# Dark matter decay in the Milky Way halo

M. R. Lovell1,2
1 Institute for Computational Cosmology, Durham University, South Road, Durham
DH1 3LE, United Kingdom
2 Department of Physics, Durham University, South Road, Durham DH1 3LE, United
Kingdom
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Dark matter may be detected in X-ray decay, including from the decay of the
dark matter particles that make up the Milky Way (MW) halo. We use a range of
density profiles to compute X-ray line intensity profiles, with a focus on the
resonantly produced sterile neutrino dark matter candidate. Compared to the
Navarro–Frenk–White density profile, we show that using an adiabatically
contracted halo profile suppresses the line intensity in the halo outskirts
and enhances it in the Galactic Centre (GC), although this enhancement is
eliminated by the likely presence of a core within 3 kpc. Comparing our
results to MW halo observations, other X-ray observations, and structure
formation constraints implies a sterile neutrino mixing angle parameter
$s_{11}\equiv\sin^{2}(2\theta)\times 10^{11}\sim[3,4]$ (particle lifetime
$\tau_{28}\equiv\tau/(10^{28}\mathrm{sec})\sim[1.0,1.3]$), which is
nevertheless is strong tension with some reported non-detections. We make
predictions for the likely decay flux that the XRISM satellite would measure
in the GC, plus the Virgo and Perseus clusters, and outline further steps to
determine whether the dark matter is indeed resonantly produced sterile
neutrinos as detected in X-ray decay.

###### keywords:

dark matter

††pubyear: 2024††pagerange: Dark matter decay in the Milky Way halo–A

##  1 Introduction

One of the key mechanisms for discovering dark matter is indirect detection.
This method uses rare instances in which dark matter emits electromagnetic
radiation – either through decay or annihilation – that can be distinguished
from astrophysical processes. For example, Hooper & Goodenough (2011) observed
a gamma-ray signal in the Galactic Centre (GC) consistent with the
annihilation of weakly interacting massive particles (WIMPs). However, these
measurements are inconsistent with non-detections in dwarf spheroidal (dSph)
satellites of our Milky Way (MW) galaxy (McDaniel et al., 2023), and it
remains to be seen whether the dark matter annihilation origin hypothesis is
still viable.

Another claimed signal is instead at the keV scale. Bulbul et al. (2014)
reported an excess in X-ray emission at an energy of 3.55 keV in stacks of
galaxy clusters, and at the same time Boyarsky et al. (2014) identified a
similar feature in the Perseus cluster and in M31. Other targets for which
3.55 keV feature detections have been reported include the GC (Boyarsky et
al., 2015; Hofmann & Wegg, 2019) and blank sky observations of the MW halo
Neronov et al. (2016); Cappelluti et al. (2018). This line was consistent with
the two-body decay of a dark matter particle of mass $7.1$ keV and a decay
lifetime $>10^{27}$ sec. Other studies have presented alternative origins for
this feature, including charge exchange with sulphur ions (Gu et al., 2015;
Shah et al., 2016) and a higher background than used elsewhere (Dessert et
al., 2020b; Dessert et al., 2023b, see also Abazajian, 2020; Boyarsky et al.,
2020). Therefore nature of this signal therefore remains contested.

