1. 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

2. 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

3. 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

4. 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

5. 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

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

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

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

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

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

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

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

13. 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

14. 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

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

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

17. 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

18. 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

19. 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

20. 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

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

22. SAMSI: Mar 2006