Reticulum II: Particle Dark Matter and Primordial Black Holes Limits

MNRAS 000, 1–10 (2021) Preprint 14 September 2021 Compiled using MNRAS LATEX style file v3.0

 Reticulum II: Particle Dark Matter and Primordial Black Holes Limits
 Thomas Siegert,1★ Celine Boehm,2 Francesca Calore,3 Roland Diehl,4 Martin G. H. Krause,5
 Pasquale D. Serpico,3 and Aaron C. Vincent6
 1 Institutfür Theoretische Physik und Astrophysik, Universität Würzburg, Campus Hubland Nord, Emil-Fischer-Str. 31, 97074 Würzburg, Germany
 2 Sydney Consortium for Particle Physics and Cosmology, School of Physics, The University of Sydney, NSW 2006, Australia
 3 Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAPTh, F-74940 Annecy, France
 4 Max-Planck-Institut für extraterrestrische Physik, Gießenbachstraße, 85741 Garching b. München, Germany
 5 Centre for Astrophysics Research, Department of Physics, Astronomy and Mathematics, University of Hertfordshire, College Lane, Hatfield, Hertfordshire AL10 9AB, UK
arXiv:2109.03791v2 [astro-ph.HE] 13 Sep 2021

 6 Department of Physics, Enigneering Physics and Astronomy, Queen’s University, Kingston, ON, K7L 3N6, Canada

 Accepted XXX. Received 14 September 2021; in original form ZZZ

 ABSTRACT
 Reticulum II (Ret II) is a satellite galaxy of the Milky Way and presents a prime target to investigate the nature of dark matter
 (DM) because of its high mass-to-light ratio. We evaluate a dedicated INTEGRAL observation campaign data set to obtain
 -ray fluxes from Ret II and compare those with expectations from DM. Ret II is not detected in the -ray band 25–8000 keV,
 and we derive a flux limit of . 10−8 erg cm−2 s−1 . The previously reported 511 keV line is not seen, and we find a flux limit of
 . 1.7 × 10−4 ph cm−2 s−1 . We construct spectral models for primordial black hole (PBH) evaporation and annihilation/decay of
 particle DM, and subsequent annihilation of + s produced in these processes. We exclude that the totality of DM in Ret II is made
 of a monochromatic distribution of PBHs of masses . 8 × 1015 g. Our limits on the velocity-averaged DM annihilation cross
 section into + − are h i . 5 × 10−28 ( DM /MeV) 2.5 cm3 s−1 . We conclude that analysing isolated targets in the MeV -ray
 band can set strong bounds on DM properties without multi-year data sets of the entire Milky Way, and encourage follow-up
 observations of Ret II and other dwarf galaxies.
 Key words: keyword1 – keyword2 – keyword3

 1 INTRODUCTION the possible annihilation of positrons ( + s) as a DM product into
 the galaxy emission spectra. Our results are presented in Sec. 4. We
 Siegert et al. (2016c) reported a 3.1 signal of a 511 keV line from
 conclude in Sec. 5.
 the direction of the dwarf galaxy Reticulum II (Ret II; distance =
 30 ± 2 kpc; Koposov et al. 2015). The line flux from this previous
 work was extraordinarily high, 511 = (1.7±0.5)×10−3 ph cm−2 s−1 ,
 which would make Ret II the most luminous 511 keV source in the
 sky. The signal has been interpreted in terms of a recent binary
 neutron star (NS) merger (Fuller et al. 2018), possibly outshining the
 entire galaxy. Ret II has also been reported to show an excess at GeV
 energies (Geringer-Sameth et al. 2015) which might point to a dark 2 NEW OBSERVATION CAMPAIGN
 matter (DM) origin of the signal. Follow-up analyses found no flux The previous analysis by Siegert et al. (2016c) focussed on the search
 enhancement above the diffuse GeV emission (Albert et al. 2017). of 511 keV signals from the 39 satellite galaxies of the Milky Way
 Intrigued by these finding, the INTEGRAL (Winkler et al. 2003) (MW) known at the time. In their data set, the entire sky was mod-
 satellite performed a dedicated observation campaign to follow up elled and the diffuse 511 keV emission from the MW itself was
 the signal. In this work, we analyse the new 1 Ms data together included. Before the new observation campaign, Ret II had never
 with archival data from the coded-mask -ray spectrometer onboard been targeted for observations by SPI because 1) the galaxy was only
 INTEGRAL, SPI (Vedrenne et al. 2003). We use our findings to discovered in 2015 (Koposov et al. 2015), and 2) other known X-
 set limits on possible DM models, including primordial black holes or -ray sources are separated from Ret II by at least 18◦ . However,
 (PBHs) and annihilation or decay of (light) DM particles. thanks to SPI’s wide fully-coded field of view (FCFOV) of 16◦ × 16◦
 This paper is structured as follows: In Sec. 2, we describe the new (30◦ × 30◦ partially coded), Ret II data were collected also before its
 observations, and how SPI data are analysed in general. We build discovery. Outside the full coding region, the effective area of the
 photon-emission models expected from the evaporation of PBHs and instrument sharply drops which placed previous Ret II observations
 annihilating/decaying particle DM in Sec. 3. Here, we also include right onto exposure edges with a cumulative on-target observation
 time of 550 ks, split over ten years. While these effect were taken into
 account in the analysis, dedicated observations, pointed directly to
 ★ E-mail: tho.siegert@gmail.com Ret II, would be suitable to validate earlier findings.

