Observational consistency and future predictions for a 3.5 keV ALP to photon line
Pedro D. Alvarez, Joseph P. Conlon, Francesca V. Day, M.C. David Marsh, Markus Rummel
PPrepared for submission to JCAP
Observational consistency and futurepredictions for a 3.5 keV ALP tophoton line
Pedro D. Alvarez, Joseph P. Conlon, Francesca V. Day,M.C. David Marsh, Markus Rummel
Rudolf Peierls Centre for Theoretical Physics, University of Oxford,1 Keble Road, Oxford, OX1 3NP, United KingdomE-mail: [email protected], [email protected],[email protected], [email protected],[email protected]
Abstract.
Motivated by the possibility of explaining the 3.5 keV line through dark matterdecaying to axion-like particles that subsequently convert to photons, we study ALP-photonconversion for sightlines passing within 50 pc of the galactic centre. Conversion depends onthe galactic centre magnetic field which is highly uncertain. For fields at low or mid-rangeof observational estimates (10–100 µ G), no observable signal is possible. For fields at thehigh range of observational estimates (a pervasive poloidal mG field over the central 150 pc)it is possible to generate sufficient signal to explain recent observations of a 3.5 keV linein the galactic centre. In this scenario, the galactic centre line signal comes predominantlyfrom the region with z >
20 pc, reconciling the results from the Chandra and XMM-NewtonX-ray telescopes. The dark matter to ALP to photon scenario also naturally predicts thenon-observation of the 3.5 keV line in stacked galaxy spectra. We further explore predictionsfor the line flux in galaxies and suggest a set of galaxies that is optimised for observing the3.5 keV line in this model.
Keywords: dark matter, axion, axion-like particle a r X i v : . [ h e p - ph ] O c t ontents → ALP → photon scenario 3 → a a → γ → a → γ scenario 5 → a → γ at the centre of the Milky Way 10 → a → γ model in other galaxies 17 Determining the nature and properties of dark matter is one of the most significant challengesof contemporary high-energy physics. Among the variety of strategies for detecting signsof dark matter is the search for unidentified emission lines in the spectra of dark matterdominated astrophysical objects. Such lines could arise in the two-body decay/annihilationof dark matter particles in which the final state contains a photon. Hence, it is not surprisingthat the recent detection of an unidentified emission line at 3.5 keV in the X-ray spectrumof galaxy clusters and the Andromeda galaxy has generated much interest.The 3.5 keV line was first observed by two distinct groups, using the detectors of twoindependent satellites, XMM-Newton and Chandra. After carefully subtracting the astro-physical background of a stacked sample of galaxy clusters, reference [1] found an additionalemission line at E ∼ .
55 keV, with no apparent astrophysical origin. The line was observedwith both the MOS and PN cameras on the XMM-Newton instrument, and also reconfirmedin the Perseus cluster using both ACIS-I and ACIS-S configurations on the Chandra satellite.In [2], a line at very similar energies was also observed in the outskirts of the Perseus cluster(using a distinct set of XMM-Newton observations than in [1]) and also in the central regionof the Andromeda galaxy (M31).These papers have triggered many subsequent observational searches for further E ∼ . observed with XMM-Newton is above the upper bound set by non-observationswith Chandra in [3].Furthermore, reference [4] also challenged the existence of a line in M31 (claimed in [2]),finding only 1 σ support for such a line, and questioned the existence of an unexplained linefrom clusters, arguing that by including K XVIII lines at 3.47 and 3.51 keV and a Cl XVIIline at 3.51 keV, no significant excess around 3.5 keV could be established after allowingfor systematic uncertainties. In response, the authors of reference [2] pointed out in [6] thatthe lower significance of the 3.5 keV line for the M31 analysis in [4] was due to a restrictionto an inappropriately narrow fitting interval of 3–4 keV, resulting in a relatively poorer fitfor the index of the power law background and an inevitably lower significance for the linesignal. In reference [7], the authors of [1] robustly disagreed with the analysis of cluster datain [4], arguing that they relied upon incorrect atomic data and inconsistent spectroscopicmodelling.Less controversial, yet equally important, are the null results reported from searches forthe line in galaxies other than M31 and the Milky Way. In [8], a stacked sample of dwarfspheroidal galaxies were observed using XMM-Newton data. Dwarf galaxies are classic darkmatter targets as they have high dark matter densities and low background light. Underthe sterile neutrino interpretation of the results of [1], a signal of the strength reported in[1] was ruled out at a level of 4.6 σ (or 3.3 σ under the most conservative assumptions aboutthe galactic dark matter column density). However, no line signal was observed. Similarly,in [9], a search for the 3.5 keV line was performed in large stacks of archived Chandra andXMM-Newton data of galaxies. No evidence of a line at 3.5 keV was found, and the sterileneutrino interpretation of the line suggested in [1] was found to be ruled out at 4.4 σ and11.8 σ for Chandra and XMM-Newton samples respectively.As these same limits apply directly to any model in which dark matter decays directlyto the 3.5 keV photon, these results appear fatal to all models consisting of dark matterdecaying directly to photons. The challenge for any dark matter interpretation of the datain [1] and [2] is to explain why,1. A signal is produced in galaxy clusters, but is absent in the spectrum of dwarf spheroidalsand stacked galaxies.2. A signal is observed in the spectrum of M31, but not in other galaxies.Our focus here is on the scenario proposed in [10] – which we elaborate on below –which can explain both these features. In this scenario, dark matter with mass ∼ . O (1) Mpc) and support substantial conversion probabilities, while regulargalaxies and dwarf spheroidals give rise to much weaker photon lines as their magnetic fieldsare only present on O (10) kpc scales.This model was first proposed in [10] to explain the morphology of the signal observed ingalaxy clusters in [1], which is itself in tension with direct dark matter decay to photons. Thesignal found in [1] is stronger in the Perseus cluster – by up to a factor of eight – than thatinferred from the full 73 cluster sample. The signal in Perseus also comes disproportionatelyfrom the central cool core region of the cluster. The signal from the sample of local brightclusters – Coma, Ophiuchus and Centaurus – is also notably stronger than for the full stackedsample. As the central regions of cool core galaxy clusters host particularly high magneticfields, an enhanced signal from central cool core regions is a natural prediction of the DM → a → γ model.Among galaxies, central observations of edge-on spiral galaxies represent the most at-tractive targets, as the regular disk magnetic field is both coherent on the scale of the galaxyand, for central observations, orthogonal to the line of sight. Observational studies [11] sug-gest that M31 is particularly attractive among edge-on spiral galaxies, as the regular magneticfield is both unusually large (with B reg ∼ B random ) and unusually coherent (with no evidenceof reversals among spiral arms). The above puzzling features are therefore consistent with –and indeed required by – this model.Following its introduction in [10], this scenario has been analysed in more detail for thecase of the Milky Way halo (excluding the central region) in [12] and for the case of cool-coreand non-cool-core clusters in [13]. The objectives of the paper are to extend the study ofthis scenario to the observationally interesting region of the Milky Way centre and to furtherclarify its predictions for observations of other galaxies.This paper is organised as follows. We start by reviewing the dark matter to ALP tophoton (DM → a → γ ) scenario of [10] in Section 2. In Section 3 we review X-ray observationsof the galactic centre and review models for the magnetic fields and free electron density inthis region. In Section 4 we study the phenomenology of ALP-photon conversion in thegalactic centre. In Section 5 we consider observations of other galaxies and provide a listof galaxies optimised for the DM → a → γ scenario and that have been observed by eitherChandra or XMM-Newton. → ALP → photon scenario In this section, we review the scenario proposed in [10] in which the 3.5 keV line is producedby dark matter particles decaying into an ALP, a , which subsequently converts into X-rayphotons in the presence of astrophysical magnetic fields. → a While the 3.5 keV line – if indeed caused by decaying dark matter – may determine the massof the dark matter particle, it does not by itself provide information on its spin and relevantdecay channels. For example, if the dark matter particle is a fermion, ψ , it may decay into aphoton and a neutrino through the effective operator ψ † σ µν νF µν . A frequently consideredcase in this category is for ψ to be a right-handed sterile neutrino with a Majorana mass m ψ and a mixing with the Standard Model neutrinos induced by an operator yH † ¯ Lψ .If such a sterile neutrino constitutes dark matter with m ψ = 7 . Z -mediated channel ψ → ν ¯ νν . At the one-loop level,– 3 –harged currents induce the decay channel ψ → νγ with the decay rate,Γ ψ → νγ = 9 α EM π sin (2 θ ) G F m ψ , (2.1)where sin θ = yv √ m ψ for the Higgs VEV v/ √
