Particle Constraints from the

C. TYLER Department of Astronomy & Astrophysics, The University of Chicago, Chicago, IL 60637

Abstract. It is widely thought that neutralinos, the lightest The cold dark matter (CDM) scenario starts with a weakly supersymmetric particles, could comprise most of the dark interacting massive particle (WIMP), created abundantly in matter. If so, then dark halos will emit radio and gamma ray the big bang and surviving today to dominate the matter den- signals initiated by neutralino annihilation. A particularly sity in galaxies. The velocities of stellar orbits imply that promising place to look for these indicators is at the center dark matter occupies galaxies with a particular radial density of the Draco, and recent profile. Over a wide range in radius, the density appears to 2 measurements of the motion of its stars have revealed it to be fall off as ρ r− , which is the same structure as would be an even better target for dark matter detection than previously found in a self-gravitating∼ isothermal sphere. But when sim- thought. We compute limits on WIMP properties for various ulations of the cosmological CDM evolution are performed, models of Draco’s dark matter halo. We find that if the halo taking into account the influence of N-body dynamics and is nearly isothermal, as the new measurements indicate, then tidal interactions, the profiles are found to be more compli- current gamma ray flux limits prohibit much of the neutralino cated at large and small r (e.g., Navarro, Frenk, and White parameter space. If Draco has a moderate magnetic field, (1996), Moore et al. (1999)). We discuss these profiles in then current radio limits can rule out more of it. These results more detail in subsection 2.1. are appreciably stronger than other current constraints, and Simulations universally generate a central cusp in every γ so acquiring more detailed data on Draco’s density profile dark halo. Slopes vary, but ρ r− remains for some becomes one of the most promising avenues for identifying positive γ, and therefore the density∼ blows up at the center. dark matter. However, a controversy ensues at this point because obser- vations often differ. 21 cm rotation curves from low surface brightness (LSB) galaxies have presented evidence for cen- 1 Introduction tral cores of approximately constant density (McGaugh and de Blok, 1998). A separate study of 16 LSB galaxies con- Despite the popularity of the neutralino as a dark matter can- tends that beam smearing effects render these measurements didate, efforts to constrain its properties have only been able unreliable, such that cusps are still fully consistent with the to rule out a small fraction of the relevant parameter space. data (van den Bosch et al., 2000). Many of the same galax- Direct detection experiments based on scattering with nuclei, ies have since been re-examined with high resolution opti- and indirect searches based on neutralino annihilation, have cal rotation curves, and that work found 30 LSB galaxies shared this difficulty. But recent results can improve the sit- evidently possessing cores, not CDM cusps (de Blok, Mc- uation. New observations of the Draco dwarf galaxy reveal Gaugh, and Rubin, 2001). Dwarf galaxies too have long that it is strongly dark matter dominated, and that its dark been considered problematic for CDM because they typi- matter distribution is at least nearly isothermal. In this paper cally have the linearly rising rotation curves of a constant we investigate the detectability of WIMP annihilation sig- density core (Navarro, Frenk, and White, 1996), but a subse- nals from several appropriate halo models. Interestingly, if quent study of 20 dwarf galaxies gives rise to interpretations 2 Draco is indeed isothermal down to very small radii, we will ranging from cores to steep r− cusps (van den Bosch and show that current gamma ray measurements rule out a signif- Swaters, 2001). These issues constitute an area of much de- icant fraction of the neutralino parameter space. If one makes bate, but in general, because many observations favor cores an additional assumption that Draco harbors magnetic fields, over cusps, it is of great interest to determine what processes then current radio measurements rule out additional param- might prevent or destroy a cusp. eter space. For example, a 1 µG field would eliminate most If a WIMP cusp were to realize, and if WIMPs exist to- of it. Other halo models yield weaker current limits, some of day in equal numbers with antiWIMPs (or if they are their which become significant with upcoming detectors. own antiparticles), then we can expect densities high enough The choice of a particular dark matter distribution is crit- for dark matter annihilation to become significant. If the an- ical to both gamma ray and radio limits, and we devote sec- nihilation products lead to observable particles, then cusps tion 2 of this paper to that consideration. The assumption become hot spots to search for WIMPs. of a magnetic field in Draco is discussed in section 4. The Perhaps the most likely WIMP candidate is the neutralino remainder of this introductory section provides the relevant (χ), which is the lightest supersymmetric particle (LSP) in background. the minimal supersymmetric extension of the standard model. Supersymmetry is a symmetry relating fermions to bosons, 2TheDarkHalo and thus it introduces a new class of particles which are “su- γ perpartners” to standard model particles. The theory helps A power law cusp’s density profile has the form ρ r− . ∼ to remedy the hierarchy problem set by the electroweak and Binney and Tremaine (1987) have put forth some reasoning Planck scales. In models with conservation of R-parity, a su- for the expectation of a singular isothermal sphere (SIS) dis- persymmetric particle is prohibited from decaying into only tribution to a first approximation, where γ =2. But do steep non-supersymmetric ones, and since the neutralino is the LSP, cusps like this really exist in nature? there is no supersymmetric state of lower energy available. Observations of galaxy centers with the Hubble Space Tele- Thus the χ is stable, making it a well motivated dark mat- scope (HST) shed some light on this question (Crane et al., ter candidate (for a review of supersymmetric dark matter, 1993; Gebhardt et al., 1996). These authors have constructed see Jungman, Kamionkowski, and Griest (1996)). three dimensional luminosity densities based on an assumed Each χχ¯ annihilation produces quarks which bind primar- spherical geometry in the target ellipticals and in the bulges ily into pions. Neutral pions usually decay to gamma rays; of spirals, further assuming that the light traces the mass charged pions usually decay to neutrinos and muons, and in these systems. Among power law galaxy centers, two subsequently electrons (and positrons), which generate radio populations emerge: bright galaxies (roughly MV 20) waves by synchrotron radiation in an ambient magnetic field. with slopes in the range 0.2 γ 1.5, and faint≤− galax- ≤ ≤ (This approach does not wholly rely on supersymmetry, but ies (MV 20) with γ ranging between 1.5 γ 2.5. rather on some DM particle which annihilates into pions.) The peaks≥− in this bimodal distribution are at 0.8≤ and 1.9,≤ re- Therefore, a cusp produces observable radio and gamma ra- spectively. Many faint galaxies in the sample therefore have diation (Berezinsky, Gurevich, and Zybin, 1992; Berezinsky, nearly isothermal density profiles at their centers. Bottino, and Mignola, 1994); the trick is just finding a galaxy with a cusp. 2.1 Halo Models The candidate galaxy should have a very high mass-to- light ratio (M/L) to ensure that its internal gravitation is DM An ordinary kinetic-molecular gas, self-gravitating and held 2 dominated, and it should be nearby to facilitate the detection isothermal, will assume the SIS density profile ρ r− .In- ∼ of a weak signal. As it turns out, the highest known M/L stead, we have a collisionless system of WIMP particles. If comes from a dwarf satellite of our own Galaxy. The Draco they are allowed to evolve under their collective gravitational dwarf spheroidal galaxy, only 79 kpc distant from (van potential, the initial infall can be expected to be clumpy, and den Bergh, 2000), was recently re-examined for detailed stel- as a result, the overall potential seen by any one WIMP is the lar velocities (Kleyna et al., 2001a). A new, higher than pre- superposition of the central field and the temporary group- viously estimated M/L was found to be 440 240M /L , ings. Trajectories are thus altered, and over a few orbital pe- with a mass of M 8.6 107 M . Draco has± the mass of a riods, a new smooth centrally weighted distribution will be dwarf galaxy and the' starlight× of a globular cluster. obtained. This process was termed “violent relaxation” by Furthermore, the candidate galaxy should have a high cen- its author, Lynden-Bell (1967), wherein the central collapse tral WIMP density, without contaminants such as massive is achieved by transporting angular momentum outward. baryonic clouds or a central black hole. The stellar veloc- The solution at the end of this relaxation turns out to be ity dispersion measurements in Draco are consistent with an the Maxwellian velocity distribution with constant velocity isothermal dark matter halo (Kleyna et al., 2001a), implying dispersion at every radius, although the population dwindles a steep central density. Its present hydrogen gas population somewhat beyond the virial radius, as those particles do not is less than 450 M (Young, 1999), with no evidence of a often interact with the lumpy center and therefore do not join central black hole. The Sloan Digital Sky Survey reported the infall. One important caveat, however, is that the violent 3 that tidal deformation can only exist at a level of 10− of relaxation process proceeds until a steady state is achieved, the central stellar surface density (Odenkirchen et≤ al., 2001), whether or not that state is the Maxwellian result of complete supporting the case for dynamical relaxation and therefore relaxation. This means that it is possible for the final state meaningful extraction of mass density from velocity infor- to retain features unique to its own collapse sequence (one mation. Given these properties, we present particle dark mat- reason why simulations are useful for this work). ter constraints enabled by observations of the stars in Draco. Supposing the end state of the process is complete relax- The next section provides more detail regarding halos with ation, we have a system of self-gravitating particles with a cusps and cores. Section 3 follows with the calculation of Maxwellian velocity distribution at every point in space, the the observable flux created by WIMP annihilations, includ- solution of which is again the SIS density profile (Binney ing electron losses by inverse Compton scattering, which is and Tremaine, 1987). If the system fails to achieve com- an important correction not usually considered in such anal- plete relaxation, then the innermost orbits may not be fully yses. Section 4 presents the results of these calculations, us- populated, and one might expect γ<2 as is often seen in ing observational bounds from EGRET and the VLA, with a numerical simulations. discussion of anticipated improvements by new experiments, Rotation curves of galaxies usually show a constant ve- and a comparison among various halo profiles. We summa- locity over a wide range of radial values. On the contrary rize in section 5. however, some observations reveal core regions of about 1 −

