# Dark Matter search in dwarf irregular galaxies with the Fermi Large Area
Telescope

V. Gammaldi<EMAIL_ADDRESS>Departamento de Física Teórica,
Universidad Autónoma de Madrid, Madrid, Spain $\&$ Instituto de Física
Teórica, UAM/CSIC, E-28049 Madrid, Spain  J. Pérez-Romero<EMAIL_ADDRESS>Departamento de Física Teórica, Universidad Autónoma de Madrid, Madrid, Spain
$\&$ Instituto de Física Teórica, UAM/CSIC, E-28049 Madrid, Spain  J.
Coronado-Blázquez<EMAIL_ADDRESS>Departamento de Física Teórica,
Universidad Autónoma de Madrid, Madrid, Spain $\&$ Instituto de Física
Teórica, UAM/CSIC, E-28049 Madrid, Spain  M. Di Mauro
<EMAIL_ADDRESS>INFN, Istituto Nazionale di Fisica Nucleare, via P.
Giuria, 1, 10125 Torino, Italy  E. V. Karukes<EMAIL_ADDRESS>AstroCeNT,
Nicolaus Copernicus Astronomical Center Polish Academy of Sciences,
ul. Rektorska 4, 00-614 Warsaw, Poland  M.A. Sánchez-Conde
<EMAIL_ADDRESS>Departamento de Física Teórica, Universidad
Autónoma de Madrid, Madrid, Spain $\&$ Instituto de Física Teórica, UAM/CSIC,
E-28049 Madrid, Spain  P. Salucci<EMAIL_ADDRESS>SISSA, International
School for Advanced Studies,
Via Bonomea 265, 34136, Trieste, Italy  INFN, Istituto Nazionale di Fisica
Nucleare - Sezione di Trieste,
Via Valerio 2, 34127, Trieste, Italy

(August 27, 2024)

###### Abstract

We analyze 11 years of Fermi-LAT data corresponding to the sky regions of 7
dwarf irregular (dIrr) galaxies. DIrrs are dark matter (DM) dominated systems,
proposed as interesting targets for the indirect search of DM with gamma rays.
The galaxies represent interesting cases with a strong disagreement between
the density profiles (core vs. cusp) inferred from observations and numerical
simulations. In this work, we addressed the problem by considering two
different DM profiles, based on both the fit to the rotation curve (in this
case a Burkert cored profile) and results from N-body cosmological simulations
(i.e., NFW cuspy profile). We also include halo substructure in our analysis,
which is expected to boost the DM signal a factor of ten in halos such as
those of dIrrs. For each DM model and dIrr, we create a spatial template of
the expected DM-induced gamma-ray signal to be used in the analysis of Fermi-
LAT data. No significant emission is detected from any of the targets in our
sample. Thus, we compute upper limits on the DM annihilation cross-section
versus mass parameter space. Among the 7 dIrrs, we find IC10 and NGC6822 to
yield the most stringent individual constraints, independently of the adopted
DM profile. We also produce combined DM limits for all objects in the sample,
which turn out to be dominated by IC10 for all DM models and annihilation
channels, i.e. $b\bar{b}$, $\tau^{+}\tau^{-}$ and $W^{+}W^{-}$. The strongest
constraints are obtained for $b\bar{b}$ and are at the level of $\langle\sigma
v\rangle\sim 7\times 10^{-26}\text{cm}^{3}\text{s}^{-1}$ at $m_{\chi}\sim 6$
GeV. Though these limits are a factor of $\sim 3$ higher than the thermal
relic cross section at low WIMP masses, they are independent from and
complementary to those obtained by means of other targets.

PACS numbers

###### pacs:

Valid PACS appear here

††preprint: APS/123-QED

##  I Introduction

Astrophysical and cosmological evidence suggests that non-baryonic cold dark
matter (DM) constitutes $84\%$ of the matter density of the Universe (Ade  _et
al._ , 2016; Aghanim  _et al._ , 2020). Although the actual nature of DM is
still unknown, weak interacting massive particles (WIMPs) are well-motivated
DM candidates. They are predicted to annihilate or decay into Standard Model
(SM) particles, whose decay and hadronization processes would produce
secondary particles, such as cosmic rays, neutrinos and gamma rays (Bertone
and Merritt, 2005). These messengers are expected to be observable in ground-
based or satellite observatories, laying the groundwork for the indirect
searches of DM. Either the expected DM signal could be disentangled from the
background (here defined as the emission from any well-known astrophysical
source) resulting in a DM hint, or the absence of an exotic signal provides
constraints on the nature of the WIMP particle, here its mass and annihilation
cross section. Among the messengers, gamma rays have represented the golden
channel as of today: they are (very-) high-energy neutral particles traveling
practically undeflected, along straight paths in the Universe. Nonetheless,
only an agreement of a few hints in several observation channels - i.e. the
multi-messenger approach - would result in a competitive claim in the sense of
the indirect detection of DM Bergström (2013); Gammaldi (2019).

DM-dominated systems - e.g. galaxy clusters, dwarf spheroidal (dSph) galaxies
as well as the Galactic center - are benchmark targets for indirect searches
of DM ( see e.g. Conrad and Reimer (2017); Charles  _et al._ (2016) and refs
therein). Among other targets, Milky Way dSph galaxies are considered to be
especially promising objects due to their relatively close position and their
appearance as point-like or marginally extended sources in gamma-ray
telescopes. Although the DM density profiles inferred from their kinematics
are often affected by large uncertainties (Hayashi  _et al._ , 2016), the
contamination from intrinsic gamma-ray sources is negligible in this type of
objects, making them particularly appealing as well Winter  _et al._ (2016).

Recently, dwarf irregular (dIrr) galaxies in the Local Volume have been
claimed to be interesting targets for gamma-ray DM searches as well: (i)
unlike pressure-supported dSph galaxies, dIrrs are rotationally-supported
galaxies, i.e. their DM profiles can be obtained from their rotation curves
(RC) and, indeed, from these they appear to be DM-dominated systems at all
radii (Oh  _et al._ , 2015, 2011; Gentile  _et al._ , 2007) \- just as dSphs;
(ii) dIrrs are isolated galaxies of the Local Volume with DM halo mass
$M_{200}\approx 10^{7}-10^{10}\,M_{\odot}$. Here, we consider dIrrs at a
distance less than $\sim 1$ Mpc. Roughly speaking, the astrophysical component
of the gamma-ray DM flux scales with $\approx M_{200}^{2}/d^{2}$, so that it
is expected to have comparable contributions for both the dIrrs and dSphs
(which typical have $M_{200}\approx 10^{5}-10^{7}\,M_{\odot}$ and $d<0.5$
Mpc), as observed in our analysis (see e.g. Fig. 5 in Gammaldi  _et al._
(2018)); (iii) Unlike aged, largely evolved dSph galaxies, dIrrs are star-
forming galaxies; yet the astrophysical gamma-ray emission from their star-
forming regions has been estimated to be several orders of magnitude lower
than the gamma-ray flux expected from DM annihilation events in the halo, for
vanilla WIMP models. In other words, the astrophysical background emission is
expected to be negligible in dIrrs similarly to the case of dSphs Gammaldi
_et al._ (2018). In fact, dIrrs have star-forming regions of very small
angular size (unresolved by most of gamma-ray observatories, e.g. Fig. 6 in
Gammaldi  _et al._ (2018)), while their DM halos appear to be extended - due
to their distances and typical values of the virial radius Gammaldi  _et al._
(2018).

