083007 (2012) Anisotropies in the diffuse gamma-ray ... · M. Roth,52 C. Sbarra,8 J. Schmitt,5 C. Sgro`,4 J. Siegal-Gaskins,3,50, ... The contribution of unresolved sources to the

Anisotropies in the diffuse gamma-ray background measured by the Fermi LAT

M. Ackermann,1 M. Ajello,2 A. Albert,3 L. Baldini,4 J. Ballet,5 G. Barbiellini,6,7 D. Bastieri,8,9 K. Bechtol,2 R. Bellazzini,4

E. D. Bloom,2 E. Bonamente,10,11 A.W. Borgland,2 T. J. Brandt,12,13 J. Bregeon,4 M. Brigida,14,15 P. Bruel,16 R. Buehler,2

S. Buson,8,9 G.A. Caliandro,17 R. A. Cameron,2 P. A. Caraveo,18 C. Cecchi,10,11 E. Charles,2 A. Chekhtman,19 J. Chiang,2

S. Ciprini,20,11 R. Claus,2 J. Cohen-Tanugi,21 J. Conrad,22,23 A. Cuoco,23,* S. Cutini,24 F. D’Ammando,25,26

F. de Palma,14,15 C.D. Dermer,27 S.W. Digel,2 E. do Couto e Silva,2 P. S. Drell,2 A. Drlica-Wagner,2 R. Dubois,2

C. Favuzzi,14,15 S. J. Fegan,16 E. C. Ferrara,28 P. Fortin,16 Y. Fukazawa,29 P. Fusco,14,15 F. Gargano,15 D. Gasparrini,24

S. Germani,10,11 N. Giglietto,14,15 M. Giroletti,30 T. Glanzman,2 G. Godfrey,2 G.A. Gomez-Vargas,31,32,33 T. Grégoire,12,13

I. A. Grenier,5 J. E. Grove,27 S. Guiriec,34 M. Gustafsson,8 D. Hadasch,17 M. Hayashida,2,35 K. Hayashi,29 X. Hou,36

R. E. Hughes,3 G. Jóhannesson,37 A. S. Johnson,2 T. Kamae,2 J. Knödlseder,12,13 M. Kuss,4 J. Lande,2 L. Latronico,38

M. Lemoine-Goumard,39 T. Linden,40,† A.M. Lionetto,31,41 M. Llena Garde,22,23 F. Longo,6,7 F. Loparco,14,15

M.N. Lovellette,27 P. Lubrano,10,11 M.N. Mazziotta,15,‡ J. E. McEnery,28,42 W. Mitthumsiri,2 T. Mizuno,29 C. Monte,14,15

M. E. Monzani,2 A. Morselli,31 I. V. Moskalenko,2 S. Murgia,2 M. Naumann-Godo,5 J. P. Norris,43 E. Nuss,21 T. Ohsugi,44

A. Okumura,2,45 M. Orienti,30 E. Orlando,2,46 J. F. Ormes,47 D. Paneque,48,2 J. H. Panetta,2 D. Parent,49 V. Pavlidou,50

M. Pesce-Rollins,4 M. Pierbattista,5 F. Piron,21 G. Pivato,9 S. Rainò,14,15 R. Rando,8,9 A. Reimer,51,2 O. Reimer,51,2

M. Roth,52 C. Sbarra,8 J. Schmitt,5 C. Sgrò,4 J. Siegal-Gaskins,3,50,§ E. J. Siskind,53 G. Spandre,4 P. Spinelli,14,15

A.W. Strong,46 D. J. Suson,54 H. Takahashi,44 T. Tanaka,2 J. B. Thayer,2 L. Tibaldo,8,9 M. Tinivella,4 D. F. Torres,17,55

G. Tosti,10,11 E. Troja,28,56 T. L. Usher,2 J. Vandenbroucke,2 V. Vasileiou,21 G. Vianello,2,57 V. Vitale,31,41,k A. P. Waite,2

B. L. Winer,3 K. S. Wood,27 M. Wood,2 Z. Yang,22,23 and S. Zimmer22,23

(Fermi LAT Collaboration)

1Deutsches Elektronen Synchrotron DESY, D-15738 Zeuthen, Germany2W.W. Hansen Experimental Physics Laboratory, Kavli Institute for Particle Astrophysics and Cosmology,

Department of Physics and SLAC National Accelerator Laboratory, Stanford University, Stanford, California 94305, USA3Department of Physics, Center for Cosmology and Astro-Particle Physics, The Ohio State University, Columbus, Ohio 43210, USA

4Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, I-56127 Pisa, Italy5Laboratoire AIM, CEA-IRFU/CNRS/Université Paris Diderot, Service d’Astrophysique, CEA Saclay, 91191 Gif sur Yvette, France

6Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, I-34127 Trieste, Italy7Dipartimento di Fisica, Università di Trieste, I-34127 Trieste, Italy

8Istituto Nazionale di Fisica Nucleare, Sezione di Padova, I-35131 Padova, Italy9Dipartimento di Fisica ‘‘G. Galilei,’’ Università di Padova, I-35131 Padova, Italy10Istituto Nazionale di Fisica Nucleare, Sezione di Perugia, I-06123 Perugia, Italy11Dipartimento di Fisica, Università degli Studi di Perugia, I-06123 Perugia, Italy

12CNRS, IRAP, F-31028 Toulouse cedex 4, France13GAHEC, Université de Toulouse, UPS-OMP, IRAP, Toulouse, France

14Dipartimento di Fisica, ‘‘M. Merlin’’ dell’Università e del Politecnico di Bari, I-70126 Bari, Italy15Istituto Nazionale di Fisica Nucleare, Sezione di Bari, 70126 Bari, Italy

16Laboratoire Leprince-Ringuet, École polytechnique, CNRS/IN2P3, Palaiseau, France17Institut de Ciències de l’Espai (IEEE-CSIC), Campus UAB, 08193 Barcelona, Spain

18INAF-Istituto di Astrofisica Spaziale e Fisica Cosmica, I-20133 Milano, Italy19Space Science Division, Naval Research Laboratory, Washington, D.C. 20375, USA,

and George Mason University, Fairfax, Virginia 22030, USA20ASI Science Data Center, I-00044 Frascati (Roma), Italy

21Laboratoire Univers et Particules de Montpellier, Université Montpellier 2, CNRS/IN2P3, Montpellier, France22Department of Physics, Stockholm University, AlbaNova, SE-106 91 Stockholm, Sweden

23The Oskar Klein Centre for Cosmoparticle Physics, AlbaNova, SE-106 91 Stockholm, Sweden24Agenzia Spaziale Italiana (ASI) Science Data Center, I-00044 Frascati (Roma), Italy

25IASF Palermo, 90146 Palermo, Italy26INAF-Istituto di Astrofisica Spaziale e Fisica Cosmica, I-00133 Roma, Italy

27Space Science Division, Naval Research Laboratory, Washington, D.C. 20375-5352, USA28NASA Goddard Space Flight Center, Greenbelt, Maryland 20771, USA

29Department of Physical Sciences, Hiroshima University, Higashi-Hiroshima, Hiroshima 739-8526, Japan30INAF Istituto di Radioastronomia, 40129 Bologna, Italy

31Istituto Nazionale di Fisica Nucleare, Sezione di Roma ‘‘Tor Vergata,’’ I-00133 Roma, Italy

PHYSICAL REVIEW D 85, 083007 (2012)

1550-7998=2012=85(8)=083007(28) 083007-1 � 2012 American Physical Society

Report Documentation Page Form ApprovedOMB No. 0704-0188Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering andmaintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information,including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, ArlingtonVA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if itdoes not display a currently valid OMB control number.

1. REPORT DATE 02 MAY 2012 2. REPORT TYPE

3. DATES COVERED 00-00-2012 to 00-00-2012

4. TITLE AND SUBTITLE Anisotropies in the diffuse gamma-ray background measured by theFermi LAT

5a. CONTRACT NUMBER

5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

6. AUTHOR(S) 5d. PROJECT NUMBER

5e. TASK NUMBER

5f. WORK UNIT NUMBER

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Naval Research Laboratory,Space Science Division,Washington,DC,20375

8. PERFORMING ORGANIZATIONREPORT NUMBER

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S)

11. SPONSOR/MONITOR’S REPORT NUMBER(S)

12. DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release; distribution unlimited

13. SUPPLEMENTARY NOTES

14. ABSTRACT

15. SUBJECT TERMS

16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT Same as

Report (SAR)

18. NUMBEROF PAGES

28

19a. NAME OFRESPONSIBLE PERSON

a. REPORT unclassified

b. ABSTRACT unclassified

c. THIS PAGE unclassified

Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18

32Departamento de Fı́sica Teórica, Universidad Autónoma de Madrid, Cantoblanco, E-28049, Madrid, Spain33Instituto de Fı́sica Teórica IFT-UAM/CSIC, Universidad Autónoma de Madrid, Cantoblanco, E-28049, Madrid, Spain

34Center for Space Plasma and Aeronomic Research (CSPAR), University of Alabama in Huntsville, Huntsville, Alabama 35899, USA35Department of Astronomy, Graduate School of Science, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan

36Centre d’Études Nucléaires de Bordeaux Gradignan, IN2P3/CNRS, Université Bordeaux 1, BP120,F-33175 Gradignan Cedex, France

37Science Institute, University of Iceland, IS-107 Reykjavik, Iceland38Istituto Nazionale di Fisica Nucleare, Sezioine di Torino, I-10125 Torino, Italy

39Université Bordeaux 1, CNRS/IN2p3, Centre d’Études Nucléaires de Bordeaux Gradignan, 33175 Gradignan, France40Santa Cruz Institute for Particle Physics, Department of Physics and Department of Astronomy and Astrophysics,

University of California at Santa Cruz, Santa Cruz, California 95064, USA41Dipartimento di Fisica, Università di Roma ‘‘Tor Vergata,’’ I-00133 Roma, Italy

42Department of Physics and Department of Astronomy, University of Maryland, College Park, Maryland 20742, USA43Department of Physics, Boise State University, Boise, Idaho 83725, USA

