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A Comparison of Energy Spectra in Different Parts of the Sky

A Comparison of Energy Spectra in Different Parts of the Sky. Carl Pfendner, Segev BenZvi, Stefan Westerhoff University of Wisconsin - Madison. Outline. Motivation New Statistical Method Tests of the Method Application to Pierre Auger Data Conclusions and Future. GZK Suppression.

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A Comparison of Energy Spectra in Different Parts of the Sky

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  1. A Comparison of Energy Spectra in Different Parts of the Sky Carl Pfendner, Segev BenZvi, Stefan Westerhoff University of Wisconsin - Madison The Ohio State University

  2. Outline Motivation New Statistical Method Tests of the Method Application to Pierre Auger Data Conclusions and Future The Ohio State University

  3. 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 • 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 The Ohio State University

  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. error 25% syst. error 25% syst. error HiRes Collaboration, PRL 100 (2008) 101101

  5. 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 The Ohio State University

  6. 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 The Ohio State University

  7. Split Sky Analysis Region A Spectrum from region A, A • 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. dN/dLog(E/eV) Spectrum from region B, B Log(E/eV) Region B The Ohio State University

  8. 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) The Ohio State University

  9. Outline Motivation New Statistical Method Tests of the Method Application to Pierre Auger Data Conclusions and Future The Ohio State University

  10. Bayesian Comparison H2 = 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 The Ohio State University

  11. 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 sincethe experiments could be observing two different fluxes. Allows any spectrum. H2 = two-“parent” hypothesis H1 = one-“parent” hypothesis dN/dLog(E)*w dN/dLog(E)*w’ The Ohio State University

  12. dN/dLog(E)*w dN/dLog(E)*w dN/dLog(E)*w dN/dLog(E)*w dN/dLog(E)*w dN/dLog(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: The Ohio State University

  13. Outline Motivation New Statistical Method Tests of the Method Application to Pierre Auger Data Conclusions and Future The Ohio State University

  14. 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 The Ohio State University

  15. 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 The Ohio State University

  16. 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 The Ohio State University

  17. 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 The Ohio State University

  18. 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 The Ohio State University

  19. 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-squared Red = Bayes factor Chi-squared produces a tail probability which biases against the null hypothesis and regularly underestimates the posterior probability The Ohio State University

  20. 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 The Ohio State University

  21. 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 The Ohio State University

  22. 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 The Ohio State University

  23. 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 The Ohio State University

  24. 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 The Ohio State University

  25. 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 The Ohio State University

  26. 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 The Ohio State University

  27. 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% The Ohio State University

  28. Outline Motivation New Statistical Method Tests of the Method Application to Pierre Auger Data Conclusions and Future The Ohio State University

  29. Detection Techniques Surface Detector (SD) 3 PMTs per tank measure Cherenkov light from charged shower particles entering the tank Fluorescence Detector (FD) Array of PMTs observes the UV light from the air showers fluorescing the nitrogen in the atmosphere The Ohio State University

  30. 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 The Ohio State University

  31. Fluorescence 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. Energy Measurements S(1000) Distance to shower core [m] The Ohio State University

  32. 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 The Ohio State University

  33. 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) The Ohio State University

  34. 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 The Ohio State University

  35. Skymap - Angular Bin Size Change Preliminary Method 1, 18.4 in Log(E/eV), 5°-30° binning The Ohio State University

  36. Skymap changes (cont.) Preliminary Method 2, 18.4 in Log(E/eV), 5°-30° binning The Ohio State University

  37. Skymap changes (cont.) Preliminary Method 1, 19.8 in Log(E/eV), 5°-30° binning The Ohio State University

  38. Skymap changes (cont.) Preliminary Method 2, 19.8 in Log(E/eV), 5°-30° binning The Ohio State University

  39. Skymap Changes (cont.) Preliminary Method 1, 18.4 – 20.4, 23 degrees The Ohio State University

  40. 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) The Ohio State University

  41. Spectrum around Cen A • 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) Outside events weighted by relative exposure w = 0.0602 The Ohio State University

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

  43. Outline Motivation New Statistical Method Tests of the Method Application to Pierre Auger Data Conclusions and Future The Ohio State University

  44. 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 The Ohio State University

  45. Backup Slides The Ohio State University

  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

  47. Gets normalized to 1.0 at  = 38° CIC() 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() • Definethe energy parameter S38 = S(1000)/CIC() • This removes the -dependence of the ground parameter. • S38is 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.

  48. 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. Auger Energy Spectrum 387 hybrid events Ground parameter Energy from FD

  49. Method 1, 18.4 in Log(E/eV), 20° binning • No signal The Ohio State University

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