University of Amsterdam Dark matter self-annihilation signals from Galactic dwarf spheroidals; intermediate mass black holes and mini-spikes Mark Wanders Supervisor: Gianfranco Bertone GRAPPA Institute, University of Amsterdam June 17, 2014 Abstract.Theformationofanintermediatemassblackholeinevitablyaffectsthedistri- bution of dark and baryonic matter in its vicinity. Under the assumption that the black hole grows adiabatically, large overdensities in the halo’s dark matter density, called spikes, can be achieved, resulting in signi(cid:12)cant ampli(cid:12)cations to the gamma-ray signal originating from self-annihilation of WIMP dark matter. Galactic dwarf spheroidals in particular are expected to contain and maintain mini-spikes, and we use (cid:12)ve years of Fermi-LAT gamma-ray data to derive upper limits on the mass of an intermediate mass black hole in the center of the subhalo. Our limits are more constraining than those previously calculated using other methods. Contents 1 Dark matter 2 1.1 Evidence 2 1.2 Candidates 5 1.3 Origin of WIMP dark matter and relic density 7 1.4 Density pro(cid:12)le 10 1.5 Gamma-rays from dark matter annihilation 10 2 Mini-spikes 14 2.1 Dwarf spheroidals 14 2.2 Intermediate mass black holes 15 2.2.1 Motivation 15 2.2.2 Formation scenarios 16 2.2.3 Dark matter mini-spikes 18 3 Analysis 23 3.1 Fermi Large Area Telescope 23 3.2 Data acquisition and preparation 23 3.3 Maximum likelihood analysis 24 3.4 Error propagation 26 4 Results 28 4.1 Flux and cross-section constraints 28 4.2 Black hole mass upper limit 29 4.3 Age of the black hole 30 4.4 Other dwarf spheroidals 30 5 Discussion and conclusion 33 5.1 Persistence of the mini-spike 33 5.2 Other limits on black hole mass 34 5.3 Conclusion 35 { 1 { 1 Dark matter Current astrophysical evidence suggests non-baryonic dark matter accounts for roughly 27% of the energy content of the present-day universe [1]. The nature of dark matter (DM) beyond its gravitational in(cid:13)uence is largely a mystery, but a weakly interacting massive particle (WIMP) is a popular candidate [2]. A principle example of such a particle is the lightest neutralino, a stable mixture of supersymmetric partners of the Z boson, photon and Higgs boson, as predicted in the Minimal Supersymmetric Standard Model (MSSM). Since constraining the properties of such a weakly-interacting particle through direct detection (i.e., looking for nuclear recoils from scattering of dark matter particles off nucleons) appears difficult, indirect detection (i.e. looking for gamma-rays resulting from dark matter self-annihilation in astrophysical objects) could prove to be a useful alternative. Nowadays, evidence for the existence of stellar mass black holes and supermassive black holes (SMBHs, M(cid:15) (cid:24) 106 (cid:0) 108M⊙), is compelling. Recently, evidence has emerged suggesting the existence of intermediate mass black holes (IMBHs), which could have masses from 102M⊙ (cid:0)106M⊙, bridging the mass gap between stellar mass black holes and SMBHs. The formation of a massive black hole inevitably affects the distribution of (both baryonic and non-baryonic) matter around it, and can lead to large overdensi- ties called spikes. As the DM annihilation rate scales with density squared, this could improve prospects for indirect detection. However, mergers, off-center formation of the seed BH and gravitational scattering off of stars are likely to disrupt and reduce such spikes [3]. These overdensities could persist in a milder manifestation, called a mini- spike, in the case of an IMBH, as they are unlikely to undergo the processes that disrupt the overdensity. This is especially true in the case of an IMBH in the center of a dwarf spheroidal (dSph); objects orbiting the Milky Way with low central stellar densities and a relatively high DM content. In this work, we shall examine the consequences of mini-spikes in the dark matter