Angular Correlation Function Of Galaxies Research Articles

LATIS: Constraints on the Galaxy–Halo Connection at z ∼ 2.5 from Galaxy–Galaxy and Galaxy–Lyα Clustering

The connection between galaxies and dark matter halos is often quantified using the stellar mass–halo mass (SMHM) relation. Optical and near-infrared imaging surveys have led to a broadly consistent picture of the evolving SMHM relation based on measurements of galaxy abundances and angular correlation functions. Spectroscopic surveys at z ≳ 2 can also constrain the SMHM relation via the galaxy autocorrelation function and through the cross-correlation between galaxies and Lyα absorption measured in transverse sight lines; however, such studies are very few and have produced some unexpected or inconclusive results. We use ∼3000 spectra of z ∼ 2.5 galaxies from the Lyα Tomography IMACS Survey (LATIS) to measure the galaxy–galaxy and galaxy–Lyα correlation functions in four bins of stellar mass spanning 109.2 ≲ M */M ⊙ ≲ 1010.5. Parallel analyses of the MultiDark N-body and ASTRID hydrodynamic cosmological simulations allow us to model the correlation functions, estimate covariance matrices, and infer halo masses. We find that results of the two methods are mutually consistent and broadly accord with standard SMHM relations. This consistency demonstrates that we are able to measure and model Lyα transmission fluctuations δ F in LATIS accurately. We also show that the galaxy–Lyα cross-correlation, a free by-product of optical spectroscopic galaxy surveys at these redshifts, can constrain halo masses with similar precision to galaxy–galaxy clustering.

The bright-star masks for the HSC-SSP survey

Abstract We present the procedure to build and validate the bright-star masks for the Hyper-Suprime-Cam Strategic Subaru Proposal (HSC-SSP) survey. To identify and mask the saturated stars in the full HSC-SSP footprint, we rely on the Gaia and Tycho-2 star catalogues. We first assemble a pure star catalogue down to GGaia < 18 after removing ∼1.5% of sources that appear extended in the Sloan Digital Sky Survey (SDSS). We perform visual inspection on the early data from the S16A internal release of HSC-SSP, finding that our star catalogue is 99.2% pure down to GGaia < 18. Second, we build the mask regions in an automated way using stacked detected source measurements around bright stars binned per GGaia magnitude. Finally, we validate those masks by visual inspection and comparison with the literature of galaxy number counts and angular two-point correlation functions. This version (Arcturus) supersedes the previous version (Sirius) used in the S16A internal and DR1 public releases. We publicly release the full masks and tools to flag objects in the entire footprint of the planned HSC-SSP observations at 〈ftp://obsftp.unige.ch/pub/coupon/brightStarMasks/HSC-SSP/〉.

The SDSS galaxy angular two-point correlation function

We present the galaxy two-point angular correlation function for galaxies selected from the seventh data release of the Sloan Digital Sky Survey. The galaxy sample was selected with $r$-band apparent magnitudes between 17 and 21; and we measure the correlation function for the full sample as well as for the four magnitude ranges: 17-18, 18-19, 19-20, and 20-21. We update the flag criteria to select a clean galaxy catalog and detail specific tests that we perform to characterize systematic effects, including the effects of seeing, Galactic extinction, and the overall survey uniformity. Notably, we find that optimally we can use observed regions with seeing $< 1\farcs5$, and $r$-band extinction < 0.13 magnitudes, smaller than previously published results. Furthermore, we confirm that the uniformity of the SDSS photometry is minimally affected by the stripe geometry. We find that, overall, the two-point angular correlation function can be described by a power law, $\omega(\theta) = A_\omega \theta^{(1-\gamma)}$ with $\gamma \simeq 1.72$, over the range $0\fdg005$--$10\degr$. We also find similar relationships for the four magnitude subsamples, but the amplitude within the same angular interval for the four subsamples is found to decrease with fainter magnitudes, in agreement with previous results. We find that the systematic signals are well below the galaxy angular correlation function for angles less than approximately $5\degr$, which limits the modeling of galaxy angular correlations on larger scales. Finally, we present our custom, highly parallelized two-point correlation code that we used in this analysis.

