Proper Motions Of VLBI Lenses, Inertial Frames and The Evolution of Peculiar Velocities

Precise determinations of the image positions in quad gravitational lenses using VLBI can be used to measure the transverse velocity of the lens galaxy and the observer. The typical proper motions are $\mu$as yr$^{-1}$, so the time scale to measure the motion is ten years. By measuring the dipole of the proper motions in an ensemble of lenses we can set limits on the deviation of the inertial frame defined by the lenses from that defined by the CMB dipole and estimate the Hubble constant. The residual proper motions after subtracting the dipole probe the evolution of peculiar velocities with redshift and can be used to estimate the density parameter $\Omega_0$. For $N$ lenses, VLBI measurement accuracies of $\sigma_\theta$, and a baseline of $T$ years, we estimate that the 2$\sigma$ limit on the rms peculiar velocity of the lens galaxies is 3100 (\sigma_\theta/10\mu\{as})({yrs}/T)/N^{1/2} \kms, and that the time required for the 2--$\sigma$ limit to reach the level of the local rms peculiar velocity $v_{0,rms}$ is approximately 10 N^{-1/2} (v_{0,rms}/600\kms)(\sigma_\theta/10\mu as) years. For a ten year baseline and $N=10$ lenses we expect the 1$\sigma$ limit on the misalignment with the CMB dipole to be $\Delta \theta=20^{\circ}$ or equivalently to obtain an upper limit of $\Delta H_0 /H_0 < 0.34$.Comment: 23 pages, figures included uuencoded gzipped ps-file, submitted to the ApJ. One correction made from the original versio

Do Spirals and Ellipticals Trace the Same Velocity Field?

We test the hypothesis that the velocity field derived from Tully-Fisher measurements of spiral galaxies, and that derived independently from Dn-sigma measurements of ellipticals and S0s, are noisy versions of the same underlying velocity field. The radial velocity fields are derived using tensor Gaussian smoothing of radius 1200 km/s. They are compared at grid points near which the sampling by both types of galaxies is proper. This requirement defines a volume of ~(50 Mpc/h)^3, containing ~10 independent subvolumes, mostly limited by the available ellipticals. The two fields are compared using a correlation statistic, whose distribution is determined via Monte-Carlo simulations. We find that the data is consistent with the hypothesis, at the 10% level. We demonstrate that the failure to reject the correlation is not just a result of the errors being big, by using the same method to rule out complete independence between the fields at the 99.8% level. The zero points of the two distance indicators are matched by maximizing the correlation between the two velocity fields. There is a marginal hint that the ellipticals tend to stream slower than the spirals by ~8%. The correlation reinforced here is consistent with the common working hypotheses that (a) the derived large-scale velocity field is real, (b) it has a gravitational origin, and (c) the large-scale velocities of spirals and ellipticals are hardly biased relative to each other. On the other hand, it does not rule out any alternative to gravity where objects of all types obtain similar large-scale velocities.Comment: 16 pages, compressed and uuencoded PostScript 0.6Mbyte, (Also anonymous ftp venus.huji.ac.il pub/dekel/es/es.ps.Z of 0.43Mbyte

Gravitational lensing of type Ia supernovae by galaxy clusters

We propose a method to remove the mass sheet degeneracy that arises when the mass of galaxy clusters is inferred from gravitational shear. The method utilizes high-redshift standard candles that undergo weak lensing. Natural candidates for such standard candles are type Ia supernovae (SN Ia). When corrected with the light-curve shape (LCS), the peak magnitude of SN Ia provides a standard candle with an uncertainty in apparent magnitude of $\Delta m\simeq 0.1-0.2$. Gravitational magnification of a background SN Ia by an intervening cluster would cause a mismatch between the observed SN Ia peak magnitude compared to that expected from its LCS and redshift. The average detection rate for SN Ia with a significant mismatch of $\ge2\Delta m$ behind a cluster at $z\simeq0.05-0.15$ is about $1-2$ supernovae per cluster per year at $J,I,R\lesssim25-26$. Since SNe are point-like sources for a limited period, they can experience significant microlensing by MACHOs in the intracluster medium. Microlensing events caused by MACHOs of $\sim10^{-4} M_\odot$ are expected to have time scales similar to that of the SN light curve. Both the magnification curve by a MACHO and the light curve of a SN Ia have characteristic shapes that allow to separate them. Microlensing events due to MACHOs of smaller mass can unambiguously be identified in the SN light curve if the latter is continuously monitored. The average number of identifiable microlensing events per nearby cluster ($z\lesssim0.05$) per year is $\sim 0.02 (f/0.01)$, where $f$ is the fraction of the cluster mass in MACHOs of masses $10^{-7} < M_{macho}/M_\odot < 10^{-4}$.Comment: Accepted for publication in the MNRA

How to Plant a Merger Tree

We investigate several approaches for constructing Monte Carlo realizations of the merging history of virialized dark matter halos (``merger trees'') using the extended Press-Schechter formalism. We describe several unsuccessful methods in order to illustrate some of the difficult aspects of this problem. We develop a practical method that leads to the reconstruction of mean quantities such as the conditional and overall mass functions as given by the Press-Schechter model. This method is convenient, computationally efficient, and works for any power spectrum or background cosmology. In addition, we investigate statistics that describe the distribution of the number of progenitors and their masses as a function of redshift.Comment: 13 pages, LaTeX, 10 postscript figures. To appear in MNRAS. Changed to MNRAS format with inlined figures. Minor changes in text and figures to match published version. No significant changes in conten