Statistical biases in extragalactic distance determinations. G. Paturel, Observatoire de Lyon In collaboration with P. Teerikorpi IHP Avril 2005

The difficult problem of Ho

When does a bias appear ? 1) When the absolute magnitude is expressed as a function of an observable parameter with a scatter Examples : Tully-Fisher Relation Period-Luminosity Relation 2) When the sample has a limiting apparent magnitude m lim

Let us explain in a simple case The « sosie » method (e.g.for TF relation) We select galaxies with the same logV M Through they should constitute « Standard Candles » with the same absolute magnitude. Are standard candles free of bias ? No !

Graphical explanation of two kinds of bias (1) Classical Malmquist bias

Graphical explanation of two kinds of bias (2) Incompleteness bias (at a constant distance modulus)

How Cepheid Period Luminosity relation looks like ? The Cepheid PL relation fill the conditions to have a bias at a constant distance

Does the bias affect the Cepheid Period-Luminosity relation ? The bias should exist but it can be small due to the small scatter of the PL relation. How to test the existence of a possible bias in the PL relation ?

A simple simulation shows that a bias can exist Is it possible to use redshift as a relative distance indicator ? YES

The Hubble law: V= H.r The original discovery: V < 1200 km/s

Small dispersion around the regular Hubble flow  In 1957 de Vaucouleurs noted that deviations from Hubble law are small (  <100 km/s)  In 1972 Sandage and coworkers (ApJ 172, 253) found still maller value (  <60 km/s)  In 1999 Ekholm et al. confirm that the Hubble law works at small scale  In 2001 Ekholm et al. and Karachentsev et al., independently found a still smaller dispersion (  <40 km/s) of the very local expansion

Comparison with models

We have a tool to have accurate, relative distances

Hubble diagram with Cepheid distances

Use of the 2-parameter bias model by Teerikorpi (1975) to check a bias diagram

A bias diagram When the absolute limiting Magnitude allows us to see faint Cepheids the bias is Negligible. The fit of the bias model Leads to : logH=1.76 ; H=56 (km/s)/Mpc

Influence of the correlation of errors The fit of the bias model Leads to : logH=1.80 ; H=63 (km/s)/Mpc The reality could be H=60 (km/s)/Mpc

Comparison of corrected and uncorrected distances, using HST anf ground-based Cepheid distances Ground-based HST

How the Hubble diagram changes

What about long range distances

The final question by A.Blanchard : Are the SN- standard candle affected in a similar way The possibility exists because the reference sample is not similar to the distant one (mixing several luminosities) Another effect could be due to evolution effect