Study of Two Metrics for Anisotropy in the Highest Energy Cosmic Rays

Transcription

1 Study of Two in the Highest Energy Cosmic Rays John D. Hague, B. R. Becker, L. Cazon, M. S. Gold, J. A. J. Matthews University of New Mexico Department of Physics and Astronomy The Impact of High Energy Astrophysics Experiments on Cosmological Physics KAVLI Institute for Cosmological Physics October 28, 2008

2 Outline of Topics 1 Two Catalogue Independent Metrics Introduction Shape-Strength 2 Hypothesis Testing Mock Signal Types 3 Methods Food for Thought

3 Introduction Shape-Strength Context The Auger Method... Science 318:938, Nov 07. ( 1/ 1/2004 -> 31/ 8/2007) E/EeV 56.0, z 0.018, ψ is statistically sound and physically intuitive. has 1% significance with 27 events. is catalogue dependent Cent. A 180 SGC GC The Two Methods Here... are applied to mock data sets. use a catalogue independent test-statistic. use doublets or triplets of events. Figure: The 27 Auger events (black), low-redshift AGN (magenta) and the signal region (color gradient) in galactic coordinates

4 Introduction Shape-Strength Uniform Test-Statistic Paradigm p-value probability that a sky is isotropic significance level Both methods follow the same steps to determine the significance of a sky s departure from isotropy. 0 Choose a sky (mock signal, isotropic or otherwise). 1 Choose a metric (2-pt or shape-strength). 2 Compare the distribution of the metric between the test sky and Monte-Carlo isotropic reference skies to obtain the pseudo-log-likelihood Σ P. 3 The ratio of the number of isotropic skies with a Σ P less than that of the data set to the total number of simulated isotropic data sets gives the p-value.

5 Introduction Shape-Strength Figure: Two events on the celestial sphere. 1 The distance between two points is cos 1 e 1 e 2. 2 The metric distribution is the collection all pairs in a sky. 2a The depth of each bin is P i (n obs n exp) = n n obs exp e nexp /n exp!. 2b The sum over these bins is the pseudo-log-likelihood, Σ P = PN bins ln P i=1 i (n obs n exp). 3 The fraction of isotropic skies with Σ P < that of the data gives the p-value.

6 Introduction Shape-Strength Figure: The number observed n obs and expected n exp pairs of events for an example sky from the quadrapole ensemble. 1 The distance between two points is cos 1 e 1 e 2. 2 The metric distribution is the collection all pairs in a sky. 2a The depth of each bin is P i (n obs n exp) = n n obs exp e nexp /n exp!. 2b The sum over these bins is the pseudo-log-likelihood, Σ P = PN bins ln P i=1 i (n obs n exp). 3 The fraction of isotropic skies with Σ P < that of the data gives the p-value.

7 Introduction Shape-Strength Figure: The Poisson probability at each angular scale for an example sky from the quadrapole ensemble. 1 The distance between two points is cos 1 e 1 e 2. 2 The metric distribution is the collection all pairs in a sky. 2a The depth of each bin is P i (n obs n exp) = n n obs exp e nexp /n exp!. 2b The sum over these bins is the pseudo-log-likelihood, Σ P = PN bins ln P i=1 i (n obs n exp). 3 The fraction of isotropic skies with Σ P < that of the data gives the p-value.

8 Introduction Shape-Strength Figure: The the test statistic for an example sky from the quadrapole ensemble is the red vertical line. The distribution of isotropic likelihoods is the black distribution. 1 The distance between two points is cos 1 e 1 e 2. 2 The metric distribution is the collection all pairs in a sky. 2a The depth of each bin is P i (n obs n exp) = n n obs exp e nexp /n exp!. 2b The sum over these bins is the pseudo-log-likelihood, Σ P = PN bins ln P i=1 i (n obs n exp). 3 The fraction of isotropic skies with Σ P < that of the data gives the p-value.

9 Introduction Shape-Strength There are 3 distances in a triplet; What metric? Let (x i, y i, z i ), with i = 1, 2, 3, be the Cartesian coordinates of a set of three arrival directions { e 1, e 2, e 3 }. Then the eigen-values/vectors of T ij = 1 P 3 k { } `ri r j k satisfy τ 1 + τ 2 + τ 3 = 1 and τ 1 τ 2 τ 3 0. Only two free parameters! Shape γ = lg { lg(τ1/τ 2) lg(τ 2/τ 3) Strength ζ = lg(τ 1 /τ 3 ) where τ 1, τ 2 and τ 3 are the moments about the principle, major and minor axis. } Figure: Three events on the celestial sphere.

10 Introduction Shape-Strength There are 3 distances in a triplet; What metric? Let (x i, y i, z i ), with i = 1, 2, 3, be the Cartesian coordinates of a set of three arrival directions { e 1, e 2, e 3 }. Then the eigen-values/vectors of T ij = 1 P 3 k { } `ri r j k satisfy τ 1 + τ 2 + τ 3 = 1 and τ 1 τ 2 τ 3 0. Only two free parameters! Shape γ = lg { lg(τ1/τ 2) lg(τ 2/τ 3) Strength ζ = lg(τ 1 /τ 3 ) where τ 1, τ 2 and τ 3 are the moments about the principle, major and minor axis. } Figure: The orientation vectors of the triplet.

