1. Deriving and fitting LogN-LogS distributions An Introduction
Andreas Zezas
University of Crete

2. Some definitions D

3. LogS -logS
Definition
Cummulative distribution of number of sources per unit intensity
Observed intensity (S) : LogN - LogS
Corrected for distance (L) : Luminosity function
CDF-N
Brandt etal, 2003
CDF-N LogN-LogS
Bauer etal 2006

4. LogN-LogS distributions
Kong et al, 2003
Definition or

5. Importance of LogN-LogS distributions
Provides overall picture of source populations
Compare with models for populations and their evolution
populations of black-holes and neutron stars in galaxies, populations of stars in star-custers, distribution of dark matter in the universe
Provides picture of their evolution in the Universe

6. How we do it Start with an image CDF-N
Alexander etal 2006; Bauer etal 2006

7. How we do it Start with an image Run a detection algorithm
CDF-N
Start with an image
Run a detection algorithm
Measure source intensity
Convert to flux/luminosity
(i.e. correct for detector sensitivity, source spectrum, source distance)
Alexander etal 2006; Bauer etal 2006

8. How we do it Start with an image Run a detection algorithm
CDF-N
Start with an image
Run a detection algorithm
Measure source intensity
Convert to flux/luminosity
(i.e. correct for detector sensitivity, source spectrum, source distance)
Make cumulative plot
Do the fit (somehow)
Alexander etal 2006; Bauer etal 2006

9. Detection
Problems
Background

10. Detection Problems Background Confusion Point Spread Function
Limited sensitivity

11. Detection Problems Background Confusion Point Spread Function
CDF-N
Brandt etal, 2003
Problems
Background
Confusion
Point Spread Function
Limited sensitivity

12. Detection Problems Background Confusion Point Spread Function
Limited sensitivity

13. Detection Statistical issues
Source significance : what is the probability that my source is a background fluctuation ?
Intensity uncertainty : what is the real intensity (and its uncertainty) of my source given the background and instrumental effects ?
Position uncertainty : what is the probability that my source is the same as another source detected 3 pixels away in a different exposure ?
what is the probability that my source is associated with sources seen in different bands (e.g. optical, radio) ?
Completeness (and other biases) : How many sources are missing from my set ?

14. Luminosity functions
Statistical issues
Incompleteness
Background
PSF

15. Luminosity functions Statistical issues Incompleteness Eddington bias
Background
PSF
Eddington bias
Other sources of uncertainty
Spectrum

16. Luminosity functions Statistical issues Incompleteness Eddington bias
Background
PSF
Eddington bias
Other sources of uncertainty
Spectrum e.g.
(Γ)
Fit LogN-LogS and perform non-parametric comparisons taking into account all sources of uncertainty

17. Fitting methods (Schmitt & Maccacaro 1986)
Poisson errors, Poisson source intensity - no incompleteness
Probability of detecting
source with m counts
Prob. of detecting N
Sources of m counts
Prob. of observing the
detected sources
Likelihood

18. Fitting methods
Udaltsova & Baines method

19. Fitting methods (extension SM 86)
Poisson errors, Poisson source intensity, incompleteness
(Zezas etal 1997)
Number of sources with
m observed counts
Likelihood for total sample
(treat each source as independent sample)
If we assume a source
dependent flux conversion
The above formulation can be written in terms of S and 

20. Fitting methods Or better combine Udaltsova & Baines with
BLoCKs or PySALC
Advantages:
Account for different types of sources
Fit directly events datacube
Self-consistent calculation of source flux and source count-rate
More accurate treatment of background
Account naturally for sensitivity variations
Combine data from different detectors (VERY complicated now)
Disantantage: Computationally intensive ?

21. Some definitions
rmax D

22. Importance of LogN-LogS distributions
Evolution of galaxy formation
Why is important ?
Provides overall picture of source populations
Compare with models for populations and their evolution
Applications :
populations of black-holes and neutron stars in galaxies, populations of stars in star-custers, distribution of dark matter in the universe
Luminosity
N(L)
Density evolution
Luminosity
N(L)
Luminosity evolution

23. A brief cosmology primer (I)
Imagine a set of sources with the same luminosity within a sphere rmax
rmax D

24. A brief cosmology primer (II)
If the sources have a distribution of luminosities
Euclidean universe
Non Euclidean universe

25. How we do it Start with an image Run a detection algorithm
CDF-N
Start with an image
Run a detection algorithm
Measure source intensity
Convert to flux/luminosity
(i.e. correct for detector sensitivity, source spectrum, source distance)
Make cumulative plot
Do the fit (somehow)
Alexander etal 2006; Bauer etal 2006

26. Luminosity functions Statistical issues Incompleteness Eddington bias
Background
PSF
Eddington bias
Other sources of uncertainty
Spectrum
Fit LogN-LogS and perform non-parametric comparisons taking into account all sources of uncertainty