Table Of ContentSearches for Cosmic-Ray Electron Anisotropies with the Fermi Large Area Telescope
M. Ackermann,1 M. Ajello,1 W. B. Atwood,2 L. Baldini,3 J. Ballet,4 G. Barbiellini,5,6 D. Bastieri,7,8 K. Bechtol,1
R. Bellazzini,3 B. Berenji,1 E. D. Bloom,1 E. Bonamente,9,10 A. W. Borgland,1 A. Bouvier,1 J. Bregeon,3
A. Brez,3 M. Brigida,11,12 P. Bruel,13 R. Buehler,1 T. H. Burnett,14 S. Buson,7,8 G. A. Caliandro,15
R. A. Cameron,1 P. A. Caraveo,16 S. Carrigan,8 J. M. Casandjian,4 C. Cecchi,9,10 O¨. C¸elik,17,18,19
E. Charles,1 A. Chekhtman,20,21 C. C. Cheung,20,22 J. Chiang,1 S. Ciprini,10 R. Claus,1 J. Cohen-Tanugi,23
J. Conrad,24,25,26 A. Cuoco,25 C. D. Dermer,20 A. de Angelis,27 F. de Palma,11,12 S. W. Digel,1 G. Di Bernardo,3
E. do Couto e Silva,1 P. S. Drell,1 R. Dubois,1 C. Favuzzi,11,12 S. J. Fegan,13 W. B. Focke,1 M. Frailis,27,28
Y. Fukazawa,29 S. Funk,1 P. Fusco,11,12 D. Gaggero,3 F. Gargano,12 S. Germani,9,10 N. Giglietto,11,12
1 P. Giommi,30 F. Giordano,11,12 M. Giroletti,31 T. Glanzman,1 G. Godfrey,1 D. Grasso,3 I. A. Grenier,4
1 J. E. Grove,20 S. Guiriec,32 M. Gustafsson,7 D. Hadasch,15 A. K. Harding,17 K. Hayashi,29 E. Hays,17
0 R. E. Hughes,33 G. J´ohannesson,1 A. S. Johnson,1 W. N. Johnson,20 T. Kamae,1 H. Katagiri,29 J. Kataoka,34
2
M. Kerr,14 J. Kno¨dlseder,35 M. Kuss,3 J. Lande,1 L. Latronico,3 S.-H. Lee,1 M. Lemoine-Goumard,36
n M. Llena Garde,24,25 F. Longo,5,6 F. Loparco,11,12 M. N. Lovellette,20 P. Lubrano,9,10 A. Makeev,20,21
a
M. N. Mazziotta,12,∗ J. E. McEnery,17,37 J. Mehault,23 P. F. Michelson,1 T. Mizuno,29 A. A. Moiseev,18,37
J
C. Monte,11,12 M. E. Monzani,1 E. Moretti,38,25 A. Morselli,39 I. V. Moskalenko,1 S. Murgia,1 T. Nakamori,34
0
1 M. Naumann-Godo,4 P. L. Nolan,1 E. Nuss,23 T. Ohsugi,40 A. Okumura,41 N. Omodei,1 E. Orlando,42
J. F. Ormes,43 D. Paneque,1 J. H. Panetta,1 D. Parent,20,21 V. Pelassa,23 M. Pepe,9,10 M. Pesce-Rollins,3
E] F. Piron,23 T. A. Porter,1 S. Profumo,2 S. Raino`,11,12 R. Rando,7,8 M. Razzano,3 A. Reimer,44,1 O. Reimer,44,1
H T. Reposeur,36 J. Ripken,24,25 S. Ritz,2 M. Roth,14 H. F.-W. Sadrozinski,2 A. Sander,33 T. L. Schalk,2 C. Sgro`,3
J. Siegal-Gaskins,33 E. J. Siskind,45 D. A. Smith,36 P. D. Smith,33 G. Spandre,3 P. Spinelli,11,12 M. S. Strickman,20
.
