It has long been speculated on the fundamental role that the angular momentum could play in determining the fate of collapsing proto-structures and several models have been proposed to correlate the galaxy type with the angular momentum per unit mass of the structure itself (Faber 1982; Kashlinsky 1982; Fall 1983). Some authors (see Barrow & Silk 1981; Szalay & Silk 1983 and Peebles 1990) have proposed that non-radial motions would be expected within a developing proto-cluster due to the tidal interaction of the irregular mass distribution around them, typical of hierarchical clustering models, with the neighbouring proto-clusters. The kinetic energy of these non-radial motions prevents the collapse of the proto-cluster, enabling the same to reach statistical equilibrium before the final collapse (the so-called previrialization conjecture by Davis & Peebles 1977, Peebles 1990). This effect may prevent the increase in the slope of the mass autocorrelation function at separations given by , expected in the scaling solution for the rise in but not observed in the galaxy two-point correlation function. The role of non-radial motions has been pointed by several authors (see Davis & Peebles 1983: Gorski 1988; Groth et al. 1989; Mo et al. 1993; van de Weygaert & Babul 1994; Marzke et al. 1995 and Antonuccio-Delogu & Colafrancesco 1995). Antonuccio-Delogu & Colafrancesco derived the conditional probability distribution of the peculiar velocity around a peak of a Gaussian density field and used the moments of the velocity distribution to study the velocity dispersion around the peak. They showed that regions of the proto-clusters at radii, r, greater than the filtering length, , contain predominantly non-radial motions.
Non-radial motions change the energetics of the collapse model by introducing another potential energy term. In other words one expects that non-radial motions change the characteristics of the collapse and in particular the turn around epoch, , and consequently the critical threshold, , for collapse. Here, we want to remind that is the time at which the linear density fluctuations, that generate the cosmic structures, detach from the Hubble flow. The turn-around epoch is given by:
where is the mean background density, z is the redshift and is the mean over-density within the non-linear region. After the turn around epoch, the fluctuations start to recollapse. As known for a spherical top hat model, the perturbation of the density field is completely collapsed when
where is the time of collapse which is twice the turn around epoch. One expects that non-radial motions produce firstly a change in the turn around epoch, secondly a new functional form for , thirdly a change in the mass function calculable with the Press-Schechter (1974) formula and finally a modification of the two-point correlation function. As we shall show in a forthcoming paper (Del Popolo & Gambera 1997b) non-radial motions can reduce several discrepancies between the SCDM model and observations: the strong clustering of rich clusters of galaxies () far in excess of CDM predictions (Bahcall & Soneira 1983), the X-ray temperature distribution function of clusters over-producing the observed cluster abundances (Bartlett & Silk 1993).
For the sake of completeness, we remember that alternative models with more large-scale power than SCDM have been introduced in order to solve the latter problem. Several authors (Peebles 1984; Efstathiou et al. 1990; Turner 1991; White et al. 1993) have lowered the matter density under the critical value () and have added a cosmological constant in order to retain a flat universe () . The spectrum of the matter density is specified by the transfer function, but its shape is affected because of the fact that the epoch of matter-radiation equality is earlier, being increased by a factor . Around the epoch the growth of the density contrast slows down and ceases after . As a consequence the normalisation of the transfer function begins to fall, even if its shape is retained.
Mixed dark matter models (MDM) (Bond et al. 1980; Shafi & Stecker 1984; Valdarnini & Bonometto 1985; Holtzman 1989; Schaefer 1991; Schaefer & Shafi 1993; Holtzman & Primack 1993) increase the large-scale power because free-streaming neutrinos damp the power on small scales. Alternatively changing the primeval spectrum several problems of SCDM are solved (Cen et al. 1992). Finally, it is possible to assume that the threshold for galaxy formation is not spatially invariant but weakly modulated ( on scales ) by large scale density fluctuations, with the result that the clustering on large-scale is significantly increased (Bower et al. 1993).
Moreover, this study of the role of non-radial motions in the collapse of density perturbations can help us to give a deeper insight in to the so-called problem of biasing. As pointed out by Davis et al. (1985), unbiased CDM presents several problems: pairwise velocity dispersion larger than the observed one, galaxy correlation function steeper than that observed (see Liddle & Lyth 1993 and Strauss & Willick 1995). The remedy to these problems is the concept of biasing (Kaiser 1984), i.e. that galaxies are more strongly clustered than the mass distribution from which they originated. The physical origin of such biasing is not yet clear even if several mechanisms have been proposed (Rees 1985; Dekel & Silk 1986; Dekel & Rees 1987; Carlberg 1991; Cen & Ostriker 1992; Bower et al. 1993; Silk & Wyse 1993). Recently Colafrancesco et al. (1995, hereafter CAD) and Del Popolo & Gambera (1997a) have shown that dynamical friction delays the collapse of low- peaks inducing a bias of dynamical nature. Because of the dynamical friction, under-dense regions in clusters (the clusters outskirts) accrete less mass than that accreted in absence of this dissipative effect and as a consequence over-dense regions are biased toward higher mass (Antonuccio-Delogu & Colafrancesco 1994 and Del Popolo & Gambera, 1996). Non-radial motions acts in a similar way to dynamical friction: they delay the shell collapse consequently inducing a dynamical bias similar to that produced by dynamical friction. This dynamical bias can be evaluated defining a selection function similar to that given in CAD and using Bardeen et al. (1986, hereafter BBKS) prescriptions.
The methods used in this paper are fundamentally some results of the statistics of Gaussian random fields, the biased galaxy formation theory and the spherical model for the collapse of density perturbations. In particular, we calculate the specific angular momentum acquired by protoclusters and the time of collapse of protoclusters using the Gaussian random fields theory and the spherical collapse model following Ryden's (1988a, hereafter R88a) approach. The selection function that we introduce is general and obtained by the only hypothesis of Gaussian density field. The approach and the final result is totally different from BBKS selection function and similar to that of Colafrancesco, Antonuccio & Del Popolo (1995). Only the biasing parameter is obtained from a BBKS approximated formula. This choice will be clarified in the following sections of the paper.
The plan of the paper is the following: in Sect. 2 we obtain the total specific angular momentum acquired during expansion by a proto-cluster. In Sect. 3 we use the calculated specific angular momentum to obtain the time of collapse of shells of matter around peaks of density having and we compare the results with Gunn & Gott's (1972, hereafter GG) spherical collapse model. In Sect. 4 we derive a selection function for the peaks giving rise to proto-structures while in Sect. 5 we calculate some values for the bias parameter, using the selection function derived, on three relevant filtering scales. Finally in Sect. 6 we discuss the results obtained.
© European Southern Observatory (ESO) 1998
Online publication: August 6, 1998