Correlation Length R0 Research Articles

Interpreting the Observed Clustering of Red Galaxies atz∼ 3

Daddi et al. have recently reported strong clustering of a population of red galaxies at z � 3 in the Hubble Deep Field-South. Fitting the observed angular clustering with a power law of index � 0.8, they infer a comoving correlation length r0 � 8 h � 1 Mpc; for a standard cosmology, this r0 would imply that the red galaxies reside in rare, M � 10 13 h � 1 Mhalos, with each halo hosting � 100 galaxies to match the number density of the pop- ulation. Using the framework of the halo occupation distribution (HOD) in a CDM universe, we show that the Daddi et al. data can be adequately reproduced by less surprising models, e.g., models with galaxies residing in halos of mass M > Mmin ¼ 6:3;10 11 h � 1 Mand a mean occupation Navg(M) ¼ 1:4(M =Mmin) 0:45 above this cutoff. The resultant correlation functions do not follow a strict power law, showing instead a clear transition from the one-halo-dominated regime, where the two galaxies of each pair reside in the same dark matter halo, to the two-halo-dominated regime, where the two galaxies of each pair are from different halos. The observed high- amplitude data points lie in the one-halo-dominated regime, so these HOD models are able to explain the observations despite having smaller correlation lengths, r0 � 5 h � 1 Mpc. HOD parameters are only loosely constrained by the current data because of large sample variance and the lack of clustering information on scales that probe the two-halo regime. If our explanation of the data is correct, then future observations covering a larger area should show that the large-scale correlations lie below a � � 1:8 extrapolation of the small-scale points. Our models of the current data suggest that the red galaxies are somewhat more strongly clustered than UV-selected Lyman break galaxies and have a greater tendency to reside in small groups. Subject heading gs: galaxies: halos — galaxies: high-redshift — large-scale structure of universe

Large-scale structure in the ELAIS S1 Survey

We present an analysis of the two-point angular correlation function of The European Large-Area Infrared Space Observatory (ELAIS) S1 survey. The survey covers 4 deg2 and contains 462 sources detected at 15 mum to a 5sigma flux limit of 0.45 mJy. Using the 329 extragalactic sources not repeated in different observations, we detect a significant clustering signal; the resulting angular correlation function can be fitted by a power law w(theta) =Atheta1-gamma with A= 0.014 +/- 0.005 and gamma= 2.04 +/- 0.18. Using the observed redshift distribution of the objects (from spectroscopic and photometric redshifts), we invert Limber's equation and deduce a spatial correlation length r0= 4.3+0.4-0.7h-1 Mpc. This is smaller than that obtained from optical surveys but it is in agreement with results from IRAS. This extends to higher redshift the observational evidence that infrared selected surveys show smaller correlation lengths (i.e. reduced clustering amplitudes) than optical surveys.

Near-Infrared Bright Galaxies at z 2. Entering the Spheroid Formation Epoch?

Spectroscopic redshifts have been measured for nine K-band luminous galaxies at 1.7 < z < 2.3, selected with Ks < 20 in the K20 survey region of the Great Observatories Origins Deep Survey (GOODS) area. Star formation rates (SFRs) of ~100-500 M? yr-1 are derived when dust extinction is taken into account. The fitting of their multicolor spectral energy distributions indicates stellar masses of M 1011 M? for most of the galaxies. Their rest-frame UV morphology is highly irregular, suggesting that merging-driven starbursts are going on in these galaxies. Morphologies tend to be more compact in the near-IR, a hint for the possible presence of older stellar populations. Such galaxies are strongly clustered, with seven out of nine belonging to redshift spikes, which indicates a correlation length of r0 ~ 9-17 h-1 Mpc (1 ? range). Current semianalytical models of galaxy formation appear to underpredict by a large factor (30) the number density of such a population of massive and powerful starburst galaxies at z ~ 2. The high masses and SFRs, together with the strong clustering, suggest that at z ~ 2 we may have started to explore the major formation epoch of massive early-type galaxies.