Clustering of galaxies around gamma-ray burst sight-lines

There is evidence of an overdensity of strong intervening MgII absorption line systems distributed along the lines of sight towards GRB afterglows relative to quasar sight-lines. If this excess is real, one should also expect an overdensity of field galaxies around GRB sight-lines, as strong MgII tends to trace these sources. In this work, we test this expectation by calculating the two point angular correlation function of galaxies within 120$^{\prime\prime}$ ($\sim470~h_{71}^{-1}~\mathrm{Kpc}$ at $\langle z\rangle \sim0.4$) of GRB afterglows. We compare the Gamma-ray burst Optical and Near-infrared Detector (GROND) GRB afterglow sample -- one of the largest and most homogeneous samples of GRB fields -- with galaxies and AGN found in the COSMOS-30 photometric catalog. We find no significant signal of anomalous clustering of galaxies at an estimated median redshift of $z\sim0.3$ around GRB sight-lines, down to $K_{\mathrm{AB}}<19.3$. This result is contrary to the expectations from the MgII excess derived from GRB afterglow spectroscopy, although many confirmed galaxy counterparts to MgII absorbers may be too faint to detect in our sample -- especially those at $z>1$. We note that the addition of higher sensitivity Spitzer IRAC or HST WFC3 data for even a subset of our sample would increase this survey's depth by several orders of magnitude, simultaneously increasing statistics and enabling the investigation of a much larger redshift space.}

Spatial clustering of high-redshift Lyman-break galaxies

We present a physically motivated semi-analytic model to understand the clustering of high redshift LBGs. We show that the model parameters constrained by the observed luminosity function, can be used to predict large scale (\theta > 80 arcsec) bias and angular correlation function of galaxies. These predictions are shown to reproduce the observations remarkably well. We then adopt these model parameters to calculate the halo occupation distribution (HOD) using the conditional mass function. The halo model using this HOD is shown to provide a reasonably good fit to the observed clustering of LBGs at both large (\theta>80") and small (\theta < 10") angular scales for z=3-5 and several limiting magnitudes. However, our models underpredict the clustering amplitude at intermediate angular scales, where quasi-linear effects are important. The average mass of halos contributing to the observed clustering is found to be 6.2 x 10^{11} M_\odot and the characteristic mass of a parent halo hosting satellite galaxies is 1.2 \times 10^{12} M_\odot for a limiting absolute magnitude of -20.5 at z=4. For a given threshold luminosity these masses decrease with increasing z and at any given z these are found to increase with increasing value of threshold luminosity. We find that approximately 40 % of the halos above a minimum mass M_{min}, can host detectable central galaxies and about 5-10 % of these halos are likely to also host a detectable satellite. The satellites form typically a dynamical timescale prior to the formation of the parent halo. The small angular scale clustering is due to central-satellite pairs and is quite sensitive to changes in the duration of star formation in a halo. The present data favor star formation in a halo lasting typically for a few dynamical time-scales. Our models also reproduce different known trends between parameters related to star formation.

Cosmological forecasts from photometric measurements of the angular correlation function

We study forecasts for the accuracy of the determination of cosmological parameters from future large-scale photometric surveys obtained using the full shape of the 2-point galaxy angular correlation function. The effects of linear redshift-space distortion, photometric redshift Gaussian errors, galaxy bias and nonlinearities in the power spectrum are included on our analysis. The Fisher information matrix is constructed with the full covariance matrix, including the correlation between nearby redshift shells arising from the photometric redshift error. We show that under some reasonable assumptions, a survey such as the imminent Dark Energy Survey should be able to constrain the dark energy equation of state parameter $w$ and the cold dark matter density ${\ensuremath{\Omega}}_{\mathrm{cdm}}$ with a precision of the order of 20% and 13%, respectively, from the full shape of the angular correlation function alone. When combined with priors from other observations the precision in the determination of these parameters improve to 8% and 4%, respectively.

