In this work, we investigate a cosmological model with the tachyon and fermion fields with barotropic equation of state, where pressure [Formula: see text], energy density [Formula: see text] and barotropic index [Formula: see text] are related by the relation [Formula: see text]. We applied the tachyonization method which allows to consider cosmological model with the fermion and the tachyon fields, driven by special potential. In this paper, tachyonization model was defined from the stability analysis and exact solution standard of the tachyon field. Analysis of the solution via statefinder parameters illustrated that our model in fiducial points with deceleration parameter [Formula: see text] and statefinder [Formula: see text] corresponds to the matter-dominated universe (SCDM) but ends its evolution at a point in the future [Formula: see text] which corresponds to the de Sitter expansion. Comparison of the model parameters with the cosmological observation data demonstrates that our proposed cosmological model is stable at barotropic index [Formula: see text].

In this work, it is shown that the equations of motion of the scalar field for spatially flat, homogeneous, and isotropic space-time Friedmann-Robertson-Walker have a form-invariance symmetry, which is arising from the form invariance transformation. Form invariance transformation is defined by linear function ρ = n 2 ρ in general case. It is shown the method of getting potential and the scalar field for the power law scale factor. The initial model is always stable at exponent of the scale factor α > 1, but stability of the transformation model depends on index n. Slow roll parameters and spectral induces is obtained and at large α they agree with Planck observation data.

The authors investigate cosmological inflation in slow roll framework for the f (R) gravity. The authors got slow roll parameters scalar spectral index n S , tensor spectral index n T and scalar-to-tensor ratio r * for minimal coupling between matter and gravity. Having built their graphs, the authors concluded that the predictions of the Starobinsky model are in good agreement with the measurement of Planck, but allow a wide range of acceptable values for r and nT. It is shown, that connection between cosmographic parameters -deceleration parameter q , jerk j , snap s and lerk l and modern values of the scale factor. The proposed analysis makes it possible to relate the model-independent results obtained from cosmography with theoretically substantiated assumptions of f (R) gravity.