# Points to Three Points

Post by Tom Kitching, MSSL

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In a recent paper, led by collaborator Dr Liping Fu the CFHTLenS survey was used to measure the “3-point correlation function” from weak lensing data. This is one of the first times this has been measured, and certainly one of the clearest detections.

A “2-point” statistic, in cosmology jargon, is one that uses two data points in some way. Usually an average over many pairs of objects (galaxies or stars) are used to extract information. In this case what is being measured is called the “two-point [weak lensing] correlation function” and what it measures is the excess probability that any pair of galaxies (separated by a particular angular distance) are aligned. This is slightly different to a similar statistic used in galaxy cluster analysis. The two-point correlation function is related to the Fourier transform of matter power spectrum and can be used to measure cosmological parameters, which is why we are interested in it. In a sense the two-point correlation function is like a scale-dependent measure of the variance of the gravitational lensing in the data: the mean orientation of galaxies is assumed to be zero (when averaged over a large enough number) because there is no preferred direction in the Universe, but the variance is non-zero. The measurement of the 2-point statistic is represented above, “sticks” (of various [angular] lengths) are virtually analysed on the data and the for each stick-length the ellipticity (or “ovalness”) of the galaxies along the direction of the sticks is measured. If the two galaxies are aligned then the multiplication of these ellipticities (e * e) will be positive, but if not then sometimes it will be positive and sometimes negative. Ellipticity can be expressed as two numbers e1 and e2 that lie on a Cartesian graph where positive and negative values represent different alignments. When multiplied together aligned ellipticities therefore produce a positive number and anti-aligned produce a negative number. When averaged over all galaxies a purely random field with no preferred alignment (equal positive and negative) the multiplication averages to zero. If there is alignment the average is positive.

Galaxies will align is there is some common material that is causing the gravitational lensing to be coherent. So when averaged over many galaxies the multiplication of the ellipticities <e*e> (the angular brackets represent taking an average) for a particular stick length tells us whether there is lensing material with a scale the same as the sticks length: a positive result means there is alignment on average, a zero result means there  is no alignment on average, a negative result would mean there is anti-alignment on average. Figure from Kitching et al. (Annals of Applied Statistics 2011, Vol. 5, No. 3, 2231-2263). Gravitational lensing from the large scale structure in the universe makes preferred aligned occur around clusters of dark matter of around voids, what we call “E-mode”. Anti-alignment is not normally caused by gravitational lensing, what we call “B-mode”.

In this new paper we not only measured the two-point correlation function but also the 3-point correlation function! This is an extension of the idea to now measure the excess probability that any 3 galaxies have preferred alignment. Now instead of a single angle and pairs of galaxies the measurement uses triangle configurations of galaxies and results in a measurement that depends on two angles. This is a much more demanding computational task, because there are many more possible ways that triangle can be drawn than a stick (for every given length of stick the other two sides of the triangle can take many different lengths). The amplitude of the 3-point correlation function tells us if there is any coherent structure on multiple-scales, and in particular allows us to test whether the simple description of large-scale structure using only the 2-point correlation function – and the matter power spectrum – is sufficient or not.

This is one of the first measurements of this statistic and paves the way for extracting much more information from lensing data sets than could be done using 2-point statistics alone.