dis- tribution caused by the adiabatic growth of IMBHs on indirect detection of WIMP dark matter using data from the Fermi-LAT space telescope. This work is organized as fol- lows: in section 1 we will give an introduction to particle dark matter, the motivation andevidence1.1, possiblecandidates1.2, WIMPrelicdensity1.3anditsdistribution1.4 before moving on to the theory of mini-spikes, including dwarf spheroidals 2.1 and in- termediate mass black holes 2.2, evidence for their existence 2.2.1, possible formation scenarios 2.2.2 and mini-spikes themselves 2.2.3, (cid:12)nishing with a description of our data analysis 3.1{3.3. In section 4 we will cover our results, before moving on to the discus- sion 5 and ending with our conclusions 5.3. 1.1 Evidence Galactic rotational velocity curves The(cid:12)rstsignsofoverlookedmassintheuniversecamefromobservationsofstarsandgas { 2 { ingalaxiesandcomparingtheirobservedvelocitiestothoseexpectedfrommovinginthe gravitational potential of the visible, luminous matter. Speci(cid:12)cally, the rotation curve of a galaxy is usually obtained through observations of the 21cm line, i.e. radio-emission fromneutralatomichydrogen(HI)intheinterstellargas, andshowsthecircularvelocity of stars and gas as a function of their distance to the center of the galaxy. If we consider simple Newtonian dynamics, we would expect the circular velocity at a radius r to be √ GM(r) v(r) = ; (1.1) r ∫ where the enclosed mass is M(r) = 4(cid:25)r2(cid:26)(r)dr and (cid:26)(r) is the density pro(cid:12)le. We p would then, from observed stellar densities, naively expect v(r) / 1= r, i.e. Keplerian behavior. However, observations revealed that the velocity curve remained (cid:13)at up to large radii (see e.g. [4]). From equation 1.1 we see that a (cid:13)at rotation curve requires M(r) / r, or(cid:26) / r(cid:0)2. Toexplainthe(cid:13)atbehavioroftherotationalvelocitycurveoutto largeradii,theconceptofthedarkmatterhalowasintroduced; alarge,roughlyspherical halo of non-luminous matter extending far beyond the visible size of the galactic disk. Gravitational lensing Another way to probe the amount of dark matter in a galaxy, given to us by Einstein’s Theory of General Relativity, is through gravitational lensing. General Relativity pos- tulates that space-time can be bent and stretched by massive objects, which in turn affects the motion of bodies, thus causing what we perceive as gravity. This space-time curvature not only affects massive bodies such as stars, which now follow geodesics in- stead of simple straight lines, the path of light traveling through curved space-time is bent as well; massive objects effectively act as a lens. In the case where a very massive object (such as a cluster of galaxies) is directly in front of a very bright object (such as a galaxy), one would see an \Einstein ring"; the closer, central object surrounded by a distorted ring-shaped image of the more distant object. However, such a perfect alignment is very rare, and so we generally observe only partial Einstein rings, called arclets. The Einstein radius (cid:18) , the radius of such an arclet in radians, gives us a way E to estimate the mass within a lensing cluster [5]: √ 4GM d LS (cid:18) = ; (1.2) E c2 d d L S where M is the mass of the lens and d , d and d are the (angular-diameter) distance LS L S between the lens and the source, distance to the lens and distance to the source, respec- tively. Deriving the mass of the cluster this way and comparing it to the luminous mass again con(cid:12)rms that a large portion of the mass must be non-luminous. For example, an analysis of the cluster Abell 370 resulted in a M=L ratio of 102{103M⊙=L⊙ [6]. Gravitational lensing has another use in the search for dark matter, through mi- crolensing, i.e. the change in brightness of a distant object due to the lensing effect of the passing of a massive object in front of it. The (cid:12)rst attempt