Despite these advantages, dIrrs have not been studied so far with data of the
Large Area Telescope on board the NASA Fermi satellite (Fermi-LAT) Atwood  _et
al._ (2013) in the context of DM searches. The Fermi-LAT is a pair conversion
telescope designed to observe the energy band from 20 MeV to greater than 300
GeV, that has been surveying the sky searching for gamma-ray sources since
2008 Atwood (2009). Several point-source Fermi-LAT catalogs have been released
that contain hundreds to thousands of gamma-ray objects, many of them
previously unknown Abdollahi  _et al._ (2020). In this work, we will perform
the first spatial analysis of 7 dIrr galaxies in the Local Volume with Fermi-
LAT data. 111 A recent work Hashimoto  _et al._ (2020) focuses on Low Surface
Brightness galaxies (LSBs) to constrain the DM model with Fermi-LAT data. Such
objects are similar to dIrrs but differ in key aspects, particularly in
morphology: both LSB and dIrr galaxies are rotationally supported systems, but
LSBs still show spiral arms, while dIrrs are much more irregular. Further, the
LSB sample in Ref. Hashimoto  _et al._ (2020) includes (i) farther objects,
with distances $>20$ Mpc and already significant redshifts, while dIrrs are
Local Volume objects with distances $\lesssim 1$ Mpc; (ii) J-factors of
$10^{14}-10^{16}\,\text{GeV}^{2}\text{cm}^{-5}$, i.e. a factor 100 - or more -
smaller than the J-factors in this work (for details see section III).
Secondly, the LSB in Ref. Hashimoto  _et al._ (2020) are point-like sources
for Fermi-LAT, while dIrrs in this work are extended sources due to their
proximity. For the same reason, there is no overlapping between the list of
objects in the LSBs analyses (e.g. Hashimoto  _et al._ (2020); Bhattacharjee
_et al._ (2021)) and our work. Instead, an overlap in the targets exists with
some preliminary analyses obtained with the High Altitude Water Cherenkov
(HAWC) observatory Cadena  _et al._ (2017); Hernández Cadena  _et al._ (2021),
as well as with the very recent study of WLM published by the High Energy
Stereoscopic System H. E. S. S. Collaboration  _et al._ , both of them at TeV
energy scale. Indeed, the results of these studies are complementary to this
work.. We will also discuss in-depth both the main theoretical advantages and
uncertainties related to this new class of sources, as briefly introduced in
points (i)-(iii) above. In more details:

(i) Due to the current uncertainty that affects the inner DM density profile
(in particular for galaxies in the range of masses considered in this paper),
we adopt - for each dIrr galaxy in our sample - both a core and a cuspy
profile, as we discuss in Section II.

(ii) For the computation of the expected gamma-ray flux, we include in our
analysis the effect of halo substructure, leading to the so-called "subhalo
boost". In fact, subhalos may play a crucial role for indirect DM searches, in
terms of an enhancement in the gamma-ray flux. In particular, the boost in the
gamma-ray flux due to substructures is expected to be more dominant in field
halos (i.e dIrrs, that are main halos of the Local Volume), instead of dwarf
satellites (e.g. dSphs, that fill subhalos in our galaxy) Ando  _et al._
(2019), as we discuss in Section III.

(iii) Due to both the apparent angular size of the dIrr DM halos as well as
the inclusion of halo substructure - that particularly boosts the annihilation
flux in the outer regions of host halos - the DM signal from dIrrs could
appear as extended for Fermi-LAT. For this reason, we perform a spatial
analysis of these objects, instead of the point-like analysis commonly adopted
in the study of dSphs, as discussed in section III.

In section IV we present the Fermi-LAT data analysis for each individual
target in our sample, by using 11 years of LAT data and adopting the
corresponding spatial template for each of the dIrrs. No significant emission
is detected from any of the targets. Indeed, the absence of a gamma-ray signal
detection in our sample of dIrrs represents one important result of this work,
which - in addition - is in agreement with the expectation Gammaldi  _et al._
(2018). Thus, we proceed and compute $95\%$ confidence level (C.L.) upper
limits in the standard annihilation cross section versus WIMP mass parameter
space, $<\sigma v>-m_{\chi}$, for each dIrr and considered DM model scenario.
The results of both the individual and combined analysis of our sample of
dIrrs are presented and discussed in Section V. Finally, we summarize and
discuss the main results of this work in Sec. VI.

##  II Kinematics and DM modelling

DIrrs are rotationally-supported galaxies and DM-dominated systems at all
radii, with masses $M_{200}\approx 10^{7}-10^{10}\,M_{\odot}$. Yet, the exact
shape of the inner DM density profile in dIrr galaxies still represents an
open issue. On the one hand, the cold HI and $H_{\alpha}$ rotating disks in
dIrrs yield high resolution and high quality RCs, which point to the existence
of cores in their centers, with halos extending well beyond their star-forming
regions (Salucci  _et al._ , 2007). On the other hand, this kind of cored-like
profiles are in contrast with the predictions of N-body, DM-only cosmological
simulations, which predict cuspy DM profiles such as the Navarro-Frenk-White
(NFW) Navarro  _et al._ (1996, 1997). Nevertheless, the current DM profiles
around galaxies could be different from the primordial ones and/or from the
results of DM-only simulations, due to e.g. baryonic feedback Gómez-Vargas
_et al._ (2013); Schaller  _et al._ (2015); Peirani  _et al._ (2017).
Unfortunately, the impact of baryons in cosmological simulations at the scales
relevant for this work is not yet fully understood. Whether or not a DM core
will form in a given dwarf galaxy may depend primarily on its stellar mass to
halo mass ratio, e.g. Read  _et al._ (2019); Di Cintio  _et al._ (2014). In
fact, bursty star formation limited to dense $H_{2}$-rich regions creates
repeated, fast outflows which break the adiabatic approximation. These fast
and repeated outflows progressively lower the central DM density of galaxy
halos and turn DM central cuspy profiles into much shallower cores Read  _et
al._ (2019); Di Cintio  _et al._ (2014); Sawala  _et al._ (2016); Governato
_et al._ (2012). Furthermore, the stars can be dynamically heated similarly to
the DM, leading to a stellar velocity dispersion that approaches the local
rotational velocity of the stars ($V/\sigma\sim 1$) within the projected half
radius of the stars. Nonetheless, some authors Bose  _et al._ (2019); Benítez-
Llambay  _et al._ (2019) found that the fast and repeated outflows are not
able to lower the central DM density of galaxy halos, and the cuspy profiles
remain unperturbed. This is probably due to a relatively low gas density
threshold for converting gas into stars which prevents the gas from becoming
gravitationally dominant on kpc scales.
In light of this debate, we prefer to include in our analysis both a core and
cuspy profile, in this way also estimating how such an open issue may affect
our final results. First, we study the RCs of the sample of seven dIrr
galaxies that will be later analyzed in gamma rays. Four of them (NGC6822,
IC10, Wolf-Lundmark Melotte (WLM) and IC1613) have been selected as the most
promising objects 222The four most promising objects have been selected based
on the results of Gammaldi  _et al._ (2018) \- i.e. the first proof-of-
concepts paper. In the latter, a point-like study based on a theoretical
approach was developed, by taking into account several key factors, e.g. not
only the mass, distance and angular dimension of the targets, but also their
position in the sky and their star-forming gamma-ray emission, expected to be
negligible with respect to both the gamma-ray emissivity expected by the DM
halos and the diffuse and isotropic components of the background. of thirty-
six dIrr galaxies, that were previously investigated as a new category of
targets for indirect DM searches Gammaldi  _et al._ (2018); the last three
galaxies (Phoenix, DDO210 and DDO216) are brand new targets, never studied
before in this context and considered as interesting targets due to their mass
and distance. The seven dIrrs are located at a distance less than $\sim 1$
Mpc, and are thus in the Local Group. Starting from the distance of $\sim 4$
Mpc, dIrrs start to outnumber dSphs, indeed the total number of known dIrrs in
the Local Volume is at present $\sim 200\%$ of that of known dSphs Gammaldi
_et al._ (2018). In this regard, we note that our sample only includes dIrrs
located in a particular region of the sky, due to availability of data Namumba
_et al._ (2017); Karachentsev  _et al._ (2013); Hunter and Elmegreen (2004);
Hidalgo  _et al._ (2013); Hunter  _et al._ (2012). The main characteristics of
these galaxies are shown in Table 1, which provides name, distance, scale
length $R_{D}$, stellar mass $M_{D}$ and position in the sky. Here the stellar
disk scale length is defined as the radius at which the surface luminosity of
a galaxy has fallen off by a factor of $e$ ($\sim$2.7).

First of all, we fit the observed RC data (see Appendix A) to a parametrized
model described below and perform the global mass modelling. We consider the
contribution to the total measured circular velocity from two components: the
luminous part (stellar disk and the gaseous component when available) and the
DM contribution, i.e.:

Name | Distance | $R_{D}$ | $\log_{10}M_{D}$ | l | b | RC & $M_{D}$
---|---|---|---|---|---|---
| [Mpc] | [kpc] | [$\rm M_{\odot}$] | [deg] | [deg] | reference
NGC6822 | 0.48 | 0.66 | 7.0 | 23.3 | -18.4 | Namumba  _et al._ (2017)
IC10 | 0.79 | 0.79 | 8.1 | 119.0 | -3.3 | Oh  _et al._ (2015)
WLM | 0.97 | 0.55 | 7.2 | 75.9 | -73.6 | Oh  _et al._ (2015)
IC1613 | 0.76 | 0.64 | 7.5 | 129.7 | -60.6 | Oh  _et al._ (2015)
Phoenix | 0.44 | 0.23 | 6.8 | 272.2 | -68.9 | Kacharov  _et al._ (2016); Shao  _et al._ (2018)
DDO210 | 0.9 | 0.17 | 5.8 | 34.0 | -31.3 | Oh  _et al._ (2015)
DDO216 | 1.1 | 0.54 | 7.2 | 61.5 | -67.1 | Oh  _et al._ (2015)
Table 1: Sample of the seven dIrr galaxies studied in this work, together with
their distances ($D_{\rm Earth}$), scale lengths, $R_{D}$, and stellar masses,
$M_{D}$, and position in the sky. In the last column we provide for each
target the reference for both the Rotation Curve (RC) data (see also Appendix
A) and stellar mass $M_{D}$ data Namumba  _et al._ (2017); Karachentsev  _et
al._ (2013); Hunter and Elmegreen (2004); Hidalgo  _et al._ (2013); Hunter
_et al._ (2012).