One key step in discerning the origin of this reported feature is to resolve
the line with a very high resolution X-ray spectrometer, both for its presence
and for its velocity dispersion. The ability of such a spectrometer to detect
any lines present will first establish whether the previously features do
indeed originate from a line or were instead part of the background, and
furthermore measuring the velocity dispersion in galaxy clusters with large
velocity dispersions enables us to discern whether the line is more likely to
originate from DM decay ($[400,800]$ $\,{\rm km}\,{\rm s}^{-1}$) or from
astrophysical processes ($<200$ $\,{\rm km}\,{\rm s}^{-1}$) (Bulbul et al.,
2014; Lovell et al., 2019). The Hitomi mission hosted the SXS spectrometer but
failed before it could take sufficient exposures to detect a line in Perseus
(Aharonian et al., 2017). The successor XRISM mission launched in September
2023 carrying the Resolve instrument, which has an energy resolution of $<7$
eV in the energy range 0.3-12 keV. This corresponds to a velocity resolution
of $<300$ $\,{\rm km}\,{\rm s}^{-1}$, and is a factor of 10 improvement on the
XMM-Newton resolution ($<80$ eV, $<3400$ $\,{\rm km}\,{\rm s}^{-1}$). The
cleanest indication of a dark matter decay origin is to measure the velocity
dispersion across multiple spectroscopic bins; however, this width makes the
line more difficult to detect than a narrow line of the same total flux.
Therefore, it is easier to investigate possible detections first in bright,
low velocity dispersion systems and then use the subsequent detection, if any,
to motivate deeper exposures on high velocity dispersion targets. Any further
evidence will then support proposals to investigate high mass-to-light
ratio/low-background galaxies such as the MW’s dwarf spheroidal satellites
(Lovell et al., 2015; Jeltema & Profumo, 2016; Ruchayskiy et al., 2016).

The best available choice for this high luminosity–low velocity dispersion
target in the near future is the GC. Its proximity to Earth leads to a very
high flux, which may have already been detected (Boyarsky et al., 2015;
Hofmann & Wegg, 2019), and the velocity dispersion is capped by the escape
velocity of the MW at $\sim 200$ $\,{\rm km}\,{\rm s}^{-1}$, although the
astrophysical background may be difficult to estimate. The GC is a
particularly important target because of its intrinsic connection to the MW
halo dark matter decay emission as a whole: the decay signal of the MW halo
must be self-consistent from the GC all the way to the opposite side of the
sky. There have been several studies of the MW halo, including the GC itself
and blank sky observations probing the outer halo. Some have reported
detections of a viable 3.55 keV line (Boyarsky et al., 2015; Cappelluti et
al., 2018; Boyarsky et al., 2018; Hofmann & Wegg, 2019) while others have
reported upper limits on the possible flux (Carlson et al., 2015; Sicilian et
al., 2022; Roach et al., 2023) or even limits so strong that they rule out any
dark matter decay at many standard deviations (Dessert et al., 2020b, a; see
Dessert et al., 2023a for their blank sky XRISM predictions.)

One of the key uncertainties in the expected signal from the MW halo is the
dark matter density profile. The spherically averaged profile measured from
$N$-body simulations of structure formation is well fit by a
Navarro–Frenk–White (NFW) density profile (Navarro et al., 1996, 1997), which
is a two parameter fit of the form:

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

where $r$ is the distance from the halo centre, $r_{\mathrm{s}}$ is the scale
radius and $\rho_{\mathrm{c}}$ is the characteristic density. This profile has
an inner cusp of $-1$ and has frequently been used to model the MW halo for
X-ray detection purposes (McMillan, 2017; Sicilian et al., 2022). These
simulations also predict that the halo should be prolate or oblate rather than
spherical (e.g. Frenk et al., 1988). On the other hand, simulations that model
hydrodynamical processes, including gas cooling, feedback from supernova and
supermassive black holes, and the redistribution of angular momentum between
stars and the dark matter can all have a measurable impact on the halo, with
cooling drawing dark matter inwards through adiabatic contraction while
feedback may drive it outwards. Different astrophysics prescriptions can lead
to very different results (Lovell et al., 2018), but in general contraction
leads to steeper halo density profiles around $[5,20]$ kpc whereas feedback
evacuates some fraction of the dark matter from within 5 kpc. Evidence for
this result arises from two complementary studies: Cautun et al. (2020) used
GAIA data to infer contraction in the outer parts of the galaxy, while Portail
et al. (2017) found evidence for a cored profile within 3 kpc with dynamical
modelling of the stellar components. Finally, it is also expected that
astrophysics makes the centre of the halo more spherical than is found in
$N$-body simulations (e.g. Chua et al., 2019).

