A Comparison of Energy Spectra in Different Parts of the Sky

13 December 2011 The Ohio State University 1

A Comparison of Energy Spectra in

Different Parts of the Sky

Carl Pfendner, Segev BenZvi, Stefan Westerhoff

University of Wisconsin - Madison

Outline

MotivationNew Statistical Method

Tests of the MethodApplication to Pierre Auger Data

Conclusions and Future


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• Cosmic rays interact with the 2.7 K microwave background.• Protons above ~ 51019 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

• 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

GZK Suppression

The Ohio State University

December 15, 2008 4

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. error25% syst. error

25% syst. error

HiRes Collaboration, PRL 100 (2008) 101101

13 December 2011 5

proton + cmb + nucleon

Particle Propagation (Toy Model)

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



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

Region A

Region B


Split Sky Analysis

• 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.

Log(E/eV)

dN/d

Log(

E/eV

)

Spectrum from region A, A

Spectrum from region B, B

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


• Model independent – no power law required• Automatically penalizes overly complex models• Naturally account for uncertainties in the data (including systematics if desired)

Outline






Bayesian ComparisonH2 = two-“parent” hypothesis

H1 = one-“parent” hypothesis

• Bayes factor is the probability ratio that the data supports hypothesis 2 over hypothesis 1

• Advantage: Can get posterior probability from the Bayes factor

• Assume P(H1)=P(H2), to get:

• 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


Bayesian Comparison

= 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.

H2 = two-“parent” hypothesis

H1 = one-“parent” hypothesis

dN/d

Log(

E)*

wdN

/dLo

g(E

)*w

’


Methods 1 & 2• 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• Result:

• 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:

dN/

dLog

(E)*

wdN

/dL

og(E

)*w

dN/

dLog

(E)*

wdN

/dL

og(E

)*w

dN/

dLog

(E)*

w

dN/

dLog

(E)*

w

Outline





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

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 eventsViolet – 3000 eventsRed – 10000 events

As events increase, the differentiation power increases dramatically


Broken Power Law Test

• Generated 20000-event sets using a broken-power-law

• Kept power law index set at a constant 2.7

• Varied the first power-law index, break energy

• Sensitivity is the width of the blue region - very sensitive

• Many events in lower energy bins


Broken Power Law Test• 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

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

Single vs Broken Power Law (cont.)


Posterior probability of the single-parent hypothesis (same shape) vs lower energy threshold

10000 events in broken- and single- power law functions

Blue = Chi-squaredRed = Bayes factor

Chi-squared produces a tail probability which biases against the null hypothesis and regularly underestimates the posterior probability

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



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



Single Power law vs broken power law with 0.05/0.95 exposure


Triple events of March 31, 2009

Even with a decreased exposure, can differentiate single and broken power law functions

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

Contamination (cont.)


Peak Bayes factor vs. contamination fraction


Horizontal line shows Bayes factor = 100

Point at which the Bayes factor reaches 100 indicated by vertical line

68% confidence bands

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

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.

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


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%

Outline





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Detection Techniques

Fluorescence Detector (FD)

Array of PMTs observes the UV light from the air showers fluorescing the nitrogen in the atmosphere

Surface Detector (SD)

3 PMTs per tank measure Cherenkov light from charged shower particles entering the tank


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Pierre Auger ObservatoryHybrid 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 MeasurementsFluorescence Detector• Measure light intensity along the track

and integrate.• Nearly calorimetric, model- and mass-

independent.• 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]


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

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)

13 December 2011 33The Ohio State University

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


Skymap - Angular Bin Size Change

Method 1, 18.4 in Log(E/eV), 5°-30° binning


Preliminary

Skymap changes (cont.)



Preliminary




Preliminary

Skymap Changes (cont.)


Method 1, 18.4 – 20.4, 23 degrees

Preliminary

• 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)


Preliminary

• 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)

Spectrum around Cen A


Outside events weighted by relative exposure w = 0.0602

• For 23 degrees around maximal point for method 1• Blue = Method 1, Red = Method 2 • Local peak at 19.8 in Log(E/eV)


Outline





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


Backup Slides


December 15, 2008 46

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


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.

Constant Intensity Cut

Gets normalized to 1.0 at = 38°

CIC()


Auger Energy SpectrumHybrid 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.

387 hybrid events

Gro

un

d p

ara

met

er

Energy from FD

• Method 1, 18.4 in Log(E/eV), 20° binning• No signal


A Comparison of Energy Spectra in Different Parts of the Sky

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