![Anisotropy in the gamma-ray annihilation signal from the subhalo distribution found in Aquarius [40] (left) and Via Lactea II [39] (right) simulations, with the former following the prescription in ref. [63]. The plots show the intensities of the substructures relative to their average intensity in the |b| > 20 • region.](https://www.researchgate.net/profile/A-Morselli/publication/271218295/figure/fig3/AS:613514469199921@1523284656710/Anisotropy-in-the-gamma-ray-annihilation-signal-from-the-subhalo-distribution-found-in_Q320.jpg)
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ULB/14-21
Limits on Dark Matter Annihilation
Signals from the Fermi LAT 4-year
Measurement of the Isotropic
Gamma-Ray Background
The Fermi LAT Collaboration
M. Ackermann1M. Ajello2A. Albert3L. Baldini4G. Barbiellini5,6
D. Bastieri7,8K. Bechtol9R. Bellazzini4E. Bissaldi10 E. D. Bloom3
R. Bonino11,12 J. Bregeon13 P. Bruel14 R. Buehler1S. Buson7,8
G. A. Caliandro3,15 R. A. Cameron3M. Caragiulo16 P. A. Caraveo17
C. Cecchi18,19 E. Charles3A. Chekhtman20 J. Chiang3G. Chiaro8
S. Ciprini21,22 R. Claus3J. Cohen-Tanugi13 J. Conrad23,24,25,26
A. Cuoco24,11,12 S. Cutini21,22 F. D’Ammando27,28 A. de Angelis29
F. de Palma16,30 C. D. Dermer31 S. W. Digel3P. S. Drell3
A. Drlica-Wagner32 C. Favuzzi33,16 E. C. Ferrara34 A. Franckowiak3,?
Y. Fukazawa36 S. Funk3P. Fusco33,16 F. Gargano16
D. Gasparrini21,22 N. Giglietto33,16 F. Giordano33,16 M. Giroletti27
G. Godfrey3S. Guiriec34,37 M. Gustafsson38,64?J.W. Hewitt39,40
X. Hou41 T. Kamae3M. Kuss4S. Larsson23,24,42 L. Latronico11
F. Longo5,6F. Loparco33,16 M. N. Lovellette31 P. Lubrano18,19
D. Malyshev3F. Massaro43 M. Mayer1M. N. Mazziotta16
P. F. Michelson3W. Mitthumsiri44 T. Mizuno45 M. E. Monzani3
A. Morselli46 I. V. Moskalenko3S. Murgia47 M. Negro3,11
R. Nemmen48 E. Nuss13 T. Ohsugi45 M. Orienti27 E. Orlando3
J. F. Ormes49 D. Paneque50,3J. S. Perkins34 M. Pesce-Rollins4
F. Piron13 G. Pivato4S. Rain`o33,16 R. Rando7,8M. Razzano4,51
A. Reimer52,3O. Reimer52,3M. S´anchez-Conde24,23,? A. Schulz1
C. Sgr`o4E. J. Siskind53 G. Spandre4P. Spinelli33,16 A. W. Strong54
D. J. Suson55 H. Tajima3,56 H. Takahashi36 J. G. Thayer3
J. B. Thayer3L. Tibaldo3M. Tinivella4D. F. Torres57,58
E. Troja34,59 Y. Uchiyama60 G. Vianello3M. Werner52 B. L. Winer61
K. S. Wood31 M. Wood3G. Zaharijas10,62,63,?
arXiv:1501.05464v1 [astro-ph.CO] 22 Jan 2015
1Deutsches Elektronen Synchrotron DESY, D-15738 Zeuthen, Germany
2Department of Physics and Astronomy, Clemson University, Kinard Lab of Physics, Clem-
son, SC 29634-0978, USA
3W. W. Hansen Experimental Physics Laboratory, Kavli Institute for Particle Astrophysics
and Cosmology, Department of Physics and SLAC National Accelerator Laboratory, Stan-
ford University, Stanford, CA 94305, USA
4Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, I-56127 Pisa, Italy
5Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, I-34127 Trieste, Italy
6Dipartimento di Fisica, Universit`a di Trieste, I-34127 Trieste, Italy
7Istituto Nazionale di Fisica Nucleare, Sezione di Padova, I-35131 Padova, Italy
8Dipartimento di Fisica e Astronomia “G. Galilei”, Universit`a di Padova, I-35131 Padova,
Italy
9Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA
10Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, and Universit`a di Trieste, I-34127
Trieste, Italy
11Istituto Nazionale di Fisica Nucleare, Sezione di Torino, I-10125 Torino, Italy
12Dipartimento di Fisica Generale “Amadeo Avogadro” , Universit`a degli Studi di Torino,
I-10125 Torino, Italy
13Laboratoire Univers et Particules de Montpellier, Universit´e Montpellier 2, CNRS/IN2P3,
Montpellier, France
14Laboratoire Leprince-Ringuet, ´
Ecole polytechnique, CNRS/IN2P3, Palaiseau, France
15Consorzio Interuniversitario per la Fisica Spaziale (CIFS), I-10133 Torino, Italy
16Istituto Nazionale di Fisica Nucleare, Sezione di Bari, 70126 Bari, Italy
17INAF-Istituto di Astrofisica Spaziale e Fisica Cosmica, I-20133 Milano, Italy
18Istituto Nazionale di Fisica Nucleare, Sezione di Perugia, I-06123 Perugia, Italy
19Dipartimento di Fisica, Universit`a degli Studi di Perugia, I-06123 Perugia, Italy
20Center for Earth Observing and Space Research, College of Science, George Mason Uni-
versity, Fairfax, VA 22030, resident at Naval Research Laboratory, Washington, DC 20375,
USA
21Agenzia Spaziale Italiana (ASI) Science Data Center, I-00133 Roma, Italy
22INAF Osservatorio Astronomico di Roma, I-00040 Monte Porzio Catone (Roma), Italy
23Department of Physics, Stockholm University, AlbaNova, SE-106 91 Stockholm, Sweden
24The Oskar Klein Centre for Cosmoparticle Physics, AlbaNova, SE-106 91 Stockholm, Swe-
den
25Royal Swedish Academy of Sciences Research Fellow, funded by a grant from the K. A.
Wallenberg Foundation
26The Royal Swedish Academy of Sciences, Box 50005, SE-104 05 Stockholm, Sweden
27INAF Istituto di Radioastronomia, 40129 Bologna, Italy
28Dipartimento di Astronomia, Universit`a di Bologna, I-40127 Bologna, Italy
29Dipartimento di Fisica, Universit`a di Udine and Istituto Nazionale di Fisica Nucleare,
Sezione di Trieste, Gruppo Collegato di Udine, I-33100 Udine
30Universitagrave Telematica Pegaso, Piazza Trieste e Trento, 48, 80132 Napoli, Italy
31Space Science Division, Naval Research Laboratory, Washington, DC 20375-5352, USA
32Center for Particle Astrophysics, Fermi National Accelerator Laboratory, Batavia, IL 60510,
USA
33Dipartimento di Fisica “M. Merlin” dell’Universit`a e del Politecnico di Bari, I-70126 Bari,
Italy
34NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
36Department of Physical Sciences, Hiroshima University, Higashi-Hiroshima, Hiroshima 739-
8526, Japan
37NASA Postdoctoral Program Fellow, USA
38Service de Physique Theorique, Universite Libre de Bruxelles (ULB), Bld du Triomphe,
CP225, 1050 Brussels, Belgium
39Department of Physics and Center for Space Sciences and Technology, University of Mary-
land Baltimore County, Baltimore, MD 21250, USA
40Center for Research and Exploration in Space Science and Technology (CRESST) and
NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
41Centre d’´
Etudes Nucl´eaires de Bordeaux Gradignan, IN2P3/CNRS, Universit´e Bordeaux
1, BP120, F-33175 Gradignan Cedex, France
42Department of Astronomy, Stockholm University, SE-106 91 Stockholm, Sweden
43Department of Astronomy, Department of Physics and Yale Center for Astronomy and
Astrophysics, Yale University, New Haven, CT 06520-8120, USA
44Department of Physics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand
45Hiroshima Astrophysical Science Center, Hiroshima University, Higashi-Hiroshima, Hi-
roshima 739-8526, Japan
46Istituto Nazionale di Fisica Nucleare, Sezione di Roma “Tor Vergata”, I-00133 Roma, Italy
47Center for Cosmology, Physics and Astronomy Department, University of California, Irvine,
CA 92697-2575, USA
48Instituto de Astronomia, Geof´ısica e Cincias Atmosf´ericas, Universidade de S˜ao Paulo, Rua
do Mat˜ao, 1226, S˜ao Paulo - SP 05508-090, Brazil
49Department of Physics and Astronomy, University of Denver, Denver, CO 80208, USA
50Max-Planck-Institut f¨ur Physik, D-80805 M¨unchen, Germany
51Funded by contract FIRB-2012-RBFR12PM1F from the Italian Ministry of Education,
University and Research (MIUR)
52Institut f¨ur Astro- und Teilchenphysik and Institut f¨ur Theoretische Physik, Leopold-
Franzens-Universit¨at Innsbruck, A-6020 Innsbruck, Austria
53NYCB Real-Time Computing Inc., Lattingtown, NY 11560-1025, USA
54Max-Planck Institut f¨ur extraterrestrische Physik, 85748 Garching, Germany
55Department of Chemistry and Physics, Purdue University Calumet, Hammond, IN 46323-
2094, USA
56Solar-Terrestrial Environment Laboratory, Nagoya University, Nagoya 464-8601, Japan
57Institute of Space Sciences (IEEC-CSIC), Campus UAB, E-08193 Barcelona, Spain
58Instituci´o Catalana de Recerca i Estudis Avan¸cats (ICREA), Barcelona, Spain
59Department of Physics and Department of Astronomy, University of Maryland, College
Park, MD 20742, USA
603-34-1 Nishi-Ikebukuro, Toshima-ku, Tokyo 171-8501, Japan
61Department of Physics, Center for Cosmology and Astro-Particle Physics, The Ohio State
University, Columbus, OH 43210, USA
62The Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, Trieste
34151 - Italy
63Laboratory for Astroparticle Physics, University of Nova Gorica, Vipavska 13, SI-5000 Nova
Gorica, Slovenia
64Institut f¨ur Theoretische Physik, Friedrich-Hund-Platz 1, D-37077 G¨ottingen, Germany
?corresponding author
E-mail: afrancko@slac.stanford.edu,
michael.gustafsson@theorie.physik.uni-goettingen.de,
sanchezconde@fysik.su.se, gabrijela.zaharijas@ung.si
Abstract. We search for evidence of dark matter (DM) annihilation in the isotropic
gamma-ray background (IGRB) measured with 50 months of Fermi Large Area Tele-
scope (LAT) observations. An improved theoretical description of the cosmological
DM annihilation signal, based on two complementary techniques and assuming generic
weakly interacting massive particle (WIMP) properties, renders more precise predic-
tions compared to previous work. More specifically, we estimate the cosmologically-
induced gamma-ray intensity to have an uncertainty of a factor ∼20 in canonical
setups. We consistently include both the Galactic and extragalactic signals under the
same theoretical framework, and study the impact of the former on the IGRB spectrum
derivation. We find no evidence for a DM signal and we set limits on the DM-induced
isotropic gamma-ray signal. Our limits are competitive for DM particle masses up to
tens of TeV and, indeed, are the strongest limits derived from Fermi LAT data at TeV
energies. This is possible thanks to the new Fermi LAT IGRB measurement, which
now extends up to an energy of 820 GeV. We quantify uncertainties in detail and
show the potential this type of search offers for testing the WIMP paradigm with a
complementary and truly cosmological probe of DM particle signals.