Most prior studies of the MW halo decay signal assume an NFW profile, such as
the fit of McMillan (2017): we hereafter refer to this particular profile as
Mc17. One exception is Roach et al. (2023), who instead use the contracted MW
halo fit of Cautun et al. (2020) at $>1$ kpc and a core $<1$ kpc, and found
this model slightly relaxed their non-detection limits relative to the NFW
model; we label the Cautun et al. (2020) profile Ca20. In this paper we build
on the contracted halo model by building in the 3 kpc core measured by Portail
et al. (2017) and placing our results in the context of both anticipated XRISM
results and of structure formation considerations. We compute a series of
spherically averaged dark matter density profiles, including NFW profiles and
empirical models, compute the flux from these models as a function of
observing angle from the GC, compare the results to observations, and show how
any future detection must be consistent with results from galaxy clusters with
XRISM. We will not consider the impact of any non-sphericity effects of the
halo.

This paper is organised as follows. In Section 2 we compute the density and
flux profiles, and present the results, and in Section 3 we draw our
conclusions.

##  2 Computation and results

###  2.1 Density profiles

One of the key inputs for the MW halo dark matter decay flux is the amplitude
and shape of the dark matter density profile, as discussed in the previous
section. We therefore begin our analysis by computing a wide range of density
profiles. Our primary profile of interest is the Ca20 empirical, contracted
profile. In order to illustrate how the shape of this profile differs from the
NFW, we compute 10 NFW profiles that have the same mass enclosed within 200
kpc as Ca20 but cover a wide range in $r_{\mathrm{s}}$, logarithmically spaced
between $r_{\mathrm{s}}=5$ kpc and $r_{\mathrm{s}}=50$ kpc. We also include
two further profiles from the literature: Mc17 and the fit to the inner 3 kpc
by Portail et al. (2017). We display all of these profiles together in Fig. 1.

Figure 1: A range of density profiles proposed to model the MW halo. The Ca20
profile is shown as a solid black line and the Mc17 NFW is shown as a grey
line. The fit to the central 3 kpc of Portail et al. (2017) is indicated with
a solid brown line within its fitting range, and continued outwards with a
dashed brown line. The series of coloured lines indicates NFW profiles with
the same mass as Cautun et al. (2020) within 200 kpc but with a range of scale
radii as indicated in the legend.

The contraction of Ca20 leads to important changes with respect to the NFW
models. At $20$ kpc its amplitude is approximately the same as the
$r_{\mathrm{s}}=23.2$ kpc model but at $3$ kpc it instead has the density of
the $r_{\mathrm{s}}=13.9$ kpc model. It matches the Mc17 density at $\sim 5$
kpc and drops faster towards the outskirts, leading to a lower overall halo
mass. The Portail et al. (2017) inner-galaxy fit is much shallower than either
model, and intersects Ca20 at just over $2$ kpc from the halo centre. We
therefore anticipate that this is the scale where one should transition
between the two different sets of fits in order to reflect where each fit best
matches the observations.

###  2.2 Line intensity profiles

Having demonstrated the range of available density profiles, we proceed to
compute their expected X-ray line intensity. For this process we adopt four
profiles: Ca20, Mc17, and both of these with the Portail et al. (2017) core
implemented from the halo centre up to the radius at which it intersects the
other profile; we will refer to these two combined-fit profiles as Ca20c and
Mc17c. We do not attempt to enforce consistency in the enclosed mass as a
function of radius: we simply stitch the two density profiles together to form
a single density profile, $\rho_{\mathrm{DM}}(r)$, then compute the mass
column density along the line of sight,
$S_{\mathrm{DM}}=\int\rho_{\mathrm{DM}}dr$.

