On solving dynamical equations in general homogeneous isotropic cosmologies with scalaron

We study general dynamical equations describing homogeneous isotropic cosmologies coupled to a scalaron $psi$. For flat cosmologies ($k=0$), we analyze in detail the gauge-independent equation describing the differential, $chi(alpha)equivpsi^prime(alpha)$, of the map of the metric $alpha$ to the scalaron field $psi$, which is the main mathematical characteristic locally defining a `portrait of a cosmology in `$alpha$-version. In the `$psi$-version, a similar equation for the differential of the inverse map, $bar{chi}(psi)equiv chi^{-1}(alpha)$, can be solved asymptotically or for some `integrable scalaron potentials $v(psi)$. In the flat case, $bar{chi}(psi)$ and $chi(alpha)$ satisfy the first-order differential equations depending only on the logarithmic derivative of the potential. Once we know a general analytic solution for one of these $chi$-functions, we can explicitly derive all characteristics of the cosmological model. In the $alpha$-version, the whole dynamical system is integrable for $k eq 0$ and with any `$alpha$-potential, $bar{v}(alpha)equiv v[psi(alpha)]$, replacing $v(psi)$. There is no a priori relation between the two potentials before deriving $chi$ or $bar{chi}$, which implicitly depend on the potential itself, but relations between the two pictures can be found by asymptotic expansions or by inflationary perturbation theory. Explicit applications of the results to a more rigorous treatment of the chaotic inflation models and to their comparison with the ekpyrotic-bouncing ones are outlined in the frame of our `$alpha$-formulation of isotropic scalaron cosmologies. In particular, we establish an inflationary perturbation expansion for $chi$. When all the conditions for inflation are satisfied and $chi$ obeys a certain boundary (initial) condition, we get the standard inflationary parameters, with higher-order corrections.

General properties and some solutions of generalized Einstein - Eddington affine gravity I

After a brief exposition of the simplest class of affine theories of gravity in multidimensional space-times with symmetric connections, we consider the spherical and cylindrical reductions of these theories to two-dimensional dilaton-vecton gravity (DVG) field theories. The distinctive feature of these theories is the presence of a massive/tachyonic vector field (vecton) with essentially nonlinear coupling to the dilaton gravity. In the massless limit, the classical DVG theory can be exactly solved for a rather general coupling depending only on the field tensor and the dilaton. We show that the vecton field can be consistently replaced by a new effectively massive scalar field (scalaron) with an unusual coupling to dilaton gravity (DSG). Then we concentrate on considering the DVG models derived by reductions of D=3 and D=4 affine theories. In particular, we introduce the most general cylindrical reductions that are often ignored. The main subject of our study is the static solutions with horizons. We formulate the general conditions for the existence of the regular horizons and find the solutions of the static DVG/DSG near the horizons in the form of locally convergent power - series expansion. For an arbitrary regular horizon, we find a local generalization of the Szekeres - Kruskal coordinates. Finally, we consider one-dimensional integrable and nonintegrable DSG theories with one scalar. We analyze simplest models having three or two integrals of motion, respectively, and introduce the idea of a `topological portrait giving a unified qualitative description of static and cosmological solutions of some simple DSG models.

An old Einstein - Eddington generalized gravity and modern ideas on branes and cosmology

We briefly discuss new models of an `affine theory of gravity in multidimensional space-times with symmetric connections. We use and generalize Einsteins proposal to specify the space-time geometry by use of the Hamilton principle to determine the connection coefficients from a geometric Lagrangian that is an arbitrary function of the generalized Ricci curvature tensor and of other fundamental tensors. 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 concrete choice of the geometric Lagrangian determines further details of the theory. The most natural geometric models look similar to recently proposed brane models of cosmology usually derived from string theory.

Two-Dimensional Dilaton Gravity and Toda - Liouville Integrable Models

General properties of a class of two-dimensional dilaton gravity (DG) theories with multi-exponential potentials are studied and a subclass of these theories, in which the equations of motion reduce to Toda and Liouville equations, is treated in detail. A combination of parameters of the equations should satisfy a certain constraint that is identified and solved for the general multi-exponential model. From the constraint it follows that in DG theories the integrable Toda equations, generally, cannot appear without accompanying Liouville equations. We also show how the wave-like solutions of the general Toda-Liouville systems can be simply derived. In the dilaton gravity theory, these solutions describe nonlinear waves coupled to gravity as well as static states and cosmologies. A special attention is paid to making the analytic structure of the solutions of the Toda equations as simple and transparent as possible, with the aim to gain a better understanding of realistic theories reduced to dimensions 1+1 and 1+0 or 0+1.

A New Integrable Model of (1+1)-Dimensional Dilaton Gravity Coupled to Toda Matter

A new class of integrable two-dimensional dilaton gravity theories, in which scalar matter fields satisfy the Toda equations, is proposed. The simplest case of the Toda system is considered in some detail, and on this example we outline how the general solution can be obtained. Then we demonstrate how the wave-like solutions of the general Toda systems can be simply derived. In the dilaton gravity theory this solutions describe nonlinear waves coupled to gravity. A special attention is paid to making the analytic structure of the solutions of the Toda equations as simple and transparent as possible, with the aim to apply the idea of the separation of variables to non-integrable theories.