Deriving and fitting LogN-LogS distributions

Andreas Zezas

Harvard-Smithsonian

Center for Astrophysics

• 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

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

• 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

Luminosity

N(L

)

Density evolution

LuminosityN(L

)Luminosity

Luminosity evolution

• 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• Confusion

Detection

• Problems• Background• Confusion • Point Spread Function• Limited sensitivity

CDF-N

Brandt etal, 2003

411 Ksec70 Ksec

•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

Spatial distribution

• Separate point-like from extended sources

• Statistical issues• IncompletenessBackground

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

• No uncertainties - no incompleteness

fitted distribution :

Likelihood :

Slope :

Fitting methods (Crawford etal 1970)

• Gaussian intensity uncertainty - no incompleteness if S is true flux and F observed flux

Likelihood

where :

Fitting methods (Murdoch etal 1973)

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

• 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)

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)

• 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

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)

• 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

Non parametric comparisonsincluding incompleteness and

biasses

The Luminosity functions : M82

• The XLF is fitted by a power-law (~-0.5) Possible break, due to background sources (~15 srcs)