The second ingredient in computing the flux is the decay rate of the dark
matter particles in the X-ray decay channel, which itself is a function of the
particle physics parameters. We therefore have a choice of using a generic
decay parameter or that of a specific particle physics model. The most popular
particle candidate for this decay signal is the resonantly produced sterile
neutrino (Dodelson & Widrow, 1994; Shi & Fuller, 1999; Abazajian et al., 2001;
Dolgov & Hansen, 2002), which has the benefit of being part of a larger
standard model extension that also explains baryogenesis and neutrino
oscillations (Asaka & Shaposhnikov, 2005; Laine & Shaposhnikov, 2008; Boyarsky
et al., 2009). The key parameters for computing the decay rate in this model
are the dark matter sterile neutrino mass, $M_{\mathrm{s}}$, and the mixing
angle with respect to standard model neutrinos, $\sin^{2}(2\theta)$, which,
following Bulbul et al. (2014); Boyarsky et al. (2014); Hofmann & Wegg (2019),
give the line intensity:

$I_{\mathrm{DM}}=\frac{0.1\mathrm{cm^{-2}s^{-1}sr^{-1}}}{3.25}\frac{\sin^{2}(2\theta)}{10^{-11}}\frac{S_{\mathrm{DM}}}{10^{9}{\,\rm
M_{\odot}}\mathrm{kpc^{-2}}}\left(\frac{M_{\mathrm{s}}}{7\mathrm{keV}}\right)^{4}.$
(2)

We can then express the decay rate as a particle lifetime, $\tau$, in the
following manner:

$\tau=7.2\times
10^{29}\mathrm{sec}\times\frac{10^{-8}}{\sin^{2}(2\theta)}\left(\frac{1}{M_{\mathrm{s}}}\right)^{5}.$
(3)

This parameter is useful in comparing to limits that are agnostic about the
underlying particle physics model111Note that the sterile neutrino model has a
dominant decay channel into three neutrinos and therefore its true lifetime is
shorter than discussed here.. On the other hand, the mixing angle can be used
to compare the X-ray results to sterile neutrino structure formation
constraints (e.g. Horiuchi et al., 2016; Lovell et al., 2016; Lovell, 2023).
We will therefore quote both quantities in the remainder of this paper.
Finally, in order to be able to quote parameter values of order $\sim 1$, we
will define $\tau_{28}\equiv\tau/10^{28}\mathrm{sec}$ and
$s_{11}\equiv\sin^{2}(2\theta)\times 10^{11}$.

We initially adopt $s_{11}=2.1$ ($\tau_{28}=1.9$) from Hofmann & Wegg (2019)
to compute line intensity profiles for the four dark matter density profiles
discussed above, and plot the results in Fig. 2 alongside three reported line
detections (Hofmann & Wegg, 2019; Boyarsky et al., 2018; Cappelluti et al.,
2018) and two sets of limits (Dessert et al., 2020b, a; Sicilian et al.,
2022). The limits set by Roach et al. (2023) are very close to the Sicilian et
al. (2022) limits and are therefore omitted for clarity.

Figure 2: Top panel: the emission line intensity profile of the MW dark matter
decay signal for four density profiles. The Cautun et al. (2020)-derived
profiles are shown in black and the McMillan (2017) equivalents are shown in
grey. Profiles modified to include the Portail et al. (2017) core (Ca20c,
Mc17c) are drawn as dashed lines and those that do not include this core
(Ca20, Mc17) are instead drawn as solid lines. The reported detections by
Cappelluti et al. (2018), Boyarsky et al. (2018) and Hofmann & Wegg (2019) are
included as red, pink, and green symbols respectively, and the limits of
Dessert et al. (2020b, a) and Sicilian et al. (2022) as cyan and orange
symbols. Bottom panel:  the four profiles normalised by their value at
$1^{\circ}$ from the GC.

The difference between the Ca20 and Mc17 profiles is significant for GC-angles
$\theta_{\mathrm{GC}}<10^{\circ}$, with a factor of three difference between
the two at $\theta_{\mathrm{GC}}\sim 1^{\circ}$. Conversely, Ca20 is
suppressed by 30 per cent relative to Mc17 at $\theta_{\mathrm{GC}}\sim
180^{\circ}$. However, the difference at small angles disappears when the
Portail et al. (2017) core is included.