 © 2021 The Authors

2 Thomas Siegert et al.
2.1 Data Set 2.3 Spectral Fitting
Our new data set includes previous observations between 2003 and The spectrum created in this way is subject to the instrument dis-
2013 and adds ∼ 1 Ms during the Ret II campaign in 2018. We persion, that is, the probability that a photon with initial energy 
selected all pointed observations (pointings) whose distance to Ret II is measured at a final energy owing to scattering in the instru-
is less than 16◦ so that our target is always in the FCFOV. This ment. In order to correct for the intrinsic dispersion, expected spec-
amounts to 603 pointings with an average observation time of 3300 s tral models are convolved with the energy-redistribution matrix
each. The mean dead time of a working detector is about 18.2 %. ( , ) that forward-folds the models into the appropriate data
Since the launch of the INTEGRAL mission in October 2002, four space. Thus, the spectral models read
of the 19 SPI detectors have failed, reducing the effective area by 
 ∫
 ; ( ; )
≈ 20 %. The total dead-time- and efficiency-corrected exposure time = ( , ) , (4)
then amounts to 1.38 Ms. We use SPI’s bandpass between 25 and 
8000 keV, and perform our analysis in 29 logarithmic energy bins, being the folded model which is compared to the extracted
 with 
plus a single narrow bin for the 511 keV line from 508 to 514 keV. 
 flux values from Sec. 2.2, and is the intrinsic source model that
 
 depends on a set of spectral parameters . We use the Multi-Mission-
 Maximum-Likelihood (3ML, Vianello et al. 2015) framework to
2.2 General Analysis Method perform spectral fits. Details about the choices of prior probabilities
 and parameter ranges are given in Appendix A.
SPI data analysis relies on the comparison of expected detector il-
 We note that, ideally, the steps explained in Secs. 2.2 and 2.3
luminations, that is, patterns in the 19-detector data space, with the
 should be performed in one single step to enhance the sensitivity of
measured count rate per pointing and energy. Fluxes are estimated
 the parameter estimation, which is, however, not yet implemented in
by maximising the Poisson likelihood,
 the current software release.
 Ö exp(− )
L ( | ) = , (1)
 
 ! 3 EXPECTATIONS IN THE MEV BAND
where are the measured counts in pointing , and is the No MeV emission has ever been detected from a (dwarve) satellite
model expectation. MeV -ray measurements are described by a two galaxy. Therefore, the expected signals have a large variety, ranging
component model, one representing the instrumental background from -ray line emission due to radioactive decays and (subsequent)
 BG , and one representing the celestial emission SKY . Instrumental + emission. Population synthesis models estimate a 1.8 MeV -ray
background originates from cosmic-ray interactions with the satellite line flux from the decay of 26 Al of the order of 10−6 ph cm−2 s−1 from
material and leads to continuum ( ) and line ( ) backgrounds (Diehl the Large Magellanic Cloud (LMC), about one order of magnitude
et al. 2018). Spatial emission models are convolved with the imaging below SPI’s sensitivity Diehl et al. (2018). Because the expected line
response function of SPI, that is, the mask coding, leading to a 19- flux is related to the star formation and supernova rate, any other
element vector of expected source counts for each pointing. The total satellite galaxy, and in particular Ret II, can be expected to show
model reads no significant excess at 1.8 MeV. Cordier et al. (2004) performed a
 search for a 511 keV line owing to dark matter annihilation/decay
 = BG + 
 SKY
 = from the Sgr dwarve galaxy, but found no excess. Siegert et al.
 ∑︁  ∑︁  
 ∑︁  (2016c) extended this search to all satellite galaxies of the MW,
 BG SKY which resulted in one 3 signal from the direction of Ret II out of 39
 
 =  
 , , + 0 
 , ; ( )
  , (2)
 
 tested galaxies. Where this signal, if true, comes from is unknown,
 
 , 0  ∈ { , } =1 
 
 and might point to different origins in terms of NS mergers (Fuller
where BG
 , is the background model (response) appropriate for point-
 et al. 2018), accreting X-ray binaries (Siegert et al. 2016a), or DM
ing , scaled by the amplitudes , for background components (Boehm et al. 2004). All these scenarios would show additional
 ∈ { , } varying on timescales , SKY
 ; 
 is the coded-mask response emission features in the MeV band besides a sole 511 keV line. In
 particular additional broad -ray lines from other nucleosynthesis
in pointing , converting the = 1 . . . sky models 
 ( ) from products in NS mergers, continuum emission from a population of
image space to the data space. X-ray binaries, or radiative corrections in the final products from DM
 The source models may have additional parameters , for ex- decay/annihilation or PBH decay. Therefore, we extract the spectrum
ample defining the position of point sources, and are scaled by the of Ret II and nearby sources using the logarithmic energy binning
amplitudes , 0 , possibly varying on another time scale 0 . Given defined in Sec. 2.1, plus a single bin for the 511 keV line. In this way,
SPI’s angular resolution of 2.7◦ , and the expected maximum extent we can assess the previously found signal from Ret II in a model-
of Ret II’s DM halo of 1◦ , we model the galaxy as point source, independent fashion, and determine model dependent parameters
 Ret II ( , ) = ( − 0 ) ( − 0 ), (3) using the additionally-expected components in Sec. 3.1.

with the Galactic coordinates of Ret II ( 0 , 0 ) = (266.30, −49.74).
 3.1 Modelling the MeV Band in Ret II
All other sources that can be expected in our data set, EXO 0748-676,
ESO 33-2, SMC X-1, and LMC X-4 are modelled as point sources We focus on the -ray spectra expected from the evaporation of PBHs
as well. The fitted parameters in Eq. (2) are , and , 0 . The in the mass range 1014 –1018 g and annihilating/decaying particle DM
fit is performed with spimodfit (Halloin 2009), which creates a in the mass range 0.05–300 MeV. Other signals, for example the -
spectrum by looping over the energy bins defined in our data set. ray emission from old NS mergers remnants, are not part of our
Details about the procedure are found in Siegert et al. (2019). predictions because the expected emission of one 10 kyr old remnant

MNRAS 000, 1–10 (2021)

