poisson equation and the non-homogeneous modified Helmholtzequations. The general solution to this system is obtained by usingthe Green function method and it is not limited to Newtonian gravity.In the case of insignificant gravitational quadrupole polarization, theequation for macroscopic gravitational potential becomes the Poissonequation with the matter density renormalized by a factor including thevalue of the quadrupole gravitational polarization of the source. Thegeneral solution to this equation obtained by using the Green functionmethod is limited to Newtonian gravity.

2. Bisnovatyi-Kogan, G. S. and Lovelace, R. V. E. and Belinski, V. A.;“A Cosmic Battery Reconsidered”; ApJ, 580 (2002).

We revisit the problem of magnetic field generation in accretion flowsonto black holes owing to the excess radiation force on electrons. Thisexcess force may arise from the Poynting-Robertson effect. Instead of arecent claim of the generation of dynamically important magnetic fields,we establish the validity of earlier results from 1977 that show that onlysmall magnetic fields are generated. The radiative force causes themagnetic field to initially grow linearly with time. However, this lineargrowth holds for only a restricted time interval that is of the order of theaccretion time of the matter. The large magnetic fields recently foundresult from the fact that the linear growth is unrestricted. A model ofthe Poynting-Robertson magnetic field generation close to the horizonof a Schwarzschild black hole is solved exactly using general relativity,and the field is also found to be dynamically insignificant. These weakmagnetic fields may however be important as seed fields for dynamos.

3. Barkov, M. V. and Belinski, V. A. and Bisnovatyi-Kogan, G. S.; “Modelof ejection of matter from non-stationary dense stellar clusters andchaotic motion of gravitating shells”; arXiv:astro-ph/0107051.

It is shown that during the motion of two initially gravitationally boundspherical shells, consisting of point particles moving along ballistic tra-jectories, one of the shells may be expelled to infinity at subrelativisticspeed vexp ≤ 0.25c. The probelm is solved in Newtonian gravity. Mo-tion of two intersecting shells in the case when they do not runawayshows a chaotic behaviour. We hope that this toy and oversimplified

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model can nevertheless give a qualitative idea on the nature of themechanism of matter outbursts from dense stellar clusters.

4. Zalaletdinov, R. M.; “Averaging out the Einstein equations”; GeneralRelativity and Gravitation, 24, 1015 (1992).

A general scheme to average out an arbitrary 4-dimensional Rieman-nian space and to construct the geometry of the averaged space isproposed. It is shown that the averaged manifold has a metric and twoequi-affine symmetric connections. The geometry of the space is charac-terized by the tensors of Riemannian and non-Riemannian curvatures,an affine deformation tensor being the result of non-metricity of oneof the connections. To average out the differential Bianchi identities,correlation 2-form, 3-form and 4-form are introduced and the differ-ential relations on these correlations tensors are derived, the relationsbeing integrable on an arbitrary averaged manifold. Upon assuminga splitting rule for the average of the product including a covariantlyconstant tensor, an averaging out of the Einstein equations has beencarried out which brings additional terms with the correlation tensorsinto them. As shown by averaging out the contracted Bianchi identi-ties, the equations of motion for the averaged energy-momentum tensordo also include the geometric correction terms. Considering the gravi-tational induction tensor to be the Riemannian curvature tensor (thenthe non-Riemannian one is the macroscopic gravitational field), a the-orem that relates the algebraic structure of the averaged microscopicmetric with that of the induction tensor is proved. Due to the theoremthe same field operator as in the Einstein equations is manifestly ex-tracted from the averaged ones. Physical interpretation and applicationof the relations and equations obtained to treat macroscopic gravity arediscussed.

5. Mars, M. and Zalaletdinov, R. M.; “Space-time averages in macroscopicgravity and volume-preserving coordinates”; Journal of MathematicalPhysics, 38, 4741 (1997).

The definition of the covariant space-time averaging scheme for theobjects (tensors, geometric objects, etc.) on differentiable metric man-ifolds with a volume n-form, which has been proposed for the formula-tion of macroscopic gravity, is analyzed. An overview of the space-timeaveraging procedure in Minkowski space-time is given and comparisonbetween this averaging scheme and that adopted in macroscopic gravityis carried out throughout the paper. Some new results concerning thealgebraic structure of the averaging operator are precisely formulated

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and proved, the main one being that the averaging bilocal operator isidempotent iff it is factorized into a bilocal product of a matrix-valuedfunction on the manifold, taken at a point, by its inverse at anotherpoint. The previously proved existence theorems for the averaging andcoordination bilocal operators are revisited with more detailed proofsof related results. A number of new results concerning the structure ofthe volume-preserving averaging operators and the class of proper coor-dinate systems are given. It is shown, in particular, that such operatorsare defined on an arbitrary n-dimensional differentiable metric mani-fold with a volume n-form up to the freedom of (n1) arbitrary functionsof n arguments and 1 arbitrary function of (n1) arguments. All the re-sults given in this paper are also valid whenever appropriate for affineconnection manifolds including (pseudo)-Riemannian manifolds.

6. Montani, G. and Ruffini, R. and Zalaletdinov, R.; “Gravitating macro-scopic media in general relativity and macroscopic gravity”; NuovoCimento B, 115, 1343 (2000).

The problem of construction of a continuous (macroscopic) mattermodel for a given point-like (microscopic) matter distribution in generalrelativity is formulated. The existing approaches are briefly reviewedand a physical analogy with the similar problem in classical macro-scopic electrodynamics is pointed out. The procedure by Szekeres inthe linearized general relativity on Minkowski background to constructa tensor of gravitational quadruple polarization by applying Kaufman’smethod of molecular moments for derivation of the polarization tensorin macroscopic electrodynamics and to derive an averaged field oper-ator by utilizing an analogy between the linearized Bianchi identitiesand Maxwell equations, is analyzed. It is shown that the procedurehas some inconsistencies, in particular, it has only provided the termslinear in perturbations for the averaged field operator which do notcontribute into the dynamics of the averaged field, and the analogy be-tween electromagnetism and gravitation does break upon averaging. Amacroscopic gravity approach in the perturbation theory up to the sec-ond order on a particular background space-time taken to be a smoothweak gravitational field is applied to write down a system of macro-scopic field equations: Isaacson’s equations with a source incorporat-ing the quadruple gravitational polarization tensor, Isaacson’s energy-momentum tensor of gravitational waves and energy-momentum tensorof gravitational molecules and corresponding equations of motion. Asuitable set of material relations which relate all the tensors is proposed.

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7. Montani, G. and Ruffini, R. and Zalaletdinov, R.; “Modelling self-gravitating macroscopic media in general relativity: Solution to Szek-eres’ model of gravitational quadrupole”; Nuovo Cimento B, 118, 1109(2003).

A model for the static weak-field macroscopic medium is analyzed andthe equation for the macroscopic gravitational potential is derived.This is a biharmonic equation which is a non-trivial generalization ofthe Poisson equation of Newtonian gravity. In case of the strong grav-itational quadrupole polarization it essentially holds inside a macro-scopic matter source. Outside the source the gravitational potentialfades away exponentially. The equation is equivalent to a system of thePoisson equation and the nonhomogeneous modified Helmholtz equa-tions. The general solution to this system is obtained by using Green’sfunction method and it does not have a limit to Newtonian gravity.In case of the insignificant gravitational quadrupole polarization theequation for macroscopic gravitational potential becomes the Poissonequation with the matter density renormalized by the factor includingthe value of the quadrupole gravitational polarization of the source.The general solution to this equation obtained by using Green’s func-tion method has a limit to Newtonian gravity.

8. Montani, G. and Ruffini, R. and Zalaletdinov, R.; “Gravitating macro-scopic media in general relativity and macroscopic gravity”; NuovoCimento B, 115, 1343 (2002).

The problem of construction of a continuous (macroscopic) mattermodel for a given point-like (microscopic) matter distribution in generalrelativity is formulated. The existing approaches are briefly reviewedand a physical analogy with the similar problem in classical macro-scopic electrodynamics is pointed out. The procedure by Szekeres inthe linearized general relativity on Minkowski background to constructa tensor of gravitational quadruple polarization by applying Kaufman’smethod of molecular moments for derivation of the polarization tensorin macroscopic electrodynamics and to derive an averaged field oper-ator by utilizing an analogy between the linearized Bianchi identitiesand Maxwell equations, is analyzed. It is shown that the procedurehas some inconsistencies, in particular, it has only provided the termslinear in perturbations for the averaged field operator which do notcontribute into the dynamics of the averaged field, and the analogy be-tween electromagnetism and gravitation does break upon averaging. Amacroscopic gravity approach in the perturbation theory up to the sec-

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ond order on a particular background space-time taken to be a smoothweak gravitational field is applied to write down a system of macro-scopic field equations: Isaacson’s equations with a source incorporat-ing the quadruple gravitational polarization tensor, Isaacson’s energy-momentum tensor of gravitational waves and energy-momentum tensorof gravitational molecules and corresponding equations of motion. Asuitable set of material relations which relate all the tensors is proposed.

