Global dynamics and asymptotics for monomial scalar field potentials and perfect fluids

Abstract

We consider a minimally coupled scalar field with a monomial potential and a perfect fluid in flat FLRW cosmology. We apply local and global dynamical systems techniques to a new three-dimensional dynamical systems reformulation of the field equations on a compact state space. This leads to a visual global description of the solution space and asymptotic behavior. At late times we employ averaging techniques to prove statements about how the relationship between the equation of state of the fluid and the monomial exponent of the scalar field affects asymptotic source dominance and asymptotic manifest self-similarity breaking. We also situate the 'attractor' solution in the three-dimensional state space and show that it corresponds to the onedimensional unstable center manifold of a de Sitter fixed point, located on an unphysical boundary associated with the dynamics at early times. By deriving a center manifold expansion we obtain approximate expressions for the attractor solution. We subsequently improve the accuracy and range of the approximation by means of Padé approximants and compare with the slow-roll approximation. * Electronic address:aalho@math.ist.utl.pt † Electronic address:blanca@math.fu-berlin.de ‡ Electronic address:claes.uggla@kau.se arXiv:1503.06994v2 [gr-qc] 4 Aug 2015 1 Some further examples of references that describe minimally coupled scalar field cosmology in dynamical systems settings are [6,7,8], with additional references therein.

Scalar field deformations ofΛCDMcosmology

“…The orbits on Z = 0 are characterized by Ω m = const. They therefore correspond to the periodic orbits found in the monomial case at late times, given in different variables in [15], but in the present case they are interrupted by being cut in half by the line of fixed points, M x 0 , which is a formal consequence of the new time variable. The physical reason for this feature is that in contrast to the monomial case there are no scalar field oscillations at late times for inverse power-law potentials.…”

“…These variables are by no means suitable for all scalar field potentials, e.g., for potentials with a zero minimum such as monomial potentials it is advisable to replace the scalar field variable Z with an H-based variable that takes into account a varying averaged oscillatory time scale at late times, as done in [15]. Nevertheless, the above variables are useful for positive monotonic potentials.…”

“…We have here introduced a notation for the fixed points where the kernel M stands for a massless scalar field state; FL stands for a Friedmann-Lemaître perfect fluid state, while dS stands for a de Sitter state (for interpretation of various types of de Sitter states in scalar field cosmology, see [15]); the kernel PL stands for power law (inflation, when λ < √ 2, since this leads to an accelerating state with q < 0), while EM stands for a scaling solution with an exponential potential and matter in the form of a perfect fluid with a linear equation of state. In addition a superscript describes the value of x, while in the full three-dimensional treatment we also add a subscript that describes the value for Z.…”

On the oscillations and future asymptotics of locally rotationally symmetric Bianchi type III cosmologies with a massive scalar field*

“…Secondly, to better illustrate our results and the utility of the averaging method, we also solve an averaged system of (42)-(44) of the form [∂ τ H, ∂ τ x] T = F 1 (x) + H F [2] (x) together with ∂ τ t = 1/H. 5 The averaging reduces to substituting q → q := 2Σ 2 + + Ω/2 cos(t − ϕ) 2 → 1/2 sin(t − ϕ) cos(t − ϕ) → 0 in (42)-(44).…”

Inflationary α -attractor cosmology: A global dynamical systems perspective

“…As discussed in [15,16], to obtain explicit expressions for future asymptotics, one can use available approximations for late stage behavior when V ∼ φ 2n , or one can use the averaging techniques developed in [16]. The overall global solution structure for the E-models is depicted in Fig.…”