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We derive an analytical formula for the maginification of an extended source with constant surface brightness lensed by a point mass. We find that the magnification for an extended source is significantly different from that for a point source when the impact parameter is comparable to the source size. In particular, the peak magnification for an extended source can be larger or smaller than the peak magnification for a point source. The results are relevant to the short-duration (approximately day) and high-magnification events in the search of massive compact halo objects in the Galactic halo by microlensing.

We point out the uncertainties in the potential models of the foreground galaxy in the gravitational lens Q2237 + 0305. The surface densities and external shears are uncertain by a factor of appproximately less than 2. Very long light curves are calculated for different models using the method developed by Witt (1993) and Lewis et al. (1993). We find that the probability of images being in a quiescent state depends strongly on the potential models and the direction of the source motion relative to the alignment of the (gravitational lens) shear term of the macroimages. For example, the probabilities of finding variations smaller than 0.1 mag and for a period of longer than 0.5 Einstein radii (approximately equals 3-5 yr) in image C and D are approximately 30%. This offers an explanation of why image C and D showed relatively small variations. The quiescent images are more likely to be demagnified, consistent with the observations. Further, the higher the magnification of the macroimages, the weaker are the microlensing events. In addition, we conclude that a simple Gaussian profile has some difficulties in explaining the asymmetric shape and the small amplitude of the observed high-magnification events if they are typical. Source models in which the emission is produced by a brightt compact core and a more extended halo can occur better explain the observed double peak events. More data are needed to clearly distinguish different galaxy potential models.

Kouveliotou et al. recently confirmed that gamma-ray bursts are bimodal in duration. In this paper we compute the statistical properties of the short (less than or = 2 s) and long (greater than 2 s) bursts using a method of analysis that makes no assumption regarding the location of the bursts, whether in the Galaxy or at a cosmological distance. We find the 64 ms channel on Burst and Transient Source Experiment (BATSE) to be more sensitive to short bursts and the 1024 ms channel to be more sensitive to long bursts. We show that all the currently available data are consistent with the simple hypothesis that both short and long bursts have the same spatial distribution and that within each population the sources are standard candles. The rate of short bursts per unit volume is about 40% of the rate of long bursts. Although the durations of short and long gamma-ray bursts span several orders of magnitude and the total energy of a typical short burst is smaller than that of a typical long burst by a factor of about 20, surprisingly the peak luminosities of the two kinds of bursts are equal to within a factor of about 2.

A Monte Carlo code is developed to calculate gravitational microlensing in three dimensions when the lensing optical depth is low or moderate (not greater than 0.25). The code calculates positions of microimages and time delays between the microimages. The majority of lensed gamma-ray bursts should show a simple double-burst structure, as predicted by a single point mass lens model. A small fraction should show complicated multiple events due to the collective effects of several point masses (black holes). Cosmological models with a significant fraction of mass density in massive compact objects can be tested by searching for microlensing events in the current BATSE data. Our catalog generated by 10,000 Monte Carlo models is accessible through the computer network. The catalog can be used to take realistic selection effects into account.

The probability distributions of time delay in gravitational lensing by point masses and isolated galaxies (modeled as singular isothermal spheres) are studied. For point lenses (all with the same mass) the probability distribution is broad, and with a peak at delta(t) of about 50 S; for singular isothermal spheres, the probability distribution is a rapidly decreasing function with increasing time delay, with a median delta(t) equals about 1/h month, and its behavior depends sensitively on the luminosity function of galaxies. The present simplified calculation is particularly relevant to the gamma-ray bursts if they are of cosmological origin. The frequency of 'recurrent' bursts due to gravitational lensing by galaxies is probably between 0.05 and 0.4 percent. Gravitational lensing can be used as a test of the cosmological origin of gamma-ray bursts.

Statistical properties of the distribution of gamma-ray bursts are calculated for a large range of different luminosity functions assumed for the sources. Surprisingly, the observable parameters are almost the same as in the 'standard-candle' scenario. In particular, no way is found to reconcile the observed distribution of gamma-ray bursts with any Galactic disk model. A Galactic halo distribution must have a core radius greater than 18 kpc to have no measurable dipole anisotropy. A cosmological distribution of the bursters is a more reasonable alternative.

Gravitational microlensing by a single star plus external shear is considered. It is shown that for a general cusp the magnification probability distribution follows pc(A)dA of about (A exp -7/2)dA for sufficiently large magnifications. An adaptive grid technique is developed to calculate the magnification probability distributions. The results could be useful for cases of microlensing where the surface-mass density is low.

Statistical properties of gamma-ray bursts are analyzed assuming that the bursters are cosmological. The model adopted is the simplest possible - it has no adjustable parameters. All the bursts are assumed to be standard candles with identical power-law spectra. Their burst rate is assumed to be constant per unit comoving volume per unit comoving cosmological time in a Friedmann universe with Omega of 1 and Lambda of 0. The calculated distribution of burst intensities is consistent with that found by the BATSE experiment for the weak events and by the PVO for the strong events. The range of observations is estimated to be about 1.5 z sub BATSE and about 0.2 z sub PVO.

Numerical simulations have uncovered a previously unrecognized 'bump' in the macroimage magnification probabilities produced by a planar distribution of point masses. The result could be relevant to cases of microlensing by star fields in single galaxies, for which this lensing geometry is an excellent approximation. The bump is produced by bright pairs of microimages formed by sources lying near the caustics of the lens. The numerically calculated probabilities for the magnifications in the range between 3 and 30 are significantly higher than those given by the asymptotic relation derived by Schneider. The bump present in the two-dimensional lenses appears not to exist in the magnification probability distribution produced by a fully three-dimensional lens.

A 2D algorithm is applied to observed sky maps and numerical simulations. It is found that when topology is studied on smoothing scales larger than the correlation length, the topology is approximately in agreement with the random phase formula for the 2D genus-threshold density relation, G2(nu) varies as nu(e) exp-nu-squared/2. Some samples show small 'meatball shifts' similar to those seen in corresponding 3D observational samples and similar to those produced by biasing in cold dark matter simulations. The observational results are thus consistent with the standard model in which the structure in the universe today has grown from small fluctuations caused by random quantum noise in the early universe.

The assumptions in the analysis of gravitational lensing statistics are examined. Special emphasis is given to the uncertainties in the theoretical predictions. It is shown that a simple redshift cutoff model, which may result from galaxy evolution, can significantly reduce the lensing probability and explain the large mean separation of images in observed gravitational lenses. This effect may affect the constraint on the contribution of the cosmological constant to producing a flat universe from the number counts of the observed lenses. For the Omega(0) = 1 (filled beam) model, the lensing probability of early-type galaxies with finite core radii is reduced roughly by a factor of 2 for high-redshift quasars as compared with the corresponding singular isothermal sphere model. The finite core radius effect is about 20 percent for a lambda-dominated flat universe. It is also shown that the most recent galaxy luminosity function gives lensing probabilities that are smaller than previously estimated roughly by a factor of 3.