3. NEUTRALINO DARK MATTER SEARCHES

The second part of these lectures is an introduction to several methods to detect non-baryonic dark matter. We will use the lightest neutralino as our guinea pig, because of the variety of techniques that can be employed to detect it, but the discussion is more general and can be applied to a generic WIMP. Thus in this second part we assume that non-baryonic dark matter is made of WIMPs (in particular, neutralinos), and we examine several observational ways to test our assumption.

Neutralino dark matter searches are traditionally divided into two main categories: (1) direct detection of Galactic dark matter in laboratory experiments, and (2) indirect detection of neutralino annihilation products. For the sake of exposition, indirect searches are further subdivided into: (2a) searches for high-energy neutrinos from the center of the Sun or of the Earth; (2b) searches for anomalous cosmic rays and gamma-rays from galactic halos, especially our own; and (2c) searches for neutrinos, gamma-rays, and radio waves from the Galactic Center. We now examine each of them in turn.

3.1. Direct detection

The idea here is that neutralino dark matter is to be found not only in the halo of our galaxy and in our solar system, but also here on Earth and in the room we are in. Thus if we could set up a detector that records the passage of dark matter neutralinos, we could hope of detecting neutralino dark matter.

A process that can be used for this purpose is the elastic scattering of neutralinos off nuclei. Inelastic scattering could also be used in principle, as could scattering off electrons, but the rate of these processes are expected to be (much) smaller.

Dozens of experiments worldwide, too numerous to be all listed here, are using or plan to use elastic scattering to search for neutralino dark matter, or WIMP dark matter in general. The small expected detection rate, and the necessity of suppressing any ionizing radiation passing through the detector, are reasons to shelter these experiments from cosmic rays, e.g. by placing them in mines or underground laboratories.

Generally, with the notable exception of directional detectors described below, only the energy deposited in the detector during the elastic scattering can be measured. This energy is of the order of a few keV, for typical neutralino masses and speeds in the galactic halo. The kinetic energy of the recoiling nucleus is converted partly into scintillation light or ionization energy (giving an electric current) and partly into thermal energy (heating up the detector).

In cryogenic detectors, a simultaneous measurement of both ionization and thermal energy allows the discrimination of nuclear recoils from electrons produced in radioactive decays or otherwise. This discrimination, however, cannot tell if the nuclear recoil was caused by a WIMP or an ambient neutron. The detector, most often a germanium or silicon crystal, needs to be cooled at liquid helium temperature so that its low heat capacity converts a small deposited energy into a large temperature increase. Only relatively small crystals can be currently used in these cryogenic detectors, with relatively low detection rates.

Detection rates can be increased by using bigger detectors operated at room temperature, at the expense of giving up a measurement of the thermal energy and loosing discrimination power against electrons. The biggest dark matter detector is currently of this type. It is a sodium iodide crystal (a scintillator) under the Gran Sasso mountain in Italy, and it belongs to the Italian-Chinese collaboration DAMA (short for DArk MAtter). Interestingly, the loss of discrimination power and the gain in target mass almost compensate each other, and the sensitivity of cryogenic and scintillation detectors is not very different.

Annual modulation

Few years ago, the DAMA collaboration reported a possible detection of WIMP dark matter (Belli, 1997; Bernabei et al., 2000, 1998, 2003, 1999). Their most recent data (Bernabei et al., 2003) span 7 years and show a 6.3 modulation in their total counting rate (signal+background) with a period of 1 year and an amplitude of ~ 0.02 events/day per kg of detector and keV of visible recoil energy (Figure 9). This kind of yearly modulation in a WIMP signal was predicted by Drukier, Freese, & Spergel (1986) and Freese, Frieman, & Gould (1988) on the basis that the velocity of the Earth around the Sun adds vectorially to the velocity of the Sun in the Galaxy to produce a yearly modulation in the average speed of the WIMPs relative to the Earth (the WIMPs are assumed on average at rest in the Galaxy). For an observer on the Earth, the WIMP `wind' arrives at a higher speed when the Earth and the Sun move in the same direction and at a lower speed when they move in opposite directions. (The two velocities are actually misaligned by ~ 60°; see Figure 10.) The WIMP flux, and the WIMP detection rate, both proportional to the relative speed of the Earth and the WIMPs, are similarly modulated.

Figure 9. Annual modulation of the total counting rate (background plus possible dark matter signal) in seven years of data with the DAMA-NaI detector. A constant counting rate has been subtracted to give the `residuals.' The significance of the modulation is 6 and its period is 1 year. The interpretation of the yearly modulation as due to a WIMP signal is controversial. (Figure from Bernabei et al., 2003.)

Figure 10. Sketch illustrating the directions of the Sun's and the Earth's motions during a year. As the Sun moves in the Galaxy (here at 232 km/s, 60° out of the plane of the Earth's orbit), the Earth moves around the Sun (here at 30 km/s). The vectorial sum of their velocities gives the velocity of the Earth with respect to the Galaxy. Assuming the WIMPs to be on average at rest in the Galaxy, it follows that the average speed of the WIMPs relative to the Earth is modulated with a period of 1 year.

While the presence of a yearly modulation in the DAMA data seems now to be established, its interpretation as due to WIMPs is controversial. Firstly, it is not hard to imagine that the background itself can undergo seasonal variations with a period of a year. The DAMA collaboration has examined many possible sources of background variation, including a yearly modulation of the cosmic ray intensity underground due to winter-summer temperature changes in the upper atmosphere. They claim to have found no annual variation in the background level that would produce an amplitude as big as the observed one. Secondly, the EDELWEISS cryogenic detector has sensitivity comparable to DAMA's but has not recorded any nuclear recoil event (Benoit et al., 2002). And the CDMS-I cryogenic detector, also of comparable sensitivity, has detected nuclear recoil events attributed to ambient neutrons in the shallow site where the detector was running (Akerib et al., 2003). Comparison of these experimental results is however not as straightforward as it may seem, because the relationship between detection rates in cryogenic and scintillation detectors depends, among other things, on the kind of WIMP-nucleus interaction and on details of the WIMP velocity distribution in the halo, which are both poorly known.

