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On the probability distributions of ellipticity
Viola, M.; Kitching, T. D.; Joachimi, B.
AA(Leiden Observatory, Leiden University, Niels Bohrweg 2, NL-2333 CA Leiden, the Netherlands ), AB(University College London, Mullard Space Science Laboratory, Holmbury St. Mary, Dorking, Surrey RH5 6NT, UK), AC(Scottish Universities Physics Alliance, Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK; Department of Physics & Astronomy, University College London, Gower Place, London WC1E 6BT, UK)
Monthly Notices of the Royal Astronomical Society, Volume 439, Issue 2, p.1909-1932 (MNRAS Homepage)
Publication Date:
Astronomy Keywords:
methods: analytical, cosmology: theory, dark matter
Abstract Copyright:
2014 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society
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In this paper we derive an exact full expression for the 2D probability distribution of the ellipticity of an object measured from data, only assuming Gaussian noise in pixel values. This is a generalization of the probability distribution for the ratio of single random variables, that is well known, to the multivariate case. This expression is derived within the context of the measurement of weak gravitational lensing from noisy galaxy images. We find that the third flattening, or ε-ellipticity, has a biased maximum likelihood but an unbiased mean; and that the third eccentricity, or normalized polarization chi, has both a biased maximum likelihood and a biased mean. The very fact that the bias in the ellipticity is itself a function of the ellipticity requires an accurate knowledge of the intrinsic ellipticity distribution of the galaxies in order to properly calibrate shear measurements. We use this expression to explore strategies for calibration of biases caused by measurement processes in weak gravitational lensing. We find that upcoming weak-lensing surveys like KiDS or DES require calibration fields of the order of several square degrees and 1.2 mag deeper than the wide survey in order to correct for the noise bias. Future surveys like Euclid will require calibration fields of order 40 square degree and several magnitude deeper than the wide survey. We also investigate the use of the Stokes parameters to estimate the shear as an alternative to the ellipticity. We find that they can provide unbiased shear estimates at the cost of a very large variance in the measurement. The PYTHON code used to compute the distributions presented in the paper and to perform the numerical calculations are available on request.
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