11 Introduction Shape-Strength There are 3 distances in a triplet; What metric? Let (x i, y i, z i ), with i = 1, 2, 3, be the Cartesian coordinates of a set of three arrival directions { e 1, e 2, e 3 }. Then the eigen-values/vectors of T ij = 1 P 3 k { } `ri r j k satisfy τ 1 + τ 2 + τ 3 = 1 and τ 1 τ 2 τ 3 0. Only two free parameters! Shape γ = lg { lg(τ1/τ 2) lg(τ 2/τ 3) Strength ζ = lg(τ 1 /τ 3 ) where τ 1, τ 2 and τ 3 are the moments about the principle, major and minor axis. } Figure: The orientation vectors of the triplet.

12 Introduction Shape-Strength General Method, Specific Implications Strength Shape Angular Scale Angular Scale Figure: As the strength ζ increases, the events become more concentrated. Figure: As the shape γ increases, the events become more stringy.

13 Introduction Shape-Strength Shape-Strength Example Sky Figure: The shape-strength p-value determination for an example sky from the quadrapole ensemble.

14 Hypothesis Testing Mock Signal Types Hypothesis Test Errors We use ensembles of mock-signal and isotropic skies to check... Sensitivity Type I Error Prob α to reject ISO, given that ISO is true. Chose a priori significance threshold, α = 1%, 0.1%. Fraction of ISO skies w/ p-value< α. Efficiency Type II Error Prob β to accept mock-signal, given that ISO is false. Fraction of mock-signal skies w/ p-value< α. β near 1.0 is good.

15 Hypothesis Testing Mock Signal Types Test Statistic Signal Region Figure: The distribution of the 2-point Σ P for the ISO (black) and quadrapole (red) ensembles. The long(short)-dashed-vertical lines correspond to α = 1%(0.1%).

16 Hypothesis Testing Mock Signal Types Signal-Background Mixture Dilute each sky with background events. Total Number of Events Most important # in all of CR-astronomy. 60 events/sky. 40 events/sky. 20 events/sky. QUADRUPOLEx 60 events/sky Efficiency α = pt SS α = pt SS Signal/Total Figure: The efficiency for detecting quadrapole as a function of signal/60-events.

17 Hypothesis Testing Mock Signal Types Signal-Background Mixture Dilute each sky with background events. Total Number of Events Most important # in all of CR-astronomy. 60 events/sky. 40 events/sky. 20 events/sky. QUADRUPOLEx 40 events/sky Efficiency α = pt SS α = pt SS Signal/Total Figure: The efficiency for detecting quadrapole as a function of signal/40-events.

18 Hypothesis Testing Mock Signal Types Signal-Background Mixture Dilute each sky with background events. Total Number of Events Most important # in all of CR-astronomy. 60 events/sky. 40 events/sky. 20 events/sky. QUADRUPOLEx 20 events/sky Efficiency α = pt SS α = pt SS Signal/Total Figure: The efficiency for detecting quadrapole as a function of signal/20-events.

19 Hypothesis Testing Mock Signal Types Mock Signal Types Figure: Galactic coordinate histograms of four mock signal types, each with 60 events/sky 10 4 skies.

20 Hypothesis Testing Mock Signal Types Diluted Efficiency DIPOLEz 60 events/sky 1.2 α = 0.01 α = pt 2pt 1 SS SS QUADRUPOLEx 60 events/sky 1.2 α = 0.01 α = pt 2pt 1 SS SS Efficiency Efficiency Signal/Total VCV events/sky 1.2 α = 0.01 α = pt 2pt 1 SS SS Signal/Total VCVflat events/sky 1.2 α = 0.01 α = pt 2pt 1 SS SS Efficiency Efficiency Signal/Total Signal/Total Figure: The efficiency for detecting 4 mock signal types as a function of signal/60-events.

21 Hypothesis Testing Mock Signal Types Signal Discrimination: Two-Point Figure: The 2-pt distance distribution for a test sky from 4 mock signal types.

22 Hypothesis Testing Mock Signal Types Signal Discrimination: Shape-Strength Figure: The shape-strength log-likelihood per bin for a test sky from 4 mock signal types.

23 Methods Food for Thought These Methods Generally... Are catalogue independent. Are statistically sound and intuitively descriptive. Use either doublets (2-pt) or triplets (shape-strength) to test for anisotropy. Mock-Signal Data Sets... Allow unbiased tuning. Shows that mock-signal type, the total number of events and diluted signals dramatically effect the efficiency. No silver bullet!

24 Food for Thought Two Catalogue Independent Metrics Multiple Metrics and Multiple Mock Types Could we identify the type of a real signal? Not yet. Event Number and Dilution Real signal detectability... IF IF Lucky i.e. Quad or VCV, then 60 events with 70% signal fraction. Not-lucky i.e. Octu or IRAS, then 60 events. Efficiency Methods Food for Thought α = pt Rayleigh Resultant SS α = pt Rayleigh Resultant SS VCVflat0020-ISO VCVflat0020 VCV0020-ISO VCV0020 IRAS0020-ISO IRAS ISO FiveEll ISO FiveEll ISO OCTUPOLEz OCTUPOLEx QUADRUPOLEz QUADRUPOLEx DIPOLEzn DIPOLEz DIPOLEx ISOTROPY Figure: Some mock-signals can be seen, some cannot! The efficiency for detecting 15 mock-signal types with four different methods. The 2-Point and Shape-Strength were detailed above. Experimentally... Unknown Signal Many More Than 60 Events, More Events + GZK-suppression = Larger Detector