# Multi-messenger High-Energy Signatures of Decaying Dark Matter and the
Effect of Background Light

B. Skrzypek,11footnotetext: Corresponding author. M. Chianese  C. A. Argüelles

###### Abstract

The IceCube Neutrino Observatory at the South Pole has measured astrophysical
neutrinos using through-going and starting events in the TeV to PeV energy
range. The origin of these astrophysical neutrinos is still largely
unresolved, and among their potential sources could be dark matter decay.
Measurements of the astrophysical flux using muon neutrinos are in slight
tension with starting event measurements. This tension is driven by an excess
observed in the energy range of 40-200 TeV with respect to the through-going
expectation. Previous works have considered the possibility that this excess
may be due to heavy dark matter decay and have placed constraints using gamma-
ray and neutrino data. However, these constraints are not without caveats
since they rely on the modeling of the astrophysical neutrino flux and the
sources of gamma-ray emission. In this work, we derive background-agnostic
galactic and extragalactic constraints on decaying dark matter by considering
Tibet ASγ data, Fermi-LAT diffuse data, and the IceCube high-energy starting
event sample. For the gamma-ray limits, we investigate the uncertainties on
secondary emission from electromagnetic cascades during propagation arising
from the unknown intensity of the extragalactic background light. We find that
such uncertainties amount to a variation of up to $\sim 55\%$ in the gamma-ray
limits derived with extragalactic data. Our results imply that a significant
fraction of the astrophysical neutrino flux could be due to dark matter and
that ruling it out depends on the assumptions on the gamma-ray and neutrino
background. The latter depends on the yet unidentified sources.

##  1 Introduction

The measurement of High-Energy Starting Event (HESE) neutrinos in the TeV-PeV
range by the IceCube Neutrino Observatory has introduced new questions in
connection to their origin and properties [1, 2, 3]. These open questions have
the possibility of being answered through multi-messenger efforts in
astroparticle physics [4, 5]. In particular, the excess observed in the
7.5-year HESE sample, with its sensitivity to the Galactic center and,
generally, to the entire sky, cannot originate entirely from the atmospheric
neutrino background, nor does it lend itself to a simple astrophysical power-
law distribution given the observations in other channels [6]. Instead, a two-
component power-law spectrum, the nature of which is in question, is favored
by the data. Moreover, the modeling of this astrophysical contribution is yet
to be determined. However, it is well-known that the production mechanisms
involved in giving rise to highly energetic astrophysical neutrinos, namely
the decay of charged pions originating from hadronic [7, 8] and photohadronic
[9, 10] interactions, also produce coincident gamma-rays at the source. Most
notably, this coincident emission was confirmed by the multi-messenger
observation of the flaring blazar TXS 0506+056 [11, 12, 13].

However, the high level of gamma-ray suppression from astrophysical sources
along with tensions among current IceCube datasets in measurements of the
astrophysical spectrum make it difficult to characterize the nature of the
source of these ultra-high-energy neutrinos. With the assumption of a single
unbroken power-law flux, the 10-year through-going (TG) muon sample, which is
sensitive to neutrinos coming only from the Northern sky and with energies
larger than $15\text{\,}\mathrm{TeV}$, measures a spectral index of
$2.37^{+0.09}_{-0.09}$ [14]. This measurement is consistent with theoretical
predictions of neutrinos originating from photohadronic processes [15] and
with measurements of TXS 0506+056. On the other hand, HESE, with sensitivity
to the entire sky and to energies starting from $20\text{\,}\mathrm{TeV}$,
obtains a spectral index $2.87^{+0.20}_{-0.19}$ [6]. This suggests that there
could be an additional component contributing to the neutrino flux that HESE
observes that dominates at lower energies and comes from [16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27].

Given this observation, heavy dark matter has been proposed as a potential
source of the neutrinos in the HESE sample [28, 29, 30, 31, 32, 33, 34, 35,
36, 37] (see Ref.s [38, 39, 40] for the IceCube analyses). Heavy decaying dark
matter also gives rise to gamma-rays [41, 42, 43] which have been searched
with gamma-ray telescopes. Diffuse gamma-ray data places strong constraints on
the dark matter lifetime for various decay channels, and these largely
disfavor the dark matter hypothesis for IceCube’s observations [44, 45, 46,
47, 48, 49, 50, 51, 52, 53, 54]. In order to properly constrain the size of
the dark matter contribution in HESE, we need to establish the robustness of
these gamma-ray bounds.

In this work, we elaborate on previous dark matter decay results by
considering the impact that the Extragalactic Background Light (EBL) could
have on the predicted gamma-ray spectrum from dark matter. The presence of
different EBL models is due to the fact that direct measurements are difficult
and involve many uncertainties and tensions among observations, such as the
disagreement between near-IR data and observations of very high-energy photons
from extragalactic sources. As a result, there are various techniques used to
model the EBL [55, 56, 57, 58, 59, 60, 61, 62, 63], some of which are based on
local galaxy properties and others which are phenomenological approaches
targeting a larger redshift range. We simulate the propagation of photons,
electrons, and positrons using three different models of the EBL distribution,
and we compute the resulting gamma-ray spectra at Earth predicted by dark
matter decay. From there, we compare the fluxes to Tibet ASγ data [64] and
Fermi-LAT diffuse measurements of the isotropic diffuse gamma-ray background
(IGRB) [65], and we obtain lower bounds on the dark matter lifetime among the
different EBL models that could have significant implications for existing
limits.

The paper is organized as follows. In Section 2, we discuss the high-energy
neutrino and gamma-ray signatures of dark matter decay. In Section 3, we study
the impact of the EBL uncertainties in the propagation of gamma-rays. In
Section 4, we describe our analyses on neutrino and gamma-ray data sets and
present our results. Finally, in Section 5 we conclude.

##  2 Heavy Dark Matter Decay and its Signatures

Generally, models of heavy dark matter allow for decay modes where the final
products are Standard Model particle-antiparticle pairs. In all the decay
modes, neutrinos and gamma-rays are directly produced through the electroweak
and hadronic cascades which immediately follow the decay as well as the
electroweak radiative corrections which become relevant for dark matter masses
larger than $100\text{\,}\mathrm{GeV}$ [41, 43]. Together, these determine the
neutrino and gamma-ray differential energy spectra, which we denote as ${\rm
d}N_{i}/{\rm d}E$ with $i=\nu,\gamma$. This is used as the first input to
compute the observed fluxes at Earth.

Because of their nature, neutrinos injected by dark matter decay reach the
Earth unperturbed, barring effects on their final state flavor configuration
due to neutrino oscillations. Therefore, the neutrino flux at the Earth can be
easily calculated from the initial neutrino differential spectrum. We can
characterize dark matter sources as either galactic (G) or extragalactic (EG)
in nature, and thus the final differential neutrino flux can be broken down
into galactic and extragalactic contributions, namely

$\frac{{\rm d}\phi_{\nu}}{{\rm d}E_{\nu}{\rm d}\Omega}=\frac{{\rm d}\phi^{\rm
G}_{\nu}}{{\rm d}E_{\nu}{\rm d}\Omega}+\frac{{\rm d}\phi^{\rm EG}_{\nu}}{{\rm
d}E_{\nu}{\rm d}\Omega}\,,$ (2.1)

where $E_{\nu}$ denotes the neutrino energy and $\Omega$ denotes the solid
angle.

