A Comparison of
Energy Spectra in
Different Parts of the Sky
Carl Pfendner, Segev BenZvi, Stefan
Westerhoff
University of Wisconsin - Madison
13 December 2011
The Ohio State University
0
Outline
Motivation
New Statistical Method
Tests of the Method
Application to Pierre Auger Data
Conclusions and Future
13 December 2011
The Ohio State University
1
GZK Suppression
•
•
Cosmic rays interact with the 2.7 K microwave background.
Protons above ~ 51019 eV suffer severe energy loss from photopion production.
•
Proton (or neutron) emerges with reduced energy, and further interaction occurs until the
energy is below the cutoff energy.
Greisen-Zatsepin-Kuz’min (GZK) Suppression: Greisen, K., (1966). PRL 16 (17); Zatsepin, G. T.;
Kuz'min, V. A. (1966). Journal of Experimental and Theoretical Physics Letters 4
•
2mN mp + mp
E th =
» 5 ×1019 eV
4e
2
•
This energy loss means that particles observed above this cutoff energy are likely to come
from sources that are relatively close by because they would travel through less of the CMB --> GZK horizon
13 December 2011
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2
GZK Suppression
•
Low flux at high energy limited
the ability to observe this cutoff.
•
The predicted “end to the
cosmic ray spectrum” was
recently observed by the High
Resolution Fly’s Eye (HiRes)
detector operated between 1997
and 2006 in Utah.
•
HiRes has ~ 5  evidence for
suppression in the spectrum.
•
Confirmed with Auger data.
25% syst. error
25% syst. error
25% syst. error
HiRes Collaboration,
PRL 100 (2008) 101101
December 15, 2008
3
Particle Propagation (Toy Model)
proton + cmb   + nucleon
No matter the initial energy, the final
energy drops to EGZK after ~100
Mpc = GZK “Horizon”
Diameter of Milky Way ~20 kpc
Galaxy cluster diameter ~ 2-10 Mpc
CenA ~ 3 Mpc
Virgo ~ 14-18 Mpc
13 December 2011
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4
Current Studies
• Most popular anisotropy method is 2-point
correlation function
• Difficulties with a 2-point correlation analysis
– Dependent on magnetic deflection, angular resolution
of detector
– Low statistics at highest energies limits the analysis
• Most energy spectrum based methods
– Are model-dependent
– Cover the whole sky
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5
Region A
dN/dLog(E/eV)
Split Sky Analysis
Spectrum from
region A, A
Spectrum from
region B, B
Log(E/eV)
Region B
•
•
•
Question posed: are spectra different in different parts of the sky?
Hypothesis test: spectra from two different regions of the sky derive from the
same “parent” spectrum (H1) or two distinct “parent” spectra (H2)
Example: the region within 20° of a single point in the sky and outside that area.
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Spectrum Comparison
• Can’t use χ2 method
– For low events statistics,
doesn’t follow χ2
distribution
• Must use different method Bayes factor
– Derivation and some tests
of this method described
in ApJ paper: BenZvi et al,
ApJ, 738:82
n 2 n1i - n1n 2i )
1
(
c =
å
n1n2 i=1
n1i + n2i
m
2
2
• Model independent – no power law required
• Automatically penalizes overly complex models
• Naturally account for uncertainties in the data (including systematics if
desired)
13 December 2011
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7
Outline
Motivation
New Statistical Method
Tests of the Method
Application to Pierre Auger Data
Conclusions and Future
13 December 2011
The Ohio State University
8
Bayesian Comparison
P(DA,DB | H 2 )
H = two-“parent” hypothesis
B=
H = one-“parent” hypothesis
P(DA,DB | H1 )
2
1
• Bayes factor is the probability ratio that the data supports
hypothesis 2 over hypothesis 1
• Advantage: Can get posterior probability from the Bayes
factor
B21
• Assume P(H1)=P(H2), to get:
P ( H 2 | D) =
1+ B21
• High B21  support for H2, Low B21  support for H1
• Example: If B21=100, P(H2|D) ≈ .99 - support for H2
• If B21=0.01, P(H2|D) ≈ .0.01 - support for H1
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Bayesian Comparison
P(DA,DB | H 2 )
B=
H2 = two-“parent”
P(DA,DB | H1 )
hypothesis
B
A
B
 = total number of hypothesized events for both regions of
the sky - marginalized thus not model dependent
w = DA exposure / (DA exposure + DB exposure) -- the relative
weight of set A
w’ marginalized - In the numerator, every possible relative
weight, w’, is permitted since the experiments could be
observing two different fluxes. Allows any spectrum.
