Cosmological functions and their relation

In the non-metric gravity proposed by K. Krasnov, this is a theory in which the scalar constraint of the Ashtekar’s formalism is modified in such way that the cosmological constant is replaced by a cosmological function that depends of the canonical variables $A_a{}^i$ and $E_i{}^a$, the General Relativity (GR) is the particular case when the cosmological function is a constant. Some years ago inspired by this theory Rosas-Rodríguez proposed two cosmological functions, one for the Ashtekar’s formalism and the other for the ADM formalism. In this paper we show that this cosmological functions are related thought the three-dimensional Ricci scalar.