Models for the clustering of far-infrared and submillimetre selected galaxies

Abstract We discuss and compare two alternative models for the two-point angular correlation function of galaxies detected through the submillimetre emission using the Herschel Space Observatory. The first, now-standard Halo Model, which represents the angular correlations as arising from one-halo and two-halo contributions, is flexible but complex and rather unwieldy. The second model is based on a much simpler approach: we incorporate a fitting function method to estimate the matter correlation function with approximate model of the bias inferred from the estimated redshift distribution to find the galaxy angular correlation function. We find that both models give a good account of the shape of the correlation functions obtained from published preliminary studies of the Herschel Multi-tiered Extragalactic Survey (HerMES) and the Herschel-Astrophysical Terahertz Large Area Survey (H-ATLAS) performed using Herschel, and yield consistent estimates of the minimum halo mass within which the submillimetre galaxies must reside. We note also that both models predict an inflection in the correlation function at intermediate angular scales, so the presence of the feature in the measured correlation function does not unambiguously indicate the presence of intra-halo correlations. The primary barrier to more detailed interpretation of these clustering measurements lies in the substantial uncertainty surrounding the redshift distribution of the sources.

Herschel-ATLAS: The angular correlation function of submillimetre galaxies at high and low redshift

We present measurements of the angular correlation function of galaxies selected from the first field of the H-ATLAS survey. Careful removal of the background from galactic cirrus is essential, and currently dominates the uncertainty in our measurements. For our 250 micron-selected sample we detect no significant clustering, consistent with the expectation that the 250 micron-selected sources are mostly normal galaxies at z<~ 1. For our 350 micron and 500 micron-selected samples we detect relatively strong clustering with correlation amplitudes A of 0.2 and 1.2 at 1', but with relatively large uncertainties. For samples which preferentially select high redshift galaxies at z~2-3 we detect significant strong clustering, leading to an estimate of r_0 ~ 7-11 h^{-1} Mpc. The slope of our clustering measurements is very steep, delta~2. The measurements are consistent with the idea that sub-mm sources consist of a low redshift population of normal galaxies and a high redshift population of highly clustered star-bursting galaxies.

The VIMOS VLT deep survey

We present a new K_s-band survey that represents a significant extension to the previous wide-field K_s-band imaging survey within the 0226-04 field of the VIMOS-VLT deep survey (VVDS). The new data add ~458 arcmin2 to the previous imaging program, thus allowing us to cover a total contiguous area of ~600 arcmin2 within this field. Sources were identified both directly on the final K-band mosaic image and on the corresponding, deep χ^2-g'r'i' image from the CFHT Legacy Survey in order to reduce contamination, while ensuring compilation of a truly K-selected catalogue down to the completeness limit of the K_s-band. The newly determined K_s-band magnitudes are used in combination with the ancillary multiwavelength data for determining accurate photometric redshifts. The final catalogue totals ~52 000 sources, out of which ~4400 have a spectroscopic redshift from the VVDS first epoch survey. The catalogue is 90% complete down to KVega = 20.5 mag. We present K_s-band galaxy counts and angular correlation function measurements down to this magnitude limit. Our results are in good agreement with previously published work. We show that using K magnitudes to determine photometric redshifts significantly lowers the incidence of catastrophic errors. The data presented in this paper are publicly available through the CENCOS database. Based on observations collected at the European Southern Observatory New Technology Telescope, La Silla, Chile, programme 075.A-0752(A), on data obtained with the European Southern Observatory Very Large Telescope, Paranal, Chile, programme 070.A-9007(A), and on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT), which is operated by the National Research Council (NRC) of Canada, the Institut National des Science de l'Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. This work is based in part on data products produced at TERAPIX and the Canadian Astronomy Data Centre as part of the Canada-France-Hawaii Telescope Legacy Survey, a collaborative project of NRC and CNRS.

The Angular Correlations of Galaxies in the COSMOS Field

We present measurements of the two-point galaxy angular correlation function w(θ) in the COSMOS field. Independent determinations of w(θ) as a function of magnitude limit are presented for both the Hubble Space Telescope (HST) ACS catalog and also the ground-based data from Subaru and the Canada-France-Hawaii Telescope. Despite having significantly different masks, these three determinations agree well. At bright magnitudes (I_(AB) 10', which we attribute to a particularly rich structure known to exist at z ≃ 0.8. For fainter samples, however, the level of clustering is somewhat higher than reported by some previous studies; in all three catalogs we find w(θ = 1') ≃ 0.014 at a median I_(AB) magnitude of 24. At these very faintest magnitudes, our measurements agree well with the latest determinations from the Canada-France Legacy Survey. This level of clustering is approximately double what is predicted by the semianalytic catalogs (at all angles). The semianalytic results allow an estimate of cosmic variance, which is too small to account for the discrepancy. We therefore conclude that the mean amplitude of clustering at this level is higher than previously estimated.