2. The total observable photon flux in the fieldof view is then given by F ψ → νγ = Γ ψ → νγ π (cid:90) FOV (cid:37) d (cid:37) d φ (cid:90) l . o . s . ρ DM ( l, (cid:37), φ ) m ψ d l, (2.2)where l is the distance along the line of sight and ( (cid:37), φ ) are cylindrical coordinates withinthe field of view, with (cid:37) the angular radial coordinate.Reference [1] showed that, if the unidentified 3.5 keV line is interpreted as arising fromdecaying sterile neutrinos, then the mixing angle is determined to be very small but non-vanishing, sin (2 θ ) ≈ × − . Since then the 3.5 keV signal has also been interpreted ina variety of models in which dark matter decays, annihilates or de-excites with the promptemission of a photon. Among these, models involving axion or ALPs as the dark matterparticle have been considered in [14–19].The scenario of [10] considered in this paper is however crucially different from thesemodels in that the photon line is a secondary, environmental effect due to the existence ofan otherwise ‘invisible’ ALP line. Decay modes for dark matter particles into ALPs exist forboth fermionic and scalar dark matter [10]. For example, fermionic dark matter can decayto an ALP and a neutrino through the operator ∂ µ a Λ ¯ ψγ µ γ ν with the decay rate,Γ ψ → νa = 116 π m ψ Λ , (2.3)which is in principle independent of Γ ψ → νγ if Λ is independent of the mixing angle θ .In the presence of magnetic fields, ALPs may convert into photons in a process akin tothat of neutrino oscillations. As we will now discuss in more detail, in this way a 3.5 keVALP line may produce an associated 3.5 keV photon line. a → γ The relevant interaction term for axion-photon conversion is the Lagrangian operator,
L ⊃ M aF µν ˜ F µν ≡ M a (cid:126)E · (cid:126)B ≡ g aγγ a (cid:126)E · (cid:126)B , (2.4)where g aγγ = M − is the ALP photon coupling. The linearised equations of motion for amode of energy ω propagating in the x -direction in the presence of a classical backgroundmagnetic field, (cid:126)B , are given by, [20] ω + ∆ γ ∆ F ∆ γay ∆ F ∆ γ ∆ γaz ∆ γay ∆ γaz ∆ a − i∂ x γ y γ z a = 0 . (2.5)Here ∆ F denotes the interaction which induces Faraday rotation between photon polarisationstates in an external magnetic field. As we will be concerned with the total photon flux fromALP-photon conversion, we will neglect these terms in the subsequent analysis.– 4 –he refractive index for photons in a plasma is given by ∆ γ = − ω / ω , where ω pl =(4 παn e /m e ) / is the plasma frequency with n e the free electron density. The axion-photonmixing induced by the Lagrangian operator of equation (2.4) is determined by the matrixelements ∆ γai = B i / M , where B i denotes the magnetic field in the directions perpendicularto the ALP direction of travel. Finally, ∆ a = − m a /ω (in this work we assume vanishingALP mass m a = 0). Formally, we may write the general solution to equation (2.5) for aninitial state, | i (cid:105) = ( γ y , γ z , a ) T (cid:12)(cid:12)(cid:12) x = − L/ propagating from x = − L/ x = L/ | f (cid:105) = T x (cid:34) exp (cid:32) − iωL − i (cid:90) L/ − L/ M ( x )d x (cid:33)(cid:35) | i (cid:105) , (2.6)where M ( x ) = ∆ γ ( x ) 0 ∆ γay ( x )0 ∆ γ ( x ) ∆ γaz ( x )∆ γay ( x ) ∆ γaz ( x ) ∆ a ( x ) . (2.7)Here, T x denotes the ‘ x -ordering’ operator. For an initially pure ALP state, the ALP-photonconversion is then given by, P a → γ = |(cid:104) , , | f (cid:105)| + |(cid:104) , , | f (cid:105)| = (cid:0) | γ y | + | γ z | (cid:1) (cid:12)(cid:12)(cid:12) x = L/ . (2.8)In this scenario, the strength of the photon line then depends on both the magnitude andcoherence of the magnetic field, and the dark matter column density along the line of sight.The total predicted photon flux is then given by: F ψ → νγ = Γ DM → a π (cid:90) FOV (cid:37) d (cid:37) d φ (cid:90) l . o . s . ρ DM ( l, (cid:37), φ ) m DM P a → γ ( l, (cid:37), φ ) d l, (2.9)As we will discuss in Section 4, ALP-photon conversion in the the Milky Way only proceedsefficiently in the very central region close to Sgr A*, if at all, and the observable photon lineflux is then well-approximated by, F DM → a → γ (cid:39) Ω FOV πτ DM (cid:104) P a → γ (cid:105) FOV (cid:90) ∞ l GC ρ DM ( l ) m DM dl , (2.10)where (cid:104) P a → γ (cid:105) FOV denotes the average conversion probability over the telescope field of view,Ω
FOV is the angular size of the field of view in steradians, and the dark matter density isaveraged over the field of view. → a → γ scenario Here we briefly review predictions made in previous work for the DM → a → γ scenario.With regards to galaxy clusters, reference [10] showed that the would-be dark matter decaytime assuming DM → photons would vary from cluster to cluster in a DM → a → γ scenario,with shorter decay times being inferred for clusters with stronger or more coherent magneticfields. Within a cluster, the line strength should approximately trace out the square of themagnetic field strength, in particularly peaking strongly in the centre of cool core clusters.The central region of cool core clusters will also give a stronger signal than the central regionof non-cool core clusters. These predictions were further discussed and quantified in [13].– 5 –eference [13] also noted that, due to the increase in magnetic field strength at the centre ofa cluster, the would-be decay time inferred from local clusters that fill the field of view willbe greater than the decay time inferred from more distant clusters, where the entire clusterfits in the field of view.With regards to galaxies, reference [12] found that the conversion probability in theMilky Way halo is too low to produce an observable signal, while the conversion probabilityin M31 is much higher, with M31 displaying highly beneficial conditions for a → γ conversion.Reference [12] also predicted a sharp decrease in the signal strength as we move away fromthe centre of M31, as the magnetic field (following the spiral arms) becomes parallel to theline of sight. Furthermore, [12] predicts that the 3.5 keV line signal in a typical galaxy ismuch weaker than in a galaxy cluster. Reference [10] predicted that, among galaxies, edge-onspirals will give the strongest line signal, as the ALP will propagate a larger distance throughthe disk. Motivated by recent analyses of X-ray line emission from the Milky Way centre in [3–5],we here determine the circumstances under which a signal from the Milky Way centre isachievable in the DM → a → γ model. ALP to photon conversion in the bulk of the MilkyWay has been studied in [12] and found to be too inefficient to contribute significantly tothe photon flux. However, before discussing the predictions of this model, we first reviewin Section 3.1 the important aspects of the observations [3–5] in some detail. In 3.2 and3.3, we review the pertinent aspects of the observational models for the electron density andmagnetic field in the central region of the Milky Way. The dynamic centre of the Milky Way is the supermassive black hole associated to the radiosource Sgr A*. We take a distance of 8.5 kpc to the galactic centre, and so 1 (cid:48) corresponds to2.47 pc at the galactic centre.As it plays an important subsequent role, we first review details of the XMM-Newtonand Chandra telescopes. The archival Chandra observations analysed in [3] involve data fromACIS-I configuration which has a square field of view of 16 . (cid:48) by 16 . (cid:48) , consisting of 4 CCDchips I0–I3. The archival XMM-Newton observations analysed in [4, 5] are with either theXMM-MOS or XMM-PN cameras. These involve slightly different geometric arrangementsof the chips, but in both cases result in a field of view with approximate radius of 15 (cid:48) .Observations with MOS1 after 2005 (2012) have reduced coverage due to the failure of one(two) CCDs following micrometeorite damage.In the Chandra observations of [3], a 95% bound on line emission at E ∼ .
