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We analyse spacetimes with a conformal scalar field source, a cosmologicalconstant and a quartic self-interaction term for the scalar field. We alsoconsider additional matter contents in the form of Maxwell and Yang-Millsfields or radiation fluids. Existence theorems for weakly asymptotically flatspacetimes are given. We give a generalisation of Bekenstein's result [Ann.Phys. 82, 535 (1974)] and use it to derive existence theorems for spacetimesthat contain an isotropic singularity. The results are combined to suggest amathematical setup for Penrose's CCC scenario using a conformal scalar fieldcosmology.

Conformal scalar fields, isotropic singularities and conformal cyclic cosmologies