Dark Matter in Reticulum II 3
 100

 Eekin × Fe [10 3]
 Model PBH Evaporation (primary)
 
  max PBH Evaporation (secondary) 101
 1→ 

 E × F [keV phcm 2 s 1 keV 1]
 PBH PBH BH  Ps , kin
 max 10 1 Positronium Decay
 Annihilation in Flight
 DM Annihilation h i 2 DM 2→ ,
  Ps kin Annihilation in Flight (weighted) 10 1
 Internal Bremsstrahlung
 10
 max
 DM Decay 1/ DM 1→ Ps , kin 2 Internal Bremsstrahlung (weighted)
 Total Spectrum 102 104
Table 1. Parameters for considered DM models in Eq. (5).
 10 3
 (MBH = 1016 g,
 D = 2.5 × 1018 GeV cm 2, Eekin [keV]
 fPs = 1.0, Eekin, max = 4.0 MeV)

 10 4
in Ret II is about one million times lower (Korobkin et al. 2020) than
the instrument sensitivity. While emission from GeV DM would also 10 5

imprint in the MeV band, SPI is most sensitive to annihilation/decay 10 6
to + − and subsequent emission processes considering the final
state radiation (FSR) of the pairs as well as the annihilation of + s 10 7
 101 102 103 104
during propagation and after thermalisation. Therefore, we restrict E [keV]
our models to the two channels + − and , and present one + − 100

 Eekin × Fe [10 3]
 PBH Evaporation (primary)
example for comparison. We follow the formalism of Fortin et al. PBH Evaporation (secondary) 101

 E × F [keV phcm 2 s 1 keV 1]
(2009) throughout the paper. 10 1 Positronium Decay
 Annihilation in Flight
 The emission from DM haloes is described by a ‘particle physics’ Annihilation in Flight (weighted) 10 1
 Internal Bremsstrahlung
 10 2 Internal Bremsstrahlung (weighted)
and an ‘astrophysics’ component as Total Spectrum 102 104
 10 3
 (MBH = 3 × 1015 g,
 D = 2.5 × 1018 GeV cm 2, Eekin [keV]
 fPs = 0.1, Eekin, max = 10.0 MeV)

 10
 
 
 ∫
 ; , , , ; 4
 = Ω halo ( , Ω) ,
 4 los
 | {z } | {z } 10 5
 particle physics: PBH or DM astrophysics: D or J
 (5)
 10 6

where , , and describe the parameters of interest for the specific
 10 7
 101 102 103 104
cases of PBH evaporation, self-conjugated DM annihilation or decay E [keV]
(see Tab. 1), and the last term is the D- ( = 1) or J-factor ( = 2) of
 Figure 1. Expected -ray spectra from PBH evaporation and subsequent an-
the galaxy. They describe the extent of the emission, calculated by nihilation of positron-electron pairs, plus internal bremsstrahlung of emitted
an integration of the galaxy’s halo profile halo ( , Ω) over the line of pairs. The two panels show different model parameters (see legends) for a D-
 ( ; )
sight (los). The photon spectrum is related to the photon factor of 2.5 × 1018 GeV cm−2 , consistent with Reticulum II. The small insets
 
 show the distribution of electrons and positrons from evaporation, together
density of states as 1 = 1 
 so that the number of emitted
 with the maximum kinetic energy kin max considered by the shaded area, and
 
 ∫ the mean kinetic energy of the electron/positron population (dashed line). The
photons per process is = . is thus normalising the
 shaded area of the electron/positron distribution corresponds to the shaded
source spectrum, Eq. (5), in terms of the totally-emitted photons. areas in the -ray spectra.
 We use the range of D- and J-factors of Ret II from the lit-
erature (Bonnivard et al. 2015; Evans et al. 2016; Albert et al.
2017), in particular ≈ (1–4) × 1018 GeV cm−2 , and ≈ (0.2– In Eq. (6), is the energy of particle and its spin. Γ ( , BH ) is
3.7) × 1019 GeV2 cm−5 . The conversion of the D- and J-factor into the ‘greybody’ factor that alters the expected blackbody distribution
other often-used units are = (0.9–3.5) × 10−2 M pc−2 = (1.9– of possible emitted particles, given the BH temperature
7.3) × 10−6 g cm−2 , and = (0.6–8.4) × 10−3 M2 pc−5 = (0.8–
 ~ 3
    16 
 10 g
11.8) × 10−29 g2 cm−5 . We use these ranges as uniform priors in our BH = = 6 × 10−8 K = 1.06 MeV,
spectral fits to include uncertainties in the DM halo as well as the 8 BH BH BH
distance to Ret II. Depending on the DM model, the parameters of (7)
interest change and are listed in Tab. 1. Here, PBH is the fraction where ~ is reduced Planck’s constant, is the gravitational constant,
of DM made of PBHs (unitless), h i is the velocity-averaged DM is the speed of light, and is Boltzmann’s constant.
annihilation cross section in units of cm3 s−1 , = Γ−1 is the decay We use BlackHawk (Arbey & Auffinger 2019) to calculate the
time of a DM particle in units of s, BH is the mass of PBHs in spectra of all relevant particles in the energy range between 1 keV and
units of g, and DM is the DM particle massn in units
 o of keV. The 1 GeV. BlackHawk allows to include secondary particle production
 max appear when
additional spectral model parameters = Ps , kin due to hadronization, fragmentation, decay, and other processes as a
 + annihilation is included (see below). result of BH evaporation, which we add to our spectra.
 The production of + s from BH evaporation has no kinematic
 threshold based on the mass of the BG. However, the number of + s
 produced that may annihilate subsequently can lead to a 20–8000 keV
3.1.1 Primordial Black Hole Evaporation
 flux stronger than the evaporation signal itself which is given by a
The spectrum of an evaporating black hole (BH) of mass BH due BH mass of . 1.1 × 1017 g. Given the particle spectrum of + s from
to Hawking radiation (Hawking 1975; Page & Hawking 1976) is BlackHawk, the differential + flux in a galaxy with D-factor is
  given by
 1 Γ ( , BH )
 =   . (6) 1 
 BH 2 exp − (−1) 2 = , (8)
 BH 4 BH 

