Studies of the evolution of X-ray emission from clusters of galaxies have so far used optically discovered distant clusters then observed at X-ray wavelengths. A different approach to the study of cluster evolution is to use clusters selected directly by their X-ray emission since X-ray selection is extremely successful at discovering high redshift clusters (up to z = 0.5). The Extension of the Medium Sensitivity Survey (Gioia, Maccacaro and Wolter, 1987), a flux limited sample of serendipitous X-ray sources discovered with the Einstein IPC, has been used to select 68 optically identified clusters and 66 possible candidates (using X-ray isointensity maps and optical POSS and ESO prints). Their X-ray fluxes are in the range 10−13 – 10−12 ergs cm−2 s−1 in the 0.3–3.5 keV energy band. Few misidentifications are expected, thus the results presented here have to be considered as preliminary. This sample can be used to study the X-ray logN-logS, the cluster luminosity function and its possible evolution with cosmic time, and also to compare the properties of X-ray selected clusters with those of samples selected by other techniques. Since the identification process is still in progress, and has not proceeded completely at random, the subset of identified sources is not representative of the entire population. With this caveat in mind we have constructed the redshift and X-ray luminosity distributions for the clusters for which we have spectroscopic observations. 68% of the clusters are at z > 0.1, 34% at z > 0.2, and 12% at z > 0.3. Their X-ray luminosities range from about 3 × 1042 to 1045 ergs s−1. At present the lack of a redshift determination for many of the cluster candidates in our sample prevents us from studying the cluster luminosity function. W e can however derive their logN(>S)-logS relation.

In a continuing redshift observation program, we have recently determined or confirmed the redshifts of 32 R ≥ 0 Abell clusters. With these data added to the previously measured cluster redshifts, there is now a sample of Abell clusters to z = 0.085 that is 92% complete in redshift measurements. The sample consists of 225 clusters in the North and South Galactic Caps, with latitude |b| > 30° to minimize obscuration effects. The longitude range 270° < l < 360° was also excluded for this sample because of an apparent large spur of galactic absorption.

Following Peebles (Ap.J. 155, 393; Astron.Astrophys. 11, 377)and Efstathiou and Silk (Fund.Cosm.Phys. 9, 1) we assume that a developing proto-galaxy of mass M=1011M⊙ gains angular momentum during a first Phase 1 exten ding from recombination to the maximum expansion and then a second Phase 2 from the maximum expansion to a relaxed virialized structure passing through an intermediate Phase I.A very simple model is proposed in which a first mechanism connected with increasing density perturbatio ons leading to a torque ∼ t2/3, is applied to a spherical homogeneus structure. The initial conditions for its dynamical evolution are taken from a cosmological flat model (Ω = 1) with an isothermal perturbation mass spectrum. Phase 2 begins when the proto-galaxy is completely separated from the background and until this Phase it has been recognized that the initial spherical symmetry turns out to be well preserved, provided the angular momentum gained does not exceed 3.6 1073 g cm2 sec−1.

The occurrence of mergers in simulated loose groups of galaxies (Barnes 1985; Mamon 1987) has generated the concern that mergers may be significantly more frequent in simulated groups than in observed groups (Ostriker 1987). I compare here quantitatively the rates of merging between a subset of 102 groups (Mamon 1986) of the CfA group catalog (Geller and Huchra 1983) and samples of 50 dynamically simulated groups of given initial density. The simulated groups start out virialized with galaxy luminosities pseudo-randomly generated from a Schechter function of index −1 and cutoff 0.5 L* (giving an average luminosity ∼ L*, close to the average in the CfA group catalog). The groups are evolved with the N-body code described in Mamon (1987). The merging time of the sample of 50 simulated groups is defined as the time at which the Tremaine-Richstone (1977) statistic T1 = σ(M1)/〈ΔM12〉 becomes smaller than 0.8. This value of T1 is significantly smaller than expected from a random realization of groups with the luminosity function given above (Mamon 1987).