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dN/dLog(E)*w
A
dN/dLog(E)*w’
=
òò P( D ,D w¢,h)q( w¢,h) dw¢dh
ò P ( D ,D w,h)q( w,h) dh
H1 = one-“parent”
hypothesis
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• Method 2 - Compares shape only
– Relative weight (w) in single parent case is
marginalized but as a standard term over all bins –
no longer a constant factor
• Result:
13 December 2011
N A + N B + 1)! N æ DA,i!DB,i! ö
(
B=
ç
÷
( N A + 1)!( N B + 1)! Õ
i=1 è ( DA,i + DB,i + 1)!ø
The Ohio State University
dN/dLog(E)
*w
dN/dLog(E)
*w
dN/dLog(E)
*w
dN/dLog(
E)*w
dN/dLog(
E)*w
• Assume: flat prior distribution, binned spectrum,
Poisson statistics
• Method 1 - Is sensitive to absolute flux differences
• Requires knowledge of the relative exposure of the two
regions
N
-1
• Result:
D
+
D
+
1
Õ( A,i B,i )
i=1
B= N
DB,i ( DA,i + DB,i )!
DA,i
w
1w
Õ i ( i ) D !D !
A,i
B,i
i=1
dN/dLog(
E)*w
Methods 1 & 2
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Outline
Motivation
New Statistical Method
Tests of the Method
Application to Pierre Auger Data
Conclusions and Future
13 December 2011
The Ohio State University
12
Power Law Spectrum Tests
Use the published fit parameters
as a model to test the
effectiveness of the two methods
Try to recreate expected
scenarios and see how the
methods respond
ICRC 2011 proceedings
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Single power law comparison
Power law comparison:
Compare large numbers of
data sets with power law
functions of different
indices over 18.4-20.4
energy range with 68%
confidence bands
Blue – 1000 events
Violet – 3000 events
Red – 10000 events
As events increase, the
differentiation power
increases dramatically
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Broken Power Law Test
•
•
•
•
•
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Generated 20000event sets using a
broken-power-law
Kept power law index
set at a constant 2.7
Varied the first powerlaw index, break
energy
Sensitivity is the width
of the blue region very sensitive
Many events in lower
energy bins
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Broken Power Law Test
•
•
•
•
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Same idea as
previous
Varied the second
power-law index,
break energy
Sensitivity drops
quickly as the
break energy
increases
Lower energy
bins can
dominate
calculation change lower
energy threshold
to better test
data
16
Single vs Broken Power Law
Comparison of single and
broken power laws with fitted
parameters
10000 events in each set
Extended the lower energy
power law index to higher
energies and compared with
the fully broken power law
spectrum
Increase the minimum
energy to scan the data
Peak around 19.5 for
method 1 and around 19.1
for method 2
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Single vs Broken Power Law (cont.)
Posterior probability of the
single-parent hypothesis
(same shape) vs lower
energy threshold
10000 events in brokenand single- power law
functions
Blue = Chi-squared
Red = Bayes factor
Chi-squared produces a
tail probability which
biases against the null
hypothesis and regularly
underestimates the
posterior probability
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More Single vs Broken Power Law
But the regions we’re
interested in are not the
same size as the rest
of the sky! Make the
relative exposure 0.05
Bayes factor vs
threshold energy for
single power law vs
broken power law with
0.05/0.95 exposure
14519 events total
Events as of March 31,
2009
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More Single vs Broken Power Law
Single Power law
vs broken power
law with 0.05/0.95
exposure
14519 X 2 events
total
Double events of
March 31, 2009 –
approximately
current number of
events
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More Single vs Broken Power Law
Single Power law
vs broken power
law with 0.05/0.95
exposure
14519 X 3 events
total
Triple events of
March 31, 2009
Even with a
decreased
exposure, can
differentiate single
and broken power
law functions
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Contamination
What happens when
the contributing signal is
mixed between broken
and single power law?