 MNRAS 000, 1–10 (2021)

4 Thomas Siegert et al.
so that the total possible + − -annihilation flux Ann in that galaxy MW-like conditions,
 ( , )
 ∝ , so that cancels.
 Coulomb
is
 The total IA spectrum for mono-energetic injection of + s thus reads
 ∫ max
 kin 
 Ann = kin . (9)
 mono
    
 0 ,kin 1 1 
 = 511 , (15)
 max is the maximum kinetic energy of a + that annihilates
Here, kin IA 1 − 34 Ps IA
within the galaxy and does not escape into the intergalactic medium. with
For the MW, kinmax has been estimated to be around 3–7 MeV (Bea-
 ( , )
 mono
   ∫ kin
com & Yüksel 2006; Sizun et al. 2006). We caution, however, that ( kin , )
 = 
 . (16)
these estimates may be inaccurate because the statistical methods to IA 2 1 ( , )
 
compare the expected spectrum with the measurements of two differ-
 max
  
ent instruments are not rigorously correct. Therefore, we leave kin ( − ) 2 + 2
 where the integration limit 1 = max ( − )+ , is
as a free parameter in our spectral fits. Because the transport and
environmental conditions in Ret II are largely unknown, this proce- bound to obey energy conservation  (Svensson
  1982; Beacom & Yük-
dure effectively takes into account these uncertainties. For reference,
 ∫ 
 sel 2006). We note that = 1 − . Since in the case
we discuss the two extreme cases of no and maximal + annihila- IA
 of PBHs, the injection energy is not mono-energetic, we calculate a
tion in our results, Sec. 4, together with the intermediate solution of weighted photon spectrum given the distribution of kinetic energies
 max .
marginalising over kin from BlackHawk as
 The total annihilation flux is composed of three components,
 ( kin ) mono
 ∫ max  
 kin
 
 
 
 0
 kin 
 Ann = 511 + oPs + IA , (10) = IA
 . (17)
 IA ∫ max 
 kin
where 511 is the flux in the 511 keV line due to direct annihilation 0
 kin 
with − s and intermediate formation of Positronium (Ps), oPs is the Radiative corrections in the production of + − -pairs can be de-
three-photon decay flux of ortho-Ps, and IA is the total flux of + s scribed as internal bremsstrahlung (IB) which can be calculated with-
annihilating in flight (IA) before stopping/thermalising. out detailed knowledge of the previous process. This final state radia-
 The ortho-Ps and the line flux are related by quantum statistics as tion (FSR) arises as a result of the production itself and is independent
 9 Ps of the astrophysical environment. The photon source spectrum of IB
 oPs = =: 32 511 , (11)
 8 − 6 Ps 511 per 511 keV line photon is given by Beacom et al. (2005),
 mono 1 mono
    
with Ps being the Ps fraction, that is, the number of + s forming 1
 = , (18)
Ps before annihilating (Leventhal et al. 1978). The scaling factor 32 IB 2 1 − 3 Ps IB
 4
ranges between 0 ( Ps = 0) and 4.5 ( Ps = 1). In general Ps depends with
on the annihilation conditions, for example the gas in which + s anni- " #
 mono 2 + ( − 2 ) 2 ( − 2 )
   
hilate, which is a function of the temperature, density, and ionisation = ln , (19)
fraction, among others (e.g., Jean et al. 2006; Churazov et al. 2005, IB 2 2 2 
2011; Siegert et al. 2016b). Since the true conditions in Ret II are where here, = kin + is the particles’ injection energy from
unknown, we leave Ps as free parameters in our spectral fits. The PBH evaporation. Because the + -injection spectrum is again not
 
ortho-Ps spectrum has been calculated by Ore & Powell mono-energetic, we weight over expected energy distribution such
 oPs
 that
  
 
(1949), and we model the 511 keV line, as a Gaussian at
 ( kin ) mono
 511 ∫ max  
511 keV with a width according to SPI’s energy resolution (Diehl
 
 
 
 0
 kin
 kin 
 IB
et al. 2018; Attié et al. 2003). = ∫ max . (20)
 IB kin
 
 Beacom & Yüksel (2006) derived a relation between IA and the 0 kin 
511 keV line flux,
 Finally, the total photon source spectrum of PBH evaporation in-
 1 1− cluding subsequent + -annihilation and IB in a galaxy is the sum the
 IA = 3
 511 =: IA 511 , (12)
 1 − 4 Ps components:
where = ( kin , ) is the probability of a + with initial kinetic tot
        
 
 PBH = PBH + IB + Ann =
energy kin to annihilate before stopping/thermalising. Therefore,      
 
the total annihilation flux can be expressed as
 Í
 = + + ∈A , (21)
 PBH IB 
 Ann = (1 + 32 + IA ) 511 . (13)  
 with A = {511, oPs, IA}. 
 
 is a function of BH , and
 PBH
The survival probability in Eq. (12) is given in general by 
 
 
 max . We show two
 PBH , and is a function of Ps and kin
 Ann
 © ∫ kin
 ( ) ª examples of the total emission spectrum from PBH evaporation in
 ( kin , ) = exp ­­− ®, (14) Ret II in Fig. 1.
 ( , ) ®
 « ¬
 ( , ) 3.1.2 Particle Dark Matter Annihilation and Decay
where ( ) is the total annihilation cross section, and is
the stopping power (energy loss rate, cooling function) of a + propa- We consider light ( DM . 300 MeV) DM particles that either anni-
gating in a medium with density . For simplicity, we only consider hilate or decay directly into standard model particles, leading to an
Coulomb losses which are expected to dominate up to 100 MeV for expected photon emission spectrum. Therefore we evaluate the cases

MNRAS 000, 1–10 (2021)

