Deriving and fitting LogN-LogS distributions An Introduction

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

2. Some definitions D

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

4. Kong et al, 2003 LogN-LogS distributions • 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 CDF-N • Start with an image Alexander etal 2006; Bauer etal 2006

7. How we do it 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 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. CDF-N Brandt etal, 2003 Detection • 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 Background PSF • Eddington bias • Other sources of uncertainty Spectrum

16. Luminosity functions • Statistical issues • Incompleteness 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. Luminosity Luminosity Density evolution Luminosity evolution N(L) N(L) Luminosity Luminosity 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

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 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 Background PSF • Eddington bias • Other sources of uncertainty Spectrum Fit LogN-LogS and perform non-parametric comparisons taking into account all sources of uncertainty