The Canada-France deep fields survey–II: Lyman-break galaxies and galaxy clustering atz $\mathsf{\sim3}$

We present a large sample of band dropout galaxies extracted from the Canada-France deep fields survey (CFDF). Our catalogue covers an effective area of ∼1700 arcmin2 divided between three large, contiguous fields separated widely on the sky. To , the survey contains 1294 Lyman-break candidates, in agreement with previous measurements by other authors, after appropriate incompleteness corrections have been applied to our data. Based on comparisons with spectroscopic observations and simulations, we estimate that our sample of Lyman-break galaxies is contaminated by stars and interlopers (lower-redshift galaxies) at no more than . We find that is well fitted by a power-law of fixed slope, , even at small () angular separations. In two of our three fields, we are able to fit simultaneously for both the slope and amplitude and find and Mpc, and and Mpc (all spatially dependent quantities are quoted for a Λ-flat cosmology). Our data marginally indicates in one field (at a level) that the Lyman-break correlation length r0 depends on sample limiting magnitude: brighter Lyman-break galaxies are more clustered than fainter ones. For the entire CFDF sample, assuming a fixed slope we find Mpc. Using these clustering measurements and prediction for the dark matter density field computed assuming cluster-normalised linear theory, we derive a linear bias of . Finally we show that the dependence of the correlation length with the surface density of Lyman-break galaxies is in good agreement with a simple picture where more luminous galaxies are hosted by more massive dark matter halos with a simple one-to-one correspondence.

The power spectrum of galaxy clustering in the APM Survey

We measure the power spectrum of galaxy clustering in real space from the APM Galaxy Survey. We present an improved technique for the numerical inversion of Limber's equation that relates the angular clustering of galaxies to an integral over the power spectrum in three dimensions. Our approach is underpinned by a large ensemble of mock galaxy catalogues constructed from the Hubble volume N-body simulations. Mock catalogues are used to test for systematic effects in the inversion algorithm and to estimate the errors on our measurement. We find that we can recover the power spectrum to an accuracy of better than 15 per cent over three decades in wavenumber. A key advantage of the use of mock catalogues to infer errors is that we can apply our technique on scales for which the density fluctuations are not Gaussian, thus probing the regime that offers the best constraints on models of galaxy formation. On large scales, our measurement of the power spectrum is consistent with the shape of the mass power spectrum in the popular ‘concordance’ cold dark matter model. The galaxy power spectrum on small scales is strongly affected by non-linear evolution of density fluctuations, and, to a lesser degree, by galaxy bias. The rms variance in the galaxy distribution, when smoothed in spheres of radius 8 h−1 Mpc, is σg8= 0.96+0.17−0.20 and the shape of the power spectrum on large scales is described by a simple fitting formula with parameter Γ= 0.19+0.13−0.04 (these errors are the 1σ ranges for a two-parameter fit). We use our measurement of the power spectrum to estimate the galaxy two-point correlation function; the results are well described by a power law with correlation length r0= 5.9 ± 0.7 h−1 Mpc and slope γ= 1.61 ± 0.06 for pair separations in the range 0.1 < r/(h−1 Mpc) < 20.

Angular and three-dimensional correlation functions determined from the Muenster Red Sky Survey

ABSTRA C T We measure the two-dimensional galaxy ‐ galaxy correlation function from the Muenster Red Sky Survey. This survey covers a slightly larger sky area compared with the APM survey and is complete to rF , 18:3 mag ﷿zmed , 0:14fi. The large dynamic range of the survey makes it possible to examine sub-catalogues with different limiting magnitudes. The positive parts of our measured angular correlation functions (after the colour correction) show excellent agreement with those measured by Maddox, Efstathiou & Sutherland from the APM galaxy catalogue. From the measured angular correlation functions, the three-dimensional two-point correlation function j(r) is calculated. We have chosen the evolution parameter e so that the change of the correlation length r0 with the depth of the sub-catalogue is minimal. This prescription gives r0 o 5:82 ^ 0:05 h 21 Mpc and a very high, positive evolution parameter e o 2:6. The position of the point where the three-dimensional correlation function breaks away from the power law depends on the depth of the sub-catalogues: the break comes earlier for lower magnitude limits, and later for higher magnitude limits.

