Scaling solutions in Robertson-Walker spacetimes

Abstract

We investigate the stability of cosmological scaling solutions describing a barotropic fluid with p = (γ − 1)ρ and a non-interacting scalar field φ with an exponential potential V (φ) = V0e −κφ . We study homogeneous and isotropic spacetimes with non-zero spatial curvature and find three possible asymptotic future attractors in an ever-expanding universe. One is the zero-curvature power-law inflation solution where Ω φ = 1 (γ < 2/3, κ 2 < 3γ and γ > 2/3, κ 2 < 2). Another is the zero-curvature scaling solution, first identified by Wetterich, where the energy density of the scalar field is proportional to that of matter with Ω φ = 3γ/κ 2 (γ < 2/3, κ 2 > 3γ). We find that this matter scaling solution is unstable to curvature perturbations for γ > 2/3. The third possible future asymptotic attractor is a solution with negative spatial curvature where the scalar field energy density remains proportional to the curvature with Ω φ = 2/κ 2 (γ > 2/3, κ 2 > 2). We find that solutions with Ω φ = 0 are never late-time attractors. PACS numbers(s): 04.20.Jb, 98.80.Hw gr-qc/9901014

Stability analysis of an autonomous system in loop quantum cosmology

“…And we can see that there is not any barotropic fluid coupled with the scalar-fielddominated scaling solution. This is different from the dynamical behavior of scalar field with exponential potential V ¼ V 0 expðÀÞ in classical cosmology [5][6][7][8][9][10][11][12][13], and also is different from the properties of the scalar field in brane cosmology [3], in which ¼ const (notice that the definition of in [3] is different from the one in this paper) and À is a function of LððaÞÞ and jVj. In these models, the Universe may enter a stage dominated by a scalar field coupled with fluid when , satisfy some conditions [3,5].…”

“…The dynamical behavior of the scalar field has been discussed by many authors (e.g., see [1,2,[5][6][7][8][9][10][11][12][13]). If one wants to get the potentials that yield the cosmological scaling solutions beyond the exponential potential, one can add a d dN term into the autonomous system [36].…”

“…If we only consider a special case of the potential, like exponential potential [5][6][7][8][9][10][11][12][13], then and À are both constants. In this case, the four-dimensional dynamical system, Eqs.…”

“…In this case, the four-dimensional dynamical system, Eqs. (8)- (11), reduces to a three-dimensional one, since is a constant. (In the classical dynamical system, the z term does not exist, and then the dynamical system is reduced from three dimensions to two dimensions.)…”

“…One always considers the dynamical behavior of a scalar field with an exponential potential in the classical cosmology [5][6][7] or modified cosmology [8,9]. And, if one considers the dynamical behavior of a scalar field coupled with a barotropic fluid, the exponential potential is also the first choice [10][11][12][13]. The exponential potential V leads to the fact that the variables À ¼ VV 00 =V 02 equal 1 and that ¼ V 0 =V is also a constant.…”