Investigating the detection of dark matter subhalos as extended sources with Fermi-LAT
LLAPTH-030/20
Investigating the detection of dark matter subhalos as extended sources with
Fermi -LAT
Mattia Di Mauro,
1, 2, ∗ Martin Stref, † and Francesca Calore ‡ NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA The Catholic University of America, Department of Physics, Washington DC 20064, USA Univ. Grenoble Alpes, USMB, CNRS, LAPTh, F-74000 Annecy, France (Dated: July 20, 2020)Cold dark matter (DM) models for structure formation predict that DM subhalos are present inthe Galaxy. In the standard paradigm of DM as weakly interacting massive particle, subhalos areexpected to shine in gamma rays and to provide a signal detectable with current instruments, notablywith the Large Area Telescope (LAT) aboard the
Fermi satellite. This is the main motivation behindsearches for DM signals towards dwarf spheroidal galaxies and unidentified
Fermi -LAT sources. Asignificant angular extension detected from unassociated sources located at relatively high latitudesis considered a “smoking gun” signature for identifying DM subhalos. In the present work, wesystematically explore, by means of state-of-the-art models of cold DM halos in the Galaxy, thedetectability of extended subhalos with
Fermi -LAT. We simulate a DM signal exploring differentassumptions of subhalos distribution in the Galaxy and DM profile, and reconstruct its flux througha realistic
Fermi -LAT analysis pipeline. In the most optimistic case, we show that a detection of extended
DM subhalos can be made for annihilation cross sections higher than 3 × − cm /s(for a 100 GeV DM mass), still compatible with existing gamma-ray constraints, and that, in thiscase, the preference for extension of the source (vs point-like hypothesis) is significant. For faintersignals, instead, halos not only do not show significant extension, but they are not even detectablesignificantly as point-like sources. I. INTRODUCTION
Unveiling the nature of dark matter (DM) remains oneof the major challenges for particle physics and cosmol-ogy. Despite the achievements on the theoretical andexperimental side, the standard paradigm of DM beingmade of weakly interacting massive particles (WIMP) [1]is challenged by null results of detection of these elusiveparticles with current experiments. In particular, indi-rect detection of DM signals with high-energy photonsstrongly constrains the WIMP DM parameter space [2].Since they are expected to have a low astrophysicalbackground and are predicted to be dynamically domi-nated by DM, dwarf spheroidal galaxies represent promis-ing targets for DM identification [3]. Besides, DM halosthat cannot form stars are predicted to exist in cold DMscenarios of structure formation. While such objects are“dark” for optical telescopes, gamma-ray instruments mayunveil DM signals emitted therein. Searches for DM insubhalos, either in the faintest detectable dwarf galaxiesor in their “dark halos” counterparts, represent a powerfultest for the WIMP paradigm.Typically, searches towards known dwarf galaxies, aswell as searches for DM subhalos have been performedunder the assumption that the emitted DM signal is ∗ [email protected] † [email protected] ‡ [email protected] point-like (with respect to the angular resolution of theinstrument). Both data-driven, e.g. [4, 5], and template-based, e.g. [6], searches towards known dwarf spheroidalgalaxies look for excess(es) of photons above the atrophys-ical background, compatible with point-like DM signal(s)from the dwarf(s) direction(s). Analogously, most of thesensitivity predictions for DM subhalo detectability [7–9], as well as searches for DM subhalos in unidentifiedsources [10, 11], treat DM subhalos as point-like objects.Nevertheless, DM subhalos may have a significant an-gular extension in the sky, depending on their positionand mass profile. The detection of angular extension ofunidentified high-energy gamma-ray sources located atlatitudes | b | > ◦ has been advocated to be a “smokinggun” signature of DM subhalos [12], and it has recentlyreceived more and more attention in the literature. Inparticular, analyses of angular extension of unidentifiedsources detected by the Large Area Telescope (LAT),aboard the Fermi satellite, were performed without muchsuccess: No extended halo was found (or finally confirmed)around
Fermi -LAT unidentified sources [10, 13–15]. Alsoa complementary approach, looking for optical counter-parts of
Fermi -LAT extended and unidentified sources inGAIA data, gave null results [16]. Also, recent sensitiv-ity predictions for DM subhalo identification with futuregamma-ray instruments included a spatial analysis of DMhalos [17, 18].Although searches for extension in real data have beencarried out, to the best of our knowledge there is a lack of acomplete analysis of the detectability of angular extension a r X i v : . [ a s t r o - ph . H E ] J u l of DM subhalos. It is not clear, for example, if the Fermi -LAT has the capability of detecting subhalos as extendedand, if yes, for which DM particle physics parameters(notably, the annihilation cross section) – there is indeedno a priori reason why the
Fermi -LAT sensitivity to DMextended subhalos should be the same as for point-likesubhalos, as for example derived in [7]. In the presentwork, for the first time we address this issue and quantifywhat is the impact of modeling DM subhalos as fullyextended objects. To this end, we rely on semi-analyticalmodels for the distribution and statistics of DM subhalosin the Galaxy, which account for Milky-Way dynamicalconstraints and include tidal effects which the subhalosare subject to when moving in the Galactic gravitationalpotential [19]. Such models do not distinguish between“dark subhalos” and optically detectable ones (i.e. dwarfgalaxies), since they do not implement any recipe forgalaxy formation. In what follows, we therefore indicateas “subhalo” every DM substructure present in the Galaxy.Our final goal is to quantify the sensitivity of
Fermi -LATto the brightest extended
DM subhalo, and, ultimately,understand how to use cold DM predictions to identifyDM subhalo candidates in unidentified sources exploitingangular information.In Sec. II, we describe models and statistics of theGalactic subhalo population. In particular, we stress theimportance of a correlation between intensity of the pre-dicted DM signal and angular extension of DM subhalos.In Sec. III A, we illustrate the setup to simulate
Fermi -LAT data and the analysis detection pipeline we follow.We present our results in Sec. IV, and conclude in Sec. V.
