Mapping Cluster Mass Distributions via Gravitational Lensing of Background Galaxies

We present a new method for measuring the projected mass distributions of galaxy clusters. The gravitational amplification is measured by comparing the joint distribution in redshift and magnitude of galaxies behind the cluster with that of field galaxies. We show that the total amplification is directly related to the surface mass density in the weak field limit, and so it is possible to map the mass distribution of the cluster. The method is shown to be limited by discreteness noise and galaxy clustering behind the lens. Galaxy clustering sets a lower limit to the error along the redshift direction, but a clustering independent lensing signature may be obtained from the magnitude distribution at fixed redshift. Statistical techniques are developed for estimating the surface mass density of the cluster. We extend these methods to account for any obscuration by cluster halo dust, which may be mapped independently of the dark matter. We apply the method to a series of numerical simulations and show the feasibility of the approach. We consider approximate redshift information, and show how the mass estimates are degraded.Comment: ApJ in press. 23 pages of LaTeX plus figs. Text & figs available by anonymous ftp from resun03.roe.ac.uk in directory /pub/jap/lens (you need btp.tex and apj.sty

Transcendental numbers and the topology of three-loop bubbles

We present a proof that all transcendental numbers that are needed for the calculation of the master integrals for three-loop vacuum Feynman diagrams can be obtained by calculating diagrams with an even simpler topology, the topology of spectacles.Comment: 4 pages in REVTeX, 1 PostScript figure included, submitted to Phys. Rev. Let

Free Form Lensing Implications for the Collision of Dark Matter and Gas in the Frontier Fields Cluster MACSJ0416.1-2403

We present a free form mass reconstruction of the massive lensing cluster MACSJ0416.1-2403 using the latest Hubble Frontier Fields data. Our model independent method finds that the extended lensing pattern is generated by two elongated, closely projected clusters of similar mass. Our lens model identifies new lensed images with which we improve the accuracy of the dark matter distribution. We find that the bimodal mass distribution is nearly coincident with the bimodal X-ray emission, but with the two dark matter peaks lying closer together than the centroids of the X-ray emisison. We show this can be achieved if the collision has occurred close to the plane and such that the cores are deflected around each other. The projected mass profiles of both clusters are well constrained because of the many interior lensed images, leading to surprisingly flat mass profiles of both components in the region 15-100 kpc. We discuss the extent to which this may be generated by tidal forces in our dynamical model which are large during an encounter of this type as the cores "graze" each other. The relative velocity between the two cores is estimated to be about 1200 km/s and mostly along the line of sight so that our model is consistent with the relative redshift difference between the two cD galaxies (dz = 0.04).Comment: 22 pages, 18 figures, 2 table

A faint galaxy redshift survey to B=24

Using the multislit LDSS-2 spectrograph on the {\it William Herschel Telescope} we have completed a redshift survey in the magnitude range $22.5<B< 24$ which has produced 73 redshifts representing a 73\% complete sample uniformly-selected from four deep fields at high Galactic latitude. The survey extends out to $z>1$ and includes the highest redshift galaxy ($z=1.108$) yet discovered in a field sample. The median redshift, \zmed=0.46, and form of the redshift distribution constitute compelling evidence against simple luminosity evolution as an explanation of the large excess of faint galaxies ($\simeq\times$2--4 no-evolution) seen in this magnitude range. Rather we identify the excess population as blue objects with $z\sim 0.4$ and $B$\, luminosities similar to local $L^*$ galaxies indicating a dramatic decrease in the density of such objects over the last Hubble time, confirming the trends found in brighter redshift surveys. We also find a marked absence of {\it very} low redshift galaxies ($z<$0.1) at faint limits, severely constraining any significant steepening of the local field galaxy luminosity function at low luminosities.Comment: uuencoded compressed postscript. The preprint are also available at URL http://www.ast.cam.ac.uk/preprint/PrePrint.htm

The nature of power corrections in large β0\beta_0 approximation

We investigate the nature of power corrections and infrared renormalon singularities in large $\beta_0$ approximation. We argue that the power correction associated with a renormalon pole singularity should appear at O(1), in contrast to the renormalon ambiguity appearing at $O(1/\beta_0)$, and give an explanation why the leading order renormalon singularities are generically poles.Comment: 6 page