h A. W. Strong,42 D. J. Suson,46 H. Takahashi,40 T. Takahashi,41 T. Tanaka,1 J. B. Thayer,1 J. G. Thayer,1
p
D. J. Thompson,17 L. Tibaldo,7,8,4,47 D. F. Torres,15,48 G. Tosti,9,10 A. Tramacere,1,49,50 Y. Uchiyama,1
-
o T. L. Usher,1 J. Vandenbroucke,1 V. Vasileiou,18,19,† N. Vilchez,35 V. Vitale,39,51 A. P. Waite,1 P. Wang,1
r B. L. Winer,33 K. S. Wood,20 Z. Yang,24,25 T. Ylinen,38,52,25 G. Zaharijas,24,25,53 and M. Ziegler2
t
s
a (The Fermi LAT Collaboration)
[ 1W. W. Hansen Experimental Physics Laboratory,
3 Kavli Institute for Particle Astrophysics and Cosmology,
v Department of Physics and SLAC National Accelerator Laboratory,
9 Stanford University, Stanford, CA 94305, USA
1 2Santa Cruz Institute for Particle Physics,
1 Department of Physics and Department of Astronomy and Astrophysics,
5 University of California at Santa Cruz,
. Santa Cruz, CA 95064, USA
8
0 3Istituto Nazionale di Fisica Nucleare,
0 Sezione di Pisa, I-56127 Pisa, Italy
1 4Laboratoire AIM, CEA-IRFU/CNRS/Universit´e Paris Diderot,
: Service d’Astrophysique, CEA Saclay,
v
91191 Gif sur Yvette, France
i
X 5Istituto Nazionale di Fisica Nucleare,
Sezione di Trieste, I-34127 Trieste, Italy
r
a 6Dipartimento di Fisica,
Universita` di Trieste, I-34127 Trieste, Italy
7Istituto Nazionale di Fisica Nucleare,
Sezione di Padova, I-35131 Padova, Italy
8Dipartimento di Fisica “G. Galilei”,
Universita` di Padova, I-35131 Padova, Italy
9Istituto Nazionale di Fisica Nucleare,
Sezione di Perugia, I-06123 Perugia, Italy
10Dipartimento di Fisica, Universita` degli Studi di Perugia,
I-06123 Perugia, Italy
11Dipartimento di Fisica “M. Merlin” dell’Universit`a e del Politecnico di Bari,
I-70126 Bari, Italy
12Istituto Nazionale di Fisica Nucleare,
Sezione di Bari, 70126 Bari, Italy
13Laboratoire Leprince-Ringuet, E´cole polytechnique,
CNRS/IN2P3, Palaiseau, France
2
14Department of Physics, University of Washington,
Seattle, WA 98195-1560, USA
15Institut de Ciencies de l’Espai (IEEC-CSIC),
Campus UAB, 08193 Barcelona, Spain
16INAF-Istituto di Astrofisica Spaziale e Fisica Cosmica,
I-20133 Milano, Italy
17NASA Goddard Space Flight Center,
Greenbelt, MD 20771, USA
18Center for Research and Exploration in Space Science and
Technology (CRESST) and NASA Goddard Space Flight Center,
Greenbelt, MD 20771, USA
19Department of Physics and Center for Space Sciences and Technology,
University of Maryland Baltimore County,
Baltimore, MD 21250, USA
20Space Science Division, Naval Research Laboratory,
Washington, DC 20375, USA
21George Mason University, Fairfax, VA 22030, USA
22National Research Council Research Associate,
National Academy of Sciences,
Washington, DC 20001, USA
23Laboratoire de Physique Th´eorique et Astroparticules,
Universit´e Montpellier 2,
CNRS/IN2P3, Montpellier, France
24Department of Physics, Stockholm University,
AlbaNova, SE-106 91 Stockholm, Sweden
25The Oskar Klein Centre for Cosmoparticle Physics,
AlbaNova, SE-106 91 Stockholm, Sweden
26Royal Swedish Academy of Sciences Research Fellow,
funded by a grant from the K. A. Wallenberg Foundation
27Dipartimento di Fisica,
Universita` di Udine and Istituto Nazionale di Fisica Nucleare,
Sezione di Trieste, Gruppo Collegato di Udine,
I-33100 Udine, Italy
28Osservatorio Astronomico di Trieste,
Istituto Nazionale di Astrofisica, I-34143 Trieste, Italy
29Department of Physical Sciences, Hiroshima University,
Higashi-Hiroshima, Hiroshima 739-8526, Japan
30Agenzia Spaziale Italiana (ASI) Science Data Center,
I-00044 Frascati (Roma), Italy
31INAF Istituto di Radioastronomia, 40129 Bologna, Italy
32Center for Space Plasma and Aeronomic Research (CSPAR),
University of Alabama in Huntsville,
Huntsville, AL 35899, USA
33Department of Physics,
Center for Cosmology and Astro-Particle Physics,
The Ohio State University, Columbus, OH 43210, USA
34Research Institute for Science and Engineering,
Waseda University, 3-4-1, Okubo,
Shinjuku, Tokyo, 169-8555 Japan
35Centre d’E´tude Spatiale des Rayonnements, CNRS/UPS,
BP 44346, F-30128 Toulouse Cedex 4, France
36Universit´e Bordeaux 1, CNRS/IN2p3,
Centre d’E´tudes Nucl´eaires de Bordeaux Gradignan,
33175 Gradignan, France
37Department of Physics and Department of Astronomy,
University of Maryland, College Park, MD 20742, USA
38Department of Physics,
Royal Institute of Technology (KTH),
AlbaNova, SE-106 91 Stockholm, Sweden
39Istituto Nazionale di Fisica Nucleare,
Sezione di Roma “Tor Vergata”, I-00133 Roma, Italy
40Hiroshima Astrophysical Science Center,
Hiroshima University, Higashi-Hiroshima,
3
Hiroshima 739-8526, Japan
41Institute of Space and Astronautical Science, JAXA,
3-1-1 Yoshinodai, Sagamihara,
Kanagawa 229-8510, Japan
42Max-Planck Institut fu¨r extraterrestrische Physik,
85748 Garching, Germany
43Department of Physics and Astronomy,
University of Denver, Denver, CO 80208, USA
44Institut fu¨r Astro- und Teilchenphysik and Institut fu¨r Theoretische Physik,
Leopold-Franzens-Universit¨at Innsbruck,
A-6020 Innsbruck, Austria
