1. Deriving and fitting LogN-LogS distributions Andreas Zezas Harvard-Smithsonian Center for Astrophysics

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

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

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

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

6. 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 A brief cosmology primer (III) Luminosity N(L) Density evolution Luminosity N(L) Luminosity Luminosity evolution

7. 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) How we do it CDF-N Alexander etal 2006; Bauer etal 2006

8. Detection Problems Background Confusion

9. Detection Problems Background Confusion Point Spread Function Limited sensitivity CDF-N Brandt etal, 2003 411 Ksec 70 Ksec

10. 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 ? Extent : is my source extended ? 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 ? Detection

11. Spatial distribution Separate point-like from extended sources

12. Statistical issues Incompleteness Background PSF Confusion Eddington bias Other sources of uncertainty Spectrum Distance Classification Luminosity functions Kim & Fabbiano, 2003 Fornax-A  cum =1.3 Fit LogN-LogS and perform non-parametric comparisons taking into account all sources of uncertainty

13. No uncertainties - no incompleteness fitted distribution : Likelihood : Slope : Fitting methods (Crawford etal 1970)

14. Gaussian intensity uncertainty - no incompleteness if S is true flux and F observed flux Likelihood where : Fitting methods (Murdoch etal 1973)

15. 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 Fitting methods (Schmitt & Maccacaro 1986)

16. If we assume a source dependent flux conversion The above formulation can be written in terms of S and  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) Fitting methods (extension SM 86)

17. Bayessian approach (Poisson errors, Poisson source intensity, incompleteness, and more…) Nondas’ method Model source and background counts as Poi(S), Poi(B) Number of sources follows Poi(  ), where  has a Gamma prior Estimate number of missing sources | observed sources, L, E Sample flux of observed and missing sources (rejection sampling given (E, L) which accounts for Eddington bias) Obtain parameters of the model

18. Nondas’ method Status Working single power-law model (need test runs) Broken power-law with fixed break-point implemented Immediate goals Complete implementation of broken power-law (fit break- point) Test code Speed-up code (currently VERY slow)

19. Spectral uncertainties Fit sources with different spectral shapes include spectral uncertainties for each source Model comparisons single power-law vs. broken power-law power-law with exp. cutoff vs. broken power-law Extend to luminosity functions Distance uncertainties Malmquist bias (for flux-limited sample the luminosity limit is a function of distance) Nondas’ method - Proposed extensions

20. Non parametric comparisons including incompleteness and biasses

21. The Luminosity functions : M82 The XLF is fitted by a power-law (  ~-0.5) Possible break, due to background sources (~15 srcs)