For the galactic contribution, we assume a generalized Navarro-Frank-White
(NFW) dark matter profile density, which takes the form

$\rho(r(s,\ell,b))=\rho_{s}\Big{(}\frac{r}{R_{s}}\Big{)}^{-\gamma}\Big{(}1+\frac{r}{R_{s}}\Big{)}^{-3+\gamma}\,,$
(2.2) $r(s,\ell,b)=\sqrt{s^{2}+R_{\odot}^{2}-2sR_{\odot}\cos\ell\cos b}\,,$
(2.3)

where $b$ and $\ell$ are the galactic coordinates. Here, we fix the parameters
$\gamma=0.4$, $\rho_{s}\sim 0.57\,{\rm GeV}/{\rm cm}^{3}$, $R_{\odot}=8.5\,\rm
kpc$, and $R_{s}\sim 83.24\,{\rm kpc}$ by minimizing the chi-squared
distributions obtained in [66].222The chi-squared distribution given in [66]
when profiled over the other parameters is very flat in $R_{s}$ allowing
values that range from 5 to 90 kpc. Another typical value that is chosen in
the literature for $R_{s}$ is approximately 24 kpc. The resulting column
densities that are relevant for this work are not significantly affected by
this choice. Varying these parameters within their allow ranges does not
significantly affect our results. This is because, in the case of decaying
dark matter, the choice of galactic profile parameters and their corresponding
uncertainties does not carry as much weight as it does for the case of
annihilating dark matter, where the flux is proportional to the square of the
density rather than just a single power of the density. Hence, the galactic
neutrino flux can be computed as

$\frac{{\rm d}\phi^{\rm G}_{\nu}}{{\rm d}E_{\nu}{\rm d}\Omega}=\frac{1}{4\pi
m_{\rm DM}\tau_{\rm DM}}\frac{{\rm d}N_{\nu}}{{\rm d}E_{\nu}}\int{\rm
d}s\rho(s,\ell,b)\,,$ (2.4)

where $m_{\rm DM}$ and $\tau_{\rm DM}$ are the dark matter mass and lifetime,
respectively. The differential neutrino spectrum includes neutrino and anti-
neutrinos of all the three flavours and is computed by means of the HDMSpectra
code [43].

The extragalatic dark matter component, on the other hand, is generally taken
to be isotropic. The resulting extragalactic flux can be written as an
integral over cosmological distances parametrized by redshift $z$, where we
also need to account for the redshift in the neutrino energies. This amounts
to a difference between the fluxes at production and detection. The
corresponding neutrino flux at detection is:

$\frac{{\rm d}\phi^{\rm EG}_{\nu}}{{\rm d}E_{\nu}{\rm
d}\Omega}=\frac{\Omega_{\rm DM}\rho_{\rm cr}}{4\pi m_{\rm DM}\tau_{\rm
DM}}\int\frac{{\rm d}z}{H(z)}\left.\frac{{\rm d}N_{\nu}}{{\rm
d}E_{\nu}}\right|_{E^{\prime}_{\nu}=E_{\nu}(1+z)}\,,$ (2.5)

where $H(z)=H_{0}\sqrt{\Omega_{\Lambda}+\Omega_{m}(1+z)^{3}}$ is the Hubble
parameter and $\rho_{\rm cr}$ is the critical density of the Universe [67].
Throughout this work we set the cosmological parameters to be: $\Omega_{\rm
DM}=0.23$, $\Omega_{\Lambda}=0.77$, $\Omega_{m}=0.27$, and $H_{0}=67.7~{}{\rm
km/s/Mpc}$. These are not the most recent values of the Planck parameters, but
small changes to these should result in neither large changes to our
conclusions nor changes to the final limits that we obtain.

Unlike in the case of neutrinos, the flux of gamma-rays produced from dark
matter decay is heavily impacted by the intermediate interactions that arise
from the subsequent propagation of photons and charged particles to Earth. At
high energies, photons interact with background radiation and produce
electron-positron pairs, preventing a subset of gamma-rays from reaching the
Earth. Primary gamma-rays therefore experience an attenuation which is encoded
in the optical depth, $\tau_{\gamma\gamma}$, which is a quantity that depends
on the photon energy as well as the distance from the source to Earth.
Additionally, electrons and positrons undergo interactions such as Inverse
Compton scattering with background photons, bremsstrahlung, and synchrotron
radiation in the galactic and intergalactic magnetic field. These processes
can occur several times over the course of propagation, thus developing
electromagnetic cascades that result in a secondary flux of low-energy photons
at Earth [68, 69, 70, 71, 72]. Hence, the final photon flux is a combination
of galactic and extragalactic components, each of which receives a prompt and
two secondary (sec.) contribution from primary photon and electrons:

$\frac{{\rm d}\phi_{\gamma}^{\alpha}}{{\rm d}E_{\gamma}{\rm
d}\Omega}=\frac{{\rm d}\phi^{\rm prompt}_{\gamma}}{{\rm d}E_{\gamma}{\rm
d}\Omega}+\frac{{\rm d}\phi^{\rm sec.}_{\gamma\to\gamma}}{{\rm
d}E_{\gamma}{\rm d}\Omega}+\frac{{\rm d}\phi^{\rm
sec.}_{e^{\pm}\to\gamma}}{{\rm d}E_{\gamma}{\rm
d}\Omega}\qquad\text{with}\quad\alpha={\rm G~{}or~{}EG}\,,$ (2.6)

where $\phi^{\rm prompt}_{\gamma}$ and $\phi^{\rm sec.}_{\gamma\to\gamma}$ are
the unscattered and cascaded components of the primary gamma-ray flux
$\phi^{\rm prompt}_{\gamma}$ respectively, while $\phi^{\rm
sec.}_{e^{\pm}\to\gamma}$ is the contribution of gamma-rays from the
interactions of the primary electrons.

The prompt gamma-ray flux can be similarly computed as Eq. 2.4 and Eq. 2.5 up
to an attenuation factor also know as gamma-ray opacity,
$e^{-\tau_{\gamma\gamma}}$. As will be discussed in Section 3, the opacity is
computed from the photon energy distribution in the background field and the
cross sections for each of the relevant processes. As a result, changes in our
modeling of the background radiation result in differences in the observed
prompt flux. Although the spectrum of electromagnetic cascades initiated by
primary photons and electrons approaches a nearly universal form consisting of
two power-law components: $E_{\gamma}^{-1.5}$ for
$E_{\gamma}\leq\mathcal{O}\left(0.1~{}{\rm GeV}\right)$, and
$E_{\gamma}^{-1.9}$ for $E_{\gamma}\leq\mathcal{O}\left(10^{5}~{}{\rm
GeV}\right)$ [72], different spectral energy distributions of the background
radiation can still slightly alter the universal gamma-ray spectrum. The
secondary gamma-ray flux can typically be computed by means of analytical and
numerical approaches [73, 74, 75, 76, 72, 77].

In this analysis, we use the public simulation software CRPropa [78, 79] to
calculate the gamma-ray flux in Eq. 2.6. This code allows us to propagate
photons and electrons over galactic and extragalactic distances with different
EBL models. In principle, such calculations require as input the differential
spectrum of primary photons and electrons directly injected by dark matter
decay as well as the dark matter distribution on galactic and extragalactic
scales. However, these calculations are computationally intensive, making the
injection of different dark matter spectra impractical. To overcome this
difficulty we have developed a reweighting scheme that allows us to use a
benchmark CRPropa simulation and transform it to yield the appropriate result.
Our benchmark simulation assumes a single unbroken power-law in energy that
ranges from the minimum energy of interest to the largest dark matter mass
considered. To be able to efficiently reweight the galactic and extragalactic
components, we produce two Monte Carlo data sets with the benchmark energy
spectra which are then used to compute the galactic and extragalactic dark
matter flux components. In both cases, we inject a spectrum that is uniformly
distributed over a sphere, where the radii are $100\text{\,}\mathrm{kpc}$ and
$10\text{\,}\mathrm{Gpc}$ for the galactic and extragalactic sets,
respectively. Further details of the simulation and our Monte Carlo
reweighting procedure can be found in Appendix A.