Peak Bayes factor vs.
contamination fraction
14519 events total with
5% in the “region of
interest” with some
fraction of those events
actually deriving from a
broken power law
function
13 December 2011
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Contamination (cont.)
Peak Bayes factor vs.
contamination fraction
14519 X 2 events total
Horizontal line shows
Bayes factor = 100
Point at which the
Bayes factor reaches
100 indicated by vertical
line
68% confidence bands
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Contamination (cont.)
14519 X 3 events
As events increase,
better and better
discrimination even
with contamination
With this number of
events, method 1
could differentiate a
50% contaminated
signal
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Change Bin Size?
Bin size changes produce
changes in the Bayes factor
One might think that
decreasing bin size would
increase information thus
increasing discriminatory
power but generally the
reverse is true.
N
B=
Õ( DA,i + DB,i + 1)
i=1
N
Õ wi
i=1
13 December 2011
DA,i
(1- w i )
DB,i
-1
( DA,i + DB,i )!
It does not matter to method
1 what the bins represent
but merely that they are
comparable values. Test
diminishes in power with
less and less bin content
DA,i!DB,i!
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Weight Dependence
• Split a data set of
40000
• Varied the weight
value (w) in the
calculation - error
in the calculated
exposure
• There is a limiting
range in which the
weight can vary
and still produce
the correct results.
• Error on exposure
is well within these
limits < 10%
13 December 2011
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Outline
Motivation
New Statistical Method
Tests of the Method
Application to Pierre Auger Data
Conclusions and Future
13 December 2011
The Ohio State University
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Detection Techniques
Surface Detector (SD)
Fluorescence Detector (FD)
3 PMTs per tank measure
Cherenkov light from charged
shower particles entering the tank
Array of PMTs observes the UV
light from the air showers
fluorescing the nitrogen in the
atmosphere
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Pierre Auger Observatory
Hybrid Detector
•
Auger combines a surface detector
array (SD) and fluorescence
detectors (FD).
•
1600 surface detector stations with
1500 m distance.
•
4 fluorescence sites overlooking the
surface detector array from the
periphery.
•
•
3000 km2 area.
Largest ground array
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Energy Measurements
Fluorescence Detector
• Measure light intensity along the track
and integrate.
• Nearly calorimetric, model- and massindependent.
• 10% duty cycle, atmosphere needs to be
monitored.
Surface Detector Array
• Particle density S at fixed distance to the
shower core is related to shower energy
via simulations.
• Choice of distance depends on array
geometry (Auger: signal @ 1000 m)
• Model- and mass-dependent, but
available for all showers.
S(1000)
Distance to shower core [m]
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Direction Reconstruction
Timing gives arrival direction
Spherical shower front arrives at
different tanks at different times
Fluorescence detector observes the
shower development itself, improves
reconstruction even more
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Application to Data
• Using Observer data through 28 Feb 2011
• Factors to consider in examining data
– Position in sky
• Scanned entire Auger skymap in ~1° steps
– Size of region used in comparison
• Circular regions of 5°-30° around each point in sky
– Lower energy threshold – low energy events dominate statistics
• From paper, for a non-GZK-attenuated spectrum, the signal is
highest at lower threshold energy of 19.6 for method 1 and
19.4 for method 2
• Changed threshold in steps of 0.1 from 18.4 to 19.8 in
Log(E/eV)
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Maximal Points
•
Method 1 : B21 = 16 at (b = 21.4°, l = -57.7°)
•
•
Angular bin size = 23°, threshold Log(E/eV) = 19.8
Method 2 : B21 = 20 at (b = 61.0°, l = -90.0°)
•
Angular bin size = 28°, threshold Log(E/eV) = 19.8
•
Conservative estimate of trial factor:
–
–
–
–
49000 bins for position
26 bins for search region size
15 bins for energy threshold
B21 = 16  ~1e-6, B21 = 20  ~1e-6
•
–
Still well below a significant signal
However, the values are highly correlated
•
Must run a trial test – Pchance, isotropy = 0.99
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Skymap - Angular Bin Size Change
Method 1, 18.4 in Log(E/eV), 5°-30°
binning
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Skymap changes (cont.)
