Particle Dark Matter Constraints: the Effect of Galactic Uncertainties
Maria Benito, Nicolas Bernal, Nassim Bozorgnia, Francesca Calore, Fabio Iocco
PPI/UAN-2016-598FTLAPTH-071/16
Prepared for submission to JCAP
Particle Dark Matter Constraints:the Effect of Galactic Uncertainties
Maria Benito, a Nicolás Bernal, a,b
Nassim Bozorgnia, c FrancescaCalore c,d and Fabio Iocco a a ICTP South American Institute for Fundamental ResearchInstituto de Física Teórica - Universidade Estadual Paulista (UNESP)Rua Dr. Bento Teobaldo Ferraz 271, 01140-070 São Paulo, SP Brazil b Centro de Investigaciones, Universidad Antonio NariñoCra 3 Este c GRAPPA Institute, University of AmsterdamScience Park 904, 1090 GL Amsterdam, The Netherlands d LAPTh, CNRS, 9 Chemin de Bellevue, 74941 Annecy-le-Vieux, France
Abstract.
Collider, space, and Earth based experiments are now able to probe severalextensions of the Standard Model of particle physics which provide viable dark matter candi-dates. Direct and indirect dark matter searches rely on inputs of astrophysical nature, suchas the local dark matter density or the shape of the dark matter profile in the target in object.The determination of these quantities is highly affected by astrophysical uncertainties. Thelatter, especially those for our own Galaxy, are ill–known, and often not fully accounted forwhen analyzing the phenomenology of particle physics models. In this paper we present asystematic, quantitative estimate of how astrophysical uncertainties on Galactic quantities(such as the local galactocentric distance, circular velocity, or the morphology of the stellardisk and bulge) propagate to the determination of the phenomenology of particle physicsmodels, thus eventually affecting the determination of new physics parameters. We presentresults in the context of two specific extensions of the Standard Model (the Singlet Scalarand the Inert Doublet) that we adopt as case studies for their simplicity in illustrating themagnitude and impact of such uncertainties on the parameter space of the particle physicsmodel itself. Our findings point toward very relevant effects of current Galactic uncertaintieson the determination of particle physics parameters, and urge a systematic estimate of suchuncertainties in more complex scenarios, in order to achieve constraints on the determinationof new physics that realistically include all known uncertainties. a r X i v : . [ h e p - ph ] M a y ontents Searches for the very nature of the elusive dark component of matter (DM) are experiencinga crucial moment in these very years: the enhanced sensitivity of direct and indirect searches,together with the latest data coming from collider experiments, allows to constrain the pa-rameter space of several extensions of the Standard Model (SM) of particle physics, in somecases strongly challenging models which have been very popular in the last years. The mul-tichannel searches for DM are seeing the dawn of a real precision era. The grandeur of thisendeavor carries the burden of precision, and it becomes timely and mandatory to properlyassess the entire budget of uncertainties that affect such an amazingly refined construction.It is very well known, and we also recall in the following in more detail, that the particleinterpretation of the data coming from direct and indirect searches depends on quantities ofastrophysical nature, such as the spatial distribution of DM in the target for indirect searches,and its phase space distribution in the solar neighborhood for direct ones. Strenuous effortsare ongoing from the side of the astrophysical community to assess these quantities, in a ma-jor endeavor involving astronomical observations of diverse nature. Yet, the determination ofthe quantities of interest are affected by often sizable uncertainties. This is also well knownin the literature, where the entire extent of these uncertainties does not always propagate itsway in the determination of new physics.In this work, we aim at presenting a case study by systematically analyzing how theuncertainties on the DM structure in our Galaxy affect the determination of new physics. Wewill use two of the simplest possible extensions of the SM: the Singlet Scalar (SSDM) and theInert Doublet (IDM) DM models. Those models have been chosen as ideal testbeds given therelatively simple dependence of their phenomenology on a limited set of parameters, whichmakes it easy to visualize the effects of astrophysical (and in this case especially, Galactic)uncertainties in the parameter space of the particle physics model itself. Our goal is toprompt the evidence for the relevance of the propagation of known, but too often overlooked,– 1 –nknowns of astrophysical nature directly into the determination of new physics. In order todo so, we first present the most recent results on the determination of the DM distributionin our Galaxy, and most relevantly its uncertainties. We then show the dependence of DMdirect and indirect searches on the Galactic uncertainties, and specify how the constraints onthe parameters of the IDM and SSDM models set by direct and indirect searches are affectedby such uncertainties.The paper has the following structure. In section 2 we describe the setups of our analysis:datasets and techniques adopted for the determination of the DM density structure of theMilky Way (MW); the adopted benchmark limits for DM direct detection; and the probeschosen as indirect detection targets. In section 3 we describe our benchmark particle physicsmodels: extensions of the SM which provide a well posed DM candidate, and have recentlybeen claimed to be strongly constrained by existing data. In section 4 we present the findingof our analysis, and the impact of uncertainties on Galactic morphologies and parameterson the determination of new physics. In our conclusions, we summarize our key results andmotivate how they prompt the extension of a similar complete analysis to more complexextensions of the SM.