We have obtained CCD images of the brightest elliptical galaxy in 27 clusters of galaxies with redshifts z < 0.1. For each galaxy we fitted elliptical isophotes to the galaxy image. We also measured the position angle of each cluster from the Palomar Observatory Sky Survey (POSS) prints. We find only marginal evidence for alignment of brightest cluster members (BCM) with their parent clusters, but we find that our CCD isophotes are well aligned with the POSS image.

We perform a test of reproducibility of the results of Argyres, et al. (A.J. 91, 3; 1986). Their conclusion is that there is marginal evidence for alignment of the brightest galaxies in rich Abell clusters in the northern galactic hemisphere with the surrounding large scale distribution of galaxies in the Lick map made by Seldner, et al. (A.J. 82, 249; 1977).

Using the Uppsala General Catalog of bright galaxies (Nilson, 1973) and the northern and southern galaxy maps (Seldner, et al., 1977) made from the Lick counts (Shane and Wirtanen, 1967), we find statistical evidence of a morphology-orientation effect. Our results indicate that elliptical galaxies are preferentially oriented along the large scale features of the Lick maps. However, position angles of spiral and SO galaxies show no clear signs of non-random behavior. The angular scale of the detected alignment effect for ellipticals is θ ∼ 2 to 4°, which at a redshift of Z ∼ .03 (which comprises the depths of both catalogs) corresponds to a linear scale of ∼ 3 to 6 h−1 Mpc.

The cluster autocorrelation function ξc and the galaxy-cluster cross correlation ξgc are used to test the biased structure formation for scale-free spectra P(k)∝kn. Following Kaiser (1984), we assume that rich clusters form only at higfr density regions with the matter distribution represented by a Gaussian random field. Then, the correlation ξ of two regions with characteristic scales R1 and R2 lying above the thresholds v1 and v2 (δ ≡ νσ), is given by the expression for the bivariate Gaussian where z(r) is the coviance function, i.e. and is the correlation of the matter distribution Gaussian filtered on the comoving scales R1 and R2. From the previous equations, one can obtain for P(k)∝ kn, τ≪1 and large r a powers-law form either for ξc or ξgc. Moreover, the amplification factor Ac - gc ≡ ξc/ξgc is

We have analysed the cross-correlations between Abell clusters and the 10′ × 10′ Lick galaxy counts. The main new features of our analysis are as follows: (a) we correct for gradients in the Lick counts that depend only on galactic latitude; (b) we check the effects of applying a filter to remove (likely artificial) gradients on scales >20°; (c) redshifts for Abell clusters are used to compute a cross-correlation function wcg(σ), where σ = vϑ/Ho is the projected separation between a cluster with recession speed v and a galaxy at angular distance ν from the cluster centre; (d) the amplitude of the spatial cross-correlation function ξcg(r) is estimated using modern determinations of the galaxy luminosity function. Our results show that the shape of ξcg is reasonably well determined on scales r ≲20h−1 Mpc. On larger scales, the results depend sensitively on the corrections for large-scale gradients in the Lick counts. The results based on wcg(σ) are well described by a power-law model ξcg(r)≃(r/8.8h−1 Mpc)−2.2, while the angular cross-correlations for Abell clusters in distance class D=5 are better described by the model ξcg(r)=(r/6h−1 Mpc)−2.3 + (r/7h−1 Mpc)−1.7. Seldner and Peebles (1977) derived a larger amplitude for ξcg because they assumed a different luminosity function. We compare these results with theoretical predictions of ξcg in the Ω=1 biased cold dark matter (CDM) model. On scales 1 h−1 Mpc ≲ r ≲ 10 h−1 Mpc, the theoretical predictions for ξcg are in surprisingly good agreement with our observational estimates. On smaller scales, the CDM model gives too high an amplitude whereas on larger scales the predicted amplitude is too low. Neither of these discrepancies is particularly serious.