Dark Matter in Reticulum II 5
 100 DM Annihilation (FSR) related spectra. Given the equivalence in Eq. (22), one can estimate
 Positronium Decay 100

 Eekin × Fe
 the total 511 keV line flux from DM particle annihilation as
E × F [keV phcm 2 s 1 keV 1]

 10 1 Annihilation in Flight
 Total Spectrum 10 1
 (MDM = 3.8 MeV, v = 3 × 10 27 cm3 s 1  
 J = 2.0 × 1019 GeV2 cm 5, 3 h i
 10 2 fPs = 1.0, Eekin, max = 10.0 MeV) 10 2 Ann
 511 = 1 − Ps , (24)
 102 104 4 2
 2 DM
 10 3 Eekin [keV]
 and from DM particle decay as
 10 4
 Dec
 
 3
 
 
 511 = 1 − Ps . (25)
 10 5 4 2 DM 
 The remaining spectral components from ortho-Ps and annihilation
 10 6
 in flight are the same as in Sec. 3.1.1, except that now the mono-
 10 7
 101 102 103 104
 energetic cases are considered. The total source photon spectrum is
 E [keV] again the sum of the components, here FSR, 511 keV line, ortho-Ps,
 and in-flight annihilation. We show examples of the expected DM
 100 DM Annihilation (FSR, e + e )
 DM Annihilation (FSR, + ) 101 annihilation spectra in Fig. 2.
 Eekin × Fe
E × F [keV phcm 2 s 1 keV 1]

 10 1 DM Annihilation ( ) ×10 5
 Positronium Decay
 Annihilation in Flight 10 1
 10 2 Total Spectrum
 (MDM = 10.3 MeV, v = 3 × 10 24 cm3 s 1
 J = 2.0 × 1019 GeV2 cm 5, 102 104 4 RESULTS
 10 3 fPs = N/A, Eekin, max = 1 MeV) Eekin [keV]
 We examine our model fits to measured data through a careful anal-
 10 4 ysis of the residuals in different data space dimensions. Herein, it
 proves valuable to apply a back-projection onto the sky of the count
 10 5 residuals. For cases of background model imperfections, we would
 discover irregular or large-scale patterns on the sky, whereas for im-
 10 6
 perfections of the sky model (e.g., missing a point source) would
 10 7
 101 102 103 104
 show up as a single peak at the source’s location. In Fig. 3, we
 show the residual count map of an instrumental-background-only fit
 E [keV]
 in three example energy bands. Only the high-mass X-ray binary
 Figure 2. Expected -ray spectra from light dark matter annihilation and LMC X-4 is detected with a significance of 21 up to an energy of
 subsequent annihilation of positron-electron pairs, similar to Fig. 1. The two ∼ 200 keV. The other sources, and in particular Ret II, are not seen
 panels show different model parameters (see legends) for a J-factor of 2 × in any energy band. The count residuals show no strong patterns as
 1019 GeV2 cm−5 . The lower panel also shows model expectations from final expected from an otherwise empty field. In the 511 keV band, the
 state radiation of the + − -channel and direct decay into two photons. residuals appear more structured, which is a result of the reduced
 count rate compared to other bands. For reference, the particularly
 bright spot between ESO 33-2 and LMC X-4 at ( , ) = (−78◦ , −37◦ )
 (i) DM + DM → e+ + e− + ( )FSR ,
 has a significance of ∼ 2.5 , and is thus considered a background
 (ii) DM + DM → + ,
 fluctuation. Details about the 511 keV line from Ret II are given in
 (iii) DM → e+ + e− + ( )FSR ,
 Sec. 4.2.
 (iv) (a) DM → + , and
 (b) DM → Y + ,
 where Y is any relativistic particle, for example a sterile neutrino that 4.1 Total Spectrum
 decays into + . The photon spectra of the FSR in cases 1. and 3. We show the spectrum measured by SPI from the direction of Ret II,
 are identical to the IB spectrum, spatially modelled as a point source, Eq. (3), in Fig. 4. Because the
 source is not detected in any of the energy bins, we plot 2 upper
    
 
 ≡ , (22) limits. For comparison, we show several models that are excluded,
 FSR IB
 given these fluxes. Detailed exclusion plots are found in the following
 however now, = 2 DM for annihilation and = 1 DM for decay. 
  
 sections. From a spectral fit with a generic power-law, ∝ 1MeV ,
 Another factor of 12 is required in Eq. (19) for Dirac DM annihila-
 tion/decay to account for distinct particle-antiparticle DM with equal we derive an upper bound on the total flux in the band 20–8000 keV
 densities. The expected photon spectrum of direct annihilation/decay of 10−8 erg cm−2 s−1 .
 into two photons (cases 2. and 4.) is a delta-function,
    
 4.2 511 keV Line
 = 2 − . (23)
 2
 We find a 511 keV line flux limit of 1.7×10−4 ph cm−2 s−1 . This value
 Case 4. (b) differs from case 4. (a) only by an additional factor of 2. is about one order of magnitude smaller than the 3 signal reported
 We model Eq. (23) as Gaussians with the energy resolution of SPI. by Siegert et al. (2016c) of (1.7 ± 0.5) × 10−3 ph cm−2 s−1 . Given
 Similar to the PBH case, the emitted + s from DM particles in a the enhanced exposure time and the source being inside the FCFOV
 galaxy’s halo might annihilate during (IA) and after (Ps) propagation all the time thanks to the new dedicated observation campaign, the
 in the galaxy, or escape into the intergalactic medium if their injection improvement in sensitivity is plausible.
 energy is larger than a threshold kinmax . The only difference is now The previous detection of Ret II might be due to an intrinsic vari-
 that the energy distribution of produced pairs is mono-energetic, so ability in time, as could be expected from the outburst of a micro-
 that either all or none of the pairs lead to additional + -annihilation quasar (Siegert et al. 2016a) or a NS merger (Fuller et al. 2018),

