What if the Friedmann equation told how the (square of) fractional growth rate of spatial volume relates to the energy density, instead of describing how the (square of) fractional growth rate of spatial distance relates to energy density? Just two slightly different ways of describing the same physical relationship--how scale and energy density interact and change over time. Maybe someone here has seen the Friedmann formulated that way, and can give us a pointer to it. I don't recall seeing it. And I think it's really just on the level of a fun exercise--by which one might get to know the Friedmann a little better. I'll just consider the spatially flat case, with matter dominant over radiation. To keep it simple and manageable, think of radiation as a negligible part of the overall energy density. The cosmological curvature constant Λ is just that, here, (not related to any imagined form of "dark energy".) The ordinary Friedmann: H(t)2 - Λ/3 = [const]ρ(t) where H = a'/a is the fractional growth rate of the (linear) scale factor, and [const] is some constant which relates the energy density ρ to the excess square expansion rate (eser) over and above the residual amount inherent in spacetime geometry. You can see that as density goes to zero the excess square expansion rate (eser) goes to zero and the growth rate H must decline and approach Λ/3 in the limit. H∞2 = Λ/3 H(t)2 - H∞2 = [const]ρ(t) The present energy density ρnow determines the present excess square expansion rate Hnow2 - H∞2 = [const]ρnow Since by assumption: H(t)2 - H∞2 = [const]ρnow/a(t)3 we have: H(t)2 - H∞2 = (Hnow2 - H∞2 )/a(t)3 this is the kind of thing we could "transpose into a different key". We'd see how it would work when, instead of being about the linear scale factor a(t) and its fractional growth rate a'/a it is adapted to be the equivalent equation about a volumetric scale factor f(t) and its fractional growth rate f'/f. Suppose we have a function f(t) that tracks volume as the U expands. What "Friedmann" equation would it satisfy? We'd better not call the fractional growth rate f'(t)/f(t) of volume by the letter H(t). Might cause confusion. Maybe we can call it J(t) the "Jubble" instead of the "Hubble"? Or perhaps we don't need to. Maybe the volume growth can be described by 3H(t) using the same growth rate function H(t) that we use for distances. The growth rate might not care whether it is used to describe growth of distances or volumes---except for a factor of 3. I'll leave this here for the time being. Nice to get some response, if any be forthcoming.