An Analytical description for the cosmological constant

From the principles of quantum cosmologies we can justify the reason for an inverse-square law for the cosmological constant with no conflict with observations. Although this general expression for $\Lambda$ is well known in the literature, in this work we introduce some analytical solutions for the scale factor different from the literature. The underlying motivation to carry out these solutions is the fact that the time variation of $\Lambda$ can lead to a creation of matter and/or radiation such as to help in the investigation of the age of the Universe. The knowledge of the scale factor behavior might shed some light on these questions since the entire evolution of a homogeneous isotropic Universe is contained in the scale factor.