44Hiroshima Astrophysical Science Center, Hiroshima University, Higashi-Hiroshima, Hiroshima 739-8526, Japan45Institute of Space and Astronautical Science, JAXA, 3-1-1 Yoshinodai, Chuo-ku, Sagamihara, Kanagawa 252-5210, Japan

46Max-Planck Institut für extraterrestrische Physik, 85748 Garching, Germany47Department of Physics and Astronomy, University of Denver, Denver, Colorado 80208, USA

48Max-Planck-Institut für Physik, D-80805 München, Germany49Center for Earth Observing and Space Research, College of Science, George Mason University, Fairfax, Virginia 22030, USA,

and Space Science Division, Naval Research Laboratory, Washington, D.C. 20375, USA50Cahill Center for Astronomy and Astrophysics, California Institute of Technology, Pasadena, California 91125, USA51Institut für Astro- und Teilchenphysik and Institut für Theoretische Physik, Leopold-Franzens-Universität Innsbruck,

A-6020 Innsbruck, Austria52Department of Physics, University of Washington, Seattle, Washington 98195-1560, USA

53NYCB Real-Time Computing Inc., Lattingtown, New York 11560-1025, USA54Department of Chemistry and Physics, Purdue University Calumet, Hammond, Indiana 46323-2094, USA

55Institució Catalana de Recerca i Estudis Avançats (ICREA), Barcelona, Spain56NASA Goddard Space Flight Center, Greenbelt, Maryland 20771, USA

57Consorzio Interuniversitario per la Fisica Spaziale (CIFS), I-10133 Torino, Italy

E. Komatsu{

Texas Cosmology Center and Department of Astronomy, The University of Texas at Austin, Austin, Texas 78712, USA(Received 2 November 2011; published 23 April 2012; publisher error corrected 2 May 2012)

The contribution of unresolved sources to the diffuse gamma-ray background could induce anisotropies

in this emission on small angular scales. We analyze the angular power spectrum of the diffuse emission

measured by the Fermi Large Area Telescope at Galactic latitudes jbj> 30� in four energy bins spanning1–50 GeV. At multipoles ‘ � 155, corresponding to angular scales& 2�, angular power above the photonnoise level is detected at>99:99% confidence level in the 1–2 GeV, 2–5 GeV, and 5–10 GeVenergy bins,

and at >99% confidence level at 10–50 GeV. Within each energy bin the measured angular power takes

approximately the same value at all multipoles ‘ � 155, suggesting that it originates from the contributionof one or more unclustered source populations. The amplitude of the angular power normalized to the

mean intensity in each energy bin is consistent with a constant value at all energies, CP=hIi2 ¼ 9:05�0:84� 10�6 sr, while the energy dependence of CP is consistent with the anisotropy arising from one ormore source populations with power-law photon spectra with spectral index �s ¼ 2:40� 0:07. We discussthe implications of the measured angular power for gamma-ray source populations that may provide a

contribution to the diffuse gamma-ray background.

DOI: 10.1103/PhysRevD.85.083007 PACS numbers: 98.70.Vc, 95.85.Pw, 98.70.Rz

I. INTRODUCTION

The origin of the all-sky diffuse gamma-ray emissionremains one of the outstanding questions in high-energyastrophysics. First detected by OSO-3 [1], the isotropicgamma-ray background (IGRB) was subsequently mea-sured by SAS-2 [2], the Energetic Gamma RayExperiment Telescope [3,4], and most recently by theLarge Area Telescope (LAT) on board the Fermi

*[email protected]†[email protected]‡[email protected]§[email protected]@roma2.infn.it{[email protected]

M. ACKERMANN et al. PHYSICAL REVIEW D 85, 083007 (2012)

083007-2

Gamma-Ray Space Telescope (Fermi) [5]. The term IGRBis used to refer to the observed diffuse gamma-ray emissionwhich appears isotropic on large angular scales but maycontain anisotropies on small angular scales. The IGRBdescribes the collective emission of unresolved membersof extragalactic source classes and Galactic source classesthat contribute to the observed emission at high latitudes,and gamma-ray photons resulting from the interactions ofultra-high-energy cosmic rays with intergalactic photonfields [6].

Confirmed gamma-ray source populations withresolved members are guaranteed to contribute to theIGRB at some level via the emission from fainter, un-resolved members of those source classes. In theEnergetic Gamma Ray Experiment Telescope era thepossibility that blazars are the dominant contributor tothe IGRB intensity was extensively studied (e.g., [7–9]),however the level of the blazar contribution remainsuncertain, with recent results suggesting differentenergy-dependent contributions from blazars, whichamount to as little as �15% or as much as �100% ofthe Fermi-measured IGRB intensity, depending on theenergy [10–13]. Star-forming galaxies [14] and gamma-ray millisecond pulsars [15] may also provide a signifi-cant contribution to the IGRB at some energies.However, substantial uncertainties in the properties ofeven confirmed source populations present a challengeto estimating the amount of emission attributable toeach source class, and currently the possibility that theIGRB includes an appreciable contribution from un-known or unconfirmed gamma-ray sources, such asdark matter annihilation or decay (e.g., [16–22]), cannotbe excluded.

The Fermi-measured IGRB energy spectrum is rela-tively featureless, following a simple power law to goodapproximation over a large energy range (�200 MeV to�100 GeV) [5]. As a result, identifying the contribu-tions from individual components based on spectralinformation alone is difficult. However, in addition tothe energy spectrum and average intensity, the IGRBcontains angular information in the form of fluctuationson small angular scales [23]. The statistical propertiesof these small-scale anisotropies may be used to inferthe presence of emission from unresolved sourcepopulations.

If some component of the IGRB emission originatesfrom an unresolved source population, rather than from aperfectly isotropic, smooth source distribution, the dif-fuse emission will contain fluctuations on small angularscales due to the varying number density of sourcesin different sky directions. Unlike the Poisson fluctua-tions between pixels in a map of a truly isotropicsource distribution (which we shall call ‘‘photon noise’’),which are due to finite event statistics, the fluctuationsfrom an unresolved source population are inherentin the source distribution and will not decrease in

amplitude even in the limit of infinite statistics. Hence,with sufficient statistics, these fluctuations could bedetected above those expected from the photon noise,and could be used to understand the origin of the diffuseemission.The angular power spectrum of the emission provides a

metric for characterizing the intensity fluctuations. For asource population modeled with a specific spatial andluminosity distribution, the angular power spectrum canbe predicted and compared to the measured angular powerspectrum; in this way an anisotropy measurement has thepotential to constrain the properties of source populations.Other approaches to using anisotropy information in theIGRB have also been considered. For example, the 1-pointprobability distribution function (PDF), i.e. the distributionof the number of counts per pixel, is an alternative metric tocharacterize the fluctuations [13,24,25]. In addition, cross-correlating the gamma-ray sky with galaxy catalogs or thecosmic microwave background can be used to constrain theorigin of the emission [26].In recent years theoretical studies have predicted the

angular power spectrum of the gamma-ray emissionfrom several known and proposed source classes.Established astrophysical source populations such asblazars [27–29], star-forming galaxies [30], andGalactic millisecond pulsars [31] have been consideredas possible contributors to the anisotropy of the IGRB.In addition, it has been shown that the annihilation ordecay of dark matter in Galactic subhalos [32–34] andextragalactic structures [23,27,29,34–39], may generatean anisotropy signal in diffuse gamma-ray emission.Interestingly, the predicted angular power spectra ofthese gamma-ray source classes in the multipole rangeof ‘� 100–500 are in most cases fairly constant inmultipole (except for dark matter annihilation and decaysignals, e.g., [23,27,38]), although the amplitude of thepredicted anisotropy varies between source classes. Thismultipole-independent signal arises from the Poissonterm in the angular power spectrum, which describesthe anisotropy from an unclustered collection of pointsources. The multipole independence of the predictedangular power spectra therefore indicates that the ex-pected degree of intrinsic clustering of these gamma-raysource populations has a subdominant effect on theangular power spectra in this multipole range. The an-gular power spectra of dark matter annihilation anddecay signals are predicted to be smooth and relativelyfeatureless, with the angular power generally falling offmore quickly with multipole than Poisson angularpower.In this work we present a measurement of the angular

power spectrum of the high-latitude emission detected bythe Fermi LAT, using �22 months of data. The data wereprocessed with the FERMI SCIENCE TOOLS [40], and binnedinto maps covering several energy ranges. Regions of the

ANISOTROPIES IN THE DIFFUSE GAMMA-RAY . . . PHYSICAL REVIEW D 85, 083007 (2012)

083007-3

sky heavily contaminated by Galactic diffuse emissionand known point sources were masked, and then angularpower spectra were calculated on the masked sky for eachenergy bin using the HEALPIX package [41], describedin [42].

To understand the impact of the instrument responseon the measured angular power spectrum, several tai-lored validation studies were performed for this analysis.The robustness of the anisotropy analysis pipeline wastested using a source model with known anisotropyproperties that was simulated to include the effects of theinstrument response and processed with the same analy-sis pipeline as the data. The data processing was cross-checked to exclude the presence of anisotropies createdby systematics in the instrument exposure calculation byusing an event-shuffling technique (as used in [43]) thatdoes not rely on the Monte-Carlo-based exposure calcu-lation implemented in the SCIENCE TOOLS. In addition,validation studies were performed to characterize theimpact of foreground contamination, masking, and inac-curacies in the assumed point spread function (PSF).

We use a set of simulated models of the gamma-ray skyas a reference, and compare the angular power spectrummeasured for the data to that of the models to identify anysignificant differences in anisotropy properties. Finally, wecompare the predicted anisotropy for several confirmedand proposed gamma-ray source populations to the mea-sured angular power spectrum of the data.

The data selection and map-making procedure aredescribed in Sec. II, and the angular power spectrumcalculation is outlined in Sec. III. The event-shufflingtechnique is presented in Sec. IV, and the details of themodels simulated to compare with the data are given inSec. V. The results of the angular power spectrum mea-surement and the validation studies are presented inSec. VI. The energy dependence of the anisotropy is dis-cussed in Sec. VII, and the implications of the results forspecific source populations are examined in Sec. VIII. Theconclusions are summarized in Sec. IX.

