Two-point Angular Correlation Function Research Articles

SAMPLE VARIANCE IN LARGE-SCALE COSMIC MICROWAVE BACKGROUND ANISOTROPY MEASUREMENTS

The effects of limited sky coverage in large-angle cosmic microwave background anisotropy experiments are investigated by computing the variance of the angular two-point correlation function with an incomplete sphere. We find that, assuming a power spectrum of density fluctuations with spectral index n = 1, the Galactic cut of half-width 20° (40°) about the Equator made by the COBE DMR experiment would induce a sample variance on the rms temperature fluctuation [(ΔT/T) rms ]2 (or equivalently, the correlation function at zero lag), which is 12% (38%) greater than the cosmic variance with a whole sky coverage. This result is about two times smaller than the naive expectation that the cosmic variance is enhanced by a factor of [Formula: see text], where A is the solid angle sampled by the experiment. We also find that the sample variance of the correlation function at nonzero lag can approach the cosmic variance limit. Our approach provides an analytic way of finding a theoretical error to the theoretical prediction for a particular experiment (either large- or small-scale), without having recourse to computationally intensive Monte Carlo or maximum likelihood methods.

Clustering of Galaxies in the Hubble Deep Field

We compute the two-point angular correlation function w(θ) for a sample of ~1700 galaxies to a magnitude limit equivalent to R ~ 29.5, using a catalog derived from the Hubble Deep Field images. A nonzero value of w(θ) is measured down to R = 29.0. The amplitude of w(θ) at the bright magnitude limit (R ~ 26) is consistent with previous ground-based observations. At fainter magnitudes the clustering amplitude continues to decrease, but at a slower rate than that predicted by the power law w(1'') ∝ 10-0.27R observed for shallower samples. The observed w(θ) over the magnitude range 20 < R < 29 is consistent with linear evolution of the clustering of a galaxy population which at present has a correlation length r0 of about 4 h-1 Mpc, close to that of local IRAS galaxies. We also investigate the impact that magnification bias induced by weak gravitational lensing may have on our results. Although the observed amplitude of w(θ) can differ from the true amplitude by up to 30%, this effect is not large enough to influence our conclusions. Finally, by using a color-selected sample, we examine whether the expected effects of magnification bias can be used for an independent determination of cosmological parameters in deep images. We conclude that the amplitude of the effect can be large and in some cases even produce an upturn of the amplitude of the correlation with limiting magnitude. However, we find that it is not possible to detect the effects of magnification bias on w(θ) from images alone. If redshift information becomes available, it is possible to measure the effects of magnification bias directly and thus constrain the density parameter Ω0 and the bias factor b.

Evidence for Galaxy Interactions/Mergers from Medium Deep Survey WFPC2 Data

We examine the morphological and statistical properties of close galaxy pairs from two sets of 28 WFPC2 fields, acquired for the Medium Deep Survey (MDS) and the Groth-Westphal Survey (GWS) in F606W (V) and F814W (I) passbands. In the GWS sample all fields have uniform 95% completeness down to I 24.3 mag, whereas in the MDS sample, fields have varying 95% completeness limits in the range I 23.6-24.8 mag. In each field ~400 galaxies per 5 square arcmin field are detected. We exploit high-resolution WFPC2 images to systematically determine morphological classifications of galaxies as disk or bulge dominated down to I ≤ 23 mag (V 24 mag) and to differentiate galaxies from stars 1 magnitude fainter. Down to I ≤ 25 mag the number of galaxy pairs with separations θ ≤ 30 is consistent with a shortward extrapolation of the angular two-point correlation function ω(θ) θ-0.8 observed from the same data; the fraction of such pairs showing morphological evidence for physical association accounts for a third of the total numbers suggested by a shortward extrapolation of ω(θ). The latter result may not be too surprising given the low surface brightness of the tidal tails resulting from galaxy interactions; i.e., much of the evidence for interactions may fall below our detection limit. Moreover, we find no trend between apparent physical association (on the basis of morphology) and (V - I) color or I-magnitude difference between pair members of the θ 30 pair sample. We use recent galaxy redshift surveys to estimate the rate of galaxy merging occurring in the MDS and GWS galaxy pair samples. From this work we find that merging has a moderate dependence on redshift: we derive an estimate for the galaxy pair fraction Pf (1 + z)m, with m = 1.2 ± 0.4 for galaxies with I ≤ 25 mag (zmed 1-2). Two scenarios are consistent both with this low value of m and for the low correlation amplitude: (a) a low-density universe with strong clustering evolution parameterized by a clustering exponent e 1.0 such that galaxy cluster-scale structures shrink relative to the proper coordinate frame; and/or (b) a weakly clustered galaxy population, the majority of which fade or dissipate below zmed 0.5 (I 20 mag), thus mimicking the apparently strong evolution in the correlation amplitude, Aω. Although not directly observed using our data, the possible flattening of the slope of ω(θ) with increasing survey depth can explain the strong decline in Aω and allow for greater pair fraction evolution limited to m ≤ 1.6.