9. Zalaletdinov, R. M.; “Towards a theory of macroscopic gravity”; Gen-eral Relativity and Gravitation, 25, 673 (1993).

By averaging out Cartan’s structure equations for a four-dimensionalRiemannian space over space regions, the structure equations for theaveraged space have been derived with the procedure being valid on anarbitrary Riemannian space. The averaged space is characterized by ametric, Riemannian and non-Rimannian curvature 2-forms, and corre-lation 2-, 3- and 4-forms, an affine deformation 1-form being due to thenon-metricity of one of two connection 1-forms. Using the procedurefor the space-time averaging of the Einstein equations produces the av-eraged ones with the terms of geometric correction by the correlationtensors. The equations of motion for averaged energy momentum, ob-tained by averaging out the contracted Bianchi identities, also includesuch terms. Considering the gravitational induction tensor to be theRiemannian curvature tensor (the non-Riemannian one is then the fieldtensor), a theorem is proved which relates the algebraic structure of theaveraged microscopic metric to that of the induction tensor. It is shownthat the averaged Einstein equations can be put in the form of theEinstein equations with the conserved macroscopic energy-momentumtensor of a definite structure including the correlation functions. Byusing the high-frequency approximation of Isaacson with second-ordercorrection to the microscopic metric, the self-consistency and compati-bility of the equations and relations obtained are shown. Macrovacuumturns out to be Ricci non-flat, the macrovacuum source being definedin terms of the correlation functions. In the high-frequency limit theequations are shown to become Isaacson’s ones with the macrovauumsource becoming Isaacson’s stress tensor for gravitational waves.

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6 APPENDICES

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A Early Cosmology

The dynamics of the actual Universe is appropriately described by a Friedmann-Robertson-Walker model for a large interval of its evolution. In particular,the isotropic solution of the Einstein equations is a valid framework to repro-duce cosmological features from the nucleosynthesis of light elements (i.e.,O(10−2 − 10−3s)) up to the recombination age O(105 years). In this widecosmological window, theoretical predictions are appropriately confirmed byobservations.

However, the very early stages of the Universe evolution are expected tobe described by more general dynamical scenarios than the high symmetricone, associated with a Friedmann-Robertson-Walker background. In fact,the isotropic dynamics is backward in time unstable with respect to tensor-like perturbations, increasing as the inverse power of the cosmic scale factor.Furthermore, near enough to the cosmological singularity, quantum effectsto the Einsteinian dynamics are expected to be important. In a quantumregime, the request of a symmetric space-time can take place at most on acausal portion of the Universe and, therefore, the quantum fluctuations mustproduce a large scale asymmetric picture.

Finally, we have to consider that at pre-inflationary scales, say at tem-peratures greater than O(1016GeV ), the particle mean free-path exceeds thecausal horizon. Since that time up to the singularity, the Universe cannot bedescribed by thermal equilibrium and significant deviations from the stan-dard cosmological picture must arise.

All these considerations provide convincing motivations for studying verygeneral cosmological models, as well as dissipative effects concerning thecosmological fluid. As far as this point of view is addressed, the singularityis approached by non-isotropic behaviors and, in the generic case, a chaoticdynamics arises. On the other hand, viscous effects and matter creationcontributions induce very different dynamical regimes to the singular pointeven if the isotropic assumption is retained. In this respect, the Universebecomes backward unstable under scalar perturbations as far as dissipativeeffects are taken into account.

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In this line of research, many different aspects of these generalized cos-mological models are faced with particular attention to chaotic cosmology inthe homogeneous sector as well as in the inhomogeneous one. The resultsof the last ten years of investigations lead to very significant results towardsthe accomplishment of a detailed picture for the chaos structure. Further-more, in the last years, our attention was concentrated on analyzing dynam-ical implications coming out from canonical and generalized approaches tothe quantization of the geometrodynamics. This sector of the CGM groupactivity involves more than five people both from undergraduate level upto post-doctoral expertises in the field. Below, we summarize some of themost remarkable results obtained in these early-cosmology investigations, bystressing some fresh and timely results in the most advanced areas.

A.1 Birth and Development of the Generic

Cosmological Solution

In 1968-1975 the question of existence of the cosmological singularity in thegeneral solution of Einstein equations have been solved and the theory ofthe chaotic oscillatory behaviour of gravitational field and matter in vicinityof this singularity have been created by V. Belinski, I. Khalatnikov and E.Lifschitz (BKL).

This problem appeared around 85 years ago when the first exactly solvablecosmological models revealed the presence of the Big Bang singularity. Sincethat time the fundamental question has arisen whether this phenomenon isdue to the special simplifying assumptions underlying the exactly solvablemodels or if a singularity is a general property of the Einstein equations.The BKL showed that a singularity is an unavoidable property of the generalcosmological solution of the gravitational equations and not a consequenceof the special symmetric structure of exact models. Most importantly theywere able to find the analytical structure of this generic solution and showedthat its behaviour is of a complex oscillatory character of chaotic type.

The detailed theory of the oscillatory cosmological regime can be foundin the following papers:

- V. Belinski and I. Khalatnikov On the Nature of the singularities in the

General Solution of the Gravitational Equations,Sov. Phys. JETP, 29,911, (1969).This was the first investigation of the homogeneous cosmological modelof Bianchi IX type and it was the first discovery of the new type ofcosmological singularity - oscillating cosmological regime. In the subse-

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quent literature this model has been given the second name ”MixmasterUniverse”.

- V. Belinski and I. Khalatnikov General Solution of the Gravitational

Equations with a physical Singularity,Sov. Phys. JETP,30, 1174, (1970).In this paper was made the first statement that the oscillating cosmo-logical regime of Bianchi IX model is the paradigm of the behaviour ofthe General cosmological Solution near singularity and that the GeneralSolution with singularity really exists. Paper investigated a number ofanalytical properties of this Solution.

- V. Belinski, I. Khalatnikov and E. Lifshitz Oscillatory Approach to a Sin-

gular Point in the Relativistic Cosmology,Adv. in Phys., 19, 525, (1970).The properties of the General Cosmological Solution near singularitywas described. It was constructed the method for qualitative descrip-tion of the oscillating cosmological evolution in terms of successivelychanging ”Kasner epochs”. It was described the statistical propertiesof the chaotic oscillating regime in ultra asymptotic region near singu-larity.

- V. Belinski and I. Khalatnikov Effect of scalar and Vector Fields on the

nature of the cosmological singularity, Sov. Phys. JETP, 36, 591, (1973).The effect of scalar and vector fields on the character of the cosmolog-ical singularity is investigated. The fields may either be gravitational(in the sense of the Brans-Dicke ideas) or extraneous physical fieldswhich are sources of an ordinary gravitational field. It is shown that inthe presence of only a scalar field the gravitational equations possess amonotonic power-law asymptotic for the general solution near the sin-gular point in place of an oscillating form. However, if a vector field isincluded on the basis of five-dimension geometry concepts, the generalsolution becomes oscillatory again.

- V. Belinski and I. Khalatnikov, On the influence of matter and Physical

Fields upon the Nature of Cosmological Singularities,Soviet Physics Re-views, Harwood Acad. Publ., 3, 555, (1981).It was investigated the influence of Yang-Mills fields and perfect liquidmatter with unusual equations of state on cosmological singularities. Itwas shown that Yang-Mills fields do not change qualitatively the oscil-lating regime near singular point. The same is correct for perfect liquidin a wide range of equations of state with only one exception, namely,the stiff matter equation of state. In this case the asymptotic near sin-gularity changes to the smooth Kasner-like (similar to the scalar field

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case) behaviour. For this case we constructed the general CosmologicalSolution near the singularity in analytical form.

- V. Belinski, I. Khalatnikov and E. Lifshitz, A general solution of the Ein-

stein equations with a time singularity, Adv. in Phys., 31, 639, (1982).This paper is a concluding review exposition of the investigations aimedat the construction of a general cosmological solution of the Einsteinequations with a singularity in time (including the description of thenew phenomenon of the rotations of Kasner axes). Thus it is a directcontinuation of the previous (1970) paper by the authors in this Jour-nal. A detailed description is given of the analysis which leads to theconstruction of such a solution, and of its properties.

These results have a fundamental significance not only for Cosmology butalso for evolution of collapsing matter forming a black hole. The last stageof collapsing matter in general will follow the BKL regime.