II. SUBHALO MODELS AND STATISTICS
In this section, we describe the DM subhalo populationmodels we use, as well as the mock subhalo catalogsgenerated from these models.
A. The subhalo models
Our analysis is based on the semi-analytical subhalomodel developed by Stref & Lavalle [19] which is referredto as SL17 from now on. SL17 is built upon the realis-tic Milky-Way mass model developed by McMillan [20]in which the Galactic dark halo is assumed to have aNavarro-Frenk-White (NFW) [21] density profile shape.Cold DM subhalos are expected to have cuspy densityprofiles, and the profile shape can be chosen freely in themodel. In the following we consider either NFW subhalosor Einasto subhalos (with α Ein = 0 .
16 based on [22]).Subhalos are subject to tidal effects as they orbit in thegravitational potential of the Galaxy and its DM halo.Two distinct effects are accounted for in SL17: The tidalmass loss due to the smooth gravitational potential ofthe Galaxy, and the effect of gravitational shocking ex-perienced by a subhalo crossing the Galactic stellar disk. Both these effects strongly impact the subhalo popula-tion by stripping off mass from these objects, sometimesdestroying them completely. The efficiency of this de-struction is still a matter of debate. Studies based oncosmological simulations find that subhalos are efficientlydisrupted in the inner parts of the Galaxy [23, 24]. Onthe other hand, recent semi-analytical studies find thatcuspy subhalos such as those predicted by cold DM arevery resilient to tides and can survive considerable masslosses [25, 26]. Also, the disruption observed in cosmo-logical simulations could be due to numerical artifacts[27]. Whether subhalos can be disrupted or not has con-sequences on predictions for DM searches, in particularindirect searches for self-annihilating DM because theannihilation rate is very high in cuspy subhalos [28]. Inthe present work, we remain agnostic about the resilienceof subhalos to tides and treat it as a theoretical uncer-tainty for our predictions. We bracket this uncertaintyby considering two extreme configurations of the SL17model. The “SL17-fragile” configuration corresponds towhat is commonly observed in cosmological simulations,i.e. subhalos are efficiently disrupted by tides. In the“SL17-resilient” configuration, on the other hand, subhaloscan lose most of their mass but the central cusp almostalways survives. More precisely, in the SL17-fragile con-figuration it is assumed that a subhalo is disrupted assoon as its tidal radius is smaller than its scale radius r t (cid:54) r s . In the SL17-resilient configuration, on the otherhand, disruption only takes place if r t (cid:54) . r s . Theseconfigurations were originally defined in [28] which werefer the reader to for additional details. B. The subhalo mock population
The SL17 model gives a statistical description of theGalactic subhalo population. More precisely, it providesa recipe to compute the probability distribution func-tion (PDF) of various subhalo parameters (mass m ,concentration c and position). This model is fully im-plemented in the CLUMPY public code [29–31], which canbe used to generate mock subhalo population catalogsstarting from these parameters’ PDFs. Each of thesecatalogs is therefore a realization of the Galactic subhalopopulation based on the SL17 model.
CLUMPY also com-putes the J -factor of each subhalo, i.e. the integral alongthe line of sight (l.o.s.) of the DM density squared: J (∆Ω) = (cid:90) ∆Ω0 (cid:90) l . o . s . ρ d l dΩ (1)where ρ DM is the subhalo mass density, and ∆Ω = 2 π (1 − cos( θ )) is the solid angle for a viewing angle θ . The J -factor appears in the expression of the gamma-ray fluxproduced by DM annihilation. The total J -factor, i.e. the J -factor integrated up to the full angular extension of asubhalo (i.e. its tidal radius), is labeled as J tot .For a subhalo of radius R and at a distance d fromthe observer, we define the total angular size as: θ tot ≡ arcsin( R/d ). The angular size of a DM subhalo is there-fore a geometric consequence of the subhalo mass profileand its distance. In the top panel of Fig. 1, we show that,in the subhalo catalogs, there exists a correlation betweenthe subhalo J tot and its total angular size on the sky, seealso [10]. While we are not interested in parameterizingsuch a correlation nor we directly use it in the following,we can generally conclude that subhalos with the highest J -factors also show a significant angular extension of upto a few degrees. This suggests that the DM subhaloswith the highest gamma-ray flux could be detected as extended sources rather than point-like objects. Previousworks have mostly focused on the analysis of DM subhalodetectability in the case of point-like sources. However, ifthe brightest subhalo is indeed extended in the sky – asthe correlation suggests – the Fermi -LAT sensitivity tosubhalos may be different. Here, we aim at quantifyingwhether or not a search for extended sources improves de-tection prospects. To do so, we consider the distributionof the brightest subhalo, i.e. the subhalo with the highest J -factor, J (cid:63) tot .We generate 1010 mock population catalogs for eachof the two model configurations, SL17-fragile and SL17-resilient. We perform a latitude cut in the catalogs, dis-carding all subhalos with | b | < ◦ (rejecting on average17% of subhalos with J > GeV / cm ), and identify,in each Monte Carlo realization, the subhalo with the high-est J -factor among the remaining ones – and so only onesubhalo for each Monte Carlo realization. We then com-pute the PDF of the J -factor of the brightest halo for bothconfigurations and show the result in the bottom panel ofFig. 