In order to determine the DM density profile of our own Galaxy, we adopt a well knowndynamical method: objects in circular orbits around the Galactic center (GC) are used astracers of the total gravitational potential, and the rotation curve (RC) thus obtained (in theplane of the disk) is compared to the circular velocity expected to be caused by the visiblecomponent of the MW alone. The mismatch between the two is accounted for by a nonvisible, dark component of matter, whose density distribution can be obtained by fitting anappropriately parametrized function to the total RC. This class of methods, often knownas global methods , offers a series of advantages with respect to local ones, which permit todetermine the DM distribution only in a small region around the location of the Sun, anda series of shortcomings. Both are carefully described in the recent review [1], and whileaddressing the reader to it and references therein for a complete overview, we highlight herethe advantages and shortcomings of relevance to this specific analysis. We follow the recentanalysis in [2], which respect to previous similar studies [3–5], offers the remarkable improve-ment to adopt a vast range of data–driven morphologies for the three visible components ofthe MW (stellar bulge, stellar disk(s), gaseous disk). As shown therein, the choice of stellarbulge/disk affects the shape of the DM profile beyond the statistical uncertainties associatedto each one of the visible components, leading to the conclusion that our ignorance on themorphology of the MW hinders our determination of the DM profile more than the uncer-tainties associated to their normalization. Also, a quantitative estimate of the effect of thecurrently quoted uncertainties on the Galactic parameters ( R , v ) is offered, showing effectscomparable with that of the Galactic morphology. This matter is certainly well known inprinciple, but again its actual magnitude is ill–known, and its effect on the determinationof new physics is equally unaccounted for in most of the literature. Unfortunately, none ofthese uncertainties are easily treatable in a statistical way, and one runs the risk to underes-timate the effect of cross–correlations between datasets, or to be affected by hidden biases inthe choice of the null hypothesis. For this reason, we aim here only to offer an estimate of– 2 –he effects in the parameter space of particle physics models , by varying different sources ofuncertainty one at a time.By combining together one by one all possible combinations of bulge, disk, and gas, weobtain a set of unique “baryonic morphologies”, i.e. a catalogue of observationally–inferredmorphologies, each one of them carrying a statistical uncertainty arising from the normaliza-tion of the density profile of each component, which is then propagated to the correspondingRC (generated by that specific configuration of visible matter) [2, 6]. For each baryonic mor-phology and each set of Galactic parameters, separately, we add the RC due to DM to the onedue to visible matter. The DM density profile is parameterised though a generalised NFW(gNFW) profile: ρ DM ( R ) = ρ (cid:18) R R (cid:19) γ (cid:18) R s + R R s + R (cid:19) − γ , (2.1)where R is the distance of the Sun from the GC, ρ the DM “local” density (i.e. at the Solarposition), R s the so-called “scale radius” of the DM density profile, and γ the so-called “profileindex” (the standard NFW profile having γ =1). We compare the resulting “total” (baryon +DM) RC to the latest compilation of observed RC data, presented in [6]. Following closely themethodology of [2], we scan the ( ρ , γ ) space, while keeping the scale radius R s constant. Wedetermine the goodness of fit of each point in the parameter space using the two–dimensionalvariable: χ = N (cid:88) i =1 d i ≡ N (cid:88) i =1 (cid:34) ( y i − y t,i ) σ y,i + σ b,i + ( x i − x t,i ) σ x,i (cid:35) , (2.2)where we have introduced the reduced variables x = R/R and y = w/w − ; ( x i ± σ x,i , y i ± σ y,i ) are the RC measurements, σ b,i is the uncertainty of the individual baryonic modelevaluated at x i and ( x t,i , y t,i ) are the points that minimize d i along the curve y t ( x ) = w t ( R = xR ) /w − . The variable w ( R ) = v c ( r ) /R is the angular velocity at the galactocentricdistance R (with w = w ( R ) ), and it is used as an independent variable as the uncertaintieson R and w are uncorrelated (see [6] and references therein). The sum runs over all the N objects in the compilation at R > R cut = 2 . kpc in order to exclude the innermost regionsof the Galaxy where axisymmetry breaks down and some tracers may present non-circularorbits. The function in eq. (2.2) has been shown to have a χ distribution for the case at handin [6], and offers the advantage of an unbinned analysis which properly takes into accountthe statistical uncertainties of the observed RC dataset (in both dimensions), and that of thebaryonic RC, propagated from the normalization of the stellar bulge and disk (respectivelyfrom microlensing optical depth in the direction of the bulge and local stellar surface density,see [2, 4] for methodology, and references therein). The “best fit” point is obtained by pickingthe point in ( ρ , γ ) space that minimizes the two dimensional χ described above, while wehave kept the scale radius constant at the value R s =20 kpc. We note that also the variationof R s is expected to have some impact. Although we have tested that the choice of a fixed R s value does not affect significantly our conclusions, a full analysis of the effect of the R s variation is beyond the scope of the present paper and we postpone it to a future work. We checked that by varying its value by a factor 2 we observe a maximal variation in the local DM densityof (cid:46) J -factor (see section 2.3 for the definition of J -factor) of (cid:46)
10% for the region of interestof the GC GeV excess. As it will be seen, this is well below the effects of the variation of baryonic morphologyor Galactic parameters. – 3 –n order to probe the effect of different sources of ignorance, we test the following un-certainties one at a time: • Statistical uncertainties; • Uncertainties on Galactic parameters, ( R , v ); • Uncertainties on the morphology of the visible, i.e. baryonic, component of the Galaxy.The numerical values adopted, the results obtained, as well as the reference for themorphologies that comply with the above conditions are presented schematically in table 1,and we summarize here the criteria behind the choices adopted, according to the aboverationale: • Standard Galactic parameters.
The “standard” Galactic parameter values are ( R , v )=(8 kpc, 230 km/s). When these values are adopted, the peak speed of the Maxwellianvelocity distribution of DM particles is taken to be equal to the local circular speed, v peak = v = 230 km / s. • Reference morphology.
The “representative” baryonic morphology is [8–11]; referred toas “
BjX ” in table 1. • Galactic parameters variation.
The extreme values for the Galactic parameters are cho-sen to vary between R =[7.5 – 8.5] kpc and v =[180 – 312] km/s, for our representativemorphology BjX . The local circular speed can range from (200 ± km/s to (279 ± km/s [12]. Hence, we take v = 180 km/s and 312 km/s as lower and higher estimates,respectively. When these values are adopted for v , we take v peak = 250 km / s, regardlessof v . • Morphology variation.
