## 1. IntroductionCurrent cosmological data indicates that the density of matter in the Universe is around (in units of critical density). Of this, big bang nuclesynthesis requires 15-20% be in the form of baryons, with the rest in some other more exotic form. Even among the baryons only has been accounted for directly. The rest could be in warm gas (Cen & Ostriker 1999), or in compact objects such as brown dwarfs, white dwarfs, or neutron stars. The nonbaryonic dark matter could be composed of a smooth microscopic component, such as axions, or alternatively it may be composed of compact objects, such as primordial black holes (Crawford and Schramm 1982). Although compact objects have been searched for in our own halo using microlensing studies (see Sutherland 1999 for a review), results are as of yet inconclusive. While the most straightforward interpretation is that a large fraction of the halo (up to 100%) is composed of compact objects of roughly , other scenarios using existing stellar populations remain viable. The bottom line is that, at present, the form of the bulk of the dark matter remains unknown. In recent years there have been tremendous advances in our ability to observe and characterize type Ia supernovae. In addition to dramatically increasing the number of SNe observed at high redshift, the intrinsic peak brightness of these supernovae is now thought to be known to within about 15%. These supernovae are thus excellent standard candles, and by measuring the spread in their observed brightnesses it may be possible to determine the nature of the lensing, and thereby infer the distribution of the lensing matter. It has long been recognized that the lensing of SNe can be used to search for the presence of compact objects in the Universe (Linder, Schneider & Wagoner 1988; Rauch 1991). Since the amount of matter near or in the beam determines the amount of magnification of the image, the magnification distribution from many SNe can probe for the presence of compact objects in the Universe. If the Universe consists of compact objects, then on very small scales most of the light beams do not intersect any matter along the line of sight, resulting in a dimming of the image with respect to the filled-beam (standard Robertson-Walker) result. On occasion a beam comes very near a compact object, resulting in a tremendous brightening of the ensuing image. In such a Universe the magnification distribution will be sharply peaked at the empty beam value, and will possess a long tail towards large magnifications. The lensing is sensitive to objects with Einstein rings larger than the linear extent of the SNe (roughly cm at their maximum, which gives a lower limit on the mass of the lenses of ). Lensing due to compact objects is not, however, the only way to modify the flux of a SN. Even smooth microscopic matter, such as the lightest SUSY particle or axions, are expected to clump on large scales. The effect on the magnification distribution depends on the clumpiness of the Universe. If the clumping of matter in the Universe is very nonlinear then all of the matter will reside in dense halos, and the filaments connecting them, and there will be large empty voids extending tens or even hundreds of megaparsecs in diameter. There will thus be a large probability that a given line of sight will be completely devoid of matter, and so will give a large demagnification (as compared to the pure Robertson-Walker result). A simple way to estimate the importance of this effect is to compare the rms scatter in magnification to the demagnification of an empty beam relative to the mean (given by the filled beam value). Both can be calculated analytically, the former as an integral over the nonlinear power spectrum, and the latter as an integral over the combination of distances. The result depends on the redshift of interest, but for most realistic models the rms is smaller than the empty beam value at . This means that at least qualitatively it should be possible to distinguish between the compact objects and smoothly distributed matter at such redshifts. In this letter we extend previous work in several aspects. First,
we provide the formalism to investigate models with a combination of
both compact objects and smooth dark matter. Although Rauch (1991) and
Metcalf and Silk (1999) have explored the use of lensing of supernovae
to detect compact objects, they concern themselves with distinguishing
between the two extreme cases: either all or none of the matter in
compact objects. Our formalism allows us to address the more general
question of how well the fraction of dark matter in compact objects
can be measured with any given SN survey. Both baryonic and dark
matter are dynamically significant and could contribute to the lensing
signal. For example, we can imagine four simplistic scenarios, in
which the baryonic and dark matter are each in one of two states:
smoothly distributed or clumped into compact objects (with masses
above 10 Second, we use realistic cosmological N-body simulations (Jain,
Seljak and White 1999) to provide the distribution of magnification
for the smooth microscopic component. Previous works (Metcalf and Silk
1999; Holz and Wald 1998) make the simplifying assumption of an
uncorrelated distribution of halos to determine this distribution. As
shown in Jain et al. (1999; JSW99), there are differences in the
probability distribution function (pdf) of magnification for models
with different shapes of the power spectrum and/or different values of
cosmological parameters. For example, open models exhibit large voids
up to , and their pdf 's extend almost
to the empty beam limit. On the other hand, flat
models have less power (being
normalized to the same cluster abundance), and are more linear at
higher © European Southern Observatory (ESO) 1999 Online publication: November 2, 1999 |