Keywords: gamma ray experiments, dark matter experiments, dark matter theory,
gamma ray theory, dark matter simulations
Contents
1 Introduction 1
2 Theoretical predictions of the isotropic dark matter annihilation sig-
nals 3
2.1 Halo-model setup 5
2.2 Power-spectrum setup 7
2.3 Comparison of the two approaches and their cosmological dark matter
signal predictions 8
2.4 Galactic dark matter signal contributions 10
3 Constraints on WIMP signals 14
3.1 A review of the new Fermi LAT IGRB measurement 14
3.2 Known source contributors to the IGRB 15
3.3 Statistical analysis 16
3.3.1 WIMP signal search 17
3.3.2 Conservative approach for setting WIMP limits and the sensi-
tivity reach of the IGRB measurement 19
3.4 Limits on WIMP annihilation cross sections 20
4 Robustness of the IGRB measurement in the presence of a Galactic
dark matter signal component 25
5 Summary 29
A Diffuse foreground models and their impact on limits 32
B Limits at different confidence levels 34
1 Introduction
The Fermi Large Area Telescope (LAT) [1] provides a unique potential to measure
gamma-ray intensities with an almost uniform full-sky coverage in the energy range
from 20 MeV to greater than 300 GeV. Due to its good angular resolution, more
than 1800 gamma-ray point sources have been reported in the second source catalog
(2FGL) [2] and more than 500 sources with a hard spectrum above 10 GeV have also
been identified in the high-energy 1FHL catalog [3]. Most of these are of extragalactic
origin. In addition, the excellent discrimination between charged particles and gamma
rays allows LAT to directly measure diffuse gamma-ray emissions too. Note that this
emission is notoriously hard to measure with Cherenkov telescopes from the ground at
higher energies (above 100 GeV) due to isotropic cosmic-ray (CR) backgrounds (for a
recent effort see [4]). As a result, the Fermi LAT is in a unique position to measure the
– 1 –
diffuse emission from the Milky Way with good angular resolution and to establish an
isotropic emission, that is presumably of extragalactic origin, at energies greater than
10 GeV [5].
First detected by the SAS-2 satellite [6] and confirmed by EGRET [7], the isotropic
gamma-ray background (IGRB) is what remains of the extragalactic gamma-ray back-
ground after the contribution from the extragalactic sources detected in a given survey
has been subtracted1. The Fermi LAT collaboration has recently published a new
measurement of the IGRB [8] based on 50 months of data and extending the analy-
sis described in [5] down to 100 MeV and up to 820 GeV. The aim of this paper is
to use this new measurement to search for evidence of a possible contribution from
Weakly Interacting Massive Particle (WIMP) annihilation. This signal depends both
on cosmological aspects of the DM clustering and the WIMP properties, and therefore
potentially encodes a wealth of information. With the new measurements presented
in ref. [8], it is possible to test DM models over a wide mass range, thereby testing
candidates up to several tens of TeV (for Fermi LAT works that used other gamma-ray
measurements for indirect DM searches see e.g., [9,10]).
It is common to consider any DM annihilation signal viewed from the Sun’s po-
sition as having three contributions with distinct morphological characteristics and
spectra: Galactic smooth DM distribution, Galactic substructures and extragalactic
(or, equivalently, cosmological) signal. The extragalactic DM annihilation signal is
expected to be isotropic to a large degree and constitutes the main topic of our work,
but we will carefully explore the relevance of the smooth Galactic and Galactic sub-
structure components as well. This is one important addition to the methodology
presented in the original Fermi LAT publication on this topic [11]. In addition to that
we use improved theoretical predictions for the extragalactic DM signal, which both
takes advantage of a better knowledge of the DM clustering and its cosmic evolution
(section 2.1), and utilizes a complementary and novel method [12,13] to calculate the
extragalactic DM annihilation rate in Fourier space (section 2.2). Very interestingly,
we find the complementary approach to agree well with the improved predictions from
the traditional method (section 2.3). We also explore the degree of anisotropy of the
DM signal that originates from Galactic substructures, and include that component
consistently with the extragalactic DM emission (section 2.4). Finally, we present our
constraints on the total isotropic DM signal in section 3, for which we make minimal
assumptions on the isotropic signal of conventional astrophysical origin: we either as-
sume this astrophysical emission to be negligible (and derive conservative limits) or
we assume that its contribution is perfectly known and makes up the measured IGRB
intensity (thereby estimating the sensitivity reach of the IGRB measurement to WIMP
annihilation signals). We then study the robustness of the IGRB against adding a non-
isotropic smooth Galactic DM halo signal in the IGRB derivation procedure of ref. [8].
Specifically, we check for consistency of the IGRB measurement with the presence of
such a Galactic DM signal in section 4. We summarize the main results of our work
1Since the origin of the emission from detected sources is known by definition, it is appropriate to
consider only the IGRB in the search for any other components, as in this analysis to search for dark
matter signals.
– 2 –
in section 5.
2 Theoretical predictions of the isotropic dark matter anni-
hilation signals
The extragalactic gamma-ray intensity dφDM/dE0produced in annihilations of DM
particles with mass mχand self-annihilation cross section hσviover all redshifts zis
given by [14–16]:
dφDM
dE0
=chσvi(ΩDMρc)2
8π m2
χZdz e−τ(E0,z)(1 + z)3ζ(z)
H(z)
dN
dE
E=E0(1+z)(2.1)
where cis the speed of light, ΩDM is the current DM abundance measured with respect
to the critical density ρc,H(z) is the Hubble parameter or expansion rate, and dN/dE
is the spectrum of photons per DM annihilation.2The function τ(E , z) parametrizes
the absorption of photons due to the extragalactic background light. The flux multiplier
ζ(z), which is related to the variance of DM density in the Universe and measures the
amount of DM clustering at each given redshift, is the most uncertain astrophysical
quantity in this problem. It can be expressed both in real space, making use of a Halo
Model (HM) approach [17], and in Fourier space by means of a Power Spectrum (PS)
approach [12,13].
In the HM framework, ζ(z) is calculated by summing the contributions to the an-
nihilation signal from individual halos3of mass Mfrom all cosmic redshifts, hF(M, z)i,
and for all halo masses, i.e.:
ζ(z) = 1
ρcZMmin
dM dn
dM M∆(z)
3hF(M, z)i,(2.2)
where Mmin is the minimum halo mass considered, and ∆(z) and dn
dM are the DM
halo-mass over-density and the halo mass function, respectively. The mean halo over-
density ∆(z) is defined with respect to the critical density of the Universe and its
value determines the virial radius of a halo, R∆, at each redshift. In this paper we
will adopt ∆(0) = 200, following previous choices in the literature, and compute it
at different redshifts as in ref. [18]. The halo mass function dn
dM is normalized by
imposing that all mass in the Universe resides inside halos (see [14] for more details).
hF(M, z)iin turn depends on the DM halo density profile and the halo size. Halo
density profiles are determined by N-body cosmological simulations, with the most
recent results favoring cuspy Navarro-Frenk-White (NFW) [19] and Einasto halos [20,
21], while some astrophysical observations favor cored halos, e.g., Burkert density
profiles [22]. The density profile κcan be easily expressed in terms of a dimensionless
variable x=r/rs,rsbeing the scale radius at which the effective logarithmic slope of
2We assume here that the thermally-averaged annihilation cross section is velocity independent
and that DM is composed of self-conjugated particles.
3The term ‘halos’ refers to all types of virialized DM clumps and structures in the Universe that
lead to a DM density enhancement over the background.
– 3 –
the profile is −2. In this prescription, the virial radius R∆is usually parametrized by
the halo concentration c∆=R∆/rsand the function Fcan be written as follows:
F(M, z)≡c3
∆(M, z)Rc∆
0dx x2κ2(x)
Rc∆
0dx x2κ(x)2.(2.3)
More realistically Fis an average over the probability distribution of the relevant pa-
rameters (most notably c∆). Note that the above expression depends on the third
power of the concentration parameter. From simulations it has been determined that
the halo mass function and halo concentration decrease with halo mass and, conse-
quently, the flux multiplier ζ(z) given by eq. (2.2) is dominated by small mass halos
(as we will discuss in section 2.1). Furthermore, DM halos contain populations of
subhalos4, possibly characterized by different mean values of the relevant parameters
(e.g., different concentrations than those of field halos). The signals from subhalos are
typically included by adding an extra term in eq. (2.2) to account for halo substructure,
see [14].
As noted in [12] the flux multiplier can also be expressed directly in terms of
the non-linear matter power spectrum PN L (the Fourier transform of the two-point
correlation function of the matter density field):
ζ(z)≡ hδ2(z)i=Zkmax d k
k
k3PNL(k, z)
2π2≡Zkmax d k
k∆NL(k, z),(2.4)
where ∆NL(k, z)≡k3PN L (k, z)/(2π2) is the dimensionless non-linear power spectrum
and kmax(z) is the scale of the smallest structures that significantly contribute to the
cosmological annihilation signal. Loosely speaking, one could define a relation M=
4/3πρh(π/k)3with ρhthe characteristic density of the DM halo. Therefore kmax is the
PS counterpart to the minimal halo mass Mmin in eq. (2.2) in the HM prescription.
The extrapolation to masses or kscales beyond the resolution of N-body simu-
lations is the largest source of uncertainty in predictions of the extragalactic signal
of DM annihilation, since the smallest scales expected for the WIMP models are far
from being probed either by astrophysical observations or simulations.5Thus, different
methods of extrapolating to the smallest masses can lead to completely different re-
sults for the relevant quantities. Typical expectations for the minimum halo masses in
WIMP models are in the range Mmin ∈[10−9,10−4] M(see [26–28] and refs. therein),
while we only have observational evidence of structures down to 107M[29] implying
that extrapolations are required to span >
∼10 orders of magnitude in halo mass (or
>
∼3 orders of magnitude in k).
Both ways of expressing ζ, eq. (2.2) and (2.4), have advantages and disadvantages.
While eq. (2.2) is given in real space and thus deals with ‘intuitive’ quantities, it
4Subhalos are halos within the radius of another halo. A halo that does not reside inside any other
halo will be referred to as a field halo or, simply, halo.
5Notable exceptions on the simulation side are the works [23,24], which simulated a few individual
∼10−6Mhalos and the recent work [25], where for the first time dozens to thousands of halos have
been resolved with superb mass resolution in the range 4 ×10−4−2×10−6M.
– 4 –
depends to a large extent on several poorly constrained parameters, most notably the
concentration and halo mass function. This is particularly true for the smallest halos,
which, as we have noted, are expected to dominate in the evaluation of ζ. The same
is applicable to the subhalo population, whose internal properties and abundance are
even less well understood. In contrast, eq. (2.4) depends only on one quantity directly
determined from N-body simulations,6which can be extrapolated based on simple
scale-invariant arguments, but lacks the intuitive interpretation of breaking structures
into individual halos and subhalos, relevant when comparing the expected signals from
Milky Way substructures with the total cosmological one.
In this work, we use both of these approaches in parallel: the HM is used to define
our benchmark model following simple but well motivated arguments for the choice of
the relevant ingredients, and the PS framework is used to calculate the associated
uncertainty due to extrapolation to small (unresolved) scales.
