Transcript Stats talk - Harvard University

Deriving and fitting
LogN-LogS distributions
Andreas Zezas
Harvard-Smithsonian
Center for Astrophysics
LogS -logS
• Definition
Cummulative distribution of number of sources
per unit intensity
Observed intensity (S) : LogN - LogS
Corrected for distance (L) : Luminosity function
CDF-N
CDF-N LogN-LogS
Brandt etal, 2003
Bauer etal 2006
LogN-LogS distributions
• Definition
or
Kong et al, 2003
A brief cosmology primer (I)
Imagine a set of sources with the
same luminosity within a sphere rmax
A brief cosmology primer (II)
If the sources have a distribution
of luminosities
Euclidean universe
Non Euclidean universe
A brief cosmology primer (III)
• Evolution of galaxy formation
N(L)
N(L)
• Why is important
? evolution
Density
Luminosity evolution
• Provides overall picture of source populations
• Compare with models for populations and their
evolution Luminosity
Luminosity
•Applications :
populations
of black-holes and neutron stars
in
Luminosity
Luminosity
galaxies, populations of stars in star-custers,
distribution of dark matter in the universe
How we do it
CDF-N
• 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)
Alexander etal 2006; Bauer etal 2006
Detection
• Problems
• Background
• Confusion
Detection
CDF-N
• Problems
• Background
• Confusion
• Point Spread Function
• Limited sensitivity
70 Ksec
411 Ksec
Brandt etal, 2003
Detection
•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 ?
Spatial distribution
• Separate point-like from extended sources
Luminosity functions
• Statistical issues
• Incompleteness
Background
PSF
Confusion
• Eddington bias
• Other sources of
uncertainty
Fornax-A
cum=1.3
Spectrum
Distance
Classification
Fit LogN-LogS and perform non-parametric
comparisons taking into account all sources of
uncertainty
Kim & Fabbiano, 2003
Fitting methods (Crawford etal 1970)
• No uncertainties - no incompleteness
fitted distribution :
Likelihood :
Slope :
Fitting methods (Murdoch etal 1973)
• Gaussian intensity uncertainty - no incompleteness
if S is true flux and F observed flux
Likelihood
where :
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Fitting methods (Schmitt & Maccacaro 1986)
• 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 (extension SM 86)
• 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)
If we assume a source
dependent flux conversion
The above formulation can be written in terms of S and 
Nondas’ method
• Bayessian approach (Poisson errors, Poisson source
intensity, incompleteness, and more…)
• 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 breakpoint)
• Test code
•Speed-up code (currently VERY slow)
Nondas’ method - Proposed extensions
• 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)
Non parametric comparisons
including 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)