The BKL analysis provides the description of intrinsic properties of theEinstein equations which can be relevant also in the quantum context. Re-cently (T.Damour, M.Henneaux, H. Nicolai et al., 2000-2007) it has beenshown that the BKL regime is inherent not only to General Relativity butalso to more general physical theories, such as the string models. This discov-ery has created an important field of research which has been continuouslyactive. During the last three decades the BKL theory of the cosmological sin-gularity has attracted the active attention of the scientific community. Thedevelopments of this theory made by many researches between 1980 and 2007(among them Ya. Sinai, J. Barrow, B.K. Berger, A.A. Kirillov, V. Moncrief,G. Montani, J. Wainwright, D. Garfinkle, H. Ringstrom, L. Andersson, A.Rendall, C. Uggla, M. Henneaux, T. Damour, H. Nicolai) was dedicated tothe foundation of its rigorous statistical description, to the numerical con-firmation of its principal statements, to the quest of its more deep hiddenmathematical structure and to its extension to the multidimensional spaceand to the string theories. The last reviews are:

- J.M.Heinzle, C.Uggla, N.Rohr, The cosmological billiard attractor,gr-qc/0702141

- L.Andersson On the relation between mathematical and numerical rela-

tivity,Class. Quant. Grav. 23, S307 (2006), gr-qc/0607065

- A.Rendall, The nature of spacetime singularities,100 Years of Relativity,Space-Time Structure: Einstein and Beyond, A. Ashtekar (ed.); gr-qc/0503112.

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- T. Damour, M. Henneaux, H. Nicolai, Cosmological billiards, Class.Quant. Grav. 20, R145 (2003), hep-th/0212256

- T. Damour and S. de Buyl, Describing general cosmological singularities

in Iwasawa variables, gr-qc/0710.5692.

Early Cosmology. ICRANet Activity In the ICRANET group theresearch on the oscillatory regime near the cosmological singularity has beenone of the principal research field starting from 1992 . The most importantpapers made in this group are

- G.Montani On the general behaviour of the Universe near the cosmological

singularity, Class. Quant. Grav. 12, 2505 (1995)

- G.P. Imponente and G.Montani, On the Covariance of the Mixmaster

Chaoticity, Phys. Rev. D63, 103501 (2001)

- R. Benini and G. Montani,Frame independence of the inhomogeneous

mixmaster chaos via Misner-Chitre-like variables, Phys. Rev. D70, 103527-1 (2004).

The research on the properties of oscillatory behaviour of the gravitationalfield and matter near the cosmological singularity is still in progress in thisgroup, the main topics are: the multidimensional generalization, influenceof viscosity, influence of quantum effects. The group is working under theleadership of G. Montani.

A.2 Classical Mixmaster

A.2.1 Chaos covariance of the Mixmaster model

The study of the subtle question concerning the covariance chaoticity ofthe Bianchi type VIII and IX model, led to important issues favourable tothe independence of the “chaos” with respect to the choice of the temporalgauge in terms of positive Lyapunov numbers. Such analysis found its basiseither on the standard approach using the Jacobi metric (a scheme allowedby the existence of an energy-like constant of motion), either by a StatisticalMechanics approach in which the Mixmaster evolution is represented as abilliard on a Lobatchevski plane and therefore admitting a Microcanonicalensemble associated to such an energy-like constant.

A detailed discussion was pursued in view of clarifying the peculiarity ex-isting to characterize chaos in General Relativity; in particular, we critically

765

analyzed the predictability allowed by the fractal basin boundary approachin qualifying the nature of the Mixmaster dynamics, getting the numeri-cal approximations limits when treating iterations of irrational numbers andoverall on the potential methods commonly adopted in the dynamical sys-tems approach. The description of chaos finds its ambiguity also in terms ofgeodesic deviation when the background metric is a pseudo-Riemannian one;a correct characterization of the Lyapunov exponents required a projectionof the connecting vector over a Fermi basis.

We develop the Hamiltonian formulation of the cosmological problemshowing how it can be reduced to the dynamics of a billiard-ball (IMPO-NENTE and MONTANI, 2005).In particular an original reformulation of the Bianchi type IX dynamics isstudied by using a set of Misner–Chitre-like variables with a generic func-tion of one coordinate, thus overcoming the ambiguities of many assessmentsfound in the literature, due to the dependence of the choice of the timeparameter (Imponente and Montani, 2001).

Our reformulation is not affected by such a possibility and permits to dis-cuss the dynamics via a standard Arnowitt-Deser-Misner (ADM) approachin the reduced phase space. The Jacobi metric obtained induces the deriva-tion of an invariant formulation of the Liouville measure (Imponente andMontani, 2002) within the microcanonical ensemble framework (Imponenteand Montani, 2005b).This new approach permits to derive, within the potential approximation, ananalytic expression for the Lyapunov exponents (Imponente and Montani,2001), independently of the choice of the temporal gauge and a discussionabout a correct formulation of the same problem in General Relativity (Im-ponente and Montani, 2004).

A.2.2 Chaos covariance of the generic cosmological so-lution

In the homogeneous Mixmaster model, it was shown that chaos is a propertyof the Einsteinian dynamics because it is not induced by particular choicesof the temporal variables as previously argued in literature. This result wasextended to the more general case of the generic cosmological solution (Beniniand Montani, 2004).

The complex dynamics of the generic cosmological solution was analyzedby means of the Hamiltonian formulation of General Relativity; in this frame-work, the gravitational degrees of freedom are twelve, the six components of

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the three dimensional metric tensor hij and their conjugate momenta Πij.Among these variables, only four are physical, while the remaining concernwith the diffeomorphism invariance of the theory. The ”embedding variables“can be eliminated solving the four constraints, the super-Hamiltonian andthe supermomentum ones, that emerge when the Legendre transformation isperformed to pass from the Lagrangian to the Hamiltonian framework.

The analysis of the Ricci scalar (that in vacuum behaves as a poten-tial term that couples the space points) showed how the time evolution ofthe space points dynamically decouple from each other while reaching theBig Bang (in accordance with the previous results of Belisnkii et al. in thefield equations framework); in each space point, a Mixmaster like evolutiontakes place. Here, the physical meaning of ”space point“ is that of a cos-mological horizon, and the obtained decoupling corresponds to deal with”super-horizon“ sized perturbation. This fact is also known as long wavelength approximation, that mathematically corresponds to the result thatthe spatial gradients in the Ricci scalar grow slower in time than the timederivatives.

We succeeded in applying the ADM technique to the embedding variableswithout choosing any particular form for the lapse function N or for the shiftvector N i; this was done with a particular but quite general choice of thecoordinates for the space-time, and using an infinite potential well structurefor the Ricci scalar. The resulting dynamics consists of the sum of infiniteMixmaster model, and the previous discussion on the covariance of the chaosin the homogeneous case was extended to the generic cosmological solution.

A.2.3 Inhomogeneous inflationary models

The investigation performed about a quasi-isotropic inflationary solution(Imponente and Montani, 2005c) showed how there is no chance for classicalinhomogeneous perturbations to survive after the de Sitter phase, stronglysupporting the idea that only quantum fluctuations of the scalar field canprovide a satisfactory explanation for the observed spectrum of inhomoge-neous perturbations, when requiring the matter to dominate the first orderof the solution (Imponente and Montani, 2005a).

We consider the inflationary scenario as the possible way to interpolatethe rich and variegate Kasner dynamics of the Very Early Universe discussedso far with an inflationary scenario (Imponente and Montani, 2004), in orderto reach the present state observable FLRW Universe, via a bridge solution.The Einstein-Hamilton-Jacobi equation is solved in presence of a real self-interacting scalar field.

Hence we show how it is possible to have a quasi-isotropic solution of

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the Einstein equations in presence of the ultrarelativistic matter and a realself-interacting scalar field. In this case, the spatial distributions of bothadmit an arbitrary form but such a small inhomogeneity is incompatiblewith structures formation of classical origin (Imponente and Montani, 2003).Furthermore, a generic inhomogeneous solution has been provided concerningthe dynamics of a real self interacting scalar field minimally coupled to gravityin a region of the configuration space where it performs a slow rolling on aplateau of its potential. During the generic inhomogeneous deSitter phasethe scalar field which dominates zero- and first-order of approximation is afunction of the spatial coordinates only. This solution specialized nearbythe Friedmann-Lemaitre-Robertson-Walker (FLRW) model allows a classicalorigin for the inhomogeneous perturbation spectrum.

A.2.4 The Role of a Vector Field

The effects of an Abelian vector field on the dynamics of a generic (n + 1)-dimensional homogeneous model has been investigated in the BKL scheme;the chaos is restored for any number of dimensions, and a BKL-like map,exhibiting a peculiar dependence on the dimension number, is worked out(R Benini and Montani, 2005). These results have also been inserted in moregeneral treatment by Damour and Hennaux.

