Jacobi stability analysis of scalar field models with minimal coupling to gravity in a cosmological background

We perform the study of the stability of the cosmological scalar field models, by using the Jacobi stability analysis, or the Kosambi-Cartan-Chern (KCC) theory. In the KCC approach we describe the time evolution of the scalar field cosmologies in geometric terms, by performing a "second geometrization", by considering them as paths of a semispray. By introducing a non-linear connection and a Berwald type connection associated to the Friedmann and Klein-Gordon equations, five geometrical invariants can be constructed, with the second invariant giving the Jacobi stability of the cosmological model. We obtain all the relevant geometric quantities, and we formulate the condition of the Jacobi stability for scalar field cosmologies in the second order formalism. As an application of the developed methods we consider the Jacobi stability properties of the scalar fields with exponential and Higgs type potential. We find that the Universe dominated by a scalar field exponential potential is in Jacobi unstable state, while the cosmological evolution in the presence of Higgs fields has alternating stable and unstable phases. By using the standard first order formulation of the cosmological models as dynamical systems we have investigated the stability of the phantom quintessence and tachyonic scalar fields, by lifting the first order system to the tangent bundle. It turns out that in the presence of a power law potential both these models are Jacobi unstable during the entire cosmological evolution.Comment: 24 pages, 14 figures, accepted for publication in Advances in High Energy Physics, special issue "Dark Physics in the Early Universe

Series solution of the time-dependent Schr\"{o}dinger-Newton equations in the presence of dark energy via the Adomian Decomposition Method

The Schr\"{o}dinger-Newton model is a nonlinear system obtained by coupling the linear Schr\"{o}dinger equation of canonical quantum mechanics with the Poisson equation of Newtonian mechanics. In this paper we investigate the effects of dark energy on the time-dependent Schr\"{o}dinger-Newton equations by including a new source term with energy density $\rho_{\Lambda} = \Lambda c^2/(8\pi G)$, where $\Lambda$ is the cosmological constant, in addition to the particle-mass source term $\rho_m = m|\psi|^2$. The resulting Schr\"{o}dinger-Newton-$\Lambda$ (S-N-$\Lambda$) system cannot be solved exactly, in closed form, and one must resort to either numerical or semianalytical (i.e., series) solution methods. We apply the Adomian Decomposition Method, a very powerful method for solving a large class of nonlinear ordinary and partial differential equations, to obtain accurate series solutions of the S-N-$\Lambda$ system, for the first time. The dark energy dominated regime is also investigated in detail. We then compare our results to existing numerical solutions and analytical estimates, and show that they are consistent with previous findings. Finally, we outline the advantages of using the Adomian Decomposition Method, which allows accurate solutions of the S-N-$\Lambda$ system to be obtained quickly, even with minimal computational resources.Comment: 20 pages, 1 table, 8 figure

Mindfulness-based cognitive therapy v. group psychoeducation for people with generalised anxiety disorder: randomised controlled trial

Background: Research suggests that an 8-week mindfulness-based cognitive therapy (MBCT) course may be effective for generalised anxiety disorder (GAD). Aims: To compare changes in anxiety levels among participants with GAD randomly assigned to MBCT, cognitive–behavioural therapy-based psychoeducation and usual care. Method: In total, 182 participants with GAD were recruited (trial registration number: CUHK_CCT00267) and assigned to the three groups and followed for 5 months after baseline assessment with the two intervention groups followed for an additional 6 months. Primary outcomes were anxiety and worry levels. Results: Linear mixed models demonstrated significant group × time interaction (F(4,148) = 5.10, P = 0.001) effects for decreased anxiety for both the intervention groups relative to usual care. Significant group × time interaction effects were observed for worry and depressive symptoms and mental health-related quality of life for the psychoeducation group only. Conclusions: These results suggest that both of the interventions appear to be superior to usual care for the reduction of anxiety symptoms