Using Galaxy Two‐Point Correlation Functions to Determine the Redshift Distributions of Galaxies Binned by Photometric Redshift

We investigate how well the redshift distributions of galaxies sorted into photometric redshift bins can be determined from the galaxy angular two-point correlation functions. We find that the uncertainty in the reconstructed redshift distributions depends critically on the number of parameters used in each redshift bin and the range of angular scales used, but not on the number of photometric redshift bins. Using six parameters for each photometric redshift bin, and restricting ourselves to angular scales over which the galaxy number counts are normally distributed, we find that errors in the reconstructed redshift distributions are large; i.e., they would be the dominant source of uncertainty in cosmological parameters estimated from otherwise ideal weak lensing or baryon acoustic oscillation data. However, either by reducing the number of free parameters in each redshift bin, or by (unjustifiably) applying our Gaussian analysis into the non-Gaussian regime, we find that the correlation functions can be used to reconstruct the redshift distributions with moderate precision; e.g., with mean redshifts determined to around 0.01. We also find that dividing the galaxies into two spectral types, and thereby doubling the number of redshift distribution parameters, can result in a reduction in the errors in the combined redshift distributions.

The Galaxy Angular Correlation Functions and Power Spectrum from the Two Micron All Sky Survey

We calculate the angular correlation function of galaxies in the Two Micron All Sky Survey. We minimize the possible contamination by stars, dust, seeing, and sky brightness by studying their cross correlation with galaxy density and limiting the galaxy sample accordingly. We measure the correlation function at scales between 1' 25. Our results are in good agreement with other measurements of the angular correlation function. We invert the angular correlation function using singular value decomposition to measure the three-dimensional power spectrum and find that it too is in good agreement with previous measurements. A dip seen in the power spectrum at small wavenumber k is statistically consistent with cold dark matter (CDM) type power spectra. A fit of CDM-type power spectra in the linear regime (k < 0.15 h Mpc-1) gives constraints of Ωmh = 0.13 ± 0.07 and σ8 = 1.0 ± 0.09 for a spectral index of 1.0. This suggests a Ks-band linear bias of 1.1 ± 0.2. These measurements are in good agreement with other measurements of the power spectrum on linear scales. On small scales the power-law shape of our power spectrum is shallower than that derived for the Sloan Digital Sky Survey. This may imply a biasing for these different galaxies that could be either waveband- or luminosity-dependent. The power spectrum derived here in combination with the results from other surveys can be used to constrain models of galaxy formation.

Evolution and Color Dependence of the Galaxy Angular Correlation Function: 350,000 Galaxies in 5 Square Degrees

When applied to deep photometric catalogs, the two-point angular correlation function, w(theta), is a sensitive probe of the evolution of galaxy clustering properties. Here we present measurements of w(theta) as a function of I_AB magnitude and (R-I) color to a depth of I_AB=24, using a sample of ~350,000 galaxies covering 5 degrees^2 in total over 5 separate fields. Using redshifts of 3319 galaxies in early DEEP2 Galaxy Redshift Survey data, we construct robust galaxy redshift distributions as a function of I_AB and R_AB magnitude and (R-I) color for galaxies between 0 1 to z=0. A model in which the comoving scale-length, x_0, evolves linearly with redshift, x_0(z)=x_0(0)(1-Bz), fits the data better than the 'epsilon' model proposed by Groth and Peebles (1977). The clustering properties depend strongly on observed (R-I) color, with both the reddest and bluest galaxies exhibiting large clustering amplitudes and steeper slopes. Different observed (R-I) color ranges are sensitive to very disparate redshift regimes. Red galaxies with (R-I)~1.5 lie in a narrow redshift range centered at z~0.85 and have a comoving scale length of clustering at z=0.85 of x_0=5.0 +/-0.3 Mpc/h. These galaxies have early-type spectra and are likely progenitors of massive local ellipticals. The bluest galaxies with (R-I)~0 appear to be a mix of star-forming galaxies, both relatively local (z~0.3-0.6) dwarfs and bright z>1.4 galaxies, and broad-line AGN. We find that local blue dwarfs are relatively unclustered, with x_0=1.6 +/-0.2 Mpc/h. The z>1.4 blue galaxies have a larger clustering scale-length, x_0>5 Mpc/h.