The extreme baryonic morphologies are chosen to be those thatrequire the maximum/minimum values of γ , ρ , in order to visualize the maximumimpact on both direct and indirect detection as we will see in the following sections.With reference to the nomenclature in table 1, we find that (for assigned, standardGalactic parameters) the baryonic morphologies that maximize/minimize • the index γ are respectively “ FkX ” and “
DiX ”; • the local DM density ρ are respectively “ CjX ” and “
FiX ”.In figure 1 we display the rotation curves corresponding to the baryonic morphologiesdescribed above, assuming fixed Galactic parameters ( R , v )=(8 kpc, 230 km/s). Statisticaluncertainties associated to the displayed central values are not shown, but they are takeninto account for the fitting procedure as described above. We also display our compilation ofdata for the observed RC and their 1 σ uncertainties, as originally presented in [6]. In order tonormalize the data to different values of the Galactic parameters, we have used the publiclyavailable tool galkin , [13]. In table 1 we report the results of the fitting procedure describedand the parameters of the selected morphologies. This choice for the peak speed of the Maxwellian velocity distribution falls in the range of [223 – 289]km / s suggested by high resolution hydrodynamic simulations [7] (see also section 2.2). We have checked thatvarying v peak does not make a visible difference in the direct detection limits in the parameter space of theSSDM and IDM models, and its effect is much smaller than the effect of variation of other Galactic parameters. – 4 – orphology R (kpc) v (km/s) M ∗ ( × M (cid:12) ) γ ρ (GeV/ cm ) ReferenceBjX 8 230 . +0 . − . . +0 . − . . ± . [8–11]BjX 7.5 312 . +0 . − . . +0 . − . . ± . [8–11]BjX 8.5 180 . +0 . − . . ± .
07 0 . ± . [8–11]FkX 8 230 . ± . . +0 . − . . +0 . − . [10, 11, 24, 25]DiX 8 230 . ± . . +0 . − . . ± . [10, 11, 26, 27]CjX 8 230 . +0 . − . . ± .
04 0 . +0 . − . [9–11, 28]FiX 8 230 . ± . . ± .
05 0 . ± . [10, 11, 24, 27] Table 1 . We adopt a gNFW density profile with R s = 20 kpc. From left to right we reportthe nomenclature adopted for each morphology, the values of Galactic parameters ( R , v ), thebaryonic mass in the Galaxy for that specific baryonic morphology, the best–fit values of index γ and ρ according to the procedure described in the text, and the references for the three-dimensionalmorphology shape. The criteria that led to the choice of these morphologies are explained in the text. When varying Galactic parameters, we obtain the total mass of the MW within 50 kpcto be in the range M( <
50 kpc) = (1 . − . × M (cid:12) . The lower limit is in agreement withprevious determinations [14–18], while the larger MW masses we obtain can not be directlycompared, as the the adopted Galactic Parameters are different than ours. When varyingbaryonic morphology, the minimum/maximum values obtain are M( <
50 kpc) = 4 . +0 . − . × M (cid:12) and M( <
50 kpc) = 7 . ± . × M (cid:12) . The former value for the variation ofmorphology is in good agreement with mass estimate from kinematics of globular clusters,satellite galaxies and halo stars [14, 15, 17, 19]. There is, however, a discrepancy at the σ level with the independent determination in [18], that used the Sagittarius stream, and isslightly smaller than the other cited determinations. All our results are in agreement at the σ level with the recent estimate of the dynamical mass [20] within the region of the Galacticbulge, as in the analysis presented in [21].In figure 2, we show the DM density profiles corresponding to the selected morphologiesin table 1. When varying the morphology, almost all DM profiles are in agreement withrecent findings for MW-like galaxies in hydrodynamical simulations [22, 23]. The upper paneldisplays the DM density as a function of the distance from the GC, while in the lower panelthe relative error with respect to the reference model is shown. As a reference we also depictthe traditionally adopted NFW profile, corresponding to a gNFW with parameters γ = 1 , ρ = 0 . GeV/cm . The aim of direct detection experiments is to measure the small recoil energy of a nucleus in anunderground detector after the collision with a WIMP arriving from the DM halo of the MW.The current status of direct detection searches is ambiguous with a few experiments reportinghints for a DM signal [29–32], while all other experiments report null-results. Currently theLUX (Large Underground Xenon) experiment [33] places the strongest exclusion limit in theplane of spin-independent DM-nucleon cross section and WIMP mass for large DM masses,while the PandaX-II (Particle and Astrophysical Xenon Detector) experiment [34] has recentlyreported competitive null results. In this paper we focus on the impact of astrophysicaluncertainties on the LUX exclusion limit in the parameter space of specific particle physics– 5 –
10 15 20
R [kpc] v [ k m / s ] R = 8 kpc , v = 230 km / s BjXDiX CjXFiX FkX
Figure 1 . RC produced by the benchmark baryonic morphologies (colored curves), reported in table 1and described in the text. RC data points and their 1 σ errors shown in gray are from the compilationpresented in [6]. − − − − − ρ D M [ G e V / c m ] NFWBjX DiXCjX FiXFkX − − R [kpc] − − ∆ ρ D M Figure 2 . The DM density, ρ DM , as a function of radial distance from the GC, R , for a standardNFW profile (black dotted line), and all baryonic morphologies in table 1 with standard Galacticparameters, as described in the text. The bottom panel refers to the relative error of ρ DM withrespect to the reference morphology BjX . The dashed gray line in the bottom panel corresponds to aperfect match between the morphology considered and the