Figure 1: The different gamma-ray components from galactic (G) and
extragalactic (EG) sources for $m_{\rm DM}=1~{}{\rm PeV}$ and $\tau_{\rm
DM}=10^{27}~{}{\rm s}$. The dashed lines show the initial spectra for gamma-
rays and $e^{+}e^{-}$ resulting from ${\rm DM}\rightarrow b\bar{b}$ decay as
they appear at the source prior to propagation. The solid lines are the final
spectra at Earth we obtain using CRPropa simulations and subsequently applying
our reweighting procedure (see Appendix A). We fix the EBL spectrum according
to the Dominguez et al. model [55].

In Fig. 1, we show the different gamma-ray components described in this
section, in the case of dark matter decaying to $b\bar{b}$ with a mass of 1
PeV and a lifetime of $10^{27}~{}{\rm s}$. The dashed lines are the initial
spectra corresponding to the gamma-rays and electron/positron final-states,
directly emitted by dark matter particles. Additionally, the solid lines
represent the final gamma-ray spectra that we see at Earth after propagation.
Among these, we note that the galactic dark matter gamma-ray spectrum (red
shades of color) is heavily dominated by an overall attenuation from the
initial spectrum, whereas secondary production is merely a subleading effect.
This is consistent with the fact that the short propagation scales contribute
to the significant suppression of gamma-ray cascade production. In contrast,
there is visible secondary production in the extragalactic components (blue
shades of color), seen as the abundance at lower energies, indicating that
secondary production is the dominant mechanism giving rise to gamma-rays from
extragalactic dark matter decay. Alluding to the main goal of this work,
secondary production is heavily informed by the spectral energy distribution
of the background radiation, suggesting that the extragalactic component will
see greater effects from changes in the modeling of that background. We
elaborate on this point in the next section.

##  3 The Extragalactic Background Light

The EBL, along with the Cosmic Microwave Background (CMB), is the primary
target for the $\gamma-\gamma$ and $e-\gamma$ interactions. It is produced by
the radiation emitted by all the galaxies over the history of cosmic
evolution. Unlike the well-measured CMB spectrum, however, there are limited
ways to measure the EBL directly, and therefore it has significant
uncertainties. In particular, there are no direct measurements of the EBL
below $\sim$100\text{\,}\mu$$m because of radiation from Zodiacal light coming
from interplanetary dust radiation, which is about two orders of magnitude
larger than the EBL flux [56]. Collective limits obtained from direct and
indirect methods show that the EBL has a two-peak structure in the spectral
energy distribution from the ultraviolet and the far-infrared [80, 81].
However, the incomplete information surrounding the structure of EBL has led
to the development of different models, which use different methods for
estimating its structure [55, 56, 57, 58, 59, 60, 61, 62, 63]. This
uncertainty propagates down to the prediction of gamma-ray fluxes from distant
point-like sources [82, 83] and the aggregated diffuse distribution of sources
as is the case of dark matter. In this analysis we consider the following
benchmark models provided by Dominguez et al. [55] and Stecker et al. [56]. We
select these as benchmark models for our analysis in part because they are
motivated by observational uncertainties in the EBL and because they are
supported by constraints from data.

Figure 2: Spectral energy distributions (SEDs) at $z=0$ predicted by the EBL
models that we consider. Shown in purple is the 68% confidence band obtained
from Stecker et al. Minimal and Maximal models [56], and the cyan curve is the
SED computed by Dominguez et al [55]. The bottom axis gives the wavelength of
the target photons, while the top axis recasts this in terms of the minimum
(threshold) energy $E_{\gamma,\rm threshold}$ required by an incoming photon
to produce an electron-positron pair. We also show several observational
results for comparison, including direct constrains on the EBL in the form of
upper and lower limits from direct measurements [84] and galaxy counts [85].
In addition, we show the most recent data provided by VERITAS [86] and H.E.S.S
[87]., and the $1\sigma$ confidence level from Fermi-LAT based on
reconstructed cosmic emissivity [81]. Note that the lower cutoff seen in the
VERITAS band is artificial.

The Dominguez et al. 2011 model [55] is an ab initio evolution model that
begins with initial conditions and evolves forward in time by using semi-
analytical models of galaxy formation. The authors adopt a novel method
whereby they use the observed evolution of the rest-frame $K$-band galaxy
luminosity function along with galaxy SED functions, which are based on a
sample of 6000 galaxies in the range of 0.2 to 1.0 in redshift from the All-
wavelength Extended Groth Strip International Survey (AEGIS). In addition to
this, the authors also use their own estimates of the SED fractions of star-
forming galaxies, starburst galaxies, and active galactic nuclei (AGN)
galaxies. From this, the authors are able to extrapolate the evolution of the
luminosity densities from the UV to the IR range in target wavelengths.
Comparing their results to observations, they conclude with the claim that the
EBL is well-constrained in the UV to mid-IR range, but that additional
constraints from IR and gamma-ray astronomy are needed in order to account for
uncertainties in the far-IR region.

On the other hand, the Stecker et al. 2016 model [56] is a backward evolution
model that begins with a present-day luminosity function and extrapolates
backwards in time. To accomplish this, they use recent deep galaxy survey data
at infrared wavelengths, which is where galaxy emission arises from dust re-
radiation. In addition to this, the authors use the gamma-ray opacity from
photons in the $2.7\text{\,}\mathrm{K}$ cosmic background radiation to extend
their results to energies beyond the TeV scale. For wavelengths in the range
$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ to
$850\text{\,}\mathrm{\SIUnitSymbolMicro m}$, they use empirical data from the
Spitzer telescope, the Herschel telescope, the Planck space telescope, and
observations from the Atacama Large Millimeter Array (ALMA) and the Balloon-
borne Large Aperture Submillimeter Telescope (BLAST). Since the authors use
only empirical data, their results are model independent, thereby taking into
consideration the observational uncertainties of the EBL without additional
assumptions about the evolution of galaxy luminosity functions, star formation
rates, or galaxy evolution. The resulting uncertainties on the EBL that they
present therefore reflect the observationally determined errors that they use.
When performing their fit to observational data, they assume that the gamma-
ray opacity arises entirely from pair-production interactions with the EBL.
Moreover, they make use of luminosity densities (LDs), which are obtained by
integrating fits to observationally determined luminosity functions, to
determine photon densities. This includes fits to broken power-law functions
describing galaxies as well as a double power-law fitting function. In order
to compute the 68% limits on the LDs, the authors perform an upper and lower
limit power-law function fit in the region representing the rate of decline of
the LDs at redshifts beyond the peak of star formation. In computing the upper
limit, they assume a $(1+z)^{-2}$ decline in LD, while for the lower limit,
they use a steeper decline of $(1+z)^{-4}$.

Figure 3: Opacities calculated for sources at distances on galactic (left) and
extragalactic (right) scales. The solid (dashed) lines correspond to optical
depths calculated including the CMB with (without) the EBL contribution for
the Dominguez et al. model [55].

These models predict significantly different photon number densities,
encompassing the ranges allowed by uncertainties in the EBL. Fig. 2 shows the
spectral energy distributions predicted by the Dominguez et al. and Stecker et
al. models as well as data points from various observations and experiments.
These target wavelengths are in one-to-one correspondence with the minimum
incoming photon energy required to produce electron-positron pairs, resulting
in absorption and the development of an electromagnetic cascade. We show this
quantity, which we call $E_{\gamma,\rm threshold}$, on the top axis. In terms
of the target wavelength, this is given by $E_{\gamma,\rm threshold}\sim
m_{e}^{2}\lambda$, where $m_{e}$ is the electron mass and $\lambda$ is the
target wavelength.

