Difference between 3D real space and 3D redshift space? With regards to cosmological simulations (e.g., N-body dark matter only simulations), I've seen people talk about the positions of objects (halos, galaxies) in the simulation in terms of 3D real space and 3D redshift space. The idea is that the positions of objects in 3D real space does not add the contribution of redshift space distortions, whereas 3D redshift space does. But since redshift space distortions are a real physical effect (i.e., the positions/velocities of objects will be affected not only by cosmological expansion but also gravitational interactions), why is 3D real space ever even considered? Is it some kind of highly idealistic cosmology-dependent only scenario, ignoring peculiar velocities?
Difference between 3D real space and 3D redshift space?
  
  Why is 3D real space ever even considered? Is it some kind of highly idealistic cosmology-dependent only scenario, ignoring peculiar velocities?


Simple explanation:

3D real space is the actual distance, position, velocity, (even spin), etc. of an object; this can be used to measure Earth's distance to our Sun or another planet's distance to its Sun.

3D redshift space is the apparent distance, a distorted view by which one can compare one object to another. The distortion is caused by varying density of the interstellar medium, gravity, motion of Earth and the objects being viewed, even dark matter, etc.

The real distance between two objects can, for example, affect each other by gravity; this alters the trajectory of the objects. The apparent distance has no effect on gravity, only the real distance.

By comparing the real distance, obtained by making calculations, with the apparent distance, obtained by making observations, we can compare what we see with what we calculate; this permits checking the theory, finding new objects (or phenomena, such as suggesting that there is dark matter). This gives us the path lengths for each direction and distance (because distance varies depending on direction and distance due to gravity).

A more complicated explanation is offered in:


"A Spherical Harmonic Analysis of Redshift Space" (Sept 13 1994), by A.F. Heavens and A.N. Taylor


  "1 INTRODUCTION
  An accurate three-dimensional map of the galaxy distribution would be enormously valuable for cosmology, but the lack of an accurate estimator of distance precludes this. Despite best efforts, the errors in the Tully-Fisher and Faber-Jackson distance indicators mean that the best one can do is to use the recession velocity $v$, or redshift, of a galaxy, and to assign its distance $s$ by using the Hubble expansion law $v = H_0s$, where $H_0$ is the Hubble parameter. The resulting `redshift space' map is of course not perfect, because galaxies are not necessarily moving precisely with the idealised expansion of a homogeneous universe. Density inhomogeneities have associated with them peculiar velocities, which introduce a radial distortion between the true (real-space) map and the redshift space map. This distortion is, of course, inconvenient: for example, extracting the power spectrum of density fluctuations from a galaxy redshift survey becomes a non-trivial task. However, the distortion itself can be exploited, as the magnitude of the distortion is dependent on the density parameter of the Universe $\Omega_0$, if fluctuations grow by gravitational instability. Constraining $\Omega_0$ is one of the main aims of this paper.".

"Design and analysis of redshift surveys" (May 27 1997), by A.F. Heavens and A.N. Taylor


  "1 INTRODUCTION
  ...  [This omitted portion: Similar to the above quote, but not identical]
  It is clear that the longest wavelength which can be measured is limited by the size of the survey, so it is attractive to consider surveys which are essentially one- or two-dimensional, to maximise at least one dimension without incurring prohibitive cost in observation time (e.g. Broadhurst et al 1990). The difficulty with such an approach as a method for measuring the power spectrum is that a low-dimensional power measurement at a given wavenumber will have a contribution (which may be dominant) from much smaller scales in three dimensions (e.g. Kaiser &amp; Peacock 1991). The interpretation of the observed power spectrum can therefore be difficult. For surveys which do not correspond to the ‘distant-observer’ approximation (cf. Kaiser 1987), the power spectrum measurement and the redshift distortion become linked, and this further complicates the analysis.
  
  The ease with which the parameters of interest may be extracted depends on the choice of coordinate system and basis functions in which the density field is expanded. For two reasons the choice of spherical polar coordinates is compelling. Firstly the survey is almost certain to be defined in terms of a fixed areal coverage (independent of depth), and secondly, a flux limited survey will have a selection function φ (or, equivalently, a mean observed density
  $ρ_0(r)$) which is dependent on distance, but not on direction. The mean density of the survey is then separable in spherical coordinates $\overline{ρ}(r) = ρ_0(r) M(θ, ϕ)$, where $M$ is either 1 or 0 depending on whether the direction $(θ, ϕ)$ is in the survey or not.
  The second reason is that, unless $β ≪ 1$, it is impossible to ignore redshift distortion effects, and since the distortion between the real-space map and the redshift-space map is purely radial, it is straightforward to include the distortion in a power-spectrum analysis in spherical coordinates (cf Zaroubi &amp; Hoffman 1996).".



More simply, this webpage describing Halton Arp's image:



Our position (the Earth) is at the bottom point in all cases. Distance (away from the Earth) is measured along the straight edges. In the top left image, we show what a galaxy cluster in Arp's universe would look like without the big bang perspective.  It is a family of galaxies and quasars and gaseous clouds of mixed redshifts (in the top diagrams, the large dots are low- redshift, the medium-sized dots are medium-redshift, and the small dots are high redshift). At the center, there is a dominant galaxy -- it's usually the largest galaxy, and the galaxy with the lowest redshift of the cluster. This galaxy is surrounded by low-to-medium redshift galaxies, and toward the edges of the cluster we find the highest redshift galaxies, HII regions, BL Lac objects and quasars.

The image to the right shows what happens if we try to force the same galaxy cluster into a redshift-equals-distance relationship. The cluster becomes distorted. What was once a sphere becomes an elongated bubble. The central dominant galaxy drops to the front of this bubble, followed by a spike of low-to-medium redshift galaxies stretching away from the earth and "bubble and void" of high redshift objects.

Every cluster in the sky does this, like fingers of god pointed at the earth from every direction. The third image is a 90 degree slice of the sky showing all galaxies arranged according to their redshift-  determined distances. The Fingers of God distortions show clearly, each representing a single galaxy cluster. (The bubbles and voids are not as clear, because this chart cuts off before it gets to high redshift.) Everything points at the Earth.

Without the redshift-equals-distance distortion, a new picture of galaxy clusters and the universe itself is revealed. The age of the universe is no longer known, because we no longer have a constant expansion to backtrack to a bang. The size is also unknown. Most quasars and some galaxies that we see are closer than we thought they were, because they have been distorted by the Fingers of God. But we have no idea how far the universe stretches beyond our telescopes' limits.

Compare the real space in the upper left with the observed image in the upper right.