BjX . – 6 –odels. However, we note that the variation of the exclusion limits set by other directdetection experiments due to astrophysical uncertainties would be similar to those discussedfor LUX.For a DM particle scattering elastically off a nucleus with atomic mass number A , thedifferential event rate (per unit energy, per unit detector mass, per unit time) in directdetection experiments for the case of spin-independent scattering can be written as, dRdE R = ρ A σ SI m DM µ p F ( E R ) η ( v min , t ) , (2.3)where E R is the nuclear recoil energy, ρ is the local DM density, m DM is the DM mass, µ p is the reduced mass of the DM-nucleon system, σ SI is the spin-independent DM-nucleonscattering cross section, and F ( E R ) is a form factor. v min = (cid:113) m A E R / (2 µ A ) is the minimumspeed needed for the DM particle to deposit a recoil energy E R in the detector. Here m A is themass of the nucleus, and µ A is the DM-nucleus reduced mass. The halo integral, η ( v min , t ) ,which together with the local DM density encompasses the astrophysics dependence of therecoil rate, is defined as, η ( v min , t ) ≡ (cid:90) v>v mim d v f det ( v , t ) v , (2.4)where f det ( v , t ) is the local DM velocity distribution in the detector rest frame.Eq. (2.3) can be written as, dRdE R = C PP F ( E R ) ρ η ( v min , t ) , (2.5)where the coefficient C PP = A σ SI / (2 m DM µ p ) contains the particle physics dependence ofthe event rate, while ρ η ( v min , t ) contains the astrophysics dependence.In the analysis of direct detection data, usually the Standard Halo Model (SHM) isadopted. In the SHM, the DM halo is spherical and isothermal, and the local DM velocitydistribution is an isotropic Maxwell-Boltzmann distribution with a peak speed, v peak equalto the local circular speed, v .The results of state-of-the-art high resolution cosmological simulations which includeboth DM and baryons indicate that a Maxwellian distribution with a best fit peak speed inthe range of 223 – 289 km / s fits well the local velocity distribution of simulated MW-likehaloes [7]. Based on the results of ref. [7], for the analysis of direct detection data in thiswork we adopt a Maxwellian velocity distribution truncated at the Galactic escape speed,and with a peak speed in the range of [223 − km / s, independent from the local circularspeed. For the local circular speed, we adopt v = 180 km / s and 312 km / s as high and lowestimates. For the peculiar velocity of the Sun with respect to the Local Standard of Rest weassume (11 . , . , . km / s [35] in Galactic coordinates. We adopt the median value ofthe local Galactic escape speed reported by the RAVE survey, v esc = 533 km / s [36].Recently, the LUX experiment has reported the results of 332 live days of data, with noevidence of a DM signal [33]. Since the exposure and detector response information is notpublicly available for each event in the recent LUX data, we perform an analysis of the 2015LUX results [37] instead. In ref. [37], the LUX collaboration presented an improved analysis oftheir 2013 data for an exposure of . × kg days. To set an exclusion limit using the LUXdata, we employ the maximum gap method [38], since we cannot reproduce the likelihoodanalysis performed by the LUX collaboration with the available information. We consider– 7 – eference modelGalactic parametervariation ��� ��� ������ - �� �� - �� �� - �� �� - �� �� - �� �� - �� �� - �� �� - �� � �� ( ��� ) σ � � ( � � � ) Figure 3 . LUX exclusion limit at 90% CL in the spin-independent DM-nucleon cross section and DMmass plane for the reference choice of Galactic parameters and local DM density (red curve) given inthe first row of table 1, and two variations of Galactic parameters and local DM density (blue curves)given in the second and third rows of table 1, see text in section 2.1 for a complete description. the events which fall below the mean of the nuclear recoil band (red solid curve in figure 2 of[37]) as signal events, and assume an additional acceptance of 0.5. As seen from figure 2 of[37], one event makes the cut. We take the detection efficiency from figure 1 of [37], and setit equal to zero below the recoil energy of E R = 1 . keV, following the collaboration. Sincewe are only interested in events at < cm radius, we multiply the efficiency by (18 / .To find the relation between E R and the primary scintillation signal S1, we find the value ofS1 at the intersection of each recoil energy contour and the mean nuclear recoil curve fromfigure 2 of [37]. Assuming a Maxwellian velocity distribution with the same parameters as in[37], we can find an exclusion limit at 90% CL which agrees well with the limit set by theLUX collaboration.In figure 3, we show the LUX exclusion limit in the m DM − σ SI plane for the standardchoice of Galactic parameters ( R , v ) and the local DM density ρ for our reference morphol-ogy given in the first row of table 1 (“reference model” in figure legend), as well as for tworepresentative variations of Galactic parameters and ρ given in the second and third rows oftable 1. The largest variation of the exclusion limit with respect to the reference limit is dueto the variation in the local DM density. Notice that in the exclusion limits shown in figure 3,we take v peak = v , while in the figures of section 4 where we vary the Galactic parameters,we adopt a peak speed value independent of the local circular speed. Indirect detection aims at detecting the flux of final stable particles produced by DM an-nihilation or decay. Among those, gamma rays are considered the golden channel for theidentification of a possible DM signal since they preserve the spectral and spatial informationof the signal itself. In the present work, we focus on the gamma-ray flux from DM annihila-tion for a twofold reason: First, we work with gamma rays since the expected flux is directlyexpressed in terms of the DM density distribution. For charged cosmic rays, instead, the– 8 –xpected flux at Earth is obtained after propagating the produced particles in the interstellarmedium and thus the dependence of the propagated flux on the DM density is less trivial. Secondly, the choice of the annihilation process – instead of the decay one – is motivatedby the fact that the predicted flux is more affected by the astrophysical uncertainties on theDM density, since it depends on the DM density squared. We stress that our aim is to giveconcrete – and intuitive – examples of the effect of the astrophysical uncertainties for DMphenomenology.Typically, the gamma-ray flux from DM particles annihilation or decay can be factorizedin terms of a particle physics term, Φ PP , which contains the information of the underlyingparticle physics theory of DM, and an astrophysical term, J , which instead encodes theinformation about the geometrical distribution of DM in space. The DM expected flux writesas: Φ DM ( E ) = Φ PP ( E ) J . (2.6)In this section, we remain agnostic about the particle physics sector, while we are interestedin quantifying the uncertainty affecting the J -factor in light of the analysis performed insection 2.1. The J -factor is defined as the integral along the line-of-sight of the DM density, ρ DM , in case of DM decay, or of the DM density squared, ρ , in case of DM annihilation.While the uncertainty