deriving and fitting logn-logs distributions an introduction andreas zezas university of crete
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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