The ROSAT-ESO Flux-Limited X-ray (REFLEX) galaxy cluster survey - II. The spatial correlation function

We report the results of the spatial two-point correlation functioncc(r) for the new X- ray galaxy cluster survey REFLEX, which comprises of 452 X-ray selected clusters (449 with redshifts) detected by the ROSAT satellite during the ROSAT All-Sky-Survey (RASS). The REFLEX cluster sample is flux limited to 3 × 10 12 erg s 1 cm 2 in the ROSAT energy band (0.1 − 2.4 keV) and spans 3 decades in X-ray luminosity (10 42 −10 45 h 2 erg s 1 ), containing galaxy groups and rich clusters out to a redshift z ≤ 0.3. Covering a contiguous area of 4.24 sr REFLEX is the largest X-ray cluster sample to date for which spatial clustering has been analysed. Correlation studies using clusters selected on the basis of their X-ray emission are particularly interesting as they are largely free from the projection biases inherent to optical studies. For the entire flux-limited sample we find that the correlation length (the scale at which the correlation amplitude passes through unity) r0 ≃ 20h 1 Mpc. For example, if a power-law fit is made to �(r) over the range 4 − 40h 1 Mpc then r0 = 18.8 ± 0.9. An indication of the robustness of this result comes from the high degree of isotropy seen in the clustering pattern on scales close to the correlation length. On larger scalescc(r) deviates from a power-law, crossing zero at ≃ 45h 1 Mpc. From an examination of 5 volume-limited cluster sub-samples we find no significant trend of r0 with limiting X-ray luminosity. A comparison with recent model predictions for the clustering properties of X-ray flux-limited samples, indicates that Cold Dark Matter models with the matter density m = 1 fail to produce sufficient clustering to account for the data, while m ≃ 0.3 models provide an excellent fit.

The pinocchio algorithm: pinpointing orbit-crossing collapsed hierarchical objects in a linear density field

ABSTRA C T PINOCCHIO (PINpointing Orbit-Crossing Collapsed HIerarchical Objects) is a new algorithm proposed recently by Monaco et al. (Paper I) for identifying dark matter haloes in a given numerical realization of the linear density field in a hierarchical universe. Mass elements are assumed to have collapsed after undergoing orbit crossing, as computed using perturbation theory. It is shown that Lagrangian perturbation theory, and in particular its ellipsoidal truncation, is able to predict accurately the collapse, in the orbit-crossing sense, of generic mass elements. Collapsed points are grouped into haloes using an algorithm that mimics the hierarchical growth of structure through accretion and mergers. Some points that have undergone orbit crossing are assigned to the network of filaments and sheets that connects the haloes; it is demonstrated that this network resembles closely that found in N-body simulations. The code generates a catalogue of dark matter haloes with known mass, position, velocity, merging history and angular momentum. It is shown that the predictions of the code are very accurate when compared with the results of large N-body simulations that cover a range of cosmological models, box sizes and numerical resolutions. The mass function is recovered with an accuracy of better than 10 per cent in number density for haloes with at least 30 ‐ 50 particles. A similar accuracy is reached in the estimate of the correlation length r0. The good agreement is still valid on the object-by-object level, with 70 ‐ 100 per cent of the objects with more than 50 particles in the simulations also identified by our algorithm. For these objects the masses are recovered with an error of 20 ‐ 40 per cent, and positions and velocities with a root mean square error of ,1 ‐2 MpcO0:5 ‐ 2 grid lengths) and ,100 km s 21 , respectively. The recovery of the angular momentum of haloes is considerably noisier, and accuracy at the statistical level is achieved only by introducing free parameters. The algorithm requires negligible computer time as compared with performing a numerical N-body simulation.

