## [1003.4531] Manifestly Covariant Gauge-invariant Cosmological Perturbation Theory

 Authors: P. G. Miedema, W. A. van Leeuwen Abstract: It is shown that a first-order cosmological perturbation theory for the open, flat and closed Friedmann-Lemaitre-Robertson-Walker universes admits one, and only one, gauge-invariant variable which describes the perturbation to the energy density and which becomes equal to the usual Newtonian energy density in the non-relativistic limit. The same holds true for the perturbation to the particle number density. Using these two new variables, a new manifestly gauge-invariant cosmological perturbation theory has been developed. Density perturbations evolve diabatically. Perturbations in the total energy density are gravitationally coupled to perturbations in the particle number density, irrespective of the nature of the particles. There is, in first-order, no back-reaction of perturbations to the global expansion of the universe. Small-scale perturbations in the radiation-dominated era oscillate with an increasing amplitude, whereas in older, less precise treatments, oscillating perturbations are found with a decreasing amplitude. This is a completely new and, obviously, important result, since it makes it possible to explain and understand the formation of massive stars after decoupling of matter and radiation. [PDF]  [PS]  [BibTex]  [Bookmark]

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Anze Slosar
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### [1003.4531] Manifestly Covariant Gauge-invariant Cosmologic

Savvas Nesseris
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### [1003.4531] Manifestly Covariant Gauge-invariant Cosmologic

Hi Anze,
Actually, I am a bit confused myself as well. I only had a quick look at the paper, but my attention fell on eqs (56) and (57) that describe the evolution of perturbations of the energy density $\delta$ in the "new" case and the standard case respectively.

There is clearly a huge difference in these two eqs and (if I understood it correctly - see Section I.B) the reason they give is that the pressure should depend both on the number n and the energy density $\epsilon$, ie $p=p(n,\epsilon)$, as they claim the perturbations actually do not evolve adiabatically. They explain why they believe this is so in a previous paper 0805.0421, see text near eq. (193)/page 30. Their result is that only in the non-relativist limit do the perturbations evolve adiabatically.

I might have missed or misunderstood something, but this is what I got after having a quick look. I can't say I am really convinced but I might have a closer look.

Cheers,
Savvas

PS By the way, the previous paper 0805.0421, doesn't seem to be published anywhere yet. I'm not sure about what this means.

rishi khatri
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### [1003.4531] Manifestly Covariant Gauge-invariant Cosmologic

My understanding is that adiabaticity is an initial condition. I guess from a cursory look, assuming the math is correct, what they are effectively doing is evolving a mixture of curvature and entropy perturbations ...

Pieter Miedema
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### [1003.4531] Manifestly Covariant Gauge-invariant Cosmologica

Hi Anze, Rishi and Savas,

The fact that density perturbations do evolve diabatically follows from $p=p(n, \varepsilon)$, thermodynamics (23) and the background Einstein equations (5).

The difference between (56) and (57) is that the general solution of the standard equation (57) contains the gauge function $\psi$. Consequently, the standard equation (57) has no physical meaning. The general solution of (56) is gauge-invariant, and $\delta_\varepsilon$ is the real, physical density contrast, as has been shown in Section IX on the non-relativistic limit. Note that in the large-scale limit, (56) yields the two well-known solutions proportional to $t^{2/3}$ and $t^{-5/3}$: see references below (45).

Regards,
Pieter

Pieter Miedema
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