In 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 n they should not explicitely depend on scale (i.e. on ) so that the approximation (27) is valid. In turn, such a picture of the non-linear density field implies that non-linear mass condensations can be described as an infinite hierarchy of increasingly small and overdense substructures (which follow the rise at small scales of ), as discussed in Valageas et al.(1999) and Valageas (1999a). On the other hand, it is customary to model collapsed objects as smooth halos with a density profile which follows a power-law or a curved profile as those obtained by Navarro et al.(1996) or Hernquist (1990). Moreover, within galactic cores where baryonic matter plays a dominant role one may expect a smooth density profile. Hence in this section we consider the case where collapsed halos are described by a power-law density profile:
which defines the slope (with ). The factor is the overdensity at the virial radius R of the object. Recently a similar description was used by Porciani & Madau (1999), with (isothermal sphere), with the Press-Schechter mass function (Press & Schechter 1974) of just-virialized objects. Here we intend to compare the results obtained from such a model to those we obtained in the previous sections.
First, we note that such a description of the density field leads to a power-law tail at high µ for the probability distribution of the magnification. Indeed, as shown in Valageas (1999a) the point correlation functions implied by (70) fol low in the highly non-linear regime the power-law behaviour:
From (26) we see that:
where R is the upper or lower cutoff of the integrals in (26). Thus, if and there is no lower cutoff (which would correspond to the radius of galactic cores where ) the cumulants and the moments diverge for . This is inconsistent with an exponential cutoff for while it is the natural outcome of a power-law tail given by:
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 i while is the contribution to the magnification of the source by a halo at redshift . From (21) we write:
comes from the integration along the line-of-sight through the halo. Here we made the approximation that the impact parameter b of the line-of-sight is much smaller than the radius R of the halo. The factor comes from the fact that here b and R (and l below) are physical scales while is a comoving scale. Note that the index denotes both the mass (or radius) of the halos and the impact parameter. Then, if we neglect the correlations between the collapsed objects we obtain in the continuous limit, using :
where we defined the comoving mass function of halos of mass M to as . We also used:
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:
and the variance of the magnification is:
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 b:
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 M and z) as:
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:
where the parameters a and describe the small x behaviour of , see (9), is the density contrast of the smallest collapsed halos we consider at the threshold and is the mean molecular weight. Indeed, the number of halos along the line of sight is dominated by the contribution of the smallest objects (the multiplicity function diverges at small mass if there is no cutoff). We use K, it corresponds to inefficient cooling and photo-heating by the UV background. However, note that does not enter the probability distribution (90) as long as it is small (the exponential factor in (86) can be put to unity) which is the case in practice. Thus, our res ults do not depend on the cutoff . We also get:
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 µ by (20). For large values of the convergence the approximation (20) breaks down and µ should be obtained from (19). However, since in this article we are mostly interested in the regime we always define µ by (20). Thus, can also be understood as with the change of variable (20). Then, large values of the magnification correspond to and to strong lensing events with multiple images. We can see in Fig. 7 that the probability distribution from (90) is of the same order as the results from (45) in the range . Indeed, we count the same mass and the same collapsed halos. Our results are similar to those obtained by Porciani & Madau (1999) who got a probability of a few to have a strong lensing event for a source at with . However, we can see that (90) is quite sensitive to the slope of the halos. Indeed, for we get while for we have . On the other hand, below the exponential cutoff the prescription developped in Sect. 3 leads to down to . However, since is small this pure power-law regime does no really appear as the corrections due to the peak at and the exponential cutoff are not negligible. Nevertheless, we can clearly see the extended large µ tail of and its non-gaussian behaviour. In particular, we recover the trend seen in numerical simulations (e.g. Wambsganss et al.1997). As we explained above the formulation (90) cannot describe the regime while for large magnifications it predicts a higher probability since it leads to a power-law tail instead of an exponential cutoff. This is directly linked to the strong growth with p of the point correlation functions at small scales implied by this model, which does not seem to be compatible with numerical simulations as argued in Valageas (1999a). On the other hand, if we only use (90) to obtain the weak lensing effect due to inner galactic halos where baryonic matter dominates the density field we would get lower probabilities since the matter content and size of these objects would be smaller, so that up to the probability distribution (45) would dominate. Thus, the formalism developped in Sect. 3 is better suited to obtain the magnification of distant sources by weak lensing effects.
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 µ come from the deflection of the light ray by a single object. Of course, as explained above this approach cannot handle low µ close to . Indeed, these lines of sight do not cross totally empty regions but low-density patches of matter so that for , as in Fig. 3, while the formulation (86) leads to a Dirac in .
© European Southern Observatory (ESO) 2000
Online publication: February 25, 2000