2.1 Halo-model setup
The DM annihilation signal from a halo depends on the third power of the concentration
(see eq. 2.3), and the results will be extremely sensitive to the way the concentration-
mass relation is extrapolated to low mass. A common practice in the past has been to
use a single power law of Mfor c(M) all the way down to the minimum halo mass (see
e.g. [30–32]). In most cases, these power-law extrapolations assign very high concentra-
tions to the smallest halos, resulting in very high DM annihilation rates. In addition,
these results are extremely sensitive to the power-law index used. However, these
power-law extrapolations are unphysical and not expected in the ΛCDM cosmology
[33]. Indeed, since natal CDM concentrations are set by the halo formation epoch and
the smaller structures collapse at nearly the same time in the early Universe, low-mass
halos are expected to possess similar natal concentrations, and therefore are expected
to exhibit similar concentrations at the present epoch as well. In other words, the
expectation that c(M) at the small mass end flattens is deeply rooted in the ΛCDM
framework. This kind of behavior is correctly predicted by the c(M) model of ref. [34],
which explicitly relates halo concentrations to the root mean square (r.m.s.) of the
matter density fluctuations, σ(M). In addition, both the ΛCDM expectations and the
model predictions at the low-mass end are supported by the (few) results that come
from N-body simulations that were specifically designed to shed light on this extreme
small-halo-mass regime [23–25]. This was also pointed out in ref. [33] where, making
use of all available N-body simulation data, the authors examine the c(M) relation at
redshift zero for all halo masses (i.e. from Earth-mass microhalos up to galaxy clus-
ters). We refer the reader to these works for further details and discussion on the c(M)
behavior at the low mass end.
In what follows we define our benchmark halo model.
6The quantity is the non-linear matter power spectrum, which is determined using only a matter
density map, without invoking the concept of halos and without relying on standard halo finders.
– 5 –
•Cosmological parameters: we assume those recently derived from Planck data
[35], i.e., ΩM= 0.315, h= 0.673, σ8= 0.834.7
•Minimum halo mass: here we make a choice that has become standard in the
DM community, i.e. Mmin = 10−6h−1M, though we caution the reader that in
supersymmetric models this value is simply expected to be in the [10−9,10−4] M
range [26–28,36]. We explore how predictions of the extragalactic DM signal are
affected by the adopted Mmin value later in Section 2.3.
•Halo concentration: we adopt the model of ref. [34] discussed above, which im-
plicitly assumes a critical over-density, ∆ in eq. (2.2), equal to 200; thus it gives
a relation between c200 and M200.
•Halo mass function: we use the state-of-the-art halo mass function as proposed
by Tinker et al., [37], but with the parameters deduced by [34] at redshift zero
for the Planck cosmology.8Inspired by the prior work by Sheth & Tormen [38],
the Tinker et al. function gives a better agreement with N-body simulations
especially at the high-mass end.
•DM density profile: we use the familiar NFW profile [19]. The predicted values
of the flux multiplier ζ(z) are fairly independent of this choice (for the standard
assumptions of the profiles considered in this work). It was shown for example,
that the quantity hF(M, z)iin eq. (2.3) changes at the 10% level when the profile
is changed from the NFW profile to the cored Burkert profile (see also figure 3
in ref. [14]).
•Contribution of the subhalo population: while both the halo mass function and
halo concentrations are reasonably well studied, the properties of the subhalos
are more uncertain. In order to estimate the DM annihilation rate produced in
subhalos and its contribution to the total extragalactic signal, here we resort to
the results of the study recently presented in ref. [33]. Two parameters that con-
trol the amount of substructure and therefore its contribution to the annihilation
flux are the minimum subhalo mass and the slope of the subhalo mass function.
Following our choice for main DM halos, we adopt a value of 10−6h−1Mfor
the first of these two ingredients. As for the slope of the subhalo mass function,
dnsub
dM ∝(msub/Mhost)−α, we take α= 2, although we will also examine the impact
of changing this exponent to 1.9 in the next section. Following results found in
high-resolution N-body cosmological simulations of Milky-Way-sized halos above
their mass resolution limits [39,40], the normalization of this subhalo mass func-
tion is such that the mass contained in subhalos down to 10−6h−1Mis ∼45%
7The σ8is the r.m.s. amplitude of linear matter fluctuations in 8 h−1Mpc spheres at z= 0 with
the Hubble constant hdefined via H(z= 0) = h×100 km/s/Mpc.
8We still respect the parameters’ z-dependence found in ref. [37] though.
– 6 –
of the total parent halo mass.9The mentioned values correspond to the refer-
ence substructure boost model in ref. [33]. We note that this model implicitly
assumes that both subhalos and field halos share the exact same internal prop-
erties, which is probably not the case. Nevertheless, as discussed in ref. [33], this
choice represents a conservative case in terms of expected gamma-ray intensity
from annihilations in subhalos.
We show the extragalactic intensity predicted by the benchmark HM described
above in figure 1. Since the uncertainty of this signal is not easy to quantify within
the HM approach given its dependence on the many variables involved (see, e.g., [30,
41,42]), we use the PS approach to define our uncertainty band, as will be detailed in
the next subsection.
2.2 Power-spectrum setup
In this section we estimate the uncertainty affecting the evaluation of the flux multi-
plier, ζ. We focus on the PS approach, i.e. eq. (2.4), which relies on the work done in
ref. [13] using the data from the Millenium-II N-body cosmological simulation (MS-II)
[43]. MS-II has the highest mass resolution among available large-scale structure sim-
ulations, but its cosmological parameters are from early WMAP 1-year [44] and 2dF
Galaxy Redshift Survey [45] data. In ref. [13], it was shown that the largest uncertain-
ties in the calculation of ζcome from modeling the behavior of the power spectrum
beyond the MS-II resolution and the exact value of the cut-off scale. Lacking a better
theory, in [12,13] the extrapolation of the data was inspired by the behavior of ∆NL (k)
found in the simulation itself. In particular, at redshift zero, it was assumed that the
true ∆NL(k) value is bracketed by two alternative extrapolations for scales smaller
than k > k1%:
∆min(k) = ∆N L (k1%) (2.5)
∆max(k) = ∆N L (k1%)k
k1% neff
(2.6)
where neff =dln ∆(k1%)/d ln kand k1% is the scale at which the shot noise contribution
to the power spectrum ∆NL is 1% and sets the resolution threshold. In other words,
∆max(k) was found by imposing that the spectral index neff found at the resolution
threshold stays constant at larger kscales. This is conservative, as a flattening of
the power spectrum is predicted by theoretical arguments (see, e.g., [13] for further
details) while the extrapolation is from the trend observed in simulations at the smallest
resolved mass scales.
At higher redshifts, where MS-II resolves only the largest co-moving scales, an
additional constraint was imposed on ∆max(k): the ratio of the nonlinear to linear
power spectrum is not allowed to decrease with time at any fixed scale k(i.e., the
9Note, however, that the substructure mass fraction which is resolved in the simulations is only of
about 10%. The 45% quoted in the text refers to the total (resolved plus unresolved) fraction which
is obtained by extrapolating the subhalo mass function down to 10−6h−1M.
– 7 –
Universe can only become clumpier in time, at every co-moving scale). Similarly, in
order to also constrain the minimal ζvalue more precisely, ∆min (k) is required to have
an effective spectral index bounded from below by the spectral index predicted in linear
theory. For details on the physical motivations for these extrapolations see [13].
The cosmological parameters used for the MS-II simulation differ significantly
from the values recently measured by Planck, which we adopted in the HM approach.
The most critical factor in this regard is probably the value of σ8, since this parameter
has a large influence over the growth of fluctuations in the early Universe and thus on
the subsequent evolution of structures. The results of the HM approach are derived
assuming the most-up-to-date value σ8= 0.835 recently given by Planck data, but
MS-II adopts σ8= 0.9. Thus, in order to make a fair comparison between the two
methods, we follow [13] to apply the Planck cosmology to the results derived from
the MS-II. We comment on the particular choice of the cut-off scale, kmax, in the next
subsection.
2.3 Comparison of the two approaches and their cosmological dark matter
signal predictions
In figure 1we compare the values of ζobtained by means of the HM and PS approaches
described above. In order to make a proper comparison and extract meaningful con-
clusions, we call attention to a few caveats.
A conceptual issue in comparing the results of the two approaches is the deter-
mination of their cut-off scales. Such a cut-off exists, for example, due to the free
streaming length of DM particles after kinetic decoupling [26,27], which gives the
highest frequency relevant Fourier mode kmax.10 In the PS approach, kmax sets a sharp
upper cut-off on the matter power spectrum at the scale corresponding to the presumed
free streaming length in the linear regime, below which structures do not contribute
to the DM annihilation signal. In the HM approach, the cut-off is instead imposed
as a minimum halo mass,11 below which no halos are formed and therefore no anni-
hilation signal is expected. Within each DM halo, however, the annihilation signal is
calculated by extrapolating the adopted DM density profile of the halo down to r→0.
Strictly speaking the HM thus includes Fourier modes all the way to infinity, even if
the largest wave numbers would not contribute much. Note that the exact shape of
the DM density profile at those small scales as well as the mass of the first virialized
objects to form is still scarcely probed in simulations. As described in section 2.1 and
2.2, we chose Mmin = 10−6h−1Mas the cut-off scale in our benchmark HM model
and we adopt kmax =π/rsas the default choice for the corresponding cut-off in the
case of the PS approach (where rsis derived assuming the ∆ = 200 value used in the
HM approach, i.e. rs=R∆(z)c∆(z) = R200(z)c200(z)). We recall that other motivated
choices for kmax are possible, and refer the reader to ref. [13] for further details.
Another caveat when comparing HM to PS in figure 1is the signal contribution
from substructures that are present in extragalactic DM halos. As discussed in the
previous section, the PS-derived ζvalues implicitly include such signal by covering
10In the linear regime, the quantity kmax is a redshift independent quantity.
11Chosen to correspond to the linear free streaming scale.
– 8 –
0
1
2
3
4
5
6
104
105
106
z
H1+zL3ΖHzLH0HHzL
HM, Mmin, ss =10-6h-1M,Α = 2
HM, Mmin, ss =10-6h-1M,Α = 1.9
PS HmaxL
PS HminL
kmax = Π rs
Figure 1. Normalized ζas a function of redshift. A value of Mmin = 10−6h−1Mwas
used in both the PS (gray) and HM predictions (red). The benchmark HM model detailed
in section 2.1 is shown by the red solid line. The red dashed line corresponds to the case in
which the slope of the subhalo mass function varies from the fiducial α= 2 to 1.9 (i.e., less
substructure). The dotted line, labeled ‘PS (min)’, shows the minimum approximation from
Equation 2.5 in the PS approach, while the dashed line, ‘PS (max)’, shows the maximum
approximation given by Equation 2.6.
contributions down to length scales ∼π/kmax. In the HM approach, by contrast, the
substructures’ contribution is calculated by introducing additional parameters. We
show in figure 1the HM prediction for two different scenarios: the ones correspond-
ing to the minimum and maximum allowed values of the (substructure-induced) boost
factor to the annihilation signal from field halos as predicted in ref. [33] for a fixed
value of Mmin,ss = 10−6h−1M. In this case, the differences in boost factors are due
to different assumptions for the slope of the subhalo mass function, α(larger αval-
ues lead to more substructure and thus to larger boosts). As a consequence of the
aforementioned limitations, we expect some uncertainty when making a quantitative
comparison between the HM and PS approaches. Nevertheless, the agreement is quite
good as can be seen in figure 1, our benchmark HM prediction being within the mini-
mum and maximum PS values at all redshifts. We have so far explored the expected
WIMP signal for a given assumed cut-off scale Mmin (or, equivalently, kmax(z) defined
by π/rs). However, this effective cut-off scale can vary significantly between various
DM candidates, depending for example on their free-streaming lengths, as discussed
in, e.g., [27]. In figure 2we explore this dependence of ζon the cut-off scale Mmin and
kmax(z).