We consider the statistical testing of models for galaxy formation against the observed peculiar velocities on 10–100 Mpc scales (the Rubin-Ford effect). If we assume that observers are likely to be sited near maxima in the primordial field of density perturbations (Peacock & Heavens 1985: MNRAS 217, 805; Bardeen et al. 1986: Ap. J. 304, 15), then the observed filtered velocity field will be biased to low values by comparison with a point selected at random. The streaming-velocity data constrain models for galaxy formation with large-scale damping (adiabatic perturbations) to have a damping length close to the Rubin-Ford scale (rD ∼ 20 h−1 Mpc) and are mildly non-linear (Ω0.6 σo ∼1), where σo is the current fractional rms density variation. The Figure illustrates the regions of parameter space allowed at the 10,5,1 & 0.1% levels; the results are nearly independent of n, the power index of the primordial spectrum. Thus, both purely baryonic universes and universes dominated by massive neutrinos can account for the observed velocities, provided 0.1 ≲ Ω ≲ 1. In particular, the canonical Ω = 1 massive-neutrino model yields the required velocities quite naturally (in contradiction to the conclusion of Vittorio et al. 1986: Nature 323, 132), while not violating the constraints on the isotropy of the microwave background. For further details of this work, see Peacock, Lumsden & Heavens 1987: MNRAS in press.

We have calculated the growth of angular momentum about local density maxima at early epochs. We find that high peaks experience higher torques than low peaks, counteracting the short collapse time during which the high peaks can acquire angular momentum. Which effect is dominant depends on the perturbation power spectrum: for power spectra characteristic of both cold dark matter and hot dark matter, the effects nearly cancel, and the total angular momentum acquired by a collapsing object is almost independent of the height of the peak. Furthermore, the distributions of angular momenta acquired by collapsing protosystems are extremely broad, for all power spectra, far exceeding any modest differences between peaks of different height.

These results indicate that it is not possible to account for the systematic differences in angular momentum properties of disk and elliptical galaxies simply by postulating that the latter arise from fluctuations of greater overdensity, contrary to some recent suggestions. The figure shows the probability distributions for the final angular momentum acquired by peaks of dimensionless height 1–4, for a power spectrum similar to cold dark matter. A fuller account of this work has been submitted to MNRAS.

We have suggested a scenario of fractal turbulence which might explain the origin of galaxies and the observed large scale structure of the universe (Liu and Deng, 1987). Under the condition of the early universe, the cosmic fluid can be regarded as incompressible. If we assume that the density perturbations in the early universe are adiabatic and have the scale-free Zeldovich spectrum, we may obtain the spectrum of the velocity perturbations. Perturbations with scales less than horizon will undergo dissipative process by Thomson scattering. So, the cosmic fluid can be considered as a viscous fluid (Peebles, 1971). We can find the largest and smallest scale of the perturbations in the cosmic fluid by taking account of the Reynold's number on given scale and the scale of horizon. Using the present values of Hubble constant and the mean density of matter, we have found that on the scale of horizon the Reynold's number is just the order of 102. This result shows that perturbations with scale a little smaller than horizon may produce Karman vortices before recombination and the vortices might form fractal turbulence due to Thomson drag.

In the former work (Xia, Deng and Zhou, 1986), we have showed by two- point correlation analysis that more luminous galaxies cluster stronger. Now we present the result of cross-correlation analysis for galaxies with different luminosity. This analysis supplies information about the relations between the distributions of galaxies with different luminosity. The analyses are based on the data given by CfA survey and have made the same corrections as in the former work. The samples are divided into three subgroups in absolute magnitude ranges: a) −21–22, b) −20–21 and c)−19–20. We make the cross-correlation analysis for each two subgroups. Fig. 1 gives the obtained cross-correlation function ξc(r) and Fig. 2 shows the log ξc(r)-log r diagram, the straight lines in Fig. 2 are given by linear regression. These results show that the two brightest subgroups have the strongest correlation. Combining with the results of former work, it follows that the probability of two brighter galaxies being close to each other is larger than that of fainter galaxies.