2 10 kpc in radius which appear to enclose an r-independent mass. The following are numerical values for these quanti- 26 3 matter density. This is evident in the rotation curves of spiral ties, taking mχ = 100 GeV and σv χχ¯ =10− cm /s, galaxies which tend to rise linearly at small radii, and those listed here for the γ = 2 case (the generalizationh i to γ<2 is of some dwarf galaxies which rise linearly for the full extent straightforward): of the observed galaxy. To model this, one typically invokes a modified isothermal profile: a spherically symmetric den- Rext =4.6kpc 14 sity profile which smoothly blends a constant density core Rmin =9.4 10 cm × 5 3 with isothermal behavior outside the core. nhalo =2.2 10− cm− In order to calculate observables from annihilation reac- × 9 3 n(Rmin)=5.7 10 cm− . tions, one must start by assuming some profile. For exam- × ple, gamma rays from annihilating neutralinos were origi- As will be discussed in subsection 3.1, the gamma ray flux nally calculated by Berezinsky, Gurevich, and Zybin (1992) observed from a DM clump depends on Rmin,soitisworth 1.8 using ρ r− for the . In Gondolo (1993), ∼ considering whether it is appropriate to use the free fall time the same signature from the was as the cusp formation time. Since violent relaxation is a com- calculated assuming ρ a2/(r2 + a2), a modified isother- ∼ plex process one might imagine some longer characteristic mal model with core radius a. Non-constant central densi- time scale. Without any particularly compelling choice for ties with radial power law exponents have been employed this time scale, we can set a conservative upper limit by using in different contexts: γ =1(NFW: Navarro, Frenk, and the age of the . For Draco, using τuniv =15Gyr, the White (1996), used for example in Gondolo (2000) and Blasi, resulting gamma ray flux is found to be about 5 times weaker Olinto, and Tyler (2002)), =15 (Moore: Moore et al. γ . than it would be with eq. (1), for any choice of mχ. (1999), used for example in Calc´aneo–Rold´an and Moore The NFW and Moore halo profiles mentioned previously (2001)), and γ =2(isothermal: used for example in Blasi, are the most popularly quoted “universal profiles” obtained Olinto, and Tyler (2002)). It is worth mentioning that γ ap- from simulations. Both employ power-law cusps internal to pears to vary with the resolution of the CDM simulation, and 3 some scale radius, outside which the density falls off as r− . at small radii, a realistic model of the high cusp density can The behavior is nearly isothermal in the vicinity of the scale be expected to require more accurate simulations. radius rs. In this work, we consider seven distinct models of Draco. In calculating Rext, we have used the customary NFW These include steep power-law cusps with γ 2, the Moore profile for our own Galaxy, profile, and isothermal halos truncated by constant≤ density 1 2 cores. r − r − A power law DM halo profile must include a small con- nhalo = n0 1+ , (3) rs rs stant density core inside the cusp for any γ>0, due to the     fact that inside some radius, the annihilation rate gets so large whose scale radius rs 30 kpc, total size Rext 300 kpc, ' 4 '3 that the over-density is destroyed as fast as new infall can and characteristic density n0 7.5 10− cm− , can be ' × fill the region. The constant density region, found by setting set by the boundary conditions imposed by the halo density 25 3 cusp forming time scale (the clump crossing time, or approx- local to our solar system (6.5 10− g/cm ), and the total 12 × imately the free fall time 1/√Gρ¯) equal to the annihi- mass of the Galaxy (10 M ), subject to an estimate that ∼ lation time scale (1 ), establishes a characteristic Rext/rs 10. As it turns out, various mod- /nχ σv χχ¯ ' inner radius (Berezinsky,h Gurevich,i and Zybin, 1992). For els roughly agree on the local dark matter density at Draco’s example, if γ =2(SIS), we have (Blasi, Olinto, and Tyler, distance from the center, so the choice of NFW as the Milky 2002): Way DM profile is not particularly important here.