55 keVwas derived as F (cid:46) × − erg cm − s − , which equates to an upper limit of F (cid:46) × − photons cm − s − . The baseline fitted background model also included atomic linesfrom K XVIII at 3.48 keV, 3.52 keV and an Ar line at 3.62 keV, and the upper bound onthe line flux is sensitive to the strengths assigned to these lines. In Table 1, we collect theline strengths for the base fit to the data. As the uncertainty in the background modelling is large, it is possible that the assignedline strengths may hide an actual dark matter signal. A caution on these line strengths is We thank Signe Riemer-Sørensen for communicating these to us. – 6 –lement Energy Strength Strength per arcmin (keV) (ph cm − s − ) (ph arcmin − cm − s − )95 % Upper bound 3.55 keV (cid:46) × − (cid:46) . × − K XVIII 3.48 2 . × − . × − K XVIII 3.52 4 . × − . × − Ar XVII 3.62 4 . × − . × − Table 1 . Observations of the galactic centre region with Chandra [3]. We give the 95 % upper boundon line emission and also fitted values for atomic lines included in XSPEC [21] (note that these fittedvalues are not necessarily statistically distinct from zero)
Detector Energy Strength Strength per arcmin (keV) (ph cm − s − ) (ph arcmin − cm − s − )XMM MOS [4] 3.5 4 . × − . × − XMM PN [4] 3.5 2 . × − . × − XMM [5] 3.53 (2 . ± . × − (5 . ± . × − Table 2 . XMM-Newton observations of the galactic centre region: line emission detected around 3.5keV that as they do not come with error bars (due to difficulties of making XSPEC converge) itis possible that there is actually no statistically significant line emission at these frequencies.For subsequent comparison with XMM-Newton observations, we re-express these in terms offlux per arcminute . In [3], the central 2 . (cid:48) radius around Sgr A* is masked. Hence, for theanalysis in Section 4, we use an effective field of view of 240 arcminute . Using archival XMM-Newton data, references [4] and [5] both detect an emission line at E ∼ . . For the MOScamera, this comes from averaging the field of view of MOS1 and MOS2 from the tables inthe appendix of [5], and we have assumed the same field of view for the PN camera.The line strength observed with XMM-Newton is at a level markedly stronger than theupper bound from Chandra observations. In terms of interpretations involving K XVIII lines,it is unclear what importance to place on this: the galactic centre environment is complexand multiphase, and it is conceivable that the regions enclosed by the XMM-Newton fieldof view involve a higher average K abundance than those within the Chandra field of view.However, this would be surprising for the case of dark matter decaying to produce photons.One aim of this paper is to explain how, in the context of the DM → a → γ scenario, thisdifference can arise naturally. In this scenario, the signal is suppressed within the galacticplane, and so the XMM-Newton field of view, which extends further vertically out of theplane, contains more signal region. To understand this we now discuss the astrophysics ofthe galactic centre. There is further reduction in field of view due to masking of point sources, corresponding to an additional7% reduction [3]. We omit this here as a similar point source masking was carried out for XMM-Newton, andwe do not know the percentage of field of view lost there. Given the other uncertainties, this error is minor. – 7 – .2 Electron density
As discussed in Section 2.2, ALP to photon conversion depends on the free electron density,with large electron densities suppressing the conversion amplitude. The electron densityin the Milky Way centre is therefore an important input into the resulting signal for theDM → a → γ scenario.We first describe the coordinates used. We use right-handed Cartesian ( x, y, z ) coordi-nates, where the origin (0 , ,
0) corresponds to the centre of galactic coordinates ( r, b, l ) =(8 . , , x -coordinate points from the Milky Way centre towards the sun, y isin direction of decreasing l and z points vertically upwards out of the galactic plane (to-wards positive b ). However note that, in these coordinates, the true dynamic marker of theMilky Way centre Sgr A* (where the majority of observations considered here are centred)is slightly offset, with a physical location of ( l, b ) = ( − . , − .