In this work, we start from the definition of fractal dimension and by the number counting of galaxies to investigate that if the large scale distribution of galaxies has really a fractal structure. A fractal dimension in a distribution of objects is defined as where N(R) is the number of objects within scale R. To find out D from number counting we have to pay attention to that the distribution of galaxies could not be a regular fractal map. The N(R) would understand as a statistical average. To avoid the edge effect appearing in the number counting we compare it with that obtained from an average by many times Monte Carlo sampling which is a uniform distribution and would have Dr = 3. From eq. (1) we get

The spatial two point galaxy correlation function, ξ(r), is, at present, the most useful statistic for comparing theoretical models to observational data. We have derived expressions for the dynamical evolution of ξ for structures arising from Gaussian initial conditions under the assumption that non-linear evolution may be described by the Zel'dovich approximation. The observed angular correlation function, w(θ), places constraints on the spectrum of initial fluctuations on large scales.

The next move forward in simulations of cosmological structure is to include the hydrodynamics and thermal history of a gaseous component. The task is not an easy one. The dynamic range is wide in all interesting quantities (density, temperature, length-scales, time-scales, etc.). Generic initial mass distributions sampled from Gaussian random fields will, for many interesting power spectra, lead to a high degree of substructure present at all stages of the evolution. Grid-based hydrodynamic techniques currently lack the resolution necessary to evolve several levels of a clustering hierarchy simultaneously. A particle-based method known as SPH (Smoothed Particle Hydrodynamics, see Monoghan (1985) for a review) appears best suited for cosmological application. I have recently imbedded the technique into the P3M N-body code, described by Efstathiou et al. (1985) and used extensively by Efstathiou and collaborators, most recently in investigations of the cold dark matter scenario.

N-body models were run in order to test the mass estimators considered by Heisler, Tremaine and Bahcall (1985, HTB), when systems with a large number of mass-points with a non-flat mass-spectrum are considered. The initial conditions of the models were a analytic King profile for the number density, a Gaussian velocity distribution function and a Schechter-type mass spectrum. The models were left to evolve from a far from virial initial configuration, so a violent collapse occurs before the system reachs equilibrium. The code we use was NBODY2 code kindly provided to us by Dr. S. Aarseth.

We present a method of estimating distances to clusters of galaxies from twodimensional catalogues. The angular diameters (or magnitudes) of galaxies are used as distance indicators. The mapping from 2-D to 3-D is done by using a ‘diameter function’ (analogous to a luminosity function), which is based on a redshift survey from a section of the sky. The problem is formulated as follows. The number of galaxies with a metric diameter D in a volume element d3r is: where n(r) is the ‘true’ number density of galaxies at position r, n& is the mean number density of galaxies in the universe and ϕ(D)dD is the diameter function. We assume that within a narrow cone n(r) = n(r) and then express N(> θ), the number of galaxies greater than a certain angular diameter θ. In a discrete form we write the relation as: where ni is the density at the i – th distance bin and Pik is our ‘point spread function’, which is a function of the diameter function and Galactic obscuration. We express (2) in terms of χ2 statistics over the measurements, and require it to be less than a certain value. The entropy of the image is expressed as:

Hot dark matter particles have large thermal velocities at teg and hence cannot be gravitationally bound on small scales (free streaming). In models of formation of structure based on linear adiabatic perturbations all inhomogeneities on scales smaller than the maximal free streaming length λj are washed out. The mass λj inside a bail of radius exceeds the galaxy mass. Hence in the above models galaxies can only lorm by fragmentation of larger-scale objects. This is a severe problem.

Cosmic Strings are one-dimensional topological defects that may be formed in the early universe during a phase transition, and which may be the source of all inhomogeneities in our universe. Their mass per unit length, μ, gives us a dimensionless parameter, μ6 ≡ 106Gμ/c2, which must be of order unity for strings to seed galaxy formation. Results to date from the ongoing CfA redshift survey suggest that galaxies are distributed on two-dimensional surfaces, whose typical separation is about 50h50−1 Mpc. The loop distribution is unlikely to imprint such large-scale patterns in the galaxy positions so we have examined whether this structure could be caused by infinite strings. Because an infinite string typically moves at a substantial fraction of the speed of light, it will leave behind a very large accretion wake in the ambient medium. Gravitational instablity causes these wakes to continue to accrete matter long after the string has moved elsewhere. These wakes form around the two-dimensional surfaces swept out by the long strings.