of ρ DM translates linearly in the uncertainty on J decay , the J annih ismore sensitive to the uncertainty on ρ DM given the squared dependence. In the case of DMannihilation: J annih = (cid:90) l . o . s ρ ( R ( l, ψ )) dl, (2.7)where ψ is the opening angle between the line of sight l and the direction towards the GC.The radial distance from the GC is R = R + l − l R cos ( ψ ) .In figure 4 we show the uncertainty on the DM annihilation J -factor (bottom panel) fordifferent morphologies as in figure 2 (cf. table 1). In the upper panel, we show the J -factor asa function of the angle ψ , comparing a standard NFW profile with our reference baryonicmorphology and other morphology configurations, cf. table 1. In the inner region, i.e. withinfew degrees, the astrophysical uncertainty on the J -factor is (cid:38) O (10).Accounting for the astrophysical uncertainty on the predicted DM flux is crucial whencomparing results from different targets. For example, a positive signal might be seen ina gamma-ray target and interpreted in terms of DM annihilation. The preferred particlephysics parameter space, typically the average velocity annihilation cross section (cid:104) σv (cid:105) vs DMmass m DM , depends on the J -factor assumed for the target considered. On the other hand,null results from other targets impose upper limits on the allowed ( (cid:104) σv (cid:105) , m DM ) parameterspace. It might occur that the constraints are in tension with the signal. However, such atension relies on the assumed J -factor and, thus, the uncertainty on the J -factor must befully accounted for before claiming a strong tension. This is what happened for example inthe case of the GC GeV excess (see e.g. [39, 40]): the DM interpretation of such an excessstarted to be challenged by the latest constraints from dwarf spheroidal galaxies (dSphs) [41].Here we demonstrate that such a tension can be alleviated (or worsened) by the un-avoidable uncertainty on the J -factor of the MW – we do not discuss here the uncertaintieson the dSphs J -factor related to the choice of the dSphs DM profile (see e.g. [42] for a dis- We note however that the “source term” of charged particles produced by DM depends directly on theDM density and thus is affected by the same uncertainties that we discuss explicitly for gamma rays. – 9 –
10 20 30 40 50 60 70 80 9010 J a nn i h [ G e V / c m ] NFWBjX DiXCjX FiXFkX ψ [ ◦ ] − ∆ J a nn i h Figure 4 . The DM annihilation J -factor as a function of the angle ψ between the line-of-sightand the GC for the same DM profiles in table 1 and figure 2 (same colors). The bottom panel refersto the relative error of the J -factor with respect to the reference morphology BjX . The dotted linecorresponds to the traditional NFW profile. The dashed gray line in the bottom panels correspondsto a perfect match between the morphology considered and
BjX . cussion), nor the possible effect of varying the Sun position, R , on the stacked dSphs limit. In figure 5, we show the latest dSphs limits [41] and the region preferred by the GC GeVexcess for DM annihilating into a pair of b-quarks [43], for our reference morphology
BjX with standard Galactic parameters (“reference model” in figure legend), and for the variationof Galactic parameters, as given in the second and third rows of table 1. For the sake ofclarity we show only the 1 and 2 σ contours rescaled to the morphology of interest from thecontours in ref. [43]. The rescaling of the original contours (in figure 5 as well as in all otherfigures showing the GeV excess rescaled contours) is made by imposing that the GeV excessflux measured at 2 GeV and 5 ◦ away from the GC is conserved, in analogy to what done inref. [43]. Indeed, at 5 ◦ away from the GC the GeV excess intensity has been shown to bealmost independent on the assumption on the gNFW slope used in the data analysis. There-fore, the rescaling of the contours only involves J ( ψ = 5 ◦ ), and not the integral over the fullregion-of-interest. The variation of the contours in (cid:104) σv (cid:105) due to the uncertainty on Galacticparameters ( R , v ) can be larger than a factor of two in both directions. We emphasise thatalso constraints on DM from the annihilation in the MW halo or the GC region would beaffected by an analogous uncertainty. For a single dSph, under the assumption of point-like source emission,
J ∝ /d , being d the distancefrom the observer and, thus, depending on the Sun position, R , in the Galactic reference frame. In thesimplest case of a dSph located at the GC: J ( R a ) / J ( R b ) ∝ ( R b /R a ) . However, when considering morerealistic geometries and the stacking of the dSphs, as done to derive the limits in ref. [41], the dependencefrom R is not anymore trivial and assessing its effect would require a re-analysis of the dSphs data which isbeyond the scope of the present paper. – 10 – m DM [GeV] − − − − h σ v i [ c m / s ] Fermi-LAT dSphs 2016Reference modelGalactic parameter variation
Figure 5 . Upper limit on DM annihilation cross section (into ¯ bb ), (cid:104) σv (cid:105) , vs DM mass, m DM , fromthe analysis of gamma rays from dwarf spheroidal galaxies [41] (black line). Best fit contours at 2 σ of the GeV excess as due to DM annihilation in b-quark pairs for the gNFW parameters preferredby our reference morphology BjX (red contours) with standard Galactic parameters, and the samemorphology by varying Galactic parameters (blue countours), as given in the second and third rowsof table 1, see text in section 2.1 for a complete description. This setup is the same as used for thevariations in figure 3.
We now apply the setup shown until now to some benchmark DM particle models, in order toshow how the astrophysical uncertainties affect the determination of the physical parametersof the specific models at hand, in the context of DM direct and indirect detection. Weconcentrate on two minimal extensions of the SM, the SSDM and the IDM, as they arearguably among the most minimal models for which it is easy to quantify and visualizein a clear way the effects described above. We do expect that our results will prompt ageneralization to more complex scenarios in which the effects are not so trivially discerniblefrom effects due to the interplay of numerous model parameters. In this section we presentthe general framework of these two simple models and depict the state-of-the-art constraintson the model parameter space. The discussion of the impact of astrophysical uncertaintieswill be the focus of section 4.