05) [22]. This corresponds to( y SgrA* , z
SgrA* ) = (8 . , − . n e, GC ( x, y, z ) = 10 cm − exp (cid:20) − x + ( y − y GC ) L (cid:21) exp (cid:20) − ( z − z GC ) H (cid:21) , (3.1)with L GC = 145 pc and H GC = 26 pc. This dominates over thin and thick disk componentsin the innermost galaxy. The centroid of the distribution is offset by y GC = 10 pc and z GC = −
20 pc. However, note that the physical offset from Sgr A* is reduced as Sgr A*is itself offset from x = y = z = 0. Also note in the NE2001 model, the electron densityin (3.1) is formally truncated to zero when the argument of the exponential is less than -1.However, this truncation reflects an abrupt change in scattering diameters for OH masers inthe galactic center, and can be omitted if we are interested only in the free electron densityrather than its fluctuations (see the discussion in Section 2.4 of [24]). We shall therefore use(3.1) as our baseline electron density model in this paper. We also include the thick diskcomponent of [23], which becomes comparable to the galactic centre component at the edgeof our region of interest.Let us enumerate the caveats on the above electron density. This electron density isderived via pulsar dispersion and emission measures, which are sensitive to integrated electrondensities along the line of sight. The electron density thus determined is a smooth function,and does not account for patchiness, or the presence of dense clouds with partial filling factorsinterspersed by voids. It is also a single simple function that will represent a fit to data forall lines of sight within O (100) pc from Sgr A*, while our interest is only in lines of sightenclosed by the fields of view of XMM-Newton and Chandra (extending to a maximum of37 pc from Sgr A*), and in particular the regions along them with large transverse magneticfields. For all these caveats, the distribution in [23] is nonetheless observationally derivedand captures genuine features of the free electron distribution in the galactic centre. Whileaware of its limitations, we shall therefore use it in our subsequent studies. The magnitude, direction and coherence of the transverse magnetic field in the galacticcentre region are clearly important for us to determine the a → γ conversion probability. More detailed studies of gas distributions within the inner 10 pc appear in [25]. – 8 –nfortunately, the magnetic field in the central 100–200 pc of the Milky Way is poorlyknown, and estimates vary by two orders of magnitude. As the galactic centre magnetic fieldin this region arises from different processes to the bulk of the galaxy, this area is excludedfrom the Milky Way magnetic field model of [26, 27]. Following [28–32], we here provide abrief summary of the observational possibilities for the galactic centre magnetic field, fromhigh to low values.There exists a longstanding case that the magnetic field within the galactic centre isdominantly poloidal (vertical) and with a uniform milligauss strength throughout the central ∼
150 parsecs [33–36]. This argument arises from the presence of nonthermal radio filamentsin the galactic centre region, orientated predominantly orthogonal to the galactic plane andemitting via synchrotron radiation. These filaments are remarkably straight and uniform,even though some are clearly interacting with molecular clouds. The apparent rigidity ofthe filaments against collisions with molecular clouds can be used to infer their magneticpressure, leading to estimates of a B ∼ B ∼ µ G averaged over a 400 pc region, with a preferred averagefield of B ∼ µ G over the entire galactic centre region.In contrast, a much lower estimate is suggested in reference [30], which argues for arelatively weak pervasive poloidal magnetic field ( B ∼ µ G) within the galactic centreregion. These lower magnetic field estimates are based on equipartition arguments [38] andon studies of short pc-scale nonthermal radio filaments [39]. On this view, the large-scaleradio filaments are localised dynamical structures with B ∼ B ∼ µ G [40]. However, Faraday rotation only probes the component of the magneticfield along the line of sight, while ALP to photon conversion relies on the field perpendicularto the line of sight. For a field with random orientation, these will be comparable, but giventhat there are significant reasons to think that the galactic centre field is strongly poloidal,Faraday rotation estimates of the parallel magnetic field cannot be said to give a reliablemeasurement of the strength of the transverse field.Magnetic field estimates have also be reported for smaller sub-regions within the central ∼
150 parsec. Dense molecular clouds are widely argued to support horizontal magnetic fieldsof the order of B ∼ mG [41, 42], with such fields being produced by shearing of the poloidalfield by cloud motions or tidal forces. At a distance of 0.1 pc from the central black hole,reference [43] found a magnetic field B > As the a → γ conversion probability scaleswith B , this maximises the resulting signal. We will see that in this maximal scenario, anobservable signal from DM → a → γ is just achievable. Due to the B dependence of thesignal, it follows that we do not need to consider the other (medium and low) scenarios indetail: they are incapable of generating an observable 3.5 keV line from the galactic centrein the DM → a → γ scenario. → a → γ at the centre of the Milky Way We are now ready to discuss the characteristics of ALP to photon conversion in the centre ofthe Milky Way. We use the maximal model for the magnetic field discussed in Section 3.3,i.e., (cid:126)B = 1 mG ˆ z , (4.1)in the central for r (cid:46)
150 pc in the galactic plane, with the electron density as given bythe galactic centre and thick disk components of the NE2001 model, c.f. Section 3.2. Inparticular, we assume (cid:126)B = 1 mG ˆ z for | x | <
150 pc along sight lines within the XMM Newtonand Chandra field of views.The predicted photon flux per unit steradian is then given by (2.10),
F (cid:39) πτ DM (cid:104) P a → γ (cid:105) FOV (cid:90) ∞ l GC ρ DM m DM dl . (4.2)As sufficiently large conversion probabilities are only obtained in the vicinity of the galacticcentre, dark matter decaying between Earth and the galactic centre does not contributesignificantly to the observed photon flux. Consequently, the integration in (4.4) is from thegalactic centre at l GC = 8 . ρ NF W ( r ) = ρ s r s r (1 + r/r s ) , (4.3)where r is the distance to the galactic centre, ρ s = 20 . × M (cid:12) / kpc (local dark matterdensity) and r s = 10 . (cid:90) ∞ l GC ρ DM m DM dl (cid:39) . × cm − , (4.4)where we have used m DM = 7 . (cid:90) ∞ l GC ρ DM m DM dl (cid:39) . × − . × cm − . (4.5)This uncertainty should be kept in mind when flux values (4.2) are calculated in the following. We note that magnetic flux conservation implies that the strength of such a field cannot fall off rapidlyon a scale of 20–40 pc above the galactic plane. – 10 – .1 ALP to photon conversion probability
We now solve equation (2.5) to derive the axion-photon conversion probability for ALPs withnegligible mass, ∆ a (cid:28) ∆ γ and propagating through the galactic centre region of the MilkyWay. We will do this in two ways: first, we will solve equation (2.6) analytically by notingthat a small-mixing perturbative approximation is appropriate, and second, we will solveequation (2.5) numerically by discretising the evolution of an initially pure ALP-state.The strength of the ALP-photon mixing can be estimated by considering the ratio,2 B ⊥ ωM m eff ≈ − × (cid:18) B ⊥ (cid:19) (cid:18) GeV M (cid:19) (cid:18) . ω (cid:19) , where we have specialised to ∆ a = 0 and taken n e = 10 cm − . This suggests that a pertur-bative, small mixing approximation of the interaction should provide a good approximationto the full solution. To linear order, we find that an initially pure ALP state travelling from x = − L/ x = L/ P a → γ ( L ) = (cid:88) i = z,y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) L/ − L/ dx e iϕ ( x ) ∆ γ i a ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (4.6)where, ϕ ( x ) = (cid:90) x − L/ dx (cid:48) ∆ γ ( x (cid:48) ) = − ω (cid:90) x − L/ dx (cid:48) ω pl ( x (cid:48) ) . (4.7)For a constant magnetic field, this expression may be further simplified to, P a → γ ( L ) = B ⊥ M (cid:90) L/ − L/ dx (cid:90) L/ − L/ dx cos (cid:18) παωm e (cid:90) x x dx (cid:48) n e ( x (cid:48) ) (cid:19) . (4.8)For an electron density with a Gaussian fall-off – such as the galactic centre component ofthe NE2001 model – we may perform the x (cid:48) -integral to obtain an argument of the cosinewhich is proportional to the difference between two error functions with arguments x i /L GC for i = 1 ,
