Statistical biases in extragalactic distance determinations.

G. Paturel, Observatoire de Lyon

In collaboration with P. TeerikorpiIHP 28-29 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 mlim

bVaM M log

bPaM log

Let us explain in a simple case

The « sosie » method (e.g.for TF relation)

We select galaxies with the same logVM

Through

they should constitute « Standard Candles » with the same absolute magnitude.

Are standard candles free of bias ? No !

bVaM M log

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 seefaint Cepheids the bias is Negligible.

The fit of the bias modelLeads to :

logH=1.76 ; H=56 (km/s)/Mpc

Influence of the correlation of errors

The fit of the bias modelLeads 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