Realspace and redshift space correlation function questions

[1ytrewq]

why is the redshift space correlation function smaller than the realspace correlation function at small scales and the opposite on large scales?

 

[marcus]

could you give a link to some online source where the correlations are defined and this inequality is exhibited? that way everybody will know what you are talking about.

 

[1ytrewq]

well the 2-point correlation function $$\xi$$(r) is simply the Fourier transform of the power spectrum and in astronomy it is defined as the excess probability of finding a pair of galaxies at separation r.
the correlation function can be separated into a function of 2 variables by decomposing r as r= sigma + pi which represent perpendicular and parallel to the line of sight respectively.

this paper goes over it a bit: http://iopscience.iop.org/0004-637X/479/1/82/pdf/0004-637X_479_1_82.pdf

i made a plot of \xi(r) vs logr and \xi(sigma,pi) vs logr and found that for small scales, the correlation function in redshift space was smaller than the correlation function in real space and the opposite was true for large scales. I was wondering why this was the case?

 

[marcus]

Thanks,
for what it's worth here is my unauthoritative reaction (eventually someone else will reply, I expect).

I'd say, if I understand you, that this is the kind of thing that happens in all kinds of contexts whenever you have something like a density and you have two ways to plot it and a nonlinear map from one variable to the other.

Change of variable.

 

[sardar]

The redshift space correlation function and the realspace correlation function are two different measures used to study the distribution of matter in the universe. The redshift space correlation function is calculated using the observed redshifts of galaxies, while the realspace correlation function is calculated using the actual distances between galaxies.

At small scales, the redshift space correlation function is smaller than the realspace correlation function because of the effects of peculiar velocities. Peculiar velocities are the random motions of galaxies within clusters and groups, caused by the gravitational interactions between them. These peculiar velocities cause an apparent displacement of galaxies along the line of sight, resulting in a smaller observed distance between galaxies in redshift space compared to their actual distance in realspace. This leads to a smaller correlation function in redshift space.

On the other hand, at large scales, the redshift space correlation function is larger than the realspace correlation function. This is because at larger scales, the effects of peculiar velocities are averaged out, and the overall distribution of galaxies is dominated by the underlying large-scale structure of the universe. In this case, the redshift space correlation function provides a better measure of the true clustering of galaxies, resulting in a larger correlation function compared to the realspace correlation function.

In summary, the difference in the redshift space and realspace correlation functions at different scales is due to the effects of peculiar velocities and the underlying large-scale structure of the universe. By studying both correlation functions, we can gain a better understanding of the distribution of matter in the universe and the processes that govern its evolution.