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.

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.

We consider the wave equation, g ψ = 0, in fixed flat Friedmann-Lemaître-Robertson-Walker and Kasner spacetimes with topology R + × T 3 . We obtain generic blow up results for solutions to the wave equation towards the Big Bang singularity in both backgrounds. In particular, we characterize open sets of initial data prescribed at a spacelike hypersurface close to the singularity, which give rise to solutions that blow up in an open set of the Big Bang hypersurface {t = 0}. The initial data sets are characterized by the condition that the Neumann data should dominate, in an appropriate L 2 -sense, up to two spatial derivatives of the Dirichlet data. For these initial configurations, the L 2 (T 3 ) norms of the solutions blow up towards the Big Bang hypersurfaces of FLRW and Kasner with inverse polynomial and logarithmic rates respectively. Our method is based on deriving suitably weighted energy estimates in physical space. No symmetries of solutions are assumed.

This is the published version of a paper published in Physical Review D: covering particles, fields, gravitation, and cosmology.Citation for the original published paper (version of record):Alho, A., Uggla, C. (2017) Inflationary alpha-attractor cosmology: A global dynamical systems perspective. We study flat Friedmann-Lemaître-Robertson-Walker α-attractor E-and 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 figures 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 particular 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 slowroll approximation asymptotically describes the tangency condition of the center manifold at the asymptotic de Sitter state. Physical

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 covering state space adapted variables. This is shown explicitly by an illustrative example, f (R) = R + αR 2 , α > 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. * Electronic address:aalho@math.ist.utl.pt † Electronic address:sante.carloni@tecnico.ulisboa.pt ‡ Electronic address:claes.uggla@kau.se 1 arXiv:1607.05715v2 [gr-qc]