Affine generalizations of gravity in the light of modern cosmology

We discuss new models of an `affine' theory of gravity in multidimensional space-times with symmetric connections. We use and develop ideas of Weyl, Eddington, and Einstein, in particular, Einstein's proposal to specify the space - time geometry by use of the Hamilton principle. More specifically, the connection coefficients are determined using a `geometric' Lagrangian that is an arbitrary function of the generalized (non-symmetric) Ricci curvature tensor (and, possibly, of other fundamental tensors) expressed in terms of the connection coefficients regarded as independent variables. Such a theory supplements the standard Einstein gravity with dark energy (the cosmological constant, in the first approximation), a neutral massive (or tachyonic) vector field (vecton), and massive (or tachyonic) scalar fields. These fields couple only to gravity and can generate dark matter and/or inflation. The new field masses (real or imaginary) have a geometric origin and must appear in any concrete model. The concrete choice of the geometric Lagrangian determines further details of the theory, for example, the nature of the vector and scalar fields that can describe massive particles, tachyons, or even `phantoms'. In `natural' geometric theories, which are discussed here, dark energy must also arise. We mainly focus on intricate relations between geometry and dynamics while only very briefly considering approximate cosmological models inspired by the geometric approach.Comment: 12 pages; several typos, eq.(37), and references [24] and [26] correcte

Integrals of equations for cosmological and static reductions in generalized theories of gravity

We consider the dilaton gravity models derived by reductions of generalized theories of gravity and study one-dimensional dynamical systems simultaneously describing cosmological and static states in any gauge. Our approach is fully applicable to studying static and cosmological solutions in multidimensional theories and also in general one-dimensional dilaton - scalaron gravity models. We here focus on general and global properties of the models, on seeking integrals, and on analyzing the structure of the solution space. We propose some new ideas in this direction and derive new classes of integrals and new integrable models.Comment: 15 pages. arXiv admin note: substantial text overlap with arXiv:1302.637

On Einstein - Weyl unified model of dark energy and dark matter

Here we give a more detailed account of the part of the conference report that was devoted to reinterpreting the Einstein `unified models of gravity and electromagnetism' (1923) as the unified theory of dark energy (cosmological constant) and dark matter (neutral massive vector particle having only gravitational interactions). After summarizing Einstein's work and related earlier work of Weyl and Eddington, we present an approach to finding spherically symmetric solutions of the simplest variant of the Einstein models that was earlier mentioned in Weyl's work as an example of his generalization of general relativity. The spherically symmetric static solutions and homogeneous cosmological models are considered in some detail. As the theory is not integrable we study approximate solutions. In the static case, we show that there may exist two horizons and derive solutions near the horizons. In cosmology, we propose to study the corresponding expansions of possible solutions near the origin and derive these expansions in a simplified model neglecting anisotropy. The structure of the solutions seems to hint at a possibility of an inflation mechanism that does not require adding scalar fields.Comment: Report to conference `Selected problems of modern theoretical physics' Dubna, Russia, June 23-27, 2008; 18 pages LaTex; sections 2.3.1 and 2.3.3, comments to Discussion added; Appendix II removed; 2 references removed, several references added for section 2.3. In version 3, typos corrected, the paragraph with equations (33), (34) somewhat extended and clarifie

A fresh view of cosmological models describing very early Universe: general solution of the dynamical equations

The dynamics of any spherical cosmology with a scalar field (`scalaron') coupling to gravity is described by the nonlinear second-order differential equations for two metric functions and the scalaron depending on the `time' parameter. The equations depend on the scalaron potential and on the arbitrary gauge function that describes time parameterizations. This dynamical system can be integrated for flat, isotropic models with very special potentials. But, somewhat unexpectedly, replacing the `time' variable by one of the metric functions allows us to completely integrate the general spherical theory in any gauge and with apparently arbitrary potentials. The main restrictions on the potential arise from positivity of the derived analytic expressions for the solutions, which are essentially the squared canonical momenta. An interesting consequence is emerging of classically forbidden regions for these analytic solutions. It is also shown that in this rather general model the inflationary solutions can be identified, explicitly derived, and compared to the standard approximate expressions. This approach can be applied to intrinsically anisotropic models with a massive vector field (`vecton') as well as to some non-inflationary models.Comment: 10 pages; added 2 pages (Sec. 5); significantly edited: Sec.4 (p.7), Abstract, Sec.1; corrected misprint