Are Hubble Deep Field Galaxy Counts Whole Numbers?

The Hubble Deep Field (HDF) offers the best view to date of the optical sky at faint magnitudes and small angular scales. Early reports suggested that faint source counts continue to rise to the completeness limit of the data, which implies a very large number of galaxies. In this Letter, we use the two-point angular correlation function and number-magnitude relation of sources within the HDF in order to assess their nature. We find that the correlation peaks between 025 and 04 with amplitude of 2 or greater and is much higher for the smallest objects. This angular scale corresponds to physical scales of order 1 kpc for redshifts z 1. The correlation must therefore derive from objects with subgalactic separations. At faint magnitudes, the counts satisfy the relation number ∝1/flux, which is expected for images that are subdivisions of larger ones. Several explanations for these observed correlations are possible, but a conservative explanation can suffice to produce our results. Since high-redshift space (z 0.5) dominates the volume of the sample, observational redshift effects are important. Rest-frame ultraviolet radiation appears in the HDF's visible and near-UV bands, and surface brightness dimming enhances the relative brightness of unresolved objects versus resolved objects. Both work to increase the prominence of compact star-forming regions over diffuse stellar populations. Thus, a normal gas-rich galaxy at high redshift can appear clumpy and asymmetric in the visible bands. For sufficiently faint and distant objects, the compact star-forming regions in such galaxies peak above undetectable diffuse stellar backgrounds. Our results do not exclude asymmetric formation or fragmentation scenarios.

The correlation function of radio sources

We investigate the large-scale clustering of radio sources in the Green Bank and Parkes-MIT-NRAO 4.85 GHz surveys by measuring the angular two-point correlation function w(\theta). Excluding contaminated areas, the two surveys together cover 70 per cent of the whole sky. We find both surveys to be reasonably complete above 50 mJy. On the basis of previous studies, the radio sources are galaxies and radio-loud quasars lying at redshifts up to z \sim 4, with a median redshift z \sim 1. This provides the opportunity to probe large-scale structures in a volume far larger than that within the reach of present optical and infrared surveys. We detect a clustering signal w(\theta) \approx 0.01 for \theta = 1\degr. By assuming an evolving power-law spatial correlation function in comoving coordinates \xi(r_c,z) = ( r_c / r_0 )^{-\gamma} (1+z)^{\gamma-(3+\epsilon)}, where \gamma = 1.8, and the redshift distribution N(z) of the radio galaxies, we constrain the r_0--\epsilon parameter space. For `stable clustering' (\epsilon = 0), we find the correlation length r_0 \approx 18 Mpc/h, larger than the value for nearby normal galaxies and comparable to the cluster-cluster correlation length.

The Clustering of Faint Galaxies and the Evolution of XI (r)

The two-point angular correlation function, ω(θ), is constructed from a catalog of 13,000 objects in 24 fields distributed over an area of 2 degrees square and complete to a limit of R = 23.5. The amplitude and slope of our correlation function on arcminute scales are in broad agreement with recent CCD results in the literature and decreases with depth. No evidence is found for a flattening of the slope of the correlation function away from δ ~ 0.8. Using the redshift distribution from the recent I-band selected Canada-Fr;ance Redshift Survey, the observed ω(θ) implies a value of the clustering length \(r_0= 1.86 \pm 0.42h^{-1} \textup{Mpc} (q_0 = 0.5)\) at:z = 0.48. This is consistent with the clustering of optically selected local galaxies, if clustering has developed with epoch. Specifically, clustering evolution to a CfA-like sample (with r 0 = 5.5h -1 Mpc ) would require growth in clustering stronger than the type seen in CDM-like models. Evolution to a less clustered IRAS-like sample (with r_0 = 4h -l Mpc) would require evolution represented by stable clusters of fixed physical size. If there was no growth in clustering (i.e galaxies are fixed in comoving space) then a local sample that is very weakly clustered (with r 0 = 3h -l Mpc) would result. These possibilities are discussed in the context of our understanding of the nature of the faint galaxy population.