$v^{2}_{tot}(r)=v^{2}_{disk}(r)+v^{2}_{gas}(r)+v^{2}_{dm}(r).$ (1)

We assume that the stellar component, $v_{disk}$, is well represented by a
Freeman disk Freeman (1970) and, thus, can be written as:

$v^{2}_{disk}(r)=\frac{1}{2}\frac{GM_{D}}{R_{D}}\left(\frac{r}{R_{D}}\right)^{2}(I_{0}K_{0}-I_{1}K_{1}),$
(2)

where $M_{D}$ is the stellar mass, $R_{D}$ is the disk scale length, and
$I_{n}$ and $K_{n}$ are the modified Bessel functions computed at
$\frac{r}{2\,R_{D}}$. The disk scale length and the distribution of the gas
component are taken from literature and listed in Table 1.333This is done for
all dIrrs but Phoenix I, for which data for the gas distribution are not
available. Thus, for this object we assume that the gas follows the Freeman
disk given by Eq. (2), yet with $R_{D}$ being three times that of the disk
Tonini  _et al._ (2006). We take the gaseous mass to be $\log_{10}M_{\rm
HI}=4.92$ from Shao  _et al._ (2018).

The stellar mass is left as a free parameter: in Table 1 we show the values
available in literature that we adopt as central values for the Gaussian
priors with the standard deviation equal to 0.1444If we adopt larger standard
deviation the implied stellar masses become unrealistically low/high..

In more detail, our model of Eq. (1) has three free parameters: $M_{D}$ and
two parameters (the core density and scale radius) describing the DM density
profile, either Burkert ($\rho_{c},\,r_{c}$) or NFW ($\rho_{0},\,r_{s}$)
respectively. We explore the model parameter space using uniform priors for
the DM parameters and Gaussian priors for $M_{D}$555The good quality of the RC
of NGC6822, allows to leave the stellar mass free.. The posterior is then
sampled using emcee, an open source affine-invariant Markov chain Monte Carlo
(MCMC) ensemble sampler Foreman-Mackey  _et al._ (2013).

The best-fit maximum likelihood (ML) values of the MCMC analysis for the
Burkert profile are presented in Table 2, namely the core density, $\rho_{c}$,
and the core radius, $r_{c}$:

Name | $r_{c}$ | $\log_{10}\rho_{c}$ | $\log_{10}M_{D}$ | $\chi^{2}_{\mathrm{red}}$ | $R_{200}$ | $\log_{10}M_{200}$ | $\theta_{200}$
---|---|---|---|---|---|---|---
| [kpc] | [$\rm M_{\odot}/kpc^{3}$] | [$\rm M_{\odot}$] | — | [kpc] | [$\rm M_{\odot}$] | [deg]
NGC6822 | $3.3^{+0.8}_{-0.7}$ | $7.5\pm 0.1$ | $7.9^{+0.2}_{-0.3}$ | $0.1$ | $62.9^{+8.4}_{-6.7}$ | $10.5^{+0.2}_{-0.1}$ | $7.5$
IC10 | $2.0^{+0}_{-1.5}$ | $8.2^{+0.4}_{-0.2}$ | ${}^{*}8.1\pm 0.1$ | $0.1$ | $71.3^{+7.1}_{-47.6}$ | $10.6^{+0.1}_{-1.5}$ | $5.2$
WLM | $1.3^{+0.2}_{-0.1}$ | $7.8\pm 0.1$ | ${}^{*}7.1^{+0.5}_{-0.9}$ | $0.1$ | $33.3^{+2.0}_{-1.5}$ | $9.6\pm 0.1$ | $2.0$
IC1613 | $7.0^{+0}_{-1.2}$ | $6.3\pm 0.05$ | ${}^{*}7.1^{+0.07}_{-0.06}$ | $0.9$ | $45.7^{+1.7}_{-6.9}$ | $10.0^{+0.05}_{-0.2}$ | $3.4$
Phoenix | $0.2$ | $7.5$ | ${}^{*}6.8$ | $1.2$ | $3.6$ | $6.7$ | $0.5$
DDO210 | $0.5^{+0.9}_{-0.2}$ | $8.0^{+0.1}_{-0.2}$ | ${}^{*}5.8\pm 0.1$ | $0.5$ | $14.2^{+23.7}_{-4.6}$ | $8.5^{+1.2}_{-0.5}$ | $0.9$
DDO216 | $0.3^{+0.3}_{-0.1}$ | $8.1\pm 0.3$ | ${}^{*}7.2\pm 0.1$ | $0.3$ | $10.8^{+3.7}_{-2.7}$ | $8.2\pm 0.3$ | $0.6$
Table 2: Best-fit ML values and corresponding 1 $\sigma$ uncertainties for the
main parameters of our sample of dIrrs, for a Burkert DM density profile.
Asterisks indicate that the estimation of the stellar mass was taken from Oh
_et al._ (2015) and the Gaussian prior is further assumed. We provide both the
core radius $r_{c}$ and density $\rho_{c}$, the stellar mass $M_{D}$, the
$\chi^{2}_{\mathrm{red}}$ value, $R_{200}$ and $M_{200}$. Finally,
$\theta_{200}$ is the angular extension of the galaxy, defined as the sky
angle subtended by $R_{200}$. Note that we do not provide the uncertainties
for the Phoenix galaxy due to very poor quality of its RC data.

$\rho_{\mathrm{Bur}}(r)=\frac{\rho_{c}\,r_{c}^{3}}{(r+r_{c})\,(r^{2}+r_{c}^{2})}.$
(3)

In Table 2 we also list values of the reduced $\chi^{2}_{\mathrm{red}}$, where
the number of d.o.f. is the number of RC data points minus the number of free
parameters (in the Burkert profile case is equal to 3). We fit the RC of each
galaxy, instead of adopting the Universal Rotation Curve hypothesis as in
Gammaldi  _et al._ (2018); Karukes and Salucci (2017). This fact generates
small differences with respect to Gammaldi  _et al._ (2018), in the resulting
parameters of the Burkert profiles and the related uncertainties.

We follow the same procedure for the NFW profile:

$\rho_{\mathrm{NFW}}(r)=\frac{\rho_{0}}{\left(\frac{r}{r_{s}}\right)\left(1+\frac{r}{r_{s}}\right)^{2}},$
(4)

where $\rho_{0}$ and $r_{s}$ are the density normalization and the scale
radius, respectively. However, even in those cases where this profile shows
reasonable fits to the RC data, the obtained best-fit values for the
concentration parameter are unrealistically low for a $\Lambda$CDM Universe
McGaugh  _et al._ (2003); Sánchez-Conde and Prada (2014); Read  _et al._
(2016). We also tried to use uniform priors based on structure formation
models Ando  _et al._ (2020), with similar results. Consequently, either the
DM in these galaxies is distributed somewhat differently from what is expected
from the DM-only $\Lambda$CDM cosmological simulations, or the data contains
unknown systematic uncertainties that should be taken into account, e.g.
systematic uncertainties related to the inclination and/or distance errors in
the RC reconstruction Oman  _et al._ (2016). Thus – by looking out for an
agreement with $\Lambda$CDM cosmology and the DM-only simulations – we still
model each dIrr galaxy with an NFW profile but with one key assumption:
$M_{200}$, the mass contained within the virial radius, $R_{200}$, and
obtained by the fit of the RCs for the Burkert profiles (Table 2), is also
taken as a starting point to build the NFW profiles. Though valid, just as a
first approximation, this assumption is good enough for our purposes,
especially when considering that, later in our work, we will define and use a
set of four different DM models (described in the next section) to
conservatively encapsulate the uncertainties coming from the modelling of the
DM distribution in our sample of dIrrs. Thus, uncertainties from the
assumption of $M^{\rm NFW}_{200}=M^{\rm Bur}_{200}$ are subdominant compared
to the level of global uncertainties from the envelope of all four considered
DM models Salucci  _et al._ (2007). From this value of $M^{\rm NFW}_{200}$ for
each dIrr, it is possible to obtain the corresponding radius $R_{200}$:

$R_{200}=\left(\frac{3M_{200}}{4\pi\Delta_{200}\rho_{crit}}\right)^{1/3},$ (5)

$\Delta_{200}$ being the overdensity with respect to the critical density of
the Universe, $\rho_{crit}\,=\,137\,$M${}_{\odot}\,$kpc-3.

