Testing a varying-$\Lambda$ model for dark energy within Co-varying Physical Couplings framework

The Co-varying Physical Couplings (CPC) framework is a modified gravity set up assuming Einstein Field Equations wherein the quantities $\{G,c,\Lambda\}$ are promoted to space-time functions. Bianchi identity and the requirement of stress-energy tensor conservation entangle the possible variations of the couplings $\{G,c,\Lambda\}$, which are forced to co-vary as dictated by the General Constraint (GC). In this paper we explore a cosmological model wherein $G$, $c$ and $\Lambda$ are functions of the redshift respecting the GC of the CPC framework. We assume a linear parametrization of $\Lambda$ in terms of the scale factor $a$. We use the ansatz $\dot{G}/G = \sigma \left( \dot{c}/c \right)$ with $\sigma =$ constant to deduce the functional forms of $c=c(z)$ and $G=G(z)$. We show that this varying-$\{G,c,\Lambda\}$ model fits SNe Ia data and $H(z)$ data with $\sigma = 3$. The model parameters can be constrained to describe dark energy at the background level.