# Realspace and redshift space correlation function questions

• 1ytrewq
In summary, the redshift space correlation function is smaller than the realspace correlation function at small scales and larger at large scales. This is due to the decomposition of the correlation function into two variables, perpendicular and parallel to the line of sight, and the nonlinearity of the map between them. A link to an online source discussing this concept is provided and a plot is referenced to support the observation. This phenomenon can be seen in various contexts involving nonlinear maps and changes of variables.

#### 1ytrewq

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

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.

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?

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.

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.

## 1. What is the difference between realspace and redshift space correlation function?

The realspace correlation function is a measure of the spatial distribution of galaxies in the universe, while the redshift space correlation function takes into account the observed redshift of galaxies due to the expansion of the universe. The redshift space correlation function is distorted compared to the realspace correlation function due to the effects of peculiar velocities.

## 2. How is the correlation function used to study large-scale structure in the universe?

The correlation function is used to study the clustering of galaxies and the distribution of matter on large scales in the universe. By measuring the clustering of galaxies at different distances from each other, scientists can infer information about the underlying cosmological model and the growth of structures in the universe.

## 3. What is the significance of the shape of the correlation function?

The shape of the correlation function can reveal information about the underlying physical processes that govern the formation and evolution of galaxies and large-scale structures. For example, a steep correlation function at small scales indicates a strong clustering of galaxies, while a flat correlation function at large scales suggests a homogenous distribution of matter.

## 4. How does the correlation function change with cosmic time?

The correlation function evolves with cosmic time as the universe expands and structures grow. In the early universe, the correlation function is expected to be more uniform due to the homogenous distribution of matter, while in later times, it becomes more clustered as gravity pulls matter together to form structures.

## 5. How is the correlation function measured?

The correlation function is typically measured by counting the number of galaxy pairs at different distances from each other and comparing it to a random distribution. This is done using large galaxy surveys such as the Sloan Digital Sky Survey or the Dark Energy Survey. The correlation function can also be obtained through theoretical models and simulations.