A generic (n + 1)-dimensional space-time coupled to an Abelian vectorfield Aµ = (ϕ,Aα), with α = (1, 2, . . . , n) in the ADM framework is describedby the action

S =

∫dnxdt

(Παβ ∂

∂thαβ + Πα ∂

∂tAα + ϕDαΠα − NH − NαHα

), (A.1)

where

H =1√h

[Πα

βΠβα − 1

n − 1(Πα

α)2 +1

2hαβΠαΠβ + h

(1

4FαβFαβ − (N)R

)],

(A.2a)

Hα = −∇βΠβα + ΠβFαβ , (A.2b)

denote the super-Hamiltonian and the super-momentum respectively, whileFαβ is the spatial electromagnetic tensor, and the relation Dα ≡ ∂α + Aα

holds. Moreover, Πα and Παβ are the conjugate momenta to the electromag-netic field and to the n-metric, respectively, which result to be a vector anda tensorial density of weight 1/2, since their explicit expressions contain thesquare root of the spatial metric determinant. The variation with respect tothe lapse function N yields the super-Hamiltonian constraint H = 0, while

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with respect to ϕ it provides the constraint ∂αΠα = 0.We will deal with a source-less Abelian vector field and in this case one canconsider the transverse (or Lorentz) components for Aα and Πα only. There-fore, we choose the gauge conditions ϕ = 0 and DαΠα = 0, enough to preventthe longitudinal parts of the vector field from taking part to the action.It is worth noting how, in the general case, i.e. either in presence of thesources, or in the case of non-Abelian vector fields, this simplification can nolonger take place in such explicit form and the terms ϕ(∂α + Aα)Πα must beconsidered in the action principle.

A BKL-like analysis can be developed R Benini and Montani (2005) aswell as done previously, following some steps: after introducing a set of Kas-ner vectors ~la and the Kasner-like expanding factors exp(qa), the dynamics is

dominated by a potential of the form∑

eqaλ2a, where λa are the projection of

the momenta of the Abelian field along the Kasner vectors. With the samespirit of the Mixmaster analysis, an unstable n-dimensional Kasner-like evo-lution arises, nevertheless the potential term inhibits the solution to last upto the singularity and, as usual, induces the BKL-like transition to anotherepoch. Given the relation exp(qa) = tpa , the map that links two consecutiveepochs is

p′1 =−p1

1 + 2n−2

p1

, p′a =pa + 2

n−2p1

1 + 2n−2

p1

, (A.3a)

λ′1 = λ1 , λ′

a = λa

(1 − 2

(n − 1) p1

(n − 2) pa + np1

). (A.3b)

An interesting new feature, resembling that of the inhomogeneous Mixmaster(as we will discuss later), is the rotation of the Kasner vectors,

~ℓ′a = ~ℓa + σa~ℓ1 , (A.4a)

σa =λ′

a − λa

λ1

= −2(n − 1) p1

(n − 2) pa + np1

λa

λ1

. (A.4b)

which completes our dynamical scheme.The homogeneous Universe in this case approaches the initial singularity

described by a metric tensor with oscillating scale factors and rotating Kas-ner vectors. Passing from one Kasner epoch to another, the negative Kasnerindex p1 is exchanged between different directions (for istance ~ℓ1 and ~ℓ2) and,at the same time, these directions rotate in the space according to the rule(A.4b). The presence of a vector field is crucial because, independently ofthe considered model, it induces a dynamically closed domain on the config-uration space.

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In correspondence to these oscillations of the scale factors, the Kasner vec-tors ~ℓa rotate and the quantities σa remain constant during a Kasner epochto lowest order in qa; thus, the vanishing of the determinant h approachingthe singularity does not significantly affect the rotation law (A.4b).There are two most interesting features of the resulting dynamics: the mapexhibits a dimensional-dependence, and it reduces to the standard BKL onefor the four-dimensional case.

A.3 Dissipative Cosmology

With respect to this line of research, peculiar topics concerning the dynamicsof the gravitational collapses are developed both in the Newtonian approachand in the pure relativistic limit, including dissipative effects mainly reas-sumed by the presence of viscosity. The physical interest in dealing with dis-sipative dynamics is related to thermodynamical properties of the analyzedsystem. In fact, both the extreme regime of a gravitational collapse and thevery early stages of the Universe evolution are characterized by a thermalhistory which can not be regarded as settled down into the equilibrium. Atsufficiently high temperatures, the cross sections of the micro-physical pro-cesses are no longer able to restore the thermodynamical equilibrium. Thus,stages where the expansion and collapse induce non-equilibrium phenomenaare generated. The average effect of having such kind of micro-physics resultsinto dissipative processes appropriately described by the presence of bulk vis-cosity ζ, phenomenologically described as a function of the energy density ρin terms of a power-law as

ζ = ζ0ρs , ζ0, s = const . (A.5)

In this approach, this kind of viscosity affects the form of the energy-momentumtensor with a corrective term:

Tµν = (p + ρ)uµuν − p gµν , p = p − ζ uρ; ρ , (A.6)

where p denotes the usual thermostatic pressure.The analysis is focused on three main models:

(i) Perturbed FRW-UniverseWe present a discussion of the effects induced by the bulk viscosity on thevery early Universe stability Carlevaro and Montani (2005), Carlevaro andMontani (2007). The matter filling the cosmological isotropic and homoge-neous background is described by a viscous fluid having an ultra-relativisticequation of state (i.e., p = ρ/3). The analytic expression of the density con-trast, obtained for s = 1/2 (i.e., in order to deal with the maximum effect

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that bulk viscosity can have without dominating the dynamics), shows twodifferent dynamical regimes characterized by intensity of the viscous effectsrelated to the critical value

ζ∗0 = 2

9√

3. (A.7)

In the case 0 6 ζ0 < ζ∗0 , perturbations increase forward in time. This

behavior corresponds qualitatively to the same picture of the non-viscousUniverse (obtained setting ζ0 = 0) in which the expansion can not imply thegravitational instability. In the case ζ∗

0 < ζ0, the density contrast is sup-pressed since it behaves like negative powers of the time variable. When thedensity contrast results to be increasing, the presence of viscosity inducesa damping of the perturbation evolution in the direction of the expandingUniverse. In this regime, density fluctuations decrease forward in time, butthe most interesting result is the instability that the isotropic and homoge-neous Universe acquires in the direction of the collapse toward the Big-Bang:the density contrast diverges approaching the cosmological singularity, thusscalar perturbations destroy asymptotically the primordial Universe symme-try.

The dynamical implication of these issues is that an isotropic and ho-mogeneous stage of the Universe can not be generated, from generic initialconditions, as far as the viscosity becomes smaller than the critical value,i.e., ζ0 < ζ∗

0 .(ii) Quasi-isotropic model

In 1963, E.M. Lifshitz and I.M. Khalatnikov first proposed this model whichis based on the idea that, as a function of time, the 3 -metric is expandablein powers of t, i.e., a Taylor expansion of the spatial metric is addressed. InCarlevaro and Montani (2007) we propose a generalization of the line elementin order to include dissipative effects:

γαβ = tx aαβ + ty bαβ , γαβ = t−x aαβ − ty−2x bαβ , (A.8)

where x > 0 (constraint for the space contraction) and y > x (consistenceof the perturbation scheme). In this approach, the pure Friedmann modelbecomes a particular case of a larger class of solutions existing only for spacefilled with matter. In the analysis, the viscous exponent is fixed s = 1/2as the only case in which viscosity plays a significant role in the singularityphysics. The parameter ζ0 is left free to define the intensity of the viscouseffects.

Following the spirit of the LK’s work, both Einstein and hydrodynamicequations, up to first- and second-order in time, are analyzed. A power-lawsolution exists only in correspondence to a restricted domain of ζ0. In fact,

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the consistence of the perturbation scheme, i.e., y > x, yields the validityconstraint

ζ0 < 3 ζ∗0 , (A.9)

in agreement with the results obtained for the pure isotropic model.(iii) Extreme gravitational collapse of a gas cloud

Aim of this analysis Carlevaro and Montani (2007), Carlevaro and Montani(2005) is to investigate whether the presence of viscous corrections to theevolution of a collapsing gas cloud can influence the top-down fragmenta-tion process. To this end, a generalization of the approach firstly presentedby C. Hunter is developed in order to include the negative pressure contri-bution associated to the bulk viscosity phenomenology in the dynamics ofthe (uniform and spherically symmetric) cloud. Within the framework ofa Newtonian approach, both the Lagrangian, and the Eulerian equation ofmotion of the system are extended to the viscous case. We construct suchan extension requiring that the asymptotic dynamics of the collapsing cloudis not qualitatively affected by the presence of viscosity: in this respect, wecan assume the viscous exponent as s = 5/6.

The adiabatic-like behavior of the gas (i.e., when the politropic index γtakes values 4/3 < γ 6 5/3) is deeply influenced by viscous corrections whenits collapse reaches the extreme regime towards the singularity. In fact, forsufficiently large viscous contributions, density contrasts acquire, asymptot-ically, a vanishing behavior that prevents the formation of sub-structures.Since, in the non-dissipative case, density contrasts diverge (except for thepurely adiabatic behavior γ = 5/3 in which they remain constant), in theadiabatic-like collapse the top-down mechanism of the structure formation issuppressed as soon as enough strong viscous effects are taken into account.Such a feature is not present in the isothermal-like case (i.e., 1 6 γ < 4/3).