II. DATA SELECTION AND PROCESSING

The Fermi LAT is designed to operate primarily asa survey instrument, featuring both a wide field of view(�2:4 sr) and a large effective area (*7000 cm2 fornormally-incident photons above 1 GeV). The telescopeis equipped with a 4� 4 array of modules, each consistingof a precision tracker and calorimeter, covered by an anti-coincidence detector that allows for rejection of chargedparticle events. Full details of the instrument, includingtechnical information about the detector, onboard andground data processing, and mission-oriented support, aregiven in [44].

We selected data taken from the beginning of scientificoperations in early-August 2008 through early-June 2010,encompassing over 56.6 Ms of live time [45]. We selected

only ‘‘diffuse’’ class [44] events to ensure that the eventsare photons with high probability, and restricted our analy-sis to the energy range 1–50 GeV where the PSF of theLAT is small enough to allow for sufficient sensitivity toanisotropies at small angular scales. The upper limit of50 GeV was chosen because the small photon statisticsabove this energy severely limit the sensitivity of theanalysis at the high multipoles of interest. The data andsimulations were analyzed with the LAT analysis softwareSCIENCE TOOLS version v9r15p4 using the standard P6_V3

LAT instrument response functions (IRFs). Detailed docu-mentation of the SCIENCE TOOLS is given in [46].In order to both promote near uniform sky exposure

and to limit contamination from gamma rays originatingin Earth’s atmosphere, the tool gtmktime was used toremove data taken during any time period when the LATrocked to an angle exceeding 52� with respect to thezenith, and during any time period when the LAT wasnot in survey mode. Beginning in its second year ofoperation (September 2009), Fermi has been operatingin survey mode with a large rocking angle of 50�, incontrast to the 35� rocking angle used during the firstyear of operation. The rocking-angle cut is used to limitthe amount of contamination from gamma rays producedin cosmic-ray interactions in the upper atmosphere byusing only data taken when the Earth’s limb was outsideof the field of view (the Earth’s limb has zenith angle�113�). However, due to the LAT’s large field of view,some Earth-limb gamma rays may be observed evenwhen the rocking angle constraint is not exceeded, thusthe gtselect tool was also used to remove each individualevent with a zenith angle exceeding 105�. We note thatall events in the data set were detected while the Fermispacecraft was outside of the South Atlantic Anomalyregion in which the cosmic-ray fluxes at the altitude ofFermi are significantly enhanced.In order to balance the need for a large effective area

with the need for high angular resolution, the LAT uses acombination of thin tracker regions near the front of theinstrument and thicker tracker regions in the back of thedetector. While the effective area of each region is compa-rable, the width of the PSF for events detected in the fronttrackers is approximately half that of events detected in theback of the instrument. For a measurement of the angularpower at high multipoles, it is thus necessary to differ-entiate between photons observed in the front and backtrackers of the Fermi LAT. In this study, we processedfront- and back-converting events separately, using thegtselect tool to isolate each set of events and calculatingthe exposure maps independently. The P6_V3_DIFFUSE:FRONT and P6_V3_DIFFUSE:BACK IRFs were used to analyze

the corresponding sets of events.Taking the selection cuts into account, the integrated live

time was calculated using gtltcube. We chose a pixel sizeof 0.125�, which produces a HEALPIX map with resolution


083007-4

parameter Nside ¼ 512. At this resolution, the suppressionof angular power from the pixel window function is sub-dominant with respect to the suppression from the LATPSF. We adopted an angular step size cosð�Þ ¼ 0:025 inorder to finely grid the exposure map for different gamma-ray arrival directions in instrument coordinates. Theexposure was then calculated using gtexpcube withthe same pixel size, for 42 logarithmic energy bins span-ning 1.04–50.0 GeV. These finely-gridded energy binswere then summed to build maps covering four largerenergy bins, as described in Sec. III A. Using theGaRDiAn package [47], both the photon counts and expo-sure maps were converted into HEALPIX-format maps withNside ¼ 512.

III. ANGULAR POWER SPECTRUMCALCULATION

We consider the angular power spectrum C‘ of an in-tensity map Iðc Þ where c denotes the sky direction. Theangular power spectrum is given by the coefficientsC‘ ¼ hja‘mj2i, with the a‘m determined by expanding themap in spherical harmonics,

Iðc Þ ¼ X‘m

a‘mY‘mðc Þ: (1)

The intensity angular power spectrum indicates the dimen-sionful size of intensity fluctuations and can be comparedwith predictions for source classes whose collective inten-sity is known or assumed (as in, e.g., [31]). The intensityangular power spectrum of a single source class is not ingeneral independent of energy due to the energy depen-dence of the mean map intensity hIi.

We can also construct the dimensionless fluctuationangular power spectrum by dividing the intensity angularpower spectrum C‘ of a map by the mean sky intensity(outside of the mask, for a masked sky map) squared hIi2.The fluctuation angular power spectrum characterizes theangular distribution of the emission independent of theintensity normalization. Its amplitude for a single sourceclass is the same in all energy bins if all members of thesource class share the same observed energy spectrum,since this results in the angular distribution of the collec-tive emission being independent of energy. Energy depen-dence in the fluctuation angular power due to variation ofthe energy spectra between individual members of thepopulation is discussed in Sec. VII.

A. Energy dependence

We calculate the angular power spectrum of the data andsimulated models in four energy bins. Using multipleenergy bins increases sensitivity to source populationsthat contribute significantly to the anisotropy in a limitedenergy range, and may also aid in the interpretation of ameasurement in terms of a detection of or constraints onspecific source populations [39,48]. In addition, the detec-

tion of an energy dependence in the fluctuation angularpower spectrum of the total emission (the anisotropy en-ergy spectrum) may be used to infer the presence ofmultiple contributing source classes [49]. In the case thata single source population dominates the anisotropy over agiven energy range, the energy dependence of the intensityangular power spectrum can indicate the energy spectrumof that contributor.Since the LAT’s angular resolution and the photon sta-

tistics depend strongly on energy, the sensitivity of theanalysis is also energy-dependent: at low energies theLAT’s PSF broadens, resulting in reduced sensitivity tosmall-scale anisotropies, while at high energies the mea-surement uncertainties are dominated by low statistics.We calculate angular power spectra in the energy bins1.04–1.99 GeV, 1.99–5.00 GeV, 5.00–10.4 GeV, and10.4–50.0 GeV. The map for each energy bin for theangular power spectrum analysis was created by summingthe corresponding maps produced in finely-gridded energybins, as described in Sec. II.

B. Angular power spectrum of a masked sky

The focus of this work is to search for anisotropies onsmall angular scales from unresolved source populations,hence the regions of the sky used in this analysis wereselected to minimize the contribution of the Galactic dif-fuse emission from cosmic-ray interactions and the emis-sion from known sources. A mask excluding Galacticlatitudes jbj< 30� and a 2� angular radius around eachsource in the 11-month Fermi LAT catalog (1FGL) [50]was applied prior to performing the angular power spec-trum calculations in all energy bins. The fraction of the skyoutside of this mask is fsky ¼ 0:325. The 2� angular radiusfor the source masking approximately corresponds to the95% containment angle for events at normal incidence at1 GeV (front/back average for P6_V3 IRFs); the contain-ment angle decreases with increasing energy. The effect ofthe mask on the angular power spectra is discussed belowand in Sec. VI F, and the impact of variations in the latitudecut is assessed in Sec. VI E. An all-sky intensity map of thedata in each energy bin is shown in Fig. 1, both with andwithout applying the default mask.The angular power spectra of the masked maps were

calculated using HEALPIX, after first removing the mono-pole and dipole terms. To approximately correct for thepower suppression due to masking, the raw angular powerspectra output by HEALPIX were divided by the fraction ofthe sky outside the mask, fsky. This correction is valid at

multipoles greater than�10, where the power spectrum ofthe signal varies much more slowly than the windowfunction, as detailed below.When a fraction of the sky is masked, the measured

spherical harmonics coefficients are related to the true,underlying spherical harmonics coefficients, atrue‘m , via amatrix multiplication, a‘m ¼

P‘0m0a

true‘0m0W‘‘0mm0 , where


083007-5

FIG. 1 (color online). All-sky intensity maps of the data in the four energy bins used in this analysis, in Galactic coordinates; the mapprojection is Mollweide. The data shown are the average of the maps of the front- and back-converting events, and are shownunmasked (left panels) and with the default mask applied (right panels). The mask excludes Galactic latitudes jbj< 30� and a 2�angular radius around each source in the 1FGL catalog. The map images shown have been downgraded in resolution to Nside ¼ 128 toimprove the visual quality of the images; however, the analysis was performed on the higher resolution maps as described in the text.


083007-6

DATA (P6_V3 diffuse), 1.0-2.0 GeV DATA (P6_V3 diffuse), 1.0-2.0 GeV

DATA (P6_V3 diffuse), 2.0-5.0 GeV DATA (P6_v3 diffuse), 2.0-5.0 GeV



W‘‘0mm0 is the so-called coupling matrix given byW‘‘0mm0 �

R�obs

d2n̂Y�‘mðn̂ÞY‘0m0 ðn̂Þ, where the integral isdone only on the unmasked sky whose solid angle is �obs.Then, HEALPIX returns a raw angular power spectrum,Craw‘ ¼ ð2‘þ 1Þ�1

Pmja‘mj2, whose ensemble average is

related to the true power spectrum, Ctrue‘ , as

hCraw‘ i ¼1

2‘þ 1X‘0Ctrue‘0

Xmm0

jW‘‘0mm0 j2: (2)

Now, for a given mask, one may calculateP

mm0 jW‘‘0mm0 j2and estimateCtrue‘ by inverting this equation. This approachis called the MASTER algorithm [51], and has been shownto yield unbiased estimates of Ctrue‘ . However, while this isan unbiased estimator, it is not necessarily a minimum-variance one. In particular, when the coupling matrix isnearly singular because of, e.g., an excessive amount ofmask or a complex morphology of mask, this estimatoramplifies noise. We observed this amplification of noisewhen applying the MASTER algorithm to our data set.Therefore, we decided to use an approximate, but lessnoisy alternative. It is easy to show (see, e.g., Eq. (A3) of[52]) that, when

Pmm0 jW‘‘0mm0 j2 peaks sharply at ‘ ¼ ‘0

and Ctrue‘ varies much more slowly than the width of thispeak, the above equation can be approximated as

hCraw‘ i Ctrue‘�obs4�

¼ Ctrue‘ fsky: (3)

This approximation eliminates the need for a matrix inver-sion. We have verified that this method yields an unbiasedresult with substantially smaller noise than the MASTERalgorithm at ‘ > 10. We adopt this method throughout thispaper.