2, respectively, where L GC is defined as in equation (3.1). For the maximal magneticfield model of equation (4.1), we take the magnetic field to be approximately constant overthe central region of the Milky Way, and for the purpose of an analytical estimate, we expectthat the leading order Taylor expansion of the error functions should well approximate thefunction over the relevant interval. Explicitly, we approximate,Erf (cid:18) x i L GC (cid:19) ≈ √ π x i L GC , (4.9)where the subleading corrections appear at cubic order of the argument. With this approx-imation, we may perform the integrals of equation (4.8) explicitly to obtain the conversionprobability, P a → γ ( L ) = B ⊥ ω m e π α M ( n (0) e ) e (cid:18) ( y − y GC)2 L + ( z − z GC)2 H (cid:19) sin (cid:32) παn (0) e Lωm e e − (cid:18) ( y − y GC)2 L + ( z − z GC)2 H (cid:19) (cid:33) . (4.10)This conversion probability agrees well with that obtained from a numerical simulations,which we show in Figure 1 for L = 300 pc, M = 10 GeV.– 11 – igure 1 . The values of P a → γ as a function of the galactocentric coordinates ( y, z ) according to anumerical simulation of (2.5) with M = 10 GeV. The outer solid circle indicates the field of viewof XMM-Newton. The field of view of Chandra is indicated by the solid square (parallel orientationto y -axis) and the dashed square (45 ◦ orientation to y -axis). As observations are centred on Sgr A*they are slightly offset from ( y, z ) = (0 , Figure 1 shows a marked suppression of the conversion probabilities at low values of z . This arises as the conversion probability is sensitive to the difference between the ALPmass and the plasma frequency - and the latter is set by the free electron density. Highelectron densities lead to a large ALP-photon mass difference and a suppression of the ALP-photon conversion probabilities. At larger galactic altitudes, the electron density is lowestand the resulting ALP-photon conversion probability is well approximated by the zeroth orderexpansion of the cosine in equation (4.8), giving P a → γ ( L ) = B ⊥ L / (4 M ). This explains theapparent constancy of the conversion probability at z (cid:38)
30 pc.At lower galactic altitude, the factor n e, GC (0 , y, z ) = n (0) e exp (cid:104) − (cid:16) ( y − y GC ) L + ( z − z GC ) H (cid:17)(cid:105) ,that encodes the effective line-of-sight electron density for a given path, is too large tojustify a zeroth order expansion. On the contrary, the most striking feature of Figure 1are the spatial oscillations in the conversion probability in the ( y, z )-plane, and these aredirectly sourced by the varying line-of-sight electron densities, as is clear from the analytical– 12 –pproximation, (4.10). In addition, the conversion probability scales with an overall factorof (1 /n (eff) e, GC (0 , y, z )) , which explains the suppression of the conversion probability near thegalactic plane.We note that while the galactic centre contribution to the electron density dominatesin the very centre of the galaxy, it quickly drops below the contribution from the ‘thick disc’component of the NE2001 model at large z and x . According to this model for the electrondensity, the thick disc contributes with n thick disc e = 0 . − in the central region, and ourfull model of the electron density should then be n e = n GC e + n thick disc e . In deriving equation(4.10), we only included the contribution from n GC e and not that from n thick disc e . This neglecthowever, is harmless: for small enough values of n e , the argument of the cosine of equation(4.8) can be Taylor expanded with the leading order contribution giving P a → γ = B ⊥ L / M .For n e ≤ . − , this ‘small angle’ approximation is in good agreement with the numericalsolution of equation (2.5). We used a full discretized simulation of equation (2.5) with thefull NE2001 electron density n e = n GC e + n thick disc e to obtain the results in Sections 4.2, 4.3and 5. The predictions from the dark matter DM → a → γ scenario are now easily obtained bycombining (4.2) and the simulation results of (2.5). We first compare the ALP-photonconversion probabilities in the total field of view of XMM-Newton and Chandra. XMM-Newton has a radial total field of view with a radius of 15 (cid:48) (although only the inner 14 (cid:48) wereused in [5]), while Chandra has a square total field of view of 16 . (cid:48) × . (cid:48) . Note that thesearches for a 3.5 keV line in the galactic centre actually use a smaller field of view than thetotal one for both XMM-Newton [5] and Chandra [3]. As discussed in Section 3.1, we use530 arcmin for the actual XMM-Newton field of view for the observations in [4, 5]and 240arcmin for the actual Chandra field of view for the observations [3].The roll-angle, α r , of Chandra was not fixed during the observations considered in [3],and hence the exact orientation of the detectors during each observation was not fixed andmay well have varied. As the average conversion probability over the Chandra field of viewis sensitive to the orientation, we here consider two extreme cases as indicated in Figure 1. Ifthe symmetry axes of the Chandra field of view are aligned parallel to the y and z coordinateaxes, hence α r = 0 ◦ in our notation, most of the region with high conversion probability fallsoutside the field of view. A slightly larger average conversion probability can be expected forthe tilted field of view with α r = 45 ◦ .The field of view of the XMM-Newton observations of the galactic centre is a factorof 2 . z >
20 pc region where the electrondensity is suppressed with respect to that of the galactic plane, the ALP-photon conversionprobability for XMM-Newton is larger than that of Chandra when averaged over the field ofview. For a magnetic field of B ⊥ = 1 mG which is constant for | x | <
150 pc within the fieldof view, the ratio of the averaged conversion probabilities is given by (cid:104) P a → γ (cid:105) XMM (cid:104) P a → γ (cid:105) Chandra = (cid:40) . × − . × − = 2 . α r = 0 ◦ . × − . × − = 2 . α r = 45 ◦ . (4.11)Combining the larger conversion probability of XMM-Newton with its larger field of view,– 13 –e find that the expected photon flux ratio between XMM-Newton and Chandra is given by, F XMM F Chandra = (cid:40) . α r = 0 ◦ . α r = 45 ◦ . (4.12)Such a substantial flux ratio is consistent with a detectable signal in XMM-Newton, anda non-detection in Chandra. For the dark matter column density given in (4.4), and for τ DM = 8 . × s, M = 10 GeV, we find an expected photon flux of, F XMM = 2 . × − photons s − cm − , (4.13) F Chandra = 6 . × − photons s − cm − , (4.14)where we have used α r = 45 ◦ to estimate the Chandra flux. The value of τ DM M here hasbeen set to match the XMM flux observed by [5].For comparison, in [10], the parameter values τ DM = 5 × s and M = 10 GeV wereused, motivated by the observed flux from galaxy clusters [1] and an estimated average ALPto photon conversion probability of ∼ − for M = 10 GeV in the stacked cluster sample.This value of 10 − comes from numerical simulations of the centre of the Coma cluster in [46].There are however significant uncertainties on this number of 10 − . Even within Coma, themagnetic field is uncertain to a factor of two, corresponding to a factor of four uncertainty inconversion probability. It is also probable that conversion probabilities in the centre of thebright cluster Coma are biased high compared to those for a stacked average of many clusters.We shall also see in Section 4.3 that this ratio of τ DM , clusters /τ DM, GC ∼
60 is highly sensitiveto the assumed electron density profile in the galactic centre, and can vary by a factor of 10for small changes in the electron scale height. Therefore, despite the large apparent differencein τ DM M , the observations of a 3.5 keV line from clusters and from the galactic centre mayboth be explainable as originating from dark matter in the DM → a → γ scenario. Finally, let us apply a masking that restricts the field of view of XMM-Newton to the ∼ − conversion probability region z >
20 pc, see Figure 1. The field of view shrinks from530 arcmin to 90 arcmin , but since the field of view averaged conversion probability issignificantly larger than for the total field of view of XMM-Newton, the flux is rather similarto (4.13): F z> XMM = 2 . × − photons s − cm − . (4.15)We see that the DM → a → γ model can reconcile the conflicting results from Chandra andXMM-Newton if the magnetic field in the galactic centre is large enough. In addition, wepredict that the clear majority of the XMM-Newton signal will remain when all but the z >
20 pc region is masked out, despite the ∼
80% reduction in the field of view. Thisprediction is easily testable and, if confirmed, would be difficult to explain within any otherdark matter model.