In the adiabatic-like collapse the bulk viscosity is also responsible for theappearance of a threshold scale (dependent on the politropic index),

k2C = f(γ) ρ

160 / ζ0 , (A.10)

beyond which perturbations begin to increase; this issue, absent in the non-viscous case, is equivalent to deal with a Jeans length.

Another peculiar research line deals with the study of the early singularityproposed in the scheme of matter creation (Montani and Nescatelli, 2007).A wide number of different proposals exist about the extreme physics char-acterizing early cosmology. Among such proposals, the attention is focused

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on those scenarios for which it is expected that the Universe has been cre-ated as a vacuum fluctuation, thus the study of the particle creation shouldbe added for a complete analysis of its dynamics. The aim of the researchline is to include in the work by E.M. Lifshitz and I.M. Khalatnikov on thegravitational stability a term of matter creation to study how it influencesthe Universe dynamics and its stability near the cosmological singularity.

A reliable framework to describe such a phenomenon was provided by Y.Prigogine, who proposed to apply the thermodynamics of the open systemsto deal with the variation of particle number. Successively, this theory wasextended to the case of flat or negative FLRW Universe by fixing a suitableansatz for the particle creation rate (Montani, 2001). In this scheme, theeffect of dealing with a time varying particle number is summarized by anadditional negative pressure term, having the form of a power-law in theenergy density. This negative pressure term leads to a re-interpretation ofthe stress-energy tensor. Particle creation, which comes out from the rapidtime variation of the gravitational field, can explain both an increase of theentropy of the Universe and a remarkable stability compared with the one ofthe Cosmological Standard Model.

In order to analyze the Universe stability, it is necessary to start study-ing a cosmological fluid with an “ad hoc” choice of the parameter β, whichcontrols the rate of particle creation (i.e., β = 1/2) and this way a completephenomenological scheme can be addressed. The advantage to use such spe-cial value of β consists in the analytical integrability of the (zeroth-order)Friedmann equation. The general case for the ansatz is furthermore con-sidered, retaining only the zeroth-order term of an expansion in the energydensity. Both cases indicate that the Universe is clearly stable in the direc-tion of expansion as in the Standard Model. On the other hand, we find, asa crucial result, an instability backward in time which does not appear inthe Lifshitz model.

We can conclude that the Universe cannot be created like an isotropicsystem and only after a certain time it becomes close to our usual concep-tion of isotropy. In this respect, this analysis encourages the idea of an earlyUniverse as characterized by a certain degree of anisotropy and inhomogene-ity. The natural backward evolution of the model here presented is expectedto be that of the so called Mixmaster Universe. Such a homogeneous modelis characterized by an oscillatory regime which, on the horizon scale, survivesalso in the generic inhomogeneous solution.

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A.4 Extended Theories of Gravity

The main interesting proposals to interpret the presence of Dark Energy canbe divided into two classes: those theories, that make explicitly presence ofmatter and the other ones, which relay on modifications of the Friedmanndynamics. We address a mixture of these two points of view, with the aim ofclarifying how the “non-gravitational” vacuum energy affects so weakly thepresent Universe dynamics. In particular, we study (Lecian and Montani,2007) the modified gravitational action

SG = − c3

16πG

∫d4x

√−g f(R). (A.11)

It is possible to demonstrate that the non-linear gravitational Lagrangian(A.11), in the Jordan frame, can be cast in a dynamically-equivalent form,i.e., the action for a scalar field φ in GR (with a rescaled metric), in the Ein-stein frame, by means of the conformal transformation gµν → eφgµν , whichprovides the on-shell condition φ ≡ − ln f ′(R).Within the scheme of modified gravity, an exponential Lagrangian densitywas considered, i.e., f(R) = 2Λexp (R/2Λ), and the corresponding scalar-tensor description was addressed for both positive and negative values of thecosmological constant.We determined the Friedmann equation corresponding to an exponential formfor the gravitational-field Lagrangian density. The peculiar feature of ourmodel is that the geometrical components contain a cosmological term too,whose existence can be recognized as soon as we expand the exponential formin Taylor series of its argument. An important feature of our model ariseswhen taking a Planckian value for the fundamental parameter of the theory(as requested by the cancellation of the vacuum-energy density). In fact, asfar as the Universe leaves the Planckian era and its curvature has a charac-teristic length much greater than the Planckian one, then the correspondingexponential Lagrangian is expandible in series, reproducing General Rela-tivity to a high degree of approximation. As a consequence of this naturalEinsteinian limit (which is reached in the early history of the Universe), mostof the thermal history of the Universe is unaffected by the generalized theory.The only late-time effect of the generalized framework consists of the reliccosmological term actually accelerating the Universe. Indeed, our model isnot aimed at showing that the present Universe acceleration is a consequenceof non-Einsteinian dynamics of the gravitational field, but at outlining howit can be recognized from a vacuum-energy cancellation. Such a cancella-tion must take place in order to deal with an expandable Lagrangian term

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and must concern the vacuum-energy density as far as we build up the ge-ometrical action only by means of fundamental units. The really surprisingissue fixed by our analysis is that the deSitter solution exists in presence ofmatter only for a negative ratio between the vacuum-energy density and theintrinsic cosmological term, ǫvac/ǫΛ. We can take the choice of a negativevalue of the intrinsic cosmological constant, which predicts an acceleratingdeSitter dynamics. Nevertheless, in this case, we would get a vacuum-energydensity greater than the modulus of the intrinsic term. This fact looks likea fine-tuning, especially if we take a Planckian cosmological constant. Thevacuum-energy density is expected to be smaller than the Planckian one by afactor O(1)×α4, where α < 1 is a parameter appearing in non-commutativeformulations of the relativistic particle, and , in particular, it is linked to themodified commutation relations

[ x, p ] = i~

(1 +

1

α2

G

c3~p2

). (A.12)

The analysis of the corresponding scalar-tensor model helped us to shed lighton the physical meaning of the sign of the cosmological term. In fact, fornegative values of the cosmological term, the potential of the scalar fieldexhibits a minimum, around which scalar-field equations can be linearized.The study of the deSitter regime shows that a comparison with the modified-gravity description is possible in an off-shell region, i.e., in a region wherethe classical equivalence between the two formulations is not fulfilled.

A.5 The interaction between relic neutrinos

and primordial gravitational waves

The presence in the Universe today of a stochastic background of gravita-tional waves (GWs) is a quite general prediction of several early cosmologyscenarios. In fact, the production of gravitons is the outcome of many pro-cesses that could have occurred in the early phases of the cosmological evo-lution. Notable examples of this kind of processes include the amplificationof vacuum fluctuations in inflationaryand pre-big-bang cosmology scenarios,phase transitions, and finally the oscillation of cosmic strings loops. In mostof these cases, the predicted spectrum of gravitational waves extends over avery large range of frequencies; for example, inflationary expansion producesa flat spectrum that spans more than 20 orders of magnitude in frequency,going from 10−18 to 109 Hz.

The detection of such primordial gravitational waves, produced in theearly Universe, would be a major breakthrough in cosmology and high energy

775

physics. This is because gravitons decouple from the cosmological plasmaat very early times, when the temperature of the Universe is of the orderof the Planck energy. In this way, relic gravitational waves provide us a“snapshot” of the Universe near the Planck time, in a similar way as thecosmic microwave background radiation (CMBR) images the Universe at thetime of recombination.

The extremely low frequency region (ν0 . 10−15Hz) in the spectrum ofprimordial gravitational waves can be probed through the anisotropies ofthe CMBR. In particular, gravitational waves leave a distinct imprint inthe so-called magnetic or B-modes of its polarization field. The amplitudeof the primordial spectrum of gravitational waves is usually parameterizedthrough the tensor-to-scalar ratio r, i.e., the ratio between the amplitudesof the initial spectra of the tensor and scalar perturbations in the metric.The Planck satellite, scheduled for launch in July 2008, is expected to besensitive to r ≥ 0.05 . The lower limit corresponds to a density parameterΩGW (ν) ≡ (1/ρc)dρGW /d log ν as faint as ∼ 3×10−16h−2 (h is the dimension-less Hubble constant) in the low frequency range. Although this value looksincredibly small, it should be noted that, in order to produce such an amountin the framework of inflationary models (that at present time represent themost promising way to produce a signal in the region under consideration), avery early (starting at t ∼ 10−38 sec) inflation is required, and this possibilitylooks, from a theoretical point of view, quite unnatural. Polarization dedi-cated experiments will enhance the sensitivity of one and maybe two ordersof magnitude .