1. If subhalos have a NFW profile, the SL17-fragilePDF peaks around J (cid:63) tot ∼ . × GeV / cm whilethe SL17-resilient PDF peaks at J (cid:63) tot ∼ GeV / cm .The lower J (cid:63) tot in the SL17-fragile case compared to theSL17-resilient case comes mainly from the distance tothe brightest object. The stellar disk is very efficient atstripping mass from subhalos. While this is fatal to mostclumps passing through the disk in the SL17-fragile sce-nario, in the SL17-resilient case subhalos can still surviveand remain close to the Solar system. If subhalos havean Einasto profile, the J -factors increase by a factor ofroughly 1 . We also compute the PDF of the angular size associatedto the brightest subhalos. The PDF of θ (cid:63) tot , i.e., of the to-tal angular size of the brightest subhalo in each simulation,is shown on the top panel of Fig. 2. In the SL17-fragilecase, the brightest subhalo typically has θ (cid:63) tot ∼ ◦ andthe PDF is rather narrow, while θ (cid:63) tot ∼ ◦ for the SL17-resilient model and the PDF is much broader. Note that Note that we did not generate subhalo catalogs for the Einastoprofile case. Instead we only generate catalogs for the NFWcase, find the brightest subhalo in each catalog and extract itsparameters (mass m , concentration c and position), thencompute the J -factor that would have a subhalo with an Einastoprofile with identical parameters. the choice of profile, NFW or Einasto, does not affectsignificantly neither the subhalo’s radial extension norits position. The PDF value of θ (cid:63) tot is thus the sameregardless of the density profile shape. On the bottompanel in Fig. 2, we show the PDF of θ (cid:63) which is definedwith respect to the radius enclosing 68% of the total J -factor. For both subhalo models, the PDF is centered onvalues smaller than 1 ◦ . This is what is expected whencomputing the radius enclosing 68% of the total J -factor.For an NFW subhalo with tidal radius r t (cid:29) r s , we have θ (cid:39) arcsin( r s / (2 d )). For a 10 M (cid:12) subhalo at a distanceof 10 kpc, this is θ (cid:39) . ◦ . For the NFW density profile, θ (cid:63) is 0 . ◦ and 0 . ◦ for the SL17-resilient and SL17-fragile models, respectively. The slightly larger extensionof SL17-resilient subhalos compared to SL17-fragile onesis, like their higher J -factor, due to their proximity andnot to their spatial extension. In fact, the brightest re-silient subhalo has in general a smaller tidal radius thanthe brightest fragile subhalo although the angular exten-sion on the sky is larger. The central value of θ (cid:63) isslightly smaller for Einasto compared to NFW.The two subhalo density profiles we consider, NFWand Einasto, are both cuspy. One can wonder what the J -factor and angle PDFs would be for subhalos with acored profile. The SL17 is tailor-made to handle cold DMsubhalos as it partly relies on results from cold DM cos-mological simulations. Since subhalos have cuspy profilesin these simulations, the model cannot deal with coredsubhalos in a consistent way, however we can point outsome expected differences. First, a cored subhalo with agiven m and c is less dense than a cuspy subhalowith the same parameters therefore its J -factor is smaller.Second, a lower density also means that cored subhalosare far more susceptible to tidal stripping and disruption,so subhalos in a cored scenario would be less numerousand less extended. We therefore leave aside any quanti-tative estimate for cored subhalo profiles, which wouldrequire to run dedicated simulations.As mentioned already in the introduction, we have nodirect information from the simulation for classifying asubhalo as dwarf galaxy or “dark satellite”. Nevertheless,we know that, to trigger star formation, a DM subhaloshould have a mass of around 10 − M (cid:12) , dependingon the hydrodynamic simulation, see for example [32].If we look at the mass PDF of the brightest subhalo,we realize that, in the SL17-fragile model, the brightesthalo has a mass typically around 10 − M (cid:12) , and so itshould definitely form a dwarf galaxy. On the other hand,in the SL17-resilient model, the mass of the brightestsubhalo can be lower (down to 10 M (cid:12) ), so the halo willnot necessarily form a dwarf galaxy. In this case, becausethe halo is quite close (closer than known dwarf galaxies),the J -factor can still be very high. Therefore, whetherthe brightest halo in the simulation is a dwarf galaxydepends on the subhalo model (SL17-fragile vs SL17-resilient). We stress that the nature of the subhalo, beingit a dwarf galaxy or optically dark, does not affect theconclusions reached in the present paper. The possibility θ tot [ ◦ ]1718192021 l og ( J t o t [ G e V / c m ] ) SL17 − resilientSL17 − fragileSL17 − resilientSL17 − fragile
18 19 20 21 22log ( J ? tot [GeV / cm ])0 . . . . P D F SL17 − resilientSL17 − fragileNFWEinastoNFWEinasto FIG. 1.