45NYCB Real-Time Computing Inc.,
Lattingtown, NY 11560-1025, USA
46Department of Chemistry and Physics,
Purdue University Calumet,
Hammond, IN 46323-2094, USA
47Partially supported by the International Doctorate on Astroparticle Physics (IDAPP) program
48Instituci´o Catalana de Recerca i Estudis Avanc¸ats (ICREA), Barcelona, Spain
49Consorzio Interuniversitario per la Fisica Spaziale (CIFS),
I-10133 Torino, Italy
50INTEGRAL Science Data Centre,
CH-1290 Versoix, Switzerland
51Dipartimento di Fisica,
Universita` di Roma “Tor Vergata”, I-00133 Roma, Italy
52School of Pure and Applied Natural Sciences,
University of Kalmar, SE-391 82 Kalmar, Sweden
53Institut de Physique Th´eorique,
CEA/Saclay, F-91191 Gif sur Yvette, France
The Large Area Telescope on board the Fermi satellite (Fermi LAT) detected more than 1.6 ×
106 cosmic-ray electrons/positrons with energies above 60 GeV during its first year of operation.
Thearrivaldirectionsoftheseeventsweresearchedforanisotropies ofangularscaleextendingfrom
∼10◦ up to90◦,and of minimum energy extendingfrom 60 GeV upto 480 GeV. Two independent
techniques were used to search for anisotropies, both resulting in null results. Upper limits on the
degreeoftheanisotropyweresetthatdependedontheanalyzedenergyrangeandontheanisotropy’s
angular scale. The upperlimits for a dipole anisotropy ranged from ∼0.5% to ∼10%.
PACSnumbers: 96.50.S-,95.35.+d
Keywords: CosmicRayElectrons,Anisotropy,Pulsar,SNR,DarkMatter,Fermi
I. INTRODUCTION 10◦) to large (90◦) angular scales (for instance see [1–
6]). Searches for such anisotropies can provide unique
information on the sources of CRs and the environment
The majority of detected high-energy (GeV–TeV) in which they have propagated.
charged primary Cosmic Rays (CRs) is believed to
be produced in our galaxy, most likely in Supernova Contrary to hadronic CRs, high-energy (>GeV)
CosmicRayElectronsandPositrons(CREs)propagating
Remnants (SNRs). During the transport from their
source of origin to our solar system, CRs scatter on in the GMF lose their energy rapidly through
synchrotronradiationand by inverse Comptoncollisions
random and irregular components of the µG Galactic
MagneticField(GMF),whichalmostisotropizetheCRs’ withlow-energyphotonsoftheinterstellarradiationfield.
CREsobservedwithenergies100GeV(1TeV)originated
directiondistribution. ThishappensbecausetheLarmor
radius for a typical value of 4µG for the GMF and for from relatively nearby locations, less than about 1.6 kpc
a 100 GeV singly-charged particle is 3 10−5 pc, (0.75 kpc) away [7]. This means that it could be
considerablysmallerthanthetypicaldist∼ance×toanearby possible that such high-energy CREs originate from a
source (of the order of a hundred pc). Nevertheless, highly anisotropic collection of a few nearby sources
several ground experiments across a wide range of (possibly pulsars and SNRs). Therefore, depending on
energies have still detected anisotropies of medium ( the propagation properties in the GMF, the detection
∼ of excess CREs with energies high enough to minimize
both the geomagnetic field and any heliospheric effects
might reveal the presence of such nearby CRE sources.
∗[email protected] Similarly, assuming a reasonable distribution of nearby
†[email protected] CRE sources, measurements of the CRE anisotropy
4
can be used to constrain the diffusion of CREs in the which diminishes quickly with rising energy, until it
galaxy. Finally, further anisotropythat is not associated becomesnegligibleatenergiesoverseveralhundredGeV.
with nearby CR sources is expected to result from the However, it is not easy to quantify the HMF’s influence
Compton-Getting (CG) effect [8], in which the relative on CREs of some energy propagating in it, since this
motion of the observer with respect to the CR plasma would require a good knowledge of the HMF’s structure
changes the intensity of the CR fluxes, with larger and of the propagation of CREs through it. Because of
intensity arrivingfromthe directionofmotionandlower theabsenceofawelldefinedenergythresholdoverwhich
intensity arriving from the opposite direction. the HMF effects can be ignored, we decided to start the
The dataset of the Large Area Telescope (LAT) on analysis from 60 GeV with the caveat that some of our
board the Fermi satellite [9] provides a unique high lower-energyresults might be affected by the HMF.
statistics sample for CRE anisotropy studies. In this To minimize the contamination from the Earth’s
paper,wereporttheresultsofasearchforanisotropiesin albedo, events with reconstructed directions near the
thereconstructeddirectionsoftheeventsinthisdataset. Earth’s limb or detected during the interval when the
Earth’s limb enters the LAT’s field of view more than
usual (when the rocking angle is greater than 43◦) were
II. THE INSTRUMENT AND THE DATA rejected. This cut removed roughly 16% of the events,
leaving a total of 1.35 million events for this analysis.