On the other hand the planned large scale interferometric GW detec-tors, although designed with the aim to detect astrophysical signals, canpossibly also detect signals of cosmological origin. They give complemen-tary information with respect of the CMBR polarization field since, even iftheir sensitivity is by no means comparable to the one than can be reachedby CMBR polarization experiments, nevertheless they probe a different re-gion in the frequency domain that would not be accessible to those ones.In particular the ground-based interferometers, such as the LIGO, VIRGO,GEO600and TAMA300experiments, operate in the range 1 Hz < ν0 < 104 Hz,and are expected to be sensitive to ΩGW h2 ≥ 10−2. Even more interestingis the LISA space interferometer, that will probably operate from 2013 to2018. Not being hampered by the Earth seismic noise, it will probe the fre-quency region between 10−4 and 1 Hz and will in principle be able to detectΩGW h2 ≥ 10−12 at ν0 = 10−3 Hz. According to theoretical predictions, alarge enough GW signal at this frequencies can be produced, with the appro-priate choice of parameters, by a pre-big-bang accelerated expansion, by theoscillation of cosmic strings, or by the electroweak phase transition occurring

776

at T = 300 GeV.In order to compare the theoretical predictions with the expected in-

strument sensitivities, one needs to evolve the GWs from the time of theirproduction to the present. It is usually assumed that gravitons propagatein vacuum, i.e., they freely stream across the Universe. In this case, theonly effect on a propagating GW is a change in frequency (correspondingto the usual redshift of the graviton energy caused by the expansion of theUniverse), while the intensity of the wave remains the same. However, GWsare sourced by the anisotropic stress part of the energy-momentum tensor ofmatter, so that the vacuum approximation is well-motivated only when thiscan be neglected. The relevant equation describing a GW propagating on aFriedmann-Robertson-Walker metric is :

∂2t hij +

(3

a

da

dt

)∂thij −

(∇2

a2

)hij = 16πGπij , (A.13)

where a(t) is the cosmological scale factor, hij is a small tensor perturbationrepresenting the GW, and πij is the anisotropic stress part of the energy-momentum tensor T µ

ν .It is already known that the anisotropic stress of free streaming relic

neutrinos acts as an effective viscosity, absorbing gravitational waves in theextremely low frequency region, thus resulting in a damping of the B-modesof CMBR. We have studied the generalization of this phenomenon to otherregions of the frequency domain (Lattanzi and Montani, 2007). In particular,we have considered GWs that enter the horizon before the electroweak phasetransition (EWPT). This corresponds to an observable frequency today ν0 &

10−5 Hz, i.e., to all waves possibly detectable by interferometers.In order to study this issue, one has to solve the Boltzmann equation for

the phase space density f of cosmological neutrinos:

L[f ] ≡ df

dλ= C[f ], (A.14)

where λ is some affine parameter over the neutrino word line, and the collisionoperator C takes into account the interaction between neutrinos and otherparticles. The two equations (A.13) and (A.14) are coupled by the followingexpression relating the energy momentum tensor and the phase space density:

T ij =

1√−g

∫f(xi, pj, t)

pipj

p0dp1 dp2 dp3. (A.15)

Manipulation of the above equation leads to an integro-differential equationfor the normalized amplitude χ(t) ≡ hij(t)/hij(t = 0) of the gravitational

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-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-1 0 1 2

Log[u]

χ(u)

mattervacuum

0.8

0.85

0.9

0.95

1

1.05

0 1 2 3 0.8

0.85

0.9

0.95

1

1.05

Log[u]

T(u)

Figure A.1: (Left panel) Time evolution of the gravitational wave amplitudeχ(u). Solid line represents a GW propagating in neutrino matter. Dashedline represents a GW propagating in vacuum. (Right panel) Time evolutionof the transmitted wave intensity T .

wave. In the limit of very short neutrino mean free path, valid in the veryearly Universe and relevant for waves well below a frequency of 108 Hz, thisequation can be cast in purely differential form:

χ +2

uχ + χ = −8fν

5u2(χ − 1) (A.16)

where u is a time variable related to conformal time, and fν is the fractionof the total density of the Universe provided by neutrinos. In the standardcosmological scenario, fν ≃ 0.4, although non-standard processes can changethis value. Thus, a numerical solution to Eq. (A.16) can be sought withstandard methods. It is found that the intensity of GWs is reduced to ∼ 90%of its value in vacuum (see Fig A.1), its exact value depending only on onephysical parameter, namely the density fraction of neutrinos. Neither thewave frequency nor the detail of neutrino interaction affect the value of theabsorbed intensity, resulting in an universal behaviour in the frequency rangeconsidered. A fitting formula for the transmitted intensity T∞ given by:

T∞ = 1 − 0.32fν + 0.05f 2ν (A.17)

The importance of our results relies in the fact that the damping affectsGWs in the frequency range where the LISA space interferometer and future,second generation ground-based interferometers can possibly detect a signalof cosmological origin. This effect is roughly of the same order of magnitudeas the one affecting GWs detectable through the B-modes of CMB polariza-tion. The damping is not so severe to make the detection of cosmological

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waves unfeasible by interferometers. However it should be taken into accountwhen testing the theoretical predictions of early Universe scenarios againstobservations. Moreover, the dependence of T∞ on fν can be exploited tomeasure the latter, and to constrain models of non-standard physics. Thiseven more important in view of the fact that in this way we could measurethe value of fν at very early times, while available constraints regard theneutrino fraction at the time of cosmological nucleosynthesis or at the timeof matter-radiation decoupling.

A.6 On the coupling between Spin and Cos-

mological Gravitational Waves

We study the influence of Spin on the dynamic concerning thermic equilib-rium of primordial universe(Lattanzi and Montani, 2007). In a homogeneousand isotropic flat universe described by the FRW metric, we consider a fluidof spinning particles. The equations of motion for such particles in the frameof general relativity were derived by Papapetrou in 1951; through a multipoleexpansion for the energy-momentum tensor he found that at dipole order adeviation from geodetic motion and an equation describing spin precessionare obtained. These equations are:

D

Dspµ = −1

2Rµ

νρσSρσuν (A.18)

D

DsSµν = pµuν − pνuµ (A.19)

where ds is the affine parameter, the vector pµ is the generalized momentum,the antisymmetric tensor Sµν is the angular momentum (Spin) and uµ =dxµ/ds. In order to close the system we impose a supplementary conditionwhich determines the center of mass of the spinning particles: Sµνuν = 0(Pirani condition).In our work we consider the case of absence of precession, so that the right-hand side of A.19 is zero and the generalized momentum is equal to thestandard momentum. In this case resolving the Papapetrou equations, weobtain the temporal dependence of Sij through the scalar factor:

Sij =1

a2Σij (A.20)

in which Σij denotes a quantity that doesn’t depend on the time variable.

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Considering the dynamic of thermal equilibrium for this case, due to thesymmetry proprieties of the metric tensor, the Boltzmann equation for theevolution of the distribution function of the spinning particles, remains inal-tered by the presence of the Spin. So we add a small tensorial perturbationhij in the metric looking for a coupling between Spin and cosmological gravi-tational waves. The resulting Boltzmann equation gives a first order variationof the distribution function that is proportional to the product between theSpin and the time derivative of hij. The form of the distribution function upto the first order in metric perturbation allows us to calculate the anisotropicstress arising by the presence of Spin:

π(S)ij =

i

2n

∫ u

0

du′K(u − u′)kmΣlm(kihjl + kjhil) (A.21)

where the Kernel K(u − u′)(s) = 164

∫ 1

−1eixs(1 − x2)x2dx is defined, u is the

conformal time, ki is the wave number connected with the Fourier compo-nents of all quantities and n is the number particle density.Even if this shows that the Spin alters some components of the anisotropicstress tensor, the final result is that these components are those that don’tcouple with the evolution of hij. This is easily understood considering thedifferential equation for tensorial perturbations:

hij(u) +2a(a)

a(u)hij = 16πG(π

(W )ij (u) + π

(S)ij (u)) (A.22)

and taking the gauge hi3 = 0, ~k = (0, 0, 1).(π(W )ij (u) is the part of anisotropic

stress that doesn’t depend by the Spin).

The final result is that there is not coupling between Spin and cosmolog-ical gravitational waves.

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B Fundamental General Rela-tivity

B.1 Perturbation Theory in Macroscopic Grav-

ity: On the Definition of Background

1. The notion of background metric adopted in the perturbation theory ingeneral relativity is analysed. A new definition of background is proposed.An existence theorem for a metric tensor which serves as the backgroundmetric for a specific scale has been proven. (G Montani and Zalaletdinov,2003). Let us consider the covariant volume averaging procedure adoptedin macroscopic gravity (Zalaletdinov, 1992)− (Mars and Zalaletdinov, 1997).The average value of a metric tensor is defined

gαβ(x) =1

VΣ

∫

Σ

gµ′ν′(x′)Aµ′

α (x′, x)Aν′

β (x′, x)√

−g′d4x′ . (B.1)

Here VΣ is 4-volume of a compact 4-region Σ, Aµ′

α (x′, x) is the averaging

operator which is idempotent, Aαβ′(x, x′)Aβ′

γ′′(x′, x′′) = Aαγ′′(x, x′′), and hence

factorized (Zalaletdinov, 1997),(Mars and Zalaletdinov, 1997) in general as

Aµ′

α (x′, x) = eµ′

i (x′)e−1i

α(x) where eµi (x) is a vector basis with constant anholo-

nomicity coefficients Ckij, i = 1, 2, 3, 4. Note that the Brill-Hartle procedure

belongs to the same class of linear averagings under some additional restric-tions on the structure of space-time (Zalaletdinov, 1992). The volume aver-ages (B.1) possess the property of idempotency (Zalaletdinov, 1992), (Marsand Zalaletdinov, 1997), that is gαβ(x) = gαβ(x). This is a fundamentalproperty which means geometrically that the average value of a tensor fieldremains invariant under action of the same averaging operator. Such an av-eraging procedure on a space-time manifold provides a natural criterium fora definition of background metric.

