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Multiscale Autocorrelation Function: a new approach to anisotropy studies Manlio De Domenico 12 , H. Lyberis 34 1 Laboratory for Complex Systems, Scuola Superiore di Catania, Catania, Italy 2 Istituto Nazionale di Fisica Nucleare, Sez. di Catania, Catania (Italy) 3 CNRS/IN2P3 - IPN Orsay, Paris (France) 4 Dipartimento di Fisica, Universit´ a di Torino, Torino (Italy) CRIS, Catania, 17 Sep 2010

empty Take Home Message A new method for anisotropy signal detection in the arrival direction distribution of particles (nuclei, ν , γ ,...) 1 It depends on one parameter, i.e. the clustering scale Θ 2 It is based on information theory and extreme value theory 3 It is unbiased against the null hypothesis of isotropy 4 It provides high discrimination power against the alternative hypothesis of anisotropy 5 It is semi-analytical 6 It requires few minutes to analyze (and penalize) data sets up to 10 4 objects Main Ref.: M.D.D, A. Insolia, H. Lyberis, M. Scuderi arXiv:1001.1666 Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 2 / 22

empty Multiscale Autocorrelation Function I Divide the observed sky into equal-area boxes. The number of boxes defines the angular scale Θ of the analysis. Procedure: ψ i : density of events falling in the box B i ψ i : expected density of events falling in the box B i from an isotropic distribution � � Def. the global deviation from isotropy: A (Θ) = D KL ψ || ψ i ψ i (Θ) log ψ i (Θ) D KL � ψ || ψ � = � ψ i (Θ) Kullback-Leibler Divergence (1951) We define the Multiscale Autocorrelation Function (MAF) s (Θ) = | A data (Θ) − � A iso (Θ) � | σ A iso (Θ) Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 3 / 22

empty Multiscale Autocorrelation Function II H 0 : null hypothesis of underlying isotropic distribution H 1 : alternative hypothesis “ H 0 is false ” For any scale Θ ′ in the parameter space P , estimate the chance probability to obtain s MC (Θ ′ ) ≥ s data (Θ): s iso (Θ ′ ) ≥ s data (Θ) |H 0 , ∀ Θ ′ ∈ P ˜ � � P (Θ) = Pr to take into account the penalization for the Θ − scan Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 4 / 22

empty Multiscale Autocorrelation Function III Fixed grid may cut existing clusters, causing loss of information or reducing the signal To avoid this: at any scale Θ each point is extended into 8 exposure-weighted points whose distance from the original one is Θ / 2 ( dynamical binning ) MAF uses the density of extended points Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 5 / 22

empty Understanding MAF Our numerical studies show that such a dynamical binning (by means of extended points) approach recovers the correct information on the amount of clustering in the data Θ ⋆ , where the significance is minimum, is the significative clustering scale : the scale at which occurs a greater number of points respect to that one occuring by chance, with no regard for a particular configuration of points, e.g. doublets or triplets. 3 skies of 60 events: 20% of events from a single source with 3 diff. smearing angles ρ = 4 ◦ , 10 ◦ , 25 ◦ ; 80% are isotropic. Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 6 / 22

empty MAF: Statistical Features Under H 0 (Isotropy) I Numerical experiments under the Null Hypothesis In general, the scale Θ ⋆ , where minimum chance probability occurs, is reported. Theorem: under H 0 , all p-values Θ { Pr ( s iso (Θ ′ ) ≥ s data (Θ) |H 0 , ∀ Θ ′ ∈ P ) } min { p (Θ) } = arg min corresponding to isotropic skies, should be equally likely . The distribution of min { p (Θ) } is flat , as expected, with no regards for the data set size = ⇒ unbiased against H 0 Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 7 / 22

empty MAF: Statistical Features Under H 0 (Isotropy) II For each Θ ∈ P , we investigate the density of s (Θ) = | A data (Θ) −� A iso (Θ) �| σ A iso (Θ) The distribution of s (Θ) is half-normal : 2 e − s 2(Θ) G 1 / 2 [ s (Θ)] = √ 2 2 π for all Θ ∈ P Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 8 / 22

empty MAF: Statistical Features Under H 0 (Isotropy) III We investigate the density of max { s (Θ) } , used for penalizing p − values µ ˜ = 1 . 737 ± 0 . 001 σ ˜ = 0 . 464 ± 0 . 001 χ 2 / ndf 10 − 4 ≈ Density is the generalized Gumbel distribution : (Fisher-Tippet Type I from Extreme Value Theory) g ( z ) = 1 z = max { s (Θ) } − ˜ µ σ exp [ − z − e z ] , ˜ σ ˜ NOT dependent on data set size! Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 9 / 22

