Conformal metric perturbations and boundary term as physical source

In the context of the Relativistic Quantum Geometry formalism, where the cosmological constant is promoted to a dynamical variable by attributing it a geometric interpretation as a result of a flux on the boundary of a manifold and establishing a connection between the perturbations of the Ricci tensor and the metric tensor, we propose an approach for the perturbations of the metric tensor. From this, imposing that Einstein's equations must hold for the tensors defined from the perturbed quantities obtained from conformal transformations, we derive a functional form for the cosmological parameter $\Lambda$ in terms of the cosmological parameter $\bar{\Lambda}$ of the perturbed manifold. We then use the obtained equations to propose a cosmological model based on the Friedmann-Lema\^{i}tre-Robertson-Walker metric with no spatial curvature, fitting the free parameters using observational data from Hubble and Type Ia Supernovae. The model is statistically comparable to $\Lambda$CDM; although, the joint analysis produces a smaller $H_{0}^{\rm Conformal}=69.80\rm\,\, Km \,s^{-1}\,Mpc^{-1}$ in contrast to the flat $\Lambda$CDM result $H_{0}^{\Lambda\rm CDM}=70.52\rm\,\, Km \,s^{-1}\,Mpc^{-1}$. An evident singularity occurs when the conformal factor $\Xi^{2}=2$, yields an early universe dominated only by matter $\rho_{m}$, which undoubtedly does not correspond to a viable history of our cosmos. Despite these limitations, a specific scenario remains feasible. This study aims to offer insights into the acceleration of the universe and addresses key questions in contemporary cosmology.