In the same figure we plot the line intensity normalized to its value at
$1^{\circ}$. The NFW fit predicts the line intensity to drop by 50 per cent
from $1^{\circ}$ at $10^{\circ}$, compared to $\sim 80$ per cent at the same
angle for Ca20 and 20 per cent for the two cored models. However, the top
panel indicates that the smallest degree of uncertainty due to the choice of
dark matter profile is at around $20^{\circ}$, where the suppression with
respect to the centre is 50 per cent for the cored models, 60 per cent for
Mc17, and $>90$ per cent for the Ca20. We therefore conclude that the emission
from the GC may vary by up to a a factor of 4 at a fixed $s_{11}$.

We now turn to what dark matter parameters match the observations for the
given density profiles. The Hofmann & Wegg (2019) ($\tau_{28}=1.9$) detection
preferred a value of $s_{11}=2.1$, and in the same plot we replicate some of
the observations discussed in Sicilian et al. (2022) and compare them to our
set of profiles for this $s_{11}$. The Sicilian et al. (2022) results are in
reasonable agreement with all of the profiles at
$\theta_{\mathrm{GC}}>10^{\circ}$, whereas the Dessert et al. (2020b, a)
constraint at the profile-agnostic point of $20^{\circ}$ is in strong tension.
Also strongly disfavoured is the reported Cappelluti et al. (2018) detection,
which is a factor of three higher than Mc17, and therefore is highly unlikely
to originate from dark matter decay in the MW halo. The insertion of the
Portail et al. (2017) core is relevant only for the detections registered
within $10^{\circ}$, leading to better agreement with the Boyarsky et al.
(2018) point at $\sim 2^{\circ}$ but worse agreement for their GC point. The
Hofmann & Wegg (2019) point is consistent with all curves except for Ca20.

We have shown how the four profiles compare to the observations when adopting
a specific value of $s_{11}=2.1$ ($\tau_{28}=1.9$). We can adopt further
priors on the value of $s_{11}$ with additional X-ray detections and limits,
and also from structure formation constraints. This is one of the most
remarkable aspects of the resonantly produced sterile neutrino: for the
$s_{11}$ of interest, X-ray non-detections constrain high $s_{11}$ and
structure formation constrains low $s_{11}$, such that the whole parameter
space may be eventually ruled out. A summary of these constraints, including
their often contradictory conclusions, is given in fig. 5 of Lovell (2023): we
state here that the galaxy cluster stack (Bulbul et al., 2014) and M31
(Boyarsky et al., 2014) detections prefer $s_{11}>4$; the galaxy stack of
Anderson et al. (2015) states $s_{11}<2$, and most structure formation
constraints point towards $s_{11}>3$. We explore how these constraints
interact the MW halo profile by adopting four values of
$s_{11}=[1.3,3,4,5](\tau_{28}=[3.1,1.3,1.0,0.8])$ and applying these to two of
the models: Ca20c and Mc17. We present the results in Fig. 3, using the same
plot configuration as Fig. 2 except for a linear $x$-axis – in order to
emphasise the constraints in the halo outskirts – and a truncated $y$-axis.

Figure 3: The emission line intensity profile of the MW DM decay flux for a
series of four different $s_{11}$ values – 1.3, 3, 4, 5 – and two dark density
profiles: Ca20c (dashed black lines) and Mc17 (solid grey lines). There is one
pair of lines for each $s_{11}$, which are used to label the lines. We also
include the observational limits and detections from Fig. 2 using the same
colour scheme.