Upper panel:
Correlation between J -factor andtotal angular size on the sky for subhalos in two differentmodels (only one realization for each model is shown). Lowerpanel: J -factor PDF of the brightest subhalo, J (cid:63) tot . for an extended gamma-ray signal to have a dwarf galaxyoptical counterpart, instead, can contribute to firmlyidentify it as DM subhalo [16]. III. SIMULATIONS OF
FERMI -LAT DATA
In this section we explain the setup we use to simulate
Fermi -LAT data, the analysis pipeline and the statisticalframework that we consider to calculate the significanceof the detected signal.
A. Data simulation, background and signal model
We run the full analysis on mock LAT data, realisti-cally simulating background models and the instrumentresponse function, and using state-of-the-art detectionpipelines.For simulating and analyzing the data, we use
FermiPy , θ ? tot [ ◦ ]0 . . . . P D F SL17 − resilientSL17 − fragileSL17 − resilientSL17 − fragile . . . . . θ ? [ ◦ ]01234 P D F SL17 − resilientSL17 − fragileNFWEinastoNFWEinasto FIG. 2.
Upper panel:
PDF of the total angular size of thebrightest subhalo, θ (cid:63) tot . Lower panel:
Same as the upperpanel for the angle containing 68% of the total J -factor, θ (cid:63) . which is a Python package that automates analyses withthe Fermitools [33] . FermiPy is designed to performseveral high-level analyses of LAT data such as generatingsimulations, detecting sources, calculating spectral energydistributions (SED) and finding the source extension. Weemploy the
Fermipy version and the
Fermitools version .We simulate 11 years of gamma-ray data, from 2008August 4 to 2019 August 4 in the energy range E =[1 , SOURCEVETO event class, and use the corresponding instru-ment response function
P8R3 SOURCEVETO V2 . When ana-lyzing the data, we select photons passing standard dataquality selection criteria . The simulations of gamma-raydata is performed with the simulate roi tool. Given a See http://fermipy.readthedocs.io/en/latest/ . https://fermi.gsfc.nasa.gov/ssc/data/analysis/documentation/Cicerone/Cicerone_Data_Exploration/Data_preparation.html model map (see below for the current model specifica-tions), this tool takes as input the predicted number ofcounts for the model, and generates simulated data binnedin energy and space. We bin the simulated data with 8energy bins and angular pixels of size 0 . ◦ . Using the op-tion randomize=True it is possible to randomize the datausing Poisson statistics. We will use randomize=False ,otherwise differently stated, because we want to test theideal case of a perfect knowledge of the background com-ponents.We generate mock data sky realizations of a given regionof interest (ROI), where we want to test the background-only hypothesis and the background plus signal hypothesis.We consider two different ROIs, representative of typicalbackground configurations at high Galactic latitudes. Wedefine an ROI of 12 ◦ × ◦ centered at ( l = 150 ◦ , b = 60 ◦ )for the simulations labelled high-latitude , and at ( l =40 ◦ , b = 20 ◦ ) for the simulations labelled low-latitude .The astrophysical background model includes the Galac-tic diffuse emission model, point-like and extended sourcesselected from the 4FGL catalog [34], and an isotropic emis-sion component. In particular, we use the Galactic diffuseemission and isotropic templates released, as official an-cillary files, with the 4FGL catalog .The signal model is represented by a DM subhalo,centered at the center of the ROI (either high- or low-latitude ). The spectrum of the DM injected signal isnormalized by the thermally averaged annihilation crosssection, (cid:104) σv (cid:105) , and depends on the mass of the DM particle(we test masses of 10, 100, 1000 GeV). We use a benchmarkannihilation channel into b -quarks. We vary the valueof the annihilation cross section from 10 − up to 10 − cm /s, to check how the detection sensitivity changeswith the brightness of the signal. The spatial distributionof the DM signal is built from Eq. (1). In order to getan estimate of the uncertainties at play, we select onehundred subhalos within 1 σ of the mean of the J -factorPDF shown in Fig. 1. The analysis is then repeated byusing the spatial template corresponding to each of these J -factors, as injected signal. We consider four differentmodels varying the impact of tidal disruption (SL17-fragileor SL17-resilient) and the subhalo density profile (NFWor Einasto).If not otherwise specified, we adopt as baseline con-figuration for signal injection the SL17-resilient subhalomodel, an NFW DM subhalo density, a DM mass of 100GeV, and the high-latitude ROI. https://fermi.gsfc.nasa.gov/ssc/data/access/lat/8yr_catalog/ https://fermi.gsfc.nasa.gov/ssc/data/analysis/scitools/source_models.html B. Signal reconstruction models
We perform a fit on simulated data using the gta.fit tool, which is a wrapper of the pyLikelihood fit methodimplemented in the
Fermitools . This tool returns thebest fit and error of SED parameters and the full covari-ance matrix of the model. From the fit, we extract thevalue of the log-likelihood to estimate the significanceof the detection of the DM signal with respect to thebackground-only hypothesis.In order to reconstruct the injected DM signal andstudy the detectability of DM subhalos, we make different(spectral and spatial) assumptions on the reconstructedsignal. Since, in real data analyses we cannot know thedistance and density profile parameters of the subhalo, wetry first to detect it as a point-like source ( PS ). The fittedSED is a power law, and the free model parameters arenormalization and spectral index. Secondly, we test thehypothesis of an extended source ( Ext ) for which we usea radial Gaussian as spatial template, and a power-lawSED. In this case, the free parameters are the same as the PS case with, in addition, the