January 16, 2012 1:11 WSPC/INSTRUCTION FILE stars2011 International JournalofModernPhysicsE (cid:13)c WorldScientificPublishingCompany 2 1 0 2 DETECTION OF POINT SOURCES IN COSMIC RAY MAPS USING THE MEXICAN HAT WAVELET FAMILY n a J 3 RAFAELALVESBATISTA,ERNESTOKEMP,BRUNODANIEL 1 Institutode Fsica “Gleb Wataghin”, Universidade Estadual de Campinas Cidade Universitria“ZeferinoVaz”, 13083-859 ] M Campinas-SP, Brazil rab@ifi.unicamp.br I . h ReceivedDayMonthYear p RevisedDayMonthYear - o r CommunicatedbyManagingEditor t s a An analysisof the sensitivity of gaussianand mexican hat wavelet familyfilters tothe [ detection of point sources of ultra-high energy cosmic rays was performed. A source embedded in a background was simulated and the number of events and amplitude of 1 this source was varied aiming to check the sensitivity of the method to detect faint v sourceswithlowstatisticofevents. 9 9 Keywords: ultra-highenergycosmicrays;pointsources;wavelets. 7 2 . 1. Introduction 1 0 The origin of the Ultra-High Energy Cosmic Rays (UHECRs) is still an unsolved 2 problem in astroparticle physics. A model for acceleration of cosmic rays, firstly 1 1 : devisedby Hillas predicts that cosmic raysare acceleratedto the highestenergies v (above 1018 eV) by electromagnetic fields of astrophysical objects. Therefore, the i X identication of possible astrophysical sources of UHECRs is possible by analysing r the arrival directions of the cosmic rays. The correlation of the positions of point- a like astrophysical objects (point sources) with the arrival directions of cosmic rays defines a small scale anisotropy. In the search of point-like sources it is a common procedure to convolve the sky maps containing arrival directions of cosmic rays with mathematical functions (the kernel of the convolution operation) aiming to optimizethesignaltonoiseratio.Inthisworkitisstudiedtheperformanceofsome kernels of the Mexican Hat Wavelet Family (MHWF) to identify point sources of cosmic rays, and disentangle genuine signals from the background. 2. Wavelets WaveletsaredefinedasmathematicalfunctionsbelongingtotheL2space.Theycan be thought as localized wave-like oscillating functions which can be operated with 1 January 16, 2012 1:11 WSPC/INSTRUCTION FILE stars2011 2 Rafael AlvesBatista, Ernesto Kemp, Bruno Daniel a givensignalandprovide informationabout it. The continuous wavelettransform (CWT) in two dimensions may be formally written as ∗ Φ(s,τ1,τ2)=Z Z f(t,u)Ψs,τ1,τ2(t)dtdu, (1) where s (s > 0, s R) is the scaling factor and τ1 and τ2 (τi R) are the ∈ ∈ translation parameters. So, the CWT decomposes a function f(t,u) in a basis of waveletΨ (t,u).Onecanscaleandtranslatea“mother-wavelet”Ψandobtain s,τ1,τ2 a wavelet Ψ (t,u), as follows: s,τ1,τ2 1 t τ1 u τ2 Ψ (t,u)= Ψ − , − . (2) s,τ1,τ2 √s (cid:18) s s (cid:19) The Mexican Hat Wavelet Family (MHWF), introduced by Gonz´alez-Nuevo et al.2, and its extension on the sphere have been widely used to detect point sources in maps of cosmic microwave background radiation3,4,5, due to the amplification of the signal-to-noise ratio (SNR) when transiting from real to wavelet space. The MHWF is obtained by successive application of the laplacian operator to the two- dimensional gaussian φ(~x). A generic member of this family, of order n, is: ( 1)n Ψ (~x)= − 2nφ(~x). (3) n 2nn! ∇ 3. Celestial Maps Celestialmapsarepixelationsofthecelestialspheretakingintoaccounttheangular resolution of the experiment. The events map is a celestial map representing the arrival directions of cosmic rays in a suitable coordinate system. Due to intrinsic limitationsofdetector,everyeventdetectedis convolvedwithaprobabilityrelated totheangularresolutionofthedetector,whichmeansthatthereisapointspreading function (PSF) associated to the detector. The convolution of celestial maps with filters is given by: M(j)Φ(r~,r~) j k j M (k)= , (4) f P Φ(r~ ,r~) j k j P whereM(j)isthenumberofcosmicrayswithinthepixelofindexj,inthedirection r~j. Φ(~r,r~0) is the used filter and r~k is the position vector representing the point where the integral is being calculated. 