Phase-space of flat Friedmann-Robertson-Walker models with both a scalar field coupled to matter and radiation

摘要

We investigate the phase-space of a flat FRW universe including both a scalarfield, $\phi,$ coupled to matter, and radiation. The model is inspired inscalar-tensor theories of gravity, and thus, related with $F(R)$ theoriesthrough conformal transformation. The aim of the chapter is to extent severalresults to the more realistic situation when radiation is included in thecosmic budget particularly for studying the early time dynamics. Under mildconditions on the potential we prove that the equilibrium points correspondingto the non-negative local minima for $V(\phi)$ are asymptotically stable.Normal forms are employed to obtain approximated solutions associated to theinflection points and the strict degenerate local minimum of the potential. Weprove for arbitrary potentials and arbitrary coupling functions $\chi(\phi),$of appropriate differentiable class, that the scalar field almost alwaysdiverges into the past. It is designed a dynamical system adequate to studyingthe stability of the critical points in the limit $|\phi|\to\infty.$ We obtainthere: radiation-dominated cosmological solutions; power-law scalar-fielddominated inflationary cosmological solutions; matter-kinetic-potential scalingsolutions and radiation-kinetic-potential scaling solutions. Using themathematical apparatus developed here, we investigate the important examples ofhigher order gravity theories $F(R) = R + \alpha R^2$ (quadratic gravity) and$F(R) =R^n.$ We illustrated both analytically and numerically our principalresults. In the case of quadratic gravity we prove, by an explicit computationof the center manifold, that the equilibrium point corresponding to de Sittersolution is locally asymptotically unstable (saddle point).