– 9 –
- --
[-⊙]
(+)ζ()/()
=
()
()
Figure 2. Variation of the value of ζat z= 0 as a function of the minimum halo mass con-
sidered, Mmin. Dotted and dashed lines represent minimum and maximum approximations
in the PS approach as described by Eqs. (2.5) and (2.6), respectively. In this work, we adopt
Mmin = 10−6h−1Mas our fiducial value. See text for further details.
Finally, we now turn to the calculation of the extragalactic gamma-ray spectrum,
by integrating ζ(z) over all redshifts and folding it with the induced spectrum from
WIMP annihilations, see eq. (2.1). To model the attenuation of gamma rays traveling
through cosmological distances, we adopt the Dom´ınguez et al. model [46] for the
extragalactic background light (EBL), which represents the state-of-the-art and is fully
consistent with the first direct detections of the EBL attenuation by the LAT [47] and
H.E.S.S. collaborations [48], and with the recently measured cosmic gamma-ray horizon
[49]. In figure 3we show a typical example for the gamma-ray flux resulting from 500
GeV DM particles annihilating through the b¯
bchannel. As we will discuss in detail
in section 3, the gray band in this figure translates directly into an uncertainty in the
DM limits.
2.4 Galactic dark matter signal contributions
Within a DM halo, the DM distribution has two distinct components: a centrally-
concentrated smooth distribution and a population of subhalos. While the smooth
component has been studied extensively over a large span of halo masses, it is only more
recently that several high-resolution simulations of Milky-Way-size halos addressed the
subhalo component in more detail. These simulations quantified the radial distribution
of subhalos inside the host halo, their abundance and their overall luminosity [39,50,
51]. The smooth and subhalo components are fundamentally different in terms of the
level of anisotropy and, in the following, we quantify their relevance for this work.
– 10 –
-
-
-
-
-
[]
/[ ---]
()
()
χ=
Figure 3. Comparison of the predicted cosmological DM-annihilation-induced gamma-ray
intensities as given by both the PS and HM approaches. The ζvalues implicitly used are
those given in figure 1. This particular example is for a 500 GeV DM particle annihilating
to b¯
bchannel with a cross section hσvi= 3 ×10−26 cm3/s. The signal range shown in gray
(which is computed within the PS approach) translates directly into uncertainties in the DM
limits of section 3.
The smooth DM density profile of the main halo, ρ(r), is found in pure DM N-body
simulations to be NFW [19] or Einasto [52,53], while more computationally-expensive
simulations which include baryonic effects have not yet converged on the profile shape in
the inner regions of the Galaxy, finding both more cored [54,55] and more cuspy profiles
[56,57]. Indeed, the solution to this issue might be substantially more complicated:
the ability for galaxies to retain their DM cusps may depend on the ratio of their stellar
and halo masses [58]. Observational tracers of the gravitational potential also cannot
be used to determine the DM profile within <
∼2 kpc from the Galactic center as it is
gravitationally dominated by baryons [59,60]. However, the considerable uncertainties
in the profile shape of the inner Galaxy are not critical for this work. We will deal
with Galactic latitudes >
∼20◦,∼3 kpc from the Galactic Center, a region in which
simulations and astrophysical evidence converge on the ρ∼r−2profile behavior.
Once the radial profile of the DM density is fixed, the remaining uncertainty
lies in its overall normalization, i.e. the value of ρ0at the solar radius, which can
take values in the range ρ0= 0.2–0.8 GeV cm−3[60,61].12 The gamma-ray signal is
proportional to the square of ρand therefore its uncertainty becomes greater than an
12In principle the values of rsand ρ0are not independently constrained, and they should be corre-
lated consistently. However, for our purposes the asserted ranges represent a fair description of the
uncertainties expected for the DM signal.
– 11 –
order of magnitude in the worst case. We will not consider any (portion) of the signal
from the smooth Galactic DM distribution to contribute to the isotropic emission (as
has been done in some previous works, e.g. [62]). Instead, we find that the whole-
sky DM template can be partially degenerate with at least one of the astrophysical
components present in the Galactic foreground emission. We will treat this signal from
the smooth Galactic DM halo as an additional component of the foreground Galactic
diffuse emission instead. We further discuss this choice and its impact on the derivation
of the IGRB spectra in section 4.
As far as the diffuse gamma-ray intensity from DM annihilations in the Galactic
subhalo population is concerned, some earlier works [11,63] found that it could appear
isotropic in our region of interest, and we explore this issue here in more detail. The
exact distribution in Galactocentric distance of subhalos is currently not well deter-
mined: in the Via Lactea II simulation [39] the subhalos follow the so-called anti-biased
relation with respect to the smooth DM density profile, i.e. ρsub (r)∝r×ρNFW/(r+ra)
with ra'85.5 kpc, while in the Aquarius simulation [40,50] they can be described
by ρsub ∝ρEinasto ∝exp n−2
αEhr
rsαE
−1io, with a particularly large scale radius
rs= 199 kpc and αE= 0.678 [63].
In figure 4(left) we show that the substructures give on average <
∼10 % anisotropy
in the relative intensity I/hIifor a DM-annihilation-induced signal when the DM
substructure distribution is described as in the original Aquarius simulation paper
[40]. However, for the same Aquarius simulation, authors in ref. [63] find a signal from
the substructures that is less isotropic.
Yet, the variations with respect to the average intensity are significantly less than
a factor of 2 for latitudes |b|>20◦. From the Via Lactea II simulation, the results
in ref. [39] give an anisotropy of I/hIithat is also less than a factor of 2, see figure 4
(right). These numbers can be compared to the Galactic smooth DM annihilation
signal intensity that varies by more than a factor 30 for latitudes |b|>20◦for the
NFW profile that we used.
We compare these findings with the sky residuals found when deriving the IGRB
spectrum in ref. [8], which are at the <
∼20% level. We conclude that, at least in the
case of the Aquarius simulation, the Galactic substructure would lead to a sufficiently
isotropic signal and thus we add it to the extragalactic signal when setting the DM
limits.
However, while the spatial distribution of the DM signal is taken directly from
the aforementioned simulations, the total signal strength is assigned self-consistently
with the total substructure signal as of a typical Milky-Way-sized DM halo (as used
in the calculation of the extragalactic signal within the HM approach). On the other
hand, while the extragalactic DM signal comes from a large ensemble of halos, the
properties of the Galactic substructure population are determined by the particular
formation history of the Milky Way galaxy. In that sense, its properties do not have to
correspond to the mean values found throughout the Universe, and in fact the Milky
Way is found to be atypical in several respects [43]. For that reason, we consider two
DM substructure prescriptions that introduce boosts of the total annihilation rate in
– 12 –
Figure 4. Anisotropy in the gamma-ray annihilation signal from the subhalo distribution
found in Aquarius [40] (left) and Via Lactea II [39] (right) simulations, with the former
following the prescription in ref. [63]. The plots show the intensities of the substructures
relative to their average intensity in the |b|>20◦region.
our Galactic DM halo by factors of 3 and 15. This range follows from the prescription
of [33] using a fixed minimum subhalo mass of Mmin,ss = 10−6h−1Mbut varying the
slope of the subhalo mass function (α= 1.9 and 2, respectively). In the next section, we
will show limits on DM annihilation cross sections from assuming these two bracketing
values on the substructure boost. Changing Mmin,ss to, e.g., 10−12 h−1Mwould not
affect the lower boost factor, but would increase the upper boost factor bound from
15 to about 40 (see ref. [33]).
Some of the largest or closest Galactic DM substructures could eventually be re-
solved as discrete gamma-ray sources. The contribution from few individual subhalos to
the total isotropic WIMP signal is not substantial, but nonetheless current constraints
on DM signals from dwarf spheroidal galaxies [10], as well as the non-detection of
DM signals from unassociated gamma-ray sources, e.g., [64–66], significantly limit the
total annihilation signal from the DM subhalos in the Milky Way. Our approach is
to include the total expected DM signals from all subhalos of all masses in our eval-
uation of the DM signal contribution to the IGRB, but when DM limits from, e.g.,
dwarf spheroidal galaxies are stronger they obviously also impose limits on the total
expected Galactic DM substructures contribution to the IGRB. Yet in these cases our
limits are still relevant, as they represent an independent probe of cross sections by
means of a conceptually different approach.
While the gamma-ray signal originating from Galactic substructure could appear
reasonably isotropic, an important difference with the extragalactic signal is in the
spectral shape: the extragalactic signal is redshifted and distorted by absorption on
the EBL (c.f. eq. (2.1)) while the Galactic signal directly reflects the injection spec-
trum of gamma rays from DM annihilations and is generally harder. For that reason
– 13 –
it is important to take this component properly into account, especially for heavy
DM candidates, for which EBL absorption can severely limit high-energy gamma-ray
intensities.
3 Constraints on WIMP signals
3.1 A review of the new Fermi LAT IGRB measurement
Before we derive constraints on DM signals, let us summarize the four main steps that
were taken in the analysis of ref. [8] to measure the IGRB, which will be used in this
section and further discussed in the next section. In total, 50 months of LAT data
were used, and dedicated cuts—creating two new event classes—were used to produce
data samples with minimal contamination of CR backgrounds. In particular, the data
were divided into a low-energy data set (events used to derive the spectrum in the 100
MeV to 12.8 GeV range) and a high-energy sample (12.8 GeV to 820 GeV). Stronger
cuts were applied in the low-energy range, where the event statistics are better but
CR contamination is higher; the cuts were loosened at high energies, where both event
statistics and CR contamination are lower. The full-sky data were then analyzed as
follows:
1. The full sky gamma-ray emission was modeled with a series of templates. Tem-
plates of the Galactic diffuse emission are produced with the GALPROP code13
[67], based on three distinct diffuse model setups, dubbed models A, B and C
(see Appendix Aand [8] for further details). The templates include maps of the
gamma-ray emission due to interactions of CRs with interstellar gas and the in-
verse Compton (IC) emission separately. In addition, templates modeling the IC
emission of CR electrons in the Solar radiation field, diffuse emission from Loop
I and point sources from the 2FGL catalog were used. After masking regions of
bright interstellar emission along the Galactic plane, the normalization of each
template was fitted individually in energy bins in the range between 100 MeV
and 12.8 GeV.
2. Above 51.2 GeV the event statistics do not allow for fitting in individual energy
bins. To handle low statistics at high energies, the low-energy data set in the
range from 6.4 GeV to 51.2 GeV was used to find the best-fit normalizations
of the Galactic templates. The normalizations were then fixed to these best-
fit values, and GALPROP’s predictions of the spectral shapes were applied up to
820 GeV in order to perform IGRB fits above 12.8 GeV using the high-energy
data sample.
3. The isotropic component thus derived is a sum of the IGRB emission and mis-
classified particle backgrounds. The IGRB is then obtained by subtracting a
13http://galprop.stanford.edu
– 14 –
model for the CR contamination, obtained from Monte Carlo studies,14 from the
isotropic emission.