1 The Moore profile (Moore et al., 1999) is more recent, de- 1 4 2 nhalo riving from higher resolution computations than NFW. We Rmin = Rext σv , (1) h iχχ¯ 4πGm therefore include it in our calculations of Draco. The Moore  χ  where the χχ¯ annihilation rate (cross section velocity) profile is σv is approximately constant in the low velocity× limit χχ¯ 1.5 1.5 1 happropriatei for galactic dark matter (Berezinsky, Gurevich, r − r − n(r)=n0 1+ , (4) and Zybin, 1992), and the time scales mentioned above are rs " rs # 8     5 10 years at Rmin. The external radius Rext of Draco’s ∼ × 2 3 halo is where it blends smoothly into the halo. If where n0 2.5 10− cm− and rs 0.35 kpc can be fixed ' × ' we define the cuspy halo density as in similar fashion, using Draco’s total mass, the Milky Way density local to Draco, and a concentration factor R /r = γ (2) ext s nχ Ar− , 10 as boundary conditions. ' then Rext and the coefficient A can be jointly set by equat- In a recent paper (Power et al., 2002), a convergence study ing the χ density at the edge of Draco’s halo to the ambient is described which attempts to resolve central halo densi- Milky Way halo density at Draco’s location, nhalo. Addition- ties in CDM simulations. The result for a galaxy-sized halo ally, integrating eq. (2) out to Rext must give Draco’s total appears to be a decreasing exponent moving inward from

3 the virial radius (comparable to Rext in this work), reach- steep cusp with complete impunity because many things can ing γ 1.2, without any particular inner power law slope alter the inner structure of the galaxy, but under a certain convergence.≤ Because this is a state-of-the-art simulation, set of conditions, such a profile becomes completely reason- the fact that the inner slope is not especially steep should not able. In the next section, we discuss ways in which a steep be ignored. However, the authors note that their innermost cusp might be softened (provided that the relaxation process resolved point is only 0.5% of the virial radius, which is out- progressed far enough to form one initially), and why Draco side our region of interest for Draco, and that the effect of should avoid this softening. poor resolution is usually to generate artificially low central densities. The reasons which cause this simulation to differ 2.3 Retaining a Steep Cusp in inner slope from others like Moore are currently unclear, but external to a few times the innermost resolved radius, the Calculated as in eq. (1), Rmin is only on the order of a few various CDM halo profiles are difficult to distinguish. In any AU (1AU=1.5 1013 cm) to a few 1015 cm, depending × × case, the central drop in γ is not evident in Draco. primarily on the value of σv χχ¯. It stands to reason that ex- ternal influences such as theh tidali force from the Milky Way 2.2 Modeling Draco should have negligible effect on a central cusp that small; tidal truncation is in fact observationally ruled out below a The stellar motion measurements from Draco are best fit by radius of 1 kpc (Kleyna et al., 2001b). But it is important γ 1.7 over 1 kpc (Kleyna et al., 2001a); strong veloc- that we consider any other theoretical motivation for disrupt- ' ∼ ity anisotropies and strong deviations from isothermality are ing cusps down to such small scales as Rmin. (The appro- ruled out over this radial scale. (Star counts and luminosity priate scale to verify for the validity of the gamma ray limits density are insufficient for determining the mass profile, be- derived herein is indeed Rmin; however, as described in sub- cause according to the same stellar velocty study, mass does section 3.2, the radio synchrotron emanates from within a not follow light in Draco.) In the central 0.2 kpc, γ appears larger radial size 0.01 pc.) Several such ways to spoil the to be 2, although other cusp slopes and∼ cores cannot be cusp are addressed∼ here. ≥ ruled out at small radii with the current data set. So although (1) It has been proposed that the supermassive black holes the observations support the SIS at intermediate radii, they found at the centers of large galaxies will change the dark don’t directly speak for radii smaller than about 10 pc, inside halo cusps, although how they will change the cusps is a which our annihilation signal originates. matter of some debate. One argument is that a cusp would Since annihilation rate goes as the square of the density of steepen into a spike due to accretion of DM onto the black 2 3 1 χ particles (nχ σv χχ¯ cm− s− ), the choice of γ is very im- hole (Gondolo and Silk, 1999; Gondolo, 2000), causing en- portant to the result.h i Observed γ values from rotation curves hanced annihilation signatures. If so, then any value of the are neither accurate enough nor consistent enough to use uni- cusp slope index γ over-produces the expected synchrotron formly for this purpose. Although a positive detection of signature beyond observational limits. Alternatively, if nu- gamma or radio signals could identify a particle dark mat- clear black holes pair up in a binary system during a galaxy ter source, the difficulty in choosing γ with any confidence merger, as one would expect after dynamical friction pulls has made it hard to realistically constrain WIMP parameters both holes to the center of the remnant but before they merge such as mass and cross section on the merit of observational into one hole, then the two-body interactions of the holes upper limits alone. would throw other masses out of the center (Milosavljevi´cet However, this situation may soon change. Draco’s mass al., 2001; Merritt and Cruz, 2001). In that case, both bary- has recently been shown to be completely dominated by dark onic and dark matter would be affected, and the central cusp matter, with stellar velocities that favor nearly isothermal or- would be softened; this result is corroborated by an HST sur- ganization, and the authors of these observations contend vey (Ravindranath, Ho, and Filippenko, 2001), in that central that the prospects are excellent for improving the velocity mass deficits in galaxies (i.e., the departure from a cusp) cor- curve data further by sampling more stars (Kleyna et al., relate with the masses of their central black holes. 2001a). This may become a particularly fruitful approach The relevance of this issue depends on whether or not the to further particle DM studies. So to agree well with current Draco dwarf harbors a central black hole. As a general trend, data on Draco, we use halo models with γ = 2.0, 1.9, 1.8, it appears that ellipticals and spheroidals brighter than ap- and 1.7, the Moore profile, and isothermal profiles truncated proximately 2 1010 L do have supermassive black holes, with cores at 1 pc and 0.1 pc. and dimmer galaxies× may not (Sparke and Gallagher, 2000), Other galaxies currently have more detailed velocity dis- although there are exceptions. For Draco, unless its history is persion or rotation data. We choose Draco for this study devoid of mergers, the observed velocity curve should reveal partly for its proximity, and partly for its lack of features a softened core if there were a supermassive central hole. In which would be expected to disrupt the cusp. Baryonic mat- the absence of direct evidence for such a hole in Draco, we ter and central black holes are factors which contaminate assume that Draco has none, so that the concerns of the pre- this kind of analysis in other galaxies including our own, but vious paragraph only afflict other galaxies. Draco is evidently little more than a spherical clump of self- (2) Could the cusp be scattered by baryonic matter? El- gravitating dark matter. We can’t assume that Draco has a Zant, Shlosman, and Hoffman (2001) have proposed that the