This dependence is apparent in the expression for the expected counting rate per recoil energy bin and unit detector mass dR / dE. We have

(40)

where N_T and M_T are the number of target nuclei and the detector mass, respectively, d / dE is the WIMP-nucleus differential cross section, and nvf (, t) is the WIMP flux impinging on the detector. Here n denotes the WIMP density, v the WIMP speed, and f (, t) d³v the WIMP velocity distribution. We write M_T / N_T = M, the nuclear mass, n = / m, and d / dE = ₀| F(q)|² / E_max, where ₀ is the total scattering cross section of a WIMP off a fictitious point-like nucleus, | F(q)|² is a nuclear form factor that depends on the momentum transfer q = (2ME)^1/2 and is normalized as F(0) = 1, and E_max = 2µ² v² / M is the maximum recoil energy imparted by a WIMP of speed v (µ = mM / (m + M) is the WIMP-nucleus reduced mass). Hence

(41)

Notice that one can only measure the product ₀ with this technique. Notice also that the event rate at energy E depends on the WIMP velocity distribution at speeds v > (ME / 2µ²)^1/2. This integration limit depends on the nuclear mass, and thus detectors with different kinds of nuclei are sensitive to different regions of the WIMP velocity space. Moreover, the cross section ₀ scales differently for spin-dependent and spin-independent WIMP-nucleus interactions. Finally, while there is a consensus on the spin-independent nuclear form factors, spin-dependent form factors are sensitive to detailed modeling of the proton and neutron wave functions inside the nucleus (see Jungman, Kamionkowski & Griest, 1996, and references therein).

For spin-independent interactions with a nucleus with Z protons and A - Z neutrons, one has

(42)

where G^p_s and Gⁿ_s are the scalar four-fermion couplings of the WIMP with point-like protons and neutrons, respectively (see, e.g, Gondolo, 1996). The last approximation holds for the case G^p_s Gⁿ_s, which is typical of a neutralino. The spin-independent event rate, proportional to ₀ / µ², scales with the square of the atomic number A (if we neglect the form factor). It is this dependence on A that allows detectors with relatively heavy nuclei to reach down to WIMP-proton cross sections typical of weak interactions.

For spin-dependent interactions, one has instead

(43)

where J is the nuclear spin, <S_p> and <S_n> are the expectation values of the spin of the protons and neutrons in the nucleus, respectively, and G^p_a and Gⁿ_a are the axial four-fermion couplings of the WIMP with point-like protons and neutrons (see Gondolo, 1996; Tovey et al., 2000). There is no increase of the spin-dependent rate with A², and spin-dependent cross sections of the order of weak cross sections are hard to reach with current detector technology.

Given all these ambiguities in the comparison of cryogenic and scintillator results, it is the author's opinion that the important issue if WIMP dark matter has been detected is not settled yet. A bigger DAMA detector and an upgraded CDMS detector running in the low-background Soudan mine are currently taking data. EDELWEISS is also improving their sensitivity, and new experiments, like CRESST-II and ZEPLIN-IV, should start taking data shortly.

Current bounds and future reach

The sensitivity of some future experiments is shown in Figure 11, together with the current best bounds from the cryogenic detectors CDMS-I (Akerib et al., 2003) and EDELWEISS (Benoit et al., 2002), and the region where DAMA claims evidence for a WIMP signal ( Bernabei et al., 2003). As it is conventional in comparing results from different experiments, the figure shows the WIMP-proton spin-independent cross section obtained from experimental data using Eq. (42) under the assumption of a Maxwellian distribution with conventional parameters for the WIMP velocity. For an historical perspective, the figure also displays the first observational bound on WIMP dark matter obtained by Ahlen et al. (1987).

Figure 11. Current best bounds (lines labeled `CDMS' and `EDELWEISS') on the dark matter WIMP-proton spin-independent cross section as a function of WIMP mass, together with the expected reach of a few of the upcoming experiments (`CRESST-II', `CDMS-II', `ZEPLIN-IV', `XENON'), some theoretical expectations (`Dirac neutrino', `Baltz & Gondolo', `CMSSM Ellis et al.'), and the first bound on WIMP dark matter (`Ahlen et al'). See text for details.

We also indicate theoretical predictions for a Dirac neutrino with standard model couplings, and for the lightest neutralino in two supersymmetric scenarios: minimal supergravity as in Ellis, Ferstl, & Olive (2000) (shaded yellow region) and weak-scale MSSM as in Baltz & Gondolo (2003, 2001) (black crosses and magenta squares). Theory models assume that the respective dark matter particles fill up the galactic halo. Dirac neutrinos are excluded as main constituents of galactic dark matter in the mass range 3 GeV - 3 PeV (the bounds continue linearly to the right of the figure). These are the bounds from dark matter searches used in Section 2.2 (and Figure 3) to conclude that a yet-undetected particle species is needed to provide cold dark matter. For the neutralino, the expected scattering cross section varies in a wide range. Even with the most restrictive assumptions of the Constrained Minimal Supersymmetric Standard Model (CMSSM) in Ellis, Ferstl, & Olive (2000), the cross section at a given neutralino mass can change by an order of magnitude when the other supersymmetric parameters are changed (shaded yellow region in Figure 11). For neutralino masses m ~ 400 GeV, the CMSSM model parameters may conspire to make the spin-independent cross section arbitrarily small (if such is the case, the total cross section will be dominated by spin-dependent terms, which are however much smaller for the heavy nuclei in current detectors). The highest CMSSM cross sections are within reach of the upcoming experiments, although outside the current best bounds. Other studies of supergravity models (e.g. Feng, Matchev, & Wilczek, 2000) are less restrictive and result in somewhat larger cross sections. Enlarging the parameter space beyond supergravity, as for example in Baltz & Gondolo (2003, 2001), opens up more possibilities for the values of the cross section. This happens basically for two main reasons; (i) The supergravity relation between the masses of squark and sleptons on one side and Higgs bosons on the other side is removed; the Higgs boson can thus be lighter, and the spin-dependent scattering cross section, which varies essentially with the fourth inverse power of the Higgs mass, can be larger; (ii) It is no longer required that the electroweak symmetry breaking is achieved radiatively, with the consequence that the gaugino and higgsino content of the lightest neutralino can be arbitrary; the scattering cross section is then enhanced because mixed neutralinos couple to nucleons stronger than pure gauginos or pure higgsinos. This explains the larger extent of the region covered by the Baltz & Gondolo (2003, 2001) models in Figure 11. Finally, the region in Figure 11 marked by the magenta squares indicates the subset of the supersymmetric models examined by Baltz & Gondolo (2003, 2001) that could be able to quantitatively explain the ~ 3 deviation between the measured value and the Standard Model value of the magnetic moment of the muon. ⁵ These models have relatively light masses for supersymmetric partners, and give neutralino-proton cross sections which are relatively large and within the reach of the most ambitious future experiments.

