Clustering (2) David Gilbank (SAAO) 1 Overview Background: Galaxies, Clusters… Clustering statistics Observables → mass: bias... 2 Clustering statistics Simplest measurement is the galaxy-galaxy spatial two point correlation function (2PCF/TPF), usually denoted (r) (r): Given a random galaxy(1), what is the excess probability (wrt a uniform random distribution) of finding another galaxy(2) a distance r away? dN(r) = 0 ( 1 + (r) )dV1dV2 Only depends on r = |r|. and V refer to the number density and volumes containing the galaxies. 3 2PCF (continued) Sometimes approximated by power law: (r) = ( r/r0 ) where r0 is known as the correlation length In the local Universe, r0 ~ 5 h-1Mpc and ~1.8 when measured over scales of ~(0.1--10) h-1Mpc. On larger scales, →0 rapidly r0 marks, v roughly, the transition between linear and non-linear regimes. Simplest estimator: (r)est = nDD/nRR - 1 with nDD count of data--data pairs, nRR count of random--randoms drawn from a random Poisson point process with same boundary and selection function as real data. More robust method is Landy-Szalay (1993) estimator: (r)est = (nDD - 2nDR +nRR) /nRR 4 The angular correlation fn It is observationally easier to measure the 2PCF in angular coordinates. Under the assumption that (r) follows a power law over some range of scales, it is possible to show that the angular correlation fn, w( ), follows: w( ) = A 1where has the same value as in the previous example, and the constant of proportionality A can be derived via projection. e.g. Marulli et al. 2013, A&A 557, A17 for a recent use 5 Other correlations Current and upcoming large surveys are now making it possible to measure the cluster--cluster correlation function reliably (e.g. Lu, Gilbank, Balogh & Bognat 2009, MNRAS 399, 4). It was the realisation of the different clustering scales of clusters of different masses which led Kaiser 1984 to propose the idea of a “... biased measure of large scale density correlations”. Higher-order correlations (than N=2) may be measured, but their calculation and interpretation are much more difficult, since they are functions of many inter-object distances. For N≥4, the order of connections must be defined: 6 The Power Spectrum Measuring the 2PCF directly can be avoided by instead measuring the Power Spectrum. The Power spectrum is the Fourier transform of the 2PCF (Wiener-Khintchine Theorem). Tegmark et al. 1998 ApJ, 499, 555 gives a very nice overview → Bruce 7 Galaxy bias (Concept suggested by Kaiser 1984) Returning to two previous ideas: ● the overdensity of matter, , smoothed over some scale ≝ ( / 0)-1 ● the spatial clustering of galaxies (~light) need not exactly mirror the underlying mass distribution, we can define a relation between the overdensities of mass, , and galaxies, b≝ / g where we call b the linear galaxy bias. In terms of the correlation functions: b≝( gal / dark matter ) 1/2 g , is usually measured from cosmological simulations and hence depends on cosmology. dark matter The most relevant cosmological parameter is 8 (standard deviation of counts in 8h-1Mpc spheres) 8 bias (continued) A galaxy population may be anti-biased (i.e. galaxies less clustered than DM) if b<1. For example, more luminous/massive galaxies are more strongly clustered. We can also measure the bias of one galaxy population relative to another (often done using ang. corr. fn), e.g. bpop1/pop2 = (wpop1/wpop2) Norberg et al. 2001 MNRAS, 328, 64 1/2 So are redder galaxies. c.f. cluster cores vs field. 9 Type-dependent clustering Zehavi et al. 2001 AJ, 571, 172 Real space Projected space (~angular correlation function) “Fingers of God” 10 Redshift-space distortions Redshift-space ( , ) from Peacock et al. 2001 Nature, 410, 169. “Kaiser flattening” , are coordinates in the transverse and line-of-sight directions, respectively. “Transverse separations are true measures of distance, but apparent radial separations are distorted by peculiar velocities” The deviations from circularity tell us about the large scale motions of “Fingers of God” galaxies. 11