4 missing cusps in galaxy cores are due to interactions between 3 Annihilation Signals dark matter and clumpy baryons, in a scheme which always transfers energy from baryons to WIMPs, causing the bary- 3.1 Gamma rays onic matter domination at small radii. Interestingly, many dwarf galaxies exhibit rising rotation curves out to scales like Supposing that Draco has a dark SIS profile populated by an- 10 kpc (see the discussion of dwarf galaxies in Navarro, nihilating neutralinos, one can calculate the gamma ray emis- Frenk,∼ and White (1996)). But Draco, with a much higher sion caused by decaying pions made in the χχ¯ annihilation: M/L ratio, has constant velocity across all observed radii. π0 γγ. The rate of gamma ray production above a thresh- → So even if baryonic matter is to blame for destroying cusps old energy E0 per time per volume is in other galaxies, it has a diminished role in Draco. 2 (3) Some nonlinear dynamical process among clumpy dark qγ = n σv χχ¯Nγ(Eγ >E0) . (5) χh i matter might expand the central region, by way of the halo forming from a swarm of smaller and denser subhalos which In this work, we adopt a method of approximation that has formed earlier. For our purposes, the effect is difficult to es- been used previously in this context (Berezinsky, Gurevich, timate, because current simulations have not resolved much and Zybin, 1992; Bergstr¨om, Edsj¨o, and Ullio, 2001) for the below Draco mass objects. However, there is no indication number of gamma rays produced above threshold per annihi- 0 of a cutoff immediately below this mass scale in the simu- lation, Nγ(Eγ >E0).Theπ production by χχ¯ annihilation lations (Moore et al., 2001). Metcalf (2001) finds that even follows very small objects (> 103 M , should they exist) are dense dNπ 8x 1.5 enough to survive tidal disruption. But these objects are too K e− /(x +0.00014) , (6) dx ' π light to experience significant dynamical friction, so they still 2 behave as collisionless DM, and probably do not disrupt the with x Eπ/mχc ,andKπ constant. For the two photon smooth halo. Objects near the mass of the host halo will decay process,≡ the probability of making a photon per range settle to the center due to dynamical friction (Metcalf and of energy Eγ is 2/Eπ.Thenwehave Madau, 2001), and be tidally stripped when they pass inside 1 8x the central region (Metcalf, 2001). So is there an intermedi- e− dx 8 ( )= (7) ate scale dark object which does ruin the 10 M halo’s Nγ Eγ >E0 Kγ 1.5 , E /m c2 (x +0.00014) x cusp? There is no immediately obvious reason∼ why not. Z 0 χ

However, it is interesting to note that globular clusters where the constant Kγ is set by requiring that one third of 6 (typically 10 M ) are not observed to contain signifi- the total energy released per neutralino annihilation go into ∼ cant amounts of dark matter (a typical globular cluster ra- gamma rays, because about one third of the particles pro- tio of 2 M /L is given by Binney and Tremaine (1987)), duced by ¯ are neutral pions, which decay via the two pho- χχ which may mean that dark matter clumps of mass scales be- ton channel. The remaining two thirds are divided equally + low those of dwarf galaxies are suppressed for some reason. among π and π− particles, whose decay products will be In fact, it has been proposed that globular clusters could be the topic of the next subsection. Berezinsky, Gurevich, and disrupted remains of larger dwarf galaxies (Cˆot´e, West, and Zybin (1992) have shown that gamma ray line flux due to Marzke, 2001), because their numbers are appropriate to ex- χχ¯ γγ makes a very minor correction to the flux pre- plain the missing dwarf galaxies expected by CDM theory. dicted→ above, and we will ignore it here. The possibility that intermediate mass dark objects don’t 2 1 To get the observable gamma ray flux (photons cm− s− ), exist in large numbers can be dramatically strengthened by integrate eq. (5) over the volume of the source, and divide by upcoming observations. If substructure DM clumps form 4πd2. This procedure is specific to the r-dependence of the on small mass scales like 106 M , then thousands of such halo model used, but for example, with an SIS halo where clumps will be observable above the cosmic microwave back- 2 nχ = Ar− , ground by their annihilation products (e.g., by the Planck ex- periment), even if they turn out to obey gently cusped NFW 4A2 1 profiles (Blasi, Olinto, and Tyler, 2002). These observa- Fγ (Eγ >E0)= 2 σv χχ¯Nγ (Eγ >E0) , (8) 3d h i Rmin tions are sensitive to a lower mass cutoff on substructure DM clumps. So if more than 1000 sources are not found, where d is the distance from the Earth. For cusps steeper then it is difficult to see how∼ a steep cusp could be inhib- than γ =1.5, the general result is that the gamma ray flux 3 2γ ited in a dwarf galaxy like Draco by dark matter alone. (This varies as Rmin− .Atγ 1.5, as in the Moore profile where holds unless the WIMPs do not annihilate like neutralinos, γ =1.5, the integral over≥ the volume of the source becomes but that result is immediately consistent with non-detections dependent on an outer radius instead of Rmin. For example, from Draco anyway.) If these CMB foreground sources are in the Moore profile, the emission region is of size r and ∼ s found, then the results in section 4 of this paper can be con- the flux depends only very weakly on Rmin. For an isother- strued as either limits on neutralino properties or evidence for mal halo with a constant density core, eq. (8) applies, with important dark substructure dynamics preventing steep cusps Rmin replaced by Rcore. Section 4 gives the results of this in the cores of dwarf galaxies. process for Draco.

5 3.2 Radio Synchrotron electron considered in this work, and for any given combina- tion of Ee and r, the lifetime is the fastest of these processes. The charged pions produced in χχ¯ annihilations decay as They are: (1) loss of energy by synchrotron radiation, (2) + + loss of energy by inverse Compton scattering (ICS) against π µ ν and π− µ−ν¯ . (9) → µ → µ the cosmic microwave background (CMB), (3) loss of en- Muons subsequently decay via ergy by ICS against the local synchrotron photons, and (4) annihilation with positrons. For pair annihilation, there is a + + µ e ν¯ ν and µ− e−ν ν¯ . (10) characteristic time scale; for the other processes, there is a → µ e → µ e loss rate dE /dt for which Electrons and positrons produced in this way will generate e synchrotron radiation if there are ambient magnetic fields. E τ e . (17) This calculation was first done by Berezinsky, Gurevich, and ' dEe/dt Zybin (1992) for the Galactic center, and is repeated in Blasi, + Olinto, and Tyler (2002) and here, with multiple new correc- At a given Ee,wefindthate e− pair annihilation can tions applied. (For some parts of this section, cgs units have be dominant in the center of the clump, followed outward been provided for clarity.) by a shell dominated by ICS against synchrotron photons, We need to know the number spectrum of electrons and followed by the outermost shell dominated by synchrotron losses and ICS against the CMB. The region dominated by positrons produced in each neutralino annihilation, dNe/dEe. We use a formulaic simplification of this function (Hill, 1983), ICS against synchrotron photons generates most of the total 3/2 flux from the clump. Synchrotron ICS and e± pair annihi- which reduces to dN /dE Ee− at low energy, and e e ∼ lation are discussed below; both have the effect of removing drops toward zero as it nears the cutoff Ee mχ.Thefull expression used is ' emitters preferentially from the dense central regions, reduc- ing the synchrotron signal. 2 ( ) ( ) dN mχc Eµ/r¯ dN π dN µ Employing a formula from Rybicki and Lightman (1979), e = W µ e dE dE , (11) dE π dE dE π µ the synchrotron power is e ZEe ZEµ µ e 2 dEsyn 15 2 2 where r¯ (m /m ) , =1.6 10− B E erg/s , (18) ≡ µ π dt × µ e 3/2 2 15 E − E ~ = π 1 π for Bµ = B in microgauss. It is convenient to approximate Wπ 2 2 , (12) | | 16 mχc − mχc the synchrotron as if all the radiation were emitted at the peak     frequency (which is different for each electron): and 12 2 (π) 2 νpeak =1.5 10 BµEe Hz . (19) dNµ 1 mπ × = 2 2 (13) dEµ Eπ m m Eqs. (15), (18), and (19) combine to give the radio signal π − µ we seek as follows: and dn dE dE erg (µ) 2 3 = e e syn (20) dNe 2 5 3 Ee 2 Ee jν 3 . = + . (14) dEe dν dt cm sHz dEe Eµ "6 − 2 Eµ 3 Eµ #     The luminosity is the integral of jν over the volume of the These last two equations, eqs. (13) and (14), give the decay DM source. products from charged pion and muon decays, respectively. At this point, some details need to be considered. The Note that dNe/dEe without any superscripting indicates the above picture is correct but not complete; three amendments number spectrum of electrons from the entire chain of decays are analyzed presently. following a single χχ¯ annihilation. (1) Inverse Compton scattering of relativistic electrons and The charged particle injection is positrons against the synchrotron photons created in the DM clump is the most important correction. ICS power is (Ry- 2 dNe bicki and Lightman, 1979) q = n σv ¯ , (15) e χh iχχ dE  e  dEics 4 2 2 3/2 2γ γ = σT cβ γ Uph , (21) so that q Ee− r− because n r− for a cusp with dt 3 e ∼ χ ∼ logarithmic slope γ. Throughout this subsection we will 25 2 − where σT =6.65 10− cm is the Thomson cross section continue to use this generic cusp. The electron distribution is × 2 appropriate in the limit hν m c , β 1,andUph is  e ' dne the energy density in the photon field. Therefore, the photon = qeτ, (16) dEe population with the greatest energy density controls the rate of ICS. For the CMB, where τ = τ(Ee,r) is the average lifetime of an electron of 4 energy Ee. There are four processes limiting the life of an Ucmb = aTcmb , (22)