to explain the miss- ing mass in galaxies was to turn to massive objects already known to exist in galaxies and comprised of normal, baryonic matter. These objects would need to be \dark", { 3 { and thus possible candidates are brown dwarfs, neutron stars, black holes and unassoci- ated planets. Together, these objects are referred to as Massive Compact Halo Objects (MACHOs). Searches for these microlensing events have been conducted by several collabora- tions, e.g. the MACHO Collaboration, the EROS-2 Survey, and their results are conclu- sive. The MACHO Collaboration analyzed 5.7 years of photometry on 11.9 million stars in the Large Magellanic Cloud, (cid:12)nding only 13 { 17 possible microlensing events from MACHOs in the Milky Way halo [7]. Other collaborations found similar or even smaller numbers, leading to the conclusion that MACHOs can not account for a signi(cid:12)cant part of the mass in DM. Cosmic microwave background anisotropies StrongevidencethatDMexistsinsteadintheformofparticlescomestousfromcosmol- ogy. During the (cid:12)rst seconds to minutes after the Big Bang, free neutrons and protons fused together to form light elements such as deuterium, helium and trace amounts of lithium and several others. As it turns out, this Big Bang Nucleosynthesis (BBN) is in fact the primary source of deuterium in the universe, as deuterium produced by stars is almost immediately lost in fusion to produce 4He. By treating the current amount of deuterium in the universe as a lower limit on the amount produced during BBN, it is possible to estimate the primordial deuterium to hydrogen ratio by determining the D=H abundance of regions with low levels of elements heavier than lithium. A major success of the BBN model is that theoretical abundances match up very well with obser- vational ranges, and one of the results of BBN modeling is that the D=H ratio depends strongly on the baryon density. The baryon density is usually written as Ω h2, where Ω b b is the baryon density relative to the critical density (cid:26) and h = H=100 km s(cid:0)1 Mpc(cid:0)1 c is the reduced Hubble constant. Depending on which deuterium measurement is used, calculations results in two numbers for the baryon density[8]: Ω h2 = 0:0229(cid:6)0:00013 b or Ω h2 = 0:0216+0:0020. For an alternative method of determining the baryon density, b (cid:0)0:0021 we now turn to the cosmic microwave background (CMB). TheCMBisa(nearly)isotropicbackgroundofphotonswithatemperatureofabout 2.73 K, (cid:12)rst discovered in the ’60s by Penzias and Wilson. The early universe was (cid:12)lled with a hot, dense plasma of charged particles and photons that cooled as the universe expanded. After 380,000 years or so, the universe had cooled sufficiently to allow for recombination; protons and electrons formed the (cid:12)rst neutral atoms. Photons, which previouslyhadbeenlockedinendlessscatteringwithprotonsandelectrons,nowtraveled freelythroughthenewlytransparentuniverse. Thelightfromthislastscatteringpersists and permeates the entire universe in the form of the CMB, now signi(cid:12)cantly redshifted to a very low temperature due to the expansion of the universe. Several space-based missions have been launched to map out the full-sky CMB, startingwith COBE in1989, WMAP in2001 and mostrecentlyESA’s Planck, launched in May 2009. As it turns out, the CMB (after subtraction of foregrounds and the dipole caused by the movement of the telescope with respect to the CMB frame) is not entirely uniform; its temperature (cid:13)uctuates on the order of one part per million. These tiny anisotropies can be attributed to two primary causes. Firstly, large scale (cid:13)uctuations are caused by the Sachs-Wolfe effect; photons lose energy traveling out of a { 4 { gravity well, and so we observe more low energy photons from parts of the universe that were more dense at last scattering. Secondly and for our argument, more importantly, small