Galaxy counts and the galaxy two-point angular correlation function to I=23

We present new results on the two-point angular correlation function of galaxies. The results have been determined from two large-format CCD surveys, covering 13 deg2 in total. At all magnitudes the two-point correlation function is well described by a power law with an index, J, equal to - 0.8. The amplitude of the correlation function decreases with increasing magnitude; however, the precise behaviour is not reproduced in models that include reasonable amounts of clustering evolution. We also present new data on the I-band galaxy number counts. A model that fits the count data well is a model that has a cosmological constant and a local galaxy number density equal to 1.76 x 1O-2(Ho/100)3 Mpc-3 •

The Clustering of Faint Galaxies: and the Evolution of ξ(R)

The two-point angular correlation function, ω(θ), is constructed from a catalog of 13,000 objects in 24 fields distributed over an area of 2 degrees square and complete to a limit of R = 23.5. The amplitude and slope of our correlation function on arcminute scales are in broad agreement with recent CCD results in the literature and decreases with depth. No evidence is found for a flattening of the slope of the correlation function away from δ ∼ 0.8. Using the redshift distribution from the recent I-band selected Canada-France Redshift Survey, the observed ω(θ) implies a value of the clustering length r0 = 1.86 ± 0.42h−1 Mpc (q0 = 0.5) at z = 0.48. This is consistent with the clustering of optically selected local galaxies, if clustering has developed with epoch. Specifically, clustering evolution to a CfA-like sample (with r0 = 5.5h−1 Mpc) would require growth in clustering stronger than the type seen in CDM-like models. Evolution to a less clustered IRAS-like sample (with r0 = 4h−1 Mpc) would require evolution represented by stable clusters of fixed physical size. If there was no growth in clustering (i.e galaxies are fixed in comoving space) then a local sample that is very weakly clustered (with r0 = 3h−1 Mpc) would result. These possibilities are discussed in the context of our understanding of the nature of the faint galaxy population.

First Results from the VLA FIRST Survey

The VLA FIRST survey is now in its second year. We have completed mapping over 1500 deg2 of the North Galactic Cap and present here the catalog of the 138,000 radio sources detected therein. We discuss the statistics of this new catalog including the two-point angular correlation function for all radio emitters, present our optical identification of 24,000 sources using the APM catalog, and report followup studies on radio variability, X-ray source identification, and our bright quasar sample.

Selection effects and tests of repetition (clustering) in the BATSE 1B catalog

A “repeater statistic” (derived from the two-point angular correlation function and accounting for location errors) is used to identify constraints on the rate of burst repetition from the BATSE 1B catalog. Because of the anisotropic sky exposure, repeating sources should create additional effects in the large-scale distribution of close burst pairs.

CBR anisotropy from inflation-induced gravitational waves in mixed radiation and dust cosmology.

We examine stochastic temperature fluctuations of the cosmic background radiation (CBR) arising via the Sachs-Wolfe effect from gravitational wave perturbations produced in the early universe. We consider spatially flat, perturbed FRW models that begin with an inflationary phase, followed by a mixed phase containing both radiation and dust. The scale factor during the mixed phase takes the form a(\ensuremath{\eta})=${\mathit{c}}_{1}$${\mathrm{\ensuremath{\eta}}}^{2}$+${\mathit{c}}_{2}$\ensuremath{\eta}+${\mathit{c}}_{3}$, where ${\mathit{c}}_{\mathit{i}}$ are constants. During the mixed phase the universe smoothly transforms from being radiation to dust dominated. We find analytic expressions for the graviton mode function during the mixed phase in terms of spheroidal wave functions. This mode function is used to find an analytic expression for the multiple moments 〈${\mathit{a}}_{\mathit{l}}^{2}$〉 of the two-point angular correlation function C(\ensuremath{\gamma}) for the CBR anisotropy. The analytic expression for the multipole moments is written in terms of two integrals, which are evaluated numerically. The results are compared to multipoles calculated for models that are completely dust dominated at last scattering. We find that the multipoles 〈${\mathit{a}}_{\mathit{l}}^{2}$〉 of the CBR temperature perturbations for l&gt;10 are significantly larger for a universe that contains both radiation and dust at last scattering. We compare our results with recent, similar numerical work and find good agreement. The spheroidal wave functions may have applications to other problems of cosmological interest.

Two-Point Angular Correlation Function for the Green Bank 4.85 GHz Sky Survey

This paper presents an angular correlation analysis of the Green Bank 4.85 GHz radio catalog (Gregory \& Condon 1991) of 54,579 sources (S \gsim 25 mJy). The Green Bank catalog is found to be complete to S $\geq$ 35 mJy over 20$^{\circ} \leq \delta < 74^{\circ}$, 0$^h \leq \alpha < 24^h$, and Galactic latitude $|b| \geq 10^{\circ}$. The 2-point angular correlation function shows evidence for the clustering of radio sources, with a power-law distribution consistent with a slope $\gamma = -0.8$. This may well provide the $first$ detection of an angular correlation in a large area, complete deep radio survey.

