Reconstructing modified gravity with holographic vacuum energy density

A reconstruction of modified gravity is proposed by establishing a correspondence between the effective density of the modified gravity and the holographic density. The non-homogeneous term in the modified Friedmann equation, generated by the vacuum (holographic) energy density, lead to reconstructed models that contain explicitly, as part of the solution, the Einstein-Hilbert term. It was shown that the $\mathrm{\Lambda }$CDM-type cosmic histories allow the general relativity Lagrangian with cosmological constant as a particular reconstructed solution. The Starobinsky term appears in some reconstructed solutions, and an approximate reconstruction of the Chaplygin gas cosmology was performed in terms of elementary functions of curvature.