Directional detection

We conclude this section by mentioning the very intriguing possibility of WIMP detectors that are sensitive to the direction of nuclear recoils (directional detectors).

The advantages of measuring the recoil direction are multiple: a more powerful background discrimination; the detection of the new modulation effects, such as a daily modulation in the arrival direction of WIMPs due to the Earth rotation around its axis; and the exciting possibility of reconstructing the WIMP velocity distribution in the solar neighborhood. The latter is possible because of a simple relation between the WIMP velocity distribution f (, t) and the nuclear recoil rate differential in both energy E and recoil direction (Gondolo, 2002),

(44)

where d is an infinitesimal solid angle around the direction , w = (ME / 2µ²)^1/2, and

(45)

is the Radon transform of the WIMP velocity distribution.

A promising development in this direction is the DRIFT detector (Snowden-Ifft, Martoff, & Burwell, 2000). This detector consists of a negative ion time projection chamber, the gas in the chamber serving both as WIMP target and as ionization medium for observing the nuclear recoil tracks. The direction of the nuclear recoil is obtained from the geometry and timing of the image of the recoil track on the chamber end-plates. A 1 m³ prototype has been successfully tested, and a 10 m³ detector is under consideration.

Directional detection is particularly powerful for detecting structure in the dark matter velocity space, as discussed in the next section on the Sagittarius stream.

Sagittarius stream

Recent observations of the stellar component of the Galactic halo show evidence of a merger history that has not yet become well mixed, and corroborate previous indications that halos form hierarchically. In particular, the Sloan Digital Sky Survey (Newberg et al., 2003) and the Two Micron All Sky Survey (Majewski, Skrutskie, Weinberg, & Ostheimer, 2003) have traced the tidal stream (Ibata et al., 2001) of the Sagittarius (Sgr) dwarf spheroidal galaxy. The Sagittarius dwarf spheroidal galaxy, of roughly 10⁹ M, is a satellite of our own much larger Milky Way Galaxy, located inside the Milky Way, ~ 12 kpc behind the Galactic Center and ~ 12 kpc below the Galactic Plane (Ibata et al., 1997). Two streams of matter are being tidally pulled away from the main body of the Sgr galaxy and extend outward from it. These streams, known as the leading and trailing tidal tails, are made of matter tidally pulled away from the Sgr galaxy. It appears that the leading tail is showering matter down upon the solar neighborhood (Majewski, Skrutskie, Weinberg, & Ostheimer, 2003). The flow is in the general direction orthogonal to the Galactic plane and has a speed of roughly 300 km/s. This speed is comparable to that of the relative speed of the Sun and the WIMPs in the general dark halo.

It is natural to expect that dark matter is associated with the detected tidal streams. Hence one can hope to detect the stream in direct detection experiments. The detectability depends on the density of dark matter in the stream. The mass-to-light ratio M / L in the stream is unknown, but is plausibly at least as large as that in the Sgr main body; in fact, the M / L in the stream may be significantly larger because the dark matter on the outskirts of the main body would be tidally stripped before the (more centrally located) stars. Various determinations of the M / L for the Sgr main body give values in the range 25 to 100 (see the discussion in Majewski, Skrutskie, Weinberg, & Ostheimer, 2003 and references therein). Freese, Gondolo, & Newberg (2003) and Freese, Gondolo, Newberg, & Lewis (2003) have estimated the density of dark matter in the stream, and find it to be in the range 0.3% to 23% of the local (smoothed) dark halo density. This agrees with a previous theoretical study on the tidal disruption of satellite galaxies falling into the halo of our own Milky Way by Stiff, Widrow, & Frieman (2001). These authors found that, with probability of order 1, the Sun should be situated within a stream of density ~ 4% of the local Galactic halo density.

The additional flux of WIMPs from the stream shows up as a 0.3-23% increase in the rate of nuclear recoils at energies below a characteristic energy E_c, the highest energy that WIMPs in the stream can impart to a target nucleus. Hence, there is a step in the energy recoil spectrum; the count rate in the detector is enhanced at low energies, but then returns to the normal value (due to Galactic halo WIMPs) at all energies above the critical energy E_c. This feature can be observed as a sharp decrease in the count rate above a characteristic energy that depends on the mass of the target nucleus, the mass of the WIMP, and the speed of the stream relative to the detector. Figures 12(a) and (b) show how the recoil spectrum dR/dE is modified by the presence of the stream for a sodium iodide detector (like in DAMA) and a germanium detector (like for CDMS and EDELWEISS). For the sake of illustration, the plots assume a stream density equal to 20% of the local halo density.

Figure 12. Effect of the presence of WIMPs in the Sagittarius leading tidal arm. Count rate of 60 GeV WIMPs in (a) an NaI detector such as DAMA and (b) a Ge detector such as CDMS and EDELWEISS, as a function of recoil energy. The dotted lines (towards the left) indicate the count rate due to Galactic halo WIMPs alone for an isothermal halo. The solid and dashed lines indicate the step in the count rate that arises if we include the WIMPs in the Sgr stream for vstr = 300 km/s in the direction dispersion (l, b) = (90°, -76°) with a stream velocity dispersion of 20 km/sec. The plot assumes that the Sgr stream contributes an additional 20% of the local Galactic halo density. The solid and dashed lines are for June 28 and December 27 respectively, the dates at which the annual modulation of the stream is maximized and minimized. (Figure from Freese, Gondolo, & Newberg, 2003.)

Excitingly, the effect of the stream should be detectable in DAMA, CDMS-II, and other upcoming detectors, and may already be present in the current DAMA data. A detail calculation (Freese, Gondolo, & Newberg, 2003) predicts the presence of stream WIMPs in the data with a significance of 100 for DAMA and 11 for CDMS if the stream density is 20% of the local halo density, and a significance of 24 for DAMA and 3 for CDMS if the stream density is 4% of the local halo density. (These significance figures are however very sensitive to the velocity assumed for the stream, cfr.  Freese, Gondolo, Newberg, & Lewis, 2003.)

