## 4. Tidal field and the selection functionAccording to biased galaxy formation theory the sites of formation of structures of mass must be identified with the maxima of the density peak smoothed over a scale (). A necessary condition for a perturbation to form a structure is that it goes non-linearly and that the linearly extrapolated density contrast reaches the value or equivalently that the threshold criterion is satisfied, being the variance of the density field smoothed on scale . When these conditions are satisfied the matter in a shell around a peak falls in toward the cluster center and virializes. In this scenario only rare high peaks form bright objects while low peaks () form under-luminous objects. The kind of objects that forms from non-linear structures depends on the details of the collapse. Moreover, if structures form only at peaks in the mass distribution they will be more strongly clustered than the mass. Several feedback mechanisms have been proposed to explain this
segregation effect (Rees 1985; Dekel & Rees 1987). Even if these
feedback mechanisms work one cannot expect they have effect
instantaneously, so the threshold for structure formation cannot be
sharp (BBKS). To take into account this effect BBKS introduced a
threshold or selection function, . The
selection function, , gives the probability
that a density peak forms an object, while the threshold level,
, is defined so that the probability that a
peak forms an object is 1/2 when . The
selection function introduced by BBKS (Eq. 4.13), is an empirical one
and depends on two parameters: the threshold
and the shape parameter If this selection function is a Heaviside
function so that peaks with
have a probability equal to 100% to form
objects while peaks with do not form objects.
If where is the comoving peak density (see BBKS Eq. 4.3). A form of the selection function, physically founded, can be obtained following the argument given in CAD. In this last paper the selection function is defined as: gives the probability that the peak overdensity is different from the average, in a Gaussian density field. The selection function depends on through the dependence of on . As displayed, the integrand is evaluated at a radius which is the typical radius of the object that we are selecting. Moreover, the selection function depends on the critical overdensity threshold for the collapse, , which is not constant as in a spherical model (due to the presence, in our analysis, of non-radial motions that delay the collapse of the proto-cluster) but it depends on . The dependence of on can be obtained in several ways, for example according to Peebles (1980) the value of depends on the ratio between the perturbation collapse time, , and its turn around time, : Non-radial motions slow down the collapse of the mass shell with
respect to the GG collapse time changing the value of
. Using the calculated time of collapse for a
given shell, and its dependence on ,
can be calculated using Eq. (27). An analityc
determination of can be obtained following a
technique similar to that used by Bartlett & Silk (1993). Using
Eq. (19) it is possible to obtain the value of the expansion
parameter of the turn around epoch, , which is
characterized by the condition . Using the
relation between where is the critical threshold for GG's model. In Fig. 6 we show the overdensity threshold as a function of . As shown, decreases with increasing : when the threshold assumes the typical value of the spherical model. This means, according to the cooperative galaxy formation theory, (Bower et al. 1993) that structures form more easily if there are other structures nearby, i.e. the threshold level is a decreasing function of the mean mass density.
Known and chosen a spectrum, the selection function is immediately obtainable through Eq. (25) and Eq. (26). The result of the calculation, plotted in Fig. 7, for two values of the filtering radius, (, 4 ), shows that the selection function, as expected, differs from an Heaviside function (sharp threshold).
The shape of the selection function depends on the values of the filtering length and on non-radial motions. The value of at which the selection function reaches the value 1 () increases for growing values of the filtering radius, . This is due to the smoothing effect of the filtering process. The effect of non-radial motions is, firstly, that of shifting towards higher values of , and, secondly, that of making it steeper. The selection function is also different from that used by BBKS (Table 3a). Finally it is interesting to note that the selection function defined by Eq. (25) and Eq. (26) is totally general, it does not depend on the presence or absence of non-radial motions. The latter influences the selection function form through the change of induced by non-radial motions itself. © European Southern Observatory (ESO) 1998 Online publication: August 6, 1998 |