The 2dF Galaxy Redshift Survey: luminosity dependence of galaxy clustering

We investigate the dependence of the strength of galaxy clustering on intrinsic luminosity using the Anglo-Australian two degree field galaxy redshift survey (2dFGRS). The 2dFGRS is over an order of magnitude larger than previous redshift surveys used to address this issue. We measure the projected two-point correlation function of galaxies in a series of volume-limited samples. The projected correlation function is free from any distortion of the clustering pattern induced by peculiar motions and is well described by a power-law in pair separation over the range 0.1 < r /h Mpc < 10. The clustering of L* galaxies in real space is well fit by a correlation length r0 = 4.9 +/- 0.3 /h Mpc and power-law slope gamma = 1.71 +/- 0.06. The clustering amplitude increases slowly with absolute magnitude for galaxies fainter than M*, but rises more strongly at higher luminosities. At low luminosities, our results agree with measurements from the SSRS2 by Benoist et al. However, we find a weaker dependence of clustering strength on luminosity at the highest luminosities. The correlation function amplitude increases by a factor of 4.0 between $M_{b_{J}} -5\log_{10}h = -18$ and -22.5, and the most luminous galaxies are 3.0 times more strongly clustered than L* galaxies. The power-law slope of the correlation function shows remarkably little variation for samples spanning a factor of 20 in luminosity. Our measurements are in very good agreement with the predictions of the hierarchical galaxy formation models of Benson et al.

Clustering Properties of Galaxies at [CLC][ITAL]z[/ITAL][/CLC] ∼ 4 in the Subaru/[ITAL]XMM[/ITAL] Deep Survey Field

We study the clustering properties of about 1200 z ~ 4 Lyman break galaxy (LBG) candidates with i' < 26 that are selected by color from deep BRi' imaging data of a 618 arcmin2 area in the Subaru/XMM-Newton Deep Field taken with Subaru Prime Focus Camera. The contamination and completeness of our LBG sample are evaluated, on the basis of the Hubble Deep Field-North (HDF-N) objects, to be 17% and 45%, respectively. We derive the angular correlation function over θ = 2''-1000'' and find that it is fitted fairly well by a power law, ω(θ) = Aωθ-0.8, with Aω = 0.71 ± 0.26. We then calculate the correlation length r0 (in comoving units) of the two-point spatial correlation function ξ(r) = (r/r0)-1.8 from Aω using the redshift distribution of LBGs derived from the HDF-N and find that r0 = 2.7 h-1 Mpc in a Λ-dominated universe (Ωm = 0.3 and ΩΛ = 0.7). This is twice as large as the correlation length of the dark matter at z ≃ 4 predicted from an analytic model by Peacock and Dodds but about twice as small as that of bright galaxies predicted from a semianalytic model by Baugh and coworkers. We find an excess of ω(θ) on small scales (θ ≲ 5''), departing from the power-law fit by over 3 σ significance levels. Interpreting this as being due to galaxy mergers, we estimate the fraction of galaxies undergoing mergers in our LBG sample to be 3.0% ± 0.9%, which is significantly smaller than those of galaxies at intermediate redshifts.

The correlation function of X-ray galaxy clusters in the ROSAT All-Sky Survey 1 Bright Sample

We analyse the spatial clustering properties of the RASS1 Bright Sample, an X-ray flux-limited catalogue of galaxy clusters selected from the southern part of the ROSAT All-Sky Survey. The two-point correlation function �(r) of the whole sample is well fitted (in an Einstein-de Sitter model) by the power-law � = (r/r0) , with r0 = 21.5 +3.4 4.4 h 1 Mpc and = 2.11 +0.53 0.56 (95.4 per cent confidence level with one fitting parameter). We use the RASS1 Bright Sample as a first application of a theoretical model which aims at predicting the clustering properties of X-ray clusters in flux-limited surveys for different cosmological scenarios. The model uses the theoretical and empirical relations between mass, temperature and X-ray cluster luminosity, and fully accounts for the redshift evolution of the underlying dark matter clustering and cluster bias factor. The comparison between observational results and theoretical predictions shows that the Einstein-de Sitter models display too low a correlation length, while models with a matter density parameter 0m = 0.3 (with or without a cosmological constant) are successful in reproducing the observed clustering. The dependence of the correlation length r0 on the Xray limiting flux and luminosity of the sample is generally consistent with the predictions of all our models. Quantitative agreement is however only reached for 0m = 0.3 models. The model presented here can be reliably applied to future deeper X-ray cluster surveys: the study of their clustering properties will provide a useful complementary tool to the traditional cluster abundance analyses to constrain the cosmological parameters.