Figure 4: Optical depths (top panels) and their ratios (bottom panels) for a
source of fixed distance within the galactic scale (left) and on the
extragalactic scale (right). In the top panels, the purple (blue) lines
corresponds to a target background consisting of the CMB with (without) the
EBL contribution for Dominguez et al. model [55]. The bottom panels shows the
magnitude by which the CMB-only optical depth deviates from the CMB+EBL
optical depth, illustrating the energy ranges where the EBL is significant in
determining the absorption of gamma-rays as they travel to Earth. The dashed
vertical line indicates the region that we define to be “EBL Dominated”, or
where the quantity plotted on the bottom panel is greater than 0.3. We call
this as the “Region of Interest” (RoI), and it demarcates the transition
between the region where the EBL is dominant according to our calculation and
the region where the CMB becomes the dominant target background. Figure 5:
Optical depths at various redshifts as predicted by the Dominguez et al. [55]
and Stecker et al. [56] EBL models. The purple band spans the range between
the Stecker et al. Minimal and Maximal models.

To understand the impact of the EBL in the gamma-ray fluxes that arrive at
Earth, we can analytically calculate the optical depth $\tau_{\gamma\gamma}$.
This quantity defines the gamma-ray opacity, $e^{-\tau_{\gamma\gamma}}$, which
describes the absorption of high-energy gamma-rays due to $\gamma-\gamma$
interactions. For photons of energy $E_{\gamma}$ emitted by a source at a
redshift $z$, this is given by

$\tau_{\gamma\gamma}=\int_{0}^{z}{\rm
d}z^{\prime}\,\frac{c}{(1+z)H(z)}\int_{0}^{2}{\rm
d}\mu\,\frac{\mu}{2}\int_{0}^{\infty}{\rm
d}\epsilon\,\sigma_{\gamma\gamma}(E_{\gamma},\epsilon)n_{\gamma}(\epsilon,z^{\prime}),\quad\mu=1-\cos\theta,$
(3.1)

where the final integral is over all possible target photon energies
$\epsilon$, $\sigma_{\gamma\gamma}$ is the pair-production cross-section [46],
and $n_{\gamma}$ is the differential number density of photons in the target
background, i.e. $n=n_{\rm CMB}+n_{\rm EBL}$. When the target background is
the CMB, the number density at redshift $z=0$ is given by a Planck
distribution, namely

$n_{\rm CMB}(\epsilon)=\frac{\epsilon^{2}}{\pi^{2}}\frac{1}{e^{\epsilon/T_{\rm
CMB}}-1},$ (3.2)

with $T_{\rm CMB}=2.348\times 10^{-4}\,{\rm eV}$. In Fig. 3, we show the
gamma-ray opacity as a function of gamma-ray energy for different distances
and redshifts with and without the EBL contribution according to the model by
Dominguez et al.. At energies of around $\sim$1\text{\,}\mathrm{PeV}$$, the
CMB dominates the development of electromagnetic cascades, whereas the EBL
starts to become relevant at lower energies. The EBL contribution relative to
the CMB, in particular, also becomes more significant with increasing
distances of the emission regions of the primary gamma-rays and electrons. As
can be seen in Fig. 3 (right), which displays opacities at the extragalactic
scale, the EBL contribution becomes considerably dominant at redshifts larger
than $z=0.01$; however, its effects can already be seen at the galactic scale
as shown in Fig. 3 (left).

The relative contributions of the EBL can also be seen in Fig. 4, where we
show the optical depths for sources at two fixed distances. The bottom panels
of this figure illustrate the portion of the combined optical depth
($\tau_{\rm CMB+EBL}$) that is due to interactions with the EBL, indicating
the energy ranges where the EBL is relevant, denoted by an arrow as the Region
of Interest (RoI). At both of the redshifts that we consider, the EBL becomes
important in the energy ranges spanned by both Fermi-LAT and Tibet
experiments, suggesting that any differences in the EBL could have
implications for dark matter decay limits obtained using these datasets.
Additionally, Fig. 5 shows the optical depths that the two EBL models we
consider predict for a range of redshifts and gamma-ray beam energies. Across
these ranges, the opacity exhibits differences among these models.

##  4 Analysis and Results

The motivation behind this work is to address the previously-obtained gamma-
ray dark matter constraints that disfavor the decaying dark matter hypothesis
for IceCube neutrino observations and to examine the effects that the
uncertainty of the EBL has on those constraints and whether this implicates
previous conclusions about dark matter. Hereafter, we obtain dark matter
limits for the $b\bar{b}$ decay channel with the latest high-energy neutrino
and gamma-ray data and compare our results with previous limits to illustrate
the EBL effect. We focus on the 7.5-year HESE sample, which has access to the
prompt dark matter neutrino flux from the whole sky (and in particular from
the Galactic center), thus providing the dominant constraints on the large-
mass range. Regarding gamma-rays, we analyze the data collected by Tibet ASγ
and Fermi-LAT experiments. The former has observed for the first time the
Galactic emission in the sub-PeV range and is therefore sensitive primarily to
the dark matter galactic component. On the other had, the latter constrains
the dark matter extragalactic component by measuring the IGRB over energies
lower than a few TeV. To derive lower limits on the dark matter lifetime
$\tau_{\rm DM}$, for each of the three experiments we consider the following
likelihood in the spirit of [88]:

$\mathcal{L}_{\rm exp}(\tau_{\rm DM},m_{\rm DM})=\prod_{i}^{n_{\rm
bins}}\left\\{\begin{array}[]{l r}\mathcal{P}(\phi_{i}^{\rm
exp}(E_{i})|\phi_{i}^{\rm DM}(E_{i}))&\quad(\phi_{i}^{\rm
exp}(E_{i})<\phi_{i}^{\rm DM}(E_{i}))\\\ 1&\quad(\phi_{i}^{\rm
exp}(E_{i})\geq\phi_{i}^{\rm DM}(E_{i}))\\\ \end{array}\right.,$ (4.1)

where $\phi_{i}^{\rm DM}(E_{i};\tau_{\rm DM},m_{\rm DM})$ is the predicted
energy-integrated flux on the $i$-th data bin, $\phi_{i}^{\rm exp}(E_{i})$ the
measured value by the experiment, and $\mathcal{P}$ is a Gaussian likelihood.
The width of this distribution is given by the sum of the squares of the
errors, which consist of statistical errors from our Monte Carlo generation
and experimental errors from the data. We then place a constraint on
$\tau_{\rm DM}$ for a given $m_{\rm DM}$ assuming Wilks’ theorem with one
degree of freedom.

The Tibet ASγ likelihood analysis is carried out by comparing our galactic
gamma-ray dark matter distributions to the data obtained in the angular window
$|b|<5^{\circ}$ and $25^{\circ}<l<100^{\circ}$ in the three energy bins
roughly defined by $[$100\text{\,}\mathrm{TeV}$,$160\text{\,}\mathrm{TeV}$]$,
$[$160\text{\,}\mathrm{TeV}$,$400\text{\,}\mathrm{TeV}$]$, and
$[$400\text{\,}\mathrm{TeV}$,$1\text{\,}\mathrm{PeV}$]$ [64]. In particular,
we calculate the likelihood for five different dark matter masses ranging from
1 PeV to 10 PeV, since for smaller masses the dark matter contribution to
Tibet ASγ data is almost negligible. Figure 6 shows an example of the final
result obtained from the combination of simulation and subsequent reweighting
for $m_{\rm DM}=$1\text{\,}\mathrm{PeV}$$ and $\tau_{\rm DM}=10^{27}~{}{\rm
s}$, using the Stecker et al. Minimal model (dashed purple line) and the
Dominguez et al. model (solid cyan line). As expected, the dominant
contribution to the change in the final dark matter spectrum is an overall
attenuation factor caused by gamma-ray absorption. As a result of the short
propagation distances, secondary gamma-rays and the inverse-Compton process
are merely subleading effects, thus implying a very small uncertainty due to
the choice for the EBL model. Hence, the contributions to the
$\Delta\chi^{2}=-2\ln\mathcal{L}_{\rm exp}$ (shown in the lower panel in Fig.
6), and consequently the lower limits on $\tau_{\rm DM}$, are essentially the
same for the different EBL models considered. This remains true across all the
dark matter masses that we consider in this work.

