Using the results obtained in the previous sections we can compute the probability distribution of the magnification for various cosmologies. We shall mainly consider three cases: a CDM critical density universe (SCDM), a low-density open universe (OCDM) and a low-density flat universe with a non-zero cosmological constant (CDM). The main cosmological parameters of these scenarios are described in Table 1. Here is the usual shape parameter of the power-spectrum and we use the fit given by Bardeen et al.(1986) for . These cosmologies are those we use in Valageas (1999b) to compare our predictions with available results from N-body simulations. The reader is refered to that paper for a detailed discussion of the accuracy of our approach and for an extension of our model to finite smoothing windows.
Table 1. Cosmological models
First, we present in Fig. 1 the redshift evolution of the amplitude of the fluctuations of the magnification µ of distant sources located at . The increase with of the interval between the mean and the minimum value is due to the more extended line of sight which gives more room for the weak lensing effects. This deviation is smaller for the low-density universes than for the critical case because of the factor which enters (22): the difference of matter between the mean and 0 is smaller as it is proportional to at low z. It is larger for the flat cosmology (with the same ) because of the detailed dependence on of the factor (see also Bernardeau et al. 1997): indeed the distances and are larger at fixed z which gives more room for weak lensing effects at a given . On the other hand, the variance of the "reduced magnification" decreases at larger redshift because the integration in (21) over the successive "slabs" of matter leads to be "closer" to a gaussian, in a fashion similar to the central limit theorem (although the latter does not apply here since the probability distribution of the magnification due to each slab evolves with z and ). It diverges for where the number of (highly non-linear) density fluctuations which intersect the line of sight declines. Note however that for large µ the probability distribution is always very different from a gaussian, whatever the value of and , since it shows a simple exponential cutoff rather than a gaussian cutoff. As a consequence, at low the variance of the magnification µ becomes of the order of, and even larger than, (thus is strongly non-gaussian and sharply peaked close to the minimum ) while at large redshift becomes significantly smaller than (thus the peak of gets closer to the mean ). In particular, we can check from (22) and (32) that:
We show in Valageas (1999b) that our results agree with the values obtained by Jain et al.(1999) using ray tracing through N-body simulations, for smoothing angles . In fact, since our prediction for the variance only relies on the weak lensing approximation and on the fits for the non-linear power-spectrum given by Peacock & Dodds (1996) this agreement mainly shows that both sets of simulations are consistent. As seen in Fig. 1 in Valageas (1999b), the variance obtained without smoothing (), which we consider here, is slightly larger than the value reached at (since a finite smoothing removes the power from small scales). For instance, for the SCDM case we get for (as in Fig. 1 here) and for .
We display in Fig. 2 the probability distribution of the "reduced magnification" , from (45). As explained in Sect. 3 it is strongly non-gaussian with a maximum below the mean and a power-law tail followed by a simple exponential cutoff at large . Its is more strongly peaked around its maximum which is closer to the mean for the CDM model, and even more for the SCDM scenario, following the decrease of shown in Fig. 1 (but of course this depends on the cosmological parameters one chooses). Note that even for very small variance of the magnification (, see Fig. 1 the probability distribution of the magnification displays clear deviations from gaussianity, as explained above. In particular, the normalized magnification clearly shows the shape of the probability distribution, which appears squeezed towards the mean when displayed as a function of µ.
This is shown in Fig. 3 where one can check that at low redshift tends to a Dirac . Due to the smaller value of and for low density universes is much more sharply peaked around its maximum than for a critical cosmology. One can clearly see the asymmetry due to the lower cutoff at and the extended large µ tail, as well as the shift of the maximum of below the mean . This is similar to the behaviour obtained from numerical simulations (e.g. Wambsganss et al.1997). We display below in Fig. 7 the curve which shows even more clearly the non-gaussian features of . For the critical density universe we also show in Fig. 3 the gaussians (dotted lines) which correspond to the same variance . We can see that the probability distribution is indeed very different from a gaussian, at both redshifts. Thus, it would be quite meaningless to model as a gaussian. For the sake of completeness, we note here how one can obtain the approximate behaviour of the locations and of the peak of the probability distributions and . From (45) and (9) one can show that for large variance the location of the maximum of is given by (Balian & Schaeffer 1989; Valageas & Schaeffer 1997):
Thus, for large the peak of the probability distribution of the magnification gets very close to the minima and . For smaller values of the variance one may write an Edgeworth expansion of the probability distribution (e.g. Bernardeau & Kofman 1995):
where is the Hermite polynomial of order 3, and . This gives for the location of the peak of the probability distributions:
Thus, for both large and small redshift, that is for small and large , the peak tends to the mean 1 (but for different reasons). Hence the deviation is maximum for an intermediate redshift of order unity.
From the probability distribution one can obtain intervals of confidence for the magnification µ. Thus, for we define as the minimal interval (i.e. with the smallest length) such that with probability . These intervals contain the location of the peak of and obey:
They are displayed in Fig. 4 for the redshifts and : any horizontal line of ordinate intersects the curves at the points and . For the length of the interval goes to 0 as nd tend to . For we have and . Thus, Fig. 4 clearly shows the range of µ one can expect, as well as the asymmetry of the underlying probability distribution. In particular, the large µ tail of is clearly seen. Moreover, one can check that there is a non-negligible probability to have on a line-of-sight ().
