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Publication# An Unbiased Method of Measuring the Ratio of Two Data Sets

Abstract

In certain cases of astronomical data analysis, the meaningful physical quantity to extract is the ratio R between two data sets. Examples include the lensing ratio, the interloper rate in spectroscopic redshift samples, and the decay rate of gravitational potential and E ( G ) to test gravity. However, simply taking the ratio of the two data sets is biased, since it renders (even statistical) errors in the denominator into systematic errors in R. Furthermore, it is not optimal in minimizing statistical errors of R. Based on Bayesian analysis and the usual assumption of Gaussian error in the data, we derive an analytical expression of the posterior probability density function P(R). This result enables fast and unbiased R measurement, with minimal statistical errors. Furthermore, it relies on no underlying model other than the proportionality relation between the two data sets. Even more generally, it applies to cases where the proportionality relation holds for the underlying physics/statistics instead of the two data sets directly. It also applies to the case of multiple ratios (R & RARR; R = (R (1), R (2), MIDLINE HORIZONTAL ELLIPSIS )). We take the lensing ratio as an example to demonstrate our method. We take lenses as DESI imaging survey galaxies, and sources as DECaLS cosmic shear and Planck cosmic microwave background (CMB) lensing. We restrict the analysis to the ratio between CMB lensing and cosmic shear. The resulting P(R) values, for multiple lens-shear pairs, are all nearly Gaussian. The signal-to-noise ratio of measured R ranges from 4.9 to 8.4. We perform several tests to verify the robustness of the above result.

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While the presence of any mass bends the path of light passing near it, this effect rarely produces the giant arcs and multiple images associated with strong gravitational lensing. Most lines of sight in the universe are thoroughly in the weak lensing regime, in which the deflection is impossible to detect in a single background source. However, even in these cases, the presence of the foreground mass can be detected, by way of a systematic alignment of background sources around the lensing mass.

Gravitational lens

A gravitational lens is a distribution of matter (such as a cluster of galaxies) or a point particle between a distant light source and an observer that is capable of bending the light from the source as the light travels toward the observer. This effect is known as gravitational lensing, and the amount of bending is one of the predictions of Albert Einstein's general theory of relativity. Treating light as corpuscles travelling at the speed of light, Newtonian physics also predicts the bending of light, but only half of that predicted by general relativity.

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