E ect of Gravitational Lensing on Measurements of the Sunyaev - Zel ' dovich E ect

The Sunyaev{Zel'dovich (SZ) e ect of a cluster of galaxies is usually measured after background radio sources are removed from the cluster eld. Gravitational lensing by the cluster potential leads to a systematic de cit in the residual intensity of unresolved sources behind the cluster core relative to a control eld far from the cluster center. As a result, the measured decrement in the Rayleigh{Jeans temperature of the cosmic microwave background is overestimated. We calculate the associated systematic bias which is inevitably introduced into measurements of the Hubble constant using the SZ e ect. For the cluster A2218, we nd that observations at 15 GHz with a beam radius of 0 0:4 and a source removal threshold of 100 Jy underestimate the Hubble constant by 6{10%. If the pro le of the gas pressure declines more steeply with radius than that of the dark matter density, then the ratio of lensing to SZ decrements increases towards the outer part of the cluster. Subject headings: cosmic microwave background { di use radiation { galaxies: clusters: general, individual (A2218) { gravitational lensing