The dynamical system approach to scalar field cosmology

Abstract

A spatially flat FLRW universe (motivated by inflation) is studied; by a dimensional reduction of the dynamical equations of scalar field cosmology, it is demonstrated that a spatially flat universe cannot exhibit chaotic behaviour. The result holds when the source of gravity is a non-minimally coupled scalar field, for any self-interaction potential and for arbitrary values of the coupling constant with the Ricci curvature. The phase space of the dynamical system is studied, and regions inaccessible to the evolution are found. The topology of the forbidden regions, their dependence on the parameters, the fixed points and their stability character, and the asymptotic behaviour of the solutions are studied. New attractors are found, in addition to those known from the minimal coupling case, certain exact solutions are presented and the implications for inflation are discussed. The equation of state is not prescribed a priori , but rather is deduced self-consistently from the field equations.

Possible end of the universe in a finite future from dark energy withw<−1

“…Only the two variables H and φ are needed to describe the dynamics of the system (10)- (12), and the phase space is a two-dimensional manifold with a rather complex structure [24]. This is best seen by rewriting the field equations as [24] …”

Energy conditions and classical scalar fields

“…≡ φ 0 , one has If instead φ 2 = φ 2 c ≡ 1/(κξ) with ξ > 0, then it must be V 0 = 0, R = 0, and V ′ 0 = 0; the potential V (φ) must have a zero and horizontal tangent in ±φ c . These critical scalar field values have been studied in scalar field cosmology for the potential .9) and for ξ = 1/6; in this case all the solutions with constant Ricci curvature are classified [41]. For Lorentzian wormholes without potential V , all the solutions corresponding to the critical scalar field values are also classified [22].…”

Untitled

“…[72], only two equations of the set (4.9)-(4.11) are independent, and the system can be reduced to a two-dimensional phase space manifold with variables (H, φ). It is then straightforward to verify that the solutions…”

“…The case φ = ±φ 1 not considered so far corresponds to a class of solutions with constant Ricci curvature containing a de Sitter representative [72]. However, the latter is clearly fine-tuned and unstable with respect to perturbations ∆φ.…”

“…[72]), quantum cosmology [90,31], classical and quantum wormholes [37], and constitutes a line of approach to the cosmological constant problem [91,27,92].…”