6 15 3 4 for a =7.56 10− erg cm− K− and Tcmb =2.728 K; For Draco, Rics,syn 0.01 pc, depending loosely on the whereas for the× synchrotron photon population, one must in- WIMP mass and the ambient' magnetic field strength. In- 2 2 tegrate jν over all possible e± energies (from mec to mχc ) side this radius, the electron population is limited by ICS γ 1/2 and over all lines of sight. against synchrotron photons and decreases as r− − ; out- Unfortunately, this double integration for Usyn requires a side this radius the electron population is limited by syn- priori knowledge of the electron distribution in energy (Ee) chrotron losses and ICS against the CMB, and decreases as 2γ and in space (r), since dne/dEe is contained in jν via eq. (20). r− . Interestingly, Rics,syn becomes the characteristic ob- The electron distribution depends on the rate of ICS losses servable size of the radio source, and it is resolvable to inter- and depends therefore, in turn, on Usyn (via eqs. (16), (17), ferometers with baselines of order 100 km. + and (21)). Consequently, our full calculation of dne/dEe (2) Pair annihilation between e− and e particles would is numerical and iterative; however, some assumptions and have an important role in determining the flux from a DM simplifications will get us the broad brush answer as well. source in the absence of the ICS described above. But in the First, assume that the solution takes the form of separable presence of ICS losses, pair annihilation makes little differ- power laws which we will solve for: ence to the resulting radio flux. The pair annihilation process is still important, because dne α β it reduces the population of the lower energy pairs, which (Ee,r) Ee r . (23) dEe ∼ would otherwise upscatter photons frequently enough to spoil Then we need a further assumption, that the dominant con- the synchrotron spectrum. Although in much of the DM tribution to the photon density at some point r comes from source, most electrons Compton scatter with photons, most the region inside r only. One might expect this to hold, since photons do not Compton scatter with electrons because they all photons generated inside r will pass through r, but only a escape first; it is these synchrotron photons whose survival small fraction of those produced outside r will. This is still we are considering now. In the case of Draco, and for the an unjustified assumption which can be checked afterward most optically thick line of sight through its diameter, the by trying alternative values for the exponent β; the numeri- optical depth for a photon against synchrotron self Compton 3 cal results do validate the assumption as well. scattering is of order 10− , making it a negligible effect on Let’s define a shell of width ∆r such that a photon spends the resulting observable spectrum. But one needs to calculate ∆r/c time in that region. Given these assumptions, and using the pair annihilation rate to discover that fact. We can estimate the e± lifetime for the case where elec- r0 as a radial variable of integration, trons annihilate before they lose a significant fraction of their r dUsyn 1 α β dEsyn ∆r 2 original energy via synchrotron or ICS as 2 Ee r0 4πr0 dr0 . (24) dEe ∼ 4πr ∆r Rmin dt c 1 Z τe+ (28) ' ne σe ve (The region inside Rmin makes its own contribution as well, h − ± − i but it turns out to be small.) Combining with eq. (18), we get with ve c, and the angle brackets indicating an average α+2 β+1 over E '. (This is equally valid if we interchange all the dUsyn/dEe Ee r . Then upon integrating over Ee, e− ∼ β+1 we get Usyn r . plus and minus signs, but we will keep track of signs in order From eq. (17),∼ to distinguish between particles and antiparticles, for clarity.) With eq. (16), we have 1 β 1 τ E− r− − , (25) e dn + q + ∼ e = e . (29) dE + n σ v and from eqs. (15) and (16), e h e− e± e− i The term in angle brackets is computed from the equations dne 3/2 2γ 1 β 1 in Svennson (1982) (also consistent with Coppi and Bland- (Ee− r− )(Ee− r− − ) . (26) dEe ∼ ford (1990)), particularly

So, simply equating the exponents in eqs. (23) and (26), we ∞ ∞ σv e = f(γ+) f(γ )(σv)e dγ dγ+ , (30) obtain α = 5/2 and β = γ 1/2,or h i ± − ± − − − − Z1 Z1 dne where f is a distribution function for e± of Lorentz factor γ , 5/2 γ 1/2 ± Ee− r− − . (27) and (σv) is the angle averaged reaction rate per pair given dEe ∼ e± explicitly in Svennson (1982); here we quote the asymptotic This holds wherever electron losses are fastest by ICS in form, although the full formula has been used in the numeri- the synchrotron photon field. In the case of losses being dom- cal work described in this paper: inated by synchrotron radiation and ICS against the CMB in- πe2 1 stead, the loss mechanism is independent of r, so we would ( ) (ln 4 2) ( 1) σv e± γ+γ γ+,γ . (31) 5/2 2γ ' mec γ+γ − − −  have dn /dE Ee− r− . Since the two forms of elec- e e ∼ − tron losses have different power laws in r (and the same Eq. (30) is adapted by replacing f(γ ) dne/dEe,and − → power law in E ), there is a transition radius Rics syn (which f(γ+) δ(γ+);thatis,forachosen positron of energy γ+, e , → is independent of Ee) between the two behaviors. its survival time is a function of the electron distribution.