size of extension, namelythe width of the radial Gaussian spatial model. For theextended models, the source extension is computed usingthe gta.extension tool that performs fits to the datawith different sizes of Gaussian and then maximizes thelog-likelihood as a function of this parameter, by leavingthe SED parameters free as well.In addition, we test a case in which we have a point-likesource plus an extended source, centered at the sameposition ( PS+Ext ). Both components have SED modeledas power laws . In this case, the free parameters are nor-malizations, spectral indices and sizes of extension of thetwo components. We summarize models and parametersin Tab. I. For all the model tested we do not vary theposition of the source.In Fig. 3, we compare the angular profiles, calculated asthe surface brightness (number of counts per solid angleas a function of the angular distance from the ROI center),of the three models adopted for the signal reconstruction,together with the angular profile of the DM injectedsignal. For each source template, we compute the numberof counts for concentric annuli in angular distance and wedivide the number of counts in each annulus by its solidangle. We obtain with this method the surface brightnessof the signal. All templates have been convoluted withthe instrument angular resolution (point spread function)and are normalized such as to match the DM injectedsignal at the peak. The PS case provides the poorest fitto the DM emission. In fact, this model produces a flux A priori the SED of the two components should be constrainedto be unique, in order to claim that this is a single source. Thisbeing of impractical implementation in
FermiPy , we checked aposteriori that, indeed, the best-fit spectral indices are compatiblewithin 1 σ . We also notice that the best-fit SED of the PS and Ext models are compatible. d N d [ / s r ] DMExtPSPS+Ext
FIG. 3. Surface brightness angular profile of the injectedDM signal (black crosses), compared with the angular profilesof the best-fit PS (blue circles), Ext (red stars), and
PS+Ext (green plus signs) models.TABLE I. Models used for the reconstruction of the DM sig-nal: Model name, SED parameterization, spatial distribution(morphology), and number of free parameters in the fit.Model Spectrum Morphology No. params PS power-law point-like 2 Ext power-law radial Gaussian 3
PS+Ext power-law point-like + radial Gaussian 5 comparable with the injected signal only in the inner 0 . ◦ ,instead at higher angular distances the surface brightnessis much smaller than the DM signal. Instead, the Ext starts to deviate significantly from the DM injected signalat distances > ◦ and also in the inner 0 . ◦ it slightlyunderestimates the flux. On the other hand, the PS+Ext case fits well the DM signal up to ∼ . ◦ . Also, it wellmatches the DM injected signal in the inner 0 . ◦ . A pointsource plus and extended source can therefore provide thebest fitting model of the DM injected signal. We will testthis possibility in the next section. C. Statistical significance
Our null hypothesis ( H ) is defined by the background-only model, when we fit the simulated data without anyadditional DM signal. The alternative hypothesis is in-stead represented by our reconstructed signal templates(as described above), through which we test the pres-ence of an additional source on top of the astrophysicalbackground.The reconstructed signal models are all nested modelsfor which Wilk’s theorem [35] usually applies. As usuallydone, we define the Test Statistics as T S = 2 (log L H − log L H ). However, Wilk’s theorem cannot be applied if TABLE II. Number of restricted and unrestricted ( n ; k ) pa-rameters used in Eq. 2, for the calculation of the statisticalsignificance . PS Ext PS+Ext H (1; 1) (1; 2) (2; 3) PS – (1; 1) (1; 2) Ext – – (1; 1) some of the parameters of the test hypothesis (in the limitof the null hypothesis) take values on the boundary ofthe allowed parameter space. As an example, for the PS case, the free fit parameters are the normalization and thespectral index of the SED. This model reduces to the nullhypothesis when the normalization tends to zero, whichcorresponds to the lower bound of its permitted value. Ifthis is the case, the T S distribution is given by a mixeddistribution, which depends on the number of parameterswhose null value is restricted to be at the boundary ofthe allowed range, and on those which are not. Following[36, 37] we can calculate the p -value for a given T S as: p ( T S ) = 2 − n (cid:32) δ ( T S ) + n (cid:88) i =1 (cid:18) ni (cid:19) χ i + k ( T S ) (cid:33) , (2)where n is the number of restricted parameters (i.e. theparameters that have a boundary condition in the limit ofthe null hypothesis) and k is the number of unrestrictedparameters.We report in Tab. II, the number of restricted andunrestricted parameters for the hypotheses we compareone with respect to the other. IV. RESULTS
As described in Sec. III A, we create mock data makingdifferent assumptions on the DM injected signal, besidestesting different source models in the fit. In Fig. 4, weshow the detection significance (significance of the sourcemodel with respect to H ) for the different signal recon-struction templates ( PS , Ext , PS+Ext ), and as a functionof the injected signal cross section. For illustrative pur-poses only, we add the case DM , in which we fit mockdata with the same DM template used to simulate them,leaving free only the overall normalization of the signal(one restricted parameter). The subhalo model adoptedhere is SL17-resilient, and we assume an NFW densityprofile for the subhalos. From this figure we can seeat first that, even in the optimistic (as well as unreal-istic) case in which we know everything about subhaloproperties and position ( DM ), we could reach a detectionsignificance larger that 3 σ (marginal hint) for annihilationcross sections above 3 × − cm /s. Such cross sections(for annihilation into b -quarks and DM mass of 100 GeV,i.e. our reference case) are still allowed by current con-straints coming from the observation of dwarf spheroidal v [cm /s]02468101214 [ s i g n i f i c a n c e ] NFW, SL17-resilientDMPS ExtPS+Ext