Method 2, 18.4 in Log(E/eV), 5°-30°
binning
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Skymap changes (cont.)
Method 1, 19.8 in Log(E/eV), 5°-30°
binning
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Skymap changes (cont.)
Method 2, 19.8 in Log(E/eV), 5°-30°
binning
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Skymap Changes (cont.)
Method 1, 18.4 – 20.4, 23 degrees
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• Method 1, 19.8 in Log(E/eV), 23° binning
• Notice “hot spot” in the vicinity of (b = 21.4°, l = -57.7°)
• Not far from Cen A (b = 19.4, l = 50.5)
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Spectrum around Cen A
Outside events
weighted by
relative
exposure w =
0.0602
• Events within 23 degrees of maximal point for Method 1
• More higher energy events this point esp. above 19.6
• Consistent with less attenuation from nearby source (e.g. Cen A)
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• For 23 degrees around maximal point for method 1
• Blue = Method 1, Red = Method 2
• Local peak at 19.8 in Log(E/eV)
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Outline
Motivation
New Statistical Method
Tests of the Method
Application to Pierre Auger Data
Conclusions and Future
13 December 2011
The Ohio State University
42
Conclusions and Future Work
• We have developed and tested two useful statistical
methods that can be used for spectral comparisons
• A signal might be slowly appearing in the region of
Cen A but still no significant signal yet
• Physically reasonable to expect a non-attenuated
spectrum from a nearby source
• Optimistically, expect another few years of data
before a significant signal can be observed
• A priori trial for future data
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Backup Slides
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Energy Determination in SD
•
S(1000) is the experimentally measured particle density at 1000 m from the shower core.
Want to use it as energy estimator, but it depends on zenith angle
– vertical shower sees 870 g cm-2 atmosphere
– showers at a zenith angle of 60° see 1740 g cm-2
– thus S(1000) is attenuated at large zenith angles
•
Zenith dependence of S(1000) can be determined empirically
– Assume that the cosmic ray flux is isotropic (has constant intensity or counts per unit
cos2) so that the only -dependence comes from the variation in the amount of
atmosphere through which the shower passes.
– Apply a constant intensity cut (CIC) to remove zenith dependence
December 15, 2008
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Constant Intensity Cut
Procedure:
 At different zenith angles, , the S(1000) spectra have different normalizations. We want
to fix this normalization.
 Choose a reference zenith angle where intensity is I0 (Auger: 38 = median of zenith
distribution).
 For each zenith angle, find the value of S(1000) such that I (>S(1000)) = I0 .
This determines the curve CIC()
 Define the energy parameter S38 = S(1000)/CIC()
This removes the -dependence of the ground parameter.
 S38 is the S(1000) measurement the shower would have produced if it had arrived at a
zenith angle of 38°. This is the REAL energy estimator of the SD.
Gets normalized to 1.0 at  = 38°
CIC()
December 15, 2008
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Auger Energy Spectrum
Hybrid Advantage
•
Getting the energy from S38 introduces
dependence on simulations; can use
hybrid events to calibrate S38 with FD
energy
•
Use golden hybrid events: events
reconstructed independently in FD and
SD
•
S38 is compared to the FD energy
measurement in hybrid events to
determine a correlation between
ground parameter and energy.
•
Hybrid data used to calibrate the
energy measurement of the surface
detector array.
December 15, 2008
Ground parameter
387 hybrid events
Energy from FD
47
• Method 1, 18.4 in Log(E/eV), 20° binning
• No signal
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