Do Gamma-Ray Burst Sources Repeat?

The demonstration of repeated gamma-ray bursts from an individual source would severely constrain burst source models. Recent reports (Quashnock and Lamb 1993; Wang and Lingenfelter 1993) of evidence for repetition in the first BATSE burst catalog have generated renewed interest in this issue. Here, we analyze the angular distribution of 585 bursts of the second BATSE catalog (Meegan et al. 1994). We search for evidence of burst recurrence using the nearest and farthest neighbor statistic and the two-point angular correlation function. We find the data to be consistent with the hypothesis that burst sources do not repeat; however, a repeater fraction of up to about 20% of the observed bursts cannot be excluded.

ANGULAR TWO-POINT CORRELATION FUNCTIONS FOR COSMOLOGICAL GAMMA-RAY BURST MODEL

We compute the angular two-point correlation functions of the gamma-ray bursts at cosmological distances. Since the gamma-ray burst emission mechanism is not yet established, we simply assume that the gamma-ray burst sources are associated with high-redshift galaxies in some way. Then on the basis of several simple models for the evolution of galaxy spatial correlations, we calculate the amplitude of angular two-point correlation functions on scales appropriate for the Compton Gamma Ray Observatory data. We find that in most cases the predicted correlations are difficult to detect with the current data rate and the angular resolution, but models in which the bursts preferentially occur at relatively low redshift (z≲0.5) predict correlation amplitudes on θ~5° which will be marginally detectable with the Gamma Ray Observatory data in several years. If future observations detect a signal of angular correlations, it will imply useful information on the correlation of galaxies at high redshifts provided that the gamma-ray bursts are cosmological.

The clustering properties of faint galaxies

The two-point angular correlation function of galaxies, \wte, has been computed from a new survey of faint galaxies covering a 2 deg$^2$ area near the North Galactic Pole. This survey, which is complete to limiting magnitudes \jmag=24 and \fmag=23, samples angular scales as large as $1\degpoint5$. Faint galaxies are found to be more weakly clustered (by a factor of at least two) compared to galaxies observed locally. Clustering amplitudes are closer to model predictions in the red than in the blue. The weak clustering of faint galaxies cannot be explained by any plausible model of clustering evolution with redshift. However, one possible explanation of the clustering properties of intermediate redshift galaxies is that they resemble those of starburst galaxies and H II region galaxies, which are observed locally to possess weak clustering amplitudes. Our clustering amplitudes are also similar to those of nearby late-type galaxies, which are observed to be more weakly clustered than early-type galaxies A simple, self-consistent model is presented that predicts the fraction of galaxies in the ``excess'' population at intermediate redshifts, and correctly matches observed color distributions. The available data on the clustering properties of faint galaxies are consistent with this model if the ``excess'' population of faint blue galaxies is also the weakly clustered population. Evidence is presented that the power-law slope of the angular correlation function becomes shallower at fainter magnitudes. A similar effect is seen locally both for dwarf galaxies and for galaxies with late morphological type;

The angular two-point correlation function of galaxies down to B[SUB]J[/SUB] less than or approximately = 26 magnitudes on less than or approximately = 10 arcmins scales