 MNRAS 000, 1–10 (2021)

6 Thomas Siegert et al.
 25-68 keV 504-514 keV 940-2000 keV
 90 90 90
 75 105 75 105 75 105
 60 120 60 120 60 120
 EXO 0748-676 EXO 0748-676 EXO 0748-676
 45 LMC X-4 135 45 LMC X-4 135 45 LMC X-4 135

 ESO 33-2 ESO 33-2 ESO 33-2
 15 15 15

 SMC X-1 Reticulum II 30 SMC X-1 Reticulum II 30 SMC X-1 Reticulum II 30

 45 45 45

 60 60 60

 75 75 75

 90 90 90

 Figure 3. Count residuals projected back to the the sky by applying the imaging response backwards. The radial and azimuthal coordinates are the Galactic
 latitude and longitude, respectively. Shown are the bands 25–68 keV, 508–514 keV, and 940–2000 keV as examples. Only LMC X-4 is significantly detected in
 the first energy band.

 Powerlaw ( = 0.65)
 103 DM Annihilation (FSR; MDM = 4 MeV,
 v = 4 × 10 25 cm3 s 1)

 F511 [phcm 2 s 1]
E 2 × FE [keV2 cm 2 s 1 keV 1]

 PBH (primaries, MBH = 2 × 1015 g)
 102 PBH (secondaries) 10 3
 NS merger (×106; @10 kyr)
 new observation campaign
 SPI ( < 2 )
 101 previous data set
 Siegert et al. (2016c)
 this work (variable source)
 this work (constant source)
 100 10 4 SPI data (3 upper limits)
 2000 3000 4000 5000 6000 7000
 IJD [MJD-51544]
 10 1
 Figure 5. Lightcurve of the 511 keV line from Ret II. No positive excess is
 10 2 found at any time. Therefore, shown are 3 upper limits in the band 508–
 514 keV. The inferred flux from Siegert et al. (2016c) is shown in comparison
 101 102 103 104 to the new INTEGRAL observation campaign. Additional serendipitous ob-
 Energy [keV] servations between IJD 4800 and 6600 are also included.

 Figure 4. SPI spectrum from the position of Reticulum II. Shown are 2 
 upper limits on the flux as a function of photon energy. A selection of excluded and estimate PBH conditioned on all the other unknown param-
 models is shown (see below).
 eters, and illustrate the limits on PBH as a function of BH , as it
 is typically done. However here we consider the 95th percentile in
 PBH -direction of the marginalised posterior ( PBH , BH | , )
 for example. We therefore split the data set into different time bins
 with and as the flux values and corresponding uncertainties of
 according to chance observations of nearby targets as well as the new
 energy bin in the spectrum. See Appendix A for details. In Fig. 6,
 observation campaign. The resulting 511 keV light curve is shown
 we show our upper bounds on PBH , and compare them to estimates
 in Fig. 5. Considering only observations in which Ret II is inside the
 from the recent literature, including the MW (Laha 2019; Laha et al.
 FCFOV results in upper limits consistent with the previous measure-
 2020), cosmic rays (Boudaud & Cirelli 2019), the cosmic microwave
 ment by Siegert et al. (2016c). We find that the significance of the
 background (CMB, Clark et al. 2017), and the cosmic -ray back-
 511 keV line rises up to ∼ 2 until the end of the older data set.
 ground (CGB Iguaz et al. 2021).
 Including more data after 2013 and in particular the additional 1 Ms
 The primary component of PBH evaporation alone cannot con-
 in 2018 (IJD ∼ 6700), leads to a greatly reduced upper flux bound as
 strain PBH in Ret II. Including the secondary particles, the upper
 well as a lower significance. Assuming the source to be variable in
 bound on PBH = 1 excludes monochromatic PBH distributions with
 time, the upper limit on the 511 keV line is ∼ 6.1 × 10−4 ph cm−2 s−1
 masses of . 0.4 × 1016 g. Assuming that all + s created by PBHs
 – barely consistent with the previous estimate. We conclude that the
 in Ret II annihilate strengthens the bound on PBH , then excluding
 3 signal from Ret II by Siegert et al. (2016c) was most likely due to
 masses . 2.9 × 1016 g. Since this assumption is too optimistic, we
 an instrumental background fluctuation, paired with short exposures
 included the limiting factor for + annihilation, that all + s above
 outside the FCFOV. We can not, however, exclude that the previously max escape from the galaxy and do not contribute to
 the threshold kin
 reported signal was of astrophysical origin.
 the annihilation spectrum. This describes our most realistic bound
 on PBH = 1, which excludes masses . 0.8 × 1016 g.
 Laha (2019) performed a similar estimate for the flux of the
 4.3 Primordial Black Holes
 511 keV line in the MW, however assumed that the entire annihi-
 We perform fits to the extracted spectrum, Fig. 4, with Eq. (21), and lation flux, from integrating over the + -distribution of PBHs, ends
 sub-cases thereof. Here we emphasise that we do not solve for the up in the 511 keV line. Since the Ps fraction in the MW is within
 interesting parameters until a certain figure of merit is above a thresh- a range 0.97–1.00, the 511 keV line flux from PBHs is at least four
 old, but determine the joint posterior distributions of all the free times smaller, because most of the annihilation flux goes into the
 parameters in Eq. (21). That means we include the uncertainties on three-photon decay of ortho-Ps. In return this means that the bounds

 MNRAS 000, 1–10 (2021)

Dark Matter in Reticulum II 7
 100 + (FSR, this work)
 10 19
 e+e (FSR, this work)

 v [cm3 s 1] (95th percentile)
 e+e (FSR + Ps, this work)
 10 21 e+e (FSR, Essig+2013; MW; SPI)
 e+e (FSR, Essig+2013; MW; COMPTEL)
fPBH (95th percentile)