7 Then, we have ne− σe± ve− expressed as an energy inte- per-unit-length absorption coefficient: gral which dependsh on the e idistribution that we’re trying ± c2 dE ∂ 1 dn to solve for. Therefore this pair annihilation calculation must α = syn e . (34) ν −8πν dt ∂ν ν dν be done numerically. The dependences here are nested in-    + finitely, which is to say that τe depends on ne− which de- SSA absorbers of a particular photon are electrons of similar + pends on τe− which depends on ne and so on. It is most energy to the original synchrotron emitter which made the efficient to try to determine the correct dependences apriori, photon. As such, SSA is only effective below some νcrit, and then integrate accordingly. where the corresponding electrons are more numerous. So if we neglect the natural logarithm factor built into The source function for an absorbing source is Sν = jν /αν , ( ) σv e± from eq. (31), because it evolves slowly compared and the optical depth is with powers of γ+ and γ , then we can proceed to find the − 0.05 energy dependence for τe+ to within a factor of about Ee±+ τν = αν dz , (35) in accuracy. Z The reaction rate for a given positron is n σ v , where z is a line-of-sight coordinate. The flux density from h e− e± e− i which varies inversely with electron energy (any other hy- such a source at distance d from the Earth is obtained by pothesis leads to a contradiction). Therefore, the positron summing over each line of sight through the DM clump: will preferentially annihilate against an electron at the low 1 Rcl erg end of the energy distribution, so that in eq. (29), σ v τν e± e− = 2 (1 ) 1 h i∼ Iν 2 πbSν e− db 2 . (36) E− ,andn is roughly constant. That is, a given positron 4πd 0 − cm sHz e+ e− Z of any energy sees essentially the same distribution of tar- The SSA cutoff frequency νcrit depends in part on the ab- get electrons, so to a first approximation, it is sufficient to sorber density, and without ICS or e± pair annihilation ef- replace the electron distribution with a single population at 2 fects, SSA would be the most important correction to the energy E m c . By that rationale, τ + E + . e− ' e e ∼ e calculated flux in a steep cusp. SSA has been included in With this last result in hand, it becomes useful to use the calculations performed here, but since the central ab- sorber population is diminished by ICS, it is of small im- Ee− + portance. Using appropriate parameters for Draco, we find τe− τe (32) ' E +  e  νcrit 10 MHz, whereas the limiting observations discussed in the' next section are at 4.9 GHz. (It is fortunate that SSA when doing the full numerical integral over target electron has little impact on the problem, because otherwise it would energies, to avoid the infinite nesting problem. At low ener- 3/2 create serious complications in the calculation of ICS against gies where dNe/dEe Ee− ,thismeansthatdne/dEe 1/2 1/2∼ ∼ the synchrotron photon field.) E− and ne E (where the subscript e refers to either an electron or a∼ positron). The distribution in space, rather than energy, is easier to 4Results determine. In order to satisfy eq. (29) with the injection qe 2γ ∼ r− set by a cuspy DM profile, we must have dne/dEe Supersymmetric dark matter has a number of variable param- γ ∼ r− . (As before, γ is the halo profile logarithmic slope, eters which affect its annihilation and clustering properties. − while γ+ and γ are Lorentz factors.) Then for a region However, these parameters boil down to only two relevant − whose e± losses are fastest by pair annihilation, we have quantities for annihilation signature searches, and those are m and σv ¯. Given external constraints on a dark mat- χ h iχχ dne 1/2 γ ter source (such as Draco’s total mass and the matter density Ee− r− . (33) dEe ∼ local to it, in the present case), the mass mχ determines the χ number density. The more massive the particle, the fewer γ So now we have another transition radius between the r− are needed, so the annihilations are less frequent. We present γ 1/2 pair annihilation distribution and the r− − ICS distribu- here the collection of m and σv ¯ values available to the χ h iχχ tion, which we designate Rann,ics.IfRann,ics >Rmin,then neutralino, derived from observations of Draco. we add a pair annihilation dominated central region to the Before proceeding to these limits, we note for compar- picture described above regarding ICS. But for most choices ison another indirect detection approach considered in the of mχ and Draco’s magnetic field strength, Rann,ics

8 −24 (2002)). This method can currently eliminate more parame- 10

−25 ter space than direct detection experiments, and less than the 10

Draco EGRET limits as calculated herein. The constraints −26 10 from Draco require the extrapolation of its halo density pro- −27 file to small radii; the constraints from neutrino telescopes 10 −28 /s instead require the additional steps associated with elastic 3 10

−29 scattering, capture, and neutrino propagation through mat- in cm 10 χχ ter. Of course, this approach and the one used in the present v) −30 σ

( 10 work are both indirect detection scenarios; for a review of −31 EGRET (E =1 GeV) 10 0 direct experimental searches, see e.g. Morales (2001). EGRET (with eg bg) −32 GLAST (E =1 GeV) 10 0 VERITAS (E =100 GeV) allowed region 0 −33 4.1 Gamma Ray Constraints 10 VERITAS (E0=1 TeV)

