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Register a new userHomogeneous cosmologies and the Maupertuis-Jacobi principle
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A recent work showing that homogeneous and isotropic cosmologies involving scalar fields are equivalent to the geodesics of certain effective manifolds is generalized to the non-minimally coupled and anisotropic cases. As the Maupertuis-Jacobi principle in classical mechanics, such result permits us to infer some dynamical properties of cosmological models from the geometry of the associated effective manifolds, allowing us to go a step further in the study of cosmological dynamics. By means of some explicit examples, we show how the geometrical analysis can simplify considerably the dynamical analysis of cosmological models.
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In this paper we return to the subject of Jacobi metrics for timelike and null geodsics in stationary spactimes, correcting some previous misconceptions. We show that not only null geodesics, but also timelike geodesics are governed by a Jacobi-Maupertuis type variational principle and a Randers-Finsler metric for which we give explicit formulae. The cases of the Taub-NUT and Kerr spacetimes are discussed in detail. Finally we show how our Jacobi-Maupertuis Randers-Finsler metric may be expressed in terms of the effective medium describing the behaviour of Maxwells equations in the curved spacetime. In particular, we see in very concrete terms how the magnetolectric susceptibility enters the Jacobi-Maupertuis-Randers-Finsler function.
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.
We show that several integrable (i.e., exactly solvable) scalar cosmologies considered by Fre, Sagnotti and Sorin (Nuclear Physics textbf{B 877}(3) (2013), 1028--1106) can be generalized to include cases where the spatial curvature is not zero and, besides a scalar field, matter or radiation are present with an equation of state $p^{(m)} = w, rho^{(m)}$; depending on the specific form of the self-interaction potential for the field, the constant $w$ can be arbitrary or must be fixed suitably.
We discuss the most general field equations for cosmological spacetimes for theories of gravity based on non-linear extensions of the non-metricity scalar and the torsion scalar. Our approach is based on a systematic symmetry-reduction of the metric-affine geometry which underlies these theories. While for the simplest conceivable case the connection disappears from the field equations and one obtains the Friedmann equations of General Relativity, we show that in $f(mathbb{Q})$ cosmology the connection generically modifies the metric field equations and that some of the connection components become dynamical. We show that $f(mathbb{Q})$ cosmology contains the exact General Relativity solutions and also exact solutions which go beyond. In $f(mathbb{T})$~cosmology, however, the connection is completely fixed and not dynamical.
We examine homogeneous but anisotropic cosmologies in scalar-tensor gravity theories, including Brans-Dicke gravity. We present a method for deriving solutions for any isotropic perfect fluid with a barotropic equation of state ($pproptorho$) in a spatially flat (Bianchi type~I) cosmology. These models approach an isotropic, general relativistic solution as the expansion becomes dominated by the barotropic fluid. All models that approach general relativity isotropize except for the case of a maximally stiff fluid. For stiff fluid or radiation or in vacuum we are able to give solutions for arbitrary scalar-tensor theories in a number of anisotropic Bianchi and Kantowski-Sachs metrics. We show how this approach can also be used to derive solutions from the low-energy string effective action. We discuss the nature, and possibly avoidance of, the initial singularity where both shear and non-Einstein behavior is important.
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