4. Analysis Procedure Thisworkisanextensionofpreviousones6,7.Thesimulateddetectorhastwosites, ◦ ◦ one located in the southern hemisphere (36 S and 65 W), and the other in the ◦ ◦ northern hemisphere (38 N and 102 W), seven times larger than the one in the south,implyingonafluxofcosmicraysseventimesgreater.Theacceptancelawhas ◦ ◦ the form sinθcosθ, where 0 θ 60 is the zenith angle. It was also considered ≤ ≤ the case of an ideal detector with uniform exposure and full sky coverage. January 16, 2012 1:11 WSPC/INSTRUCTION FILE stars2011 Detectionof Point Sources in Cosmic Ray Maps Using the MexicanHat Wavelet Family 3 ◦ ◦ It was simulated a point source located at (l,b)=(320 ,30 ) (galactic coordi- nates). This source was modeled by a gaussian: A ~x2 exp , (5) 2π (cid:18)−2σ2(cid:19) and was embedded in a background consisting on a superposition of four differ- ent patterns of arrival directions of cosmic rays. The simulated patterns were: (i) an isotropic distribution of events; (ii),(iii) dipoles, modeled according to Φ(uˆ) = Φ4π0 1+αDˆ.uˆ , where Φ is the flux of cosmic rays, Φ0 is related to the (cid:16) (cid:17) isotropic flux, 0 α 1 is the amplitude of the dipole (α = 0.07 for (ii) and ≤ ≤ α = 0.005 for (iii)), Dˆ is the vector which points towards the dipole ((l,b) = ◦ ◦ ◦ ◦ (0 ,0 ))for(ii)and(l,b)=(166.5 , 29 ))for(iii)) anduˆisaunitvectorpointing 8 − in an arbitrary direction ; (iv) severalsources modeled accordingto equation 5, in ◦ ◦ ◦ ◦ ◦ ◦ the directions (l,b): (0 ,0 ) [σ = 7.0 , A = 1.00], (320 ,90 ) [σ = 1.5 , A = 0.05], ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ (320 , 40 ) [σ = 0.5 , A = 0.01], (220 ,10 ) [σ = 3.0 , A = 0.05], (100 , 70 ) −◦ ◦ ◦ ◦ ◦ ◦ − ◦ [σ = 2 , A = 0.10], (240 ,50 ) [σ = 20 , A = 0.05], (350 , 80 ) [σ = 6.0 , ◦ ◦ ◦ ◦ ◦ −◦ A= 0.005], (100 ,50 ) [σ =30 , A =0.50], (140 , 40 ) [σ = 4.0 , A= 2.00] and ◦ ◦ ◦ − (60 ,50 ) [σ = 3.0 , A = 0.02]. This last background pattern was included in the simulation because unknown sources might be present during an analysis tagged on a given source, and their effects must be evaluated. From this combination of source (signal) and background patterns (noise), it was obtained the events map. The events map resulting from the sum of the background patterns and the simulated source was convolved with gaussian and MHWF (orders 1, 2 and 3) filters. The amplification of the SNR (λ) is calculated by the expression w /σ f f λ= , (6) w0/σ0 wherew0 isthevalueofthecentralpixelassociatedtothesourceinthenon-filtered sourcemap,wf isthe correspondingvalueinthefilteredsourcemap,σ0 isthe root mean square (RMS) of the non-filtered backgroundmap and σ is the RMS of the f filtered backgroundmap. Aiming to study the impact of the number of events from the source and its intensity upon the filter, different number of events (N ) were simulated in the evt direction of the source, ranging from 10 events up to 1000. The amplitude of the gaussian source (A) was also varied, from 10−4 to 1. 5. Results Infigures1and2itisshownthemaximumamplificationofthe SNR,asafunction ofthe correspondingscale.In the caseoffigure1, N is fixedandAis varied.For evt figure 2, A is fixed and N from the source is varied. evt From figure 1 it can be noted that the uniform exposure acts like a constraint for λ, and that the gaussian has a slightly better performance than the MHWF filters. The maximum amplification for nonuniform exposure is clearly achievedby January 16, 2012 1:11 WSPC/INSTRUCTION FILE stars2011 4 Rafael AlvesBatista, Ernesto Kemp, Bruno Daniel λ 3.45 MGA=aHu0W.s1s%iFa1n -- uunniiffoorrmm MHWF2 - uniform MHWF3 - uniform 3 Gaussian - nonuniform MHWF1 - nonuniform 2.5 MHWF2 - nonuniform MHWF3 - nonuniform 2 A=100% Gaussian - uniform MHWF1 - uniform 1.5 MHWF2 - uniform MHWF3 - uniform 1 Gaussian - nonuniform MHWF1 - nonuniform 0.5 MHWF2 - nonuniform MHWF3 - nonuniform 0 0 1 2 3 4 5 6 7 σ (degrees) λ 67 MGA=aHu0W.s1s%iFa1n -- uunniiffoorrmm f MHWF2 - uniform MHWF3 - uniform 5 Gaussian - nonuniform MHWF1 - nonuniform MHWF2 - nonuniform 4 MHWF3 - nonuniform A=100% 3 Gaussian - uniform MHWF1 - uniform MHWF2 - uniform 2 MHWF3 - uniform Gaussian - nonuniform MHWF1 - nonuniform 1 MHWF2 - nonuniform MHWF3 - nonuniform 0 0 1 2 3 4 5 6 7 σ (degrees) f Fig.1.MaximumamplificationoftheSNR(λ)asafunctionofthecorrespondingscale(σf).The graphsdisplayedrefertoafractionofeventsbetweenthesourceandthebackgroundof6×10−6 (top)and6.25×10−4 (bottom). January 16, 2012 1:11 WSPC/INSTRUCTION FILE stars2011 Detectionof Point Sources in Cosmic Ray Maps Using the MexicanHat Wavelet Family 5 7 Nsrc=10 λ Gaussian - uniform 6 MHWF1 - uniform MHWF2 - uniform MHWF3 - uniform 5 Gaussian - nonuniform MHWF1 - nonuniform MHWF2 - nonuniform 4 MHWF3 - nonuniform N =1000 src 3 Gaussian - uniform MHWF1 - uniform MHWF2 - uniform 2 MHWF3 - uniform Gaussian - nonuniform MHWF1 - nonuniform 1 MHWF2 - nonuniform MHWF3 - nonuniform 0 0 1 2 3 4 5 6 7 σ (degrees) f λ 5 GMNasHruc=Wss1iF0a1n -- uunniiffoorrmm MHWF2 - uniform 4 MHWF3 - uniform Gaussian - nonuniform MHWF1 - nonuniform 3 MHWF2 - nonuniform MHWF3 - nonuniform N =1000 src Gaussian - uniform 2 MHWF1 - uniform MHWF2 - uniform MHWF3 - uniform Gaussian - nonuniform 1 MHWF1 - nonuniform MHWF2 - nonuniform MHWF3 - nonuniform 0 0 1 2 3 4 5 6 7 σ (degrees) f Fig.2.MaximumamplificationoftheSNR(λ)asafunctionofthecorrespondingscale(σf).The graphsdisplayedrefertoanamplitudeA=10−4 (top)andA=1(bottom). January 16, 2012 1:11 WSPC/INSTRUCTION FILE stars2011 6 Rafael AlvesBatista, Ernesto Kemp, Bruno Daniel using MHWF filters, whereasthe gaussianfilter has amplificationclose to 1,which meansnoamplification.Comparingthetwographsinfigure1,inthebottomgraph the amplification is greater, which seems reasonable since the number of events in this case is 100 times greater. In figure 2 it can be seen the behavior of the filters when A is varied. For the uniformexposurethe amplificationofboththe gaussianandthe MHWF filters are low, but the gaussian has a slightly better performance. Comparing the top and bottom graphs in figure 2, it is clear that the amplification achieved by the filters is proportional to N . Also, when there is an acceptance, the gaussian filter does evt not provide a good amplification of the SNR, which can be achieved by using the MHWF. 6. Conclusions In this work it was analyzed the performance of the gaussian and the MHWF filters to detect point sources of cosmic rays embedded in a non uniform back- ground, whose features are modulated both by the acceptance of the detector and the backgroundpatterns imposedtothe incomingparticles.Someparametersfrom thesourcesuchastheamplitudeAandthenumberofeventsN werevaried,and evt the effects of theses changes on amplification of the SNR was studied. The trivial conclusion is that the amplification achieved by these kernels is proportional to the source intensity parameters (A and N ). It is interesting to evt notice that for a realistic case, i. e., a source with low amplitude and only a few eventscomingfromitsdirection,theamplificationsachievedarelowforbothfilters. The MHWF filters are more robust to these parameters and can provide a greater amplification of the SNR even if N is small. evt Regardingthecontributionoftheacceptanceoftheexperimentforthedetection, it can be clearly seen that for uniform exposure the amplifications are smaller. In this case, the gaussian filter provides a slightly better amplification compared to the MHWF. However,for a more realistic case taking into accountthe nonuniform exposure of the detector,the MHWF filters alwaysachieve a greateramplification. Also,theyaremorerobusttolowstatisticofevents,whichmakesthemparticularly useful for cosmic ray studies. References 1. A.M. Hillas, Nature 312 (1984) 50. 2. J. Gonz´alez-Nuevo et al., MNRAS 369 (2006) 1603. 3. A.Cayo´n et al., MNRAS 313 (2000) 757. 4. P.Vielva et al., MNRAS 326 (2001) 181. 5. P.Vielva et al., MNRAS 344 (2003) 89. 6. R.A. Batista et al., PhysicæProceedings 1 (2011). 7. R.A.Batista, E.Kemp,B.Daniel., toappearin theProceedings of the 32nd Interna- tional Cosmic Ray Conference., (2011). 8. J. Aublin and E. Parizot, Astron. Astrophys. 441 (2005) 407.