The next step to build the NFW profile is to obtain the halo concentration.
For this purpose we use the parametrization for main halos Sánchez-Conde and
Prada (2014):

$c_{200}(M_{200},z=0)=\sum_{i=0}^{5}c_{i}\times\left[\ln\left(\frac{M_{200}}{h^{-1}M_{\odot}}\right)\right]^{i},$
(6)

which has proven to work especially well for objects in the mass range between
dSphs and galaxy clusters, and includes the flattening of $c(M)$ at lower halo
masses, first pointed out by these authors and now widely accepted Ludlow  _et
al._ (2016); Wang  _et al._ (2020). Together with the value of $R_{200}$ given
by Eq. 5, it is now easy to obtain the scale radius, $r_{s}$:

$r_{s}\equiv R_{200}/c_{200}.$ (7)

Finally, we impose the condition

$M_{200}=\int_{0}^{R_{200}}\rho_{\rm NFW}(r)r^{2}drd\Omega$ (8)

and we get

$\rho_{0}=\frac{2\leavevmode\nobreak\ \Delta_{200}\leavevmode\nobreak\
\rho_{crit}\leavevmode\nobreak\ c_{200}}{3\leavevmode\nobreak\ f(c_{200})},$

where
$f(c_{200})=\frac{2}{c_{200}^{2}}\left(\ln{(1+c_{200})}-\frac{c_{200}}{1+c_{200}}\right)$.
The variable that we will use to define the spatial extension of the dIrrs is
$\theta_{200}$, i.e. the sky angle subtended by $R_{200}$:

$\theta_{200}=\arctan\left(\frac{R_{200}}{D_{\rm Earth}}\right).$ (9)

The resulting NFW profile parameters are given in Table 3. The RCs
corresponding to both the data-consistent Burkert profile and the
$\Lambda$CDM-consistent NFW profile are left for Appendix A. Despite the
adopted theoretical model, the NFW profile remains a consistently worse than
the Burkert profile, for reasons we gave above, i.e. the main discrepancies
between the model and the data for some objects may be associated with (i) an
incomplete knowledge of baryonic effects in numerical simulations and/or (ii)
systematic uncertainty in the determination of the RC data, e.g., the
estimated inclination angle of the galaxy. Due to these unknowns, the quality
of the RC fit was not considered as primary when choosing the list of Irrs
from Gammaldi  _et al._ (2018) for this analysis. Instead, we preferred the
objects with the most promising J-factors (see Gammaldi  _et al._ (2018) and
Section III ). Nonetheless, 5 of 7 RCs are well fitted with a cored profile,
by including the baryon component.

Name | $c_{200}$ | $r_{s}$ | $\log_{10}\rho_{0}$ | $R_{200}$ | $\theta_{200}$
---|---|---|---|---|---
|  | [kpc] | [$\rm M_{\odot}/kpc^{3}$] | [kpc] | [deg]
NGC6822 | 10.7 | 5.9 | 6.9 | 62.6 | 7.4
IC10 | 10.4 | 6.8 | 6.8 | 70.3 | 5.1
WLM | 12.2 | 2.8 | 7.0 | 33.6 | 2.0
IC1613 | 11.4 | 4.0 | 6.9 | 45.7 | 3.4
Phoenix | 18.7 | 0.2 | 6.9 | 3.5 | 0.5
DDO210 | 14.5 | 1.0 | 7.2 | 14.5 | 0.9
DDO216 | 15.3 | 0.7 | 7.3 | 10.8 | 0.6
Table 3: NFW density profile parameters for our sample of dIrrs, obtained
assuming $M_{200}^{\rm Bur}=M_{200}^{\rm NFW}$; see the text for details, and
Eqs. (4–9) for the definition of each parameter. The concentration $c_{200}$
corresponds to the one provided by the concentration-mass ($c-M$) relation in
Sánchez-Conde and Prada (2014) for $\mathrm{\Lambda CDM}$ halos.

##  III Annihilation signal and inclusion of halo substructure

The DM modelling performed in the previous section is used as the starting
point to obtain the induced DM annihilation gamma-ray flux from these objects.
Assuming that the DM is composed of WIMPs Bertone  _et al._ (2005); Hooper
(2010), we can compute this expected flux as

$\frac{d\phi_{\gamma}}{dE}(E,\Delta\Omega,l.o.s)=\frac{1}{4\pi}\frac{\langle\sigma
v\rangle}{\delta\,m^{2}_{\chi}}\frac{dN_{\gamma}}{dE}(E)\times
J(\Delta\Omega,l.o.s),$ (10)

where: $\frac{d\phi_{\gamma}}{dE}$ is the DM annihilation gamma-ray flux;
$\langle\sigma v\rangle$ is the thermally-averaged annihilation cross-section;
$\delta=2$ if we assume Majorana DM particles and $\delta=4$ if Dirac
particles; $m_{\chi}$ is the DM mass; $\frac{dN_{\gamma}}{dE}$ is the WIMP
annihilation spectrum; $J(\Delta\Omega,l.o.s)$ is the so-called astrophysical
J-factor (computed along the line of sight (l.o.s.), and within a given solid
angle $\Delta\Omega$). It is worth to notice that we can identify two main
dependencies in the flux: first, the energy dependence, that appears only in
the so-called particle physics term (which contains all the information about
the mass of the DM candidate and the possible annihilation channels);
secondly, the spatial dependence, appearing only in the J-factor. This allows
to factorize these two terms independently, and implies that the spatial
distribution of the DM is independent of the energy. The J-factor is then
defined as

$J(\Delta\Omega,l.o.s)=\int_{0}^{\Delta\Omega}d\Omega\int_{l.o.s}\rho^{2}(r)dl,$
(11)

where $\Delta\Omega=2\pi(1-\cos\alpha_{int})$, being $\alpha_{int}$ the
integration angle and $\rho(r)$ the DM density profile.
From N-body cosmological simulations - which follow the structure formation
processes according to $\Lambda$CDM (e.g., Kuhlen  _et al._ (2012); Zavala and
Frenk (2019) and references therein) - , we expect the smallest structures or
halos to form first. Then, via accretion or collapse, the bigger structures
are formed, leaving a population of small structures - known as subhalos -
within the main halos. Given the typical masses and sizes of dIrrs, we expect
them to host a significant number of subhalos. The population of subhalos in a
main halo can be parametrized as

$\frac{d^{3}N}{dVdMdc}=N_{tot}\frac{dP_{V}}{dV}(R)\frac{dP_{M}}{dM}(M)\frac{dP_{c}}{dc}(M,c),$
(12)

where: $N_{tot}$ is the total number of subhalos; $P_{i}$ with $i=V,M,c$ is
the probability distribution in each of the domains normalized to 1; $V$
referring to the subhalo distribution in the volume of the main halo
666However, the only real dependence is on the relative distance of the
subhalos to the center of the host, so in the following we will refer to this
distribution in volume as the Subhalo Radial Distribution.; $M$ to the subhalo
mass and $c$ to subhalo concentration. This allows to model the subhalo
distribution for each domain independently from one another (spatial
distribution, mass and concentrations). The effect of taking into account all
these small structures is to enhance the expected DM flux, which is usually
quantified in terms of the so-called substructure boost factor, $B$. This
boost can range from $B=0$, where the contribution of the substructure is
absent, up to $\sim 2$ orders of magnitude in the expected annihilation DM
signal, depending on the details of the subhalo population and of the host
halo mass (e.g. Moliné  _et al._ (2017); Ando  _et al._ (2020)). Given the
impact that the substructure can have in the J-factors and its still uncertain
nature (e.g., minimum mass to form clumps Green  _et al._ (2005); Bringmann
(2009); Cornell  _et al._ (2013), tidal stripping, subhalo survival or precise
shape of subhalo DM density profiles Zavala and Frenk (2019)), it becomes
convenient to cover the range of different but possible scenarios. Hereafter
we proceed to describe the models that we use in this work to model the
substructure within our sample of dIrrs:

  * •

$\frac{dP_{V}}{dV}$: Known as the Subhalo Radial Distribution (SRD), there is
some variety of SRD models available in the community. The most used are the
ones described in Springel  _et al._ (2008) and Diemand  _et al._ (2008), both
based to N-body simulation results. Yet, as seen in section II, dIrr galaxies
agree better with data-driven DM profiles, like the Burkert profile, rather
than with simulation-motivated profiles like the NFW. Because of this, in this
work we decide to adopt a hybrid approach, for which, as in Springel  _et al._
(2008), we choose the SRD to follow the profile of the main halo, i.e. Burkert
or NFW depending on the considered model.