∼
It should be noted that all the events in the dataset
The LAT is a pair-conversion gamma-ray telescope
weredetectedwhiletheFermi spacecraftwasoutsidethe
designed to measure gamma rays in the energy range
South Atlantic Anomaly.
from 20 MeV to more than 300 GeV. In this paper a
The Fermi satellite is usually operated in “sky-survey
brief description of the LAT is given, while full details
mode”. Inthis operationmode, the sky is fully observed
can be found in [10].
every 2 orbits ( 3 hours). This ensures a uniform (up
The LAT is composed of a 4 4 array of 16 ∼
× to 15%) all-sky exposure and allows us to search
identical towers designed to convert incident gamma- ∼
rays into e+e− pairs, and to determine their arrival for anisotropies of any angular scale (including dipole
anisotropy) and from any direction in the sky.
directions and energies. Each tower hosts a tracker
The angular resolution of the LAT for E>60 GeV
module and a calorimeter module. Each tracker module
electron events is about 0.1◦ or better and the energy
consists of 18 x-y planes of silicon-strip detectors,
resolution (1σ) is 10%. The contamination of the
interleaved with tungsten converter foils, for a total ∼
analyzed dataset with other species, such as photons
on-axis thickness equivalent to 1.5 radiation lengths
or protons, can result in some systematic uncertainties.
(r.l.). Each calorimeter module, 8.6 r.l. on-axis thick,
The fractionofhadroneventsin the datasetrangesfrom
hosts 96 CsI(Tl) crystals, hodoscopically arranged in
4% at 20 GeV to 20% at 1 TeV, while the photon
8 perpendicular layers. The instrument is surrounded ∼ ∼
contamination is negligible (<0.1%). The full details of
by a segmented anti-coincidence detector that tags the
Fermi’s CRE data analysis can be found in [11].
majority of the charged-particle background.
AlthoughtheLATwasdesignedfordetectingphotons,
it was recognized that it can also work as an excellent
detector of high-energy CREs. The calorimeter behaves III. METHOD
in the same way for electrons and gamma rays, and
helps with discriminating electromagneticfromhadronic ThewholeskywassearchedforanisotropiesinGalactic
showers. The early part of the shower can be coordinateswithnoaprioriassumptionsonthedirection,
reconstructedindetailinthetracker,contributingtothe the angular scale, or the energy spectrum of a potential
hadron/electronseparation. signal. The dataset was analyzed in its entirety, without
The analyzed dataset corresponds to the first year trying to divide it in time to performa searchoptimized
of LAT science operation and starts on August 2008. for detecting transient anisotropies.
Because the Earth’s magnetic field can affect the One way to search for anisotropies is to first calculate
directionsofincomingCREevents,introducingorhiding the flux of CRE particles from each direction in the sky
anisotropies, we have selected events with an energy (equal to the ratio of the number of detected events
high enough (E>60 GeV) to minimize the geomagnetic from some direction over the exposure towards the same
field’s influence yielding 1.6 million events. While this direction), and then examine its directionaldistribution.
∼
60 GeV energy threshold is significantly higher than the The flux calculation, which requires knowledge of the
geomagnetic cut-off in any part of Fermi’s orbit, the exposure, depends on the effective area of the detector
geomagnetic field could still introduce some deflections and the accumulated observation livetime. The effective
even abovethis energy,possibly giving rise to a spillover area, calculated from a Monte Carlo simulation of the
effect in adjacent regions of the sky. Furthermore, the instrument, could suffer from systematic errors, such
Heliospheric Magnetic Field (HMF) can also affect the as a dependence on the time or on the location of the
directions of particles propagating in it, exhibiting an spacecraft, or any miscalculations of the dependence of
appreciable effect on CREs with energies . 100 GeV, the effective areaonthe instrument coordinates(off-axis
5
and azimuthal angle). Naturally, any systematic errors direction distribution. The final no-anisotropy sky
involvedinthecalculationoftheexposurewillpropagate mapisproducedbytakingthe averageofthesesky
to the flux, possibly affecting its directionaldistribution. maps. By this construction, the simulated data
Ifthemagnitudeofthesesystematicerrorsiscomparable set preserves exactly the energy and angular (with
to or larger than the statistical power of the available respect to the LAT reference frame) distributions,
dataset,theireffectsontheflux’sdirectionaldistribution and also accounts for the detector dead times. To
mightmasqueradeasarealdetectableanisotropy. Aswill minimize the possible effects of a varying CRE
be shown below, the statistics of the available dataset eventrate,thedatasetisfirstsplitinto10segments
allow us to search for anisotropies as small as a fraction with equal number of events, and the technique is
ofapercent. Becausethe effectiveareaofthe detector is applied to each of these sets separately.