Directional detection will be a fantastic means of recognizing the presence of a dark stream through the Solar system. The recoil distribution due to WIMPs in a stream is very much different from the recoil distribution due a Maxwellian velocity distribution. The corresponding Radon transforms that appear in Eq. (44) are: for a stream of velocity ,

(46)

which is non-zero only on the surface of a sphere in (w _x, w _y, w _z) space; for a Maxwellian of bulk velocity and velocity dispersion ,

(47)

which is a smooth gaussian distribution. For our Sgr stream, consider a next-generation DRIFT detector of 30 m³ (DRIFT-2). The difference in the recoil direction distributions of stream and isothermal WIMPs is apparent in Figure 13, where we plot the differential detection rate E^-2dR / dEd for DRIFT-2 under the assumption of a 20% stream density. Aa seen in the figure, a large DRIFT detector will have the capability of clearly identifying WIMPs in the Sgr stream.

Figure 13. Count rate of 60 GeV WIMPs in a CS2 detector (DRIFT) as a function of recoil energy E and direction of the nuclear recoil (X, Y, Z). Here X points toward the galactic center, Y toward the direction of galactic rotation, and Z toward the North Galactic Pole. On the left is a density plot of the count rate in the 2-dimensional slice EY = - 20 keV. On the right is the 1-dimensional section through EY = - 20 keV and X = 0. The horizontal axis represents recoils in the direction of the Galactic center (left) and Galactic anticenter (right); the vertical axis represents recoils in the direction of the North Galactic Pole (upward) and South Galactic Pole (downward). The gray scale indicates the count rate per kilogram of detector per day and per unit cell in the 3-dimensional energy space E. Lighter regions correspond to higher count rates. The white band in the upper part is the location of nuclear recoils due to WIMPs in the Sgr stream. The fuzzy gray cloud at the center contains recoils due to WIMPs in the local isothermal Galactic halo. The two WIMP populations can in principle be easily separated, given a sufficiently long exposure. (Figure from Freese, Gondolo, & Newberg, 2003.)

3.2. Indirect detection

Besides the direct detection of galactic neutralino dark matter in the laboratory, we can search for dark matter neutralinos by looking for the products of their annihilation. We distinguish three types of searches according to the place where neutralino annihilations occur. The first is the case of neutralino annihilation in the Sun or the Earth, which gives rise to a signal in high-energy neutrinos; the second is the case of neutralino annihilation in the galactic halo, or in the halo of external galaxies, which generates gamma-rays and other cosmic rays such as positrons and antiprotons; the third is the case of neutralino annihilations around black holes, in particular around the black hole at our Galactic Center.

All these annihilation signals share the property of being proportional to the square of the neutralino density. This follows from the fact that the neutralino is a Majorana fermion, i.e. is identical to its antiparticle. Two neutralinos can annihilate to produce standard model particles. Simple stoichiometry then tells us that the annihilation rate, being proportional to the product of the densities of the initial particles, is proportional to the square of the neutralino density. In more detail, we have

(48)

where _ann is the neutralino annihilation rate per unit volume (i.e. the number of neutralinos that are annihilated per unit volume and unit time), _ann is the neutralino-neutralino annihilation cross section, v is the relative speed of the two annihilating neutralinos, is the neutralino mass density, and m is the neutralino mass. Recall that the annihilation cross section _ann goes as 1/v at small speeds, as required by kinematical arguments, and thus the product _annv does not vanish linearly with v (and is not small at the relatively small speeds of neutralinos in galactic halos). Notice also that the number of annihilations per unit volume and unit time is given by 1/2_ann, where the factor of 1/2 correctly converts between the number of annihilation events and the number of neutralinos that are annihilated (2 per annihilation). It is easy to get confused with this factor of 1/2.

If in the annihilation rate _ann we insert a typical weak interaction cross section and a typical value for the average dark matter density in the Universe, the annihilation rate we obtain gives undetectably small signals. Indirect detection is possible because dark matter is not distributed uniformly in space. Galaxies and clusters of galaxies are overdensities in the dark matter field, as is any possible substructure in galactic halos. Furthermore, dark matter may be concentrated gravitationally around massive objects, and may even get trapped inside planets and stars. The neutralino annihilation rate, proportional to the square of the neutralino density, increases substantially in these dark matter concentrations, sometimes to the point of giving observable signals.

High energy neutrinos from the core of the Sun or of the Earth

Neutralinos floating around the solar system can occasionally collide with nuclei in the Sun and in the planets (the Earth, in particular). In these collisions, they may loose enough kinetic energy to end up with a speed smaller than the escape speed, thus becoming gravitationally trapped. After some time, the trapped neutralinos will sink to the core of the celestial body in which they are captured, and will possibly reach a condition of thermal equilibrium (Figure 14).

Figure 14. Illustration of indirect detection of WIMPs using high-energy neutrinos emitted in WIMP annihilations in the core of the Sun or of the Earth.

Once concentrated in the center, neutralinos annihilate copiously. The annihilation rate is maximal when all captured neutralinos annihilate (a condition called equilibrium between capture and annihilation). Whether this condition is satisfied depends on the relative strength of the annihilation and scattering cross sections, and ultimately on the parameters of the particle and halo models. (See Jungman, Kamionkowski & Griest, 1996 and references therein for complete formulas.)

Of the annihilation products produced in the center of the Earth and the Sun only the neutrinos make it to the surface; all the other products are absorbed or decay within a short distance of production. All three flavors of neutrinos are produced for neutralino masses which are currently allowed. Direct production of a neutrino pair is however strongly suppressed in neutralino annihilation, due to the Majorana nature of the neutralino. Annihilation neutrinos are instead produced as secondaries in the decay chains of the primary particles produced in the neutralino-neutralino annihilation. As a consequence, the neutrino energy spectrum is a continuum, and the typical energy of neutrinos from neutralino annihilations is about a tenth of the neutralino mass. Given the current constraints, this means a neutrino energy between few GeVs and few TeVs.

Neutrinos of this energy can be detected in Cherenkov neutrino telescopes, whose principle of operation is depicted in Figure 15. A charged-current interaction in the material surrounding the detector (rock, ice, water) converts the neutrino into its corresponding charged lepton, which then radiates Cherenkov light in the detector medium (ice or water). Several neutrino telescopes are currently operational (among them the Super-Kamiokande detector in Japan and the AMANDA detector at the South Pole), and others are under construction or development (IceCube at the South Pole, ANTARES and NESTOR in the Mediterranean). Other neutrino telescopes have played a role in dark matter searches in the past, such as the IMB, the Fréjus, the MACRO, and the Baksan experiments.