  * •

$\frac{dP_{M}}{dM}$: The distribution in mass of the population of subhalos is
known as the Sub-Halo Mass Function (SHMF). Once again, the main input for
these models are the N-body simulations, which usually follows

$\frac{dN}{dM}\propto M^{-\alpha}.$ (13)

In the literature the most common values are $\alpha=1.9$ Springel  _et al._
(2008) and $\alpha=2.0$ Diemand  _et al._ (2008). As shown in Moliné  _et al._
(2017), the slope plays a crucial role on the boost factor calculation, since
higher boosts are obtained for higher $\alpha$ values, i.e. for the case of a
more numerous population of small subhalos. In order to cover all possible
physical scenarios, we consider both cases, noting that the $\alpha=1.9$
probably yields the more realistic J-factors as it is more in line with the
latest results in the N-body simulation side Zavala and Frenk (2019), while
$\alpha=2.0$ will provide an upper bound to the substructure boost values.

  * •

$\frac{dP_{c}}{dc}$: Even though in the literature it is possible to find
state-of-the-art $c-M$ subhalo relations for the subhalos themselves - by
including a radial dependence accounting for the location of the subhalos
within the main halo Moliné  _et al._ (2017) \- in the following we adopt the
one for main halos Sánchez-Conde and Prada (2014). The main reason for this
choice is that the latter is already available in the CLUMPY code, we use to
compute the J-factors, while the model by Moliné  _et al._ (2017) is not yet.
Since the predicted concentrations for subhalos are higher than for main halos
of the same mass Moliné  _et al._ (2017), the parametrization of Sánchez-Conde
and Prada (2014) will ensure that our integrated J-factors for the dIrrs will
be conservative (by a factor 2-3). This choice also implies to implicitly
adopt the standard NFW density profile to model the DM profile in subhalos,
$\rho_{clumps}$. However, compared to main halos, subhalos are known to
exhibit profiles that are similar to the NFW in the inner parts but decay much
more rapidly towards the outskirts due to tidal stripping Springel  _et al._
(2008); Kazantzidis  _et al._ (2004); Peñarrubia  _et al._ (2010); Errani and
Navarro (2020). Yet, for our purposes the NFW still represents a very good
approximation, as most ($\sim 90\%$) of the annihilation flux is originated
within the scale radius, well inside the subhalo. Once we assume a ($c-M$)
relation, there is also the possibility to consider an intrinsic scatter on
the concentrations Bullock (2001); Moliné  _et al._ (2017). However, it
becomes computationally extremely expensive to take into account this scatter
in CLUMPY. Thus, for the purposes of this work, we decide to neglect it: as we
are in any case considering very diverse models for the rest of variables, we
note that the effect of the scatter in concentrations would lie well within
the spread of the obtained J-factors from these different models.

After we have discussed and selected the main parameters describing the
subhalo population, we will define four different benchmark models for the
computation of the J-factors, each of them with a particular level of subhalo
boost:

  * •

MIN: the main halo is modelled with a Burkert profile and only takes into
account the smooth DM distribution within the main halo, i.e. this model
neglects the effect of substructure.

  * •

MED: the main halo is modelled with a Burkert profile. The SRD follows the
Burkert profile of the host. We adopt $\alpha=1.9$ for the slope of the SHMF
(Eq. 13).

  * •

MAX-Bur: this model is similar to MED but adopts $\alpha=2.0$ for the slope of
the SHMF.

  * •

MAX-NFW: the main halo is modelled with an NFW profile. The SRD follows the
NFW profile of the host. We adopt $\alpha=2.0$ for the slope of the SHMF.

We note that with the definition of the above benchmark models, we are
bracketing a wide range of different possible substructure scenarios, this way
also providing a bracketing for the values of the J-factors. We summarize all
these scenarios in Table 4. Let us stress that, although the labels MIN, MED,
MAX already give a rough idea of the ranking of J-factors values, in a few
cases this is not strictly true due to the different DM profiles adopted for
the main halos in the different models.

Model | $\rho_{host}$ | $\rho_{subs}$ | SRD | $\alpha$
---|---|---|---|---
MIN | Burkert | - | - | -
MED | Burkert | NFW | Burkert | 1.9
MAX-Bur | Burkert | NFW | Burkert | 2
MAX-NFW | NFW | NFW | NFW | 2
Table 4: Summary of the DM models we use in this work; $\alpha$ is the slope
in Eq. (13), see Section III for full details.

The computation of the J-factors is performed using the CLUMPY code
Charbonnier  _et al._ (2012); Bonnivard  _et al._ (2016); Hütten  _et al._
(2019). CLUMPY allows us to easily implement our four benchmark models for the
sample of dIrr galaxies. Other general parameters we use to perform these
computations are the minimum subhalo mass, that we set to $M_{min}=10^{-6}$M⊙,
the maximum subhalo mass in terms of the mass of the host,
$M^{\%}_{max}=0.01$, and the number of substructure levels to be considered,
$N_{subs}=2$ (subhalos inside subhalos). The obtained integrated J-factors for
each galaxy and for each benchmark model are summarized in Table 5.

Name | $\log_{10}J_{\rm MIN}$ | $\log_{10}J_{\rm MED}$ | $\log_{10}J_{\rm MAX-BUR}$ | $\log_{10}J_{\rm MAX-NFW}$
---|---|---|---|---
| $\rm GeV^{2}cm^{-5}$ | $\rm GeV^{2}cm^{-5}$ | $\rm GeV^{2}cm^{-5}$ | $\rm GeV^{2}cm^{-5}$
NGC6822 | 17.86 | 18.40 | 18.62 | 18.63
IC10 | 18.21 | 18.40 | 18.53 | 18.33
WLM | 16.72 | 17.10 | 17.27 | 17.24
IC1613 | 16.16 | 17.48 | 17.74 | 17.84
Phoenix | 14.40 | 15.04 | 15.16 | 15.16
DDO210 | 15.90 | 16.20 | 16.32 | 16.27
DDO216 | 15.45 | 15.72 | 15.83 | 15.73
Table 5: Total J-factor values integrated up to $R_{200}$ for each dIrr in our
sample, and for the different benchmark models summarized in Table 4.

We point out that the obtained range of J-factors for each dIrr, based on the
different substructure scenarios in Table 4, is wider than the one we would
have obtained from just having considered the uncertainty on $M_{200}$ (see
Table 2) in the J-factor computation for each dIrr. For this reason we neither
provide J-factor uncertainties in Table 5 nor include them in our data
analysis (see next Section IV).

Attending to the obtained values in Table 5, we note that they are
distributed, for all the benchmark models, according to the expected ratio
$M_{200}^{2}/D^{2}$. From them, we can also easily understand the impact of
taking into account different models for the subhalo population inside dIrrs.
In particular, we obtain boost values ranging between $B=0.6-3.4$ for the MED
model, and $B=1.1-4.8$ for MAX-Bur, depending on the considered dIrr (we
recall that $B=0$ means no boost in our definition). The only exception is
IC1613, for which we obtain $B_{\rm MED}=19.9$ and $B_{\rm MAX-Bur}=37.0$. For
the rest of the objects, we can check how these values compare to the ones
obtained in the literature. We can compare with Ref. Sánchez-Conde and Prada
(2014), who also use a $c-M$ parametrization for main halos applied to the
subhalos. For masses between $10^{6}-10^{10}$ M⊙, they obtain $B=1.2-2.0$ for
$\alpha=1.9$ and $B=2.0-7.0$ for $\alpha=2.0$. This is comparable to our
computations, despite the fact that authors in Sánchez-Conde and Prada (2014)
implicitly adopted NFW profiles instead of Burkert as we do. The same
difference is also found in Bonnivard  _et al._ (2016), where the authors
reproduce the same substructure modelling as in Sánchez-Conde and Prada
(2014). They obtain a good agreement, some small differences only arising for
high mass values ($M_{200}>10^{8}$ M⊙) and in the case of adopting $\alpha=2$
for the slope of the SHMF. This divergence, that we also find, can be
explained by the following: in the case of Sánchez-Conde and Prada (2014), the
total mass in the form of subhalos is added to the original $M_{200}$ of the
object, leading to slightly overestimated boost values for higher masses.
Instead, in Bonnivard  _et al._ (2016) the mass in the form of subhalos is
subtracted from $M_{200}$ of the host, leading to more realistic boost values,
as we also obtain. Finally, we can also compare our values to the ones
obtained using a $c-M$ relation for subhalos Moliné  _et al._ (2017) (who also
adopted NFW profiles for subhalos). In this case, for the same mass range,
they obtain $B=2.1-3.7$ for $\alpha=1.9$ and $B=3.7-10.3$ for $\alpha=2.0$. As
expected, these are a factor up to $\sim 2$ higher values, especially for the
$\alpha=2.0$ case. The reason being that, as said, is that Moliné  _et al._
(2017) predicts higher concentrations for subhalos when compared to halos.
Again, this means that our subhalo-boosted J-factors are conservative.