C. Window functions

The angular power spectrum calculated from a map isaffected by the PSF of the instrument and the pixelizationof the map, encoded in the beam window function Wbeam‘and the pixel window function Wpix‘ respectively, both ofwhich can lead to a multipole-dependent suppression ofangular power that becomes stronger at larger multipoles.Depending upon whether the power spectrum originatesfrom signal or noise, corrections for the beam and pixelwindow functions must be applied to the measurementdifferently. For our application, we must not apply anycorrections to the photon shot noise (Poisson noise) term,while we must apply both the beam and pixel windowfunction corrections to the signal term from, e.g., unre-solved sources. While it is obvious why one must not applythe beam window correction to the photon noise term, itmay not be so obvious why one must also not apply thepixel window correction to that same term. In fact, thisstatement is correct only for the shot noise, if the data arepixelized by the nearest-grid assignment (which we haveadopted for our pixelizing scheme). This has been shown

by Ref. [53] (see Eq. [20] of that work) for a three-dimensional density field, but the same is true for a two-dimensional field, as we are dealing with here. We haveverified this using numerical simulations.In this paper, although we use maps at Nside ¼ 512 (for

which the maximummultipole is ‘max ¼ 1024), we restrictthe analysis to C‘ up to ‘max � 500 where we have areasonable signal-to-noise ratio. For these multipoles theeffect of the pixel window function is negligible, and thuswe shall simplify our analysis pipeline by not applying thepixel window correction to the observed power spectrum[54]. Therefore, our signal power spectrum estimator isgiven by

Csignal‘ ¼

Craw‘ =fsky � CNðWbeam‘ Þ2

; (4)

where CN ¼ hN�;pixih1=A2pixi=�pix is the photon noiseterm, with N�;pix, Apix, and �pix the number of observed

events, the exposure, and the solid angle, respectively, ofeach pixel, and the averaging is done over the unmaskedpixels. We approximate the photon noise term by CN ¼hIi24�fsky=N�, with N� denoting the total number ofobserved events outside the mask. This approximation isaccurate at the percent level. Note that while Craw‘ is alwaysnon-negative, it is possible for our estimator for the signal

power spectrum Csignal‘ to be negative due to the subtraction

of the noise term.The beam window function in multipole space associ-

ated with the full non-Gaussian PSF is given by

Wbeam‘ ðEÞ ¼ 2�Z 1�1

d cos�P‘ðcosð�ÞÞPSFð�;EÞ; (5)

where P‘ðcosð�ÞÞ are the Legendre polynomials andPSFð�; EÞ is the energy-dependent PSF for a given set ofIRFs, with � denoting the angular distance in the distribu-tion function. The PSF used corresponds to the average forthe actual pointing and live time history of the LAT andover the off-axis angle, as given by the gtpsf tool. Wecalculate the beam window functions for both the front-and back-converting events.The PSF of the LAT, and consequently the beamwindow

function, varies substantially over the energy range used inthis analysis, and also non-negligibly within each energybin. We treated this energy dependence by calculating anaverage window function hWbeam‘ ðEiÞi for each energy binEi, weighted by the intensity spectrum of the events in eachbin,

hWbeam‘ ðEiÞi ¼1

Ibin

Z Emax;iEmin;i

dEWbeam‘ ðEÞdN

dE; (6)

where Ibin �REmax;iEmin;i

dEðdN=dEÞ and Emin;i and Emax;i arethe lower and upper edges of each energy bin. The differ-ential intensity dN=dE outside the mask in each map for


083007-7

the finely-gridded energy bins described in Sec. II was usedto approximate the energy spectrum for this calculation.

D. Measurement uncertainties

The 1� statistical uncertainty �C‘ on the measured

angular power spectrum coefficients Csignal‘ is given by [55]

�C‘ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

ð2‘þ 1Þfsky�‘s �

Csignal‘ þ

CNðWbeam‘ Þ2

�; (7)

where �‘ is the width of the multipole bin (for binneddata).

After implementing the corrections for masking and forthe beam window function to estimate the signal angular

power spectrum via Eq. (4), the coefficients Csignal‘ were

binned in multipole with �‘ ¼ 50 and averaged in eachmultipole bin, weighted by the measurement uncertainties,

hC‘i ¼P‘

C‘=�2C‘P

‘

1=�2C‘; (8)

with C‘ ¼ Csignal‘ as calculated by Eq. (4) and �C‘ given byEq. (7) with �‘ ¼ 1 and Wbeam‘ ¼ hWbeam‘ ðEiÞi for thecorresponding energy bin Ei. As expected, we find thatthe statistical measurement uncertainties calculated at thelinear center of each multipole bin via Eq. (7) with �‘ ¼50 agree well with the scatter within each multipole bin.The value ofC‘ at multipoles 2 ‘ 4was found in mostcases to be anomalously large [56], indicating the presenceof strong correlations on very large angular scales, such asthose that could be induced by the shape of the mask andby contamination from Galactic diffuse emission. To avoid

biasing the value of the average Csignal‘ in the first bin by the

values at these low multipoles, the multipole bins begin at‘ ¼ 5.

Finally, the angular power spectra of the front- and back-converting events were combined by weighted averaging,weighting by the measurement uncertainty on each datapoint. Because of the larger PSF associated with back-converting events, the measurement errors on the angularpower spectra of the back-converting data set tend to belarger than those of the front-converting data set, particu-larly at low energies and high multipoles where the sup-pression of the raw angular power due to the beam windowfunction is much stronger for the back-converting data set.The difference between the measurement uncertaintiesassociated with the front and back data sets is less promi-nent at higher energies.

IV. EVENT-SHUFFLING TECHNIQUE

Oneway to search for anisotropies is to first calculate theflux of particles from each direction in the sky (equal to thenumber of detected events from some direction divided by

the exposure in the same direction), and then examine itsdirectional distribution. The flux calculation, which re-quires knowledge of the exposure, depends on the effectivearea of the detector and the accumulated observation livetime.The effective area, calculated from a Monte Carlo simu-

lation of the instrument, could suffer from systematicerrors, such as miscalculations of the dependence of theeffective area on the instrument coordinates (off-axis angleand azimuthal angle). Naturally, any systematic errorsinvolved in the calculation of the exposure will propagateto the flux, possibly affecting its directional distribution. Ifthe magnitude of these systematic errors is comparable toor larger than the statistical power of the available data set,their effects on the angular distribution of the flux mightmasquerade as a real detectable anisotropy. For this reason,we cross-check our results using an alternative method toconstruct an exposure map that does not rely on the Monte-Carlo-based calculation of the exposure implemented inthe SCIENCE TOOLS.The starting point of this method is the construction of a

sky map that shows how an isotropic sky would look asseen by the Fermi LAT. This sky map, hereafter called the‘‘no-anisotropy sky map,’’ is directly proportional to theexposure map.One method of generating a no-anisotropy map is to

randomize the reconstructed directions of the detectedevents (as in [43]). In the case that the angular distributionof the flux is perfectly isotropic, a time-independent inten-sity should be detected when looking in any given detectordirection. Possible time variation of the intensity would bedue only to changes in the operating conditions of theinstrument. A set of isotropic events can be built by ran-domly coupling the times and the directions of real eventsin local instrument coordinates. The randomization in thisanalysis was performed by exchanging the direction of agiven real event in the LAT frame with the direction ofanother event selected randomly from the data set withuniform probability. Using this information, the sky direc-tion is reevaluated for the two events. By construction, therandomized data set preserves the exposure, the energy andangular (with respect to the LAT reference frame) distri-butions, and also accounts for the detector dead times.As already discussed in Sec. II, for this analysis a cut of

52� on the rocking angle was applied to limit possiblephoton contamination from the Earth’s albedo. For theshuffling technique, the analysis was performed with areduced field of view of the instrument, namely, the eventsused were selected to have an off-axis angle less than 50�.In this way, events with zenith angle exceeding 102� wereremoved. This selection cut avoids introducing asymme-tries in the exposure across the field of view due to cuttingevents based on zenith angle.The randomization was performed using the masked sky

map described in Sec. III, so that only real events with sky


083007-8

coordinates outside the masks were used, and the reeval-uated sky direction for each event was required to be in theunmasked region of the sky. This randomization processwas repeated 20 000 times, separately for the front- andback-converting events, each time producing a shuffled skymap that is compatible with an isotropic source distribu-tion. The final no-anisotropy sky map for each energy binwas produced by taking the average of these 20 000shuffled sky maps. For the available event statistics, aver-aging 20 000 shuffled maps was reasonably effective atreducing the Poisson noise associated with the averagenumber of events per pixel. To reduce the number ofrequired shuffled maps by increasing the average numberof events per pixel, the shuffled maps were constructed atslightly lower resolution (Nside ¼ 256) than was used in thedefault analysis. When analyzing the anisotropy with theseexposure maps from the shuffling technique, count maps atNside ¼ 256 were used to construct the intensity maps. Ano-anisotropy sky map is shown in Fig. 2. This sky mapdoes not appear entirely uniform because the sky was notobserved with uniform exposure.