not known to a better accuracy, it follows that it is not
Direct-integration technique:
safetoperformanisotropysearchesofsuchsensitivityby •
This technique is based on [15]. In general, the
examiningthedirectionaldistributionoftheparticleflux.
rate of events detected in some narrow solid angle
For that reason, we employed two alternative methods
around a given direction (θ, φ) [16] at some time t
that are free of systematic errors larger than a fraction
is equal to the all-sky rate at that time R (t)
of a percent. allsky
times the probability P(θ,φ,t) of an event being
reconstructed inside that same solid angle. Given
a dataset and the LAT’s pointing information,
A. No-anisotropy map creation
we can calculate the values of the two functions
R (t)andP(θ,φ,t). Similarly,giventhevalues
allsky
The starting point of this analysis is the construction of these two functions and the LAT’s pointing
of a sky map that shows how the sky as seen by the information for some observation we can predict
Fermi-LAT would look on average if the CRE direction how the sky would look for that same observation
distribution were perfectly isotropic. This sky map, (i.e. construct a sky map). The main idea of this
hereafter called “no-anisotropysky map”, represents the method is to first extract from the CRE dataset
null hypothesis for the existence of an anisotropy. A the set of values of R (t) and P(θ,φ,t) that
allsky
comparison of the no-anisotropy sky map to a sky map corresponds to an isotropic CRE distribution, and
generated by the actually detected CRE events (the then construct the associated no-CRE-anisotropy
“actual” or “signal” sky map) was used to reveal the sky map. The presence of any anisotropies in
presence of any anisotropies in the data. the data would create transient fluctuations in
As a cross-check of the systematic errors involved in the instantaneous values of these functions, as
this analysis, two techniques were used to build the no- these anisotropies passed through the LAT’s field
anisotropy sky map, producing similar results. The two of view. However, these anisotropies would have
techniques are described below. no effect on the longer-term average values of
these functions, since any transient fluctuations
Shuffling technique: would be averaged out. In this application, we
•
One way of generating the no-anisotropy map is usedaconstant-over-timeP (θ,φ)producedafter
ave
to randomize the reconstructed directions of the averagingP(θ,φ,t) over the whole dataset. For an
detected events [12–14]. In case the direction isotropicsky,theonlytimedependenceofP(θ,φ,t)
distribution of the CR flux is perfectly isotropic, would come only from temporal variations of the
a time-independent intensity should be detected dependence of the detector’s effective area on θ
when looking at any given detector direction. and φ. We did not detect any such variations
Possible time variation of the intensity would be in the data, therefore using an average over the
due only to changes in the operating conditions of whole dataset P (θ,φ) was valid. Similarly,
ave
the instrument. A setofisotropicsimulatedevents any temporal variations of the effective area of
can be built by randomly coupling the times and the detector would create fluctuations of the
the directions of real events in local instrument instantaneous value and the longer-term averages
coordinates. The randomization is performed of R (t), depending on the time scales of the
allsky
starting with the position of a given event in the effective-area variations. Unlike P(θ,φ,t), which
LAT frame and exchangingit with the directionof remained sufficiently constant for this purpose,
another event, which was selected randomly from the all-sky rate exhibited fluctuations on multiple
the data set with a uniform probability. Starting time scales caused by varying background rates
with this information, the sky direction is re- occurring as the spacecraft was moving through
evaluated for the simulated event. In this way the regions of different geomagnetic coordinates and
random coupling preserves the exposure and the by changes in the instrument’s hardware settings.
total number of events. This process is repeated Any such instrumental effects affecting the all-sky
multiple times (100), with each time producing a rate were parametrized, and given an averaged-
sky map that is compatible with an isotropic CRE over-multiple-orbits value of the all-sky rate
6
[17], the all-sky rate at some shorter-duration around that bin. Using such “integrated sky maps”,
segment could be accurately predicted. These it is very likely that there will be at least one bin
parametrizations are necessary because the direct- with its center roughly aligned with the direction of the
integration method constructs the no-anisotropy- centerofapotentialanisotropy,reducingspillovereffects
skymap incrementally, adding the results from and increasing sensitivity. In general, the sensitivity
observations of short enough duration that the for detecting an anisotropy of given angular scale is
LAT’s pointing can be assumed quasi-constant greater when an integration radius close to that scale
(e.g., 30 s long). As a cross-check, this technique is chosen. If the integration radius is too small or too
was also applied with R (t) being given by large compared to the angular scale of the prospective
allsky
the instantaneous rate of actually detected CRE anisotropy, the sensitivity becomes sub-optimal since
events, instead of an averaged value. This choice eitherthesignalcanbesplitamongseveraladjacentbins
has the benefit of automatically taking care of ortherecanbetoomuch“background”(isotropicsignal)
any temporal variations of the effective area, contamination. In this work, to search for anisotropies
avoiding the need to apply any corrections and, of various angular scales we compared multiple pairs
most importantly, avoiding any systematic errors of integrated no-anisotropy and actual sky maps, with
introduced by such corrections. It should be noted eachpair correspondingto a different integrationradius.