Figure 6: Final galactic dark matter decay spectra for a chosen dark-matter
mass and a lifetime of $10^{27}$ seconds. The two lines correspond to two
different EBL models. The bottom panel shows the $\Delta\chi^{2}$ calculated
by comparing our Monte Carlo simulations to Tibet ASγ data [64].

For Fermi-LAT likelihood analysis, we use the P7REP IGRB HI sample, whose data
are given as raw energy and positions in the sky [65]. Here, we ignore the
spatial distribution and bin the data into 40 energy bins, equally spaced in
logarithmic space from $200\text{\,}\mathrm{MeV}$ to $2\text{\,}\mathrm{TeV}$.
The Fermi-LAT likelihood is then calculated for eleven different dark matter
masses ranging from $10^{5.25}~{}{\rm GeV}$ to 10 PeV. As can be see in Fig.
1, where we show an example of the final result obtained from the simulation
and subsequent reweighting for a dark matter mass of $1\text{\,}\mathrm{PeV}$
decaying to $b\bar{b}$, there is a visibly significant production of secondary
gamma-rays as photons propagate to Earth, indicating that the dominant
production mechanisms for visible gamma-rays from extragalactic sources are
the $\gamma\rightarrow\gamma$ and $e^{+}e^{-}\rightarrow\gamma$ secondary
contributions.333The comparison of these two different components to data and
the contributions to the $\Delta\chi^{2}$ for different dark matter masses and
EBL models are reported in Appendix B. This results into a significant
uncertainty due to the EBL models on the dark matter constraints as shown in
Fig. 7. The three lines represent the lower bounds at 95% confidence level on
the dark matter lifetime for the $b\bar{b}$ decay channel obtained for
different EBL models: Stecker et al. Maximal model produces the weaker lower
bound (blue line), while the Minimal one produces the strongest limit (light
purple line); the model by Dominguez et al. lies in between (dark purple
line).

Figure 7: Fermi-LAT lower bounds at 95% CL obtained for the three different
EBL models considered in this work.

Figure 8: Lower limits at 95% CL on the dark matter lifetime derived from
IceCube 7.5-year HESE flux data (blue lines), Tibet ASγ data (light purple
line) and Fermi-LAT diffuse measurements (dark purple band). The left and
right plots refer to galactic and extragalactic gamma-ray data, respectively.
The dark purple band represents the uncertainty due to different modeling of
the EBL spectrum. The lower panels show the ration between the neutrino HESE
limits and the galactic (left) and extragalactic (right) gamma-ray limits we
obtain. Also shown with gray thin lines are previous bounds [35, 36, 48, 52].

Our results obtained using neutrinos and gamma-rays are shown in Fig. 8 for
the galactic (left plot) and extragalactic (right plot) dark matter
components, separately. In particular, the dark blue lines represent the
constraints placed by IceCube data, while the light (dark) purple lines in the
left (right) plot corresponds to Tibet ASγ (Fermi-LAT) data. In the right
plot, the purple band represents the allowed range within which the EBL models
we consider give predictions for the lower limit of the dark matter lifetime.
The purple dashed line within this band reflects the average taken over these
EBL models. We find that the gamma-ray limits derived with the galactic data
are weaker with respect to the ones from Fermi-LAT observations (for $m_{\rm
DM}$ smaller than 10 PeV), and that in general neutrino limits are stronger
than gamma-ray ones.

Along with our constrains, we also plot other neutrino and gamma-ray
constraints obtained outside of this work. We emphasize that, in contrast to
previously reported analyses in the literature, our analysis is background
agnostic; i.e., we do not consider fitting to an astrophysical component for
both neutrino and gamma-ray data. However, regarding neutrino limits, we find
improved constraints with respect to the works by Bhattacharya et al. [35] and
Chianese et al. [36]. This is despite the fact that both of these analyses
attempt to model the astrophysical background flux by means of a power-law
distribution. Aside from this, however, here are several additional
differences in our analyses, data collection, and likelihood treatments that
could account for the overall improvement in our neutrino constraints. First,
[35] uses six years of runtime with energies above $10\text{\,}\mathrm{TeV}$
and [36] uses the preliminary 7.5 years of HESE event data, whereas our
analysis uses the true 7.5 year HESE flux data. Second, for the galactic dark
matter distribution we rely on a generalized NFW profile with parameters that
were obtained by minimizing over the chi-squared distributions provided by the
recent work [66]. Finally, to obtain the predicted dark matter fluxes, our
analysis uses the recent code HDMSpectra which extends the calculations of the
electroweak radiative corrections to Planck-scale dark matter masses. For the
$b\bar{b}$ decay channel, these processes significantly change the energy
distribution of the neutrinos produced, yielding a harder neutrino spectrum
compared to the ones obtained by means of other software such as PYTHIA 8.2
[89] and PPPC [90](used in the analyses [35] and [36], respectively), which
in-turn produces a stronger result.

Regarding the gamma-ray constraints, our analysis on galactic data obtains
similar bounds to the ones obtained by Esmaili et al. [52]. On the other hand,
our results based on extragalactic data differ from the analysis performed by
Cohen et al. [48]. The latter considers peculiar regions around the Galactic
center (see Fig. S4 in [48]), where the dominant contribution from decaying
dark matter comes from the prompt galactic component and the secondary
galactic $e^{+}e^{-}\to\gamma$ component from Inverse-Compton scattering. We
instead focus on the IGRB data from the whole sky and find that the
extragalactic dark matter flux is the dominant contribution. Moreover, Cohen
et al. model the photon background, whereas we take a model agnostic approach.
By construction, our resulting constraints are weaker than those that are
background informed, as can be seen in the right plot of Fig. 8, where
previous constraints are significantly stronger for masses smaller than
$10^{6.5}~{}$\mathrm{GeV}$$. However, at the largest masses, we find good
agreement with [48], particularly when comparing to our mean prediction.444The
almost flat behaviour of our limit is indeed in agreement with one placed by
Cohen et al. for the extragalactic dark matter component only (see Fig. S13 in
[48]). The width of the EBL uncertainty band allows for even stronger
constraints that potentially extend to higher dark matter masses. At the
highest masses that we consider, this uncertainty gives rise to deviations of
up to 40% in lifetime. For $m_{\rm DM}=1~{}{\rm PeV}$, the EBL uncertainty
amounts to a $\sim 55\%$ variation for the dark matter lower bound on the
lifetime.

Finally, we comment on the relative strengths of the gamma-ray and neutrino
constraints obtained in this work. These are shown in the bottom panels of
Fig. 8, where we plot the ratio between the lifetime neutrino constraints and
the gamma-ray constraints. Since the flux is inversely proportional to the
lifetime, this ratio is proportional to the ratio of the dark matter fluxes
between neutrinos and photons. We find that this ratio is always greater than
one in the mass range given, even when accounting for the EBL uncertainties.
This implies that in a background agnostic treatment a significant fraction of
the neutrino flux observed in HESE can be accommodated with a dark matter
component. However, regarding the extragalactic gamma-ray limit, when the
background is modeled, the constraint from Cohen et al. applies and then we
find that this ratio is less than one below approximately $m_{\rm
DM}=10^{6.5}~{}$\mathrm{GeV}$$, and greater than one for higher masses. Our
results highlight the importance of properly modeling the gamma-ray background
in the Fermi-LAT data to derive proper constraints and, moreover, that at the
highest masses, the neutrino constraints dominate over the gamma-ray limits.