Although the no-anisotropy sky map is directly propor-tional to the exposure map, this method does not allow usto determine the absolute level of the exposure. We there-fore constructed intensity maps (with arbitrary normaliza-tion) by dividing the real data counts maps in each energybin by the no-anisotropy map for that energy bin, after firstsmoothing the no-anisotropy map with a Gaussian beamwith � ¼ 1� to reduce the pixel-to-pixel fluctuations dueto the finite number of events available to use in therandomization. This smoothing beam size removes noisein the no-anisotropy sky map above ‘� 200, and waschosen because we focus our search for anisotropies inthat multipole range. Angular power spectra were then

calculated from these intensity maps as in Sec. III.Because of the arbitrary normalization of these intensitymaps, we calculate only fluctuation angular power spectraof the data when using the exposure map produced by thisshuffling technique.

V. SIMULATED MODELS

Detailed Monte Carlo simulations of Fermi LAT all-skyobservations were performed to provide a reference againstwhich to compare the results obtained for the real data set.The simulations were produced using the gtobssim tool,which simulates observations with the LAT of an inputsource model. The gtobssim tool generates simulated pho-ton events for an assumed spacecraft pointing and live-timehistory, and a given set of IRFs. The P6_V3_DIFFUSE IRFsand the actual spacecraft pointing and live-time historymatching the observational time interval of the data wereused to generate the simulated data sets.Two models of the gamma-ray sky were simulated. Each

model is the sum of three components:(1) GAL—A model of the Galactic diffuse emission.(2) CAT—The sources in the 11-month catalog (1FGL)

[50].(3) ISO—An isotropic background.Both models include the same CAT and ISO compo-

nents, and differ only in the choice of the model for theGAL component. GAL describes both the spatial distribu-tion and the energy spectrum of the Galactic diffuse emis-sion. The GAL component for the reference sky modelused in this analysis (hereafter, MODEL) is the recom-mended Galactic diffuse model for LAT data analysis,GLL_IEM_V02.FIT [57], which has an angular resolution of

0.5�. This model was used to obtain the 1FGL catalog; adetailed description can be found in Ref. [58].An alternate sky model (ALT MODEL) was simulated

for comparison, in order to test the possible impact ofvariations in the Galactic diffuse model. This model isinternal to the LAT collaboration, and was built using thesame method as GLL_IEM_V02.FIT, but differs primarily inthe following ways: (i) this model was constructed using21 months of Fermi LAT observations, whileGLL_IEM_V02.FIT was based on 9 months of data, and

(ii) additional large-scale structures, such as the Fermibubbles [59], are included in the model through the useof simple templates.The sources in CATwere simulated with energy spectra

approximated by single power laws, and with the locations,average integral fluxes, and photon spectral indices asreported in the 1FGL catalog. All 1451 sources wereincluded in the simulation. ISO represents the sumof the Fermi-measured IGRB and an additional isotropiccomponent presumably due to unrejected chargedparticles; for this component the spectrum templateISOTROPIC_IEM_V02.TXT was used.

FIG. 2 (color online). No-anisotropy sky map created by sum-ming 20 000 shuffled maps using front- and back-convertingevents with E > 1 GeV, binned into a HEALPIX map with Nside ¼256. The map projection is Hammer-Aitoff. The features in theno-anisotropy sky map result from the fact that the sky was notobserved with uniform exposure.


083007-9

For both the MODEL and the ALT MODEL, the sum ofthe three simulated components results in a description ofthe gamma-ray sky that closely approximates the angular-dependent intensity and energy spectrum of the all-skyemission measured by the Fermi LAT. Although the simu-lated models may not accurately reproduce some large-scale structures, e.g., Loop I [60] and the Fermi bubbles,these features are not expected to induce anisotropies onthe small angular scales on which we focus in this work.

VI. RESULTS

In this section we present the measured angular powerspectra of the data, followed by the results of validationstudies which examine the effect of variations in the de-fault analysis parameters, and by a comparison of theresults for the data with those for simulated models. Wesummarize the main results of the angular power spectrummeasurements of the data and of key validation studiesdescribed in this section in Table I.

Unless otherwise noted, the results are shown for dataand models with the angular power spectra calculated afterapplying the default source mask which excludes sourcesin the 1FGL catalog and Galactic latitudes jbj< 30�.Because of the arbitrary normalization of the intensitymaps calculated using the exposure map from shuffling,we show fluctuation angular power spectra for this data set.Intensity angular power spectra are presented for all otherdata sets.

In the figures we show our signal angular power spec-

trum estimator Csignal‘ [Eq. (4)], which represents the signal

after correcting for the power suppression due to masking,

subtracting the photon noise, and correcting for the beamwindow function. A measurement that is inconsistent withzero thus indicates the presence of signal angular power.

The Csignal‘ shown is the weighted average of this quantity

for the maps of front and back events. The fluctuation

angular power spectra Csignal‘ =hIi2 were calculated by di-

viding Csignal‘ of the front and back events by their respec-

tive hIi2, and then averaging the angular power spectra. Forconciseness, in the figure labels C‘ ¼ Craw‘ =fsky is the rawangular power spectrum output by HEALPIX corrected forthe effects of masking. The error bars on points indicate the1� statistical uncertainty in the measurement in each mul-tipole bin as calculated by Eq. (7) with �‘ ¼ 50 and withthe bins beginning at ‘ ¼ 5. The binned data points arelocated at the linear center of each multipole bin.

A. Angular power spectrum of the data

We now present the results of the angular power spec-trum analysis of the data. We measure the angular powerspectrum of the data after applying the default latitude cutand source mask, and refer to this as our default dataanalysis (DATA). We also measure the angular powerspectrum of the data using the same masking and analysispipeline after performing Galactic-foreground cleaning,described below, and refer to this as the cleaned dataanalysis (DATA:CLEANED). These two measurementsconstitute our main results for the data, and so we discussthe energy dependence of the measured angular power(Sec. VII) and present constraints on specific source pop-ulations (Sec. VIII) for the results of both the default andcleaned data analyses.

TABLE I. Best-fit values of the angular power CP and fluctuation angular power CP=hIi2 in each energy bin over the multipole range155 ‘ 504. Results are shown for the data processed with the default analysis pipeline, the foreground-cleaned data, the dataanalyzed with the 2FGL source mask, and the default simulated model. Significance indicates the measured angular power expressedin units of the measurement uncertainty �; the measurement uncertainties can be taken to be Gaussian.

Emin [GeV] Emax [GeV] CP [ðcm�2 s�1 sr�1Þ2 sr] Significance CP=hIi2 [10�6 sr] SignificanceDATA 1.04 1.99 7:39� 1:14� 10�18 6:5� 10:2� 1:6 6:5�

1.99 5.00 1:57� 0:22� 10�18 7:2� 8:35� 1:17 7:1�5.00 10.4 1:06� 0:26� 10�19 4:1� 9:83� 2:42 4:1�10.4 50.0 2:44� 0:92� 10�20 2:7� 8:00� 3:37 2:4�

DATA:CLEANED 1.04 1.99 4:62� 1:11� 10�18 4:2� 6:38� 1:53 4:2�1.99 5.00 1:30� 0:22� 10�18 6:0� 6:90� 1:16 5:9�5.00 10.4 8:45� 2:46� 10�20 3:4� 8:37� 2:41 3:5�10.4 50.0 2:11� 0:86� 10�20 2:4� 7:27� 3:36 2:2�

DATA:2FGL 1.04 1.99 5:18� 1:17� 10�18 4:4� 7:23� 1:61 4:5�1.99 5.00 1:21� 0:28� 10�18 5:3� 6:49� 1:22 5:3�5.00 10.4 8:38� 2:72� 10�20 3:1� 7:67� 2:54 3:0�10.4 50.0 8:00� 9:57� 10�21 0:8� 2:28� 3:52 0:6�

MODEL 1.04 1.99 1:89� 1:08� 10�18 0:7� 2:53� 1:47 1:7�1.99 5.00 1:92� 2:10� 10�19 0:9� 0:99� 1:12 0:9�5.00 10.4 3:41� 2:60� 10�20 1:3� 3:04� 2:34 1:3�10.4 50.0 0:62� 9:63� 10�21 0:1� 0:24� 3:02 0:1�


083007-10

Tominimize the impact of Galactic foregrounds we haveemployed a large latitude cut. However, Galactic diffuseemission extends to very high latitudes and may not exhibita strong gradient with latitude, and it is thus important toinvestigate to what extent our data set may be contaminatedby a residual Galactic contribution. For this purpose weattempt to reduce the Galactic diffuse contribution to thehigh-latitude emission by subtracting a model of theGalactic foregrounds from the data, and then calculatingthe angular power spectra of the residual maps. For theangular power spectrum analysis of the residual maps(cleaned data) we note that the noise term CN is calculatedfrom the original (uncleaned) map, since subtracting themodel from the data does not reduce the photon noise level.

In the following we use the recommended Galactic dif-fuse model GLL_IEM_V02.FIT, which is also the default GALmodel that we simulate, as described in Sec. V. To tailor themodel to the high-latitude sky regions considered in thiswork, the normalization of the model was adjusted by refit-ting the model to the data only in the regions outside thelatitude mask. For the fit we used GaRDiAn which con-volves the model with the instrument response (effective

area and PSF). The normalization obtained in this way is,however, very close to the nominal one,within a fewpercent.We present the angular power spectra of the data before

and after Galactic-foreground cleaning in Fig. 3; expandedversions of the angular power spectra for the 1–2 GeVand2–5 GeV bins focusing on the high-multipole data areshown in Fig. 4. In both analyses, angular power at ‘ �155 is measured in the data in all energy bins considered,and the angular power spectra for the default and cleaneddata are in good agreement in this multipole range. In thedefault data, the large increase in angular power at ‘ < 155in the two energy bins spanning 1–5 GeV is likely due tocontamination from the Galactic diffuse emission whichfeatures correlations on large angular scales, but may alsobe attributable in part to the effects of the source mask (seeSec. VI F).At ‘ � 155 the measured angular power does not exhibit

a clear scale dependence in any energy bin. The results offitting the unbinned signal angular power spectrum estima-tor for 155 ‘ 504 in each energy bin to a power lawCsignal‘ / ð‘=‘0Þn with ‘0 ¼ 155 are given in Table II for thedefault data analysis. In each energy bin, the angular power

100 200 300 400 500

Multipole l

−2 10−17

0

2 10−17

4 10−17

6 10−17

8 10−17

1 10−16

(Cl −

CN)/

W2

[(c

m−

2 s−

1 sr

−1 )

