As a photon travels from a distant source towards the observer its trajectory is deflected by density fluctuations along the light path. This produces an apparent displacement of the source as well as a distortion of the image. In particular, the convergence (defined as the trace of the shear matrix) will magnify (or demagnify) the source as the cross section of the beam is decreased (or increased). One can show (Bernardeau et al.1997; Kaiser 1998) that the convergence is given by:
where is the radial comoving coordinate (and corresponds to the redshift of the source):
The relation (15) assumes that the metric perturbations are small () but the density fluctuations can be large (Kaiser 1992). The magnification µ is linked to the convergence and the intensity of the shear by:
Of course, because of flux conservation the mean shift of the flux over all lines of sight is zero: . We can see directly in (21) that there is a minimum value for the magnification of a source located at redshift :
This corresponds to an "empty" beam between the source and the observer ( everywhere along the line of sight).
Next we wish to obtain the probability distribution of the magnification µ from (21). To this order we simply need to derive the cumulants . This will provide the parameters , similar to (3), and the generating function , similar to (2). However, it is convenient to introduce first a "reduced magnification" and its variation by:
From (21) and (24) we obtain the cumulant of order p of :
From (13) we see that the integral over the points along the line of sight is dominated by the comoving scale such that the local slope of the two-point correlation function is . This corresponds to a local linear power-spectrum index , see (14). Indeed, for realistic power-spectra like CDM the slope of decreases from 0 at small scales down to -4 at large scales. On the other hand, power-spectra which would be pure power-laws would lead to divergences. Since for the power-spectra we shall use corresponds to galactic scales ( kpc) within the highly non-linear regime (), we must indeed use the non-linear many-body correlation functions . Thus we can use the scaling laws (13). Of course, at large redshifts the scale will enter the linear regime, however since we shall restrict ourselves to smaller z we can always use (13). Now we must estimate the contribution of the integral over the points along the line of sight. Although the points are drawn on a line and not within a sphere we shall use the approximation (compare with (1)):
where the coefficients are obtained for (i.e. ) from the numerical results described in Valageas et al.(1999). Thus, we define the quantity by:
It is convenient to express in terms of the non-linear power-spectrum which is directly provided by the fits obtained in Peacock & Dodds (1996). Thus, using the Fourier transform:
where we defined the power-spectrum by:
Note that our result (32) for the rms fluctuation does not use the approximation (27). Moreover, the expressions (32) clearly show that most of the contribution to weak gravitational lensing effects comes from the scale where is maximum. This corresponds to or where is the slope of the non-linear power-spectrum . From (14) we see that it also corresponds to where n is the slope of the linear power-spectrum. Then, from (31) we can define the generating function , similar to (2), by:
The expansion of in verifies:
which implies, as it must, that:
The relations (34) and (35) provide the probability distribution of the magnification of a source located at redshift . Note that the generating function depends on . We can also check in (35), using (34), that for as it should (since for we can push the integration path in (35) towards the right, Re, where the exponential vanishes). Thus, the approximation (27) has preserved the fact that for . In a fashion similar to (5) we can define the function:
From (39) and (9) we can obtain the parameters which govern the asymptotic behaviour of :
where is the point where is maximum. Thus, we see that the slope of at low x is the same as while the exponential cutoff is slightly modified (). From (23) and (32) we see that so we are led to the approximation:
which is still properly normalized and satisfies for . The practical advantage of the approximation (45) is that to compute the p.d.f. we can directly use the functions and obtained in numerical simulations. Note that formally can be expressed in terms of the probability distribution of the density contrast at the scale such that as:
using (4). Note however that even for the probability distribution used in (46) corresponds to the highly non-linear regime. Thus in the case the probability distribution in (46) is not the one measured at the time such that . Nevertheless, the expressions (45) and (46) clearly show that the measure of the p.d.f. , or of , provides a direct estimate of the p.d.f. of the density contrast through (see also the study of in Valageas 1999b).
