Abstract
Scalar field cosmologies with a generalized harmonic potential and a matter fluid with a barotropic equation of state (EoS) with barotropic index gamma for the locally rotationally symmetric (LRS) Bianchi I and flat Friedmann–Lemaître–Robertson–Walker (FLRW) metrics are investigated. Methods from the theory of averaging of nonlinear dynamical systems are used to prove that time-dependent systems and their corresponding time-averaged versions have the same late-time dynamics. Therefore, the simplest time-averaged system determines the future asymptotic behavior. Depending on the values of gamma , the late-time attractors of physical interests are flat quintessence dominated FLRW universe and Einstein-de Sitter solution. With this approach, the oscillations entering the system through the Klein–Gordon (KG) equation can be controlled and smoothed out as the Hubble parameter H – acting as time-dependent perturbation parameter – tends monotonically to zero. Numerical simulations are presented as evidence of such behavior.
Highlights
Mathematical methods have been widely used in cosmology
In reference [1] the method of Lie symmetries was applied to Wheeler–De Witt equation in Bianchi class A cosmologies for minimally coupled scalar field gravity a e-mail: genly.leon@ucn.cl b e-mail: sebastian.cuellar01@ucn.cl c e-mail: esteban.gonzalezb@usach.cl d e-mail: samuel.lepe@pucv.cl e e-mail: claudio.ramirez@ce.ucn.cl f e-mail: alfredo.millano@alumnos.ucn.cl and hybrid gravity in general relativity (GR)
In reference [2] a model-independent criterion based on first integrals of motion was used; and in reference [3] dynamical symmetries of the field equations were used to classify dark energy (DE) models in the context of scalar field Friedmann– Lemaître–Robertson–Walker (FLRW) cosmologies
Summary
Mathematical methods have been widely used in cosmology. For example, in reference [1] the method of Lie symmetries was applied to Wheeler–De Witt equation in Bianchi class A cosmologies for minimally coupled scalar field gravity a e-mail: genly.leon@ucn.cl (corresponding author) b e-mail: sebastian.cuellar01@ucn.cl c e-mail: esteban.gonzalezb@usach.cl d e-mail: samuel.lepe@pucv.cl e e-mail: claudio.ramirez@ce.ucn.cl f e-mail: alfredo.millano@alumnos.ucn.cl and hybrid gravity in general relativity (GR). The method is useful to study the conformal algebra of Kasner spacetime and other Bianchi type I matter solutions of GR. In reference [42] LRS Bianchi type III cosmologies with a massive scalar field were studied by means of the theory of averaging of nonlinear dynamical systems. Interacting scalar field cosmologies with generalized harmonic potentials for flat and negatively curved FLRW, and for Bianchi I metrics were studied. 2. For Kantowski–Sachs metric (see references [44,45,46,47,48,49]) in paper III [21] was proved that latetime attractors of full and time-averaged systems are: two anisotropic contracting solutions if 0 ≤ γ < 2, a non-flat. In Appendix A the proof of our main theorem is given and in Appendix B numerical evidence supporting the results of Sect. 4 is presented
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