Finally, in Fig. 5 we display the third moment of the probability distribution of the magnification. From (1) the parameter (skewness) is given by:
since . It measures the third moment of the probability distribution of the density contrast realized in spherical cells of radius R. As explained in Sect. 2 it is constant with time in the non-linear regime at the scale such that the local slope of the linear power-spectrum is . From the results obtained by Valageas et al.(1999) from numerical simulations we have . From (45), also measures the third order moment of :
Finally, we also define the parameter by:
which may be seen as a more convenient measure of the deviation of from a gaussian. We can check in Fig. 5 that the error introduced by the approximation (44) is quite small. In particular, it is negligible as compared to the inaccuracy due to the results of numerical simulations of structure formation. Indeed, while Valageas et al.(1999) get , Colombi et al.(1997) obtain and Munshi et al.(1999) find . Thus, the approximation (44) is quite sufficient in view of the accuracy of the scaling function obtained from numerical results. However, one should note that the functions measured in simulations are reasonably close in the range where they have been tested against numerical data. Indeed, the uncertainty which affects the parameters (and increasingly so for large p) comes from the large density tail of the probability distribution of the density contrast while the behaviour of around its maximum (i.e. at ) is fairly well constrained, see Valageas et al.(1999) for a detailed discussion. Note that the measure of from ray-tracing in N-body simulations would of course bear the same inaccuracy, which is due to the dispersion of the properties of the non-linear density field itself obtained from different numerical simulations.
However, we can note that at we get , 131 and 95 for the critical density, open and low-density flat universes, while Jain et al.(1999) obtain , 107 and 75 (we have ). For the critical density universe both values agree quite well, while for the low-density universes our values are somewhat larger than those obtained by these authors (although they show the same trends). This could be due to the slow rise of the skewness with smaller smoothing angle (due to the fact that is larger for non-linear scales than for quasi-linear scales, see Colombi et al.1997). Indeed, as seen in Jain et al.(1999) the skewness may not have reached its asymptotic value at yet. Moreover, as discussed in Valageas (1999b) the errorbars on the measure of from numerical simulations may be larger than the dispersion of the estimator used to compute the skewness may suggests because two p.d.f. with a rather different skewness can still agree very well, as shown by the good agreement of our prediction for with the results from these N-body simulations. In particular, although this comparison may suggest a small dependence of on the dispersion of numerical results (which provide values for which can vary by a factor 1.8 as seen above) prevents us from drawing definite conclusions. Of course, it would be interesting to measure both and in the same numerical simulation to directly check the accuracy of our relation (64). On the other hand, the main improvement to our calculation would be to directly measure (or obtain from first principles) the parameters defined by the integrals on the line-of-sight of the many-body correlation functions, i.e. the l.h.s. terms in (27), rather than the ratios of the averages defined in (1). Then one could still apply our method and simply use the new function (or ) defined by these new parameters as in (2).
As explained above, we can check in Fig. 5 that at larger redshift the probability distribution of the magnification becomes "closer" to a gaussian as the parameters and decrease. However, it is interesting to not that these parameters can be fairly large (higher than ) and even diverge for . Thus, at low z the probability distribution of the magnification is strongly non-gaussian. In particular, we obtain:
Thus, if the intrinsic magnitude dispersion of the sources is sufficiently small, one might observe these non-gaussian features and check that they agree with the usual models of the density field. Moreover, from (64) we see that one could get an estimate of the properties of the underlying density field itself (e.g. its skewness ) from . However, as seen in Fig. 3 the width of the probability distribution is quite small at low z which would make such a study rather difficult, so that intermediate redshifts may provide better results.
We show in Fig. 6 the dependence on and of the fluctuations of the magnification µ of a source located at redshift and . We vary the normalization of the power-spectrum with as which roughly accounts for the change of needed in order to reproduce as well as possible large-scale structure observations (abundance of clusters, velocity fields) with different cosmologies (of course, some cosmologies match such observations better than others, whatever their choice of ). Note that the shape of also depends on through . We can see in Fig. 6 that increases for larger . This is due to the factor which appears in (22). It translates the fact that for higher there is more matter in the universe hence there is more room for deviations from the mean , for instance in the case where everywhere along the line of sight ("empty beam") which leads to . In particular, we have:
for both flat and open universes, since for an empty universe all beams are empty. The variances and increase for because of the rise of the two-point correlation function which compensates the slower growth of the linear growth factor, see Peacock & Dodds (1996). The amplitude of the fluctuations of the density contrast, hence of the magnification µ, is smaller for a flat universe than for the open case with the same because of the detailed form of this linear growth factor. Overall, the variation of the quantities , and is rather small over the usual range of cosmological parameters . Note that the moments of the probability distribution also depend on the cosmological parameters and (mainly on ), as can be seen from (63). In particular, we have:
so that the variation of can be directly seen in Fig. 6 from the evolution of .
© European Southern Observatory (ESO) 2000
Online publication: February 25, 2000