scale (cid:13)uctuations are attributed to something called acoustic oscillations. These oscillations emerge as the photon-baryon (cid:13)uid (cid:12)lling the early universe goes through repeated contractions and expansions, as it compresses under gravity and expands due to radiation pressure. This creates acoustic waves, whose propagation is abruptly cut off when recombination severely decreases the speed of sound in the universe. Clearly, these (cid:13)uctuations are dependent on both the baryonic mass (as it is the interaction between baryons and photons that causes the expansion) and the total mass (thegravityofwhichcausesthecontraction). Oneofthemostinterestingresultsofstud- iesoftheCMBisthatthesenumbersarenotthesame. Infact,theydiffersigni(cid:12)cantly[1]: Ω h2 = 0:14300(cid:6)0:0029; Ω h2 = 0:02207(cid:6)0:00033; (1.3) m b leadingtotheconclusionthatover85%oftheuniverse’smass-energyexistsintheformof matter that interacts gravitationally but not electromagnetically; particle dark matter. 1.2 Candidates Standard model and sterile neutrinos It is prudent to begin our search for candidates of particle dark matter amongst those particles we already know to exist, in other words, the Standard Model (SM). The standard model of particle physics is the quantum (cid:12)eld theory that describes three out of the four fundamental forces; electromagnetism, the weak force and the strong force, and their interactions with the constituents of matter, fermions, through force carriers called bosons. The model contains 17 con(cid:12)rmed particles, of which 8 were predicted by the model before being con(cid:12)rmed experimentally, the (cid:12)nal one being the Higgs, discovered in July 2012, which completed the model. The fermions are divided into 6 quarks; up (with symbol u), down (d), top (t), bottom (b), charm (c) and strange (s), and 6 leptons; the electron (e), muon ((cid:22)) and tau lepton ((cid:28)) and their corresponding neutrinos ((cid:23) ). Unlike leptons, quarks are not found individually in nature, instead they l are the building blocks for hadrons, which each contain 3 quarks (like the proton; uud), and mesons, each containing 2 quarks (e.g. the pion; ud(cid:22)). The (cid:12)ve bosons of the SM are the photon ((cid:13)), responsible for electromagnetism, (cid:6) the gluons (g), force carriers of the strong force, W and Z bosons, mediating the weak force and the Higgs (H), which gives mass to the other SM particles through the Higgs (cid:12)eld. Particles in the SM are de(cid:12)ned by their quantum numbers (i.e. charge, spin etc.), for instance, all fermions have half-integer spin, whereas bosons have integer spin. Additionally, each particle has an anti-particle, with the same mass but opposite quantum numbers, usually denoted with a bar (e.g. a quark q and an anti-quark q). When a particle collides with its anti-particle they annihilate, converting their mass- energy into photons (favored in the low-energy limit) or other particles (favored in the high-energy limit). Some particles, such as the photon and the Z boson, are their own anti-particle, thus they self-annihilate. Interactions of SM particles obey the standard laws of conservation of energy and momentum as well as internal gauge symmetries, i.e. conservation of the respective quantum numbers. { 5 { Charge Spin 1=2 Spin 0 Charge 2=3 u c t g 0 Quarks 1=3 d s b (cid:13) 0 B o (cid:0)1 e (cid:22) (cid:28) Z 0 s o Leptons n 0 (cid:23) (cid:23) (cid:23) W (cid:6)1 s e (cid:22) (cid:28) Table 1: Particles of the standard model. The only particle in the SM that, at (cid:12)rst glance, ful(cid:12)lls all the criteria for a DM particle (electrically neutral, stable, massive and weakly-interacting) is the neutrino. However, neutrinos are a form of \hot" dark matter; meaning their velocities are relativistic, causing them to erase density (cid:13)uctuations below the scale of their free-streaming