##  5 Conclusions

In this work, we present newly-obtained neutrino and gamma-ray limits on
galactic and extragalactic dark matter decay. We initialize dark matter decay
spectra for neutrinos, gamma-rays, electrons and positrons, and we propagate
these to Earth using three different models for the EBL: Dominguez et al. and
the Stecker et al. Minimal and Maximal models. We then compare our spectra to
the 7.5-year IceCube HESE sample, the Tibet ASγ 2021 data, and the Fermi-LAT
diffuse data. The resulting background-agnostic analysis that we carry out
yields constraints that are, taken by themselves, consistent with a decaying
dark matter component explaining the HESE data, therefore providing some
motivation for considering this as a partial explanation for the tension
between the HESE analysis and the Northern tracks data analysis regarding the
nature of the astrophysical neutrino spectrum. Figure 8 summarizes our
results, showing that the lower bounds estimated for decaying heavy dark
matter can vary by up to $\sim 55\%$ (for $1\text{\,}\mathrm{PeV}$) and up to
$\sim 40\%$ at the larger mass end, due to the current uncertainty on the EBL
spectrum. In comparison to previously-obtained limits using gamma-rays and
neutrinos, our constraints are in some respects better, but in other respects
they can be made more robust by incorporating a background model.

Our work motivates performing a similar investigation of the effects of the
EBL on multi-messenger signals from other kinds of astrophysical sources.
Finally, modeling a gamma-ray and neutrino background is an important step in
determining the extent to which high-energy neutrinos observed in samples such
as HESE or the through-going muon tracks come from heavy dark matter decay.
However, this astrophysical background comes from as yet unidentified sources.

## Acknowledgments

We thank Aaron C. Vincent and Sergio Palomares-Ruiz for useful comments and
discussions, and Rafael Alves Batista for clarifications about CRPropa. CAA
and BS are supported by the Faculty of Arts and Sciences of Harvard
University. Additionally, CAA thanks the Alfred P. Sloan Foundation for their
support. MC is supported by the research grant number 2017W4HA7S “NAT-NET:
Neutrino and Astroparticle Theory Network” under the program PRIN 2017 funded
by the Italian Ministero dell’Università e della Ricerca (MUR) and by the
research project TAsP (Theoretical Astroparticle Physics) funded by the
Istituto Nazionale di Fisica Nucleare (INFN).

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##  Appendix A Simulation Details

To propagate photons, electrons, and positrons over galactic and extragalactic
scales, we use CRPropa [78], a tool developed to simulate ultra-high-energy
nuclei within galactic and extragalactic environments. For the propagations
that we simulate in CRPropa, we enable the following processes: pair
production, double pair production, triplet pair production, Inverse-Compton
scattering, and synchrotron radiation. For galactic propagation, we set the
magnetic field to be the JF12 field model [91], while for extragalactic
propagation we assume no additional magnetic field contribution. In both
cases, we inject an initial power-law spectrum that follows
$\phi\sim(E/E_{c})^{-1}$ and which we define uniformly over a solid sphere.
For the galactic case, propagation takes places over distances of 0 to 100
kpc, while for the extragalactic case, we propagation between 100 kpc and 10
Gpc. CRPropa creates this initial spectrum via Monte Carlo generation, where
we set the number of Monte Carlo simulations to be roughly $1.5\times 10^{6}$
per injected dark matter mass. Regarding the injected spectrum, the minimum
and maximum $(E_{c})$ energies vary with the dark matter mass that we consider
such that $E_{c}=m_{\rm DM}/2$ and $E_{\rm min}=(m_{\rm DM}/2)\times 10^{-6}$.

To then obtain the expected dark-matter decay spectrum from the CRPropa output
following propagation, we apply a reweighting procedure where we group the
photons that arrive at Earth according to their corresponding, unique Monte
Carlo generated events. The reweighting scheme then assigns a new weight to
each of these events that depends on that event’s initial energy and position,
such that the new weights correspond to a dark matter distribution.
Schematically, the end result of this is that the our reweighting scheme takes
the initial power-law distribution that is fed into CRPropa and returns the
expected final, propagated spectrum that results from dark-matter decay for a
given dark matter mass, lifetime, and decay channel. Although we apply this to
the case of dark matter decay, this procedure could be generalized for any
arbitrary, well-behaved distribution that needs to be propagated over some
distance.

The weight that we assign to each event can be decomposed into a position
weight, and energy weight, and a multiplicative prefactor that depends on dark
matter parameters and that converts the result into units of flux.

###  A.1 Position Reweighting

The position reweighting that we apply to each Monte Carlo event assigns a
multiplicative factor replaces the Monte Carlos generation weights for a
uniformly distributed source with those corresponding to the desired dark
matter profile. Schematically, these position weights can be expressed as

$w_{r}=\frac{w_{\rm physical}}{w_{\rm generated}}=\Big{(}\rho_{\rm
DM}(r)\Big{)}\Bigg{(}\frac{D(r)}{\int_{r_{\rm min}}^{r_{\rm
max}}D(r)}\Bigg{)}^{-1}=\rho_{\rm DM}(r)(r_{\rm max}-r_{\rm min})\,,$ (A.1)

where $D(r)$ is the initial distribution, $\rho_{\rm DM}(r)$ is the dark
matter distribution, and the denominator of the quantity in parentheses
corresponds to an overall normalization included in the generated weights,
which is a uniform distribution for our simulations. The physical position
weights depend on the profile that we consider. For galactic reweighting, for
example, we use the generalized NFW profile $\rho_{\rm DM}(r)$ defined in Eq.
(2.2). For the extragalactic position reweighting, we use $\rho_{\rm
cr}\Omega_{\rm DM}/H(z)$, where we set the parameters to the values mentioned
in Section 2.

###  A.2 Energy Reweighting

In a similar way, the energy reweighting acts as a multiplicative factor that
replaces the Monte Carlo simulation weights for the power-law spectrum
$\phi(E)\sim E^{-1}$ generated by CRPropa and returns a physical weight for
each event that corresponds to the dark matter decay spectrum $\phi_{\rm
DM}\propto{\rm d}N/{\rm d}E$ computed by HDMSpectra. These energy weights can
be expressed as

$w_{E}=\frac{w_{\rm physical}}{w_{\rm generated}}=\Big{(}\phi_{\rm
DM}(E)\Big{)}\Bigg{(}\frac{\phi(E)}{\int_{E_{\rm min}}^{E_{\rm
max}}\phi(E)}\Bigg{)}^{-1}=\phi_{\rm DM}(E)(E\times\ln(E_{\rm max}/E_{\rm
min}))\,.$ (A.2)

For the extragalactic energy reweighting, there is also a redshift in the
energy of the initial spectrum which resulting in an additional factor of
$(1+z)$.

##  Appendix B Spectra of Photons Arriving at Earth

The different IGRB predictions of the gamma-ray flux for the three EBL models
and for different DM masses can be seen in Fig. 9 for the $\gamma\to\gamma$
extragalactic secondary component, and in Fig. 10 for the
$e^{+}e^{-}\rightarrow\gamma$ one.

Figure 9: Final dark matter gamma-ray flux from the extragalactic secondary
$\gamma\to\gamma$ component, for different dark matter masses at a lifetime of
$10^{27}$ seconds. Different colors correspond to different EBL models
considered for the propagation through the extragalactic space. The bottom
panels show the $\Delta\chi^{2}$ calculated by comparing our Monte Carlo
simulations to Fermi-LAT measurements of the IGRB spectrum, displayed by the
triangular points. The circular data points show the mean prediction and the
corresponding errors from the Monte Carlo simulations according to the
Dominguez et al. EBL model.