In Section 4.2 we noted that for the default model of the electron density, XMM-Newton ob-tains a higher averaged conversion probability by observing regions at large z where the Decreasing τ DM by an order of magnitude from 5 × s would make the prediction in [12] of no observablesignal in the Milky Way slightly less strong, but only by changing the predicted signal from the general MilkyWay halo to two (instead of three) orders of magnitude weaker than that from galaxy clusters. – 14 – igure 2 . Left: The values of log ( (cid:104) P a → γ (cid:105) ) for XMM-Newton (blue) and Chandra (red) as afunction of the off-set of the electron density in the z -direction. Here M = 10 GeV. Right: Theratio (cid:104) P a → γ (cid:105) XMM / (cid:104) P a → γ (cid:105) Chandra as a function of the off-set. The vertical dashed line indicates thedefault NE2001 value of z GC − z SgrA ∗ = − . Figure 3 . Left: The values of log ( (cid:104) P a → γ (cid:105) ) for XMM-Newton (blue) and Chandra (red) as afunction of the scale height H GC of the electron density. Here M = 10 GeV. Right: The ratio (cid:104) P a → γ (cid:105) XMM / (cid:104) P a → γ (cid:105) Chandra as a function of the scale height. The vertical dashed line indicates thedefault NE2001 value of H GC = 26 pc. electron density is suppressed. The significance of this effect is highly dependent on the off-set of the electron density from the galactic centre and the vertical scale height, the valuesof which appear in [23] without error bars. Here, we consider the effects of deviations of thevertical offset of the electron density by 100%, and deviations of the vertical scale height by50% from their default values.The dependence of the averaged conversion probabilities on the off-set of the electrondensity are shown in Figure 2. Regardless of the off-set, XMM-Newton captures regions athigh z with smaller electron densities and thus higher conversion probabilities. The ratioof conversion probabilities – after averaging over the corresponding field of views for XMM-Newton and Chandra – is O (2–5). This corresponds to a line flux ratio of F XMM / F Chandra ∼O (4–11).For a scale height H GC smaller than the default value of 26 pc, more of the field ofviews of both XMM-Newton and Chandra capture low electron density regions. This leads to– 15 –igher averaged conversion probabilities, as is evident from Figure 3. The ratio of conversionprobabilities are again in the range O (1–6). We note that the predicted conversion probabilityis substantially increased by small increases in the electron density offset or small decreasesin the scale height. This would correspond to a much lower predicted value of τ DM M . Whilethe off-set in the y -direction and the in-plane suppression length L GC are also factors of theelectron density, a variation of their values by an O (1) factor has a negligible effect on theconversion probabilities.As mentioned above, only the uppermost end of the observational estimates for thegalactic centre field lead to an obervable signal. To quantify this, we also consider the fol-lowing magnetic field model: An ambient magnetic field of 0.1 m G with N rf cylinders withradius 0.5 pc stretching from z = −
100 pc to z = 100 pc with magnetic field 1 m G corre-sponding to observed radio filaments. We organize the cylinders on a grid in the x - y planewith gridlength 3 pc over a disk with radius 50 pc. The averaged conversion probabilities forthis magnetic field model are (cid:104) P a → γ (cid:105) XMM (cid:104) P a → γ (cid:105) Chandra = (cid:40) . × − . × − = 2 . α r = 0 ◦ . × − . × − = 2 . α r = 45 ◦ , (4.16)while the predicted fluxes become F XMM = 2 . × − photons s − cm − , F Chandra = 6 . × − photons s − cm − , (4.17)where again we have used α r = 45 ◦ to calculate the Chandra flux. As expected, these fluxesare a factor of 100 lower than in the case of a pervasive 1 mG field, corresponding to reducingthe magnetic field strength by a factor of 10. The radio filaments are much too narrow tocontribute to the conversion probability. In this field model, the signal from the Milky Waycentre cannot be explained by DM → a → γ . In this section, we discuss the search for the 3.5 keV X-ray line in galaxies other than the MilkyWay and the inferred constraints on dark matter models. We argue that currently publishedstudies of the X-ray line in other galaxies are consistent with the DM → a → γ scenario,and we also provide a list of target galaxies in the XMM-Newton and Chandra archives withsignificant exposures for which a detection of the 3.5 keV signal is more likely in this scenario. The first search for the 3.5 keV line in a galaxy was that of [2], who detected the line in thecombined XMM-Newton spectrum of the central region of M31. We reviewed this result inSection 1.In [9], stacked Chandra and XMM-Newton observations of galaxies were used to con-strain the proposed sterile neutrino origin of the line. For Chandra and XMM-Newton re-spectively, archived data from a sample of 81 and 89 galaxies with a total exposure of 15.0Ms and 14.6 Ms was considered. The stacking was made so as to optimise sensitivity tolines from decaying dark matter by minimising the X-ray background. To avoid an ICMbackground, no galaxies in clusters or groups with temperature T (cid:38) r ∈ [0 . R vir . , R vir ], where R vir . denotes the estimated virial radius of each galaxy. The resulting X-ray spectra werethen argued to be dominated by instrumental background. To avoid prominent instrumentallines, the search was then restricted to the ranges 2 . . . . σ for the Chandra spectrum and at 11.8 σ for the XMM-Newton spectrum. When the amplitude of the line was left to freely vary, thepreferred amplitude was found to be consistent with zero, thus favouring the model withoutthe additional line.In [8], a search for the 3.5 keV line in stacked XMM-Newton data from 0 . σ or 4.6 σ depending on assumptions.These results of [8, 9] provide strong constraints on scenarios with dark matter decayingdirectly to photons. We now explain how in the scenario of DM → a → γ , no detectable signalwould arise for these searches. → a → γ model in other galaxies In contrast to scenarios in which dark matter decays or annihilates into photons, the observedstrength of the X-ray line in the DM → a → γ scenario depends on the magnetic field along theline of sight to the point of decay. The prospects of observing a signal from galaxies thereforedepend on the structure and strength of the magnetic field in galaxies, which differs stronglybetween different morphological types of galaxies (for a review, see e.g., [47]).While the origin of galactic magnetic fields is not well known, enhancement of smallseed magnetic fields through dynamo mechanisms provide a plausible explanation for theirdevelopment. The magnetic fields in galaxies can be split into contributions from threecomponents. The random field is a short scale, tangled magnetic field, with coherence length ∼
100 pc (the typcial size of a supernova outflow). The random field may be enhanced to µ G strengths by turbulence within the galaxy. Many spiral galaxies also have a regularfield that is coherent over large distances. These may be generated by a mechanism like themean-field dynamo, which produced spiral magnetic fields coherent over significant distancesthrough differential rotation. Indeed, the magnetic field in the disc of a spiral galaxy generallyfollows the pattern of the spiral arms. Finally, galaxies may have striated fields, in which thedirection of the field is coherent over large distances, but the sign of the field is randomisedwith a short coherence length. Striated fields may be generated by the levitation of hotplasma bubbles and their associated random fields, or may arise from the random field bydifferential rotation. In most cases, it is only the regular field component that leads to– 17 –ignificant a → γ conversion – as the regular fields have the largest coherence scales and P ( a → γ ) ∝ L .The observed strong correlation between the total radio continuum emission at cen-timetre wavelengths and the far-infrared luminosity of star forming galaxies, i.e., the ‘radio-infrared correlation’, can be interpreted as a correlation between the field strength of theturbulent component and the star formation rate of the galaxy. Regions with high star for-mation tend to have strong turbulent magnetic fields (indeed, the highest turbulent magneticfields are observed in highly star-forming starburst galaxies). The regular component how-ever is not believed to be positively correlated with the star formation rate: in spiral galaxiesthe ordered magnetic field, combining the regular and striated fields, is in fact strongest inthe inter-arm regions.Dwarf spheroidal galaxies lack both ordered rotation and significant star formation, andconsistent with expectations from the known dynamo mechanism, do not support significantmagnetic fields [47]. Taking a turbulent magnetic field B ∼ µ G (as in [48]) with coherencelength L ∼