Definition. Given an averaging space-time procedure (B.1) with anidempotent averaging kernel, a metric tensor gαβ(x) is called a background

781

metric ifgαβ(x) = gαβ(x). (B.2)

Such a background metric is invariant with respect to the class of averag-ings, and it works in the framework of the perturbation theory as describedabove. An averaged metric is always the background one according to thedefinition (B.2).

The following important theorem considers the existence of a metric ten-sor which serves as the background metric for a specific scale.

Theorem.Given an averaging space-time procedure (B.1) with an idem-potent averaging kernel of the class of bounded and continuous functionson a space-time manifold M, there always exists a continuous and boundedbackground metric gαβ(x) (B.2) for a characteristic scale d = VΣ where Σ isa compact 4-region of M.

B.2 On Schouten’s Classification of the non-

Riemannian Geometries with an Asym-

metric Metric

Application of non-Riemannian geometries with an asymmetric metric ten-sor to the problem of geometric unification is discussed. An approach to aclassification for such kind of geometries in spirit of Schouten is proposed(Casanova et al., 1999). By adopting Schouten’s classification approach tothe affine connection geometries with an asymmetric metric the structure andvariety of such geometries can be investigated in a fully geometrical formal-ism without adopting a variational principle. It may also give the possibilityto generalize the scheme to more general geometries including spinor fieldson manifolds.

In the case of an asymmetric metric tensor gµν , gµν 6= gνµ, similar to thecase of the symmetric metric, analysis of the incompatibility between metricand connection gµν|ρ = Nµνρ brings about the following expression for theconnection Πθ

κλ

Πθκλ(δ

σθ δκ

ν δλρ + gσλδκ

ρaθν + gσκδλν aρθ) = Γσ

νρ + ∆σνρ + Cσ

νρ − Dσνρ , (B.3)

with the standard metric connection coefficients Γσνρ, the metric asymmetric-

ity object ∆σνρ = 1

2sσµ(aµν,ρ + aρµ,ν − aνρ,µ), the generalized contorsion ten-

sor Cσνρ = 1

2

[sσµ(T ε

νµgερ + T ερµgεν) + T ε

νρg.σε

]and the non-metricity tensor

Dσνρ = 1

2sσµ(Nµνρ + Nρµν − Nνρµ). The determinant of the ”hypercubic”

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structure matrix Jσκλθνρ = δσ

θ δκν δλ

ρ + gσλδκρaθν + gσκδλ

ν aρθ is related to the ex-istence of solutions of the system of inhomogeneous linear algebraic equa-tions (B.3) for the unknowns Πα

βγ similar to the case of usual quadraticmatrixes. When the determinant is not equal to zero the system has non-trivial solutions which can be expressed through the inverse structure matrixJανρ

σβγ = (J−1)ανρ

σβγ, JανρσβγJσβγ

µελ = δαµδν

ε δρλ. The espression of the Riemannian

curvature tensor Mαβρσ from Mα

βρσ is given by

Rαβρσ = M ε

νρλJανλεβσ + Σα

βρσ(Aαβσ, ∆

αβσ, J

ανρσβγ ), (B.4)

where Σαβρσ is a tensor constructed from generalised affine deformation ten-

sor, metric asymmetricity object and the inverse structure matrix and theirderivatives. The determinant of Jσκλ

θνρ has been calculated in a perturbationexpansion in terms of small asymmetric metric, | aµν |≪| sµν |. Then inlinear approximation the matrix Jσκλ

θνρ = δσθ δκ

ν δλρ + sσλδκ

ρaθν + sσκδλν aρθ has its

inverse as Jανρσβγ = δα

σδνβδρ

γ − sανδρβaγσ − sαρδν

γaσβ. The expressions (B.3) and(B.4) are the main relations describing the structure of the affine connectiongeometries with asymmetric metric.

B.3 Approximate Symmetries, Inhomogeneous

Spaces and Gravitational Entropy

The problem of finding an appropriate geometrical/physical index for mea-suring a degree of inhomogeneity for a given space-time manifold is posed. In-terrelations with the problem of understanding the gravitational/informationalentropy are pointed out. An approach based on the notion of approximatesymmetry is proposed (Zalaletdinov, 2000),(Montani et al., 2000). A num-ber of related results on definitions of approximate symmetries known fromliterature are briefly reviewed with emphasis on their geometrical/physicalcontent. A definition of a Killing-like symmetry is given and a classificationtheorem for all possible averaged space-times acquiring Killing-like symme-tries upon averaging out a space-time with a homothetic Killing symmetryis proved.

The main idea of the Killing-like symmetry is to consider the most generalform of deviation from the Killing equations. Let us consider the equationfor a Killing-like vector ξα(xµ)

ξα;β + ξβ;α = 2ǫαβ (B.5)

where a symmetric tensor ǫαβ(xµ) measures deviation from the Killing sym-metry. The tensor can be small in order to enable a continuous limit to the

783

case ǫαβ → 0.The equation (B.5) covers the cases of semi-Killing, almost-Killing and

almost symmetries with additional equations for the tensor ǫαβ(xµ). Alsocovered are standard generalizations of Killing symmetry such as conformaland homothetic Killing vectors The algebraic classification of the symmetrictensor ǫαβ gives an invariant way to introduce a set of scalar indexes mea-suring the degree of inhomogeneity of the space-time with (B.5) comparedwith that with isometries, or even weaker symmetry, for example, conformalKilling’s. For the most general case A1[111, 1] in Segre’s notationǫαβ has theform

ǫµν = λgµν + ρxµxν + σyµyν + τzµzν (B.6)

where gµν if the space-time metric, λ(xµ), ρ(xµ), σ(xµ) and τ(xµ) are eigen-values of ǫαβ and

tµ, xµ, yµ, zµ

is the eigentetrad . If all eigenvalues vanish

the space-time has an isometry (B.5), if ρ = σ = τ = 0 then there is aconformal Killing vector for λ(x) 6= 0 and a homothetic Killing vector forλ = const. For other algebraic types of Killing-like symmetry the space-timehas the following sets of eigenvalues: two complex conjugated to each otherand two real scalars for A2[11, ZZ∗], three real scalars for A3[11, 2] and tworeal scalars for B[1, 3].

B.4 Gravitational Polarization in General Rel-

ativity: Solution to Szekeres’ Model of

Gravitational Quadrupole

A model for the static weak-field macroscopic medium is analyzed and theequation for the macroscopic gravitational potential is derived (Montaniet al., 2003). This is a biharmonic equation which is a non-trivial gener-alization of the Poisson equation of Newtonian gravity. In case of the stronggravitational quadrupole polarization it essentially holds inside a macroscopicmatter source. Outside the source the gravitational potential fades away ex-ponentially. The equation is equivalent to a system of the Poisson equationand the nonhomogeneous modified Helmholtz equations. The general solu-tion to this system is obtained by using Green’s function method and it doesnot have a limit to Newtonian gravity. In case of the insignificant gravi-tational quadrupole polarization the equation for macroscopic gravitationalpotential becomes the Poisson equation with the matter density renormalizedby the factor including the value of the quadrupole gravitational polariza-tion of the source. The general solution to this equation obtained by usingGreen’s function method has a limit to Newtonian gravity.

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Calculation of the equation for the macroscopic gravitational potential ϕfrom the macroscopic gravity equations for the macroscopic tensor g

(0)µν brings

the equation

∆ϕ = 4πGµ +4πGǫg

3c2∆2ϕ (B.7)

where ∆2ϕ ≡ ∆(∆ϕ) is the Laplacian of the Laplacian of ϕ. This is a non-trivial generalization of the Poisson equation for the gravitational potentialϕ of Newtonian gravity. This is a biharmonic equation due the presence ofthe term ∆2ϕ. The equation (B.7) involves a singular perturbation, since incase of the vanishing gravitational dielectric constant, ǫg = 0, this equationbecomes the Poisson equation, but if ǫg 6= 0, this equations change its oper-ator structure to be of the fourth order equation in partial derivatives of ϕas compared with the Poisson second order partial differential equation.

