1. Galaxy clustering II 2-point correlation function 5 Feb 2013

2. The 2-point correlation function The two-point correlation function ξ (r ): One way to describe the tendency of galaxies to cluster together If we make a random choice of two small volumes V 1 and V 2, and the average spatial density of galaxies is n per cubic megaparsec, then the chance of finding a galaxy in V 1 is just nV 1. If galaxies tend to clump together, then the probability that we then also have a galaxy in V 2 will be greater when the separation r 12 between the two regions is small. We write the joint probability of finding a galaxy in both volumes as if ξ (r ) > 0 at small r, then galaxies are clustered, whereas if ξ (r ) < 0, they tend to avoid each other. Sparke & Gallagher 2007

3. We generally compute ξ (r ) by estimating the distances of galaxies from their redshifts, making a correction for the distortion introduced by peculiar velocities. Observationally it has been found that on scales r<=10h −1 Mpc, the 2- point correlation function takes roughly the form ξ (r ) ≈ (r/r 0 ) −γ, γ > 0 r 0 is the correlation length When r < r 0, the probability of finding one galaxy within radius r of another is significantly larger than for a strictly random distribution. Since ξ (r ) represents the deviation from an average density, it must at some point become negative as r increases. Sparke & Gallagher 2007

4. The two-point correlation function ξ (r ) for galaxies in the 2dF survey. The correlation length r 0 ≈ 5h −1 Mpc – 6h −1 Mpc for the ellipticals, which are more strongly clustered, – smaller for the star-forming galaxies The slope γ ≈ 1.7 For r 0 >~50h −1 Mpc, which is roughly the size of the largest wall or void features, ξ (r ) oscillates around zero: the galaxy distribution is fairly uniform on larger scales. Ellis et al. 2002, MNRAS The correlation function is not very useful for describing the one-dimensional filaments or two-dimensional walls. If our volume V 1 lies in one of these, the probability of finding a galaxy in V 2 is high only when it also lies within the structure. Since ξ (r ) is an average over all possible placements of V 2, it will not rise far above zero once the separation r Exceeds the thickness of the wall or filament (use of three-point and four-point correlation functions?) We do not yet have a good statistical method to describe the strength and prevalence of walls and filaments.

5. Power spectrum The Fourier transform of ξ (r) is the power spectrum P(k) so that small k corresponds to a large spatial scale. Since ξ (r ) is dimensionless, P(k) has the dimensions of a volume. The function sin(kr)/kr is positive for |kr| < π, and it oscillates with decreasing amplitude as kr becomes large so, very roughly, P(k) will have its maximum when k−1 is close to the radius where ξ (r ) drops to zero.

6. Variance Another way to describe the non-uniformity of the galaxy distribution is to ask how likely we are to find a given deviation from the average density. We can write the local density at position x as a multiple of the mean level ρ(x) = [1 + δ(x)] Let δ R be the fractional deviation δ(x) averaged within a sphere of radius R When we take the average δR over all such spheres, this must be zero. Its variance measures how clumpy the galaxy distribution is on this scale.

7. Three simple model calculations of 2-point correlation function A more practical way of defining the 2-point correlation function is where N p (r) is the number of pairs of galaxies whose separations r lie in the interval (r -Δr, r + Δr), and N p Poisson (r) is the number of pairs corresponding to a Poisson distribution for the same volume considered. when a distribution of N points of density n(r) in a spherical volume is analysed, the number of pairs (for small distances) separated by r + dr will be ~ To obtain ξ, therefore, one divides by the number of pairs of the Poisson distribution having the same number of points in the same volume. Combes et al. 2004

8. Simple model for a “pancake” Suppose we start with a homogeneous distribution of points in a sphere of radius R and density ξ(r) = 0 Imagine now that all the particles are 'sampled' to form an infinite, flat disc (a pancake) of the same radius and surface density Then, from the definition of ξ(r): Simple model for a “filament” An analogous calculation in the case where the particles are 'sampled' according to a diameter (a filament) of density θ and radius R Power-law Combes et al. 2004

9. Simple model for hierarchical structure Consider a sphere of radius R Place in this N spheres of radius R/λ In each of these place Ν spheres of radius R/ λ 2, and so on to L levels At the final level L there are thus N L points in N L-1 spheres each of radius r 0 =R/λ L-1 The scale of clusters at the level K (counted from the scale r o ) will be r~r ο λ Κ and contain Ν Κ particles. The mean density in each of these clusters is thus Power law Combes et al. 2004