Systematic errors in weighted two-point correlation functions: an application to interaction-induced star formation

Weighted correlation functions are an increasingly important tool for understanding how galaxy properties depend on their separation from each other. We use a mock galaxy sample drawn from the Millennium Simulation, assigning weights using a simple prescription to illustrate and explore how well a weighted correlation function recovers the true separation dependence of the input weights. We find that the use of a weighted correlation function results in a dilution of the magnitude of any separation dependence of properties and a smearing out of that dependence in radius, compared to the input behaviour. We present a quantitative discussion of the dilution in the magnitude of separation dependence in properties in the special case of a constant enhancement at r < r(c). In this particular case where there was a star formation rate (SFR) enhancement at small radius r < r(c) = 35 kpc, the matching of one member of an enhanced pair with an unenhanced galaxy in the same group gives an artificial enhancement out to large radius similar to 200 kpc. We compare this with the observations of SFR enhancement from the Sloan Digital Sky Survey, finding very similar behaviour - a significant enhancement at radii < 40 kpc and a weak enhancement out to more than 150 kpc. While we explore a particular case in this paper, it is easy to see that the phenomenon is general, and precision analysis of weighted correlation functions will need to account carefully for this effect using simulated mock catalogues.