Figure 15. Principle of operation of a water Cherenkov neutrino telescope. An incoming muon neutrino (here a muon neutrino) is converted into a charged lepton (a muon) in the material surrounding the detector (in the rock at the bottom of the sea). The charged lepton moves faster than light in water and thus Cherenkov light (blue cone) is emitted along its trajectory. The Cherenkov light is collected by the array of photomultipliers suspended on strings. Cherenkov neutrino telescopes in ice work on the same principle. (Background figure by François Montanet, ANTARES Collaboration; tracks and Cherenkov light by the present author.)

The current experimental situation for this indirect detection method is summarized in Figure 16(a) for neutrinos from WIMPs in the Sun, and Figure 16(b) for neutrinos from WIMPs in the Earth. The figures show the current best bounds from the MACRO, Baksan, Super-Kamiokande, and AMANDA experiments, as well as the first bound obtained using this technique by the IMB collaboration in 1987. Also shown is the reach of the IceCube experiment after an exposure of 10 km² yr, and the ultimate applicability of this method for the Sun, which is set by the emission of high-energy neutrinos in cosmic ray interactions with nuclei on the surface of the Sun.

Figure 16. Indirect searches for neutralino dark matter using high-energy neutrinos from (left) the Sun and (right) the Earth. On the vertical axis is the flux of neutrino-induced muons that traverse the neutrino telescope, on the horizontal axis is the neutralino mass. `IMB87' is the historically first upper limit, `MACRO,' `BAKSAN,' `SuperK,' and `AMANDA' are the limits from the corresponding experiments. The regions marked by × on the left and by dots on the right are the predictions of supersymmetric models defined at the weak scale (Ahrens et al., 2002; Bergström, Edsjö, & Gondolo, 1998). The + signs on the left and the shaded region on the right indicate the regions where there are models that have been excluded by direct dark matter searches. The line labeled `10km2yr' shows the maximum reach of such an exposure in IceCube, and the line labeled `Sun background' marks the level of high-energy neutrino emission due to cosmic ray interactions on the surface of the Sun, which is the ultimate applicability limit of this method. (Figure on the left from Gondolo, 2000; figure on the right from Ahrens et al., 2002.)

Expectations from theoretical models in Figure 16 (MSSM with seven weak-scale parameters) range from cases which are already excluded by this method to cases which this indirect method will be unable to explore. In comparison, direct searches have a different coverage of theoretical models. The reach of direct searches is indicated in the Figure 16 by + signs in the left panel and by the shading in the right panel (direct searches exclude only some of the theoretical models that are projected onto these regions from the higher-dimensional supersymmetric parameter space). There are models that can be explored by direct searches and not by indirect searches of high-energy neutrinos from the Sun and the Earth. And vice versa, there are theoretical models that cannot be explored by direct searches but can be explored by indirect searches of high-energy neutrinos from the Sun or the Earth. The latter aspect is illustrated in Figure 17, where models that can be reached by indirect searches for neutrinos from neutralinos in the Sun are marked by dots in the scattering cross section-mass plane, and compared with the sensitivity of several current and future direct search experiments. Several models fall beyond the reach of even the most ambitious direct dark matter searches. This shows the (everlasting) complementarity between direct and indirect neutralino searches.

Figure 17. Complementarity of direct and indirect neutralino dark matter searches. The figure shows that several supersymmetric models that are within the (expected) reach of a big neutrino telescope of 10 km2 yr exposure are beyond the reach of current and future direct detection experiments. The vertical axis is the product of the neutralino-proton spin-independent scattering cross section -p and the local neutralino density in units of 0.3 GeV/cm3 fCCDM. The horizontal axis is the neutralino mass m. (Figure from Duda et al., 2003.)

Gamma-rays and cosmic rays from neutralino annihilation in galactic halos

We shift now our attention to signals originating in neutralino annihilations which occur in the halo of our galaxy or in the halo of external galaxies.

The annihilation products of importance are those that are either rarely produced in astrophysical environments or otherwise have a peculiar characteristic that make them easily recognizable. In the first category are rare cosmic rays such as positrons, antiprotons, and antideuterons. In the second category are gamma-rays, whose spectrum is expected to contain a gamma-ray line at an energy corresponding to the neutralino mass (besides a gamma-ray continuum; see Figure 18). The gamma-ray line is produced directly in the primary neutralino annihilation into or Z. Positrons, antiprotons, deuterons, and the gamma continuum are generated in the particle cascades that follow the decay of the primary annihilation products. Their spectra are therefore broad, with a typical energy which is only a fraction of the neutralino mass, and a shape whose details depend on which annihilation channels are dominant. Two neutralinos can in fact annihilate into a variety of primary products, depending on their masses and compositions: fermion pairs f, Higgs boson pairs H_iH_j, gauge boson pairs W ⁺ W ^-, ZZ, etc.

Figure 18. Two examples of gamma-ray spectrum from neutralino annihilation in the galactic halo. The numerical values refer to a specific model for the dark halo of our galaxy and for observations in the direction of the galactic center, but the spectral characteristics are general. Neutralino annihilations produce a gamma-ray line at an energy corresponding to the mass of the neutralino, and a gamma-ray continuum generated in the particle cascades following the primary annihilation. The figure illustrates that the shape of the continuum spectrum depends on the neutralino mass, but notice that it also depends on the neutralino composition. (Figure from Bergström, Ullio, & Buckley, 1998.)

Detection of the gamma-ray line would be a smoking-gun for neutralino dark matter, since no other astrophysical process is known to produce gamma-ray lines in the 10 GeV - 10 TeV energy range. Good energy resolution is crucial to detect the neutralino gamma-ray line. The simulation in Figure 19 shows that the upcoming GLAST detector should have an adequate energy resolution.

Figure 19. Simulation of a gamma-ray annihilation line from the annihilation of ~ 48 GeV neutralinos, superimposed on a gamma-ray background of astrophysical origin. The simulation includes the finite energy resolution of the upcoming GLAST detector. (Figure from the GLAST Science Brochure.)

Dependence on the density profile

Historically, the density profile of galactic dark halos has been given in terms of empirical density profiles whose density is constant in a central region and decreases as r^-2 at large radii. The latter is the main ingredient in obtaining a flat rotation curve in the outer regions, which is a primary evidence for dark matter in galaxies. Central among these functions is the cored isothermal profile

(49)

where a is called the core radius (Bahcall & Soneira, 1980). This parametrization is so simple and so much used that it is sometimes called the `canonical' density profile. Another interesting empirical parametrization is the density profile of Persic, Salucci, & Stel (1996),

(50)

which provides good fits to rotation curves of hundreds of spiral galaxies (which are not as flat as one might think!).

