Transcript ppt - SLAC

Fluctuation Analysis:
Shadows of Invisible Sources
What’s happening here?
GLAST for Lunch 2004 Sept. 23
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The Basic Idea
There is background emission between the sources that GLAST will detect. Some of
it comes from gas in our galaxy. Some comes from outside our galaxy, and it seems to
be isotropic. Its intensity is ~ 10-5 photons/cm2 sec ster (E>100 MeV). It is composed
of, partially or in total, of contributions from point sources too faint to detect
individually. There’s a trick that allows us to learn something about the flux
distribution of these sources.
This is taken entirely from Tom Willis’s dissertation (1996).
Measure the “diffuse” intensity in pixels of solid angle  containing no
detectable point sources. If there are N faint sources per steradian,
the relative fluctuation in intensity measurements from pixel to pixel is
(N)-1/2.
It’s not really that simple if the sources are not the same brightness. The number of
sources per steradian with flux between S and S+dS is called dN/dS.
GLAST for Lunch 2004 Sept. 23
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The Hairy Math Part
1 K
For a given pixel, the total intensity X   Si
 i 1
Where K is the number of sources in the pixel. The joint probability distribution of
K and the Si is
F
(K )
(4) K 4 K 1 dN
( S1 ,, S K ) 
e
( Si )

K!
i 1 4 dS
Using this to get the probability distribution of X is a convolution problem, best
done with Fourier transforms. The characteristic function
 ( )  e
2 i S


  dN 2i S


 exp   dS
(e
 1) 
dS
0


1  dN
 dS
4  dS
0
Finally the distribution of pixel fluxes is
1
P( X ) 
2

 2 i X
e
 ( )d


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Diffuse + Point Sources
One question of considerable interest is whether there is a truly diffuse
extragalactic background. That will show up as a reduction in the
fluctuations, as if there was a huge density of very weak sources.
Here’s a simulated result. The three curves represent P(X) for populations
with a mixture of point sources and truly diffuse emission: 10%, 50%, and
100% sources.
10%
50%
90%
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How Reality Makes Life Harder
• There is always a non-negligible contribution from the Galactic diffuse flux, no
matter how far away from the Galactic plane.
• Each measurement has a noise contribution from Poisson photon-counting
statistics.
• It is necessary to choose a pixel size. Large pixels have less Poisson noise, but
they can contain so many sources that the fluctuation signal is washed out. The
optimum choice depends on the signal you expect to see.
• Some theoretical dN/dS choices violate the Olbers’ Paradox constraint: The
average “diffuse” intensity is
S max
 S dN dS

 dS
0
which must not diverge. Smax is the threshold for detecting
individual sources. If dN/dS doesn’t cooperate, it’s necessary
to put in an infrared cutoff at Smin. In fact, the intensity
must be set equal to the observed value.
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EGRET Results (Power Law)
Tom Willis applied this method to EGRET data with three different choices of
dN/dS.
The simplest choice is a truncated power law with Smin = 10-10 photons/cm2 sec. The
results aren’t sensitive to Smin if it’s this small. Information from fluctuations is
almost orthogonal to the information obtained from the detected sources.
In the decade of flux below the EGRET threshold, there are 100-300 point
sources, contributing 4-15% of the isotropic background.
Constraints
from detected
sources
Constraints
from
fluctuations
Combined
constraints
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EGRET Results (Other Models)
The other choices of dN/dS have some physics in them.
Salamon & Stecker (1994) proposed that the gamma ray luminosity of a blazar could
be obtained from the radio luminosity by a simple scaling law with two adjustable
parameters:
Lr = 10 L
(Luminosity scaling)
= (/r)2
(Jet opening angle  abundance)
The radio dN/dS is a broken power law with slopes of about –1.96 and –0.83.
Changing the parameters  and  modifies the position of the knee and the
normalization.
Chiang et al. (1995) considered the luminosity function, dN/dL, of blazars, assuming
a simple form of luminosity evolution. With a cosmological model this can be
translated into dN/dS. They assumed a broken power law shape for dN/dL. From
the detected blazars they obtained only a rough constraint for the low-L part of
the function (slope = -2.9  2.0). Willis’s fluctuation analysis tightens this to –1.0 >
slope > -1.9.
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EGRET Results (Summary)
Here’s a summary of the EGRET results. It shows the number of point sources
with flux > S at high galactic latitude as a function of S. With GLAST’s
detection threshold of about 10-9, it should find something like 1000-10,000
sources. Caveat: This is probably believable only down to about 10-8.
Observed
blazars
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