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Astron. Astrophys. 337, 96-104 (1998)

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4. Tidal field and the selection function

According to biased galaxy formation theory the sites of formation of structures of mass [FORMULA] must be identified with the maxima of the density peak smoothed over a scale [FORMULA] ([FORMULA]). 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 [FORMULA] or equivalently that the threshold criterion [FORMULA] is satisfied, being [FORMULA] the variance of the density field smoothed on scale [FORMULA]. 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 [FORMULA] peaks form bright objects while low [FORMULA] peaks ([FORMULA]) 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, [FORMULA]. The selection function, [FORMULA], gives the probability that a density peak forms an object, while the threshold level, [FORMULA], is defined so that the probability that a peak forms an object is 1/2 when [FORMULA]. The selection function introduced by BBKS (Eq. 4.13), is an empirical one and depends on two parameters: the threshold [FORMULA] and the shape parameter q:

[EQUATION]

If [FORMULA] this selection function is a Heaviside function [FORMULA] so that peaks with [FORMULA] have a probability equal to 100% to form objects while peaks with [FORMULA] do not form objects. If q has a finite value sub-[FORMULA] peaks are selected with non-zero probability. Using the given selection function the cumulative number density of peaks higher than [FORMULA] is given, according to BBKS, by:

[EQUATION]

where [FORMULA] 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:

[EQUATION]

where the function

[EQUATION]

gives the probability that the peak overdensity is different from the average, in a Gaussian density field. The selection function depends on [FORMULA] through the dependence of [FORMULA] on [FORMULA]. As displayed, the integrand is evaluated at a radius [FORMULA] which is the typical radius of the object that we are selecting. Moreover, the selection function [FORMULA] depends on the critical overdensity threshold for the collapse, [FORMULA], 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 [FORMULA]. The dependence of [FORMULA] on [FORMULA] can be obtained in several ways, for example according to Peebles (1980) the value of [FORMULA] depends on the ratio [FORMULA] between the perturbation collapse time, [FORMULA], and its turn around time, [FORMULA]:

[EQUATION]

Non-radial motions slow down the collapse of the mass shell with respect to the GG collapse time changing the value of [FORMULA]. Using the calculated time of collapse for a given shell, and its dependence on [FORMULA], [FORMULA] can be calculated using Eq. (27). An analityc determination of [FORMULA] 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, [FORMULA], which is characterized by the condition [FORMULA]. Using the relation between v and [FORMULA], in linear theory (Peebles 1980), we can find C that substituted in Eq. (19) gives at turn around:

[EQUATION]

where [FORMULA] is the critical threshold for GG's model. In Fig. 6 we show the overdensity threshold as a function of [FORMULA]. As shown, [FORMULA] decreases with increasing [FORMULA]: when [FORMULA] 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.

[FIGURE] Fig. 6. The critical threshold, [FORMULA] versus [FORMULA]

Known [FORMULA] 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, ([FORMULA], 4 [FORMULA]), shows that the selection function, as expected, differs from an Heaviside function (sharp threshold).

[FIGURE] Fig. 7. The selection function, [FORMULA], for [FORMULA] (the solid line plots the selection function obtained without taking into account the effects of non-radial motions; the dotted line plots the selection function obtained taking into account the effects of non-radial motions) and for [FORMULA] (the short dashed line plots the selection function obtained without taking into account the effects of non-radial motions; the long dashed line plots the selection function obtained taking into account the effects of non-radial motions).

The shape of the selection function depends on the values of the filtering length [FORMULA] and on non-radial motions. The value of [FORMULA] at which the selection function [FORMULA] reaches the value 1 ([FORMULA]) increases for growing values of the filtering radius, [FORMULA]. This is due to the smoothing effect of the filtering process. The effect of non-radial motions is, firstly, that of shifting [FORMULA] towards higher values of [FORMULA], 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 [FORMULA] induced by non-radial motions itself.

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© European Southern Observatory (ESO) 1998

Online publication: August 6, 1998
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