2 sr

]

1.0−2.0 GeV

DATA

DATA:CLEANED

100 200 300 400 500

Multipole l

−5.0 10−18

0

5.0 10−18

1.0 10−17

1.5 10−17

2.0 10−17

2.5 10−17

(Cl −

CN)/

W2

[(c

m−

2 s−

1 sr

−1 )

2 sr

]

2.0−5.0 GeV

DATA

DATA:CLEANED

100 200 300 400 500Multipole l

−2 10−19

0

2 10−19

4 10−19

6 10−19

(Cl −

CN)/

W2

[(c

m−

2 s−

1 sr

−1 )

2 sr

]

5.0−10.4 GeV

DATA

DATA:CLEANED

100 200 300 400 500Multipole l

−1.0 10−19

−5.0 10−20

0

5.0 10−20

1.0 10−19

1.5 10−19

(Cl −

CN)/

W2

[(c

m−

2 s−

1 sr

−1 )

2 sr

]

10.4−50.0 GeV

DATA

DATA:CLEANED

FIG. 3 (color online). Comparison of intensity angular power spectra of the data and Galactic-foreground-cleaned data. For ‘ � 155the measured power at all energies is approximately constant in multipole, suggesting that it originates from one or more unclusteredsource populations. The large increase in angular power in the default data at ‘ < 155 in the 1–2 and 2–5 GeV bins is likely attributablelargely to contamination from Galactic diffuse emission. In these two energy bins, foreground cleaning primarily reduces angularpower at ‘ < 155, with the most significant reductions at ‘ < 105. At energies greater than 5 GeV the effect of foreground cleaning issmall for ‘ � 55. Expanded versions of the top panels are shown in Fig. 4.


083007-11

spectrum for 155 ‘ 504 is consistent with a Poissonspectrum (constant in multipole, i.e., n ¼ 0 falls within the95% confidence level (CL) range of the best-fit power-lawindex), as expected for the angular power spectrum of oneor more unclustered source populations. However, weemphasize that the uncertainty in the scale dependence isappreciable, particularly for the 10–50 GeV bin.

In light of the scale independence of the angular powerat ‘ � 155, we associate the signal in this multipole rangewith a Poisson angular power spectrum and determine thebest-fit constant value of the angular power CP and thefluctuation angular power CP=hIi2 over 155 ‘ 504 ineach energy bin, by weighted averaging of the unbinnedmeasurements. These results for the default and cleaneddata are summarized in Table I, along with the resultsobtained for the data using an updated source catalog todefine the source mask and for a simulated model, whichwill be discussed in Sec. VIG and VIH respectively.

We note that the associated measurement uncertaintiescan be taken to be Gaussian, in which case the reportedsignificance quantifies the probability of the measuredangular power to have resulted by chance from a trulyuniform background.We consider a 3� or greater detectionof angular power (CP) in a single energy bin to be statis-tically significant. For the default data, the best-fit values ofCP indicate significant detections of angular power in the1–2, 2–5, and 5–10 GeV bins (6:5�, 7:2�, and 4:1�, re-spectively), while in the 10–50 GeV bin the best-fit CPrepresents a 2:7�measurement of angular power.We furthernote that the best-fit value of the fluctuation angular powerover all four energy bins (see Sec. VII) yields a detectionwith greater than 10� significance for the default data.

For the 1–2 GeVand 2–5 GeVenergy bands the cleaningprocedure results in a significant decrease in the angularpower at low multipoles (‘ < 105), and a smaller reductionat higher multipoles. However, the decrease is small for‘ � 155, and angular power is still measured at all ener-gies, at slightly lower significances (see Table I). Weemphasize that the detections in the three energy binsspanning 1–10 GeV remain statistically significant, and

the best-fit fluctuation angular power over all energy binsis detected at greater than 8� significance. For energiesabove 5 GeV the foreground cleaning does not stronglyaffect the measured angular power spectrum for ‘ � 55. Atall energies the decrease in angular power at low multi-poles can be attributed to the reduction of Galactic fore-grounds which feature strong correlations on large angularscales. We conclude that contamination of the data byGalactic diffuse emission does not have a substantial im-pact on our results at the multipoles of interest (‘ � 155).This conclusion is in agreement with that of Ref. [39],which found that the Galactic foregrounds have a rapidlydeclining angular power spectrum above ‘� 100.To further study the expected angular power spectrum of

Galactic foregrounds, we analyzed the angular power spec-trum of the E(B-V) emission map of Ref. [61] (hereafterSFD map), which is proportional to the column density ofthe interstellar dust, after masking jbj< 30� as in ourdefault analysis. The SFD map is a good tracer of theGalactic interstellar medium (ISM) away from theGalactic plane, the spatial structure of which should bereflected in the diffuse gamma-ray emission produced byinteractions of cosmic rays with the ISM. It has an angularresolution of 6 arcminutes, much smaller than the intrinsicresolution of the GAL model map (� 0:5�), and smallerthan the map resolution used in this study, and so itaccurately represents the small-scale structure of the ISMon the angular scales accessible to this analysis. We found

TABLE II. Multipole dependence of intensity angular powerin the data (default analysis) for 155 ‘ 504 in each energybin. The best-fit power-law index n in each energy bin is givenwith the associated �2 per degree of freedom (d.o.f.) of the fit.

Emin Emax n �2=d:o:f:

1.04 1.99 �1:33� 0:78 0.381.99 5.00 �0:07� 0:45 0.435.00 10.4 �0:79� 0:76 0.3710.4 50.0 �1:54� 1:15 0.39

100 200 300 400 500

Multipole l

−1 10−17

0

1 10−17

2 10−17

3 10−17

(Cl −

CN)/

W2

[(c

m−

2 s−

1 sr

−1 )

2 sr

]

1.0−2.0 GeV

DATA

DATA:CLEANED

100 200 300 400 500

Multipole l

−2 10−18

0

2 10−18

4 10−18

6 10−18

(Cl −

CN)/

W2

[(c

m−

2 s−

1 sr

−1 )

2 sr

]

2.0−5.0 GeV

DATA

DATA:CLEANED

FIG. 4 (color online). Expanded versions of top panels of Fig. 3, focusing on the high-multipole angular power.


083007-12

that the SFD map produces an angular power spectrumwith a slightly harder slope than the default GAL model,and consequently features more angular power at highmultipoles. However, like the GAL model, the SFD mapangular power spectrum falls off quickly with multipolecompared to a Poisson spectrum, and the amplitude of theSFD map angular power is below that measured in the datafor ‘ * 100. This further reinforces the conclusion thatGalactic foreground contamination cannot explain the ob-served high-multipole angular power in the data.

B. Validation with a simulated point source population

To ensure that our analysis procedure accurately recov-ers an input angular power spectrum, and, in particular, thatthe result is not biased by instrumental effects, we comparethe angular power spectrum calculated for a simulatedpoint source population with the theoretical predictionfor that population. It is straightforward to calculate theexpected angular power spectrum of unclustered pointsources, once a flux distribution function, dN=dS (in unitsof cm2 s sr�1), and a source detection flux threshold, Sc(in units of cm�2 s�1), are provided. The angular powerspectrum of an unclustered point source population is thePoisson component of the angular power CP, which takesthe same value at all multipoles and is given by

CP ¼Z Sc0

dSS2dN

dS: (9)

For our source population model we adopt the best-fitflux distribution for the high-latitude Fermi sources, re-ported in [10], which describes dN=dS with a brokenpower-law model:

dN

dS¼ AS��1 ; S � Sb;¼ AS��1þ�2b S��2 ; S < Sb; (10)

which contains four free parameters, A,�1,�2, and Sb. Forthis form of dN=dS, the source power spectrum can befound analytically (for Sc > Sb):

CP ¼ A S3��1c

3� �1�1� �1 � �2

3� �2�SbSc

�3��1�

: (11)

A fit for the simulated source population for 1.04–10.4 GeV yields A ¼ ð1:90� 0:48Þ � 10�13ð180=�Þ2,�1 ¼ 2:213� 0:073, �2 ¼ 1:533� 0:007, and Sb ¼1:41� 10�9 cm�2 s�1. Note that the errors are correlated.Figure 5 shows the predicted CP for this source model forthreshold fluxes in the range Sc ¼ 0:8� 10�7 � 4:2�10�7 cm�2 s�1. The error bars are calculated from thefull covariance matrix of the above parameters. Althoughwe have used zero as the lower limit of the integral inEq. (9), using the actual lower limit of the flux distributionadopted for the simulated population results in a negligibledifference in the predicted CP.

We simulated this source population model with gtobs-sim using the same procedure as described in Sec. V. Thesimulated population comprises nearly 20 000 pointsources distributed randomly across the entire sky, witheach source’s flux drawn from the flux distribution speci-fied above. The photon spectrum of each source is modeledas a power law with a spectral index � (dN=dE / E��)drawn from a Gaussian distribution with mean of 2.40 anda standard deviation of 0.28. The simulated events wereprocessed and the angular power spectrum of this sourcemodel calculated using the same procedure as was used forthe data and other simulations in this study, except that theenergy range of the map was chosen to be 1.04–10.4 GeV,and no mask was applied.The fluxes of the�20 000 simulated sources were drawn

from a flux distribution in which the maximum possibleflux (E> 100 MeV) that could be assigned to a source was10�5 cm�2 s�1, however the maximum flux of any sourcein the simulation, which represents a single realization ofthis source population, was�3� 10�6 cm�2 s�1. We takethese values as the upper and lower bound on the sourcedetection threshold flux (E> 100 MeV) corresponding tothe simulated model, since we do not impose a sourcedetection threshold by masking or otherwise excludingsimulated sources above a specific threshold flux. Aspectral index � ¼ 2:4 is assumed to determine the thresh-old fluxes in the 1.04–10.4 GeV energy band. From thesethreshold fluxes we calculate the corresponding upper andlower bound on the predicted CP in the 1.04–10.4 GeVenergy band.The angular power spectrum for the simulated source

population calculated via the analysis pipeline used in thisstudy is presented in Fig. 6, with the shaded region indicat-ing the predicted range of CP (the mean values of CP at theupper and lower flux threshold); for a given model CP isindependent of multipole, thus we expect the recoveredangular power spectrum to be independent of multipole

1.0 1.5 2.0 2.5 3.0 3.5 4.0

Threshold Flux, Sc [10−7 cm−2 s−1]

2

4

6

8

10

CP

(1.0

4 −

10.