Angular Clustering of Galaxies at 3.6 Microns from theSpitzerWide‐area Infrared Extragalactic (SWIRE) Survey

We present the first analysis of large-scale clustering from the Spitzer Wide-area Infrared Extragalactic legacy survey (SWIRE). We compute the angular correlation function of galaxies selected to have 3.6 μm fluxes brighter than 32 μJy in three fields totaling 2 deg2 in area. In each field we detect clustering with a high level of significance. The amplitude and slope of the correlation function is consistent between the three fields and is modeled as w(θ) = Aθ1-γ with A = (0.6 ± 0.3) × 10-3,γ = 2.03 ± 0.10. With a fixed slope of γ = 1.8, we obtain an amplitude of A = (1.7 ± 0.1) × 10-3. Assuming an equivalent depth of K ≈ 18.7 mag we find that our errors are smaller but our results are consistent with existing clustering measurements in K-band surveys and with stable clustering models. We estimate our median redshift z 0.75, and this allows us to obtain an estimate of the three-dimensional correlation function ξ(r), for which we find r0 = 4.4 ± 0.1 h-1 Mpc.

Galaxy Clustering Evolution in the UH8K Weak‐Lensing Fields

We present measurements of the two-point galaxy angular correlation function as a function of apparent magnitude, color, and morphology. We present new galaxy number counts to limiting magnitudes of I=24.0 and V=25.0. We find $\omega(\theta)$ to be well described by a power-law of slope -0.8. We find the amplitude of the correlation function to decrease monotonically with increasingly faint apparent magnitude. We compare with predictions utilizing redshift distributions based on deep spectroscopic observations. We conclude that simple redshift-dependent models which characterize evolution by means of the epsilon parameter inadequately describe the observations. We find a strong clustering dependence on V-I color because galaxies of extreme color lie at similar redshifts and the angular correlation functions for these samples are minimally diluted by chance projections. We then present the first attempt to investigate the redshift evolution of clustering, utilizing a population of galaxies of the same morphological type and absolute luminosity. We study the dependence of $\omega(\theta)$ on redshift for Lstar early-type galaxies in the redshift range 0.2<z<0.9. Although uncertainties are large, we find the evolution in the clustering of these galaxies to be consistent with stable clustering [epsilon=0]. We find Lstar early-type galaxies to cluster slightly more strongly (rnought = 5.25\pm0.28 \hMpc assuming epsilon=0) than the local full field population. This is in good agreement with the 2dFGRS value for Lstar early-type galaxies in the local universe (abridged).

The Angular Correlation Function of Galaxies from Early Sloan Digital Sky Survey Data

The Sloan Digital Sky Survey is one of the first multicolor photometric and spectroscopic surveys designed to measure the statistical properties of galaxies within the local universe. In this paper we present some of the initial results on the angular two-point correlation function measured from the early SDSS galaxy data. The form of the correlation function, over the magnitude interval 18 < r* < 22, is shown to be consistent with results from existing wide-field, photographic-based surveys and narrower CCD galaxy surveys. On scales between 1' and 1° the correlation function is well described by a power law with an exponent of ≈-0.7. The amplitude of the correlation function, within this angular interval, decreases with fainter magnitudes in good agreement with analysis from existing galaxy surveys. There is a characteristic break in the correlation function on scales of approximately 1°-2°. On small scales, θ < 1', the SDSS correlation function does not appear to be consistent with the power-law form fitted to the 1' < θ < 05 data. With a data set that is less than 2% of the full SDSS survey area, we have obtained high-precision measurements of the power-law angular correlation function on angular scales 1' < θ < 1°, which are robust to systematic uncertainties. Because of the limited area and the highly correlated nature of the error covariance matrix, these initial results do not yet provide a definitive characterization of departures from the power-law form at smaller and larger angles. In the near future, however, the area of the SDSS imaging survey will be sufficient to allow detailed analysis of the small- and large-scale regimes, measurements of higher order correlations, and studies of angular clustering as a function of redshift and galaxy type.