Numerical simulations of structure formation in the Universe have discovered that pure cold dark matter halos do not follow the previous empirical density profiles but instead have a universal shape whose parameters depend on the mass (or age) of the system. Navarro, Frenk, & White (1996) have found this universal profile to have the form

(51)

where the parameter r_s is the radius at which the radial dependence of the density changes from r^-1 to r^-3. The empirical dependence r^-2 is then seen as an approximation in the transition region. Moore et al. (1998) suggest instead that the universal profile may be steeper at the center,

(52)

Which of these two profiles better represents the results of numerical simulations is a question that must be answered by higher resolution simulations (which seem to be pointing to an inner slope that depends on the mass of the system).

The four profiles mentioned above are plotted in Figure 20 for a galaxy that could be our own, with total mass M_vir = 2 × 10¹² M and virial radius r_vir = 428 kpc.

Figure 20. Dark matter density profiles for a galaxy resembling our own. Models `BS' (Bahcall & Soneira, 1980) and `PS' (Persic, Salucci, & Stel, 1996) are empirical parametrizations which possess a central region with constant density (core). Models `NFW' (Navarro, Frenk, & White, 1996) and `Moore et al' (Moore et al., 1998) are derived from numerical simulations of structure formation in the Universe, and in them the density in the central region increases as a power law of radius (cusp). All four models are normalized to the same total mass and virial radius.

The considerable differences in indirect neutralino signals between a cored and a cuspy density profile are illustrated in Figure 21 for a gamma-ray signal from neutralino annihilation from the direction of the Galactic Center (without the spike contribution discussed in the next Section). There is a factor of 500 difference between assuming an NFW profile or a cored isothermal profile in the theoretical calculation of the gamma-ray flux in the seven-parameter MSSM model in Bergström, Ullio, & Buckley (1998). Superposed on the plot are the sensitivities to gamma-ray point sources of various gamma-ray telescopes, both current and upcoming (as of 2000). The comparison with the theoretical expectations in the figure is not direct, however, because the neutralino emission may not be point like in some of the telescopes. The sensitivity curves are thus only meant to provide an approximate comparison.

Figure 21. Expected gamma-ray flux in the gamma-ray line from neutralino annihilation in our galactic halo, coming from the direction of the Galactic Center (without the spike discussed in next Section). The photon energy E is to good approximation equal to the neutralino mass. The upper set of points is for an NFW model (Bergström, Ullio, & Buckley, 1998), the lower set for a cored isothermal model. The gamma-ray fluxes differ by a factor of about 500. Also shown for approximate comparison are the sensitivities of current and upcoming gamma-ray telescopes to gamma-ray point sources. (Figure from Gondolo, 2000.)

It must be mentioned here that EGRET has detected gamma-ray emission from the direction of the Galactic Center (Mayer-Hasselwander et al., 1998). However, in a reanalysis of the EGRET data by Hooper & Dingus (2002), the EGRET signal seems to originate from a source which is displaced with respect to the Galactic Center. Hooper & Dingus (2002) use their reanalysis to place constraints on the gamma-ray flux from neutralino annihilations, coming to an upper limit on the gamma-ray flux above 1 GeV which is about a factor of 2 higher than the theoretical predictions for neutralino masses m ~ 50 GeV, assuming an NFW profile. See Hooper & Dingus (2002) for details.

Effect of halo substructure

The same numerical simulations that predict a cuspy dark halo profile also predict the existence of numerous dark clumps in galactic halos. These clumps are a natural outcome of the hierarchal formation of structure in the Universe. Small structures form first, and larger structures, like galaxies, grow by attracting and swallowing smaller structures. This process continues to the present day. Clumps that fall into a galaxy are pulled apart by tidal interactions: the material pulled out forms tidal streams that crisscross the galactic halo. The central parts of some of the clumps may survive for a long time, and become a galactic satellite.

The overall picture of hierarchical structure formation has found confirmation in a variety of context, from observations of galaxy clusters and of merging galaxies to the halo substructure detected in our own galaxy (see the discussion on the Sagittarius stream in Section 3.1 above, for example). A numerical discrepancy should however be mentioned. A counting of visible satellites of our own galaxy gives a number of luminous satellites that is much smaller than the expected number of dark satellites. A resolution to this discrepancy may be that only a small fraction of dark satellites becomes luminous. It is not clear why this should happen, but on the other hand we do not fully understand how galaxies become luminous in the first place. Thus for what concerns signals from neutralino annihilation, it makes sense to examine the effect of adding substructure, i.e. clumps and streams, to galactic dark halos.

Substructure in the halo tends to increase the annihilation signals because of the dependence of the annihilation rate on the square of the dark matter density (see Eq. (48)). The enhancement factor is linearly proportional to the density enhancement ' / . To understand why the dependence is linear instead of quadratic in , consider a box of volume V containing a total mass M. The density in the box is = M / V and the annihilation rate integrated over the whole box is

(53)

Now let all the mass be concentrated equally into N small boxes, each of volume V', so that each box contains a mass M' = M / N and thus has a density ' = M' / V'. The density enhancement is then = V / NV'. The annihilation rate from the whole box is now given by a sum over the N small boxes as

(54)

One power of the density in _ann is compensated by a decrease in the volume where annihilations occur. Hence the signal enhancement is linear in the density increase.

An interesting consequence of the annihilation signal enhancement has been explored by Bergström, Edsjö, & Ullio (2001) and Ullio, Bergström, Edsjö, & Lacey (2002). These authors have found that the density enhancements produced during the formation of the large scale structure in the Universe may lead to a substantial increase in the isotropic gamma-ray background from neutralino annihilations in the early Universe. Moreover, they found that this increase in the gamma-ray signal is not very sensitive to details of the galactic density profile. Expected gamma-ray spectra may in some models be close to the measured gamma-ray background, as illustrated in Figure 22. The gamma-ray spectra include both a continuum part and gamma-ray lines (two for each neutralino case: one for , the other for Z ). The gamma-ray lines are asymmetrically broadened because photons emitted at earlier times have a larger redshift. The gamma-ray background due to neutralino annihilations should be searched for at high galactic latitudes, where the galactic emission is expected to be minimal. Detection of the line features depicted in Figure 22 would not require an energy resolution much better than the present one.

