## 4. Dependence on cosmology and redshiftUsing 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.
## 4.1. Fluctuations of the magnificationFirst, we present in Fig. 1 the redshift evolution of the
amplitude of the fluctuations of the magnification and: 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 .
## 4.2. Probability distributionsWe 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
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
which implies:
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: and 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
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
## 4.3. SkewnessFinally, 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
From the change of variable (24) we also obtain: This clearly shows the dependence on cosmology of the coefficients . Moreover, from (31) we can also derive the parameters without the approximation (44). This leads to: 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
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 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 and 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 ## 4.4. Influence of cosmological parametersWe show in Fig. 6 the dependence on
and
of the fluctuations of the
magnification 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 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 |