The Phoenix radio survey: The angular correlation function

Using the Phoenix radio survey, a homoge- neous survey selected at 1.4 GHz and covering an area of 3d eg 2 , we analyse the clustering of the sub-mJy ra- dio population using angular correlation function analy- sis. Extensive simulations are carried out to investigate the signicance of the estimated angular correlation am- plitudes. Our analysis show that for the S1:4 > 0: 5m Jy sub-samples the radio source distribution is anisotropic at the 2 signicance level. Additionally, we estimate up- per limits for the angular correlation amplitudes that, de- spite the large uncertainties, are in good agreement with the amplitude estimates for sources brighter than 1 mJy, detected in the FIRST radio survey (Cress et al. 1997). Adopting a radio luminosity function and assuming an evolving spatial correlation function of the form (r )= (r=r0) (1 + z) (3+) , with the evolution parametrised by , we nd an upper limit for the angular correlation length r0 9h 1 Mpc for S1:4 > 0: 5m Jy and =2 :1. This agrees well with the value r0 6 8h 1 Mpc esti- mated from the FIRST radio survey for sources brighter than 1 mJy. Additionally, we quantify the characteristics, in terms of areal coverage and limiting flux density, of fu- ture deep radio surveys to yield a signicant correlation amplitude detection and to explore possible changes of the correlation amplitude with flux density.

Measuring and modelling the redshift evolution of clustering: the Hubble Deep Field North

ABSTRA C T The evolution of galaxy clustering from za 0t oz . 4:5 is analysed using the angular correlation function and the photometric redshift distribution of galaxies brighter than IAB < 28:5 in the Hubble Deep Field North. The reliability of the photometric redshift estimates is discussed on the basis of the available spectroscopic redshifts, comparing different codes and investigating the effects of photometric errors. The redshift bins in which the clustering properties are measured are then optimized to take into account the uncertainties of the photometric redshifts. The results show that the comoving correlation length r0 has a small decrease in the range 0 & z & 1 followed by an increase at higher z. We compare these results with the theoretical predictions of a variety of cosmological models belonging to the general class of Cold Dark Matter scenarios, including Einstein‐de Sitter models, an open model and a flat model with non-zero cosmological constant. Comparison with the expected mass clustering evolution indicates that the observed high-redshift galaxies are biased tracers of the dark matter with an effective bias b strongly increasing with redshift. Assuming an Einstein‐de Sitter universe, we obtain b . 2: 5a tz . 2 and b . 5a tz . 4. These results support theoretical scenarios of biased galaxy formation in which the galaxies observed at high redshift are preferentially located in more massive haloes. Moreover, they suggest that the usual parameterization of the clustering evolution as jOr; zUajOr; 0UO1 1 zU 2O31eU is not a good description for any value of e . Comparison of the clustering amplitudes that we measured at z . 3 with those reported by Adelberger et al. and Giavalisco et al., based on a different selection, suggests that the clustering depends on the abundance of the objects: more abundant objects are less clustered, as expected in the paradigm of hierarchical galaxy formation. The strong clustering and high bias measured at z . 3 are consistent with the expected density of massive haloes predicted in the frame of the various cosmologies considered here. At z . 4, the strong clustering observed in the Hubble Deep Field requires a significant fraction of massive haloes to be already formed by that epoch. This feature could be a discriminant test for the cosmological parameters if confirmed by future observations.

The redshift evolution of clustering in the Hubble Deep Field

We present a correlation function analysis for the catalogue of photometric redshifts obtained from the Hubble Deep Field image by Fernandez-Soto, Lanzetta & Yahil. By dividing the catalogue into redshift bins of width Δz=0.4 we measured the angular correlation function w(θ) as a function of redshift up to z∼4.8. From these measurements we derive the trend of the correlation length r0. We find that r0(z) is roughly constant with look-back time up to z≃2, and then increases to higher values at z≳2.4. We estimate the values of r0, assuming ξ(r,z)=[rr0(z)]−γ, γ=1.8 and various geometries. For Ω0=1 we find r0(z=3)≃7.00±4.87 h−1 Mpc, in good agreement with the values obtained from analysis of the Lyman break galaxies.

