Deriving and fitting LogN-LogS distributions

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

2. 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

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

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

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

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

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) • Make cumulative plot • Do the fit (somehow) Alexander etal 2006; Bauer etal 2006

8. Detection • Problems • Background • Confusion

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

10. 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 ? • 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 ?

11. Spatial distribution • Separate point-like from extended sources

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

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

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

15. 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

16. 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 

17. Nondas’ method • Bayessian approach (Poisson errors, Poisson source intensity, incompleteness, and more…) • 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. Nondas’ method - Proposed extensions • 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)

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)