FIG. 4. Detection significance as a function of the injectedsignal annihilation cross section, for different signal reconstruc-tion models ( PS in blue, Ext in red,
PS+Ext in green, DM inblack). The cross sections used for signal injection correspondto the abscissas of the black points; for the other cases theshift along the x axis is for visual ease only (this is true forall other plots in the paper). The subhalo model adopted isSL17-resilient, and we assume an NFW density profile for thesubhalos. galaxies [4, 38], as well as of the Galactic halo at high lat-itudes [39, 40]. Moreover, given the similar results we getfor point-like and extended templates, such a sensitivityestimate is compatible with what is found for the Fermi -LAT sensitivity to point-like DM subhalos, e.g. [7, 9]. Afirm detection (above 5 σ , without accounting for look-elsewhere effects) would instead need cross sections atleast as high as 5 − × − cm /s – which, again, is notexcluded by current gamma-ray constraints. Below the3 σ DM detection significance threshold, all models providecomparable evidence for DM subhalos, as expected. Forlow cross sections the log-likelihood for the PS and Ext cases are very similar. Since the PS template has lessparameters, it gives a slightly higher detection signifi-cance. Above cross sections of 3 × − cm /s insteadthe extended template, Ext , starts to provide the best fitamong the three reconstructed signal models, with the
PS+Ext model giving comparable detection significance.If not stated otherwise, in what follows, we presentresults for the
Ext template. The
PS+Ext case wouldproduce very similar results – with < σ improvementof the fit when adding a point source component to theextended source for cross sections below 2 × − cm /s.As presented in Sec. II B the SL17-resilient and SL17-fragile models bracket the uncertainty in the modelingof tidal disruption of Galactic DM subhalos. In Fig. 5we compare the detection significance obtained with thetwo subhalo models (for an NFW DM subhalo densityprofile). The SL17-resilient model provides a much higherdetection significance regardless of the injected signalcross section. The difference in significance at fixed cross v [cm /s]02468101214 [ s i g n i f i c a n c e ] NFW, ExtSL17-resilient SL17-fragile
FIG. 5. Detection significance for the
Ext signal reconstructionmodel comparing SL17-resilient (black) and SL17-fragile (blue)subhalo models. v [cm /s]02468101214 [ s i g n i f i c a n c e ] Ext, SL17-resilient
NFW Einasto
FIG. 6. Detection significance for the
Ext signal reconstructionmodel comparing NFW (black) and Einasto (blue) subhalodensity profiles, for the SL17-resilient subhalo model. section is roughly a factor of ∼
3. Indeed, for a crosssection of 4 × − cm /s the SL17-fragile model gives 1 σ detection significance while the SL17-resilient almost 4 σ .This difference in detection significance can be understoodby looking at the difference in the J -factor distribution,cf. Fig. 1 (bottom panel), and can have an impact inthe interpretation of the results of real data analyses. Inlight of this result, detecting subhalos with Fermi -LAT inthe SL17-fragile scenario, while respecting the constraintsfrom other targets, seems quite challenging.Fig. 6 shows the comparison between NFW and Einastosubhalo density profiles, for the SL17-resilient subhalomodel. For a given cross section, the detection significanceobtained with an Einasto profile is always larger thanthe one found with the NFW, by roughly a factor of 2, v [cm /s]02468101214 [ s i g n i f i c a n c e ] NFW, Ext, SL17-resilient
High latitude Low latitude
FIG. 7. Detection significance for the
Ext signal reconstructionmodel comparing subhalos located at two different positionsin the Galaxy (see the text for further details). v [cm /s]02468101214 [ s i g n i f i c a n c e ] NFW, Ext, SL17-resilient m DM = 10 GeV m DM = 100 GeV m DM = 1000 GeV FIG. 8. Detection significance for the
Ext signal reconstructionmodel comparing different DM masses for the injected signal. cf. Fig. 1 (bottom panel).Finally, we show the results obtained by placing the DMsubhalo at the center of the low-latitude
ROI, cf. Fig. 7.As expected, it is much easier to detect a subhalo (even ifextended) at high latitudes than at lower latitudes, wherethe background from interstellar emission is more intense.The DM subhalo should therefore have larger (cid:104) σv (cid:105) toproduce the same significance as the high-latitude ROIcase.In Fig. 8, we instead compare the detection sensitivityfor different choices of the DM mass. In this case, con-sidering different masses shifts the results along the (cid:104) σv (cid:105) values. In particular, for a fixed cross section the lower isthe mass the higher is the significance for the detectionof a subhalo. This is explained by the fact that a less(more) massive DM with respect to the benchmark case v [cm /s]02468101214 E X T [ s i g n i f i c a n c e ] NFW, ExtSL17-resilient SL17-fragile v [cm /s]02468101214 E X T [ s i g n i f i c a n c e ] SL17-resilient, ExtNFW Einasto
FIG. 9. Significance of extension ( σ EXT ) of the
Ext templatew.r.to the PS one, for the subhalos of our simulations. Toppanel : Comparison between the SL17-resilient (black) andSL17-fragile (blue) subhalo models, for an NFW DM subhaloprofile.
Bottom panel : Comparison between an Einasto(blue) and NFW (black) DM subhalo profile, for the SL17-resilient subhalo model. (100 GeV) produce a gamma-ray spectrum with a peakat lower (higher) energies.