 10 23 e+e (CMB, Slatyer+2016; Planck)

 PBH (prim.)
 10 25

 PBH (sec.)
 10 1 PBH (sec.) + Ann. 10 27
 PBH (sec.) + lim. Ann. + AiF + IB
 Laha+2020; MW; SPI
 Boudaud+2019; MW; Voyager 1 10 29
 Iguaz+2021; CGB
 Clark+2017; CMB 10 31
 1016 100 101 102
 1015 1017 mDM [MeV]
 MBH [g]
 Figure 7. DM annihilation cross section into + − with subsequent FSR and
 Figure 6. Fraction of DM that can be constituted of a monochromatic PBH possible annihilation of the pairs in Ret II (case 1. in Sec. 3.1.2; gray lines),
 distribution with mass BH in Ret II (solid lines), compared to literature esti- compared to literature limits. The shaded regions are excluded.
 mates. The circles mark the masses when the 95th percentile of the posterior
 ( PBH , BH | , ) in each case is reached. Stars mark the literature limits
 of 100 % PBH DM. 10 20
 (this work)
 (Laha et al. 2020; SPI, MW)
 on PBH from Laha (2019) should be shifted by about a factor of four v [cm3 s 1] (95th percentile) 10 22
 (Ng et al. 2019; NuSTAR, M31)
 ‘vertically’, which decreases the mass at PBH = 1 from ∼ 8.9×1016 g (Slatyer et al. 2016; Planck, CMB)
 to ∼ 6.4 × 1016 g. We test this by making the same assumptions 10 24

 with our Ret II spectrum, and find PBH = 1 is excluded for masses
 BH . 4.0 × 1016 g. 10 26

 Our conservative and realistic bounds are about one order of mag-
 nitude in PBH mass below previous estimates. Nevertheless, our data 10 28

 set is considerably shorter than all the other data sets, often compris-
 ing more than a decade of observation time. Thanks to the much 10 30

 smaller region of interest in the case of Ret II, basically constituting
 a point source of SPI, our systematic uncertainties can be consid- 10 32
 ered smaller compared to studies of the MW. The entire Galactic
 (DM) profile is rather uncertain, and also the Galactic centre itself 10 2 10 1 100 101 102
 bears problems when DM models are applied, as is often the case
 mDM [MeV]
 for diffuse emission in the GeV band. While our bounds are not yet
 Figure 8. Similar to Fig. 7 but for DM + DM → + .
 as constraining, we show that reasonable progress can be made with
 even a small data set. In particular, we show that when all uncertain-
 ties on the spectral parameters are included also more conservative a general trend of the annihilation cross section at tree level of
 and realistic bounds can result. h i . 5 × 10−28 ( DM /MeV) 2.5 cm3 s−1 , based on the shape of
 the measured limits as a function of energy in Fig. 7. For complete-
 ness, we show the limits on DM annihilation into + − , being hardly
 4.4 Particle Dark Matter
 constrained by Ret II data.
 Similar to the PBH case, we perform a fit to the extracted Ret II The cross section limit for the case DM + DM → + is shown
 spectrum and marginalise over the uncertain parameters to determine in Fig. 8. Our limits on DM annihilation into two photons are compa-
 the posteriors (h i, DM | , ) and ( , DM | , ) for DM rable to those from NuSTAR observations of M31 (Ng et al. 2019),
 annihilation and decay, respectively. In what follows, the excluded and about one order of magnitude worse than from MW observations
 regions consider the 95th percentile of the posteriors in h i- and with SPI (Laha et al. 2020). Because our data set extends to the full
 -direction, respectively. energy range of SPI, we can set however limits out to DM ≤ 8 MeV,
 varying between 10−28 and 10−25 cm3 s−1 . However, limits from the
 CMB (Slatyer 2016) are still more stringent at there energies.
 4.4.1 Annihilation
 In Fig. 7, we show the excluded region of the velocity-averaged
 4.4.2 Decay
 DM annihilation cross section into + − , followed by FSR and
 subsequent + annihilation. Without additional + annihilation, The properties of decaying DM particles cannot be inferred directly
 the bounds from Ret II are barely comparable to literature val- from the previous case of annihilating DM. First, the kinematic
 ues from other -ray experiments (Essig et al. 2013) or the CMB thresholds for standard model particle production is shifted by a
 (Slatyer 2016). When limited + annihilation is included, we can factor of two which leads to altered (shifted) spectral shapes in par-
 improve the limits in the DM mass range 0.5–1.0 MeV, and find ticular for FSR and IA. Second, the J-factor of Ret II is about twice as

 MNRAS 000, 1–10 (2021)

8 Thomas Siegert et al.

 e+e (FSR, this work) 5 SUMMARY AND CONCLUSION
 1027
 e+e (FSR + Ps, this work) Ret II is not seen in soft -rays between 0.025 and 8 MeV, up to
 1026 e+e (FSR, Essig+2013; MW; SPI)
 e+e (FSR, Essig+2013; MW; COMPTEL) an upper flux limit of 10−8 erg cm−2 s−1 (5 × 10−10 erg cm−2 s−1
[s] (95th percentile)