−34 10 2 3 4 Gamma ray data specifically on Draco are lacking, so the 10 10 10 current limiting observations come from an all-sky survey. mχ in GeV The EGRET (Energetic Gamma Ray Experiment Telescope) Fig. 1. The mχ σv χχ¯ plane for neutralino dark matter. Curves instrument, flying on the CGRO satellite (Compton Gamma indicate results where−h i the SIS halo profile has been used. Shown Ray Observatory), is a pair production telescope (see Mattox as solid lines are the gamma ray constraints from non-inclusion in et al. (1996) for an introduction to EGRET and its data analy- the EGRET all-sky survey, where the thick line is the current bound sis). As such, its useful beam width is around 30o,makingit and the thin line is a potential improvement using an estimate of well suited for a full sky survey; its energy range is 30 MeV the extragalactic gamma ray background (see text). Both employ ∼ to below 100 GeV (Sreekumar et al., 1998). a threshold E0 =1GeV. Also shown are two dashed lines show- ∼ EGRET sources are usually counted by their photons above ing upcoming VERITAS constraints with cutoffs at 100 GeV and 1 TeV, and a dotted line for GLAST at GeV (sensitivities 100 MeV; however, Lamb and Macomb (1997) have per- E0 =1 for all of these taken from Weekes et al. (2001)). The region below formed a separate binning of the data, making a catalog of the curves is allowed by gamma ray observations. Uncertainties in gamma ray sources above 1 GeV, which is of particular inter- Draco’s mass and distance lead to at most a factor of 3 change in est here. The least significant detection (4σ) in the catalog, the vertical axis position of these curves. The dots denote typical with the least flux, is the Large Magellanic Cloud (LMC), supersymmetric dark matter properties (Gondolo et al., 2000). 8 2 1 at (1.1 0.4) 10− photons cm− s− above 1 GeV. We ± × 8 2 1 therefore take 10− photons cm− s− as the flux limit for detection, and require that Draco emit less. are shown: the current bound (thick solid line), and the ex- In fact, since this limit is only based on non-inclusion ample bound with the newer extragalactic background (thin in the GeV catalog, specific knowledge of the exposure on solid line). Points above the curves are prohibited by these Draco could result in a bound somewhat better that the one observational constraints from Draco. derived herein. A subsequent analysis of the EGRET data Two notes should be made regarding accuracy. First, the 7 provides an estimate of the instrument’s extragalactic gamma curves in the figure assume that MDraco =8.6 10 M , × ray background (Sreekumar et al., 1998). With Draco located which is measured using observations of its luminous mat- well outside the Galactic disk (l =86.37o, b =34.71o (van ter (Kleyna et al., 2001a). So this mass samples the inner den Bergh, 2000)), this extragalactic component is the pri- halo, and the total mass could be higher. In this sense, the mary background source. The extragalactic flux, adapted to curves plotted should be considered conservative. Second, the GeV catalog’s 30 arcmin pixel size, recall from the discussion of Rmin in subsection 2.1 that a higher estimate of Rmin would result from a slower relax- 1.1 11 E0 − photons ation process. At most, this would make these curves less F (E >E0)=8.6 10− (37) γ γ × GeV cm2 s constraining (i.e., raise them) by a factor of about 5.   The dots in the figure survey the possible combinations of is well below the level of one photon per minimum EGRET mχ and σv χχ¯ for neutralino dark matter (Gondolo et al., exposure. We therefore separately consider this background 2000). Theh dottedi region is characteristic of supersymmetric in our calculations, conservatively assuming that EGRET’s parameters often considered in the literature (e.g., Gondolo exposure on Draco is their minimum exposure (roughly true), and Silk (1999); Baltz et al. (1999)). They are constrained by for the case of no photons detected within 30 arcmin. We the requirement that the relic density be cosmologically inter- present this as an example of the type of improvement possi- esting; that is, large enough to account for a significant com- ble without collecting new data. ponent of the matter density without exceeding it. The choice In figure 1, we plot the neutralino parameter space. In of these “interesting” values is somewhat arbitrary and au- 2 this figure, we have presented limits derived by use of the thors differ slightly, but a restriction like 0.05 Ωχh 0.5 SIS halo profile with different curves giving different present as depicted in the figure is typical. It should≤ be noted≤ that and future observational bounds; a comparison of halo mod- values of mχ < 38 GeV are disfavored by accelerator exper- els will follow in subsection 4.3. Both EGRET constraints iments (Serin, 2000).

9 −24 10 Two additional assumptions beyond those discussed in sec-

−25 10 tion 2 are needed in order to place a radio limit, because syn-

−26 chrotron emission requires magnetic fields. 10 First, we must assume a magnetic field strength B~ for −27 10 the core of Draco. No such estimates exist in the literature,| |

/s −28 3 10 so this paper must be viewed as constraining either the neu-

−29

in cm 10 tralino or Draco’s magnetic properties. To give some bearing χχ

v) −30 ~ σ on the estimation of B , we can consider measurements for ( 10 | | −31 other dwarf galaxies. The , for example, 10 B = 0.01 µG each carry fields of 5 µG (Pohl, 1993). A survey of low sur- −32 10 B = 0.1 µG face brightness dwarf∼ galaxies (similar to Draco) leads Klein allowed region B = 1 µG −33 10 B = 10 µG et al. (1992) to infer typical field strengths between 2 and −34 4 µG, although this survey was conducted near 5 GHz and 10 2 3 4 10 10 10 Draco specifically gives no measured signal there. The dwarf m in GeV χ irregular NGC 4449 was found to harbor 14 µG fields de- ∼ Fig. 2. The mχ σv χχ¯ plane for neutralino dark matter. Curves spite the lack of ordered rotation (Chy˙zy et al., 2000), sug- indicate results where−h thei SIS halo profile has been used. Shown as gesting that some non-dynamo process may be sufficient to thin lines are the radio synchrotron constraints from VLA observa- generate microgauss fields. Based on these data, it seems tions. From top to bottom, these lines correspond to magnetic fields appropriate to expect 0.1 1 µG in Draco, although at of 0.01, 0.1, 1.0, and 10.0 µG in the core of Draco. The region present there is no compelling∼ − evidence. below these curves is allowed by radio observations. Uncertainties Second, not only must there exist magnetic fields, but they in Draco’s mass and distance lead to at most a factor of 3 change also must be sufficiently constricted, so as to trap electrons in the vertical axis position of these curves. The dots denote typical and positrons long enough for them to radiate. Implicit in supersymmetric dark matter properties. the computation of flux from DM sources in this paper is the assumption that e± particles never stray far from their birth places an assumption of strong magnetic confinement. To Future gamma ray telescopes offer better constraints. The − upcoming satellite mission GLAST (Gamma ray Large Area begin contemplating this, consider that the gyroradius of any Space Telescope), will offer much improved constraints as electron relevant to this problem is significantly smaller than the smallest length scale applicable; that is, Rgyro Rmin. shown in the figure (Gehrels and Michelson, 1999). It is ex-  pected to launch in 2005 for a two year all-sky survey, effec- So if we assume that electrons are reflected back and forth along a field line every 10 100 Rgyro, then strong con- tive between 20 MeV and 300 GeV, with a sensitivity about − 20 times better than EGRET at 1 GeV. As can be seen in finement is justified. Klein et al. (1991) notes that mag- the figure, GLAST has the potential to rule out most of the netic fields do in fact appear to be highly disordered in some neutralino DM parameter space. dwarf galaxies (Draco not considered), based on lack of ra- Apart from satellite missions, atmospheric Cherenkov tele- dio polarization observed. The confinement time needs to be scopes (ACTs) are ideal for measuring at high energy. Rather longer than the inverse Compton time scale against the syn- than detecting the photons via interactions inside the tele- chrotron photon field in order to validate our model. This duration is typically around 107 108 years for key emitting scope, ACTs count gamma rays by the air showers they pro- − duce in the Earth’s atmosphere. Specifically, they respond to regions in Draco’s halo, depending on neutralino parameters, Cherenkov light produced by charged particles in the shower for 4.9 GHz radiation as limited by the VLA. traveling faster than the speed of light in air. The next gener- These caveats noted, the synchrotron constraints are plot- ation gamma ray telescope VERITAS (Weekes et al., 2001) ted in figure 2, where again the points above each curve can can provide significant benefits for this study with a typical be ruled out by it. Limits for 0.01, 0.1, 1, and 10 µG fields exposure on Draco (we quote here an exposure of 50 hours). are shown here for the SIS halo; this choice will be varied in Curves for VERITAS with threshold energies of 100 GeV section 4.3. As with figure 1, the possibility that MDraco was (long dashes) and 1 TeV (short dashes) are depicted in the underestimated makes these curves conservative. figure. VERITAS is sensitive between about 50 GeV and Although the radio limits are less direct because of as- ~ 50 TeV; it is expected to see first light in 2002 and become sumptions that must be made about the B field, they can be fully operational in 2005. stronger than current gamma ray limits. For example, if the field is found to be at least 1 µG and adequately confining, 4.2 Radio Constraints then σv χχ¯ for a 100 GeV neutralino is restricted to be be- h i 31 3 28 3 low 4 10− cm /s, as compared with 10− cm /s The applicable radio continuum limits for Draco were per- in gamma∼ × rays. Heavier neutralinos are strongly∼ constrained formed by Fomalont and Geldzahler (1979), using the VLA at 0.1 µG, but they aren’t limited at all by current gamma (Very Large Array) facility in New Mexico. At 4.9 GHz, they measurements. On the other hand, if the magnetic fields are report no flux from Draco above 2 mJy, at the 3σ level. Their weak, then synchrotron offers no constraint on light χ parti- search spanned a radius of 4 arcmin around the galaxy center. cles. For example, a 100 GeV neutralino’s annihilation prod-