length, (cid:24) 40Mpc. This implies structure formation in the early universe followed a top-down process, whereby large-scale structures form before small-scale ones, something which seems excluded from the observations of galaxies, considered small-scale in the context of the structure of the universe, already having formed at high z. More convincing, perhaps, are the constraints on neutrinos as DM coming from their expected relic abundance. Considering that there are three neutrino eigenstates, their total relic density will be [2]: ∑3 m Ω h2 = i : (1.4) (cid:23) 93 eV i=1 Given the upper limits on total neutrino mass arising from cosmology [1], we (cid:12)nd an upper limit on the neutrino relic density of Ω h2 (cid:20) 0:002; (1.5) (cid:23) showing us that, while neutrinos constitute a small part of the DM in the universe, they can not comprise the majority of it. As the Standard Model does not contain a proper candidate, we will now examine extensions of the SM. The sterile neutrino is a hypothetical, right-chiral counterpart of the left-chiral SM neutrino and receives its name from the fact that it does not interact through the electromagnetic, the strong or the weak interaction. The neutrino sector is a good candidate for physics beyond the SM, as, in theory, the SM neutrino is massless, which con(cid:13)icts with observations that neutrinos in fact do have mass, and the intro- duction of a sterile neutrino would provide an explanation for neutrino (cid:13)avor oscillation anomalies. The sterile neutrino was (cid:12)rst considered as a DM candidate in 1993 [9] and, depending on its mass, can function as either cold or warm dark matter. However, current cosmological searches have as of yet found no evidence of sterile neutrinos [1]. Supersymmetry and WIMPs Another possible extension of the SM is supersymmetry (SUSY). SUSY, in essence, pro- poses an additional symmetry beyond the Lorentz invariance and gauge symmetries of { 6 { the SM, namely a symmetry between fermions and bosons. In the SM, such a symmetry is forbidden by the Coleman-Mandula theorem, which states that space-time and inter- nal symmetries can not be combined in a non-trivial way [10]. This explicitly forbids particlesfromchangingspin,i.e. fermionsturningintobosonsorvice-versa. Theallowed symmetries in SM have generators belonging to Lie algebras, and SUSY circumvents the restriction by introducing graded Lie algebras, whose operators anti-commute. This new symmetry means that every boson in the SM now has a fermionic superpartner and every fermion has a bosonic superpartner, effectively doubling the number of particles in the model. This might seem like an unnecessary complication, but there are several theoretical motivations for introducing SUSY, such as its role in solving the problem of the low mass of the Higgs boson [2]. Speci(cid:12)cally, we will focus on one incarnation of SUSY, the Minimally Supersym- metricStandardModel(MSSM),containingthesmallestpossible(cid:12)eldcontentnecessary to reproduce the SM (cid:12)elds. This is done by introducing fermionic superpartners for all gauge (cid:12)elds, called gauginos, introducing squarks and sleptons as bosonic superpartners to the fermion sector and adding one additional Higgs (cid:12)eld with one Higgsino to each of the now 5 Higgs bosons. Additionally, the MSSM assumes R-parity (where SM particles have +1 R-parity and SUSY partners have -1 R parity) is conserved, one consequence of which being that the lightest supersymmetric particle is cannot decay and is therefor stable. The MSSM in principle contains three candidates for WIMP dark matter; the sneu- e trino ((cid:23)~, superpartner of the neutrino), the gravitino (G, superpartner of the theoretical graviton) and the neutralino ((cid:31), actually the lightest of four mass eigenstates of a mix of the bino, wino and two neutral higgsinos) [5]. However, sneutrinos annihilate so rapidly in the early universethat their relic densityis far too lowto be the WIMP[11], and grav- itinos act like hot dark matter [12], something already ruled out (as discussed earlier). We are then left with one WIMP candidate from MSSM; the lightest neutralino. 