Cosmological flows on hyperbolic surfaces

Abstract

We outline the geometric formulation of cosmological flows for FLRW models with scalar matter as well as certain aspects which arise in their study with methods originating from the geometric theory of dynamical systems. We briefly summarize certain results of numerical analysis which we carried out when the scalar manifold of the model is a hyperbolic surface of finite or infinite area.

Modular inflation at higher levelN

“…The phenomenological analysis shows that this model is consistent with the data obtained from the CMB satellites [7,18], and the reheating epoch after j-inflation was considered in [19]. Subsequent work em-phasized the hyperbolic geometry underlying modular inflation in the context of α-attractors [20,21,22,23,24], and hyperbolic inflation [25,26,27,28], while the discussion of reheating after j-inflation was continued in ref. [29].…”

“…In the present paper we consider such groups in the context of inflationary models. In principle one might consider other classes of groups, such as Fuchsian groups for which a rich geometric theory exists that has been considered in discussions concerned with attractor theory in [22,23,24]. For such groups however the theory of modular forms and functions has not been developed sufficiently to lend itself to the framework of inflation with modular invariant potentials.…”

Two-field cosmological α-attractors with Noether symmetry

“…Due to the non-holonomic character of this equation, the resulting geometric system of ODEs for φ I does not generally admit a non-constrained Lagrangian formulation. This system of ODEs defines a dissipative geometric dynamical system on the tangent bundle of the scalar manifold, which can be studied with the methods of dynamical systems theory[36]. In particular, symmetries of the cosmological model could be studied directly at this level using Lie's theory of symmetries of systems of ODEs, which in this setting has an elegant geometric formulation.…”

Hidden symmetries of two-field cosmological models

“…It would also be interesting to explore whether the present work can be useful for a wider program (which was touched upon briefly in reference [16]) aimed at studying multifield cosmological models with methods from the geometric theory of dynamical systems (see [19] for an introduction). As pointed out in [16], the dynamics of such models is quite rich and in particular it is amenable to certain methods originating in asymptotic analysis. It would be interesting to gain a deeper understanding of the simplifications which the presence of a Hessian symmetry may afford in that context.…”

“…Scalar multifield models were considered in cosmology from various points of view (see, for example, [5][6][7][8][9][10][11]), including numerically [12][13][14] and to some extent phenomenologically [15]. However, current insight into their dynamics (which, as outlined in [16][17][18], is quite interesting from the perspective of the geometric theory of dynamical systems [19]) is far more limited than for one-field models. In particular, numerous questions regarding the behavior of such models have not been studied systematically.…”