100 pc for a dSph of diameter 1 kpc, the small angle approximation gives, P a → γ,dSph ∼ . × − (cid:18) GeV M (cid:19) . (5.1)In the DM → a → γ scenario, decaying dark matter in dwarf spheroidal galaxies will giverise to a 3.5 keV ALP line, but no associated photon line will be observable. The dominantcontribution to the flux from dSphs is therefore from conversion in the Milky Way, which asshown in [12] is too low for an observable signal.Spiral galaxies tend to support ordered (regular and striated) fields. The ordered fieldsare strongest between spiral arms where typical values are B ordered ∼ µ G [47]. Notethat this value includes the contribution from both the regular and the striated fields, whilewe are typically only concerned with the regular field. Within the spiral arms, the magneticfield is mostly tangled and randomly oriented. M31 is unusual in that it has an unusuallycoherent regular magnetic field B reg ∼ µ G whose strength remains constant across thespiral arms. Spiral galaxies, like the Milky Way, may also support magnetic fields in the halosurrounding the disc. Star burst galaxies can support very strong magnetic fields, however,these tend to be tangled over short scales.In the DM → a → γ scenario, we expect no line to be observable from elliptic andirregular galaxies which lack large-scale regular magnetic fields [10]. Spiral magnetic fieldsmay in principle give rise to an observable signal if the regular magnetic field is sufficientlystrong along the path of an ALP arising from dark matter decay. As ALP-photon conversionis suppressed by the plasma frequency, a stronger signal is expected from regions with smallelectron density and significant regular magnetic field. This suggests the inter-arm regions oftypical spiral galaxies will typically contribute more to the photon line than the arm regions.Moreover, edge-on galaxies for which a large fraction of the ALPs travel through a significantfraction of the disc magnetic field should yield a larger signal than face-on spiral galaxies.The magnetic field direction follows the direction of the spiral arms. Therefore, for pathswithin a few kpc of the centre the field is generally transverse to the line of sight, whereasfor paths further from the centre the field becomes parallel to the line of sight and so doesnot contribute to conversion. We therefore predict that the line flux is much stronger foron-centre observations, but may indeed be unobservable off-centre.To quantify this, we simulated the expected signal for hypothetical galaxies with electrondensity and magnetic field such as those of the Milky Way and M31, observed at inclination– 18 – igure 4 . Expected flux vs inclination angle for an M31-like Galaxy angle θ i . We will refer to these models ‘Milky Way-like’ and ‘M31-like’. We assumed thatthese galaxies were located a 1 Mpc from Earth and observed with a circular field of viewwith a 15 (cid:48) radius pointed at the centre of a galaxy (so that the central 4.4 kpc of the galaxyare included in the observation).For the Milky Way-like galaxy we used the recent magnetic field model by [26] withthe central 1 kpc sphere (not considered in [26]) filled in with a 5 µ G poloidal field with anexponential vertical scale height of 1 kpc. We use the electron density given in the thick andthin disc components of [23] (imposing a minumum value of n e = 10 − cm ) and a NFWdark matter distribution [44] with the parameters given in [49]. We note that the MilkyWay magnetic field includes a significant halo component in addition to the disk component,whereas there is no evidence for such a halo component in M31.For the M31-like galaxy, based on [11] we assume a constant azimuthal field of 5 µ G inthe disk cut off at a cylindrical radius of 20 kpc. We assume an exponential fall off above andbelow the disk with a scale height of 2 kpc. This is clearly a vastly simplifed representationof the true field in M31, underestimating the field in the centre and overestimating the fieldon the outskirts, but is sufficient to predict the qualitative relationship between inclinationangle and flux. We use the electron density n e = 0 .
09 cm − × e − (r − . × sech (cid:18) z0 .
14 kpc (cid:19) . (5.2)This is an adapted version of the thin disk component of [23], chosen by considering theelectron density values given in [11]. For the Milky Way-like case, we impose a minimumelectron density of n e = 10 − cm . We assume an NFW dark matter distribution withparameters from [50].For the M31-like galaxy in Figure 4, the peak flux is expected at inclination angle θ i = 90 ◦ (edge-on). The flux for such an edge-on galaxy is over 10 times the flux for anequivalent galaxy with θ i = 0 ◦ (face-on). Note that in this case the magnetic field modelused is symmetric above and below the disc, and so the expected flux will be symmetricaround θ i = 90 ◦ . For the Milky Way-like galaxy in Figure 5, the expected flux is lowerprimarily due to the smaller and less coherent field. Furthermore, rotating the galaxy from θ i = 0 ◦ to θ i = 90 ◦ only increases the flux by a factor of ∼
3. This is due to the significanthalo component of the Milky Way field. The halo component of the Milky Way field is– 19 – igure 5 . Expected flux vs inclination angle for a Milky Way-like galaxy not symmetric above and below the disc, and so the expected flux is not symmetric about θ i = 90 ◦ , and in fact the maximum flux occurs at an inclination angle somewhat above θ i = 90 ◦ .In a search for the DM → a → γ model, Figures 4 and 5 make it clear that we shouldconsider a stacked sample of close to edge-on spiral galaxies. To demonstrate that sucha search is feasible, in Appendix A we provide a list of spiral galaxies with an apparentdiameter of at least 1 (cid:48) with θ i ≥ ◦ that all have significant exposures in either the XMM-Newton or Chandra archives. There are 125 and 143 such galaxies in the Chandra andXMM-Newton archives with total raw exposures of 7.1 Ms and 8.7 Ms, respectively. Asignificant fraction of these exposure times may well be used to search for the 3.5 keV line.In a search optimised for the DM → a → γ model, the masking of the field of view woulddiffer from that used in [9]. For distant galaxies, the whole galaxy might fit in the field ofview, whereas we only expect an observable signal from the central region. In our case, theouter regions of galaxies should be masked and observations instead focused on the centralregions. If it were observationally possible, galaxies with high regular magnetic fields shouldbe preferred. However for more distant galaxies their regular magnetic field is unknown andthis would not be practical. The DM → a → γ scenario represents an attractive and testable proposal to explain the 3.5keV line emission, assuming it is of dark matter origin. At the current time this scenario isconsistent with all observations, and can explain discrepancies that cannot be accounted forin models of dark matter directly decaying or annihilating into photons. In this paper wehave further elucidated the phenomenology of this scenario.In the galactic centre region, we have studied the conditions under which this scenariocan generate a 3.5 keV line of the strength observed in [4, 5]. This turns out to be just We consider galaxies for which the sum of XMM-Newton and Chandra exposure is at least 5 ks. We also note that the lack of precise knowledge on the galactic magnetic fields imply that one cannotdecisively rule out the model based on such a search; a definitive exclusion would require knowledge of themagnetic fields. As discussed in both [4] and [5], it is of course possible that the galactic centre line is simply an astro-physical K XVIII line and there is no dark matter signal. – 20 –ossible – provided the magnetic field in the galactic centre is at the highest end of observa-tional estimates. It also requires the ‘average’ ALP-to-photon conversion probability for the73 cluster sample of [1] to be slightly smaller than assumed in [10]. In this case, the scenariogenerates a highly distinctive morphology, in which the signal is highly suppressed within 20pc of the galactic plane. This morphology can be easily tested by re-analysing the data usedin [4, 5] and masking the region close to the galactic plane.We have also considered samples of distant galaxies, and have further quantified thequalitative statement in [10] that edge-on spiral galaxies are the most attractive galaxiesfor dark matter searches in this scenario. To this end we have also provided a list of tar-get galaxies with significant archival observational time in the XMM-Newton and Chandraarchives.
