Dept of Physics and Astronomy University of Glasgow, UK The best of both worlds? Towards combining design-based and parametric approaches to analysing galaxy surveys Martin Hendry, Dept of Physics and Astronomy University of Glasgow, UK SAMSI: Mar 2006

Previous session: introduction to Malmquist bias and how it affects estimating galaxy distances ‘Malmquist Correction’ methods (very) parametric Optimal approach depends on what we are using galaxy distances for (there may be other systematic biases besides Malmquist anyhow). Example: using galaxy distances to estimate peculiar velocity field, and constrain the mean density of dark matter on large scales. Peculiar velocity = motion of a galaxy over and above the Hubble expansion, due to the gravitational influence of its surroundings. i.e. galaxy peculiar velocities trace the local density field of all matter – not just the luminous matter. How are the galaxy and mass distribution related?… SAMSI: Mar 2006

b = linear bias parameter, related to peculiar velocities via Simplest model: linear biasing b = linear bias parameter, related to peculiar velocities via So galaxy distances  peculiar velocities  to constrain Underlying density field of all matter Density field of luminous matter, (smoothed version of galaxy distribution) Dimensionless mean matter density of the Universe SAMSI: Mar 2006

Problem: Through the late 90s – early 00s: dichotomy of inferred values of and . Could discrepancy be down to problems with correcting for Malmquist bias?… Could a ‘design-based’ approach (c.f. Efron & Petrosian) help? Rauzy & Hendry (2000) - ROBUST method for fitting peculiar velocity field models SAMSI: Mar 2006

Robust Method Assumption: luminosity function is Universal Spatial distribution Selection effects Luminosity function Null hypothesis (Rauzy 2001) Angular and radial Selection function Step function SAMSI: Mar 2006

Robust Method: Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distance modulus m . . . . . . . . . . . . . . . . . . . . Mlim(mi ) . . . . . . . . . . . . . . . (Mi, mi) . . . . . . . . . . . mlim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Absolute magnitude M SAMSI: Mar 2006

Robust Method: Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distance modulus m . . . . . . . . . . . . . . . . . . . . Mlim(mi ) . . . . . . . . . . . . . . . (Mi, mi) . . . . . . . . . . . mlim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S1 . S2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Absolute magnitude M SAMSI: Mar 2006

Robust Method: Completeness m* > mlim SAMSI: Mar 2006

Robust Method: Completeness Define:- where Can show:- P1: P2: uncorrelated SAMSI: Mar 2006

Robust Method: Completeness Also:- but only for SAMSI: Mar 2006

Robust Method: Completeness Also:- but only for SAMSI: Mar 2006

Robust Method: Velocity Field Model Assuming define Can show:- P3: uncorrelated Estimate b via SAMSI: Mar 2006

Robust Method Strength: Robust support for VELMOD analysis: validity of inhomogeneous Malmquist corrections Weakness: Completeness requirement may restricts sample size and depth From Rauzy & Hendry 2000 SAMSI: Mar 2006

If we have the wrong LF model, P1 and P2 are not satisfied Rauzy, Hendry & D’Mellow (2001) P1: P2: and are independent If we have the wrong LF model, P1 and P2 are not satisfied SAMSI: Mar 2006

Define e.g. Schechter model, For each use K-S statistic to test P1: use sample correlation coefficient to test P2: Define

Define e.g. Schechter model, For each use K-S statistic to test P1: use sample correlation coefficient to test P2: Define

If the LF model is a good descriptor of the true distribution, c.f ML approach: If the LF model is a good descriptor of the true distribution, 2[Lmax – L(a,M*)] ~ c2 SAMSI: Mar 2006

If the LF model is a good descriptor of the true distribution, c.f ML approach: If the LF model is a good descriptor of the true distribution, 2[Lmax – L(a,M*)] ~ c2 ‘Toy’ evolution model: Gaussian LF, SAMSI: Mar 2006

If the LF model is a good descriptor of the true distribution, c.f ML approach: If the LF model is a good descriptor of the true distribution, 2[Lmax – L(a,M*)] ~ c2 ‘Toy’ evolution model: Gaussian LF, SAMSI: Mar 2006 Model rejected

Questions / Issues How sensitive are robust methods to measurement errors? How to extend to ‘fuzzy’ selection / truncation? How to extend to multi-dimensional cases? (e.g. bivariate distribution of luminosity, surface brightness) How to extend the K-S test to >1d? Can ROBUST be useful as a diagnostic of systematic errors? (e.g. where separability assumption breaks down) (What’s wrong with 2dF?) SAMSI: Mar 2006

Volume-limited subset of data Millennium galaxy catalog of Driver et al. (2003) Volume-limited subset of data

SAMSI: Mar 2006