4. Using a baseline model for the Galactic diffuse emission (model A in ref. [8]),
the above procedure was then repeated for different values of the relevant CR
parameters. The scatter among the different IGRB spectra derived in this way,
together with those derived by assuming foreground models B and C in [8],
represents the systematic error band, indicated in figure 12 of ref. [8].
The result of this procedure is a measure of the spectrum of the IGRB in the range
from 100 MeV to 820 GeV [8]. It should be noted that this measurement is performed
without including any Galactic smooth DM signal template, and the effects of such
non-isotropic DM signal will be discussed in section 4.
3.2 Known source contributors to the IGRB
The Fermi LAT has detected many extragalactic sources: among the 1873 sources
in the 2FGL catalog, there are 672 blazars (all classified according to the Roma BZ-
CAT15), 8 radio galaxies, 3 normal galaxies, 3 starburst galaxies and 2 Seyfert galaxies
[2].16 The contribution to the IGRB from unresolved members of these extragalactic
source classes has been studied over the years, e.g., [69–80]. In addition, some classes
of Galactic sources, most notably millisecond pulsars, could contribute to the isotropic
emission at large scale height in the Milky Way [81]. Their contribution to the IGRB
is however severely constrained by the strong gamma-ray angular anisotropy signal
expected for this source class [82,83]. There are other truly diffuse emission processes
that are expected to contribute to the IGRB as well, although probably only at a few
percent level, e.g. structure formation shocks in clusters of galaxies [84] and giant radio
lobes of FR II radio galaxies [85].
Overall, the origin and composition of the IGRB are still open questions. Because
of the large number of blazars detected by the LAT, direct population studies are now
feasible using gamma rays and there is arguably a guaranteed contribution from the
blazar population [69–73]. The minimum contribution below 100 GeV from unresolved
blazars has been estimated in ref. [70] to be close to 10%, the best estimate being
22 −34% in the 0.1-100 GeV range (which agrees well with previous findings, e.g.
[72,86,87]). The blazar contribution to the IGRB at the highest energies has only
recently been studied. In ref. [72] they used a preliminary version of the new IGRB
measurement reported in [8] and concluded that blazars can naturally explain the total
measured IGRB above 100 GeV.
For the other known source classes, however, we lack this kind of direct informa-
tion, and cross correlations with radio (in the case of radio galaxies, see, e.g., [74–77])
14The Monte Carlo studies include a simulation of the relevant charged particle species and in-
tensities present in the near-Earth environment as well as a phenomenological model for gamma-ray
emission from the Earth limb.
15v4.1, http://www.asdc.asi.it/bzcat/
16There are 354 additional sources all associated in the 2FGL that appear to have blazar-like
temporal or spectral characteristics but for which the lack of optical spectra did not allow a precise
classification, most of them being labeled as AGN of uncertain type [68].
– 15 –
or infrared data (for star-forming galaxies, e.g., [78]) have been used to determine the
luminosity functions and infer the expected intensities in the Fermi LAT energy range.
In a companion Fermi LAT paper [88], the contribution of blazars in the full energy
range has been reevaluated using an updated luminosity function and spectral energy
distribution model, taking advantage of recent follow-up observations [71]. When sum-
ming the contribution from star-forming galaxies [78], radio galaxies [76] and blazars,
ref. [88] shows that these three contributors could account for the intensity of the EGB
across the 0.1 - 820 GeV range sampled by Fermi LAT. In ref. [88], the methodology
of this work was adopted to derive DM limits taking advantage of the aforementioned
new estimates of the astrophysical contributions. Yet, since intensity estimates for
each of these potential IGRB contributors are uncertain (or under study) at the mo-
ment, in this work we stay agnostic about the precise contribution of the astrophysical
populations to the IGRB and instead aim for a more general approach.
3.3 Statistical analysis
We form a test statistic (T S) with a presumed χ2distribution:
T S =χ2=X
ij
(Di−Mi)V−1
ij (Dj−Mj),(3.1)
where Diis the measured IGRB intensity in energy bin i,V−1is the inverse of the
variance-covariance matrix and Miis the IGRB model prediction (see below).
On top of statistical uncertainties, the IGRB measurement inherits significant
systematic uncertainties from the effective-area and the CR-contamination determina-
tion. These systematic uncertainties, combined with the IGRB measurement procedure
(summarized in section 3.1), can induce correlations between the IGRB measurements
in the different energy bins. This should be reflected by a proper variance-covariance
matrix V, and we made a study to estimate the variance-covariance matrix and check
its impact on DM limits compared to the common approximation of taking it to be
diagonal.
To establish the expected correlation matrix for the data, 1000 Monte Carlo-
generated pseudo experiments were created. Gamma-ray sky maps were generated
with the help of HEALPix17 by taking the number of events in each sky pixel to be
Poisson-distributed around the observed number of events. The effective area and
the CR contamination were drawn from normal distributions around their nominal
values in [8]. To account for bin-to-bin correlations of the systematic uncertainty of
the effective area we included correlations on the scale of three adjacent energy bins
[89]. The CR contamination in the low- and high-energy data samples were taken
to be fully uncorrelated. However, within each of these data samples, the systematic
uncertainty (taken from [8]) was used to induce a randomized overall shift factor for the
CR contamination rate level. The remaining (subdominant) statistical uncertainties
for the CR contamination were taken to be fully uncorrelated.
17http://healpix.jpl.nasa.gov/
– 16 –
Each Monte Carlo-generated data set was then used to perform IGRB measure-
ments exactly as done with the real data (using model A for the Galactic diffuse
emission) and taking the CR and effective area determinations as described above.
From the sample of 1000 IGRB pseudo measurements, the correlation matrix can then
be directly calculated [90]. Subsequently, the variance-covariance matrix is determined
by using the IGRB total variances (i.e. the sum of the variances from statistical, CR
contamination and effective area uncertainties) in each energy bin as they were given
in ref. [8]. The correlations are seen to be the strongest between neighboring energy
bins at low energies, while energy bins further away and at the highest energies have
negligible correlations. The derived variance-covariance matrix was then directly used
in our statistical DM analysis (i.e. we included our calculated Vin the T S calculation
of eq. (3.1)), where it typically induced about a factor two increase in χ2but less effects
on ∆χ2. From this study we concluded that the impact on DM limits from including a
proper variance-covariance matrix can have a sizeable effect but are typically smaller
than the shift in DM limits coming from changing the diffuse foreground modeling
shown in Appendix A.
For our final analysis, we therefore decided to treat all data points Dias uncor-
related with Gaussian probability distributions in our χ2evaluations. The variance-
covariance matrix Vis thus approximated as diagonal with elements σi2. The system-
atic uncertainties of the effective area and the charged CR contamination as well as
the statistical errors are added in quadrature and their sum is the σi2that enter in the
covariance matrix V. The IGRB spectrum data points Diand the just specified 1-σ
errors σican be found in the supplementary tables of ref. [8].18
In addition, there is a significant systematic uncertainty due to the assumed Galac-
tic foreground emission model in the IGRB derivation. The investigated range of
Galactic foreground emission assumptions can induce correlated IGRB data points,
and uncertainties are presumably very asymmetric. Galactic foreground uncertainties
will therefore not be included in the evaluation of the χ2in eq. (3.1), but their impact
was taken into account or estimated separately in our procedures, as we detail below.
3.3.1 WIMP signal search
Before setting any limits, we will use a single power law with an exponential cut-off as
a null-hypothesis background,
dφbkg
dE0
=AE−α
0exp −E0
Ec,(3.2)
and search for any significant detection if a WIMP signal is added on top of this. This
background should be viewed as the effective IGRB spectrum from all conventional as-
trophysical sources discussed in section 3.2, where the overall normalization A, spectral
index αand exponential cut-off energy Ecare free parameters.
For the DM signal search we use the IGRB derived with model A for the Galactic
diffuse emission and only the σierrors are included in the χ2calculations. The simple
18https://www-glast.stanford.edu/pub_data/845/
– 17 –
background model of eq. (3.2) gives an excellent best-fit to the 26 data points of the
IGRB spectrum; with a χ2of 13.7 for 23 degrees-of-freedom.19 Naively this leaves little
need for including an additional DM signal. To still search for a statistically significant
DM signal we use the DM set-up described in section 2, where the isotropic DM signal
is the sum of the contributions from cosmological signal and Galactic substructures.
For the cosmological calculation we use the HM result, shown by the red solid line
in figure 1. Also, since the ratio of DM extragalactic to Galactic substructure signals
affects the total DM signal spectral shape, we investigate the theoretical uncertainty
range in the extragalactic signal strength by using the lowest and highest results from
the PS approach (given by the limits of the gray band in figure 1). As for the Galactic
substructures, in combination with either the HM or PS cosmological signals, we use
two different overall signal strengths (corresponding to boosts of 3 and 15 for the Galaxy
DM halo signal as a whole). The different WIMP annihilation channels we test are
the same as those specified in section 3.4, where we present our WIMP annihilation
cross-section limits and derive our sensitivity reach.
The χ2difference between the best-fit including such additional WIMP signal
on top of the background (with the WIMP annihilation cross section as one extra
degree of freedom) and the null hypothesis with zero DM signal, reveals that none of
the DM hypotheses we tested showed a fit improvement by more than ∆T S = 8.3.
Assuming that our T S has a χ2distribution, the local significance is 2.9σ(before
including any trials factor). The largest significance was in our minimal setup for both
the extragalactic signal and the Galactic substructure signal, with a WIMP mass of
500 GeV and an annihilation cross section into µ+µ−pairs of 1.1×10−23 cm3/s. Note
that this significance is not large, especially since a trials factor has not been yet applied
(among 192 models we tested we had more than 2σdetections for 16 different models).
More importantly, uncertainties in the IGRB related to the selected Galactic diffuse
emission model were not included. In fact, we also performed the analysis with IGRB
derived with Galactic diffuse model B and C and confirmed the DM non-detection
obtained with model A. For example, the ∆T S = 8.3 mentioned above drops to 3.9
when the IGRB derived with the Galactic diffuse model B is used.
We test at most nine WIMP masses for each annihilation channel: 10, 50, 100, 500,
1000, 5000, 10000, 20000, and 30000 GeV. This leaves a possibility that a somewhat
more significant detection could be found if smaller steps in WIMP mass were used.
For the tested WIMP masses we therefore conclude that there is no clear statis-
tically significant evidence of WIMP signals in the IGRB, and we proceed to calculate
upper limits. This is a rather naive approach and, indeed, a better understanding of
the astrophysical contributions to the IGRB could help in revealing potential anomalies
or even point toward the need for a DM contribution to the measured IGRB.
19This small χ2value is presumably related to the fact that LAT’s systematic effective area and
CR contamination uncertainties are included in the σiwhile their induced correlations are ignored in
the variance-covariance matrix. If we include the variance-covariance matrix discussed in section 3.3
then χ2become 21.0; which is in good agreement with what should be expected.
– 18 –
3.3.2 Conservative approach for setting WIMP limits and the sensitivity
reach of the IGRB measurement
We will focus on deriving i) conservative DM limits, derived by making assumptions
neither on the contributions from unresolved astrophysical populations to the IGRB
nor on a specific choice of a Galactic diffuse emission model, and ii) sensitivity reach,
which assumes both that the total contribution from conventional astrophysical sources
fully explains the measured IGRB at all energies by eq. (3.2), and that we can entirely
rely on a specific Galactic diffuse foreground model to derive the IGRB.
