Results obtained in the previous section indicate that high-z SNe can enable an accurate determination of the fraction of matter in compact objects. If the estimates of the errors used here are reasonable, around 100 SNe at are required to make a 20% determination of the fraction of dark matter in compact objects. Although such numbers of SNe are not currently available, they can be expected in the near future as the high-redshift supernovae surveys continue their observations. 1 Our results are in broad agreement with the results of Metcalf and Silk (1999), who find that about 50-100 SN are required to distinguish from with confidence. An important limitation will be systematic effects caused by uncertainties in the magnification distributions. For the smooth component these can be obtained using high resolution N-body simulations. We have argued that our results are reliable in the regime of application and can in any case be verified with higher resolution simulations in the future. Lack of knowledge of the underlying cosmological model may also introduce systematic effects, but again this can be expected to be better determined in the future.
Another important systematic error is our ignorance of the intrinsic dispersion in brightness of the observed SNe. Our simple assumption of a Gaussian noise profile in the measurement of the peak luminosities of SN Ia's is certain to break down at some level. This noise estimate is based upon phenomenological observations of SNe, and needs to be borne out both by further observations (especially at low redshift, where lensing effects are negligible and independent calibrations are available) and by theoretical models. Deviations from Gaussian noise are most likely to be important in the tail of the magnification pdf 's, which can be excluded in the analysis. The lack of a precise knowledge of the intrinsic peak values of the SNe is less likely to cause a shift in the peak of the magnification pdf 's, which is the main signature of the lensing effect.
A further source of concern is redshift evolution of the intrinsic properties of the SNe. It is possible that both the mean and the higher moments of the distribution of peak luminosities of type Ia SNe varies with redshift, and this could pose significant challenges to an accurate measurement of lensing effects. Improvements in the determination of such effects will occur as the size of the data sets at both low and high redshifts are increased, and direct comparisons of observations are available. An important consistency check will be to demonstrate that the shift of the peak of the distribution as a function of redshift is consistent with theoretical expectations (under the assumed value of ).
A complimentary method to obtain the pdf due to the background smooth matter component is to use weak lensing observations. This would provide the pdf directly from data, avoiding entirely the need for cosmological simulations. As discussed in Sect. 2 most of the power is on scales larger than 3´, so the pdf will almost converge to the correct one even if one smoothes the beam at this scale. Weak lensing surveys can provide maps of the magnifications by reconstructing the projected mass density from the shear extracted using galaxy ellipticities. The distribution of magnifications gives a pdf convolved with the random noise from galaxy ellipticities. Given an rms ellipticity of 0.4, we find that about 50 galaxies per square arcminute are required to give an rms noise comparable to the noise in the SNe. This number is similar to the density of galaxies at deep () exposures. Such galaxies have a mean redshift () comparable to the SNe under discussion. We can therefore choose the size of a patch such that its rms noise agrees with the noise in the SN data. Such a pdf can then be directly compared to that from a SN sample (provided the differences in the redshift distribution are not too large), and any differences between the weak lensing and SN pdf 's would indicate the presence of compact objects or some other source of small scale fluctuation in magnification. Note that one cannot test for compact objects by simply using cross-correlation of the two magnifications, as compact objects do not change the mean magnification in a given line of sight, and thus the cross-correlation coefficient remains unchanged. The increased scatter, however, could provide the desired signature.
In conclusion, the magnification distribution from several hundred high redshift type Ia SNe has the statistical power to make a 10-20% determination of the fraction of the dark matter in compact objects. It remains to be seen whether this statistical power can be exploited at its maximum, or whether systematic effects will prove to be too daunting.
© European Southern Observatory (ESO) 1999
Online publication: November 2, 1999