Figure 10: Final dark matter gamma-ray flux from the extragalactic secondary
$e^{+}e^{-}\to\gamma$ component, for different dark matter masses at a
lifetime of $10^{27}$ seconds. Different colors correspond to different EBL
models considered for the propagation through the extragalactic space. The
bottom panels show the $\Delta\chi^{2}$ calculated by comparing our Monte
Carlo simulations to Fermi-LAT measurements of the IGRB spectrum, displayed by
the triangular points. The circular data points show the mean prediction and
the corresponding errors from the Monte Carlo simulations according to the
Dominguez et al. EBL model.


# EADNet: Efficient Asymmetric Dilated Network for Semantic Segmentation

###### Abstract

Due to real-time image semantic segmentation needs on power constrained edge
devices, there has been an increasing desire to design lightweight semantic
segmentation neural network, to simultaneously reduce computational cost and
increase inference speed. In this paper, we propose an efficient asymmetric
dilated semantic segmentation network, named EADNet, which consists of
multiple developed asymmetric convolution branches with different dilation
rates to capture the variable shapes and scales information of an image.
Specially, a multi-scale multi-shape receptive field convolution (MMRFC) block
with only a few parameters is designed to capture such information.
Experimental results on the Cityscapes dataset demonstrate that our proposed
EADNet achieves segmentation mIoU of 67.1% with smallest number of parameters
(only 0.35M) among mainstream lightweight semantic segmentation networks.

Index Terms—  semantic segmentation, lightweight network, efficient
convolution, multi-scale multi-shape receptive field

##  1 Introduction

Semantic segmentation is a fundamental task in computer vision, and it can be
broadly applied to a variety of scenarios. Particularly, such task has strict
requirement for model size and response time when deployed in computing power-
restrained devices. Figure 1 shows the parameter sizes and accuracies (mIoUs)
for existing state-of-the-art semantic segmentation networks on the Cityscape
urban dataset [1]. It can be seen that the high-accuracy semantic segmentation
networks [2, 3, 4] tend to have a larger number of parameters, causing heavier
computational cost and often are difficult to meet the real-time segmentation
requirement on edge devices. In contrast, some lightweight networks [5, 6, 7,
8] have smaller number of parameters indicating fast inference speed, yet
sacrificing the prediction

Fig. 1: Accuracy vs. Model parameters on Cityscapes.

accuracies. Our proposed EADNet is on the left top of the figure and achieves
the best trade-off between accuracy and parameter size amongst peer networks.

Many high precision semantic segmentation networks develop various convolution
blocks such as Pyramid Pooling Module in PSPNet [2], ASPP in Deeplab series
[9], Object Context Module in OCNet [10] to extract the multi-scale context
information to improve the network performance. However, these operations are
unsuitable to lightweight networks duo to their massive computation cost.

Lightweight segmentation networks aim to generate high-quality pixel-level
segmentation results with limited computational power. ENet [6] reduces the
times of downsamling and uses depthwise convolution in pursuit of an extremely
tight framework. ERFNet [5] uses a non-bottleneck 1D structure to make trade-
off between precision and computation and achieves high precision with simple
structure. ESPNetv2 [8] introduces EESP Unit to pursuit larger receptive field
with less parameters. BiSeNet [11] proposes spatial path and context path to
extract spatial and context information. ICNet [12] uses multi-scale images as
input and cascade network to improve efficiency.

Fig. 2: (a) shows the 9 anchors in Faster-RCNN [13]. (b) shows the four
receptive field rectangles of MMRFC block. (c) shows the corresponding image
areas for the four receptive field rectangles in (b).

To extract context information with lower computation cost, in this work, we
focus on the design of lightweight convolution block to capture multi-scale
multi-shape context information and further build a rapid and accurate
semantic segmentation network by employing the designed convolution block.
Inspired by faster R-CNN [13], which utilizes anchor boxes with different
aspect ratios to generate region proposals for covering objects of different
shapes, we proposed a multi-scale multi-shape receptive field convolution
(MMRFC) block, which has receptive fields of different aspect ratios to well
match with objects presenting different shapes and scales in the feature maps.
Figure 2 shows the 9 anchor boxes used in Faster R-CNN [13] and the 4
receptive field boxes in our MMRFC block.

Unlike the lightweight networks [6, 5, 11, 12] which commonly use encoder-
decoder structure, by improving the UNet [14] structure, we use a specially
designed skip connection to combine detailed information in shallow layers and
abstract information in deep layers at a small computational cost without
decoder stage. The MMRFC block developed as above is incorporated into this
structure to form an extreme tight semantic segmentation network, EADNet.
EADNet has a very small number of parameters and involves minimum
computational operations, and is suitable for deployment on embedded edge
devices.

In summary, our contributions are summarized as follows:

1) We propose multi-scale multi-shape receptive field convolution (MMRFC)
block, to capture multi-scale context information which is sensitive to
various shapes of objects in input images.

2) Based on MMRFC block, we develop a lightweight semantic segmentation
network, EADNet, which has an extreme tight and efficient structure to combine
shallow layer features and deep layer features without decoder stage.

3) We achieve competitive segmentation accuracy (67.1% mIoU) with smallest
network parameter size (only 0.35M) on the Cityscapes test dataset.

##  2 PROPOSED METHOD

As described before, we propose MMRFC block and further adopt MMRFC block in
the specially designed structure to form EADNet. EADNet achieves the optimized
point of semantic segmentation in terms of the accuracy, the number of
parameters and FLOPs. Following subsections will give detailed analysis of the
MMRFC block and EADNet.

Fig. 3: The internal structure of MMRFC block, $dr$ denotes the base dilation
rate of current MMRFC block.

###  2.1 MMRFC Block

MMRFC is designed to capture mulit-scale multi-shape context information with
low computation cost and is suitable for lightweight semantic segmentation
networks. MMRFC block consists of two parts: branch-merge part and transform-
fusion part as shown in Figure 3. The branch-merge part contains four
convolution branches, where each branch is composed of three sequential
convolution operations: one 1$\times$1 pointwise convolution to reduce the
input dimension and extract features sensitive to this branch, one 3$\times$1
asymmetric (depthwise) convolution and one 1$\times$3 (depthwise) asymmetric
convolution. The two asymmetric convolutions in each branch have same or
different dilation rates to achieve variable shapes and sizes of receptive
fields, and a parametric rectified linear unit (PRelu) [15] is adopted between
them. The output of four branches will be concatenated as the input of
transform-fusion part.

Transform-fusion part is inspired by GhostNet [16], which shows that redundant
information is important to improve the accuracy of semantic segmentation and
redundant features can be easily generated by linear transform operation.
Transform-fusion part is composed of one 3$\times$3 depthwise convolution, a
parallel skip connection, a feature concatenation operation and a 1$\times$1
pointwise convolution, to double the channel of input feature maps and fuse
their channel information. Lots of new redundant information is generated by
the 3$\times$3 depthwise convolution with little additional cost, and finally
goes through feature fusion by 1$\times$1 pointwise convolution.

Fig. 4: Overview of our EADNet. In figure, ”C” means concatenation, $*n_{i}$
represents the number of MMRFC blocks, left bottom is the internal structure
of ConcatConv blocks, SeqDwConv block and DwConv block.

In asymmetric dilated convolution, when dilation rate gets larger, e.g.,
beyond 24, the 1$\times$3 or 3$\times$1 filter may degenerate to a 1$\times$1
filter since only the center weight is effective as analyzed in [9]. According
to this, we adopt dilation rate in four branches of branch-merge part as
(1,1), ($dr$, $dr$), ($dr*2$, $dr*4$) and ($dr*4$, $dr*2$) respectively to
generate receptive fields like Figure 2 (b), where $dr$ represents the base
dilation rate of the MMRFC block and the value of $dr$ usually increases
gradually as the layer goes deeper. To ensure every dilation convolution
kernel works, the maximum value of $dr$ is set to 6, yielding a maximum
dilation rate of 24 for each asymmetric convolution in all MMRFC blocks.