Clustering at High Redshift: Precise Constraints from a Deep, Wide‐Area Survey

We present constraints on the evolution of large-scale structure from a catalog of 710,000 galaxies with IAB ? 24 derived from a KPNO 4 m CCD imaging survey of a contiguous 4? ? 4? region. The advantage of using large contiguous surveys for measuring clustering properties on even modest angular scales is substantial: the effects of cosmic scatter are strongly suppressed. We provide highly accurate measurements of the two-point angular correlation function, ?(?), as a function of magnitude on scales up to 15. The amplitude of ?(?) declines by a factor of ~10 over the range 16 ? I ? 20 but only by a factor of 2-3 over the range 20 20, our best-fit values shift toward lower r0 and more negative . A strong covariance between r0 and prevents us from rejecting > 0 even at faint magnitudes, but if > 1, we strongly reject r0 4 h-1 Mpc (comoving). The above expression for ?(r, z) and our data give a correlation length of r0(z = 0.5) ? 3.0 ? 0.4 h-1 Mpc, about a factor of 2 larger than the correlation length at z = 0.5 derived from the Canada-France Redshift Survey (CFRS). The small volume sampled by the CFRS and other deep redshift probes, however, makes these spatial surveys strongly susceptible to cosmic scatter and will tend to bias their derived correlation lengths toward the low end. Our results are consistent with redshift distributions in which ~30%-50% of the galaxies at I = 23 lie at z > 1. The best-fit power-law slope of the correlation function remains independent of I magnitude for I ? 22. At fainter limits, there is a suggestive trend toward flatter slopes that occurs at fluxes consistent with similar trends seen by Neuschaffer & Windhorst and Campos and coworkers. The galaxy counts span 11 magnitudes and provide an accurate calibration of the galaxy surface density. We find evidence for mild galaxy evolution: about 1 mag of brightening or a doubling of the density by I = 23 relative to an ?0 = 1 no-evolution model, about 0.5 mag of brightening or a factor of 1.5 increase in surface density relative to an open model. Our galaxy counts agree well with those from the Hubble Deep Field survey and, thus, argue against a significant inclusion of subgalactic components in the latter census for I < 24.

Redshift Evolution of the Nonlinear Two‐Point Correlation Function

This paper presents a detailed theoretical study of the two-point correlation function ξ for both dark matter halos and the matter density field in five cosmological models with varying matter density Ωm and neutrino fraction Ων. The objectives of this systematic study are to evaluate the nonlinear gravitational effects on ξ, to contrast the behavior of ξ for halos versus matter, and to quantify the redshift evolution of ξ and its dependence on cosmological parameters. Overall, ξ for halos exhibits markedly slower evolution than ξ for matter, and its redshift dependence is much more intricate than the single power-law parameterization used in the literature. Of particular interest is that the redshift evolution of the halo-halo correlation length r0 depends strongly on Ωm and Ων, being slower in models with lower Ωm or higher Ων. Measurements of ξ to higher redshifts can therefore be a potential discriminator of cosmological parameters. The evolution rate of r0 for halos within a given model increases with time, passing the phase of fixed comoving clustering at z~1-3 toward the regime of stable clustering at z~0. The shape of the halo-halo ξ, on the other hand, is well approximated by a power law with slope -1.8 in all models and is not a sensitive model discriminator.

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.

The ground-state three-body correlation function in a dilute boson gas with hard-sphere interaction

The three-body correlation function is evaluated for the ground state of a dilute boson gas with hard-sphere interaction and the validity of the superposition approximationD(r12,r13,r23)=D(r12)D(r13)D(r23) is discussed. The main result of this paper is that the superposition approximation holdsonly whenall the relative co-ordinates are much greater than the correlation lengthr0=(8παϱ)−1/2 (α and ϱ are the hard sphere radius and number density of the gas, respectively). One also finds that the system shows unexpected long-distance correlations since one hasD(r12, ∞, ∞) ≠D(r12).