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

Some definitions D

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

Definition or LogN-LogS distributions Kong et al, 2003

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 Importance of LogN-LogS distributions

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

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

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

Detection Problems Background

Detection Problems Background Confusion Point Spread Function Limited sensitivity

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

Detection Problems Background Confusion Point Spread Function Limited sensitivity

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 ? Detection

Statistical issues Incompleteness Background PSF Luminosity functions

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

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

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)

Udaltsova & Baines method Fitting methods

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)

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 ? Fitting methods

r max D Some definitions

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 Importance of LogN-LogS distributions Luminosity N(L) Density evolution Luminosity N(L) Luminosity Luminosity evolution

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

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

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

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