empty MAF: Statistical Features Under H 0 (Isotropy) IV Summary MAF is unbiased against the null hypothesis 1 Penalization procedure can be analytically performed: 2 � � max { s (Θ) } − ˜ µ �� p (max { s (Θ) } ) = 1 − exp − exp , ˜ σ We find excellent agreement between estimation through Montecarlo realizations and through analytical estimation Analytical computation is ≈ 15 times faster Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 10 / 22

empty Generating Anisotropic Skies I Ultra High-Energy Cosmic Rays Detecting anisotropy of UHECR is important for understanding creation and propagation mechanisms, for indirectly investigating extra-galactic magnetic fields, ... We test MAF against anisotropic mock maps of UHECR according to some physical constraints: Reference catalog of candidate sources: Active Galactic Nuclei 1 (AGN) with known redshift z < 0 . 047 ( ≈ 200 Mpc) from Palermo SWIFT-BAT hard X-ray catalogue [Cusumano, G. et al, Astron. Astrop. 510 (2010)] # of events proportional to source flux Φ and to z − 2 2 Magnetic deflections 3 Isotropic background contamination 4 Distribution of events weighted by exposure of world-wide 5 surface detectors Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 11 / 22

empty Why AGN & SWIFT-BAT? AGN are candidate sources G.R.Farrar and P.L.Biermann, PRL 81 (1998) 3579 P.G.Tinyakov and I.I.Tkachev, JETP Lett. 74 (2001) 445 V.Berezinsky et al , astro-ph/0210095 (2002) D.F.Torres et al , ApJ 595 (2003) P.Auger Coll., Science 318 (2007) 938 P.Auger Coll., Astrop. Phys., 29 (2008) 188 I.Zaw, G.R.Farrar, J.E.Greene ApJ (2009) 696 [P.Auger Coll., In Press (2010), arXiv:1009.1855] P.Auger Coll., In Press (2010), arXiv:1009.1855 SWIFT-BAT It provides the most complete and uniform all-sky hard X-ray survey up to date top: VCV; bottom: SWIFT-BAT Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 12 / 22

empty Surface Detectors Full-time operating and fully efficient SD do not observe the sky uniformly. Effective detection area depends on the relative exposure ω ( δ ) ∝ cos φ 0 cos δ sin α m + α m sin φ 0 sin δ, (1) where φ 0 is the detector latitude and  0 ξ > 1  α m = π ξ < − 1 (2) cos − 1 ξ otherwise SD detect Extended Air Shower of  particles produced by UHECR, by mean of a large array of individual stations with ξ ≡ cos θ max − sin φ 0 sin δ . cos φ 0 cos δ [P. Sommers, Astrop. Phys. 14, 271 (2001)] Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 13 / 22

empty World-Wide Surface/Hybrid EAS Detectors Map Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 14 / 22

empty World-Wide Surface/Hybrid EAS Detectors Info Exp. ( m 2 s sr) Experiment φ 0 θ max λ # Ev. 35 . 15 ◦ N 70 ◦ 0 . 2 × 10 16 1.000 ⋆ Volcano R. 6 61 . 60 ◦ N 60 ◦ 1 . 8 × 10 16 0.625 † Yakutsk 20 53 . 97 ◦ N 74 ◦ 1.000 ⋆ H. Park − 7 35 . 78 ◦ N 45 ◦ 4 . 0 × 10 16 0.750 † AGASA 29 30 . 43 ◦ S 70 ◦ 5 . 3 × 10 16 0.500 ⋆ SUGAR 13 35 . 20 ◦ S 60 ◦ 28 . 4 × 10 16 1.200 † P. Auger 27 ⋆ [M. Kachelrieß and D. Semikoz, Astrop. Phys. 26, 10 (2006)] † [V. Berezinsky, Nucl. Phys. B - Proc. Supp. 188, 227 (2009)] Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 15 / 22

empty Current Data I UHECR 102 UHECR with rescaled energy E ′ ≥ 4 . 0 × 10 19 eV from world-wide SD Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 16 / 22

empty Current Data II Catalog+UHECR Nearby AGN within 200 Mpc ( z < 0 . 047) and UHECR Catalog + UHECR With flux-weighted catalog Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 17 / 22

empty Hypothesis Testing: Statistical Errors Test accepts H 0 Test rejects H 0 H 0 is true OK: 1 − α CL α : Type I Error H 1 is true β : Type II Error OK: 1 − β Power Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 18 / 22