Generally, the spatial and channel correlations between adjacent pixels in an
image are stronger than those between non-adjacent pixels. Therefore, we adopt
asymmetric convolution to extract spatial and cross-channel features from
adjacent pixels in the first branch in branch-merge part. In the second, third
and fourth branch which has larger dilation rates, depthwise asymmetric
convolutions with less computation cost are used to reduce the computation
cost. MMRFC block follows similar strategy as bottleneck structure to first
compress the number of input channels to $1/8$ in each of the four branches by
a 1$\times$1 convolution, perform concatenated convolution in each branch for
feature extraction in branch-merge part, and then combine them and restore the
channel number to the original in transform-fusion part. These strategies can
significantly reduce the number of FLOPs and parameters in MMRFC block.
Following will give the detailed formulas of the FLOPs and parameter size
computation.

Assume the shape of input feature is $C*W*H$, where $C$, $W$, $H$ denote the
channel number, width and height of the feature, respectively. We denote by
$P_{bi}(i=1,2,3,4)$ as the convolution operation parameters in the four
branches of branch-merge part. $P_{t}$ and $P_{total}$ are the convolution
operation parameters of transform-fusion part and whole MMRFC block.
$F_{total}$ denotes the total FLOPs in the convolution operations of MMRFC
block. They are computed as follows:

$P_{bi}\\!=\\!=\left\\{\begin{array}[]{r}\\!(C\\!+\\!1){\times}\frac{C}{8}\\!+\\!(3{\times}\frac{C}{8}\\!+\\!1){\times}\frac{C}{8}{\times}2\;for\;i=1\\\
\\\
\\!(C\\!+\\!1){\times}\frac{C}{8}\\!+\\!(3\\!+\\!1){\times}\frac{C}{8}{\times}2\;for\;i=2,3,4\end{array}\right.$
(1)

$P_{t}=(3{\times}3+1){\times}\frac{C}{2}+(C+1){\times}C$ (2)

$P_{total}=\sum_{i=1,2,3,4}^{i}P_{bi}+P_{t}$ (3)

$F_{total}=P_{total}{\times}W{\times}H$ (4)

The maximum number of input channels for MMRFC block is set at 128 in our
network. Therefore, in the case of $C\\!=\\!128$, ignoring a small number of
parameters and FLOPs in PRelu [15], batch normalization [17] and other non-
convolution operations, MMRFC block has a parameter size of 27360 and FLOPs of
$27360*W*H$, which is $1/5$ of that for the common 3$\times$3 convolution with
same input and output dimension.

Table 1:  FPS, FLOPs, parameter size and mIoU comparison on Cityscapes test set. Method | Pretrain | Input Size | Inference Time(ms) | FPS | FLOPs(G) | Parameters(M) | mIoU(%)
---|---|---|---|---|---|---|---
SegNet [18] | ImageNet | $1024*2048$ | 152.68 | 6.55 | 1310 | 29 | 57.0
SQ [19] | ImageNet | $1024*2048$ | 88.19 | 11.34 | 501 | 16 | 59.8
ERFNet [5] | - | $1024*2048$ | 43.71 | 22.88 | 103 | 2.1 | 68.0
ENet [6] | - | $1024*2048$ | 36.9 | 27.11 | 22 | 0.37 | 58.3
DFANet A[20] | ImageNet | $1024*2048$ | 26.91 | 37.16 | 28 | 2.0 | 71.3
ESPNetv2 [8] | - | $1024*2048$ | 24.58 | 40.69 | 23.5 | 1.3 | 66.2
Ours | - | $1024*2048$ | 23.98 | 41.7 | 18 | 0.35 | 67.1

###  2.2 EADNet Structure

The design of EADNet pursuits extreme lightweight, and its structure is clear
and simple as illustrated in Figure 4. We employ three types of downsampling
convolution blocks in EADNet: ConcatConv block, SeqDwConv block and DwConv
block. It is seen that we insert $n_{i}$ MMRFC blocks (e.g., denoted by
$n_{1}$, $n_{2}$ in the figure) after the second and third ConcatConv block,
respectively, for efficient feature extraction. The $dr$ of MMRFC blocks are
gradually increased as the layer goes deeper to enlarged the receptive field
of EADNet and we set $n_{1}=6$, $n_{2}=9$ experimentally in our network.

SeqDwConv block and DwConv block are designed to retain more high-resolution
information when downsampling. They make the feature map sizes from shallow
layers consistent, which are then concatenated with the final MMRFC block
output to compensate the missing of detailed information in deep layers. The
concatenated features contain detailed information in shallow layers and
semantic information in deep layers and these information is important for
network prediction. We use pointwise convolution to predict segmentation map
and perform bilinear interpolation to resize the segmentation map to the
original resolution of the input image. Compared with similar U-shape
structure with computationally complex decoder stage, the EADNet saves a lot
of parameters and FLOPs as the decoder is replaced by the pointwise
convolution. These strategies mentioned above make our network to have just
0.35M parameters, and 18 GFLOPs when the input resolution is $1024*2048$
pixels.

Table 2: Experiment result on CamVid test set.  Method | Input Size | FLOPs(G) | mIoU(%)
---|---|---|---
SegNet [18] | $960*720$ | 427.34 | 46.4
DFANet A[20] | $960*720$ | 9.03 | 64.7
ours | $960*720$ | 5.99 | 68.3

##  3 EXPERIMENTS

###  3.1 Experiment Settings

We evaluated our EADNet in Cityscapes dataset [1] and trained using the Adam
[21] optimization of stochastic gradient decent. The “poly” learning rate
policy is adopted, meaning the current learning rate is base learning rate
multiplied with $(1-\frac{iter}{maxiter})^{0.9}$, where $iter,maxiter$
correspond to current iteration index and the maximum number of iterations
respectively. The loss function is the sum of cross-entropy terms for each
spatial position in the output score map, and segmentation performances are
measured by the common Intersection-over-Union (IoU) metric.

###  3.2 Comparison with State-of-the-arts

Fig. 5: Example results of ENet (b) and EADNet (c). (a) is the images in
Cityscapes dataset and (d) is their ground truth.

The comparison results on overall FPS, FLOPs, parameter size and mIoU for
state-of-the-art methods on Cityscapes test set are shown in Table 1.
“Pretrain” in the table denotes if the model has been pretrained on external
data like ImageNet [22], FPS is measured on a single Tesla V100. As shown in
the table, compared with the existing smallest network ENet [6] which has
0.37M parameters and 22 GFLOPs, our model has fewer network parameters of
0.35M and less FLOPs of 18G, but yields $67.1\%$ mIoU that is $8.8\%$ higher
than ENet [6] without using any extra data and testing augmentation. The
visualized segmentation results are presented in Figure 5. It can be seen that
compared with ENet [6], the edges of objects segmented by EADNet are clearer.
This shows that our network has stronger ability to segment the edges of
objects.

###  3.3 Comparison on Other Dataset

We also evaluate our EADNet on CamVid dataset. We adopt the same setting as
Cityscapes dataset, and the results are illustrated in Table 2, which
consistently show that our proposed EADNet achieves higher mIoU with
significantly less FLOPs than state-of-the-art DFANet [20].

##  4 CONCLUSIONS

In this work, we propose a multi-scale multi-shape receptive field convolution
block and further apply this block to develop a lightweight semantic
segmentation network called EADNet, which is suitable to deploy on computing
resource constrained embedded devices. As a result, the EADNet reaches
competitive segmentation accuracy with fewer parameters and less computation
FLOPs than state-of-the-art works. Quantitative and qualitative experimental
results on Cityscapes well demonstrate the effectiveness of our proposed
method.

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