Constructing a family of conformally flat scalar field models

Using purely geometrical methods we present a mechanism to solve the scalar field equations of motion (non-minimally coupled with gravity) in a spherically symmetric background. We found that the full set of spacetimes, which are of Petrov type O (conformally flat) and admit a gradient conformal vector field, can be determined completely. It is shown that the full group of scalar field equations reduced to a single equation that depends only on the distance w=r2−t2 leaving the metric function (equivalently the functional form of the scalar field or the potential) freely chosen. Depending on the structure of the metric or the potential V (as a function of φ) a solution can be found either analytically or via numerical integration. We provide physically sound examples and prove that (Anti)-de Sitter fits this scheme. We also reconstruct a recently found solution (Strumia and Tetradis 2022 J. High Energy Phys. JHEP09(2022)203) representing an expanding scalar bubble with metric that has a singularity and corresponds to what is termed as Anti-de Sitter crunch.