4 G

eV)

[(10

−15

cm

−2

s−1

sr−

1 )2

sr]

FIG. 5. Predicted amplitude of the source angular power spec-trum, CP [see Eq. (9)], for energies of 1.04–10.4 GeV as afunction of a source detection threshold flux, Sc.


083007-13

with amplitude within the shaded region. The angularpower spectrum recovered from the simulated data is inexcellent agreement with the prediction up to multipoles of‘� 800. Above ‘� 800, the upturn in the measured an-gular power spectrum is likely due to inaccuracies in themodeling of the beam window function, which can intro-duce features on very small angular scales. In the remain-der of this study, we present results only for the multipolerange ‘ ¼ 5 to ‘ ¼ 504.

C. Sensitivity to the exposure map calculation

To investigate the possibility that potential inaccuraciesin the exposure map calculation for the default analysismight generate spurious anisotropy in the intensity maps,we compare the fluctuation angular power spectra of thedata using our default analysis pipeline with the resultsobtained after replacing the default exposure map with thatgenerated by the event-shuffling technique described inSec. IV. This is shown in Figs. 7 and 8. In these two figuresonly, the results from the default data analysis were ob-tained from maps of HEALPIX resolution Nside ¼ 256 to

0 200 400 600 800 1000Multipole l

0

2 10−15

4 10−15

6 10−15

8 10−15

1 10−14

(Cl −

CN)/

W2

[(c

m−

2 s−

1 sr

−1 )

2 sr

]

1.0−10.4 GeV

FIG. 6 (color online). Intensity angular power spectrum ofa simulated observation of the source population model,compared with the theoretical prediction (shaded band). Theangular power spectrum of the simulated population is inexcellent agreement with the prediction over a large multipolerange.

0 100 200 300 400 500

Multipole l

0

5 10−5

1 10−4

(Cl −

CN)

/ (〈I

〉2 W2 )

[sr

]

1.0−2.0 GeV

DATA

DATA: SHUFF EXP

0 100 200 300 400 500

Multipole l

−2.0 10−5

0

2.0 10−5

4.0 10−5

6.0 10−5

8.0 10−5

1.0 10−4

1.2 10−4

(Cl −

CN)

/ (〈I

〉2 W2 )

[sr

]

2.0−5.0 GeV

DATA

DATA: SHUFF EXP

0 100 200 300 400 500

Multipole l

−2 10−5

0

2 10−5

4 10−5

6 10−5

(Cl −

CN)

/ (〈I

〉2 W2 )

[sr

]

5.0−10.4 GeV

DATA

DATA: SHUFF EXP

0 100 200 300 400 500

Multipole l

−4 10−5

−2 10−5

0

2 10−5

4 10−5

6 10−5

8 10−5

(Cl −

CN)

/ (〈I

〉2 W2 )

[sr

]

10.4−50.0 GeV

DATA

DATA: SHUFF EXP

FIG. 7 (color online). Fluctuation angular power spectra C‘=hIi2 calculated using the default analysis pipeline compared with thoseobtained using the exposure map from the event shuffling technique described in Sec. IV. Angular power is measured in all four energybins by both analysis methods. The lack of significant differences at the multipoles of interest between the angular power spectrayielded by the two methods demonstrates that any inaccuracies in the exposure map have a negligible impact on the measured angularpower spectra. Expanded versions of the top panels are shown in Fig. 8.


083007-14

match the resolution of the maps using the exposure de-termined from the shuffling technique. All other resultspresented in this study were obtained from Nside ¼ 512maps. Because of the reduced map resolution, the pixelwindow function has a small effect on the angular powerspectra shown in Figs. 7 and 8, however it affects theresults of the default analysis and the analysis using theshuffled exposure map in the sameway, and so these resultscan still be directly compared for the purpose of checkingthe effect of the exposure map calculation.

The results of the two analysis methods are in goodagreement at all energies and multipoles considered, ex-cept for slight deviations at ‘ < 55 for 1–5 GeV. Wecaution that at these low multipoles the measured angularpower spectra may be strongly affected by the mask, whichhas features on large angular scales. The slight differencesin the data selection cuts for the analysis using the expo-sure map from the shuffling technique compared to thosefor the default data analysis could lead to the observed

differences in the low-multipole angular power spectra.The differences could also result from systematics in theMonte-Carlo-based exposure calculation implemented inthe SCIENCE TOOLS, leading to inaccuracies in the exposuremap which vary on large angular scales. As we do notfocus on the low-multipole angular power in this study, wedefer a full investigation of this issue to future work. Theagreement at ‘ � 55 demonstrates that any potentialspatially-dependent inaccuracies in the SCIENCE TOOLSexposure calculation have a negligible impact on the an-gular power spectra in the multipole range of interest. Inparticular, from the consistency of the two methods weconclude that using the Monte-Carlo-based exposure cal-culation does not induce spurious signal anisotropy in ourresults.

D. Dependence on the PSF model

We examine the impact of variations in the assumed PSFon the results of the analysis by comparing the beam

0 100 200 300 400 500Multipole l

−2 10−5

−1 10−5

0

1 10−5

2 10−5

3 10−5

4 10−5

(Cl −

CN)

/ (〈I

〉2 W2 )

[sr

]

1.0−2.0 GeV

DATA

DATA: SHUFF EXP

0 100 200 300 400 500Multipole l

−1 10−5

0

1 10−5

2 10−5

3 10−5

(Cl −

CN)

/ (〈I

〉2 W2 )

[sr

]

2.0−5.0 GeV

DATA

DATA: SHUFF EXP

FIG. 8 (color online). Expanded versions of top panels of Fig. 7.

0 100 200 300 400 500Multipole l

0.0

0.2

0.4

0.6

0.8

1.0

Wl2

P6_V3_FRONTP6_V11_FRONT

10−50 GeV5−10 GeV2−5 GeV1−2 GeV

0 100 200 300 400 500Multipole l

0.0

0.2

0.4

0.6

0.8

1.0

Wl2

P6_V3_BACKP6_V11_BACK

FIG. 9 (color online). Comparison of the beam window functions for the P6_V3 and P6_V11 IRFs; the P6_V3 IRFs are the default usedin this analysis. The quantity W2‘ , which is the factor by which the angular power is suppressed due to the finite angular resolution ofthe instrument, is shown for the front-converting (left panel) and back-converting (right panel) events, evaluated at the log-center ofeach energy bin used in this analysis. The differences between the W2‘ of these two IRFs are small ( & few percent) at all energiesconsidered, indicating that our results are insensitive to the differences between the PSF models implemented in these IRFs.


083007-15

window functions (Eq. (5)) for the PSF implemented in theP6_V3 IRFs used in this analysis to those for the PSF in the

more recently updated P6_V11 IRFs. The P6_V11 IRFs use amodified functional form for the PSF, and for energiesabove 1 GeV the PSF implemented in P6_V11 was cali-brated using in-flight data, while in P6_V3 the PSF wasbased on Monte Carlo simulations. Figure 9 shows thebeam window functions for the PSF associated with thefront- and back-converting events for each set of IRFs, atthe log-center of each energy bin used in this analysis. Thesmall variation between the window functions of the twoIRFs confirms that differences between the PSF models inthese two IRFs are not large enough to affect the anisotropymeasurement on the angular scales to which this analysis issensitive.

E. Dependence on the latitude mask

In this analysis we apply a generous latitude mask toreduce contamination of the data by Galactic diffuse emis-sion. The mask is intended to remove enough contamina-

tion so that the measured angular power can be attributed tosources whose distribution is statistically isotropic in thesky region we consider, i.e., a distribution which does notshow any preferred direction on the sky. In particular, wewish to exclude sources whose angular distribution exhib-its a strong gradient with Galactic latitude. The effective-ness of the mask at reducing the contribution to the angularpower from a strongly latitude-dependent component canbe evaluated by considering the angular power spectrum ofthe data as a function of latitude cut. The results are shownin Figs. 10 and 11.At low multipoles (‘ & 100), increasing the latitude cut

significantly reduces the angular power, indicating that inthis multipole range the contamination by a stronglylatitude-dependent component, such as Galactic diffuseemission, is considerable. For 155 ‘ 254 at 1–2 GeVand 2–5 GeV, the angular power measured using the 30�latitude mask is noticeably smaller than when using the 20�latitude mask. However, at all energies there are no signifi-cant differences in the angular power measured for ‘ � 155

0 100 200 300 400 500Multipole l

−2 10−17

0

2 10−17

4 10−17

6 10−17

8 10−17

1 10−16

(Cl −

CN)/

W2

[(c

m−

2 s−

1 sr

−1 )

2 sr

]

1.0−2.0 GeV

|b| > 40 deg

|b| > 30 deg

|b| > 20 deg

0 100 200 300 400 500Multipole l

−1.0 10−17

−5.0 10−18

0

5.0 10−18

1.0 10−17

1.5 10−17

2.0 10−17

2.5 10−17(C

l − C

N)/

W2

[(c

m−

2 s−

1 sr

−1 )

2 sr

]

2.0−5.0 GeV

|b| > 40 deg

|b| > 30 deg

|b| > 20 deg

0 100 200 300 400 500Multipole l

−2 10−19

0

2 10−19

4 10−19

6 10−19

(Cl −

CN)/

W2

[(c

m−

2 s−

1 sr

−1 )

2 sr

]

5.0−10.4 GeV

|b| > 40 deg

|b| > 30 deg

|b| > 20 deg

0 100 200 300 400 500Multipole l

−1.0 10−19

−5.0 10−20

0

5.0 10−20

1.0 10−19

1.5 10−19

(Cl −

CN)/

W2

[(c

m−

2 s−

1 sr

−1 )

2 sr

]

10.4−50.0 GeV

|b| > 40 deg

|b| > 30 deg

|b| > 20 deg

FIG. 10 (color online). Intensity angular power spectra of the data calculated with different latitude cuts. The point source mask wasapplied in addition to the latitude mask in all cases. The differences between the results masking jbj< 30� (the default latitude cut)and jbj< 40� are small for ‘ � 155 for all four energy bins, demonstrating that the power observed in the data at these multipoles isnot strongly correlated with a component that has a strong latitude dependence in the range 30� < jbj< 40�, such as the Galacticdiffuse emission. At energies above 5 GeV convergence is seen for multipoles ‘ � 155 even when masking only jbj< 20�. Pointsfrom different data sets are offset slightly in multipole for clarity. The lowest multipole data point for the jbj< 20� mask in each panelis above the range shown in the figure. Expanded versions of the top panels are shown in Fig. 11.