Another output we can easily obtain using CLUMPY are two-dimensional
templates, reproducing the spatial morphology of the expected DM annihilation
signal. We created maps for each dIrr galaxy and for each benchmark model in
Table 4, in total producing a compilation of 28 spatial templates. It should
be noted that we do not expect any of these subhalos to be individually
spatially resolved by Fermi-LAT, due to both their extremely small angular
extent at the distance of the dIrrs and the LAT instrumental resolution; thus
we decided to adopt an averaged description of the whole subhalo population
for drawing their contribution to the total signal. An example of these
templates is shown in Fig.1. Very importantly, we will use these maps in the
next section as the inputs for our Fermi spatial and spectral analysis, given
that they are the reference models to be fitted to the actual data.

Figure 1: Two-dimensional templates of the expected spatial morphology of the
DM annihilation fluxes from the IC10 dIrr, as obtained with CLUMPY for each of
the four benchmark DM substructure models in Table 4; see text for details.
The z-axis represents the values of the differential J-factors
($\frac{dJ}{d\Omega}$). Because the values of $\frac{dJ}{d\Omega}$ vary by
many orders of magnitude in these plots, we chose the colour scale in such a
way that we still could appreciate details in the MED and MAX models. However,
for the MIN case, this colour scale implies that the outer regions of the
object are not visible, as their expected fluxes already lie below the minimum
J-factor value shown by the color scale.

##  IV Fermi-LAT data analysis

Once the DM modeling of the dIrrs in our sample is complete, we can perform a
search for gamma-ray signals in Fermi-LAT data. To do so, we use Fermipy, a
Python package that automates the
ScienceTools777https://Fermi.gsfc.nasa.gov/ssc/data/analysis/documentation/
analysis. Fermipy v0.19.0 and ScienceTools v1.3.7 are used.

The first step is the photon event selection. We will use 11 years of LAT
data, from 2008 August 4 to 2019 August 8. The class event is Pass 8
SOURCEVETO Atwood  _et al._ (2013), with the corresponding P8R3_SOURCEVETO_V2
instrumental response function. We choose an energy range from 500 MeV to 1
TeV,888Although the LAT is sensitive to photons with energies as low as
$\sim$20 MeV, the Galactic diffuse emission is much more intense at these
energies, and so we decided to start the analysis at higher energies to avoid
possible contamination. with a zenith cut of $\theta_{z}>105^{\circ}$.

The data is binned using 8 energy bins per decade in energy and $0.08^{\circ}$
pixel size, defining a region of interest (ROI) of $12^{\circ}\times
12^{\circ}$, centered at the position of each dIrr. The Galactic diffuse
emission is modeled with the latest LAT template, gll_iem_v07, while the
isotropic contribution is modeled with the corresponding template,
iso_P8R3_SOURCEVETO_V2.txt.

We first perform a baseline fit to each individual ROI using the corresponding
CLUMPY template for each of the models. The spectral energy distribution (SED)
parameters of all the sources in the ROI, the normalization of the Galactic
diffuse emission and the isotropic template are left free. As the analysis
uses 3 years more data than the 4FGL, the pipeline searches for new sources.
These eventual sources would then be added to the model to optimize the fit of
the ROI; in this analysis, no new sources have been found.

Then, we run a fit and compute the likelihood profile as a function of energy
and energy flux of DM. In each energy bin, the only free parameter is the
normalization, which is computed independently from other bins.999By analyzing
each energy bin separately, we avoid selecting a single spectral shape to span
the entire energy range, at the expense of introducing additional degrees of
freedom in the fit. We then scan for each energy bin the likelihood as a
function of the flux normalization for the assumed DM signal, which depends
both on the annihilation channel and the WIMP mass, and adopt those parameters
which maximize the likelihood.

In the following, we will consider three annihilation channels: $b\bar{b}$,
$\tau^{+}\tau^{-}$, and $W^{+}W^{-}$, and WIMP masses ranging from 5 GeV up to
10 TeV. For each channel and mass, we extract the expected flux,
$\phi_{\gamma,j}$, in each energy bin. The upper limits to the flux for every
bin and target are shown in Appendix C. Then, we compute the likelihood of
observing $\phi_{\gamma,j}$, and the log-likelihood in each energy bin is
summed to get the overall log-likelihood, given by,

$\mathrm{log}\mathcal{L}\left(\mu,\theta|\mathcal{D}\right)=\sum_{j}\mathrm{log}\mathcal{L}_{j}\left(\mu,\theta_{j}|\mathcal{D}_{j}\right)$
(14)

where $\mathcal{L}$ is the likelihood, $j$ is the index of each energy bin of
the Fermi-LAT data ($\mathcal{D}$), $\mu$ are the DM parameters
($\langle\sigma v\rangle$ and $m_{\chi}$), and $\theta$ are the parameters in
the background model, i.e., the nuisance parameters. These include the
uncertainty of the J-factors, set to 0.3 dex (i.e.,
$\mathrm{log}_{10}\leavevmode\nobreak\ \sigma_{J}=0.3$) in our analysis for
every target in the sample. This value was chosen following the typical size
of J-factor uncertainties reported in the literature Bonnivard  _et al._
(2016); Chiappo  _et al._ (2016); Boddy  _et al._ (2020) as well as previous
choices by the LAT collaboration Ackermann  _et al._ (2015); Drlica-Wagner
_et al._ (2015); Albert  _et al._ (2017), associated with systematic
uncertainties in the determination of the profile parameters, i.e. the fit of
the RC.101010The uncertainty on individual J-factor values is subdominant
compared to the one coming from the use of different DM models for each
dwarfs, i.e., MIN, MED and MAX models. Nonetheless, we decided to include it
in our analysis even if this choice will not affect the results, being the
systematics associated to the different DM models dominant. The significance
of the DM hypothesis can be evaluated via the test statistics
(TS)111111$TS\sim\sigma^{2}$, this is, a $5\sigma$ detection would be
equivalent to $TS\sim 25$. Strictly speaking, $TS\sim\sigma^{2}$ only applies
in the asymptotic case with nested hypotheses and 1 additional degree of
freedom. Furthermore, the null hypothesis can’t be degenerate.,

$TS=2\leavevmode\nobreak\ \Delta\mathrm{log}\mathcal{L}=2\leavevmode\nobreak\
\mathrm{log}\left[\frac{\mathcal{L}\left(\mu,\theta|\mathcal{D}\right)}{\mathcal{L}_{null}\left(\theta|\mathcal{D}\right)}\right]$
(15)

where $\mathcal{L}_{null}$ is the likelihood in the case of null hypothesis,
i.e., no DM, and $\mathcal{L}$ is the likelihood for the DM hypothesis.

Once the individual targets have been studied, one can perform a combined
analysis simultaneously using all the targets in our sample. This is performed
by simply summing the individual log-likelihood profiles for each of the
targets, to obtain a global likelihood. As seen in Figure 2, where the MED
model is used, no significant emission is detected from any of the targets and
any of the channels, with the WLM galaxy having the largest observed TS,
$\sim$9–11, depending on the annihilation channel. We also show the combined
likelihood when considering a joint analysis. It is interesting to look at the
likelihood profile of each dIrr as a function of $m_{\chi}$, i.e., the TS
preference for each of the masses we are scanning. The wider the profile, the
more uncertain is the determination of the potential signal, while the height
measures the overall preference.

Figure 2: Likelihood profiles as a function of the WIMP mass, for each of the
dIrrs and the combined targets, assuming the MED model. The $\langle\sigma
v\rangle$ is left free for each mass bin. Top, middle and bottom panels are
for the $b\bar{b}$, $\tau^{+}\tau^{-}$, and $W^{+}W^{-}$ annihilation
channels, respectively.

