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.

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