Fermi -LAT has a peak of thesensitivity at about 2-4 GeV. Instead at higher energiesthe sensitivity increases monotonically . Therefore, DMcandidates with a peak of the spectrum at a few GeV,such as b ¯ b annihilation channel with m DM = 10 GeV, aredetected with the highest significance while candidateswith the peak at higher energies have low significancevalues.Up to this point we have demonstrated that the bright-est DM subhalo can be detected with the highest sig- See this page for the description of the LAT sensitivity as afunction of energy v [cm /s]10 [] NFW, Ext SL17-fragile, DM model SL17-resilient, DM model SL17-resilientSL17-fragile v [cm /s]10 [] SL17-resilient, Ext NFW, DM model Einasto, DM model NFWEinasto
FIG. 10.
Top panel:
Reconstructed 68% containment radius( θ ) as a function of the injected annihilation cross section,comparing the SL17-resilient (black) and SL17-fragile (blue)models for an NFW subhalo density profile. Bottom panel:
Same as the top panel, but for the SL17-resilient model com-paring NFW (black) and Einasto (blue) DM density profiles.In both panels, we also overlay theoretical predictions for theaverage value of θ (cid:63) , i.e. θ (cid:63) . nificance when fitted with an extended source template,either Ext or PS+Ext . Nevertheless, a legitimate questionto ask is: Is the evidence for extension significant?We quantify the significance for the extension of oursimulated signal. We consider the case of one extendedsource (
Ext ) fitted to the DM signal. Similar results arefound if we consider the case with one point source andone extended source (
PS+Ext ).In Fig. 9 we show the significance for the extension ofthe source ( σ EXT ). This is calculated by considering the PS case as null hypothesis and the extended source case, Ext , as test hypothesis. We compute the significance forextension following the procedure highlighted in Sec. III C.We vary several assumptions on the simulated DM sig-nal model: We compare SL17-resilient and SL17-fragile subhalo models (top panel), and the choices of differentDM subhalo density profiles for the SL17-resilient case(bottom panel). The SL17-fragile model provides muchlower σ EXT with respect to the SL17-resilient case. Forexample, at a cross section of 10 − cm /s the SL17-fragile subhalo model gives an average significance forextension of 1.8 σ while SL17-resilient gives 4.2 σ . On theother hand, the results obtained for Einasto and NFWprofiles are comparable. Indeed, for (cid:104) σv (cid:105) = 10 − cm /swe get, on average, a significance for extension of 7.5 σ and 4.2 σ , respectively. In all cases, but the SL17-fragile,a marginal detection for extension ( ∼ σ ) is achievedfor cross sections 3 − × − cm /s, which are valuesstill permitted by current constraints, as seen above. Forthe SL17-fragile case a 3 σ detection of extension requires,instead, cross sections of about 3 × − cm /s, whichstarts to be in tension with current constraints from dwarfspheroidal galaxies.In Fig. 10 we show the reconstructed 68% containmentradius ( θ , also equivalent to the standard deviation ofthe radial Gaussian template) for different hypotheses onthe injected DM signal. In the same plot, we also overlaythe theoretical values θ (cid:63) corresponding to the mean overthe θ (cid:63) distribution of the sampled halos, cf. Sec. II B.We note that the reconstructed θ increases with thebrightness of the injected signal until it reaches a plateau,which is compatible (within the 1 σ error band) with thetheoretically predicted value. Indeed, if the DM subhalosignal is too faint the analysis picks up only the morecentral part of the emission and thus the size of extensionis lower than the simulated one. This trend is visible forall the cases considered in this analysis implying that fora faint DM signal the size of extension is underestimated.We stress, however, that, as shown above, even in thecase of faint signals where the extension may be under-estimated the evidence for the extension is significant –namely above 3 σ for cross sections above 3 × − cm /sin the SL17-resilient case. At the plateau, the size ofextension is roughly 0 . ◦ for the SL17-resilient and 0 . ◦ for the SL17-fragile cases and NFW profile, while thetheoretical values are 0 . ◦ and 0 . ◦ , respectively. TheEinasto and NFW density profiles give very similar resultswith the Einasto profile which produces slightly smallervalues for θ (0 . ◦ for the SL17-resilient subhalo model).Finally, we study how the signal reconstruction is af-fected by randomizing simulated data counts using Pois-son statistics, i.e. randomize=True . The result is shownin Fig. 11 for the Ext case. We see that the detectionsignificance is not affected by randomization and, there-fore, all conclusions reached above still hold in the caseof added random Poisson noise. We stress however that the theoretically predicted value is com-puted without convolving the DM template with the point spreadfunction of the instrument, and so it is expected that the measured θ is slightly larger than the predicted θ (cid:63) . v [cm /s]02468101214 [ s i g n i f i c a n c e ] NFW, SL17-resilient, Ext
Randomize = False Randomize = True
FIG. 11. Same as Fig. 4 for the case in which we randomize(blue) or not (black) the number of counts in each pixel,according to Poisson statistics.