083007-16

using the 30� and 40� latitude masks, and for energiesgreater than 5 GeV the 20� latitude mask also yields con-sistent results. We conclude that applying the 30� latitudemask is sufficient to ensure that no significant amount of themeasured angular power at ‘ � 155 originates from theGalactic diffuse emission or from any source class thatvaries greatly in the region 30� < jbj< 40�.

F. Effects of masking on the power spectrum

To verify that the results do not depend sensitively on theangular radius of the source mask, in Figs. 12 and 13 wecompare the results when masking a 1� angular radiusaround each source with those when masking the 2� radiusused as the default in this work.

0 100 200 300 400 500Multipole l

−1 10−17

0

1 10−17

2 10−17

3 10−17

(Cl −

CN)/

W2

[(c

m−

2 s−

1 sr

−1 )

2 sr

]

1.0−2.0 GeV

|b| > 40 deg

|b| > 30 deg

|b| > 20 deg

0 100 200 300 400 500Multipole l

−2 10−18

0

2 10−18

4 10−18

6 10−18

(Cl −

CN)/

W2

[(c

m−

2 s−

1 sr

−1 )

2 sr

]

2.0−5.0 GeV

|b| > 40 deg

|b| > 30 deg

|b| > 20 deg


0 100 200 300 400 500

Multipole l

−2 10−17

0

2 10−17

4 10−17

6 10−17

8 10−17

1 10−16

(Cl −

CN)/

W2

[(c

m−

2 s−

1 sr

−1 )

2 sr

]

1.0−2.0 GeV

DATA 2DEG

DATA 1DEG

0 100 200 300 400 500

Multipole l

−1.0 10−17

−5.0 10−18

0

5.0 10−18

1.0 10−17

1.5 10−17

2.0 10−17

2.5 10−17

(Cl −

CN)/

W2

[(c

m−

2 s−

1 sr

−1 )

2 sr

]

2.0−5.0 GeV

DATA 2DEG

DATA 1DEG

0 100 200 300 400 500

Multipole l

−2 10−19

0

2 10−19

4 10−19

6 10−19

(Cl −

CN)/

W2

[(c

m−

2 s−

1 sr

−1 )

2 sr

]

5.0−10.4 GeV

DATA 2DEG

DATA 1DEG

0 100 200 300 400 500

Multipole l

−1.0 10−19

−5.0 10−20

0

5.0 10−20

1.0 10−19

1.5 10−19

(Cl −

CN)/

W2

[(c

m−

2 s−

1 sr

−1 )

2 sr

]

10.4−50.0 GeV

DATA 2DEG

DATA 1DEG

FIG. 12 (color online). Intensity angular power spectra of the data calculated with a mask excluding a 1� or 2� angular radius aroundeach source; excluding a 2� angular radius is the default in this analysis. The default latitude mask excluding jbj< 30� was applied inaddition to the source mask in all cases. At all energies the angular power spectra obtained using the different source mask radii areconsistent at ‘ � 155 (the multipole range of interest), and above 2 GeV the results are consistent at ‘ � 55. Expanded versions of thetop panels are shown in Fig. 13.


083007-17

In the 1–2 GeV energy bin the results show significantdifferences at ‘ < 155, however for ‘ � 155 (the multipolerange of interest) the angular power spectra for the 1� and2� source mask cases agree within the error bars. In thehigher energy bins the angular power spectra in all exceptthe first multipole bin (5 ‘ < 55, well below the range ofinterest) agree within the error bars. Since varying theangular size of the region masked around each sourcedoes not significantly change the measured angular powerat ‘ � 155, we conclude that any features that may beinduced in the angular power spectra by the morphology ofthe source mask are confined to low multipoles and there-fore do not affect the measurements of CP reported in thiswork.

In addition, we have confirmed that the angular powerspectra of the front- and back-converting events are in goodagreement within each energy bin in the multipole range ofinterest (‘ � 155), and are generally consistent at ‘ < 155even in the 1–2 GeV energy bin where the 95% contain-ment radius of the PSF of the back-converting events iscomparable to the angular radius used for the source mask.Consequently, although the PSF associated with the back-converting events is larger than that of the front-convertingevents, the consistency of their angular power spectraimplies that the source masking is sufficiently effectiveeven at low energies.

The sharp latitude cut used in this analysis also has thepotential to induce features in the angular power spectrum,although these would be expected to appear on the largeangular scales characteristic of the morphology of themask. We therefore note that the stability of the angularpower spectra at ‘ � 155 for latitude cuts masking at leastjbj 30�, discussed in Sec. VI E and demonstrated inFigs. 10 and 11, indicates that the latitude mask does notinduce features in the power spectrum at the angular scalesof interest.

The analysis of the simulated isotropic component, pre-sented in Sec. VIH, provides another means of assessingthe impact of the mask on the angular power spectra. Sincethe isotropic component should only contribute to the

monopole (‘ ¼ 0) term of the power spectrum, statisticallysignificant deviations from zero power at ‘ > 0 can beattributed to the use of the mask. We emphasize that theconsistency of the angular power of the isotropic compo-nent with zero at ‘ � 155 indicates that, despite the com-plex morphology of the total mask, the mask does notinduce features in the angular power spectrum at the multi-poles of interest (‘ � 155).

G. Dependence on the set of masked sources

The recently-released second Fermi LAT source catalog(2FGL) [62] is an update to the 1FGL catalog used todefine the default source mask adopted in this work. The2FGL catalog reports the detection of 1873 sources, com-pared to the 1451 included in the 1FGL catalog.We briefly comment that one motivation for using the

1FGL catalog, rather than the 2FGL catalog, to define thesource mask in our default analysis is that the 1FGLcatalog was also used in the Fermi LAT source countdistribution analysis [10]. The results of that study areclosely related to the interpretation of the results of thecurrent analysis, and so our choice to mask that samesource list in our default analysis allows the results of thetwo analyses to be used together straightforwardly.However, it is natural to ask to what extent the measuredangular power reported in the data may be attributable tothe additional sources resolved in the 2FGL catalog.We address this question by analyzing the data using a

source mask defined by the 2FGL sources and comparingthe results to those obtained using the 1FGL source mask.We repeat the analysis of the data using the default pipe-line, changing only the source mask; the total mask isdefined by the source mask combined with the defaultlatitude cut masking jbj< 30�. When combined with thedefault latitude cut, the 2FGL source mask results in anunmasked sky fraction fsky ¼ 0:295, a small decreasecompared to fsky ¼ 0:325 when using the 1FGL sourcemask.The angular power spectra of the data analyzed using the

2FGL catalog to define the source mask are shown in

0 100 200 300 400 500

Multipole l

−1 10−17

0

1 10−17

2 10−17

3 10−17

(Cl −

CN)/

W2

[(c

m−

2 s−

1 sr

−1 )

2 sr

]

1.0−2.0 GeV

DATA 2DEG

DATA 1DEG

0 100 200 300 400 500

Multipole l

−2 10−18

0

2 10−18

4 10−18

6 10−18

(Cl −

CN)/

W2

[(c

m−

2 s−

1 sr

−1 )

2 sr

]

2.0−5.0 GeV

DATA 2DEG

DATA 1DEG



083007-18

Figs. 14 and 15, compared with the results of the defaultdata analysis which uses the 1FGL catalog. The angularpower CP measured in the data using the 2FGL sourcemask is reduced relative to the 1FGL case (see Table I),while the measurement uncertainties remain roughly thesame as in the 1FGL case. The decrease in CP is�20–30%in the 1–2, 2–5, and 5–10 GeV energy bins, however

significant detections (> 3�) are still found in these threebins. A� 70% decrease in CP is seen in the 10–50 GeVbin, and due to the large measurement uncertainty thesignificance of the measurement in this bin falls from2:7� to 0:8�. The significance of the detected fluctuationangular power over all four energy bins remains greaterthan 7�.

0 100 200 300 400 500Multipole l

−1 10−17

0

1 10−17

2 10−17

3 10−17

(Cl −

CN)/

W2

[(c

m−

2 s−

1 sr

−1 )

2 sr

]

1.0−2.0 GeV

DATA 1FGL

DATA 2FGL

0 100 200 300 400 500Multipole l

−2 10−18

0

2 10−18

4 10−18

6 10−18

(Cl −

CN)/

W2

[(c

m−

2 s−

1 sr

−1 )

2 sr

]

2.0−5.0 GeV

DATA 1FGL

DATA 2FGL


0 100 200 300 400 500

Multipole l

−2 10−17

0

2 10−17

4 10−17

6 10−17

8 10−17

1 10−16

(Cl −

CN)/

W2

[(c

m−

2 s−

1 sr

−1 )

2 sr

]

1.0−2.0 GeV

DATA 1FGL

DATA 2FGL

0 100 200 300 400 500

Multipole l

−1.0 10−17

−5.0 10−18

0

5.0 10−18

1.0 10−17

1.5 10−17

2.0 10−17

2.5 10−17

(Cl −

CN)/

W2

[(c

m−

2 s−

1 sr

�

Documents

083007 (2012) Anisotropies in the diffuse gamma-ray ... · M. Roth,52 C. Sbarra,8 J. Schmitt,5 C. Sgro`,4 J. Siegal-Gaskins,3,50, ... The contribution of unresolved sources to the