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

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Some definitions D

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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

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Definition or LogN-LogS distributions Kong et al, 2003

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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

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

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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

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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

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Detection Problems Background

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Detection Problems Background Confusion Point Spread Function Limited sensitivity

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

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Detection Problems Background Confusion Point Spread Function Limited sensitivity

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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

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Statistical issues Incompleteness Background PSF Luminosity functions

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

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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)

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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)

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Udaltsova & Baines method Fitting methods

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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)

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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

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r max D Some definitions

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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

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

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

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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

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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

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