Deriving and fitting LogN-LogS distributions

An Introduction

Andreas Zezas

University of Crete

Some definitions

D

€

Source flux : S ν 0( ) =L ν1( )

4πD2

• DefinitionCummulative 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• IncompletenessBackground

PSF

Luminosity functions

• Statistical issues• IncompletenessBackground

PSF

• Eddington bias • Other sources of uncertainty

Spectrum

Luminosity functions

• Statistical issues• IncompletenessBackground

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 €

S E( ) = E −Γ +1 exp −NHσ (E)( )(Γ)

• Poisson errors, Poisson source intensity - no incompleteness

Probability of detecting source with m counts

Prob. of detecting NSources of m counts

Prob. of observing thedetected 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

rmax

D

Some definitions

€

Source flux : S ν 0( ) =L ν1( )

4πD2

• 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

Luminosity

N(L

)

Density evolution

LuminosityN(L

)Luminosity

Luminosity evolution

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

rmax

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• IncompletenessBackground

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