V. DISCUSSION AND CONCLUSIONS
A general “belief” is that among
Fermi -LAT uniden-tified sources may shine DM subhalos, although the ma-jority of those should be active galactic nuclei or othergalaxies that lack, at the moment, detection in otherwavelengths. With the present work, we re-assessed thesensitivity of the LAT to signals from the brightest DMsubhalo, in the light of the fact that subhalos with thehighest J -factor show a significant extension in the sky– as supported by a correlation between subhalo angularextension and J -factor.We quantified the sensitivity of Fermi -LAT to thebrightest extended
DM subhalo, by performing realis-tic simulations of the DM injected signal and analysisreconstruction. We tested different assumptions for theDM subhalo model (SL17-resilient and SL17-fragile) anddensity profile (NFW and Einasto), as well as differentDM masses for the DM injected signal. We fit the DMsubhalo source with three different signal reconstructiontemplates: PS , Ext and
PS+Ext .Our results show that: • For both the SL17-resilient and SL17-fragile models,above 3 σ detection significance the extended tem-plate, Ext , always provides the best fit among thethree reconstructed signal models, and also givesa detection significance comparable to the one wewould get by fitting the DM injected signal witha perfectly known DM template. A firm detec-tion (above 5 σ , without accounting for look else-where effects) of DM extended subhalos for theSL17-resilient model can be made for cross sectionsat least as high as 5 − × − cm /s (100 GeV DMmass), which are not excluded by other gamma-rayconstraints yet. • The values of the annihilation cross section increaseby about a factor of four if we consider, instead,the SL17-fragile subhalo model. This implies thataccounting for uncertainty on the subhalo model isa crucial step towards a correct interpretation ofDM searches in real data, and that the detectionof extended subhalos in the SL17-fragile scenariowould be challenging, while fulfilling other gamma-ray bounds on the annihilation cross section. Onthe other hand, the results are not very sensitive tochanging the DM density profile within subhalos. Inparticular using an Einasto or NFW profile providescompatible detection significance. • The evidence for extension is always significant forcross sections above 3 × − cm /s (SL17-resilientcase, NFW and Einasto profiles). In particular, thereconstructed extension for bright signals is compat-ible with the theoretical expectation from subhalosimulations, while it is slightly underestimated forfaint signals.In the most optimistic case, we showed that for crosssections still allowed by other gamma-ray constraints wecan detect DM subhalos with a significance of about 5 σ ,that the size of extension would be roughly 0 . ◦ , and thatthe significance of extension would be about 4 σ .As for systematic uncertainties, we studied the casewhere our simulated data are randomized following Pois-son statistics. Adding Poisson noise did not affect theresults and the same conclusions as above hold true incase of counts randomization. Other systematics thatcan possibly alter the signal detection and extension re-construction are, for example, a mismatch between the true Galactic diffuse model and the one used in the fitand/or the presence of unmodeled sources or backgroundcomponents close to the subhalo. The systematic uncer-tainty due to imperfection of Galactic diffuse modeling isalleviated when considering latitudes | b | > ◦ and ener-gies > for fluxes above 100MeV higher than 10 − ph/cm /s. For fluxes of the orderof 10 − ph/cm /s the T S for detection of a source istypically lower than 25. Assuming this number as anestimate of the density of extragalactic sources that shinebelow the
Fermi -LAT detection threshold, we see thatthe presence of unmodeled and faint sources could berelevant for the search of DM subhalos, since there shouldbe at least one faint extragalactic object in the innermost1 deg around the subhalo.Although we do not address these systematics here,we expect them to be relevant in real data analyses andshould be therefore properly taken care of when perform-ing DM subhalo searches in real data.Our analysis relies on subhalos having a cuspy densityprofile (NFW or Einasto) in agreement with the predic-tions of the cold DM scenario. If subhalos have coredprofiles instead, they would be fainter and more suscep-1tible to tidal effects, which would decrease their numberand spatial extent. We expect this to decrease the detec-tion significance associated to the extension, however thesubhalo model we used is not designed to handle coredobjects, and dedicated simulations are needed in order tocorrectly estimate the impact of subhalo distribution andstatistics.We stress that detecting one DM subhalo is a necessarycondition for the discovery of DM. However, this maynot be of course sufficient to attribute the signal to DM.To this end, it would be of interest, in future, to showwhat is the LAT sensitivity to the simultaneous detectionof two or more subhalos. While we limit ourselves tothe detectability of the brightest subhalo, we checkedwhat is the statistics of J -factor and angular extension forthe second- and third-brightest subhalos. We find thatboth of them have an extension comparable to that ofthe brightest subhalo and that the corresponding mean J -factor are less than 1 σ away from J (cid:63) tot . In particular, byrescaling our results for the mean J -factors ratios, we canestimate that, in order to detect the second- and the third-brightest subhalos with the same detection significance asthe brightest one, we would need an increase of the crosssection of a factor of 1.82 (1.70) and 2.63 (2.24) for theSL17-resilient (SL17-fragile) subhalo model.We expect the general conclusions reached in the present work to apply also to searches for DM subhalowith the upcoming Cherenkov Telescope Array (CTA).CTA will be mostly sensitive to DM masses above 100GeV. In this DM mass range, given the significant im-provement in angular resolution with respect to the LAT,CTA will provide a much better sensitivity to point-likeand extended sources, and therefore improved perspec-tives for detection of the extension of DM subhalos. Aquantitative estimate of such prospects is left for futureanalysis.Finally, while our work focused on sensitivity predic-tions, we foresee application to real Fermi -LAT datato look for extended
DM subhalos, extend previoussearches, and possibly set constraints on the DM pa-rameter space [42].
ACKNOWLEDGMENTS
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