Note that these likelihoods present each mass bin with its own best-fit
$\langle\sigma v\rangle$. From Figure 2, a $\sim 3\sigma$ excess is seen in
the cases of WLM and NGC6822. This apparent excess is present in the three
considered annihilation channels, while its position shifts to slightly larger
WIMP masses in the case of $W^{+}W^{-}$ and lower ones in $\tau^{+}\tau^{-}$.
The shift is just due to the fact that the $b\bar{b}$ channel peaks at roughly
$E_{peak}^{b\bar{b}}\sim m_{\chi}/20$, while $E_{peak}^{W^{+}W^{-}}\sim
m_{\chi}/30$ and $E_{peak}^{\tau^{+}\tau^{-}}\sim m_{\chi}/3$. It is
interesting to note that both the excesses in WLM and NGC6822 peak at the same
masses (as expected from an universal DM signal), and that in the $b\bar{b}$
channel the peak is at $m_{\chi}\sim 250$ GeV, which is still allowed by the
DM constraints obtained from the dSphs Oakes  _et al._ (2019). Yet, these
excesses are most likely due to the Galactic diffuse emission, which is not
perfectly modeled in any LAT analysis 121212We note that authors in Ref.
Gammaldi  _et al._ (2018) computed the expected astrophysical gamma-ray
emission originated in these objects to be well below the LAT detection
threshold.. In any case, these TS values are not considered to be significant,
as they are pre-trials and therefore are expected to decrease significantly in
a more complete statistical analysis including the “look-elsewhere” effect.
Thus, as no gamma-ray emission is conclusively observed from any of the
targets, in the next subsection we will proceed to set limits to the WIMP mass
vs. annihilation cross section parameter space.

##  V Dark matter limits on the annihilation cross section

From $\mathcal{L}\left(\mu,\theta|\mathcal{D}\right)$ we can evaluate the one-
sided 95% confidence level (CL) exclusion limit on the flux, which is the
value at which the log-likelihood decreases by 2.71 (as we consider one-sided
limits) with respect to its maximum value. Then, from this value and Eq. (10)
we can compute 95% C.L. upper limits in the $\langle\sigma v\rangle-m_{\chi}$
parameter space for each dIrr and DM modeling scenario described in Section
III. In Figure 3 the individual limits are plotted for the four considered DM
models and the $b\bar{b}$ annihilation channel. The individual limits for
$\tau^{+}\tau^{-}$ and $W^{+}W^{-}$ are deferred to Appendix D.

Figure 3: Upper limits for $\langle\sigma v\rangle$ for each individual
source, for the $b\bar{b}$ annihilation channel. The different DM models
considered are, from top to bottom, MIN, MED, MAX-Bur and MAX-NFW; see Table 4
for details on each of them.

As seen in the figure, none of the targets is able to reach the canonical,
thermal relic cross section, the best limits from IC10 still being a factor
$\sim$10 away. Indeed, the best results are obtained for IC10 in all DM models
which, interestingly, have almost no sensitivity to the change of the DM
density models, as the four considered models yield very similar results. This
is not true for the rest of dIrrs though, for which the change of DM model can
vary the limits up to a factor 10 (e.g., IC1613, for which a significantly
different behavior of its extended emission under the different DM scenarios
was obtained).

Our joint likelihood analysis allows also to derive combined DM constraints
using all objects in the sample at once. These constraints are plotted in
Figure 4 for the four considered DM models and the three annihilation
channels.

Figure 4: Limits for the combined signal in the four considered scenarios, for
$b\bar{b}$ (top panel), $\tau^{+}\tau^{-}$ (middle panel) and $W^{+}W^{-}$
(bottom panel) annihilation channels.

The combined limits are dominated by the brightest (i.e., with largest
J-factor) object, IC10, whose limits we recall only changed marginally between
the different DM models. Therefore, the combined limits present a very small
deviation from each other as well, and for the $b\bar{b}$ channel are at the
level of $\mathrm{\sim 10^{-25}cm^{3}s^{-1}}$ for $m_{\chi}=10$ GeV and
$\mathrm{\sim 5\cdot 10^{-23}cm^{3}s^{-1}}$ for $m_{\chi}=1$ TeV.

We note that the observed local excesses in the case of NGC6822 and WLM (see
Figure 2) weaken the joint limits at the corresponding masses in each
annihilation channel. As the combined analysis is dominated by the brightest
objects, WLM has the most relevant contribution to this weakening, which
occurs for the masses at which the excess has the largest TS values, i.e,
around 200–300 GeV for $b\bar{b}$, 30–70 GeV for $\tau^{+}\tau^{-}$ and
500–600 GeV for $W^{+}W^{-}$.

##  VI Discussion and Conclusions

We analyzed 11 years of Fermi-LAT data from the sky regions corresponding to 7
dwarf irregular galaxies, i.e. NGC6822, IC10, WLM, IC1613, Phoenix, DDO210 and
DDO216 in the context of DM searches. DIrrs are rotationally-supported star-
forming galaxies, yet DM dominated systems, thus suitable targets for indirect
DM searches. Nevertheless, they represent a clear example of the cusp-core
tension between observations and N-body simulations. For this reason, in our
work we considered both a data-driven core-like Burkert and an N-body
simulation motivated cuspy NFW DM density profile.
We used CLUMPY in order to calculate the J-factors for each target and DM
profile, which include the effect of subhalos in the annihilation flux under
different configurations of the subhalo population in these objects. The
values obtained for the subhalo boost for our benchmark DM models reach a
factor $\sim 5$, i.e. a factor $\sim 2$ lower than the ones obtained in Moliné
_et al._ (2017) for this same mass scale. This means that we derived both
conservative J-factors and DM constraints. For each DM model, we created two-
dimensional spatial templates of the expected DM annihilation signal with
CLUMPY. In fact, the angular extension of dIrr galaxies makes it mandatory to
consider it in the data analysis should we want it to be state-of-the-art and
realistic.
We have performed a search for gamma-ray signals in Fermi-LAT data in each of
the targets’ ROI. After our analysis, these objects stay undetected in gamma-
rays. No significant emission is detected, with the highest TS values $TS\sim
9-11$ corresponding to the WLM galaxy ($TS\sim 8$ for NGC6822), depending on
the considered annihilation channel ($b\bar{b}$, $\tau^{+}\tau^{-}$ and
$W^{+}W^{-}$). These TS values are not considered to be significant, mainly
because they are pre-trials and thus are expected to decrease significantly in
a more complete statistical analysis. Also, the presence of little excesses at
a few GeV is common in this type of analysis due to our imperfect knowledge of
the Galactic foregrounds, which may contribute at this TS level. Nevertheless,
this fact could point to a potential DM emission in the two objects. Indeed,
based on previous theoretical studies, we expect a negligible astrophysical
flux from SFRs in these galaxies and the DM-induced emission should be the
dominant one (Gammaldi  _et al._ , 2018).131313Assuming the significance
scales as the square root of time in this energy regime, many additional years
of LAT data would still be required to confirm a signal, if such a signal is
actually present.
Since no gamma-ray emission is conclusively observed from any of the targets,
we use our flux upper limits to set constraints on the WIMP mass vs.
annihilation cross section parameter space. The most stringent constraints are
obtained for IC10 and NGC6822, independently of the adopted DM profile, and
are at the level of $<\sigma v>\sim
10^{-25}-10^{-22}\,\text{cm}^{3}\text{s}^{-1}$ for $m_{\chi}\sim 10-10^{4}$
GeV, respectively. Differences between limits are mainly due to the different
contribution that the halo substructure boost has for each object in the
sample. Finally, we obtained combined DM limits from the joint likelihood data
analysis performed. IC10 also dominates these combined limits for each model
and three annihilation channels, i.e. $b\bar{b}$, $\tau^{+}\tau^{-}$ and
$W^{+}W^{-}$. The strongest constraints are obtained for the $b\bar{b}$
annihilation channel and are at the level of $<\sigma v>\sim 7\times
10^{-26}\,\text{cm}^{3}\text{s}^{-1}$ at $m_{\chi}\sim 6$ GeV.

Figure 5: Comparison between different limits for the $b\bar{b}$ annihilation
channel. We include the ones derived in this work, assuming the spatial
template (solid blue), the theoretical predictions from Gammaldi  _et al._
(2018) (dashed orange), the null extended simulations 95% containment band
(see Appendix B), and the LAT dSphs Albert  _et al._ (2017) in dot-dashed
green.

