Ben Gold wrote:

Ah, I think I see. If you play with a(t) you're effectively changing Einstein's equations and you need to also go and change the equations for the metric perturbations. But if I'm understanding this correctly, the Boltzmann equations should be still be fine. [tex]\dot\delta_{cdm} = -\frac12 \dot h[/tex] in the synchronous gauge regardless, right? But the equation for [tex]\dot h[/tex] is different if [tex](\dot a/a)^2 \ne 8\pi G \rho /3[/tex], and your point is that if you leave the [tex]\dot h[/tex] equation unchanged you'll get inconsistent and non-gauge-invariant results?

Exactly!

One way to see this explicitly is by looking at the superhorizon perturbations. For these perturbations, due to causality, the evolution within a Hubble patch can be obtained by pertubing the Friedmann equations. Therefore, if you change the Friedmann equations (through introducing dark energy or modified gravity), you will also change the equations for super-horizon perturbations. For adiabatic perturbations, this is equivalent to saying that the Bardeen parameter:

[tex]\zeta \equiv \phi + {2(H^{-1}(z)\dot{\phi}+\phi)\over 3(1+w(z))}[/tex]

is constant on super-horizon scales, where \phi is the metric perturbation in the longitudinal gauge.

**Notice that the perturbation equation that you arrive at by taking the derivative of [tex]\zeta[/tex], contains w(z) and its derivative, and thus, does not only depende on H(z).**
While this gives a unique way of modifying perturbation equations when you modify the background evolution, it cannot be uniquely extended to sub-horizon scales. One may do this phenomenologically by introducing a speed of sound [tex]c^2_{s}[/tex] as the coefficient of the k^2 correction to the perturbation equation. However, things can be much more complicated, as perturbations won't remain adiabatic, and thus there will be additional degrees of freedom, and/or a more general dependence on k.

As Josh was suggesting, may be the easiest self-consistent thing to do is to set [tex]c^2_s=0[/tex], which is equivalent to extending the super-horizon equation (which is derived uniquely) to sub-horizon scales without any change. However, this is physically different from the limit of switching off DE perturbations. Instead, this may be the maximal amount of DE perturbations that you may hope to get, as DE clusters similar to dark matter. [tex]c^2_s=1[/tex], might be the closest physical limit to no perturbations.

There is also the [tex]c^2_s \rightarrow \infty[/tex] limit, but intuitively I think it should break causality.