_{a}and comoving density perturbation X

_{a}, Eq. 6. This checks out, and it's certainly very pretty. Note the definition Eq. 5 is not frame invariant, in that both W

_{a}and X

_{a}must be evaluated in the comoving frame for the combination to be frame invariant.

I think the result generalizes trivially to any conserved non-interacting component of a multi-component fluid, where W

_{a}and X

_{a}are evaluated in the rest frame of that fluid.

Linearizing, but putting in the explicit dependence on the heat flux q

_{a}I get

[tex]\zeta_a^L{}' = \frac{-H}{(\rho+p)}\Gamma + \frac{S^2D_aD^b q_b}{3(\rho+p)}[/tex]

where I defined ζ

_{a}

^{L}= e

^{α}ζ

_{a}used in the paper, H is the comoving local Hubble rate, and the local scale factor is S≡ e

^{α}. Fine.

First question, can you also write W

_{a}explicitly including the q

_{a}dependence? More generally, is W

_{a}actually a real useful observable? It depends on the choice of initial hypersurface, and usually it only enters the equations via its locally observable time derivative (which describes the rate of change of local volume elements). In linear theory, under a change of frame u

_{a}→ u

_{a}+ v

_{a}

[tex]h_a' \rightarrow h_a' + \frac{1}{3}S^2 D_a D^bv_b - (Hv_a)'[/tex]

where I defined the comoving h

_{a}≡ S W

_{a}, so the transformation law for h

_{a}(and hence W

_{a}) seems to depends on a nasty integral of the new velocity. So constructing frame invariant quantities from it seems to be difficult locally. (and this is just in linear theory!)

Moving on, how do the authors justify calling their quantity a 'curvature perturbation'? For a non-perturbative generalization of the uniform-density-hypersurface curvature perturbation, I would have thought a minimal requirement would be for the linearlized quantity (with zero vorticity), to reduce to something proportional to the (linear-theory) frame-invariant curvature perturbation

[tex]\hat{\eta}_a\equiv \frac{1}{2}SD_a {}^{(3)}R + \frac{2 SD_a D^b X_b}{3(\rho+p)}[/tex]

where [tex]{}^{(3)}R[/tex] is the 3-Ricci scalar (=[tex]\mathcal{K}[/tex] in this paper). However it does not, and the evolution equation for [tex]\hat{\eta}[/tex] differs from Eq. 6 by a linear term involving the comoving shear. So what is the relation between ζ

_{a}

^{L}and the curvature tensor, if any?

Similarly, their definition of R

_{a}does not reduce to the comoving curvature perturbation

[tex]\bar{\eta}_a \equiv \frac{1}{2}S D_a {}^{(3)}R -\frac{2H}{\rho+p} D_a D^b q_b.[/tex]

Note that you do have to be a bit careful about using W

_{a}for conservation laws. For example on large scales (no matter flows), W

_{a}evaluated in the frame of constant number density (of a conserved species) is exactly conserved. This is true, but probably not very useful since it's true by definition!

One other potentially interesting thing to do is look at the the h

_{a}= W

_{a}= 0 gauge - the frame of constant number densities. Defining [tex]\mathcal{R}\equiv 2\kappa\rho - 2\theta^2/3[/tex], neglecting vorticity and anisotropic stress, in the h

_{a}= W

_{a}= 0 frame I get

[tex]S^{-2}(S^2 \mathcal{R})\dot{} = -2 D^a D^b \sigma_{ab} + 4 A^a D^b \sigma_{ba} + \frac{4}{3}\theta\sigma_{ab}\sigma^{ab}[/tex]

which generalizes the result that in linear theory the large scale curvature perturbation in the constant number density frame is conserved (c.f. astro-ph/0208055). Here [tex]A_a = \dot{u}_a[/tex], [tex]\mathcal{R}[/tex] is zero in an exact flat FRW universe, and the vorticity and anisotropic stress terms can be put back in if desired.

PS. equations above I derived with u

_{a}u

^{a}= 1 signature, so some signs may differ...