The $s_{11}=1.3$ model $(\tau_{28}=3.1)$ is the highest value of $s_{11}$
(lowest value of $\tau_{28}$)that is consistent with all of the reported non-
detections, including Dessert et al. (2020b, a), for both of the considered
density profiles, but it is incompatible with the reported X-ray detections
and with structure formation arguments. At the other end of the chosen
$s_{11}$ scale, $s_{11}=5$ $(\tau_{28}=0.8)$ is inconsistent with almost all
of the reported non-detections while still predicting intensity profiles
significantly lower than the Cappelluti et al. (2018) feature. It is also in
strong tension with Hofmann & Wegg (2019) result for the Mc17 profile, but
much weaker tension for Ca20c due to the core in the latter. The difference
between the two profiles is most significant for $s_{11}=3$ $(\tau_{28}=1.3)$,
as it is this mixing angle for which Ca20c is consistent with the Sicilian et
al. (2022) constraints at $\theta_{\mathrm{GC}}>30^{\circ}$ while the
shallower Mc17 curve is instead in tension with their results. The two
profiles both predict 25 per cent more line intensity than is consistent with
the Sicilian et al. (2022) limits at $20^{\circ}$. $s_{11}=4$
$(\tau_{28}=1.0)$ leads to worse agreement at this angle, yet even for this
value Ca20c enables agreement with Sicilian et al. (2022) at
$\theta_{\mathrm{GC}}>50^{\circ}$ whereas Mc17 does not. In conclusion, the
combination of these profiles with external constraints prefers
$s_{11}\sim[3,4]$, and by extension $\tau_{28}\sim[1.0,1.3]$.

###  2.3 XRISM flux predictions

One unavoidable test of any reported dark matter decay detection in the GC is
that it must be consistent with detections or non-detections from other
targets. Two such objects to be observed by XRISM in its performance
verification (PV) phase are the Virgo and Perseus galaxy clusters. Lovell
(2023) argued that the PV exposure of Virgo may be sufficient to return a
detection without any further time allocation, whereas the Perseus observation
time may be only a quarter of that required. We will therefore make a rough
estimate of the expected flux from both of these galaxy clusters, with the
goal of first comparing the models to the brighter GC and Virgo targets to the
total flux, then motivating further observations to measure the line width in
Virgo and make a detection in Perseus, and finally obtaining enough Perseus
exposure time to measure its characteristic velocity dispersion of $\sim 650$
$\,{\rm km}\,{\rm s}^{-1}$ (Lovell et al., 2019).

We proceed by first computing the GC flux $(\theta_{\mathrm{GC}}=0^{\circ})$
within the XRISM FoV for $s_{11}=[2,5]$ $\tau_{28}=[0.8,2.0]$ and for the two
dark matter profiles considered in Fig. 3. We assume that these two results
bracket the range of expected GC flux given the uncertainty in the density
profile. We infer counterpart fluxes for the two galaxy clusters by rescaling
the results of 500 randomly orientated XRISM mock observations for analogue
cluster hydrodynamical simulations in Lovell et al. (2019) by mixing angle: we
use the breadth of the flux distributions across the 500 sightlines to
estimate very rough uncertainties on the dark matter distribution. We then
compute the range of on-centre Virgo flux and Perseus flux as a function of GC
flux, and plot the results in Fig. 4.

Figure 4: Predictions for the flux to be measured by XRISM for the GC, Virgo
cluster, and Perseus cluster. The flux for the GC is given on the x-axis and
for the two clusters on the y-axis. The bright blue and bright red regions
indicate the anticipated flux parameter space for GC-Virgo and GC-Perseus
pairs respectively when assuming the Ca20c profile for the GC, and the faded
regions show the same GC-cluster pairs when instead assuming the Mc17 profile.
In both cases we show regions assuming $s_{11}=[2,5]$ ($\tau_{28}=[0.8,2]$),
with the specific parameter space for $s_{11}=2,3,4,5$ indicated with vertical
lines.