Integrable Models of Horizons and Cosmologies

A new class of integrable theories of 0+1 and 1+1 dimensional dilaton gravity coupled to any number of scalar fields is introduced. These models are reducible to systems of independent Liouville equations whose solutions must satisfy the energy and momentum constraints. The constraints are solved thus giving the explicit analytic solution of the theory in terms of arbitrary chiral fields. In particular, these integrable theories describe spherically symmetric black holes and branes of higher dimensional supergravity theories as well as superstring motivated cosmological models.Comment: 15 page

Paragrassmann Algebras with Many Variables

This is a brief review of our recent work attempted at a generalization of the Grassmann algebra to the paragrassmann ones. The main aim is constructing an algebraic basis for representing `fractional' symmetries appearing in $2D$ integrable models and also introduced earlier as a natural generalization of supersymmetries. We have shown that these algebras are naturally related to quantum groups with $q = {\rm root \;of \; unity}$. By now we have a general construction of the paragrassmann calculus with one variable and preliminary results on deriving a natural generalization of the Neveu--Schwarz--Ramond algebra. The main emphasis of this report is on a new general construction of paragrassmann algebras with any number of variables, N. It is shown that for the nilpotency indices $(p + 1) = 3, 4, 6$ the algebras are almost as simple as the Grassmann algebra (for which $(p + 1) = 2$). A general algorithm for deriving algebras with arbitrary p and N is also given. However, it is shown that this algorithm does not exhaust all possible algebras, and the simplest example of an `exceptional' algebra is presented for $p = 4, N = 4$.Comment: 12 page

Does There Really Exist the Problem of the Dark Matter in Spiral Galaxies?

A simple model for the dust media describing evolution of the system like spiral galaxy is considered. In contrast to previous considerations we show that the initial density fold should be quasi-one-dimensional (bar-like) instead of disc-like. The disc component of the galaxies appears only during the evolution. The model naturally reproduces some essential features of the galaxies. In particular, it reproduces all the observed typical forms of the rotation curves for the spiral galaxies with a characteristic minimum and plateau. It appears that the plateau corresponds to escaping the matter (external spiral arms, due to initial conditions, have too large velocities to be confined by the gravitational field of the galaxy). Such a scenario of the galaxy evolution leads to the conclusion that the hypothesis of the dark matter is not necessary (at least, for the spiral galaxies).Comment: 10 pages, standard LaTeX, figures are available from the Authors upon reques

Integrable Low Dimensional Models for Black Holes and Cosmologies from High Dimensional Theories

We describe a class of integrable models of 1+1 and 1-dimensional dilaton gravity coupled to scalar fields. The models can be derived from high dimensional supergravity theories by dimensional reductions. The equations of motion of these models reduce to systems of the Liouville equations endowed with energy and momentum constraints. We construct the general solution of the 1+1 dimensional problem in terms of chiral moduli fields and establish its simple reduction to static black holes (dimension 0+1), and cosmological models (dimension 1+0). We also discuss some general problems of dimensional reduction and relations between static and cosmological solutions.Comment: 27 page

Charge ordering, phase separation and charge pairing in layered 3D systems

The processes of Coulomb gas ordering in 3D layered system are studied by means of Brownian dynamics approach. It is found that at different densities of the carriers the 3D lattice of charges as well as new specific structures are possible in the system. In particular, at some small density the particles inside the layers can associate into droplets that collectively repulse between neighbouring layers. These droplets possess local stripe structure which orders spontaneously along arbitrary chosen direction. At higher densities a specific ordering of the charges into the tetragonal-like or hexagonal-like structures is observed visually and described numerically. Specific ''pairing'' of the charges from different layers plays an essential role in formation of all above structures. It is essentially many-body effect in 3D system which can be a reason for unusual properties of layered crystals.Comment: 11 pages, 6 figures, manuscript, ReVTe

A New Class of Integrable Models of 1+1 Dimensional Dilaton Gravity Coupled to Scalar Matter

Integrable models of 1+1 dimensional gravity coupled to scalar and vector fields are briefly reviewed. A new class of integrable models with nonminimal coupling to scalar fields is constructed and discussed.Comment: LaTeX, 8 pages, no figures. Talk given at the VIII International Conference on Symmetry Methods in Physics (Dubna 1997), to be published at Phys. of At. Nucl. 1998, vol. 61, #1