The SSDM [44, 45] is one of the minimal extensions of the SM that can provide a viable DMparticle candidate. In addition to the SM, this model contains a real scalar S , singlet underthe SM gauge group, but odd under a Z symmetry in order to guarantee its stability. Themost general renormalizable scalar potential is given by V = µ H | H | + λ H | H | + µ S S + λ S S + λ HS | H | S , (3.1)where H is the SM Higgs doublet. It is required that the Higgs gets a non-vanishing vacuumexpectation value, v H = 246 GeV, while the singlet does not, (cid:104) S (cid:105) = 0 . At tree level, the singlet– 11 –ass is m S = 2 µ S + λ HS v H . The phenomenology of this model is completely determined bythree parameters: the DM mass m S , the Higgs portal coupling λ HS and the quartic coupling λ S . Note however that λ S plays a minor role in vanilla WIMP DM phenomenology, andthus hereafter we will focus only on the parameters m S and λ HS . These two parameters arethe ones that determine the strength of both the direct and the indirect detection signals.There has been a large amount of research on the SSDM, most of it focused on theWIMP scenario, where the singlet S mixes relatively strongly with the Higgs and undergoesa thermal freeze-out. This scenario has been highly constrained by collider searches [47–52],DM direct detection [53–56] and indirect detection [57–63].We show the current constraints in figure 6. In both panels, the black thick line corre-sponds to the points that generate a DM relic abundance in agreement with the measurementsby Planck [64], and the gray region below the line is excluded because it produces a too largeDM abundance, thus overclosing the Universe. The hatched light blue region in the upperleft corner is forbidden by current constraints on the strength of the Higgs portal. Indeed,for m S < m h / ∼ GeV, the Higgs can decay into a pair of DM particles, thus the currentlimits on the invisible Higgs branching ratio (BR inv (cid:46) [65]) and the Higgs total decaywidth ( Γ tot h (cid:46) MeV [66]) constrain the Higgs coupling with the dark sector, λ HS .In the left panel of figure 6, we display the exclusion limit on the spin-independent elasticWIMP-nucleon cross section at 90% CL from the 2015 LUX results [37], which is translatedinto the dark red region in the top part of the figure. We plot here the limit derived fromthe red curve in figure 3 described in section 2.2. We recall that this limit has been derivedassuming the parameters in the first row of table 1, i.e. a Maxwellian velocity distributionwith v peak = v = 230 km/s, v esc = 533 km/s, and ρ = 0 . GeV/cm . In the right panel,we show instead the current limits from the analysis of dwarf spheroidal galaxies (dSphs)with Fermi -LAT [41]. The region in blue represents the parameter space favoured by theinterpretation of the GC excess (at 2 σ ), and corresponds to the red contour in figure 5, asdescribed in section 2.3.From figure 3, we see that the LUX limit strongly depends on the astrophysical un-certainties on the Galactic parameters, and especially on the uncertainty in the local DMdensity. Therefore, the available parameter space of the SSDM will depend on the actualconfiguration of Galactic parameters. On the other hand, in the case of indirect searches,we do not explore uncertainties on the limits imposed by dSphs, but we investigate how theregion favoured by the DM interpretation of the GeV excess will move because of Galacticuncertainties, as already shown in figure 5. We will show the response of the constraints toastrophysical uncertainties, and its implications in section 4. The IDM [67] is another minimal extension of the SM that contains a second complex scalardoublet. The model contains an exact Z symmetry under which all SM particles –includingone of the scalar doublets– are even, and the second scalar doublet is odd. Since this discretesymmetry prevents mixing between the scalars, one of the doublets ( H ) is identified with theSM Higgs doublet. The second doublet ( Φ ), odd under the Z parity, is inert in the sensethat it does not couple to the SM particles. The most general renormalizable scalar potential Whereas λ S has a crucial role in DM phenomenology, in scenarios where DM is a SIMP, with sizableself-interactions [46]. – 12 – m S [GeV] − − − − λ H S L U X m S [GeV] − − − − λ H S d S p h s Figure 6 . Singlet Scalar Model . In both panels, the black line corresponds to the points that generatea DM relic abundance in accordance to the measurements by Planck [64]; the lower gray region over-closes the Universe. The upper left region (hatched light blue) is ruled out by the invisible decay ofthe Higgs [65, 66]. The upper dark red region in the left panel corresponds to the LUX exclusion limiton the spin-independent elastic WIMP-nucleon cross section at 90% CL for the choice of parametersin the first row of table 1, while the one in the right panel is derived from the limits on the averagedvelocity annihilation cross section from the combined analysis of dwarf spheroidal galaxies in theMW [41]. The parameter space favoured by the GeV excess data [43] (at 2 σ ) is depicted by the blueregion in the right panel. of the IDM is given by V = µ | H | + µ | Φ | + λ | H | + λ | Φ | + λ | H | | Φ | + λ | H † Φ | + λ (cid:104) ( H † Φ) + h . c . (cid:105) . (3.2)In the general case, the λ i are complex parameters. Although considering this possibility canhave interesting consequences for CP-violation and electroweak baryogenesis [68–70], in thiswork we limit ourselves to the case of real values. Upon electroweak symmetry breaking, thetwo doublets can be expanded in components as H = √ (cid:0) v H + h (cid:1) , Φ = H +1 √ (cid:0) H + i A (cid:1) . (3.3)The h state corresponds to the physical SM-like Higgs-boson. The inert sector consists ofa neutral CP-even scalar H , a pseudo-scalar A , and a pair of charged scalars H ± . The Z symmetry guarantees the stability of the lightest state of the dark sector. If it is neutral(either H or A ), this state can play the role of the DM.At the tree level, the scalar masses are m h = µ + 3 λ v H , (3.4) m H = µ + λ L v H , (3.5) m A = µ + λ S v H , (3.6) m H ± = µ + 12 λ v H , (3.7)where λ L ≡ ( λ + λ + λ ) and λ S ≡ ( λ + λ − λ ) . The IDM scalar sector can be fullyspecified by a total of five parameters: three masses ( m H , m A and m H ± ) and two couplings– 13 – m H [GeV] − − − − − | λ L | L U X m H [GeV] − − − − − | λ L | Figure 7 . Inert Doublet Model . The dots correspond to the parameter space that generates the correctDM relic abundance. The light gray points give rise to a too large invisible Higgs decay, in tensionwith LHC