however that not performing an averaging in the The integration radii were chosen to cover the range of
all-sky rate weakens the power of this method to anisotropyangular-scalesunderconsiderationandranged
smear out the presence of any anisotropies in the from 10◦ to 90◦.
data, with the result of any stronger anisotropies The comparison was performed by calculating the
possibly leaking in the no-CRE-anisotropy sky statistical significance of the difference between the
map. Nevertheless, for this application, the results contents of the integrated actual and the no-anisotropy
of the direct-integration technique when using sky maps. To take into account the statistical errors
the instantaneous event rate and when using the involved in construction of the no-anisotropy sky maps
averaged-over-multiple orbits event rate plus the by the event-shuffling technique, the significances for
necessary corrections were consistent with each that method were evaluated using the prescription
other. in [19], by using the likelihood method [20]. It
should be noted that the results of the event-shuffling
For our sky maps, we adopted the HEALPix [18] technique were stable even after a couple tens of
pixelization scheme to ensure that all pixels across the repetitions. Forthisapplication,weused100repetitions,
sky have the same area (or solid angle). The resolution rendering the associated statistical errors negligible. For
of the HEALPix grid is expressed by the parameter the case of the direct-integration technique, since its
Nside, which defines the total number of pixels: Npix = produced no-anisotropysky map is the result of a direct
1to2×12N,2s28i8de. O3udregm2appisxehlas.dTNhseidsek=ym32a,pwshpircehsecnotrerdesipnotnhdiss caanldcutlhaetisoing,nitfihcearnecewweraesncaolcauslsaotceidatuesdinsgtasitmistpilcealPoeirsrsoorns
∼
paper are all in Galactic coordinates. The no-anisotropy probabilities.
maps were compared to the actual sky map using two
As was mentioned above, this is a search for
independent methods, described below.
anisotropies from any direction in the whole sky. As
such, it involves a large number of trials (independent
tests), that have to be accounted for when judging the
B. Direct bin-to-bin comparison statistical significance of its results. For a whole-sky
search performed using independent-bin sky maps, the
The first method is a simple direct bin-to-bin numberoftrialsisequaltothenumberofpixelsinthesky
comparison of the two maps, in which a search for map. Ontheotherhand,searchesthatusecorrelated-bin
statistically significant deviations between the number sky maps, such as this one, involve a number of effective
of actually detected and the number of expected events trials that is in general smaller than the number of bins
under the assumption of isotropy is performed. in one such sky map. The larger the integration radius,
One way to search for anisotropies of some angular the higher the degree of correlation between adjacent
scale, is to use sky maps composed of independent bins bins and the smaller the number of effective trials per
withbinsizesimilartotheangularscaleoftheanisotropy evaluated bin.
under search. However, when using independent bins, The number of effective trials involved in evaluating
a potential anisotropic signal might become too weak the contents of an integrated map was evaluated using
to be detected since it will probably be distributed a Monte Carlo simulation. Fake significance maps were
among multiple adjacent bins. A more sensitive way built corresponding to various integration radii and to
to perform the search, adopted in the present analysis, a perfectly isotropic signal. By counting the fraction of
is to use sky maps consisting of a large number of suchsky mapsP thatcontainedatleastonebin with
post
correlated bins. The content of a correlated bin is equal aprobabilitysmallerthansomevalueP ,wecalculated
pre
to the integrated number of events in a circular region theeffectivenumberoftrialsinvolvedinthesearchofsky
7
l
1
104 Cˆ = a 2. (3)
l lm
2l+1 | |
m=−l
s X
al
of tri103 The power spectrum characterizes the intensity
er fluctuations as a function of the angular scale. An
mb increased power Cˆ at a multipole l corresponds to
u l
ve N102 an anisotropic excess of angular scale ∼ 180◦/l. The
cti spherical harmonic analysis was performed using the
e
Eff anafast code provided with the HEALPix tools [18].
10 Thedatacanbetreatedasthesumoftwoindependent
components: an anisotropic component, which we are
trying to detect, and an isotropic component, which
10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 1 is known and equivalent to white noise. Because
Pre−trials probability
these two components are not correlated, the observed
angular power spectrum can be similarly split into two
FIG. 1: Curves: Numberof effective trials (Teff) involvedin components: Cˆl = Cˆlaniso + CˆlN. The quantities Cˆl,
evaluating the contents of a single significance map. From Cˆaniso andCˆN arerandomvariables,ingeneraldifferent
l l
top to bottom: trials for an independent-bins significance thantheirtrueunderlyingquantitiesC ,Caniso,andCN
map, and trials for integrated maps of 10◦, 30◦, 45◦, 60◦, respectively. Specifically, the observedlqualntities follolw
and90◦ integration radius. Thehorizontaldashedlineshows a χ2 distribution centeredat their correspondingtrue
the number of bins in these maps (12,288). As expected, 2l+1
values. The true value of the isotropic (white noise)
the number of effective trials for an independent-bins sky
component is [22]:
map (top curve) is equal to the number of bins in the map
(horizontalline). Theeffectivenumbersoftrialsforintegrated
mapsareingeneralsmallerthanthenumberofbins,withthe 4π
integrationradiusbeinginverselycorrelatedtothenumberof ClN = N , (4)
effective trials. The error bars show the statistical error that
arises from thefinitenumberof simulated sky maps. where N is the total number of observed events.
