On Efficiency and Localisation for the Torsion Function

We consider the torsion function for the Dirichlet Laplacian −Δ, and for the Schrödinger operator −Δ + V on an open set ${\Omega }\subset \mathbb {R}^{m}$ Ω ⊂ ℝ m of finite Lebesgue measure $0<|{\Omega }|<\infty $ 0 < | Ω | < ∞ with a real-valued, non-negative, measurable potential V. We investigate the efficiency and the phenomenon of localisation for the torsion function, and their interplay with the geometry of the first Dirichlet eigenfunction.

Cosmological inflation driven by a scalar torsion function

A viable model for inflation driven by a torsion function in a Friedmann background is presented. The scalar spectral index in the interval $$0.92\lesssim n_{s}\lesssim 0.97$$ 0.92 ≲ n s ≲ 0.97 is obtained in order to satisfy the initial conditions for inflation. The post inflationary phase is also studied, and the analytical solutions obtained for scale factor and energy density generalizes that ones for a matter dominated universe, indicating just a small deviation from the standard model evolution. The same kind of torsion function used also describes satisfactorily the recent acceleration of the universe, which could indicate a possible unification of different phases, apart form specific constants

Acceleration in Friedmann cosmology with torsion

A Friedmann like cosmological model in Einstein–Cartan framework is studied when the torsion function is assumed to be proportional to a single $$\phi (t)$$ϕ(t) function coming just from the spin vector contribution of ordinary matter. By analysing four different types of torsion function written in terms of one, two and three free parameters, we found that a model with $$\phi (t)=- \alpha H(t) \big ({\rho _{m}(t)}/{\rho _{0c}}\big )^n$$ϕ(t)=-αH(t)(ρm(t)/ρ0c)n is totally compatible with recent cosmological data, where $$\alpha $$α and n are free parameters to be constrained from observations, $$\rho _m$$ρm is the matter energy density and $$\rho _{0c}$$ρ0c the critical density. The recent accelerated phase of expansion of the universe is correctly reproduced by the contribution coming from torsion function, with a deceleration parameter indicating a transition redshift of about 0.65.