2. Magnification probability distribution
We wish to derive the magnification probability distribution function (pdf) in a Universe with both compact objects and smooth dark matter. We begin with the magnification pdf for smoothly distributed matter, , since this background distribution is present in all cases. For convenience we define the magnification, µ, to be zero at the empty beam value. We use the pdf 's computed from N-body simulations in JSW99, obtained by counting the number of pixels in a map that fall within a given magnification bin. We explicitly include the dependence of the pdf on the redshift of the source. The two main trends with increasing z are an increase in the rms magnification, and an increasing Gaussianity of the pdf (see Fig. 15 in JSW99). As we increase z we are superimposing more independent regions along the line of sight, and the resulting pdf approaches a Gaussian by the central limit theorem.
A possible source of concern is that the resolution limitations of the N-body simulations might corrupt the derived pdf 's in ways which crucially impact our results. In principle we would like to resolve all scales down to the scale of the SN emission region ( cm in linear size). Fortunately such high resolution is unnecessary, as there is very little power in the matter correlation function on such small scales. The contribution to the second moment of the magnification peaks at an angular scale of (see Fig. 8 in JSW99); scales smaller than this do not significantly change the value of the second moment. These smaller scales, however, are relevant for the high magnification tail of the distribution, and even the largest N-body simulations are resolution limited in the centers of halos. As high magnification events are very rare in the small SN samples being considered, limitations in the resolution of the high magnification tail of the distribution are not of great concern. The simulations are very robust around the peak of the magnification pdf, with lower resolution PM simulations giving results in good agreement with higher resolution simulations (Fig. 20 in JSW99). As these simulations converge for the 2nd moment we may conclude that, aside from the high-magnification tail, the pdf 's obtained from these simulations do not suffer from the limitations of finite numerical resolution. This is also confirmed by smoothing the map by a factor of two, and comparing the pdf 's before and after the smoothing. The resulting pdf 's are very similar for all models, indicating that small scale power has little effect on the region of most interest, near the peak of the pdf.
The pdf 's are shown in Fig. 1, plotted against deviations from the mean magnification, ( corresponds to the mean (filled beam) value). The mean magnification, , is given by the difference between the empty beam and the mean (filled) beam values, which for SN at is for , models such as standard CDM model (with ) or CDM model (with ), for CDM model with , and for open (OCDM) model with , .
In contrast, in a Universe filled with a uniform comoving density of compact objects the pdf depends on a single parameter, the mean magnification , or equivalently, the mean convergence . The two are related via ( if ). The pdf rises sharply from , and drops off as for high µ (Paczynski 1986). Based on Monte-Carlo simulations, Rauch (1991) gives a fitting formula for the pdf:
where and is chosen so that the pdf integrates to unity. Note that this expression is only valid for , and can only be used for SN with -2, depending on the cosmology.
To combine the two distributions we consider a model where a fraction of the matter is in compact objects, and where these compact objects trace the underlying matter distribution. Suppose a given line of sight has magnification µ in the absence of compact objects. In the presence of compact objects the mean magnification along this line of sight remains unchanged. Since the smooth component contributes a SN magnification of , the effect of compact objects is described with a pdf that gives a mean magnification of . The combined pdf, , is given by integrating over the whole distribution,
The middle panel in Fig. 1 shows the magnification pdf for a range of values of , for a cosmological model with , , and . The larger the value of , the closer the peak of the distribution is to the empty beam value. As increases from zero the pdf becomes wider, as the compact objects increase the large magnification tail. As increases beyond , however, the distribution begins to narrow, since more and more lines of sight are empty and thus closer to the empty beam value. Note in particular the similarity between the pdf and the CDM model with (upper panel in Fig. 1).
We need to further convolve these distributions with the measurement noise and the scatter in the intrinsic SN luminosities. Current estimates for these is 0.07 magnitudes for rms observational noise and 0.12 magnitudes for intrinsic scatter. The two combined give an additional rms scatter in magnification of 0.14 (Hamuy et al. 1996). To model this noise we convolve all the pdf 's with a Gaussian of width 0.14. The resulting pdf 's are shown in the bottom panel of Fig. 1. The distinction between the different values of , although small, is still apparent. In the next section we calculate how many SNe are required to distinguish between the different curves in the bottom panel, and thereby measure .
© European Southern Observatory (ESO) 1999
Online publication: November 2, 1999