The total range of flux inferred for the GC is $[3,14.5]\times
10^{-8}\mathrm{cts/cm^{2}/sec}$, while for the preferred $s_{11}=[3,4]$ range
($\tau_{28}=[1.0,1.3]$) it is $[5,11.5]\times 10^{-8}\mathrm{cts/cm^{2}/sec}$
including the uncertainty in the dark matter density profile. In the same
preferred $s_{11}$ range, the Virgo cluster flux is $[6,13]\times
10^{-8}\mathrm{cts/cm^{2}/sec}$ and the Perseus flux is $[3.5,7.5]\times
10^{-8}\mathrm{cts/cm^{2}/sec}$. If three emission line signals are detected,
they will need to each be consistent with these three flux ranges in order for
dark matter decay of resonantly produced sterile neutrino dark matter to be a
viable explanation.

We also note that there will be some contribution to the signals at Virgo and
Perseus from the outskirts of the MW halo, which are not included in the flux
estimates of Fig. 4. The Virgo cluster centre is $\sim 87^{\circ}$ from the
GC, where the MW halo flux contribution is 10 per cent (8 per cent) of the GC
flux when assuming the Ca20c (Mc17) model; Perseus is at
$\theta_{\mathrm{GC}}\sim 146^{\circ}$ with MW-halo contributions of 7 per
cent (5 per cent) of the GC for the Ca20c (Mc17) model. Given that the
expected Perseus flux is no less than 35 per cent of the GC and Virgo is at
least 60 per cent, the MW halo contribution to the cluster-directed signal is
$<20$ per cent for Perseus and $<16$ per cent for Virgo.

##  3 Conclusions

The detection of dark matter annihilation or decay products remains one of the
most promising avenues for establishing its particle nature. In the case of
the resonantly produced sterile neutrino dark matter candidate, the simplicity
of its detection channel – monochromatic X-ray emission with a resolvable
velocity dispersion – is complemented by structure formation constraints (e.g.
Lovell, 2023). The recent launch of the XRISM mission has provided a unique
opportunity to test this model, with the prior report of a candidate line at
an energy of 3.55 keV (Boyarsky et al., 2014; Bulbul et al., 2014) providing a
compelling starting point for decay searches.

In this paper we focus on the MW dark matter halo as a target, due to the
anticipated brightness of the GC, the wide range of existing constraints and
detections associated with the MW halo (Boyarsky et al., 2015; Hofmann & Wegg,
2019; Dessert et al., 2020b, a; Sicilian et al., 2022; Roach et al., 2023),
and the uncertainty in the MW dark matter density profile. We compute a series
of spherically averaged density profiles, including an NFW fit (Mc17), an
empirical fit that includes adiabatic contraction (Ca20), and a cored fit to
the inner 3 kpc (Fig. 1). We show that the contracted Ca20 profile strongly
enhances the flux intensity profile at $\theta_{\mathrm{GC}}<20^{\circ}$
relative to the Mc17 NFW, and by contrast suppresses it at larger angles (Fig.
2). The introduction of a Portail et al. (2017) core instead lowers the
intensity within $\theta_{\mathrm{GC}}<10^{\circ}$.

We explore the impact of changing the mixing angle $s_{11}$, and thus the
measured particle lifetime $\tau_{28}$, alongside the dark matter profile with
respect to the observations. The switch from Mc17 to Ca20 enables some
agreement with observations for $s_{11}=4$ ($\tau_{28}=1.0$) at
$\theta_{\mathrm{GC}}>50^{\circ}$, while the core applied in the Mc17c and
Ca20c profiles is crucial to preserving the agreement with the Hofmann & Wegg
(2019) GC detection (Fig. 3). When combining the results with detections and
constraints from other sources (Lovell, 2023), we find a rough preference for
$s_{11}\sim[3,4]$ ($\tau_{28}\sim[1.0,1.3]$). The greatest challenge to this
interpretation is at $\theta_{\mathrm{GC}}\sim 25^{\circ}$, where the dark
matter curves are in some tension with the Sicilian et al. (2022) constraints
and are not compatible with Dessert et al. (2020b, a).

