## 3. Magnification by weak lensing## 3.1. DefinitionAs 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: with 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 Thus, for small values of we have: and we write the flux perturbation as: 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). ## 3.2. Probability distributionNext we wish to obtain the probability distribution of the
magnification From (21) and (24) we obtain the cumulant of order Since the correlation length (beyond which the many-body
correlation functions are negligible) is much smaller than the Hubble
scale (where
is the Hubble constant at redshift
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 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: we obtain (note that , Next, using the approximation (27) in the relation (26) we get: In particular, this yields for the variances and of the fluctuations of the amplification of a source located at redshift : Note that our result (32) for the where is the generating function of the density contrast defined in (2). Next, in a fashion similar to (4) since , we obtain for the probability distribution of : 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, with 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 This leads to the approximate which is still properly normalized and satisfies
for
. The practical advantage of the
approximation (45) is that to compute the 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
As can be seen from (39) or (45) the probability distributions
and
are non-gaussian. Indeed, they are
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 ## 3.3. Scaling functions andIn 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 : with: 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): and 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
© European Southern Observatory (ESO) 2000 Online publication: February 25, 2000 |