10 −24 −24 10 10

−25 −25 10 10

−26 −26 10 10

−27 −27 10 10

/s −28 /s −28

3 10 3 10

−29 −29

in cm 10 in cm 10 χχ χχ

v) −30 SIS EGRET (E =1 GeV) v) −30 σ 0 σ ( 10 ( 10 SIS GLAST (E =1 GeV) γ = 1.9 cusp 0 γ = 1.9 cusp −31 γ = 1.8 cusp −31 10 10 γ = 1.8 cusp γ = 1.7 cusp γ = 1.7 cusp −32 Moore halo profile −32 10 10 Moore halo profile SIS with 1 pc core allowed region allowed region SIS with 1 pc core −33 SIS with 0.1 pc core −33 10 10 SIS with 0.1 pc core

−34 −34 10 2 3 4 10 2 3 4 10 10 10 10 10 10

mχ in GeV mχ in GeV

Fig. 3. Parameter space constraints from the EGRET 1 GeV catalog Fig. 4. Same as figure 3, but now showing modifications to the with seven different halo models for Draco. The halos are: cusps GLAST line from figure 1, rather than EGRET. with γ = 2.0 (solid line), 1.9 (dot-dash), 1.8 (dot-dot-dash), and 1.7 (dot-dash-dash), Moore (dotted line), SIS with constant density core 24 3 at 1 pc (thick dashed line), and SIS with constant density core at σv χχ¯ below 10− cm /s; one can do somewhat better 0.1 pc (thin dashed line). The thickest solid line is repeated from the inh thei future with∼ GLAST (figure 4). current EGRET bound given in figure 1. Lines not shown indicate For the radio case, the corresponding results are presented that the corresponding halo yields no constraints on this plot. in figure 5 for B~ =0.1 µG, and in figure 6 for B~ =1µG. In each figure,| the| same seven halo models are| shown| as in ucts aren’t energetic enough to radiate at 4.9 GHz (where the figures 3 and 4, including the appropriate SIS curve from figure 2. VLA limit is) if the field is 0.1 µG or below. As in figure 1, the dots indicate∼ the SUSY parameter space. The cusps γ = 2.0, 1.9, 1.8, and 1.7 are calculated with the full inverse Compton scattering corrections described in sub- Here we see the importance of the magnetic field. At 1 µG one can exclude the majority of neutralino dark matter can- section 3.2. Applying eq. (27) yields dne/dEe dependences which go as 2.5, 2.4, 2.3,and 2.2, respectively. didates; at 0.01 µG one can exclude almost none of it. r− r− r− r− One further point worth noting is that the gamma ray and Inverse Compton scattering does not play a significant role radio analyses, taken together, permit us to constrain the mag- in the SIS core cases or in the Moore case, because in each of netic field properties of Draco. If a future gamma ray detec- these halos, the primary emitting region is outside of Rics,syn. The Moore case turns out to create a diffuse enough collec- tion is made, then the corresponding line in the mχ σv χχ¯ plane can be transposed to figure 2, with the result that−h somei tion of electrons and positrons that synchrotron self absorp- values of B~ will not be allowed. tion is also negligible, although SSA does significantly affect | | the results for the isothermal core models. 4.3 Other Halo Profiles It is interesting to note that for both gamma ray and radio constraints, the Moore profile yields more flux than a γ = Figures 1 and 2 solely employ the SIS to generate the limiting 1.7 cusp, even though the inner Moore halo only has γ = curves. In this subsection, we consider the sensitivity of our 1.5. This is a result of the fact that the Moore profile does 1.5 results to the choice of halo profile. not extend as r− indefinitely, but rather is concentrated In figure 3, we present modified gamma ray results cor- into rs. This concentration effect makes up for the lower γ responding to the current EGRET bound shown as the thick exponent. line in figure 1. All curves on this plot correspond to the same From figures 3 through 6, it is clear that the choice of halo EGRET observations above the E0 =1GeV threshold, but model is of great importance in determining the resulting 1.9 for different halo models. Figure 4 presents the same halos, dark matter constraints. Even ρ r− yields weak lim- ∼ but for the GLAST curve from figure 1 with E0 =1GeV. its with EGRET data. With GLAST, many of the steeper The halo models considered all assume spherical symme- models are valuable, but less dense ones are not. In the ra- try. They are: cusped profiles with γ = 2.0 (SIS), 1.9, 1.8, dio, a magnetic field of 0.1 µG is roughly as constraining and 1.7, the Moore profile as described in subsection 2.1, at high mχ values as GLAST is at lower mχ values. At and the SIS truncated with constant density cores of radius 1 µG, the situation is much better, with all examined pro- 1 pc and 0.1 pc (the velocity dispersion data extend down files generating strong constraints. In this case, the Moore to 10 pc). Figures which omit some of these curves do halo too constrains WIMPs well, although the Moore profile so∼ because those curves do not constrain any of the param- is not particularly constraining in any of the other cases con- eter space shown. For example, with Draco modeled by the sidered. The primary lesson from these example cases is that Moore profile, the EGRET data are insufficient to contrain the results do vary appreciably even among relatively steep

11 −24 −24 10 10

−25 −25 10 10

−26 −26 10 10

−27 −27 10 10

/s −28 /s −28

3 10 3 10

−29 −29

in cm 10 in cm 10 χχ χχ

v) −30 v) −30 σ µ σ µ ( 10 SIS with B = 0.1 G ( 10 SIS with B = 1 G γ = 1.9 cusp γ = 1.9 cusp −31 −31 10 γ = 1.8 cusp 10 γ = 1.8 cusp γ = 1.7 cusp γ = 1.7 cusp −32 −32 10 Moore halo profile 10 Moore halo profile allowed region SIS with 1 pc core allowed region SIS with 1 pc core −33 −33 10 SIS with 0.1 pc core 10 SIS with 0.1 pc core

−34 −34 10 2 3 4 10 2 3 4 10 10 10 10 10 10

mχ in GeV mχ in GeV

Fig. 5. Parameter space constraints from the VLA with seven dif- Fig. 6. Same as figure 5, but now showing modifications to the ferent halo models for Draco, assuming a magnetic field strength B~ =1µG line from figure 2. | | B~ =0.1 µG. The halo modifications are the same as those in figures| | 3 and 4. and relevant conversations. This work was supported by the NSF through grant AST-0071235 and DOE grant DE-FG0291 ER40606. cusps, underscoring the importance of pinning down Draco’s central density structure accurately. References

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