We compute the expected flux to be measured by XRISM from the GC for the Mc17
and Ca20c, and quoted these results alongside inferred fluxes for on-centre
observations of the Virgo and Perseus clusters (Fig. 4). Using the two
profiles to bookend the possible flux and assuming $s_{11}=[3,4]$, we predict
a GC flux of $[5,11.5]\times 10^{-8}\mathrm{cts/cm^{2}/sec}$, a Virgo cluster
flux of $[6,13]\times 10^{-8}\mathrm{cts/cm^{2}/sec}$ and a Perseus cluster
flux of $[3.5,7.5]\times 10^{-8}\mathrm{cts/cm^{2}/sec}$. All of these values
must be measured in order to support the dark matter decay hypothesis.

The XRISM PV observations may be sufficient to detect these fluxes in the GC
and Virgo but not in Perseus, although measuring the line velocity dispersions
in the clusters would provide an especially strong piece of evidence in favour
of a dark matter decay origin. Important follow up observations would include
detections in low background targets such as the Draco dwarf galaxy (Lovell et
al., 2015) (see Appendix A for a discussion of the recently discovered Ursa
Major III/UNIONS 1 object) and the Bullet cluster (Boyarsky et al., 2008), and
detection of the MW halo line centroid shift induced by the Earth’s motion
(Speckhard et al., 2016). We therefore anticipate the following work flow for
X-ray analysis, where each step must return a positive detection in order to
proceed to the next step. We do not attempt to compute the XRISM time required
to make any of these detections, instead we quote the PV time allocations
alongside a rough expectation of the order in which targets would become ripe
for detection efforts.

  * •

PV: measure GC flux $[5,11.5]\times 10^{-8}\mathrm{cts/cm^{2}/sec}$ (200 ksec
Priority A plus 100 ksec in Priority C) and Virgo cluster flux $[5,11.5]\times
10^{-8}\mathrm{cts/cm^{2}/sec}$ (500 ksec in Priority A).

  * •

Extended PV: measure Virgo cluster velocity dispersion $\sim 450$ $\,{\rm
km}\,{\rm s}^{-1}$ and Perseus cluster flux $[3.5,7.5]\times
10^{-8}\mathrm{cts/cm^{2}/sec}$ (280 ksec in priority A, plus 100 ksec in
priority C).

  * •

Further XRISM observations: observe Perseus velocity dispersion $\sim 650$
$\,{\rm km}\,{\rm s}^{-1}$.

  * •

Further general instrument observations: observe Draco dSph, Bullet cluster.

The final stage of using the Draco dSph and the Bullet cluster may be well
suited to the larger FoV associated with the NewATHENA and proposed Lynx
missions. Coupled to attempts to better measure the astrophysical backgrounds
we therefore anticipate an excellent opportunity to test the resonantly
produced sterile neutrino as dark matter within the next 15 years.

## Acknowledgements

MRL would like to thank Oleg Ruchayskyi for comments on the text and Chris
Done for useful conversations.

## Data Availability

Researchers interested in using these results should contact the corresponding
author.

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##  Appendix A Ursa Major III/UNIONS 1

The recently discovered Ursa Major III/UNIONS 1 object (Smith et al., 2023)
would, if confirmed as a dwarf galaxy, be one of the closest and highest
density satellites known (Errani et al., 2023). Its high density makes it a
compelling target for dark matter annihilation studies (Crnogorčević & Linden,
2023; Errani et al., 2023), given that the annihilation signal scales as the
density squared. By contrast, the decay signal is only sensitive to the mass
within the field-of-view (eqn. 2). Errani et al. (2023) determine that its
total mass is $10^{4}$ ${\,\rm M_{\odot}}$; if we assume that the mass fits
entirely within the XRISM-Resolve field-of-view, the expected flux is
$1.3\times 10^{-8}(s_{11}/4)$ $\mathrm{cts/cm^{2}/sec}$.