We adopt the following two procedures:
•Conservative limits: The χ2in eq. (3.1) is calculated only for bins where the
DM signal alone exceeds the IGRB intensity:
χ2
cons =X
i∈{i|φDM
i>Dmax
i}Dmax
i−φDM
i(hσvi)2
σ2
i
,(3.3)
where φDM
iis the integrated DM-induced intensities in energy bin ias a function
of hσvifor a given WIMP candidate. Effectively, this corresponds to having a
(non-negative) background model that is free in normalization in each energy bin
independently. To take into account the fact that various Galactic foreground
emission models can alter the IGRB, we shift the centers of the IGRB data points
derived with Galactic emission model A (while we keep the size of the errors σi)
to the upper edge of the 1-σenvelope of all tested Galactic diffuse models used
in ref. [8]. This yields our data points Dmax
i. The mentioned shift of the IGRB
bins reflects the conservative approach of considering all tested diffuse emission
models in ref. [8] equally likely for determining the IGRB (maximal) intensity
in every energy bin. The 2σ(3σ) DM limits are then defined to be the cases in
which the DM signal component gives χ2
cons equal to 4 (9).20
•Sensitivity reach: In this procedure, we use the IGRB derived with Galactic
diffuse model A and include the DM signal on top of the background model in
eq. (3.2). The χ2of eq. (3.1) is then evaluated over all energy bins:
χ2
sens =X
i
hDi−φbkg
i(A, α, Ec)−φDM
i(hσvi)i2
σ2
i
,(3.4)
where φDM
i+φbkg
iis the model prediction Mias a function of hσvifor a given
WIMP candidate. Note that this represents a scenario in which i) the Galactic
diffuse foreground used to derive the IGRB is fixed, ii) the contribution from
20To avoid potential issues with correlations among the data points, we also performed the exercise
of setting our limits by using one single bin (the one in which DM signal to the IGRB flux ratio is
maximal) instead of a sum over bins as in eq. (3.3). We found that the two approaches lead to limits
which differ by only <
∼10%.
– 19 –
conventional astrophysical sources to the IGRB is described by the parametric
form of eq. (3.2) and iii) the parameters A,α,Ecin eq. (3.2) are fixed to their
best-fit values found in ref. [8] (given in their table 4). The 2σ(3σ) limits are
then defined to be the cases in which the DM signal component forces the χ2
sens
to increase by more than 4 (9) with respect to the best-fit χ2
sens with a free DM
signal normalization.
We have also checked how this sensitivity reach changes by varying the adopted
Galactic foreground model, namely by comparing limits when the IGRB is derived
with the alternative foreground models A, B and C in ref. [8]. With this exercise, we
gauge the impact of some systematic uncertainties associated with the modeling of the
Galactic diffuse emission. We find differences that can be substantial especially for
low WIMP masses; see Appendix Afor further details. Yet, it should be noted that
these tests are far from comprehensive and, as such, might not address the full range
of uncertainties.
The sensitivity reach derived here can also be taken as limits under the given
assumptions. However, strictly speaking they should be interpreted as DM constraints
only if the astrophysical background was independently predicted to the spectrum of
eq. (3.2) with parameters equal to the best-fit values from the current IGRB measure-
ment.
The case where the total contribution to the IGRB from conventional astrophysics
is derived as accurately as possible leads to DM constraints that typically lie between
the conservative limit and the sensitivity reach derived in this work. Indeed, this is
what is obtained in a companion work [88], where unresolved astrophysical source
populations were modeled and used to set new DM limits on DM annihilation cross
sections.
In figures 5and 6we show illustrative examples of DM-induced spectra which
have DM annihilation cross sections at the size of our 95% CL exclusion limits by our
conservative approach and our sensitivity reaches, respectively.
3.4 Limits on WIMP annihilation cross sections
In this work, we stay agnostic about the nature of the DM particle and consider
generic models in which DM annihilates with 100% branching ratio into b¯
b,W+W−,
τ+τ−or µ+µ−channels. For the first three channels we consider the prompt emission
only, because for these channels generally less energy goes to electrons with respect
to the production of gamma rays, and secondary emission from IC electron scattering
on cosmic microwave background (CMB) photons is expected to be subdominant.
On the other hand, annihilation to muons induces a hard gamma-ray spectrum due
to the Final State Radiation (FSR) emission and a large amount of hard electrons
results in a pronounced intensity enhancement at lower energies due to the electron
scattering on the CMB, which is present with increased intensity at higher redshifts.
For that reason this annihilation channel proves to be especially strongly constrained
by the IGRB measurement [11]. We calculate the DM annihilation prompt spectra
using the publicly available PPPC4DMID code [61], which takes into account electroweak
– 20 –
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Figure 5. Examples of DM-produced gamma-ray spectra which are at the border of being
excluded by our 2σconservative limits. The WIMP mass and its annihilation channel is
given in the upper left corner of each panel. The normalizations of the extragalactic signal
and of the Galactic substructure signal are given by our benchmark HM model, as defined
in section 2.1. Data points are in black, and the black lines show the upper and lower
envelopes of the systematic uncertainties defined as the scatter among the different IGRB
spectra derived in ref. [8].
bremsstrahlung corrections, which are particularly relevant for heavy DM candidates.
For the calculation of the IC emission from the muon channel we follow the calculation
presented in [11].
For the four annihilation channels under consideration, we present the conserva-
tive limits and cross-section sensitivity reach at the 2σconfidence level in figures 7and
8, respectively. In all cases, the DM limits were obtained by adopting the cosmologi-
cal DM annihilation induced gamma-ray intensities given by the HM setup described
in section 2.1, as well as a theoretical uncertainty range as estimated within the PS
approach of section 2.2 (gray band in figure 1). In addition, two configurations for the
Galactic substructure contribution—which is assumed to be isotropic in this work—are
adopted: i) the reference case, labeled as ”SS-REF” in figures 7and 8, where substruc-
tures boost the total Galactic annihilation signal by a factor of 15, and ii) the minimal
case, labeled ”SS-MIN” in the figures, where the boost from Galactic substructure is
3. Conservative DM limits and cross-section sensitivities at the 3σlevel for the b¯
band
τ+τ−channels were also derived, and can be found in Appendix B.
From theoretical considerations, various DM particle candidate masses span a
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Power Law +Exp cut-off
Sum
Μ+Μ-,1 TeV
Figure 6. Examples of DM-produced gamma-ray spectra which are at the border of being
excluded at 2σlevel in our procedure to calculate the sensitivity reach of the IGRB data.
The WIMP mass and its annihilation channel is given in the upper left corner of each panel.
The normalizations of the extragalactic signal and of the Galactic substructure signal are
given by our benchmark HM model, as defined in section 2.1. Data points from ref. [8].
huge range. For thermally produced WIMPs, however, the Lee-Weinberg limit restricts
the mass to be above few GeV [93] and unitarity considerations bound it to be below
∼100 TeV [94]. Interestingly, we are able to constrain signals for WIMP masses
up to ∼30 TeV because the IGRB measurement now extends up to 820 GeV. For
DM particle masses above ∼30 TeV, we start to probe the low-energy tail of the
DM spectra and thus we lose constraining power rapidly. Furthermore, extragalactic
WIMP signals are heavily suppressed at the highest energies as the optical depth is
very large for such gamma rays.
It is interesting to compare the conservative limits of figure 7to the cross-section
sensitivities in figure 8, at least for the case of our fiducial HM scenario and the reference
contribution from the Galactic subhalo population (‘HM, SS-REF’ case in the panels).
For the b¯
b(τ+τ−) channel, the differences are of about factors 9, 25, 11, 3 (26, 9, 4, 3)
at 10 GeV, 100 GeV, 1 TeV, 10 TeV.
For low WIMP masses, the full spectral shape of the IGRB is affected by the
WIMP signal, and hence the sensitivity reach, assuming a known spectral shape for
the astrophysical contributions to the IGRB, places stronger limits, whereas for the
largest WIMP masses only the last point(s) in the IGRB spectrum is affected and the
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Figure 7. Upper limits (95% CL) on the DM annihilation cross section in our conservative
procedure. From top to bottom and left to right, the limits are for the b¯
b,W+W−,τ+τ−and
µ+µ−channels. The red solid line shows limits obtained in our fiducial HM scenario described
in section 2.1, and assumes the reference contribution from the Galactic subhalo population;
see section 2.4 (‘HM, SS-REF’ case). The broad red band labeled as ‘PS (min→max), SS-
REF’ shows the theoretical uncertainty in the extragalactic signal as given by the PS approach
of section 2.2. The blue dashed line (‘HM, SS-MIN’) , with its corresponding uncertainty
band (‘PS (min→max), SS-MIN’), refers instead to the limits obtained when the Milky
Way substructure signal strength is taken to its lowest value as calculated in ref. [33]. For
comparison, we also include other limits derived from observations with Fermi LAT [9,10]
and imaging air Cherenkov telescopes [91,92].
two approaches are more similar.21
For the largest WIMP masses considered, the signal from Galactic substructures
is stronger than that from the extragalactic DM, with the effect that the uncertainty
range of the extragalactic WIMP signal becomes irrelevant when setting DM limits
and calculating cross-section sensitivities. This is typically the case for gamma-ray
energies above 100 GeV, where extragalactic signals are effectively attenuated due to
EBL attenuation. The effect can be clearly seen in figures 5and 6for several annihi-
lation channels (see, e.g., the spectra of a 5 TeV mass DM particle annihilating to the
21If we omit the last data point, we find that both conservative limits and cross-section sensitivity
for the b¯
bchannel worsen by <
∼30% at 5 TeV mass going up to a factor of ∼2 for masses between 10
and 30 TeV. In the case of the harder τ+τ−channel, limits and sensitivity reach progressively weaken
by a factor ∼2 to 4 between 2 and 30 TeV, respectively.
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Figure 8. DM annihilation cross section sensitivity reach (95% CL). Green solid line shows
sensitivity obtained in our fiducial HM scenario described in section 2.1, and assumes the
reference contribution from the Galactic subhalo population; see section 2.4 (‘HM, SS-REF’
case in the panels). The broad green band labeled as ‘PS (min→max), SS-REF’ shows
the theoretical uncertainty in the extragalactic signal as given by the PS approach of sec-
tion 2.2. The orange dashed line (‘HM, SS-MIN’), with its corresponding uncertainty band
(‘PS (min→max), SS-MIN’), refers instead to the cross-section sensitivity obtained when the
Milky Way substructure signal strength is taken to its lowest value as calculated in ref. [33].
For comparison, we also include other limits derived from observations with Fermi LAT [9,10]
and imaging air Cherenkov telescopes [91,92].
W+W−). As a result, as the WIMP mass increases in figures 7and 8, the cross-section
limit uncertainties get narrower (for a given Galactic substructure signal strength). For
the same reason, the uncertainty band for the minimal Galactic substructure scenario
(’SS-MIN’ case in figures 7and 8) is typically wider than the one for the reference
Galactic substructure case (‘SS-REF’), especially at the largest WIMP masses consid-
ered. This is less pronounced for the muon channel, because in that case the high-mass
limits are still set by the inverse Compton peak of the emission which contributes at
low energies.