The clustering of colour-selected galaxies

We present measurements of the angular correlation function of galaxies selected from a B_J=23.5 multicolour survey of two 5 degree by 5 degree fields located at high galactic latitudes. The galaxy catalogue of approximately 400,000 galaxies is comparable in size to catalogues used to determine the galaxy correlation function at low-redshift. Measurements of the z=0.4 correlation function at large angular scales show no evidence for a break from a power law though our results are not inconsistent with a break at >15 Mpc. Despite the large fields-of-view, there are large discrepancies between the measurements of the correlation function in each field, possibly due to dwarf galaxies within z=0.11 clusters near the South Galactic Pole. Colour selection is used to study the clustering of galaxies z=0 to z=0.4. The galaxy correlation function is found to strongly depend on colour with red galaxies more strongly clustered than blue galaxies by a factor of 5 at small scales. The slope of the correlation function is also found to vary with colour with gamma=1.8 for red galaxies while gamma=1.5 for blue galaxies. The clustering of red galaxies is consistently strong over the entire magnitude range studied though there are large variations between the two fields. The clustering of blue galaxies is extremely weak over the observed magnitude range with clustering consistent with r_0=2 Mpc. This is weaker than the clustering of late-type galaxies in the local Universe and suggests galaxy clustering is more strongly correlated with colour than morphology. This may also be the first detection of a substantial low redshift galaxy population with clustering properties similar to faint blue galaxies.

Measuring Omega/b with Weak Lensing.

A correlation between the surface density of foreground galaxies tracing the large-scale structure and the position of distant galaxies and quasars is expected from the magnification bias effect. We show that this foreground-background correlation function wfb can be used as a straightforward and almost model-free method to measure the cosmological ratio Omega/b. For samples with appropriate redshift distributions, wfb~Omegadeltag, where delta and g are respectively the foreground dark matter and galaxy surface density fluctuations. Therefore, Omega&solm0;b~wfb&solm0;w, where w identical withgg is the foreground galaxy angular two-point correlation function, b is the biasing factor, and the proportionality factor is independent of the dark matter power spectrum. Simple estimations show that the application of this method to the galaxy and quasar samples generated by the upcoming Sloan Sky Digital Survey will achieve a highly accurate and well-resolved measurement of the ratio Omega/b.

Strong Clustering of Faint Galaxies at Small Angular Scales

The 2-point angular correlation function of galaxies, \wt, has been computed on equatorial fields observed with the CTIO 4m prime focus, within a total area of 2.31 deg$^2$. In the magnitude range $19\le m_R \le 21.5$, corresponding to $ \approx 0.35$, we find an excess of power in \wt at scales $2''\le\theta \le6''$ over what would be expected from an extrapolation of \wt measured at larger $\theta$. The significance of this excess is $\approx 5\sigma$. At larger scales, $6''< \theta \le 24''$, the amplitude of \wt is 1.6 times smaller than the standard no evolutionary model. At these scales there is remarkable agreement between the present data and Infante \& Pritchet (1995). At large angular scales ($6''< \theta \le 24''$) the data is best described by a model where clustering evolution in $\xi(r,z)$ has taken place. Strong luminosity evolution cannot be ruled out with the present data. At smaller scales, $2''\le \theta \le 6''$, our data are formally fit by models where $\epsilon=-2.4 (\Omega=0.2, r_o=5.1h^{-1}$Mpc) or $r_o = 7.3h^{-1}$Mpc $(\Omega=0.2, \epsilon=0)$. If the mean redshift of our sample is 0.35 then our data show a clear detection of the scale ($\approx 19h^{-1}kpc$) where the clustering evolution approaches a highly non linear regime, i.e., $\epsilon \le 0$. The rate at which galaxies merge has been computed. If this rate is proportional to $(1+z)^m$, then $m=2.2 \pm 0.5$.

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 •