## 5. Galactic halosIn the previous sections, we have obtained the probability
distribution of the magnification due to weak lensing by assuming the
density field follows a specific scaling model described by (13),
which is consistent with numerical results (Valageas et al.1999;
Munshi et al.1999; Colombi et al.1997). More precisely, although the
parameters and the function
may show some slow scale-dependence
through the variation of the local power-spectrum index which defines the slope (with
). The factor
is the overdensity at the virial
radius First, we note that such a description of the density field leads
to a power-law tail at high From (26) we see that: where We also note from the definition (1) that the description (70) means that the parameters are scale-dependent: For reasonable values of () this implies a strong growth of the coefficients () at small scales which contradicts numerical results as discussed in Valageas (1999a). Note on the other hand that leads to a small slope for the two-point correlation function. Thus, we think this is not a good model for the dark matter density field. However, we briefly present below the results obtained by such a description as it may be valid for baryonic cores of collapsed objects and it allows us to compare our previous results with other approaches. We restrict ourselves to . Following the method used in Sect. 3, in order to derive the probability distribution we first obtain the cumulants . We divide the line-of-sight from the observer to the redshift of the source into small physical elements so that they contain at most one halo and from (21) we write for each realization the flux perturbation as: where (resp.
) if there is (resp. there is not) a
halo of type within the length
element where: comes from the integration along the line-of-sight through the
halo. Here we made the approximation that the impact parameter
where we defined the comoving mass function of halos of mass
Some of the integrals over the impact parameter diverge for but we could add for these intermediate steps of the calculation an ad-hoc cutoff which we would put to 0 later on. In a fashion similar to (33) we can define the generating fu nction and we get: with: and the variance of the magnification is: and: Thus, is the mean number of
halos along the line of sight while
is the minimum value of the magnification which shows that for
. For large where is Dirac's function. Of
course, we could have written (86) directly, as it corresponds to the
case where there is only one object along the line of sight. In
practice and we obtain after
integration over We can check that we recover the power-law behaviour of (73). Next,
in order to compare with (45) we need to estimate the halo mass
function . As discussed in details
in Valageas & Schaeffer (1997) and Valageas & Schaeffer
(1999), from the description of the non-linear density field presented
in Sect. 2 one can write the comoving mass function of halos
defined by the density contrast
(which may depend both on where is the present mean universe density and is the function obtained from counts-in-cells statistics defined in (5). A comparison of (88) with numerical results is described in Valageas et al.(1999). In order to get a direct comparison with (45) we now make the approximation: This is possible because the halos we consider here correspond to galactic masses hence to the scale as noticed in Sect. 3, where the local index of the initial linear power-spectrum is . Here and . Thus we eventually get: where we defined: as in (6). In a similar fashion, we obtain from (83): where the parameters where is the exact minimum value of the magnification obtained in (22). As expected, we recover the factor since the physical process is the same. The term of order unity comes from the approximations involved in our calculation (assumption of small impact parameter and approximation ). From the results obtained in the previous section, we know that so that we still have and . Moreover, the probability distribution (90) is only valid for as explained above, that is for , hence the value of plays no role as long as it remains small as compared to 1. We compare in Fig. 7 the probability distribution
obtained from (90) for
and
with the results obtained in Sect.
3 from (45). Note that we have defined throughout the magnification
We show in Fig. 8 the mean number of halos
along the line of sight up to the
redshift of the source, from (92).
We use K. We can check that
which justifies the approximation
(90). Thus, large magnifications
© European Southern Observatory (ESO) 2000 Online publication: February 25, 2000 |