Conformal structure of singularities in some varying fundamental constants bimetric cosmologies

In this paper, we explore the conformal structure of singularities arising from varying fundamental constants using the method of Penrose diagrams. We employ a specific type of bimetric model featuring two different metrics. One metric describes the causal structure for matter, while the other characterizes the causal structure for gravitational interactions, which is related to variations in fundamental constants such as the gravitational constant and the speed of light. For this reason, we focused on the gravitational metric to calculate the conformal transformation and compose Penrose diagrams for the singularities arising from the varying fundamental constants. We have shown that, in one case, the parameter such as the scale factor, the density and the pressure resemble those of the finite scale factor singularity (FSF). Despite singularity appears in constant conformal time in our case and in the case of FSF the Misner-Sharp horizon looks different. Our another case is similar to sudden future singularity (SFS), but there are differences in the conformal structures. We have also shown that in our cases the behavior of Misner-Sharp horizon strongly depends on initial conditions. The last analytical solution which we introduced is identical to conformal structure of the standard exotic singularity for the matter.