Cosmological acceleration in models with dynamic $\Lambda$-member for the Lyra diversity

Background. Models with a dynamic cosmological $\Lambda(t)$ member have attracted scientists' attention since the cosmological constant problem received a valid confirmation of $\Lambda(t)$ decline with time. The purpose of this paper is to construct and study cosmological models in the Lyra geometry assuming the minimal interaction of matter with the displacement vector field and dynamic $\Lambda$-member. Materials and methods. To derive the modified Friedmann equations of homogeneous and isotropic cosmological models in the Lyra geometry, the assumption of the time dependence of the cosmological member is used. Mathematical ansätze and phenomenological laws of the cosmological member evolution are applied to obtain the exact solutions. Results. New dynamic equations and their exact solutions in cosmological models for the Lyra diversity are obtained. Exact solutions of the equations of model dynamics under different equations of the state of the matter filling the universe and certain assumptions regarding the cosmological member evolution are found. It is shown that the interaction of the dynamic cosmological member with the displacement vector is able to preserve the continuity equations for the ordinary unchanged matter. The possibility of accelerated expansion in our models is established. Conclusions. The obtained results show the new properties of cosmological models with dynamic $\Lambda$-member for the Lyra diversity compared to standard models and open up new possibilities in the study of the accelerated universe expansion phenomenon in the modern era. The results are expected to be used in theoretical cosmology and astrophysics.