We place constraints on the evolution of galaxy clustering from deep CCD frames in Gunn gri from two widely separated, intermediate Galactic latitude fields (b ~&gt; 30^deg^), each roughly square in shape and ~0.5^deg^ in extent. More than 90% of the galaxies brighter than gri ~24.5 mag are detected. For each field in each pass-band, a total of ~&gt;23,000 objects are found. Neuschaefer, Windhorst &amp; DressIer (1991) (the first paper in the series) and Neuschaefer &amp; Windhorst (1994) (the second paper in the series) discuss the details of the photometric and astrometric calibrations, object classification and galaxy counts. The current paper, the third in this series, presents the faint galaxy angular two-point correlation function (as a function of median redshift and apparent magnitude) and assesses the relative importance of effects due to sample completeness, galaxy surface brightness, evolution in galaxy spectra, evolution in clustering, as well as the overall matter density in the universe. Galaxies undergo a moderate bluing trend with increasing sample depth, consistent with previous work. We establish two important trends in the correlation function, w(θ) is proportional to θ<SUP>delta</SUP>^ (with {delta] ~-0.7), as function of increasing survey depth: (1) w(θ)'s amplitude declines moderately with survey depth down to gri ~ 24.5 mag (where our galaxy catalogs are more than 90% complete); (2) w(θ)'s slope marginally flattens from δ ~ -0.8 in all-sky surveys to δ ~ -0.5 at g_med_ = 25 mag in our Four-Shooter data. The decline of the correlation amplitude with fainter magnitude is stronger than expected for an {OMEGA}_0_ = 1 universe that contains only galaxies with nonevolving spectral energy distributions. Better agreement with the observations is obtained through a combination of the following explanations: (1) w(θ)'s slope δ(z) grows steeper with cosmic time roughly as γ(z) is proportional to γ(0)(1 + z)^0.2+/-0.2 with γ(0) = 1 - δ(0) = 1.75, confirming predictions of N-body CDM galaxy clustering simulations; (2) galaxy clustering has increased somewhat in amplitude since z &lt;~ 1, parameterized by a clustering exponent ɛ ~&gt;0 (3) field galaxies have undergone mild spectral evolution out to redshifts z &lt;~ 1; and/or (4) low surface brightness galaxies are more weakly clustered than the general field galaxy population.

Galaxy Clustering in the Southern Hemisphere

AbstractIn this paper some of the major results from the COSMOS and APM digitised galaxy surveys are presented. The main motivation behind these catalogues was to study large-scale structure in the universe. We begin by outlining the importance of such studies to cosmology and discussing the early results from the visually compiled galaxy catalogues. The impact of the digitised catalogues is demonstrated by focussing on three key areas of research; the galaxy-galaxy two-point angular correlation function, the cluster-cluster spatial correlation function, and galaxy number counts.

An infrared cutoff revealed by the two years of COBE observations of cosmic temperature fluctuations.

We show that a good fitting to the first two years of observations by the Cosmic Background Explorer Differential Microwave Radiometers of the two-point angular correlation function of cosmic background radiation (CBR) temperature is given by models with a nonzero infrared cutoff ${k}_{min}$ in the spectrum of the primordial density perturbations. If this cutoff comes from the finiteness of the universe, say, a topological T3 model, we find ${k}_{min}\ensuremath{\sim}(0.3\ensuremath{-}1.1)\ensuremath{\pi}\frac{{\mathrm{H}}_{0}}{c}$ with confidence level 95%. Such a nonzero ${k}_{min}$ universe would also give a better match to the observations both of the rms quadrupole anisotropy of CBR and of galaxy clustering.

Evolution in the clustering of galaxies to r = 26

We present results for the two-point angular correlation function of galaxies, ω(θ) to a limiting magnitude of r=26. our catalogue is constructed from deep imaging using the COMIC imaging spectrograph on the hale 5-m telescope. The final sample is 95 percent complete to r=26.0, yielding ~5730 galaxies over a 90.1-arcmin2 field. Our analysis shows ω(θ) for faint galaxies can be parametrized by a power law, Aωθ–0.8, in agreement with the angular clustering statistics of shallower catalogues. The derived amplitude, Aω, for our catalogue is small, but non-zero. We combine this measurement with the latest statistical constraints on faint galaxy redshifts from gravitational lensing studies, which imply that the bulk of the r ≲26 field galaxies should be at redshifts z ~ 1. We show that Our derived Aω is significantly lower than that predicted from the local bright, optically selected galaxy correlation function using the lensing-determined galaxy redshift distribution and modest growth of clustering. This simplistic model does not, however, allow for different morphological classes. This is important as we would expect to observe a varying morphological mix as a function of apparent magnitude in our sample. At our faintest limits we reach sufficiently high redshifts that differential K- corrections will result in the observed galaxy mix being dominated by star-bursting dwarf and low- surface-brightness galaxies and assuming modest clustering evolution, we obtain reasonable agreement between our model and observations. This model supports the scenario in which the high number density and low clustering amplitude of faint galaxies are produced by normally clustered star forming dwarf galaxies at modest redshifts.

Galaxy Clustering to B = 27M

The angular two-point correlation function, ω(θ), for galaxies can be used as a probe of their redshift distribution N(z) and, therefore, of galaxy luminosity evolution. Without redshift data, we can still observe the projection onto the two-dimensional sky of the three-dimensional clustering of galaxies. The autocorrelation of this projected distribution is described by ω(θ). Observations have indicated that ω(θ) follows a θ−0.8 power-law (Peebles 1980) and that the index of the power-law remains approximately constant to the faintest limits of photographic surveys (Jones, Shanks &amp;amp; Fong 1987).