measurements. The dark red region in the left panel displays the LUX exclusion limit onthe spin-independent elastic WIMP-nucleon cross section at 90% CL for the choice of parameters inthe first row of table 1. In the right panel, we show, in blue, the points of the parameter space whichcan successfully explain the GeV excess (at 2 σ ), corresponding to the red contour in figure 5. Thepoints in light brown are those in tension with the analysis of dSphs with Fermi -LAT [41]. ( λ L and λ ). However, in this analysis the role of λ will be disregarded, as it appears onlyin quartic self couplings among dark particles and does therefore not enter in any physicallyobservable process at the tree level. The phenomenology of the IDM has been largely studied since the model allows togenerate a population of WIMP DM particles in the early Universe via a thermal freeze-outand it induces potentially observable signals in direct and indirect DM searches [67, 72, 72–84],collider searches [67, 71, 85–91] and electroweak precision tests [67, 92].For this analysis, we perform a numerical analysis scanning randomly over GeV Figure 8 . Singlet Scalar Model : Effects of Galactic uncertainties on the LUX exclusion limit in theSSDM parameter space. We display the effect of: statistical uncertainty, for our reference morphology BjX (left panel); varying the Galactic parameters for the same reference morphology (central panel);adopting different morphologies that maximize/minimize the local DM density ρ , CjX and FiX (rightpanel). Criteria are discussed in section 2.1 and values are reported in table 1. In figure 7 we study the constraints on the IDM parameter space coming from a specificGalactic model (i.e. the reference morphology model BjX ). In section 4 we show the responseof the constraints to astrophysical uncertainties, and its implications for both direct andindirect DM searches. With all elements at hand, we now turn to show the effect of astrophysical uncertaintiesdirectly onto the parameter space of particle physics models discussed in the previous section. We start by comparing the limit imposed by direct detection on the SSDM for differentcases of variation of astrophysical uncertainties: In figure 8 we show how the LUX exclusionlimit shown in the left panel of figure 6 varies by including uncertainties arising from a) thestatistical uncertainty on our reference morphology, b) the variation of the Galactic parametersfor the reference morphology, and c) the baryonic morphologies that maximize/minimize thelocal DM density ρ , as discussed in section 2.1.As it can be seen, the statistical uncertainty related to the determination of the local DMdensity ρ affects the determination of model parameters very little, thus justifying the factthat most of the literature neglects it. On the other hand, the uncertainty arising by eitherthe ignorance about the exact value of Galactic parameters or the morphology of the visiblecomponent has sizable effects in shifting the constrained region in the parameter space. Noticethat the largest uncertainty on the exclusion limit arises from the variation of the Galacticparameters. The reason is that this variation leads to a large variation in the value of ρ (seethe second and third rows of table 1) which is larger than the variation in ρ due to eitherstatistical uncertainties or the choice of morphology, with the latter still being quite sizable,as we will discuss in the following.The situation is different if one looks at the GeV excess favored region versus the con-straint imposed by dSphs. In figure 9, we show the effect of Galactic uncertainties on theGeV excess favored region in the SSDM parameter space. Blobs of different color shadingare the regions that explain the GeV excess at 2 σ confidence level, shown as a red contour infigure 5, moving as a consequence of statistical, Galactic parameters, or baryonic morphology– 15 – m S [GeV] − − − − λ H S d S p h s Statistical 30 60 100 200 m S [GeV] − − − − λ H S d S p h s Galactic 30 60 100 200 m S [GeV] − − − − λ H S d S p h s Morphology Figure 9 . Singlet Scalar Model : Effects of Galactic uncertainties following the GeV excess interpre-tation. We display the effect of: statistical uncertainty, for our reference morphology BjX left panel);changing the Galactic parameters, for the same reference morphology (central panel); adopting dif-ferent morphologies that maximize/minimize the index γ , FkX and DiX (right panel). Criteria arediscussed in section 2.1 and values are reported in table 1. uncertainty. In this figure, for the baryonic morphology uncertainty, we choose the morpholo-gies which maximise/minimise γ . As it can be easily seen, again the statistical uncertaintyon a single morphology plays little role, not affecting conclusions, but the adoption of dif-ferent Galactic parameters and morphologies sizably shift the favored region, relieving (orworsening) tension with dSphs constraints, as it was already seen in figure 5. It is interestingto notice that although the variation of Galactic parameters produces the most sizable alter-ation of the index γ and of ρ (for assigned morphology, see table 1), these effects are partiallycompensated in the computation of the J -factor, and the largest variation of the latter isobtained as a consequence of varying morphologies (for assigned Galactic parameters). Following closely the procedure described in the previous section, now we compare the limitimposed by direct detection on the IDM parameter space for different cases of variation ofastrophysical uncertainties: In figure 10 we show how the LUX limit shown in the left panelof figure 7 varies by including uncertainties arising from a) the statistical uncertainty on ourreference morphology, b) the variation of the Galactic parameters for the reference morphol-ogy, and c) the baryonic morphologies that maximize/minimize the local DM density ρ , asdiscussed in section 2.1. For DM direct detection, the effects of the systematic uncertaintieson the Galactic parameters on the IDM are similar to the ones on the SSDM: On one hand,the statistical uncertainty related to the determination of the local DM density ρ affectsmildly the determination of model parameters. On the other hand, the uncertainty arisingby either the ignorance about the exact value of Galactic parameters or the morphology ofthe visible component has sizable effects in shifting the constrained region in the parameterspace. As discussed before, the largest uncertainty in direct detection limits arises from thevariation in the local DM density. The variation of the Galactic parameters for the referencemorphology leads