To search for anisotropies in the data, we examined
the validity of the null hypothesis: Caniso = 0 or
l
maps of some integration radius [21]: equivalently Cl = ClN. This was accomplished by
checking whether the observed power spectrum Cˆ was
l
statisticallycompatiblewith the knowntruevalue ofthe
log(1 P ) isotropic power spectrum CN.
T = − post . (1) l
eff Theresultingpowerspectracanalsobeusedforsetting
log(1 P )
pre
− upper limits on the degree of anisotropy:
The effective number of trials for each integration
radiusandforthecaseofanindependent-binsskymapis Imax Imin
δ − , (5)
showninFig.1. Usingthiseffectivenumberoftrials,the ≡ I +I
max min
post-trialsprobabilityP correspondingtoapre-trials
post
where I and I are the maximum and minimum
P probability can be calculated as: max min
pre
values of the CRE intensity. Consider a dataset
consisting of the sum of a perfectly isotropic signal
P =1 (1 P )Teff . (2)
post pre of constant intensity I and of a dipole anisotropy of
− − 0
maximum intensity I . The overall intensity at an
1
angular distance θ from the maximum of the dipole
anisotropywill be I(θ)=I +I cos(θ). For this dataset,
C. Spherical harmonic analysis 0 1
the degree of its dipole anisotropy is
A more robust method involves a spherical harmonic
I
analysis of a “fluctuations sky map” equal to the δ = 1. (6)
ratio of the actual and no-anisotropy sky maps minus I0
one. Initially, the fluctuations sky map is expanded
The fluctuation map describing this dataset is:
in the basis of spherical harmonics, producing a set of
coefficients a . Then, an angular power spectrum is
lm
constructed by calculating the average variance of the I(θ) <I(θ)> I(θ) I0 I1
f(θ)= − = − = cos(θ). (7)
2l+1 alm coefficients at each multipole l as <I(θ)> I0 I0
8
Since Y0(θ,φ) = 3 cos(θ), it follows from Eq. 7 that
1 4π
f(θ)= II10 43π q×Y10 or that a10 = I1 4π. (8)
(cid:16) q (cid:17) I0r 3
All but the l = 1 term of the power spectrum vanish.
The non-zero term is:
1 1 a2 I 2 4π
C a 2 = 10 = 1 . (9)
1 1m
≡ 3 | | 3 I 9
m=−1 (cid:18) 0(cid:19)
X
It should be noted that since C (as all the C ) is a
1 l
rotationally invariant quantity, the above expression for
C , derived for the reference frame in which a = 0
1 11
and a = 0, is valid in general for every dipole and
1−1
reference frame. From Eq.s 6 and 9 we derive a relation
between the degree of dipole anisotropyand the value of
the dipole power:
C
1
δ =3 . (10)
4π
r
To set an upper limit on δ we start by calculating
the probability distribution function (pdf) of δˆ= 3 Cˆ1
4π
by a change of variable on the probability distribuqtion
function of Cˆ (χ2 centered on C ). The probability of
1 3 1
observing a certain Cˆ given the true dipole power C
1 1
(i.e. the probabilitydensity functiontoobserveCˆ given
1
the true dipole power C ) is:
1
3√3 Cˆ 3Cˆ
P(Cˆ ;C )= 1exp 1 . (11)
1 1
√2πC1sC1 −2C1!
With a change of variables we obtain:
3√6δˆ2 3δˆ2
P(δˆ;δ)= exp . (12)
√π δ3 −2δ2!
Theupperlimitofthetrueanisotropyδ isthevalue
UL
of δ for which the integrated probability of measuring
a value of δˆ at least as large as the one we measured is
equaltotheconfidencelevel. Specifically,foraconfidence
levelCL,theupperlimitonthe dipoleanisotropycanbe
evaluated using the frequentist approach[23] by solving:
δˆmeas
P(δˆ;δ)dδˆ=1 CL, (13)
−
FIG. 2: From top to bottom: no-anisotropy sky map Z0
for E>60 GeV; sky map of actually detected CREs with
E>60 GeV; significance map produced by comparing the where δˆ 3 Cˆ1,meas is the dipole anisotropy that
meas ≡ 4π
above maps. The shape of the actual and no-anisotropy sky corresponds to oqur single measurement of the dipole
mapsresultsfromthefactthattheskywasnotobservedwith power spectrum Cˆ . Integrating and solving for δ
1,meas
uniform exposure. we obtain δ .
