Background independent exact renormalization group for conformally reduced gravity

Within the conformally reduced gravity model, where the metric is parametrised by a function f(phi) of the conformal factor phi, we keep dependence on both the background and fluctuation fields, to local potential approximation and O(partial derivative(2)) respectively, making no other approximation. Explicit appearances of the background metric are then dictated by realising a remnant diffeomorphism invariance. The standard non-perturbative Renormalization Group (RG) scale k is inherently background dependent, which we show in general forbids the existence of RG fixed points with respect to k. By utilising transformations that follow from combining the flow equations with the modified split Ward identity, we uncover a unique background independent notion of RG scale, (k) over cap . The corresponding RG flow equations are then not only explicitly background independent along the entire RG flow but also explicitly independent of the form of f. In general f (phi) is forced to be scale dependent and needs to be renormalised, but if this is avoided then k-fixed points are allowed and furthermore they coincide with (k) over cap -fixed points.