We investigate the $f(R)$ theory of gravity with broken diffeomorphism due to the change of the coefficient in front of the total divergence term in the ($3+1$) decomposition of the scalar curvature. We perform the canonical analysis of this theory and show that its consistent form, i.e. with no unphysical degrees of freedom, is equivalent to the low-energy limit of the nonprojectable $f(R)$ Hořava-Lifshitz theory of gravity. We also analyze its cosmological solutions and show that the de Sitter solution can be obtained also in the case of this broken symmetry. The consequences of the proposed theory on the asymptotic solutions of a few specific models in the cosmological context are also presented.

• Received 18 December 2015

DOI:https://doi.org/10.1103/PhysRevD.93.104020