Another feature worth mentioning is that, in the case of DM annihilation into
µ+µ−, figures 7and 8show a dip in cross-section limits for DM particle masses around
1 TeV. This dip is present because the part of the gamma-ray spectrum induced by
FSR peaks at energies where the IGRB intensity has dropped exponentially above
a few hundred GeV (see figure 5). For larger WIMP masses, the FSR peak is well
– 24 –
above the energy range covered by the LAT IGRB measurement and, as the WIMP
mass increases, the limits get progressively weaker until lower-energy gamma rays—
induced by IC upscattering of CMB photons from DM-induced high-energy electrons—
eventually govern the constraints.
Finally, it is also interesting to compare the conservative limits and cross-section
sensitivities obtained in this work to other DM limits recently reported in the litera-
ture. At low masses, <
∼100 GeV, the derived sensitivity reach is comparable to the
limits derived from a stacking analysis of 25 dwarf spheroidal satellites of the Milky
Way [10], and to the limits derived from considering diffuse emission at intermediate
Galactic latitudes [9]. We note that the present analysis uses LAT data up to very high
energies (820 GeV), which represents a novelty with respect to previous Fermi LAT
DM searches. In order to put in perspective the results derived here, we also compare
to DM constraints derived from ground-based atmospheric Cherenkov telescope obser-
vations. Figures 7and 8show the limits derived by the H.E.S.S. Collaboration from
observing the Galactic center halo [91] and the MAGIC Collaboration’s limits derived
from deep observations of the Segue 1 dwarf spheroidal galaxy [92]. Our conservative
limits are comparable to the latter ones in the TeV energy range, but weaker than those
obtained by H.E.S.S.22 As for the cross-section sensitivity reach, this is substantially
better than the mentioned MAGIC limits, and comparable to the ones from H.E.S.S.
Our work, which uses the IGRB’s total intensity to set constraints on the nature of
DM, is connected to studies focused on small-scale angular anisotropies in the high-
latitude gamma-ray sky [41,95,96]. Indeed, they both test for the presence of the
same gamma-ray source class. In this regard the limits are directly comparable, and
their DM signal limits are currently of similar magnitude [95,96]. It has also been
shown that a cross-correlation of an anisotropic signal with the positions of galaxies in
the 2MASS survey [97] or with weak lensing surveys [98,99] could increase the ability
to detect a DM component.
4 Robustness of the IGRB measurement in the presence of a
Galactic dark matter signal component
The largest component of the systematic uncertainty in the measurement of the IGRB
spectrum stems from the modeling of the Galactic diffuse emission [8]. A signal from
the smooth Milky Way DM component would contribute to Galactic diffuse emission
and, as seen in ref. [9], this signal can in part be degenerate with conventional as-
trophysical emissions. Indeed, the DM component is morphologically similar to the
Inverse Compton (IC) astrophysical emission and, in ref. [8], it is seen that uncer-
tainties in the IC template have the most significant impact on the measured IGRB
spectrum.
In order to study this issue we repeat the fitting procedure in [8] and model the
Galactic diffuse emission, but with the addition of a Galactic smooth DM template.
22We caution the reader that the H.E.S.S. limits are derived under the assumption of a cuspy profile,
whereas DM limits from IGRB data are only moderately sensitive to the inner slope of the DM halo
density profiles.
– 25 –
Our aim is twofold: i) to verify that DM and Galactic diffuse templates are partially
degenerate with each other in the fitting procedure, and ii) to a posteriori check for self-
consistency of our procedure, i.e. for those DM annihilation cross sections constrained
in section 3.4, we test whether the corresponding Galactic DM counterpart emission
alters the IGRB measurement that was used to set the DM limits themselves.
Following ref. [8], we perform low- and high-energy template fits separately (as
described in section 3.1). We use templates of 10, 50, 250 and 500 GeV, and 1, 5,
20 and 30 TeV DM particle masses annihilating to b¯
bquarks and τ+τ−leptons, as
representative gamma-ray signals from WIMP annihilation.23 We produce full-sky
templates using the GALPROP code version 54 [100] into which we have incorporated
the DM signal. We then evaluate the maximum value of the DM annihilation cross
section (for each DM mass) which still leaves the IGRB measurement unchanged. The
IGRB is taken as ‘unchanged’ when, after the inclusion of the Galactic smooth DM
template, the new IGRB measurement for all energy bins falls inside either i) twice the
width of the systematic uncertainty band derived from foreground model variations
or ii) twice the 1σ‘statistical’ error bars of the IGRB measurement of ref. [8]. This
maximal normalization can be translated into a maximal cross section once a particular
Galactic DM density distribution is adopted. To be conservative—in the sense of
finding the corresponding DM annihilation cross-section values in our Galaxy which
are ‘guaranteed’ to modify the IGRB—we set the local DM density to a low value of
ρ0= 0.2 GeV cm−3while keeping the Milky Way scale radius at the standard value of
rs= 20 kpc, see e.g. [61].24
Interestingly, while repeating the above procedure with different normalizations
of the Galactic smooth DM templates we found that, for DM masses below ∼1 TeV,
an increase of the Galactic smooth DM template normalization does not translate in a
proportional reduction of the IGRB. Instead, this change gets compensated by a lower
IC template which, as the DM normalization increases, can go down to almost zero
in the corresponding energy bins before the IGRB measurement is altered. From this
exercise, we conclude that a potential smooth Galactic DM signal could be (partially)
modeled/absorbed by the IC templates, which thus may unintentionally subtract a DM
signal in the IGRB measurement. This is also mostly the reason why we decided not to
add any portion (e.g. the high-latitude emission) of the Galactic smooth DM template
to the isotropic signal, as it was often done in literature, e.g. in refs. [62,79]. For high-
mass DM templates, however, degeneracies between the DM and the IC templates are
limited, because the normalizations of the IC templates at these energies are fixed from
gamma-ray data at intermediate energies (see point 2 in section 3.1).
Note that this prescription does not guarantee that morphological residuals are
kept small, since the methodology only investigates deviations in the IGRB spectrum.
By also using the morphological information we could impose tighter constraints on a
Galactic DM annihilation signal because large-scale anisotropic residuals could poten-
23Note that the morphology of the Galactic DM templates is independent of the DM annihilation
channel considered, as long as one refers to the photons induced ‘directly’ in the annihilation process.
24Strictly speaking, rsand ρ0are not independent, but for our purposes lowering ρ0alone serves
as a reasonable way to adopt a low but realistic Galactic DM signal.
– 26 –
102103104105
E
[MeV]
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[MeV s
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modified IGRB
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102103104105
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10-3
E
2
φ
[MeV s
−
1cm
−
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−
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original IGRB
modified IGRB
one sigma stat. error
Figure 9.Left: The new IGRB measurement, after the inclusion of the Galactic smooth
DM template, when the measurement for some energy bins falls outside twice the systematic
uncertainty band, defined as the scatter among the different IGRB spectra derived in ref. [8]
(the case to be compared with our conservative limits). Right: The modified IGRB, after
the inclusion of the DM template, when the measurement for some energy bins falls outside
two times the 1σstatistical error band of the IGRB measurement originally presented in
ref. [8] (to be compared with our calculation of the sensitivity reach). A 5 TeV WIMP
which annihilates promptly into b¯
bwas used in both panels, which also explains why the
maximum differences between the original and modified IGRB are found around 200 GeV in
these particular examples. Note that we do not show the statistical error bar of the modified
IGRB because it is not relevant for our determination of the modified IGRB.
tially start to appear in the IGRB measurement. In figure 9, we show two examples of
a changed IGRB spectrum for the two cases mentioned above in the case of a 1 TeV
WIMP annihilating into b¯
b.
Figure 10 shows the largest possible DM annihilation cross sections to the b¯
band
τ+τ−channels which do not change the IGRB spectrum, together with our conservative
limits on the cross section and sensitivity reach derived in section 3.25 The non-gray-
shaded areas in figure 10 roughly indicate the regions where our method of deriving
limits on an isotropic DM signal would not lead to significantly altered results due to
the modified IGRB measurement from the presence of the assumed smooth Galactic
DM signal.
Notably, there are regions of the parameter space where DM limits overlap with
the shaded areas of our conservative limits in figure 10. Inclusion of the Galactic
smooth DM template can lead to both smaller and larger IGRB intensities around
the DM signal peak than the one reported in ref. [8]. For some DM masses <
∼250
GeV the IGRB can e.g. get higher by up to ∼40% after the inclusion of the DM
template, which would naively weaken the limits by roughly this amount. For larger
DM mass ( >
∼1 TeV) the IGRB spectrum is typically lowered. This is a consequence
of our procedure in which the normalizations of the Galactic foreground spectra are
25Only model A is used in the figure, but we note that similar results are obtained with models B
and C.
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Figure 10. The gray regions above the dotted lines indicate the DM annihilations cross
sections which would alter the measured IGRB spectra significantly due to the signal from
smooth DM halo component of the Milky Way; see section 4. Top and bottom panels are for
b¯
band τ+τ−channels, respectively. The DM limits shown are the same as those presented
in figure 7(left panels) and figure 8(right panels).
determined at energies lower than the DM signal peak and then kept fixed at the higher
energies. The measured IGRB intensity, the normalization of which is free in all energy
bins, is therefore lowered around the energies where the DM signal peaks in order to
accommodate the presence of the Galactic DM signal.
Cross sections at the level of the sensitivity reach of the IGRB measurement are
found to lie below their gray shaded region. We check the impact of using modified
IGRB models derived under the assumptions that a Galactic DM signal is present.
These alternate IGRB models are derived as above, with the Galactic DM signal fixed
by the annihilation channel and cross section (the DM density profile is kept to the
same as before). We adopt the cross-section values at the upper edge of the orange
band in the top right panel of figure 10 (the ‘PS(min), SS-MIN’ case) and then apply
our procedure to find the sensitivity reach: we find that the cross-section sensitivity
curve is basically unchanged by the inclusion of the Galactic DM component. For cross
sections within the gray shaded area the IGRB is sometimes no longer described well
by the adopted background model, so the method is no longer expected to behave well.
Note that while our DM limits depend on the substructure signal strength and
the assumed minimal DM halo mass, the shaded gray region in figure 10 is independent
– 28 –
of it, so the relative position of the gray region and the limits would be different for a
different choice of these parameters.
In order to exhaustively explore the impact of Galactic smooth DM templates on
the derivation of the IGRB, a larger number of Galactic astrophysical emission models
should be studied. In this way it would be possible to probe in detail the IGRB along
with various Galactic DM signals. However, such studies are beyond the scope of this
work, which is tied to the methodology used in ref. [8]. The initial study performed in
this section shows the importance of including the Galactic DM annihilation with its
proper morphology in a detailed study of isotropic intensities.26 Note that this issue
is more severe for decaying DM models (studied in this context in e.g. [101,102]),
since the DM Galactic component is more isotropic compared to the annihilating DM
case.27 Also, the Galactic DM signal at high latitudes is typically larger than the
corresponding cosmological one [103], when compared to the annihilating DM case
(see figure 5), and therefore a careful study of the degeneracy between the DM and IC
templates becomes mandatory before robust upper limits can be determined for any
potential cosmological decaying DM signal.
5 Summary
We utilize the recent measurement of the IGRB spectrum, based on 50 months of
Fermi LAT data, to set limits on the isotropic DM annihilation signals, i.e. the gamma
rays originating from DM annihilation in halos over all of cosmic history as well as
from the Galactic subhalos. Thanks to the broad energy range covered by the new
IGRB measurement presented in ref. [8], which extends up to 820 GeV, we are able
to effectively constrain signals from annihilation of DM particles with masses ranging
between a few GeV and a few tens of TeV.
