[astro-ph/0503416] Evolution of non-linear cosmological perturbations

Authors:  David Langlois, Filippo Vernizzi
Abstract:  We define fully non-perturbative generalizations of the uniform density and comoving curvature perturbations, which are known, in the linear theory, to be conserved on sufficiently large scales for adiabatic perturbations. Our non-linear generalizations are defined geometrically, independently of any coordinate system. We give the equations governing their evolution on all scales. Also, in order to make contact with previous works on first and second order perturbations, we introduce a coordinate system and show that previous results can be recovered, on large scales, in a remarkably simple way, after restricting our definitions to first and second orders in a perturbative expansion.
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Antony Lewis
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[astro-ph/0503416] Evolution of non-linear cosmological pert

Post by Antony Lewis » March 24 2005

This paper derives an exact expression for the evolution of a combination of the comoving scale factor perturbation W_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 \zeta_a^L = e^\alpha \zeta_a used in the paper, H is the comoving local Hubble rate, and the local scale factor is S\equiv e^\alpha. 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 \rightarrow 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 \equiv 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 \zeta_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...

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