Conformal and pure scale-invariant gravities in d dimensions

We consider conformal and scale-invariant gravities in d dimensions, with a special focus on pure $R^2$ gravity in the scale-invariant case. In four dimensions, the structure of these theories is well known. However, in dimensions larger than four, the behavior of the modes is so far unclear. In this work, we explore this question, studying the theories in conformally flat spacetimes as well as anisotropic backgrounds. First, we consider the pure theory in d-dimensions. We show that this theory propagates no degrees of freedom for flat space-time. Otherwise, we find the theory in the corresponding Einstein frame and show that it propagates a scalar field and two tensor modes, that arise from Einstein's gravity. We then consider conformal gravity in d dimensions. We argue on the number of degrees of freedom for conformally flat space-times and show that for $d>4$, there exists a frame in which this theory can be written as the Weyl-squared gravity with a cosmological constant, and also generalize this formulation to the $f\left(W^2\right)$ theories. Then, we consider the specific model of conformal gravity in five dimensions. We find the analytical and numerical solutions for the anisotropic Universe for this case, which admits super-Hubble and exponential expansions. Finally, we consider the perturbations around these solutions and study the number of the degrees of freedom.