It is convenient to introduce the factor

1

k2=

4πGǫg

3c2(B.8)

with k having a physical dimension of inverse length, [k−2] = length2. Thenthe equation (B.7) takes the form

∆ϕ = 4πGµ +1

k2∆2ϕ. (B.9)

By using the definitions of the gravitational dielectric constant ǫg, the char-acteristic oscillation frequency of molecule’s constituents ω2

0, macroscopicmatter density µ = 3m/4πA3 and the average number of molecules per unitvolume N = 4πD3/3 with D as a mean distance between molecules, thefactor k−2 can be shown to have the following form

1

k2=

1

4θ

(A3

D3

)A2. (B.10)

Here the dimensionless factor θ,

θ =ω2

0

4πGµ/3, (B.11)

reflects the nature of field responsible for bounding of discrete matter con-stituents into molecules. If θ ≈ 1, the molecules of self-gravitating macro-scopic medium are considered to be gravitationally bound. For instance,considering a macroscopic model of galaxy as a self-gravitating macroscopicmedium consisting of gravitational molecules taken as double stars, θ ≈ 1 assuch galactic molecules are gravitationally bound. If one takes the molecules

785

to be of electron-proton type, like atoms, the factor θ ≈ 1040, which makesthe factor k−2 essentially insignificant.

The dimensionless ratio A/D reflects the structure of macroscopic medium.If (A/D) ≈ 1, the macroscopic medium behaves itself like a liquid or solid.If (A/D) < 1, the macroscopic medium behaves itself like a gas. For themacroscopic galactic model for the present epoch the macroscopic mediumis like a gas, since (A/D) ≈ 10−1 − 10−2, which makes the factor A3/D3 tobe of order of 10−3 − 10−6. However, for earlier times of galaxy formulationthis factor can be expected to be of much greater order of magnitude up to1 − 10.

B.5 Averaging Problem in Cosmology and Macro-

scopic Gravity

The Averaging problem in general relativity and cosmology is discussed. Theapproach of macroscopic gravity to resolve the problem is presented. Theaveraged Einstein equations of macroscopic gravity are modified on cosmo-logical scales by the gravitational correlation tensor terms as compared withthe Einstein equations of general relativity. This correlation tensor satisfiesan additional set of structure and field equations. Exact cosmological solu-tions to the equations of macroscopic gravity for spatially homogeneous andisotropic macroscopic space-times are presented. In particular, it has beenfound that for a flat geometry the gravitational correlation tensor terms inthe averaged Einstein equations have the form of a spatial curvature termwhich can be either negative or positive. Thus macroscopic gravity providesa cosmological model for a flat spatially homogeneous and isotropic Universewhich obeys the dynamical law for either open or closed Universe geometry.

For a flat spatially homogeneous, isotropic macroscopic space-time

ds2 = a2(η)(−dη2 + dx2 + dy2 + dz2) (B.12)

the averaged Einstein equations for the case of a constant macroscopic grav-itational correlation tensor Zα

βγµ

νσ = const read

(a

a

)2

=κρ

3+

ε

3a2, (B.13)

2a

a+

(a

a

)2

= −κp +ε

3a2, (B.14)

786

or in terms of ρgrav and pgrav

(a

a

)2

=κ

3(ρ + ρgrav) , (B.15)

2a

a+

(a

a

)2

= −κ (p + pgrav) . (B.16)

with the equations of state p = p(ρ) and pgrav = −13ρgrav . They look similar

to Einstein’s equations of General Relativity for either a closed or an openspatially homogeneous, isotropic FLRW space-time, but they do have differ-ent mathematical and physical, and therefore, cosmological content since

ε

3=

κρgrava2

36= −k (B.17)

in general.The macroscopic (averaged) Einstein’s equations for a flat spatially ho-

mogeneous, isotropic macroscopic space-time have macroscopic gravitationalcorrelation terms of the form of a spatial curvature term

ε

3a2=

κρgrav

3. (B.18)

Thus, the theory of Macroscopic Gravity predicts that constant macroscopicgravitational correlation tensor Zα

βγµ

νσ = const for a flat spatially homo-geneous, isotropic macroscopic space-time takes the form of a dark spatialcurvature term it interacts only gravitationally with the macroscopic gravi-tational field it does not interact directly with the energy-momentum tensorof matter it exhibits a negative pressure pgrav = −1

3ρgrav which tends to

accelerate the Universe when ρgrav > 0.Only if one requires 12Z3

23332 = −ε to be ε = −3k the macroscopic (av-

eraged) Einstein’s equations become exactly Einstein’s equations of GeneralRelativity for either a closed or an open spatially homogeneous, isotropicspace-time for the macroscopic geometry of a flat spatially homogeneous,isotropic space-time.

This exact solution of the Macroscopic Gravity equations exhibits a verynon-trivial phenomenon from the point of view of the general-relativistic cos-mology: the macroscopic (averaged) cosmological evolution in a flat Universeis governed by the dynamical evolution equations for either a closed or anopen Universe depending on the sign of the macroscopic energy density ρgrav

with a dark spatial curvature term κρgrav/3.From the observational point of view such a cosmological model gives

a new paradigm to reconsider the standard cosmological interpretation andtreatment of the observational data.

787

Indeed, this macroscopic cosmological model has the Riemannian geome-try of a flat homogeneous, isotropic space-time. Therefore, all measurementsand data are to be considered and designed for this geometry. The dynamicalinterpretation of the obtained data should be considered and treated for thecosmological evolution of either a closed or an open spatially homogeneous,isotropic Riemannian space-time.

B.6 Astrophysical Topics

- M. V. Barkov, V. A. Belinskii and G.S. Bisnovatyi-Kogan Model of

ejection of matter from non-stationary dense stellar clasters and chaotic

motion of gravitating shells, Mon. Not. R.A.S., 334, 338, (2002)(astro-ph/0107051).A model of ballistic ejection effect of matter from spherically symmetricstellar clusters it is investigated. The problem is solved in newtoniangravity but with cutoff fixing the minimal radius of selfgravitating mat-ter shell by its relativistic gravitational radus. It is shown that duringthe motion of two initially gravitationally bound spherical shells, con-sisting of point particles moving along ballistic trajectories, one of theshell may be expelled to infinity at subrelativistic expelling velocity ofthe order of 0,25c. Also it is shown that the motion of two intersectingshells in the case when they do not runaway reveal a chaotic behaviour.

- M. V. Barkov, V.A. Belinskii and G.S. Bisnovatyi-Kogan An exact Gen-

eral Relativity solution for the Motion and Intersections of Self-Gravitating

Shells in the Field of a Massive Black Hole, JETP 95, 371, (2002) (astro-ph/0210296).It is found the complete exact solution in the General Relativity forthe intersection process of two massive selfgravitating spherically sym-metric shells (in general with tangential pressure). It is shown how onecan calculate all shell’s parameters after intersection in terms of the pa-rameters before the intersection. The result is quite new, the solutionof this kind was known only for the massless shells (Dray and t’Hooft,1985). The solution was applied to the analysis of matter ejection effectfrom relativistic stellar clusters. It is shown that in relativistic case thematter ejection effect is stronger than in newtonian gravity.

- G.S. Bisnovatyi-Kogan, R.V.E. Lovelace and V.A. Belinskii A cosmic

battery reconsidered, ApJ 580, 380, (2002) (astro-ph/0207476).The problem of magnetic field generation in accretion flows onto blackholes owing to the excess radiation force on electrons is revisited. This

788

excess force may arise from the Poynting-Robertson effect. Insteadof a recent claim of the generation of dynamically important magneticfields, we show only small magnetic fields are generated. A model of thePoynting-Robertson magnetic field generation close to the horizon of aSchwarzschild black hole is solved exactly using General Relativity, andthe field is found to be dynamically insignificant. These weak magneticfields may however be important as seed fields for dynamos.

- M.V.Barkov, V.A.Belinskii, G.S.Bisnovatyi-Kogan and A.I.NeishtadtModel of Ejection of Matter from Dense Stellar Cluster and Chaotic Mo-

tion of Gravitating Shells, in Galaxies and Chaos, page 357, Eds.G.Contopoulos and N.Voglis, Lecture Notes in Physics, Springer (2003).It is shown that during the motion of two initially gravitationally boundspherical shells, consisting of point particles moving along ballistic tra-jectories, one of the shells may be expelled to infinity at subrelativisticspeed of order 0.25c. The problem is solved in Newtonian gravity. Mo-tion of two intersecting shells in the case when they do not runawayshows a chaotic behaviour. We hope that this simple toy model cangive nevertheless a qualitative idea on the nature of the mechanism ofmatter outbursts from the dense stellar clusters.

- M. V. Barkov, G. S. Bisnovatyi-Kogan, A. I. Neishtadt and V.A. Be-linski On chaotic behavior of gravitating stellar shells, Chaos, 15, 013104(2005).Motion of two gravitating spherical stellar shells around a massive cen-tral body is considered. Each shell consists of point particles with thesame specific angular momenta and energies. In the case when one canneglect the influence of gravitation of one (”light”) shell onto another(”heavy”) shell (”restricted problem”) the structure of the phase spaceis described. The scaling laws for the measure of the domain of chaoticmotion and for the minimal energy of the light shell sufficient for itsescape to infinity are obtained.

789

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