to the largest variation in ρ , and hence the largest uncertainty seen in thecentral panel of figure 10.In figure 11, we show how the GC excess interpretation shown in the right panel offigure 7 varies with Galactic uncertainties. The dark gray dots show the constraint imposedby dSphs. Colored dots (green, blue and red) correspond to the regions of the parameterspace that explain the GeV excess, moving as a consequence of statistical, Galactic, or mor-phology uncertainties, respectively. For the baryonic morphology uncertainty, we choose themorphologies which maximise/minimise γ . It can be seen from the figure that the regions that– 16 – m H [GeV] − − − − − | λ L | L U X Statistical m H [GeV] − − − − − | λ L | L U X Galactic m H [GeV] − − − − − | λ L | L U X Morphology Figure 10 . Inert Doublet Model : Effects of Galactic uncertainties on the parameter constraints. Wedisplay the effect of: statistical uncertainty, for our reference morphology BjX left panel); changingGalactic parameters, for the same reference morphology (central panel); adopting different morpholo-gies such as they maximize/minimize the local DM density ρ , CjX and FiX (right panel). Criteriaare discussed in section 2.1 and values are reported in table 1. 30 60 100 200 m H [GeV] − − − − − | λ L | Statistical 30 60 100 200 m H [GeV] − − − − − | λ L | Galactic 30 60 100 200 m H [GeV] − − − − − | λ L | Morphology Figure 11 . Inert Doublet Model : Effects of Galactic uncertainties following the GC excess interpre-tation. We display the effect of: statistical uncertainty, for our reference morphology BjX (left panel);changing Galactic parameters, for the same reference morphology (central panel); adopting differentmorphologies such as they maximize/minimize the index γ , FkX and DiX (right panel). Criteria arediscussed in section 2.1 and values are reported in table 1. The colored dots (green, blue and red)correspond to the regions of the parameter space that explain the GC GeV excess; the dark gray dotsare in tension with the constraint imposed by dSphs. can simultaneously reproduce the measured DM relic abundance and explain the GC excessare quite reduced and typically in tension with the dSphs observations. Only marginal regionsare allowed by the dSphs constraint when taking into account Galactic and morphologicaluncertainties.It is to be noticed that figure 11 displays only one region of the parameter space, as fa-vored by the GC excess interpretation for both the Galactic parameters and the morphology,differently than in the case of SSDM. The variation of both morphology and Galactic param-eters impose a change in the IDM parameters (similar to what happens with the SSDM), butin both cases this shift ends up in a region which cannot reproduce the DM relic abundance.The “shifted” region is not visible in the figure as it is forbidden by the cosmological con-straint, which henceforth practically sets limits on the DM interpretation of the GC excess,but only for some combinations of the Galactic parameters. In this work we have studied how the uncertainties associated to Galactic core quantities, suchas the local galactocentric distance, local circular velocity, and the morphology of the stellardisk and bulge, affect the determination of DM distribution, and eventually propagate when– 17 –onstraining new physics scenarios. We have set up a systematic scan of the major sourcesof uncertainty in the determination of the DM distribution in the MW, testing (a) statisticaluncertainties; (b) variation of Galactic parameters; (c) variation of baryonic morphology.While the purely statistical uncertainties affecting the observed RC and the normalizationof the visible mass component do not sizably affect the constraints on new physics modelparameters, a significant impact on the allowed model parameter space is due to the currentignorance on the morphology of the baryonic component, and on the determination of Galacticparameters.We have shown that the latter significantly affect the constraints of two specific models,the SSDM and the IDM, which we have chosen as testbeds for the relatively simple dependenceof their phenomenology on the key parameters. Our main findings, which we summarizebelow, show the need for the study of these uncertainties in more complex scenarios, and anincreased communication between the particle physics and the astronomy communities in avirtuous interplay.The largest effects on the SSDM and IDM parameter space are obtained as a consequenceof varying the Galactic parameters ( R , v ): The variation of ( R , v ) between its currentlyestablished extreme values pushes the determination of the local DM density ρ beyondthe usually adopted bounds (which are taken for assigned Galactic parameters, and includestatistical uncertainty only, in most cases), with major effects especially on direct detectionresults. Interestingly, the remarkable changes imposed by the Galactic parameter variationalso on the index γ mitigate the effect on the determination of the J -factor, which seesthe uncertainty on the baryonic morphology as a primary source of uncertainty for indirectdetection.As an example of the above, we recall here the case of SSDM: The region of the parameterspace which permits an interpretation of the GC excess in terms of DM annihilation is allowedwith a given set of Galactic parameters, but it could be also entirely ruled out by constraintson the relic density if the other extreme values for ( R , v ) are adopted.Accounting for the astrophysical uncertainties described above, will be even more crucialin the case a tantalising DM signal will be discovered in the next–generation of direct andindirect experiments. In that case, the accurate reconstruction and interpretation of thesignal in the context of concrete particle physics models will require the full treatment of allastrophysical uncertainties presented in our work.On the other hand, future astronomical data will help in reducing significantly thoseuncertainties. In particular, the Gaia mission is expected to improve the determination ofthe Oort constants, and yield a reduction of uncertainties on the determination of ( R , v ),as already shown possible with the first year data release [98]. Acknowledgments. We thank P. D. Serpico for fruitful discussion and comments on themanuscript. N. Bernal is supported by the São Paulo Research Foundation (FAPESP) un-der grants 2011/11973-4 and 2013/01792-8, by the Spanish MINECO under Grant FPA2014-54459-P and by the ‘Joint Excellence in Science and Humanities’ (JESH) program of the Aus-trian Academy of Sciences. N. 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