Global dynamics and inflationary center manifold and slow-roll approximants

Abstract

We consider the familiar problem of a minimally coupled scalar field with quadratic potential in flat Friedmann-Lemaître-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 Padé 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.

Inflationary α -attractor cosmology: A global dynamical systems perspective

“…Arguably, the most interesting fixed point is dS, which is a center saddle, with a one-dimensional center manifold corresponding to the "attractor solution" or the "inflationary trajectory." To describe this interior state space solution, which originates from dS, we follow [15,16] and perform a center manifold analysis. Since it is more convenient to useT than T, we use the system (15), which results in the following center manifold expansion (without loss of generality, we choose θ ¼ 0 for dS):…”

“…We do so by following the treatment of monomial potentials V ∝ φ 2n in [15] and [16], but adapting the formulation to the particular features of the potentials (2a) and (2b). 2 We derive three complementary dynamical systems since the global one is not optimal for quantitative descriptions in all parts of the state space; instead, the global picture should be seen as a collecting ground which gives the overall picture.…”

“…In contrast, the dynamical systems formalism presented below is applicable to more general situations than the present one, which rather acts as a pedagogical example. With this in mind we will now develop a formalism that addresses the above issues and, at a glance, describes the global situation rigorously and also makes techniques available for describing quantitatively correct approximations for solutions in the past and future regimes (indeed, they can be combined to even give global approximations for the solutions to arbitrary accuracy if one is so inclined, see [15]). …”

Chameleon effect in the Jordan frame of the Brans-Dicke theory

“…In order to make clear what the difference is between both de Sitter solutions, 6 let us note that the Friedmann constraint (36), evaluated at the BD-de Sitter point above, can be written as…”

“…In this case the dynamical systems tools come to our rescue. The dynamical systems theory provides powerful tools which are commonly used in cosmology to extract essential information on the dynamical properties of a variety of cosmological models, in particular, those models where the scalar field plays a role [16,[24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43].…”