At the lowest WIMP masses, our conservative DM limits in figure 7reach thermal
cross-section values for b¯
band τ+τ−channels. For the case of our benchmark values for
both the DM cosmological and Galactic substructure signals, the sensitivity reach of
the IGRB measurement shown in figure 8is comparable to the limits recently obtained
using LAT observations of dwarf spheroidal galaxies [10,104] as well as those derived
from low Galactic latitude data [9].
For WIMP masses above 5 TeV, our conservative limits calculated for the bench-
mark values of the DM signal, are a factor of a few better than the ones presented in
the Fermi LAT Collaboration works cited above. At these high WIMP masses ( >
∼1
TeV), the benchmark conservative limits are comparable to those obtained from ob-
servations of dwarf spheroidal galaxies by ground-based gamma-ray telescopes (more
26Note that the morphology of a DM template is relevant when performing the full-sky fits, since
it influences the normalization of the isotropic template, i.e., the IGRB spectrum. This should not
be confused, however, with the fact that we define the normalization of the DM template by the
requirement that solely the IGRB spectrum gets changed, and not when the whole-sky residuals
worsen significantly.
27More precisely, the smooth Galactic DM signal varies by factors of ∼16 and ∼4 for annihilation
and decay, respectively, between Galactic latitudes 20◦and 90◦.
– 29 –
precisely, the recent observation of Segue 1 by both VERITAS [105] and MAGIC [92]),
but weaker than the limits derived from the Galactic center halo by H.E.S.S. [91]. The
potential sensitivity to DM annihilation signals with the current IGRB measurement
might reach an order of magnitude lower cross sections in this same WIMP mass
range, in the case of optimal knowledge of some still uncertain (non DM) astrophysical
factors.
Our derived predictions for the strength of an isotropic DM annihilation signal
can be considered realistic and not over estimated: the extrapolations performed below
the resolution of current N-body cosmological simulations—necessary to account for
the smallest halos—have been done in a physical and theoretically well-motivated way,
and uncertainties in the expected DM signal were estimated using a well suited and
complementary approach based on the non-linear matter power spectrum which is
measured in N-body simulations. Furthermore, for the first time we have quantified
how the IGRB measurement is affected by a Galactic foreground DM signal and thus
when the latter starts to impact the derived constraints on an isotropic DM signal.
When compared to the earlier Fermi LAT Collaboration work [11], which derived
DM limits from the first-year IGRB measurement, the conservative limits are now
about a factor of two stronger in the WIMP mass range 1 GeV to 1 TeV for the same
value of the flux multiplier ζ(z). This improvement can be attributed to the new
IGRB data used and, most notably, to the fact that we did not take into account the
(isotropic) signal from the Galactic substructure in the previous work.
Moreover, the uncertainties of the flux multiplier ζ(z) have considerably shrunk
in the present work, and it now has a factor ∼20 uncertainty when the minimal
halo mass cut-off is set to 10−6h−1M. We note that this theoretical uncertainty
range is a factor ∼5 smaller than in ref. [11]. We did not consider extreme power-law
extrapolations of the many relevant quantities in the Halo Model framework, which
previously resulted in an over estimation of the predicted strength of the isotropic DM
signal. In this work, the theoretical uncertainty range for the predicted DM signal
strength (for a given WIMP annihilation cross section) therefore covers only the lower
and more physically motivated part of the previously considered range in ref. [11].
This in turn implies that our limits are generally consistent with the most conservative
estimates derived in other works [11,79,80].
In our work, we identified and addressed three main sources of uncertainty affect-
ing the derived limits to the DM annihilation cross section, which can be summarized
as follows: i) theoretical predictions for the strength of the DM annihilation signal,
which stem on one hand from the modeling of DM clustering at small scales, and which
translate into an uncertainty of a factor ∼20 (see section 2.3) and, on the other, from
the precise amount of substructure in the Galaxy, which has a factor ∼3 uncertainty,
ii) modeling of the contribution of unresolved extragalactic sources to the IGRB, re-
flected in the difference between our conservative limits and our derived sensitivity
reach, which for b¯
band τ+τ−channels and our reference prediction of the isotropic
DM annihilation signal ranges by a factor ∼3−26, depending on the WIMP mass
range considered, and iii) modeling of the Galactic diffuse emission, which can lead to
variations in the limits by a factor of up to ∼3.
– 30 –
We also studied the impact on the IGRB measurement of a DM signal from the
Milky Way DM halo, and defined a region in the cross section versus DM mass plane
for which the IGRB measurement would be sufficiently changed by the presence of
Galactic DM as to potentially affect the limits derived here.
With these considerable uncertainties in mind, at present the IGRB does not
represent a clean target to search for a DM signal. At the same time, we showed in
figure 8that the method has the great potential to test the ‘vanilla’ WIMP paradigm
(i.e. the thermal cross-section value) up to masses of a few tens of GeV, making this
approach competitive with other DM probes. An additional strength lies in offering
a complementary and truly cosmological probe for any potential DM signal hint that
might be claimed from another indirect, direct or collider search.
In the coming years of the Fermi mission, the LAT sensitivity to point-like sources
will improve due to the increased exposure and improved event classification and re-
construction, which, depending on the energy band, will translate into a lower IGRB
intensity and better DM limits. This can be the case at energies above 100 GeV,
where the total IGRB might ultimately be attributable to Fermi LAT-detected point-
like sources (as suggested in ref. [8], where the cumulative intensity from 2FGL sources
and the IGRB are seen to be comparable at the highest energies). On the other hand,
at energies below 100 GeV, resolving more point sources in the next several years is not
expected to significantly lower the IGRB flux (since, e.g., new blazars will be extremely
faint and thus their contribution will be hard to detect by the LAT, see [70]). Most
importantly, developments are foreseen to help in lowering the most critical uncertain-
ties: i) the Euclid satellite [106] and next generation N-body cosmological simulations
(e.g. [107,108]), by shedding light on the small-scale DM clustering properties, ii) new
constraints on the contribution of astrophysical sources to IGRB, from anisotropy and
cross correlation studies of high latitude gamma-ray emission, and iii) more precise
cosmic-ray measurements (with e.g. AMS-02 [109]) as well as detailed Planck dust
maps, that will help in refining the modeling of the Galactic foreground.
Acknowledgments
The Fermi LAT Collaboration acknowledges generous ongoing support from a number
of agencies and institutes that have supported both the development and the operation
of the LAT as well as scientific data analysis. These include the National Aeronau-
tics and Space Administration and the Department of Energy in the United States,
the Commissariat l’Energie Atomique and the Centre National de la Recherche Scien-
tifique, Institut National de Physique Nucl´eaire et de Physique des Particules in France,
the Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in Italy,
the Ministry of Education, Culture, Sports, Science and Technology (MEXT), High
Energy Accelerator Research Organization (KEK) and Japan Aerospace Exploration
Agency (JAXA) in Japan, and the K. A. Wallenberg Foundation, the Swedish Re-
search Council and the Swedish National Space Board in Sweden. Additional support
for science analysis during the operations phase is gratefully acknowledged from the
– 31 –
Istituto Nazionale di Astrofisica in Italy and the Centre National d’Etudes Spatiales
in France.
M.G. is supported by the Belgian Science Policy (IAP VII/37), the IISN and the
ARC project. M.A.S.C. acknowledges support from NASA grant NNH09ZDA001N for
the study of the extragalactic background. G.Z. is grateful to SLAC for hospitality
during part of the realization of this work. The authors are thankful to Emiliano
Sefusatti for help with producing some of the figures. We also thank Mattia Fornasa
for useful discussions and comments.
Some of the results in this paper have been derived using the HEALPix code [110].
A Diffuse foreground models and their impact on limits
In this Appendix, we investigate how the limits on the DM annihilation cross section
depend on the particular foreground model that is used to derive the IGRB in ref. [8].
In section 3.4 the baseline foreground emission model A in ref. [8] was assumed for
computing the DM limits. However, two more foreground models, B and C, were also
defined and considered in [8], which led to slightly different derived spectra of the
IGRB. These three reference models differ in propagation and CR injection scenarios,
and have been derived from a customized version of GALPROP. Model A is the basic
reference model, whereas model B includes, e.g., an additional population of electron-
only sources near the Galactic center and model C allows the CR diffusion and re-
acceleration to vary significantly throughout the Galaxy. Furthermore, variations of
model A have been studied, by, e.g., changing the size of CR halo between 4 kpc and
10 kpc, modifying its CR-source distribution, turning off re-acceleration and adding a
‘Fermi Bubbles’ template to assess the systematic uncertainties in the derived IGRB.
In figure 11, we show the effect on the limits when assuming models B and C
for the foreground diffuse emission instead of model A, for the particular case of our
sensitivity reach estimate and the b¯
band τ+τ−annihilation channels.
The limits are substantially modified for low mass WIMPs, i.e., below ∼300 GeV,
although the limits are again rather comparable at the lowest DM masses considered,
namely below ∼20 GeV. The maximum difference between the limits is found at about
50 GeV, where model A makes the limits a factor ∼2.7 more stringent than the ones
deduced using model B (assuming our fiducial Galactic substructure scenario). In all
cases, these differences can be explained as the interplay between the measured IGRB
points obtained in ref. [8] for each foreground model and the WIMP mass considered.
For example, in the particular case of a 50 GeV WIMP, for which the emission peaks
at roughly a few GeV, there is a downward trend of the IGRB data between 2 GeV
and 10 GeV in models A and C, which is not present in model B, making the limits
substantially stronger for models A and C compared to B.
It is worth emphasizing that, in contrast, the conservative limits ‘by construction’
take into account the variation induced on the IGRB measurement from using the
different foreground models. In this case, we recall that, in order to set the limits, we
shifted the IGRB data points to the maximum allowed intensity values among those
– 32 –
given by the various foreground models, which always results in the most conservative
bounds.
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Figure 11. DM limits for the three IGRB measurements derived assuming foreground
emission models A, B and C in ref. [8]. The limits shown in section 3.4 implicitly assume
foreground model A in ref. [8]. The limits are substantially modified when assuming model B
instead, especially at low masses. This figure is for the particular case of the sensitivity reach
procedure (as it was described in section 3.3.2) for DM particles annihilating into b¯
bquarks
(top) and τ+τ−(bottom), and for the two scenarios of the Galactic substructure contribution
introduced in section 2.4. The case of our reference substructure model is shown on the left
panels, while the minimal substructure case is on the right.
– 33 –
B Limits at different confidence levels
In this Appendix we compare 2σand 3σupper limits on the DM annihilation cross
section for b¯
band τ+τ−channels (figures 12 and 13, respectively), for both the conser-
vative limits and the sensitivity reach, and for the six representative cases of the DM
signal strength considered in our work.
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Figure 12. Upper limits on the self-annihilation cross section obtained in our conservative
(left) and sensitivity reach (right) procedure. The limits are on the annihilation cross section
into b¯
bquarks and the thicker (thinner) lines show the 2σ(3σ) limits. The figures in each
row show the limits in different DM setups: (top) the minimal extragalactic signal in the PS
approach, (middle) the benchmark extragalactic signal in the HM approach, and (bottom) the
maximal extragalactic signal in the PS approach. In each figure, the solid line corresponds to
the benchmark Galactic substructure intensity, while the dashed line represents the minimal
assumed Galactic substructure signal; see section 2.4. The minimal scale for DM structures
corresponding always to a halo mass cut-off of Mmin = 10−6h−1M.
– 34 –
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Figure 13. Same as figure 12, but for the τ+τ−channel.
– 35 –
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