Reconstruction, thermodynamics and stability of the ΛCDM model in f (T, τ) gravity

We reconstruct the Lambda CDM model for f(T, tau) theory, where T is the torsion scalar and. the trace of the energy-momentum tensor. The result shows that the action of Lambda CDM is a combination of a linear term, a constant (-2 Lambda) and a nonlinear term given by the product root TFg [(T-1/3/16 pi G) (16 pi G tau + T + 8 Lambda)], with F-g being a generic function. We show that to maintain conservation of the energy-momentum tensor, we should impose that F-g [y] must be linear on the trace tau. This reconstruction decays in f (T) theory for F-g Q, with Q a constant. Our reconstruction describes the cosmological eras to the present time. The model present stability within the geometric and matter perturbations for the choice F-g y, where y (T-1/3/16 pi G)(16 pi G tau + T+ 8 Lambda), except for the geometric part in the de Sitter model. We impose the first and second laws of thermodynamics to Lambda CDM and find the condition where they are satisfied, that is, T-A, G(eff) > 0, however where this is not possible in the cases that we choose, this leads to a breakdown of positive entropy and Misner-Sharp energy.