In CMB papers, I often find that "perturbations" can be decomposed into scalar, vector, and tensor perturbations, with equations looking something like $$ \Pi_{ab} = \Pi_{ab}^{(S)} + \Pi_{ab}^{(V)} + \Pi_{ab}^{(T)}. $$ These are then referred to as density fluctuations, vorticity, and gravitational waves. I understand that they are just defined based on their rotational properties and can have various sources. However, I've struggled to find what these are perturbations to.
The fact that scalar perturbations are density fluctuations make me think they are perturbations to the stress-energy tensor. But if tensor perturbations are gravitational waves, that sounds like perturbations to the metric. I imagine that the use of the perturbation is either $$ g_{ab} = g_{ab,0} + \Pi_{ab} $$ or $$ T_{ab} = T_{ab,0} + \Pi_{ab}. $$ It seems unintuitive to me that it would be possible to add perturbations to the metric to perturbations to the stress-energy tensor.