We study flat FLRW $alpha$-attractor $mathrm{E}$- and $mathrm{T}$-models by introducing a dynamical systems framework that yields regularized unconstrained field equations on two-dimensional compact state spaces. This results in both illustrative fig
ures and a complete description of the entire solution spaces of these models, including asymptotics. In particular, it is shown that observational viability, which requires a sufficient number of $e$-folds, is associated with a solution given by a one-dimensional center manifold of a past asymptotic de Sitter state, where the center manifold structure also explains why nearby solutions are attracted to this `inflationary attractor solution. A center manifold expansion yields a description of the inflationary regime with arbitrary analytic accuracy, where the slow-roll approximation asymptotically describes the tangency condition of the center manifold at the asymptotic de Sitter state.
We discuss dynamical systems approaches and methods applied to flat Robertson-Walker models in $f(R)$-gravity. We argue that a complete description of the solution space of a model requires a global state space analysis that motivates globally coveri
ng state space adapted variables. This is shown explicitly by an illustrative example, $f(R) = R + alpha R^2$, $alpha > 0$, for which we introduce new regular dynamical systems on global compactly extended state spaces for the Jordan and Einstein frames. This example also allows us to illustrate several local and global dynamical systems techniques involving, e.g., blow ups of nilpotent fixed points, center manifold analysis, averaging, and use of monotone functions. As a result of applying dynamical systems methods to globally state space adapted dynamical systems formulations, we obtain pictures of the entire solution spaces in both the Jordan and the Einstein frames. This shows, e.g., that due to the domain of the conformal transformation between the Jordan and Einstein frames, not all the solutions in the Jordan frame are completely contained in the Einstein frame. We also make comparisons with previous dynamical systems approaches to $f(R)$ cosmology and discuss their advantages and disadvantages.
This paper treats nonrelativistic matter and a scalar field $phi$ with a monotonically decreasing potential minimally coupled to gravity in flat Friedmann-Lema^{i}tre-Robertson-Walker cosmology. The field equations are reformulated as a three-dimensi
onal dynamical system on an extended compact state space, complemented with cosmographic diagrams. A dynamical systems analysis provides global dynamical results describing possible asymptotic behavior. It is shown that one should impose emph{global and asymptotic} bounds on $lambda=-V^{-1},dV/dphi$ to obtain viable cosmological models that continuously deform $Lambda$CDM cosmology. In particular we introduce a regularized inverse power-law potential as a simple specific example.
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 equation
s 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 one-dimensional 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 Pade approximants and compare with the slow-roll approximation.
We consider the familiar problem of a minimally coupled scalar field with quadratic potential in flat Friedmann-Lema^itre-Robertson-Walker cosmology to illustrate a number of techniques and tools, which can be applied to a wide range of scalar field
potentials and problems in e.g. modified gravity. We present a global and regular dynamical systems description that yields a global understanding of the solution space, including asymptotic features. We introduce dynamical systems techniques such as center manifold expansions and use Pade approximants to obtain improved approximations for the `attractor solution at early times. We also show that future asymptotic behavior is associated with a limit cycle, which shows that manifest self-similarity is asymptotically broken toward the future, and give approximate expressions for this behavior. We then combine these results to obtain global approximations for the attractor solution, which, e.g., might be used in the context of global measures. In addition we elucidate the connection between slow-roll based approximations and the attractor solution, and compare these approximations with the center manifold based approximants.