Acknowledgments
JC thanks the Royal Society for a University Research Fellowship. JC, FD, DM, MR arefunded by the ERC Starting Grant ‘Supersymmetry Breaking in String Theory’. PDA issupported by CONICYT Beca Chile 74130061. Portions of this work have been presentedat the ‘Particle Cosmology after Planck’ workshop at DESY in September 2014. We thankStephen Angus, Alexey Boyarsky, Thorsten Bringmann, Jeroen Franse, Carlos Frenk, TeslaJeltema, Andrew Powell, Stefano Profumo, Signe Riemer-Sørensen, Oleg Ruchayskiy fordiscussions and correspondence. DM is grateful to Birzeit University for kind hospitalitywhile finishing the paper.
A List of nearly edge-on spiral galaxies with long X-ray exposures
We here list a set of galaxies with a large apparent diameters, which have inclination angles θ i ≥ ◦ and exposures with XMM-Newton and Chandra of at least 5 ks. These galaxieswould constitute natural targets for a search for the 3.5 keV line from the DM → a → γ scenario. In compiling this list, we only considered observations centred within 2 (cid:48) of thetarget galaxy. Here, n CXO and n XMM denotes the number of such observations available inthe archives of Chandra and XMM-Newton , respectively.
Table 3 : List of nearly edge-on spiral galaxies with long X-rayexposures
Galaxy Type θ i n CXO t CXO [ks] n XMM t XMM [ks]
ESO602-031 SBb
IC2163 Sc
IC2560 SBb
IC2574 SABm
IC2810 SBab
NGC0224 Sb
NGC0253 SABc
NGC0520 Sa
NGC0625 SBm
NGC0660 Sa
Continued on next page Using the function logdc of the Hyperleda database, http://leda.univ-lyon1.fr , [51]. – 21 – able 3 –
Continued
Galaxy Type θ i n CXO t CXO [ks] n XMM t XMM [ks]
NGC0891 Sb
NGC0931 Sbc
NGC1808 Sa
NGC2683 Sb
NGC2798 SBa
NGC2799 SBm
NGC2841 Sb
NGC2903 SABb
NGC2992 Sa
NGC3034 Scd
NGC3079 SBcd
NGC3221 Sc
NGC3396 SBm
NGC3623 SABa
NGC3627 SABb
NGC3628 Sb
NGC3877 Sc
NGC3972 SABb
NGC4013 Sb
NGC4039 SBm
NGC4224 Sa
NGC4244 Sc
NGC4258 SABb
NGC4388 Sb
NGC4395 Sm
NGC4490 SBcd
NGC4565 Sb
NGC4569 SABa
NGC4631 SBcd
NGC4666 SABc
NGC4698 Sab
NGC4945 SBc
NGC5005 SABb
NGC5170 Sc
NGC5253 SBm
NGC5506 Sa
NGC5746 SABb
NGC5775 SBc
NGC5793 Sb
NGC5907 SABc
NGC6118 Sc
NGC7090 Sc
NGC7212 Sb
NGC7331 Sbc
NGC7582 SBab
NGC7590 Sbc
NGC7771 Sa
PGC014370 Sc
PGC037477 Sb
PGC044990 Sc
Continued on next page – 22 – able 3 –
Continued
Galaxy Type θ i n CXO t CXO [ks] n XMM t XMM [ks]
PGC046710 SBb
PGC093080 Sc
PGC1110773 Sab
UGC12915 SBc
ESO069-006 SBb
ESO137-001 SBc
ESO244-030 SABb
ESO293-034 SBc
ESO415-029 Sbc
ESO430-020 SABc
ESO432-006 Sbc
IC0564 Scd
NGC0024 Sc
NGC0055 SBm
NGC0988 Sc
NGC1589 Sab
NGC1741 Sm
NGC2552 SABm
NGC2748 Sbc
NGC2770 SABc
NGC2783B Sb
NGC3190 Sa
NGC3198 Sc
NGC3287 SBd
NGC3556 SBc
NGC3621 SBcd
NGC3683 SBc
NGC3718 Sa
NGC4088 SABc
NGC4178 Scd
NGC4216 SABb
NGC4217 Sb
NGC4236 SBd
NGC4355 SABa
NGC4419 Sa
NGC4438 Sa
NGC4527 SABb
NGC4772 Sa
NGC4848 Sc
NGC4939 Sbc
NGC5394 SBb
NGC5395 SABb
NGC5674 SABc
NGC6027C SBc
NGC6503 Sc
NGC6872 SBb
NGC6925 Sbc
NGC7541 SBc
NGC7591 SBbc
NGC7673 Sc
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Continued
Galaxy Type θ i n CXO t CXO [ks] n XMM t XMM [ks]
NGC7753 SABb
PGC001221 SBc
PGC019078 E?
PGC027508 SBab
PGC038430 Sd
PGC046114 Sbc
PGC046133 Sbc
PGC086247 Sbc
PGC100170 SBbc
PGC2793298 Sa
UGC01934 Sbc
UGC02238 Sm
UGC02626 Sa
UGC03326 Sc
UGC03995 Sbc
ESO121-006 Sc
ESO140-043 SBb
ESO154-023 SBm
ESO195-005 Sa
ESO208-034 SBab
ESO209-012 Sa
ESO365-001 Sc
ESO365-016 SBab
ESO471-006 SBm
ESO491-021 SBab
IC1504 Sb
IC1537 Sc
IC1959 SBm
IC4518A Sc
IC4518B Sc
IC5052 SBcd
NGC0092 Sa
NGC0192 SBa
NGC0675 Sa
NGC0716 Sa
NGC0784 SBd
NGC1134 Sb
NGC1311 SBm
NGC1320 Sa
NGC1511 Sab
NGC1512 Sa
NGC2369 Sa
NGC2613 Sb
NGC3044 SBc
NGC3227 SABa
NGC3281 Sab
NGC3735 Sc
NGC3746 Sab
NGC3753 Sab
NGC3786 SABa
Continued on next page – 24 – able 3 –
Continued
Galaxy Type θ i n CXO t CXO [ks] n XMM t XMM [ks]
NGC3788 SABa
NGC3976 SABb
NGC4157 SABb
NGC4173 SBcd
NGC4235 Sa
NGC4302 Sc
NGC4319 SBab
NGC4330 Sc
NGC4437 Sc
NGC4536 SABb
NGC4605 SBc
NGC4634 SBc
NGC4686 Sa
NGC4700 SBc
NGC4845 Sab
NGC5073 SBc
NGC5356 SABb
NGC5899 Sc
NGC6045 SBc
NGC6323 Sab
NGC6810 Sab
NGC6814 SABb
NGC6926 Sc
NGC7314 SABb
PGC006966 Sc
PGC012596 Sb
PGC013944 SBab
PGC014121 S?
PGC023515 Sa
PGC026440 SABc
PGC037282 Scd
PGC044532 Sm
PGC053471 Sc
PGC061664 Sb
PGC063176 SABa
PGC064775 Sd
PGC065349 SABa
PGC066146 Sc
PGC074302 SABa
PGC402573 Sbc
UGC00717 Sb
UGC00987 Sa
UGC08515 Sab
UGC09944 Sbc
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