Figure 22. Expected isotropic gamma-ray background from neutralino annihilations in the early Universe. Dotted lines are the signal from neutralino annihilations; solid lines are the sum of the neutralino signal and a gamma-ray background of astrophysical origin. Two neutralino models are shown, together with the current EGRET measurements of the isotropic extragalactic background. The assumptions used for the dark matter profile are indicated. (Figure from Ullio, Bergström, Edsjö, & Lacey, 2002.)

One possible explanation is that the positron excess is due to the positrons generated in neutralino annihilation in the galactic halo. If the neutralinos are produced thermally in the early universe, which is the most common assumption, the annihilation cross section _ann is forced to be small by the requirement that the neutralino relic density is large enough for neutralinos to be the dark matter. Using the average value of the local dark matter density, of the order of 0.3 GeV/cm², then leads to a positron signal which is more than an order of magnitude smaller than the excess measured by HEAT. Increasing the annihilation cross section _ann does not make the signal higher, because the density decreases inversely with _ann, and hence the annihilation signal, being proportional to _ann ², decreases. Kane, Wang, & Wells (2002) suggested that the neutralinos may not have been produced thermally in the early Universe, and were thus able to decouple the annihilation rate in the halo from the constraint coming out of the relic density requirement. Alternatively, Baltz, Edsjö, Freese, & Gondolo (2002) have suggested that substructure in the galactic halo may provide the necessary boost factor to the positron signal.

Enhancing the positron signal through clumps also enhances other annihilation signals, such as antiprotons and gamma-rays. Baltz, Edsjö, Freese, & Gondolo (2002) have performed a detailed analysis of these enhancements, and have concluded that it is possible to explain the HEAT positron excess with boost factors as small as 30, but typically higher, without obtaining too many antiprotons or gamma-rays. Figure 23 shows two examples of neutralino models that provide a good fit to the positron excess. On the left, the neutralino has mass m = 340 GeV and is an almost pure gaugino (gaugino fraction Z_g = 0.98); the boost factor is 95 and other parameters are listed in the figure. On the right, the neutralino is mixed (gaugino fraction Z_g = 0.70) with a mass of 238 GeV; the boost factor is 116.7. The ² per degree of freedom is quite good for both fits, 1.34 and 1.38 respectively.

Figure 23. Two examples of neutralino models that provide a good fit to the excess of cosmic ray positrons observed by the HEAT collaboration. The two sets of data points (open and filled squares) are derived from two different instruments flown in 1994-95 and 2000. The lines represent: (i) the best expectation we have from models of cosmic ray propagation in the galaxy (`bkg. only fit'), which underestimate the data points above ~7 GeV; (ii) the effect of adding positrons from neutralino annihilations (lines `SUSY component', `SUSY+bkg. fit', and `bkg. component', the latter being the resulting background component when the data are fitted to the sum of background and neutralino contributions). (Figures from Baltz, Edsjö, Freese, & Gondolo, 2002).

The ultimate test of the explanation of the positron excess by means of neutralino clumps will be the detection of a signal in gamma-rays. Gamma-ray production would in fact be enhanced by the same mechanism that would enhance positron production. Baltz, Edsjö, Freese, & Gondolo (2002) have found that almost all neutralino models that can explain the positron excess are within the sensitivity reach of upcoming gamma-ray telescopes (see Figure 24). The realistic possibility of confirming (or disproving) the neutralino origin of the positron excess is fascinating.

Figure 24. Sensitivity of upcoming gamma-ray telescopes to neutralino models that can explain the HEAT positron excess with neutralino clumps in the galactic halo. Model points are indicated by crosses; circles denote those models that in addition can also account for the measured deviation in the muon magnetic moment. The upper set of sensitivity curves corresponds to the high latitude gamma-ray line flux (scale on the left); the lower set of curves to the direction toward the galactic center (scale on the right; no steep spike around the central black hole is assumed). (Figure from Baltz, Edsjö, Freese, & Gondolo, 2002.)

Signals from neutralino annihilation at the Galactic Center

The last indirect neutralino signals we consider are neutrinos, gamma-rays and radio waves from a possible dark matter concentration around the black hole at the Galactic Center.

Evidence for the presence of a black hole at the center of our galaxy comes from studies of the motion of stars in orbit around the center. The speeds of these stars decrease from the center as the inverse square root of the radius, which is the primary indication for the existence of a point mass at the center. The mass of the central object is measured to be ~ 4 × 10⁶ solar masses, which are contained within a sphere of less than ~ 0.05 pc (Eckart & Genzel, 1997; Ghez, Klein, Morris, & Becklin, 1998; Ghez et al., 2003). No stellar or gas system can be so dense, indicating that the central object is most probably a black hole. The position of the black hole happens to coincide with the position of a strong radio source called Sagittarius A^*, which is thus identified with the central black hole.

The radio emission from Sgr A^* is easily explained by thermal emission of hot matter falling into the black hole. However, contrary to many of the similar black holes observed at the center of external galaxies, our galactic black hole does not emit intensely in the X-ray band, and it is controversial if it emits gamma-rays. Models for such `quiet' black holes do exist, however, such as those involving advection-dominated accretion flows (ADAFs).

Further evidence for a black hole at the center of the Milky Way comes from the 2001 observation of a X-ray and infrared flares from the Galactic Center (Genzel et al., 2003b; Porquet et al., 2003; Baganoff et al., 2001). The flare time scale and intensity can nicely be explained if the flare is produced near a black hole (see, e.g., Aschenbach, Grosso, Porquet, & Predehl, 2004).

Dark matter may be driven near the black hole gravitationally, and may form a dense concentration around it (Gondolo & Silk, 1999). We will call this concentration a spike, so as to distinguish it from the dark matter cusps of Section 3.2.2. The formation of a spike is gravitational phenomenon similar to but less efficient than accretion of matter, in that the latter involves dissipation of energy and angular momentum and can thus produce concentrations which are smaller and denser.

How strong is the dark matter concentration around Sgr A^*, or around a generic black hole? This is still a matter of investigation. The simplest case is that of adiabatic compression, and was analyzed by Gondolo & Silk (1999).

Adiabatic compression of an initial dark matter distribution produces two kinds of spikes. If the initial distribution before the black hole forms is thermal, the spike is shallow, with density profile r^-3/2. If the initial phase-space distribution is a power-law in energy, the spike is steep, i.e. r^- with > 2. The physical reason for the higher concentration in the second case is the presence of many dark matter orbits with low speed, which are more easily driven into bound orbits when the black hole forms.

