Abstract
The paper considers a cosmographical approach to analyze cosmological models. Cosmography is a method to describe the kinematics of the cosmological expansion based only on the cosmological principle. We consider a method of treating free parameters of a cosmological model in terms of the directly observable cosmographic values related to the time-derivatives of the Hubble parameter (deceleration, jerk, snap). The method is applied to analyze two cosmological models involving the time-dependence of the cosmological constant in the form Λ(t)→Λ(H) when this approach is especially efficient. Both models interpret the dark energy in the form of the cosmological constant as energy of physical vacuum, which is currently the most supported treatment. The first one means being proportional to the Hubble parameter, and the second one involves a constant and a quadratic term. As a result, the free parameters of both models are expressed in terms of the currently observed values of the Hubble parameter, deceleration, and jerk. The obtained expressions for model parameters are exact, as the method does not involve any additional assumptions. Furthermore, it leads to deal with algebraic equations instead of differential ones. After this procedure, solutions of the evolution equations are obtained in the form of the time-dependence of the Hubble parameter. The obtained model parameters are substituted to the solutions, which are analyzed for a typical range of cosmographic scalars taken from recent observations. Finally, the proposed approach is used to eliminate free parameters from both models and to obtain constraints for the directly observable cosmographic values that can be tested to correspond to present observations data. For the considered cases, such constraints are received respectively for the jerk and the snap parameters with respect to the deceleration. The constraint for the linear model is compared with current observational value ranges for the deceleration and the jerk parameters.
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