Examples with minimal number of brake orbits and homoclinics in annular potential regions

We use a geometric construction to exhibit examples of autonomous Lagrangian systems admitting exactly two homoclinics emanating from a nondegenerate maximum of the potential energy and reaching a regular level of the potential having the same value of the maximum point. Similarly, we show examples of Hamiltonian systems that admit exactly two brake orbits in an annular potential region connecting the two connected components of the boundary of the potential well. These examples show that the estimates proven in [R. Giamb\`o, F. Giannoni, P. Piccione, Arch. Ration. Mech. Anal. 200, (2011) 691-724] are sharp.Comment: 12 pages, 1 figur

Qualitative analysis of collapsing isotropic fluid spacetimes

The structure of the Einstein field equations describing the gravitational collapse of spherically symmetric isotropic fluids is analyzed here for general equations of state. A suitable system of coordinates is constructed which allows us, under a hypothesis of Taylor-expandability with respect to one of the coordinates, to approach the problem of the nature of the final state without knowing explicitely the metric. The method is applied to investigate the singularities of linear barotropic perfect fluids solutions and to a family of accelerating fluids.Comment: 16 pages; to appear on Class. Quantum Gra

Expanding universes in the conformal frame of f(R)f(R) gravity

The late time evolution of Friedmann-Robertson-Walker (FRW) models with a perfect fluid matter source is studied in the conformal frame of $f(R)$ gravity. We assume that the corresponding scalar field, nonminimally coupled to matter, has an arbitrary non-negative potential function $V(\phi)$. We prove that equilibria corresponding to non-negative local minima for $V$ are asymptotically stable. We investigate all cases where one of the matter components eventually dominates. The results are valid for a large class of non-negative potentials without any particular assumptions about the behavior of the potential at infinity. In particular for a nondegenerate minimum of the potential with zero critical value we show that if $\gamma$, the parameter of the equation of state is larger than one, then there is a transfer of energy from the fluid to the scalar field and the later eventually dominates.Comment: Talk given in "The Invisible Universe", June 29 - July 3, 2009, Pari

Genericity of nondegenerate geodesics with general boundary conditions

Let M be a possibly noncompact manifold. We prove, generically in the C^k-topology (k=2,...,\infty), that semi-Riemannian metrics of a given index on M do not possess any degenerate geodesics satisfying suitable boundary conditions. This extends a result of Biliotti, Javaloyes and Piccione for geodesics with fixed endpoints to the case where endpoints lie on a compact submanifold P of the product MxM that satisfies an admissibility condition. Such condition holds, for example, when P is transversal to the diagonal of MxM. Further aspects of these boundary conditions are discussed and general conditions under which metrics without degenerate geodesics are C^k-generic are given.Comment: LaTeX2e, 21 pages, no figure

Effective field description of the Anton-Schmidt cosmic fluid

The effective theory of the Anton-Schmidt cosmic fluid within the Debye approximation is investigated. In this picture, the universe is modeled out by means of a medium without cosmological constant. In particular, the Anton-Schmidt representation of matter describes the pressure of crystalline solids under deformations imposed by isotropic stresses. The approach scheme is related to the fact that the universe deforms under the action of the cosmic expansion itself. Thus, we frame the dark energy term as a function of scalar fields and obtain the corresponding dark energy potential $V(\varphi)$. Different epochs of the universe evolution are investigated in terms of the evolution of $\varphi$. We show how the Anton-Schmidt equation of state is capable of describing both late and early epochs of cosmic evolution. Finally, numerical bounds on the Anton-Schmidt model with $n=-1$ are derived through a Markov Chain Monte Carlo analysis on the combination of data coming from type Ia Supernovae, observations of Hubble parameter and baryon acoustic oscillations. Statistical comparison with the $\Lambda$CDM model is performed by the AIC and BIC selection criteria. Results are in excellent agreement with the low-redshift data. A further generalization of the model is presented to satisfy the theoretical predictions at early-stage cosmology.Comment: 13 pages, Accepted for publication in Phys. Rev.

De Sitter-like configurations with asymptotic quintessence environment

We examine a spherically-symmetric class of spacetimes carrying vacuum energy, while considering the influence of an external dark energy environment represented by a non-dynamical quintessence field. Our investigation focuses on a specific set of solutions affected by this field, leading to distinct kinds of spacetime deformations, resulting in regular, singular, and wormhole solutions. We thoroughly discuss the underlying physics associated with each case and demonstrate that more complex deformations are prone to instability. Ultimately, we find that our results lead to an \emph{isotropic de Sitter-like solution} that behaves as a quintessence fluid. To achieve this, we investigate the nature of the corresponding fluid, showing that it cannot provide the sound speed equal to a constant equation of state near the center. Consequently, we reinterpret the fluid as a slow-roll quintessence by investigating its behavior in asymptotic regimes. Further, we explore the potential implications of violating the isotropy condition on the pressures and we finally compare our findings with the de Sitter and Hayward solutions, highlighting both the advantages and disadvantages of our scenarios.Comment: 14 pages, 4 figures, 1 tabl

Global variational methods on smooth nonholonomic constraints

AbstractWe develop a variational theory for critical points of integral functionals in a space of curves on a manifold M, between a fixed point and a one-dimensional submanifold of M, and satisfying a nonholonomic constraint equation φ=0, where φ is a C2 function defined on TM×R.We obtain existence, regularity and multiplicity results, writing the integro-differential equations satisfied by critical points. Moreover, we present some results concerning a sort of exponential map relative to the integro-differential equations and some examples