UL
9
Multiple pairs of signal and no-anisotropy sky maps
were produced by integrating the independent-bin sky
maps (e.g. such as those shown in the first two panels
of Fig.2) over circular regions of radii ranging from
n 10◦ to 90◦. Similarly to the above, significance maps
s/bi 102 were constructed by comparing the integrated signal
el and no-anisotropy sky maps. Figure 4 shows some of
x
pi the significance sky maps obtained by comparing the
of integrated no-anisotropy sky maps produced with the
r
e 10 shuffling technique to the actual sky maps. The same
b
m figure also shows significance sky maps produced by the
u
N direct-integration technique and for a 45◦ integration
radius.
1
The significances shown in these maps are pre-trials,
−4 −2 0 2 4 i.e. they do not take into account the fact that we
Significance (σ) performed multiple trials while evaluating them. Using
the effective numbers of trials shown in Fig. 1 and
Eq. 2 we have calculated the post-trials significances
FIG.3: Distributionofthesignificancevaluesoftheskymap corresponding to the single most significant bin in each
in the third panel of Fig. 2 (histogram) together with a best of the integrated maps (one bin per integration radius
fit with a Gaussian function (solid line). and minimum energy), shown in Fig. 5. Also shown
in the same figure is the correspondence between pre-
and post-trials significances for each integration radius.
IV. RESULTS As can be seen, our most-significant bins are all post-
trials insignificant [24]. It should be noted that there
Inthis sectionwe reportthe resultsofour searchfor a is one more trial factor in this search corresponding
steady excess of CREs from any direction of the sky. to searching in multiple integration radii and minimum
The two techniques, event shuffling and direct energies. This factor is of the order of few and because
integration, produced similar no-anisotropy sky maps, itisnegligiblecomparedtothe trialsfactorsofsearching
and any differences were considerably smaller than the in all directions, it was not included in the calculations.
smallest amplitude of a detectable signal. For brevity The inclusion of this trial factor would just reinforce the
and unless otherwise noted, all the sky maps and upper null result of this search.
limits presented in this paper were produced using the
We have also performed the evaluation of our
no-anisotropy sky map of the event shuffling technique.
significance maps towards the direction of a few
selected sources and regions. Such a search involves a
considerablysmallernumber oftrials since hereonly few
A. Bin to bin comparison specificdirectionsareevaluatedinsteadofthewholesky.
We have searched for excesses towards the directions of
Our no-anisotropy sky maps are initially produced the Vela (l,b = 264◦, -3◦), Geminga (205◦, -1◦), and
consisting of independent bins. To use them for Monogem (201◦, 8◦) pulsars, towards the Virgo (300◦,
anisotropy searches we first integrate their bin contents 60◦) and Cygnus (75◦, 0◦) regions, and towards the
over a range of angular scales ranging from 10◦ to 90◦. Galactic and Galactic anticenter. To be conservative
One of the produced independent-bins no-anisotropy and to avoid accumulating too many trials, we searched
sky maps is shown in the first panel of Fig. 2. The for two different minimum energies, E>60 GeV and
corresponding actual map showing the actually detected E>240 GeV, and for four different integration radii:
events for the same energy range E>60 GeV is shown in 10◦, 30◦, 60◦, and 90◦. The evaluated sample did not
the second panel of the same figure. This sky map does show any significant anisotropies; the highest pre-trials
not appear uniform because the sky was not observed significance value was from the direction of the Galactic
with uniform exposure. The third panel shows the anticenter: 3.03σ for E>240 GeV and 10◦ integration
significanceskymapproducedaftercomparingtheother radius. Thenumberoftrialsforthissearchisclosetothe
two sky maps. The distribution of the values of this sky totalnumber ofsearches(7 2 4=56)andmostlikely
× ×
map with the Gaussian best-fit superimposed is shown lower since these searches are also correlated. Using the
in the top panel of Fig. 3. As expected, assuming the maximumnumberoftrials(56),wefindabestpost-trials
absence of any strong anisotropies, the best-fit function probability of 6.6 10−2, or equivalently of 1.5σ, which
×
was statistically consistent with a Gaussian distribution is not significant. We also applied a stacking analysis in
of mean zero and unit variance. The same was also whichweaddedthemeasurementsfromthethreepulsars
true for the E>120 GeV, E>240 GeV, E>480 GeV (not forthetwominimumenergiesandintegrationradii(total
shown) distributions. four trials). This search also produced no significant
10
FIG. 4: Significance maps for E>60 GeV and for different integration radii. The first five plots were produced by the event
shuffling technique and correspond to integration radii 10◦, 30◦, 45◦, 60◦, and 90◦. The bottom two sky maps correspond
to a 45◦ integration radius and were produced by the direct integration technique with R (t) being given by the actual
allsky
instantaneous CRE event rate (left) and theaveraged-over-multiple-orbits CRE event rate (right).
