The Two-Point Correlation Function of the Hawaii-Hubble Deep Field-North

Jonathan Blazek, Harvard University

Mentors: David Sanders and Peter Capak

The two-point correlation function is an important measure of structure in the universe. In its angular form, w(θ), it is defined by the expression δP = N[1+wθ)]δΩ where δP is the probability of finding a second object at an angular separation of θ from a given object within an area of δΩ, and N is the mean object density (per steradian). The spatial correlation function can be obtained by converting from angular to spatial separations. The correlation function represents an "excess probability" above what would be expected for a random distribution of equivalent density. Accurate determination of the two-point correlation function provides a test for different components of theoretical cosmological models. In particular, it can provide limits on dark matter distribution, mean baryon density, and formation processes of galaxies and clusters. The power spectrum, used to characterize the Cosmic Microwave Background, is related to the correlation function by the Fourier transform. Our aim was to test and develop a new method for calculating the correlation function and to search for trends in various parameters such as galaxy luminosity, spectral type, and redshift. While the original goal was to utilize data from the new COSMOS survey, delays in data reduction, a process in which I was separately able to take part, prevented such an analysis. Instead, data from the Hawaii-Hubble Deep Field-North (H-HDF-N) was used. Although not as large as the COSMOS survey, the H-HDF-N field is both broad (0.19 square degrees) and quite deep, (z_ab < 25), allowing us to reach a redshift of 5. Its size makes it ideal for correlation analysis, reducing error from cosmic variance, permitting us to test larger separations, and making it possible to search further back in time for redshift evolution. It is intended that the methods tested and developed in this project will be used on COSMOS data once it becomes available.

The central aspect of this project was exploring a new method for binning objects for correlation function calculation. In studies of this type, the standard binning technique has been to group galaxies primarily by magnitude. Magnitude distributions then allow a reasonable statistical conversion from magnitude to redshift. This process is complicated even for shallow surveys and can make tracking redshift evolution challenging. Our goal was to develop a technique that would work easily for deep surveys, such as H-HDF-N or COSMOS. Using redshift, calculated photometrically for the entire catalog, we created bins of equal comoving volume. These bins divided the catalog into about 40 redshift slices out to z = 5, although for this analysis the data was only usable to z = 2 due to photometric depth. The correlation function was then calculated for bins in magnitude and galaxy type. The actual calculation of the correlation function was done using the Euclidian Spatially Inhomogenous Correlation Estimator (eSpICE - Szapudi et al. 2001). This program employs Fourier transforms to greatly increase the speed of the process without losing accuracy. Thus, our binning method, which would likely have required prohibitive amounts of computing time using older correlation estimators, became viable. Once the correlation was estimated, a power law was fit to the data. To measure "shot noise" error, we calculated the correlation for catalogs with 10% of the data points randomly removed. A more thorough error analysis should be done in the future.

Results show that our methods are quite promising. Although the new binning technique makes comparison with previous data somewhat challenging, there is reasonable agreement with several previous studies. We observed an overall trend for stronger correlation with increasing redshift, especially among older (redder) galaxies. We also found that older galaxies tend to cluster more than younger ones. The results of our magnitude binning were less conclusive and should be examined more carefully. The literature on this topic corroborates our findings regarding galaxy age and magnitude but is less supportive regarding redshift evolution. It is our hope that using deeper surveys and new binning technique allows us to more accurately probe this redshift evolution and perhaps find trends missed in previous studies. Unfortunately, while our binning process simplifies many issues, it also reduces the number of objects used in each calculation, increasing the errors in the result. Thus, the power-law fits are not ideal, making comparison between different magnitude and galaxy-type groups less conclusive. Nevertheless, with some refinement and improved error analysis, we are hopeful that applying these methods to the COSMOS survey will yield valuable results.

Figure 1 - The angular correlation function is shown for the redshift bin z = 0.41 to 0.53. The data is fit with a power law. Error bars calculated with random sampling are shown.

Figure 2 - The comoving spatial correlation functions for the redshift bins out to z = 2.03 are plotted together. Redshift is indicated by color (color changes from black-blue-green-yellow-red as redshift increases). Note how correlation strength increases with redshift. A false data point with representative error is placed in the lower left of the plot.

Figure 3 - The amplitude of the correlation function (the intercept of the best-fit line) is plotted for two different spectral type bins. The x-axis corresponds to old (red) galaxies, while the y-axis correseponds to young (blue) galaxies. Each point represents a particular redshift bin, following the same color scheme as in Figure 2. Points would fall along the solid line if correlation amplitude were the same for both groups and would fall along the dashed line if amplitude were ten times larger for old (red) galaxies. Note that old (red) galaxies appear to be more strongly correlated and that their correlation increases with redshift.