As can be seen from (39) or (45) the probability distributions and are non-gaussian. Indeed, they are strictly zero for and , they show an exponential cutoff (or ) for large (or large µ), and they have their maximum at a value smaller than the mean .
We compare in details the predictions of our approach with available numerical results from N-body simulations (Jain et al.1999) for the convergence smoothed on small angular scales () in Valageas (1999b). This comparison shows that our approach (which can be extended to finite smoothing windows in a straightforward fashion) provides very good results for all three cosmologies we consider here (e.g., see Fig. 4 and Fig. 5 in Valageas 1999b). Moreover, we show in that paper that the approximation (44) gives reasonable results which are quite close to the more accurate expression (34). Indeed, we shall check below that the correction to the third moment (for instance) due to (45) is quite small and well within the errorbars of the value of obtained by counts-in-cells calculations from numerical simulations. Note that the evaluation of the probability distribution of the magnification µ by ray tracing through N-body simulations would of course suffer from the same uncertainty, which affects the moments of the density distribution itself (because of numerical inaccuracy).
In order to perform numerical calculations, we need to choose the function , or equivalently the parameters defined in (1). As we explained above, most of the contributions to the weak lensing effects come from the scales where the local slope of the linear power-spectrum is , see also Fig. 2 in Valageas (1999b). Hence we use in the following the scaling function obtained from numerical simulations by Valageas et al.(1999) for the case of a critical universe with an initial linear power-spectrum which is a power-law :
In fact, the curvature of the CDM power-spectrum may slightly change the parameters from the value they would have for a pure power-law . However, in order to improve meaningfully this approximation one would need to measure the parameters realized on a line rather than in a sphere, see (27). Thus, we think our approach is the best analytical tool one can currently build. The scaling function shown in (47) defines the generating function through (12). In particular, one obtains (see Gradshteyn & Ryzhik 1965, Sect. 9.211, p.1058):
where is Kummer's function which can be expressed in terms of the difference between two confluent hypergeometric functions . Next, to obtain from (45) one simply needs to perform an integration in the complex plane. In order to make the integral (45) converge sufficiently fast, it is convenient to define the integration path by the constraint . However, in practice it is sufficient to use (we replace by a power-law with the right behaviour for and the right location of the singularity at ) which gives for the integration path:
Note that it is better to define from rather than trying to use a fit for itself. Indeed, from (1) and (2) we see that:
and moreover, using Schwarz' inequality and the scalar product , one can see that the coefficients must obey:
These constraints are automatically verified if one defines from . If one uses a fit for which does not obey these constraints one may get negative probabilities (since in this case has to be negative in some range).
The fit (47) for the function was obtained for a critical universe. In the case of a low density universe, we use the same function although there are no numerical results available (from counts-in-cells statistics) to validate (or invalidate) this choice. However, we note that the fact the dependence on cosmology of the two-point correlation function is accurately given by the simple term described in Peacock & Dodds (1996) suggests that the structure of the non-linear clustering pattern is the same for low as for a critical universe, once the effect of the slow-down of the linear growth factor is taken into account. Indeed, most highly non-linear structures formed when the universe was close to critical () since at later times the slow-down of the linear growth factor prevents additional new structures to form. Of course, this break in the hierarchy of scales which successively turn non-linear may also lead with time to some difference with the case of a critical universe (at least for the scales which were the last to collapse). Detailed numerical studies are needed to investigate more precisely this point. However, for reasonable cosmologies our model provides a good approximation as shown by a direct comparison of with results from N-body simulations, as described in Valageas (1999b).
Note that, except for this possible dependence of the parameters , all our results are valid for any realistic power-spectrum such that on small scales, on large scales and the scale where is non-linear. In particular, note that for such power-spectra all moments of the convergence and of the magnification µ converge: there is no need to introduce a cutoff at small scales.
© European Southern Observatory (ESO) 2000
Online publication: February 25, 2000