These two kinds of spike are illustrated in Figure 25 for the same four halo models discussed in Section 3.2.2. Models with a core, like those by Bahcall & Soneira (1980) and Persic, Salucci, & Stel (1996), give rise to a shallow spike around the central black hole, while models with a cusp, like those of Navarro, Frenk, & White (1996) and Moore et al. (1998), produce steep spikes. In the very inner regions, the density may become so high that neutralino-neutralino annihilations may have had the time to deplete the number of neutralinos and an `annihilation plateau' is formed. The typical radius of a spike around a black hole is determined by the radius of influence of the black hole, r_infl ~ GM / _v², which is the radius at which the gravitational potential energy becomes comparable to the typical kinetic energy in the dark matter gas (M is the black hole mass and _v is the gas velocity dispersion). For the black hole at the Galactic Center, r_infl ~ 1 pc.

Figure 25. Density profiles of spikes that form adiabatically around the black hole at the center of our Galaxy. The position of the Sun is indicated by a cross. Four models for the halo profile are shown: two with cores (`PS' by Persic, Salucci, & Stel (1996) and `can' by Bahcall & Soneira (1980)) and two with cusps (`NFW' by Navarro, Frenk, & White (1996) and `M&' by Moore et al. (1998)). The spikes form within the radius of influence of the black hole, rinfl ~ 1 pc. In the `annihilation plateau' neutralino annihilations have been so rapid as to deplete the number of neutralinos. (Figure from Buckley et al., 2001.)

Neutralino annihilation is enhanced in the spike, because of the dependence of the annihilation rate on square of the density. The Galactic Center then becomes a source of neutrinos, gamma-rays, radio waves, etc. from neutralino annihilation (Figure 26). The intensity of the emission depends on the steepness of the spike. If the spike is shallow, neutralino annihilation is generally undetectable. On the contrary, a steep spike at the Galactic Center produces interesting signals.

Figure 26. Artistic conception of emission from a dark matter spike around a black hole at the galactic center. Neutralino annihilation in the spike produces intense fluxes of neutrinos, gamma-rays, radio waves, etc. Some of these signals may be detectable.

For example, if, disregarding Hooper & Dingus's reanalysis mentioned above, we attribute the EGRET gamma-ray emission from the Galactic Center to neutralino annihilation in a spike born out of an NFW profile, we obtain a high-energy neutrino flux that is either excluded or mostly detectable in a km³ neutrino telescope (Figure 27). The flux of neutrino-induced muons above 25 GeV would be detectable over the atmospheric neutrino background for neutralino masses between ~ 100 GeV and ~ 2 TeV, while heavier neutralinos would already be excluded from the current limit on the neutrino emission from the Galactic Center (Habig et al., 2001).

Figure 27. Predicted neutrino signal from neutralino dark matter annihilation in a steep adiabatic spike at the Galactic Center expressed as number of neutrino-induced muons per km2-yr in a neutrino telescope. The spike corresponds to an NFW profile. Each dot in the figure corresponds to a point in a seven-parameter weak-scale MSSM, and is normalized so that the gamma-ray flux from the spike coincides with the gamma-ray signal from the Galactic Center observed by EGRET. Models with heavy neutralinos are excluded by the current limit on the neutrino emission from the Galactic Center; models with neutralinos as light as ~ 100 GeV could be detected above the atmospheric neutrino background. (Figure from Gondolo, 2002b.)

As a second and more dramatic example (Gondolo, 2000b), electrons and positrons from neutralino annihilation would emit synchrotron radiation as they spiral in the magnetic field that plausibly exists around the central black hole. While this synchrotron radiation is innocuous for a shallow adiabatic spike, it may exceed the observed radio emission by several orders of magnitude if the spike is steep. The radio synchrotron emission at 408 MHz is shown in Figure 28 for an adiabatic spike born out of an NFW profile, under two assumptions for the radial dependence of the magnetic field (a constant field of 1 mG and a field in equipartition with the infalling gas). In both cases, all dark matter neutralinos in the seven-parameter MSSM models considered are strongly excluded.

Figure 28. Synchrotron emission at 408 MHz expected from neutralino dark matter annihilation in a steep adiabatic spike at the Galactic Center. The spike corresponds to an NFW profile, and the synchrotron radiation is emitted by electrons and positrons produced in neutralino annihilation. Upper panel: constant magnetic field; lower panel: equipartition magnetic field. All dark matter neutralinos in the seven-parameter weak-scale MSSM considered are excluded by several orders of magnitude. (Figure from Gondolo, 2000b.)

For example, if the central black hole formed through the merging of two black holes of similar mass, Merritt, Milosavljevic, Verde, & Jimenez (2002) have shown that the spike would become shallow at the end of the merging, because dark matter particles would be kicked out of the black hole region via a gravitational sling-shot effect. The final shallow spike would not give dramatic signals from neutralino annihilation at the Galactic Center. In a realistic scenario of this type, the two merging black holes of similar mass would be accompanied by their host galaxies whose mass would also be similar. The merging would then constitute what is known as a major merging. A major merging is capable of destroying the Galactic disk, and so must not have occurred after the disk formed about 10¹⁰ years ago. Thus, for this scenario to work, black holes of millions of solar masses should already have been in place at very early times. Is this possible? While hard to explain theoretically, supermassive black holes have indeed been observed in very distant quasars, at redshift z > 6, so the scenario may be plausible.

In these considerations, it must be kept in mind that there is a stellar spike around the black hole at the Galactic Center. The steepness of this stellar spike is however not very well know. With large uncertainties, Genzel et al. (2003) estimate the slope of the stellar spike to be _stars ~ 1.3-1.4. This means that the current stellar spike is probably shallow. We may think that the stellar spike is our best proxy for the dark matter spike. If so, also the dark matter spike would also be shallow, and thus inconsequential for neutralino signals. However, the dark matter and stellar spikes follow very different evolution histories, because contrary to the dark matter, binary collisions of stars and coalescence of two stars into one at collisions effectively relax the stellar system to a shallower spike.

The final word has not yet been said regarding the distribution of dark matter at the Galactic Center, or around black holes and other compact objects in general. This is one of the exciting points of contact between the study of dark matter and the study of the formation and evolution of galactic nuclei.

⁵ The size of the deviation has been hard to determine conclusively because of the difficulty of the non-perturbative QCD calculations involved, in particular their dependence on the data used as input. There is also a story on sign errors in some of the theoretical calculations.... Back.