Thawing models in the presence of a generalized Chaplygin gas

Sergio del Campo, Carlos R. Fadragas, Ramón Herrera, Carlos Leiva, Genly Leon, and Joel Saavedra

Phys. Rev. D 88, 023532 – Published 26 July 2013

Abstract

In this paper we consider a cosmological model whose main components are a scalar field and a generalized Chaplygin gas. We obtain an exact solution for a flat arbitrary potential. This solution have the right dust limit when the Chaplygin parameter $A \to 0$ . We use the dynamical systems approach in order to describe the cosmological evolution of the mixture for an exponential self-interacting scalar field potential. We study the scalar field with an arbitrary self-interacting potential using the “Method of $f$ -devisers.” Our results are illustrated for the special case of a coshlike potential. We obtain that the usual scalar-field-dominated and scaling solutions cannot be late- time attractors in the presence of the Chaplygin gas (with $α > 0$ ). We recover the standard results at the dust limit ( $A \to 0$ ). In particular, for the exponential potential, the late-time attractor is a pure generalized Chaplygin solution mimicking an effective cosmological constant. In the case of arbitrary potentials, the late-time attractors are de Sitter solutions in the form of a cosmological constant, a pure generalized Chaplygin solution or a continuum of solutions, when the scalar field and the Chaplygin gas densities are of the same orders of magnitude. The different situations depend on the parameter choices.

Received 5 April 2013

DOI:https://doi.org/10.1103/PhysRevD.88.023532

Authors & Affiliations

Sergio del Campo^1,*, Carlos R. Fadragas^2,†, Ramón Herrera^1,‡, Carlos Leiva^3,§, Genly Leon^1,∥, and Joel Saavedra^1,¶

¹Instituto de Física, Pontificia Universidad de Católica de Valparaíso, Casilla 4950, Valparaíso, Chile
²Departamento de Física, Universidad Central de Las Villas, 54830 Santa Clara, Cuba
³Departamento de Física, Universidad de Tarapacá, Avenida General Velásquez 1775, Código Postal 1000007, Arica, Chile

^*sdelcamp@ucv.cl
^†fadragas@uclv.edu.cu
^‡ramon.herrera@ucv.cl
^§cleivas@uta.cl
^∥genly.leon@ucv.cl
^¶Joel.Saavedra@ucv.cl

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Vol. 88, Iss. 2 — 15 July 2013

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Images

Figure 1
(a) The value of $ω / s_{0}^{2}$ vs $Ω_{ϕ}$ assuming a nearly flat potential and $ω \sim - 1$ . The dotted (red) line corresponds to the approximated solution discovered in 12 (Eq. (23) in 12) and the dash-dotted (dark) line corresponds to the approximated solution, (28), for $χ = 0.1$ , presented here. (b) EOS parameter of the scalar field for the exponential potential with slope $s_{0} = 0.8029$ . We have considered the initial conditions $γ (1) = 0.1$ , $Ω_{ϕ} (1) = 0.7$ , $s (1) = 0.8029$ . The scale factor is normalized to 1 at present. The continuous (green) line corresponds to the exact value of $ω (a)$ for model with solely a scalar field; the dashed (blue) line corresponds to the exact value of $ω (a)$ for model that includes the Chaplygin gas; the dotted (red) line corresponds to the approximated solution found in 12 (Eq. (23) in 12) and the dash-dotted (dark) line corresponds to the approximated solution, (28), presented here (assuming $χ = 0.1$ ).Reuse & Permissions
Figure 2
The phase space of the system (33). Without lack of generality we use $α = 0.5$ . (a) $λ = 0$ . $B$ and $C$ are local sources; $A$ is a saddle. $F$ coincides with $D$ (contained in the curve $G$ ) and it is stable but not asymptotically stable for $f (0) \geq 0$ ; otherwise, it is a saddle (see Appendix ). $E$ does not exist. Any arbitrary point in the curve $G$ (that exists only for $λ = 0$ ) is stable, attracting an open set of orbits. The center manifold of $K$ is stable, as is $K$ . (b) $λ = 1$ . $B$ and $C$ are local sources; $A$ and $D$ are saddles and $K$ is the attractor. $D$ is a local attractor in the invariant set $z = 0$ . The solutions at the $x − y$ plane correspond to those in Fig. 2 in 34. (c) $λ = 2$ . $B$ and $C$ are local sources; $A$ , $D$ , and $E$ are saddles, and $K$ is the attractor. $E$ is a local attractor in the invariant set $z = 0$ . The solutions at the $x − y$ plane correspond to those in Fig. 3 in 34. (d) $λ = 3$ . $B$ (the kinetic-dominated solution with $λ x > 0$ ) is a saddle point, $C$ (the kinetic-dominated solution with $λ x < 0$ ) is the local source (unstable node), $D$ does not exist, $A$ and $E$ are saddles, and $K$ is the attractor. $E$ is a local (spiral) attractor in the invariant set $z = 0$ . The solutions at the $x − y$ plane correspond to those in Fig. 4 in 34.Reuse & Permissions
Figure 3
Projections of some orbits in the phase space of the system (47) for $ξ = 1$ . Without lack of generality we use $α = 0.5$ . Observe that $B (\pm ξ)$ and $C (\pm ξ)$ are local sources and the point $F$ located at the curve $G$ is a local attractor. The curve $K$ is stable, but not asymptotically stable. The scalar-field-dominated solution $D (\pm ξ)$ are of the saddle type, as well as the rest of the (curves of) fixed points. The scaling solutions $E (\pm ξ)$ do not exist.Reuse & Permissions
Figure 4
Projections of some orbits in the phase space of the system (47) for $ξ = 2$ . Without lack of generality we use $α = 0.5$ . Note that $B (\pm ξ)$ and $C (\pm ξ)$ are local sources and the point $F$ located at the curve $G$ is a local attractor. The curve $K$ is stable, but not asymptotically stable. The scalar-field-dominated solution $D (\pm ξ)$ and the scaling solution $E (\pm ξ)$ are of the saddle type, as well as the rest of the (curves of) fixed points.Reuse & Permissions
Figure 5
Projections of some orbits in the phase space of the system (47) for $ξ = 3$ . Without lack of generality we use $α = 0.5$ . $B (\pm ξ)$ and $C (\pm ξ)$ are local sources and the point $F$ located at the curve $G$ is a local attractor. The curve $K$ is stable, but not asymptotically stable. The scalar-field-dominated solution $D (\pm ξ)$ and the scaling solution $E (\pm ξ)$ are of the saddle type, as well as the rest of the (curves of) fixed points.Reuse & Permissions

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