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I would like to clarify something that mixes cosmology and relativistic effects. Maybe I'm not understanding something or maybe there a difference of vocabulary between the cosmological and the relativistic people.

If you ask a cosmologist at which scale the relativistic effects appears (for example in N-body simulations), he will probably answer you that a problem may occur at the scale of the horizon (so at the scale of the observable Universe). If you ask the same question to a specialist of the backreaction or a specialist of general relativity I suspect that he will answer you that a problem may occur at the scale of matter inhomogeneities (so at the scale of the cosmic web, filaments and voids).

My question is : why are there these 2 point of views ?

I suspect that is because a cosmologist hypothesizes an homogeneous space-time background and apply a perturbation theory on this background whereas for a people from general relativity there is no such background. Is it the answer or I don't understand something ?

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Most of the confusion comes from the use of the word horizon. In cosmology, when you deal with the motion equations for the perturbations you have in it two different scales. For example, in the Mukhanov-Sasaki variable $v_k$ one has $$v_k^{\prime\prime}+\left(c_s^2k^2-\frac{a^{\prime\prime}}{a}\right)v_k = 0,$$ where $c_s$ is the speed of sound $a$ the scale factor, $k$ is the mode and ${}^\prime$ the derivative with respect to the conformal time. Unless $c_s^2k^2 \gg a^{\prime\prime}/a$ perturbation "see" the expansion. In this case what controls if the scale is "relativistic" or not is the value of $a^{\prime\prime}/a$.

In some simple cases $a^{\prime\prime}/a \propto (c/H)^{-2}$ so the scale in question is the Hubble radius $c/H$, the problem is that in these cases the horizon also coincides with $c/H$. For this reason some people use the term horizon to describe the "relativistic" scales.

In general, there is no characteristic scale for problems in cosmology. You have to deal with each motion equation to study which scale is appropriated in each case.

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I don't think it is that useful to characterize this issue as a difference between the relativity and cosmology communities as there is quite a substantial overlap between those two communities. There are published arguments on both sides of the debate. To illustrate my point here is a paper by a "relativist" arguing for the N-body simulations: http://arxiv.org/abs/1111.2997 and here is a paper by a "cosmologist" arguing against them: http://arxiv.org/abs/0901.4566

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