In Figure 5 we showed the main results of this work: the combined DM limits
obtained from the spatial data analysis for the MED model and the $b\bar{b}$
annihilation channel (blue line). We note that these limits are slightly
different from the yellow band shown in the same plot, which represents the
$95\%$ C.L. containment band after having performed 100 control simulations
assuming no DM content in the targets, yet modeled with their corresponding
spatial templates (null simulations; see App. B for further details). The
observed mismatch can be easily attributed to the small local TS excesses
found for some objects in our sample at the relevant energies.

Interestingly, our combined DM limits are in remarkable agreement with the
constraints obtained in the previous theoretical work (yellow-dotted line)
Gammaldi  _et al._ (2018), where the Universal Rotation Curve was assumed as
the basis for the DM modelling and the targets were considered to be point-
like just as a first approximation. For comparison, in Figure 5 we also show
the constraints obtained by the Fermi-LAT collaboration from the combined
analysis of tens of dSph galaxies Albert  _et al._ (2017). The latter allows
to rule out light thermal WIMP masses below $\sim$60 GeV Calore  _et al._
(2018). In contrast, the limits from our combined analysis are not able to
reach the canonical, thermal relic cross section. The best limits are reached
at the lightest considered WIMP masses, still being a factor $\sim$3 above the
thermal value.
Let us stress that our work represents the first extended analysis of dIrr
galaxies with 11 years of Fermi-LAT data in the context of DM searches. By
collecting more and more spectral data and RC measurements, it will be
possible to obtain a better estimation of the DM halo mass associated to these
objects, in this way further reducing the corresponding uncertainty in this
type of analysis. Also the use of future, more refined models for both the DM
density profile and the annihilation boost due to substructures will help in
this direction. Certainly, a better understanding of baryonic physics by means
of hydrodynamical simulations and its comparison with the available
observational data Kirby  _et al._ (2014) would also help in order to reach a
complete understanding of the kinematics of these objects.
To conclude, the increasing number of dIrr galaxies that has been recently
detected and studied in their kinematics (see e.g. Oh  _et al._ (2015, 2011);
Yu  _et al._ (2020); Barat  _et al._ (2020)) , make them interesting targets
for gamma-ray DM searches and further efforts should be pursued to the study
of this class of dwarfs. Among others, the Study of H$\alpha$ from Dwarf
Emissions (SH$\alpha$DE) Barat  _et al._ (2020) is a high spectral resolution
integral field survey of 69 dwarf galaxies14141449 star-forming galaxies
selected form the Sloan Digital Sky Survey Data Release Alam (2015) as well as
20 targets from the SAMI survey Croom (2012) as a control sample. with stellar
masses $10^{6}<M_{D}<10^{9}\,\rm{M_{\odot}}$. SH$\alpha$DE is designed to
study the kinematics and stellar populations of dwarf galaxies using
consistent methods applied to massive galaxies and at matching level of
detail, connecting these mass ranges in an unbiased way.

###### Acknowledgements.

The work of VG, JCB, JPR and MASC was supported by the Spanish Agencia Estatal
de Investigación through the grants PGC2018-095161-B-I00 and IFT Centro de
Excelencia Severo Ochoa SEV-2016-0597, the Atracción de Talento contract no.
2016-T1/TIC-1542 granted by the Comunidad de Madrid in Spain, and the
MultiDark Consolider Network FPA2017-90566-REDC. VG’s contribution to this
work has been supported by Juan de la Cierva-Formación FJCI-2016-29213 and
Juan de la Cierva-Incorporación IJC2019-040315-I grants. JPR also acknowledges
the support by MINECO via the SEV-2016-0597-17-2 grant, financed jointly with
Fondo Social Europeo. EK work was supported by the grant “AstroCeNT: Particle
Astrophysics Science and Technology Centre" carried out within the
International Research Agendas programme of the Foundation for Polish Science
financed by the European Union under the European Regional Development Fund.
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 Aeronautics and Space Administration and the Department of Energy
in the United States, the Commissariat à l’Energie Atomique and the Centre
National de la Recherche Scientifique / Institut National de Physique
Nucléaire 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 Research
Council and the Swedish National Space Board in Sweden. Additional support for
science analysis during the operations phase is gratefully acknowledged from
the Istituto Nazionale di Astrofisica in Italy and the Centre National
d’Études Spatiales in France. This work performed in part under DOE Contract
DE-AC02-76SF00515.

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##  Appendix A Rotation curves

In this Appendix, we show the fits to the rotation curves adopting a Burkert
DM density profile; see Figure A.6. For all panels, green solid lines in the
zoom-in regions show the total fit, i.e., DM component (Burkert profile) and
baryons (gas plus stars in the disk). Blue solid lines show the DM-only,
Burkert component. We also show as blue dashed lines the result of
reconstructing, for each case, the rotation curves with the $\Lambda$CDM
consistent (DM-only) NFW profile introduced in section II with some
assumptions. The main discrepancies between the model and the data for some
objects may be associated with (i) an incomplete knowledge of baryonic effects
in numerical simulations and/or (ii) systematic uncertainty in the
determination of the RC data, e.g., the estimated inclination angle of the
galaxy. (see section II for further details).

Figure A.6: Rotation curve data plotted along with the fit results taking only
the DM component. All the observed data are based on the HI measurements with
the exception of Phoenix galaxy for which optical observations are used. Blue
solid line represents the Burkert profile while blue dashed the NFW predicted
profile (see Section II for further details). For each galaxy, the main plots
show the RC till $R_{200}$, indeed an indirect representation of consistency
of the $M_{200}^{\text{Burk}}=M_{200}^{\text{NFW}}$ approximation; in the
zoom-in panel we show the results of the total fit (DM+baryons) for the
Burkert profile, indeed the discrepancy with the DM-only profile at small
radius.

##  Appendix B Validation of the pipeline via simulations

To validate the pipeline, we also run controlled simulations as a check. These
are performed assuming i) no DM content in the targets, yet modeled with their
spatial templates (null simulations) and ii) random sky pointings (blank
fields) assuming point-like sources (i.e., without the spatial DM template).
Both scenarios are repeated 100 times, in the first case using 100 pure null
simulations and in the second with 100 random sky pointings. The goal of these
checks is to compute the expected DM limits in the absence of a DM signal, and
to check the robustness of the targets. Both simulations are compatible with
our results in Section V, and are shown in Figure B.7 for the MED case as an
example.There seems to be a potentially significant mismatch at around few
hundred GeV between our results with the extended template and the
corresponding 95% c.l. band for the case of using an extended template as
well. Yet, we recall that this specific band was computed from null
simulations, i.e., no actual data, which is possibly the cause of the observed
difference.

Figure B.7: Control simulations for the $b\bar{b}$ annihilation channel in the
MED model. 100 simulations are run to compute the expected DM limits, shown
here as the 95% containment bands for both the null simulation using the
spatial template (cyan band) and the point-like analysis in random sky
positions (orange band). The actual data constraints are given by the dot-
dashed red and dotted blue lines, that were obtained, respectively, for the
point-like and extended (spatial template) cases.

In the figure, we see that the results obtained with actual data are contained
within the null simulation 95% c.l. uncertainty band, except for the 100–1000
GeV region, where the actual limits are degraded due to the found excesses
reported and discussed in Section IV. On the other hand, the random pointing
results are compatible with the point-like simulations at 95% c.l., which
validates our analysis pipeline.

##  Appendix C Flux upper limits

In this Appendix, we show the 95% C.L. flux upper limits as measured by the
LAT instrument, with the analysis setup explained in Section IV. The limits
are shown in Figure C.8.

Figure C.8: 95% C.L. upper limits to the flux and likelihood values found for
the dIrrs in the gamma-ray analysis performed in section IV with the Fermi-
LAT. Color code traces the change in log-likelihood. The arrows indicate upper
limits, as no signal is detected in any of the bins. Panels show, from left to
right and top to bottom, DDO210, DD0216, IC10, IC1613, NGC6822, Phoenix, and
WLM.

##  Appendix D Individual DM limits for $\tau^{+}\tau^{-}$ and $W^{+}W^{-}$
channels

In Figure 3 of the main text, we showed individual DM limits for each dIrr in
our sample and DM model for the case of annihilations to $b\bar{b}$. In this
Appendix, we also show DM limits for the other two considered channels,
$\tau^{+}\tau^{-}$ and $W^{+}W^{-}$, in Figure D.9.

Figure D.9: Left column: Upper limits for $\langle\sigma v\rangle$ to each
individual source, for the $\tau^{+}\tau^{-}$ annihilation channel. The
different models considered are, from top to bottom, MIN, MED, MAX-Bur and
MAX-NFW. Right column: Same as left column but for the $W^{+}W^{-}$
annihilation channel.