 e+e (CMB, Slatyer+2017; Planck) within 20–80 keV). The previously reported 511 keV line signal in
 1025 Ret II has not been consolidated in this work, and we provide an
 upper limit on its flux of 1.7 × 10−4 ph cm−2 s−1 . Given the distance
 1024 to the galaxy, this constrains the + -annihilation rate (including Ps
 formation and annihilation in flight) to 0.9–3.7 × 1043 e+ s−1 , at most
 1023 the annihilation rate of the MW1 .
 1022 We calculated parametrised spectral models for PBH evaporation
 and DM annihilation/decay with subsequent + annihilation and fitted
 1021 these models to the total spectrum from Ret II. Our conservative
 limit on the monochromatic PBH mass distribution constituting the
 1020 100 101 102
 entirety of DM in Ret II is 0.8 × 1016 g. We improve the limits on the
 mDM [MeV] velocity-averaged DM annihilation cross section of decay time of DM
 particles into + − in the range 0.5–2 MeV. For annihilation or decay
 Figure 9. Similar to Fig. 7 but for DM → e+ + e− + ( )FSR . into two photons, our limits are weaker than previous estimates. We
 provide the currently only limits on the cross section h i in the
 range 3–8 MeV, ranging between 10−28 and 10−25 cm3 s−1 .
 While our estimates are at most on the same level as previous
 constraints, we arrive at our conclusions with a very small data set
 (two weeks) compared to the multi-year or even decade-long obser-
 1029 vations. With our detailed spectral modelling including the effects of
 possible + annihilation and the proper statistical treatment of known
 1027 unknowns, we show that dedicated observations of selected targets
[s] (95th percentile)

 provide valuable information about the DM conundrum. We encour-
 1025 age follow-up observations of Ret II and other dwarf galaxies with
 INTEGRAL, and foresee a bright future for soft -ray instruments to
 1023 come, such as COSI (Tomsick et al. 2019), to contribute significantly
 (this work) in the research for the nature of DM.
 (Laha+2020; MW; SPI)
 1021 (Essig+2013; MW; HEAO)
 (Essig+2013; MW; SPI)
 1019 (Essig+2013; MW; COMPTEL)
 (Ng+2019; M31; NuSTAR) SOFTWARE
 (CMB, Slatyer+2017; Planck) OSA/spimodfit (Halloin 2009), BlackHawk (Arbey & Auffinger
 101710 2 10 1 100 101 102 2019), numpy (Oliphant 2006), matplotlib (Hunter 2007), astropy
 mDM [MeV] (Collaboration et al. 2013), scipy (Virtanen et al. 2019), 3ML
 (Vianello et al. 2015), MultiNest (Feroz & Hobson 2008; Feroz et al.
 Figure 10. Similar to Fig. 7 but for DM → + . For the case DM → + Y, 2009, 2019).
 the limits are smaller by a factor of 2.

 ACKNOWLEDGEMENTS
 Thomas Siegert is supported by the German Research Foundation
 uncertain as its D-factor, which allows to populate different regions (DFG-Forschungsstipendium SI 2502/3-1). We thank Ranjan Laha,
 in the full posterior. We therefore perform spectral fits for the DM Kenny Ng, Shunsaku Horiuchi, Marco Cirelli, Sam McDermott,
 decay case separately to infer bounds on the DM lifetime . This also Tracy Slatyer, and Joaquim Iguaz for providing limits on dark matter
 results in slightly worse bounds when considering decay compared properties, and Oleg Korobkin for -ray spectra from neutron star
 to annihilation (cf. NuSTAR limits). mergers. We thank John Beacom and Hassan Yüksel for details on
 In Fig. 9 we show the the exclusion region of as a function of the in-flight annihilation spectrum.
 the DM mass. Similar to the DM annihilation case, FSR alone is
 barely constraining, but can be pushed by including + annihilation.
 We find that DM particles decaying into + − pairs must have a
 lifetime longer than 1–5 × 1024 s between 1 and 2 MeV. In the mass DATA AVAILABILITY
 range 1–2 MeV, our bounds are as strong as the limits from the CMB The data underlying this article will be shared on reasonable request
 Slatyer & Wu (2017). Above this mass range, previous limits from to the corresponding author.
 SPI and COMPTEL in the MW (Essig et al. 2013), and in particular
 those from the CMB are more constraining.
 Finally, our DM decay time limits considering the direct decay 1 We note that in Siegert et al. (2016c), there is a typo, reducing the estimate
 into two photons is not improving beyond previous constraints. This for Ret II by a factor of ten – the actual annihilation rate in the previous work
 is expected from the pure narrow line sensitivity of SPI. should be on the order of 1044 e+ s−1 .

 MNRAS 000, 1–10 (2021)

Dark Matter in Reticulum II 9
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 MNRAS 000, 1–10 (2021)

10 Thomas Siegert et al.

 v ub(mDM)
 10 21
 MultiNest Samples

 10 23
v ee [cm3 s 1]

 10 25

 10 27

 10 29

 103 104 105 106
 mDM [keV c 2]
 Figure A1. Marginalised posterior distribution ( h i, DM | , ) (red
 dots), together with upper bounds of the annihilation cross section as a func-
 tion of the DM mass (black line).

 in 3ML to evaluate the joint posterior distributions, marginalisations,
 and upper bounds. We show the samples of this example in Fig. A1.
 We list our prior distributions and parameter ranges for all considered
 models in Tab. A1.

 This paper has been typeset from a TEX/LATEX file prepared by the author.

 MNRAS 000, 1–10 (2021)

Dark Matter in Reticulum II 11

 Model Ps max
 kin
 PBH (prim.) U (0, 1) log U (1014 , 1018 ) U (1018 , 4 × 1018 ) – – –
 .. .. ..
 PBH (sec.) . . . – – –
 . . .
 PBH (sec.) + Ann. .. .. .. – U (0, 1) –
 .. .. .. ..
 PBH (sec.) + lim. Ann. . . . – . log U (103 , 105 )
 2DM → e+ e− log U (3 × 10−34 , 3 × 10−20 ) log U (511, 106 ) – U (2 × 1018 , 4 × 1019 ) U (0, 1) log U (103 , 105 )
 .. ..
 2DM → 2 . log U (25, 8000) – . – –
 DM → e+ e− log U (1015 , 1024 ) log U (1022, 106 ) U (1018 , 4 × 1018 ) – U (0, 1) log U (103 , 105 )
 ..
 DM → 2 log U (1015 , 1029 ) log U (50, 16000) . – – –

Table A1. Prior distributions for considered DM models in Eq. (5).

 MNRAS 000, 1–10 (2021)