Deriving and fitting LogN-LogS distributions
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Deriving and fitting LogN-LogS distributions
Andreas Zezas
Harvard-Smithsonian
Center for Astrophysics
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• 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
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• Definition
or
LogN-LogS distributions
Kong et al, 2003
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A brief cosmology primer (I)Imagine a set of sources with the same luminosity within a sphere rmax
rmax
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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• 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
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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• Confusion
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Detection
• Problems• Background• Confusion • Point Spread Function• Limited sensitivity
CDF-N
Brandt etal, 2003
411 Ksec70 Ksec
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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 ?• 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
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Spatial distribution
• Separate point-like from extended sources
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• 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
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• No uncertainties - no incompleteness
fitted distribution :
Likelihood :
Slope :
Fitting methods (Crawford etal 1970)
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• 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.
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• 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)
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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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• 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
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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)
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• 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
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Non parametric comparisonsincluding incompleteness and
biasses
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The Luminosity functions : M82
• The XLF is fitted by a power-law (~-0.5) Possible break, due to background sources (~15 srcs)