1.3 Origin of WIMP dark matter and relic density The neutralino, and WIMPs in general, are expected to have masses at the weak scale, m (cid:24) 100 GeV { 1 TeV. The (cid:3)-Cold Dark Matter ((cid:3)CDM) model predicts that, in (cid:31) the very early universe, the temperature of the universe was high enough that such very massiveparticleswerecreatedand,throughprocesseslikepair-productionandcollisions, existed in thermal equilibrium with SM particles: (cid:31)+(cid:31) ⇋ SM+SM: (1.6) However, the expansion and subsequent cooling of the universe contributed to the end of this equilibrium in two ways; (cid:12)rstly, as the temperature decreased, lighter particles no longerhadthekineticenergynecessarytoproducemassiveparticlesthroughinteractions, and secondly, the expansion of the universe itself diluted the number density of the particles, thus decreasing the interaction rate. When the interaction rate (cid:0) equals the expansion rate i.e. the Hubble constant H, (cid:0) (cid:17) n⟨(cid:27)v⟩ = H (cid:17) a_=a, massive particles will ‘freeze-out’ and their comoving number density remains constant. This freeze-out density is called the relic density. { 7 { WecanusetheBoltzmannequationtoderiveanapproximationto⟨(cid:27)v⟩,thethermal average of the annihilation cross section (cid:27) times the relative velocity v of the particles, whichinthefollowingwewillsimplycallthecrosssection. Forsupersymmetricparticles, the Boltzmann equation will have four terms, due to: 1) the expansion of the universe, 2) coannihilation of two SUSY particles into a standard model particle, 3) particle decay and4)scatteringoffthethermalbackground. WecanthuswritetheBoltzmannequation for N SUSY particles as: ∑N ( ) ∑[ ( )] dn i =(cid:0)3Hn (cid:0) ⟨(cid:27) v ⟩ n n (cid:0)neqneq (cid:0) (cid:0) (n (cid:0)neq)(cid:0)(cid:0) n (cid:0)neq dt i ij ij i j i j ij i i ji j j j=1 j̸=i ∑N [ ( ) ( )] (cid:0) ⟨(cid:27)′ v ⟩ n n (cid:0)neqneq (cid:0)⟨(cid:27)′ v ⟩ n n (cid:0)neqneq : Xij ij i X i X Xji ji j X j X j̸=i (1.7) We can simplify this equation by assuming R-parity (already an inherent feature of the MSSM) and by assuming the decay rate of SUSY particles is very high compared to the age of the universe, meaning that all SUSY particles present at the early universe have since decayed into the lightest stable particle, the neutralino (i.e. n = n +:::+n ). i N This means the third and fourth terms in equation 1.7 cancel. If we now additionally simplify the second term by excluding coannihilations between SUSY particles, we are left with ( ) dn = (cid:0)3Hn(cid:0)⟨(cid:27)v⟩ n2(cid:0)n2 : (1.8) dt eq In the non-relativistic limit of the Maxwell-Boltzmann distribution, we can express the number density of particles in thermal equilibrium as: ( ) mT 3=2 n = g e(cid:0)m=T; (1.9) eq 2(cid:25) where T is the temperature and m the particle mass. If we now introduce the new variable n Y (cid:17) ; (1.10) s where 2(cid:25)2g(cid:3)T3 s = (1.11) 45 is the entropy density of the universe (and g(cid:3) denotes the number of relativistic degrees of freedom), we can, under the assumption of conservation of entropy per comoving volume, write dY dn nds dn s = (cid:0) = +3Hn: (1.12) dt dt s dt dt Substituting this into equation 1.8, we get ( ) dY s = (cid:0)s2⟨(cid:27)v⟩ Y2(cid:0)Y2 ; (1.13) dt eq { 8 { We will now de(cid:12)ne a new variable m x (cid:17) ; (1.14) T and, since T / a(cid:0)1, we can write dx = xH. Using this, we can rewrite equation 1.13 as: dt dY dt dY (cid:0)s⟨(cid:27)v⟩ ( ) = = Y2(cid:0)Y2 : (1.15) dt dx dx Hx eq For heavy states, ⟨(cid:27)v⟩ can be approximated with the non-relativistic expansion in v2 [2]: ( ) ⟨(cid:27)v⟩ = a+b⟨v2⟩+O ⟨v4⟩ (cid:25) a+6b=x: (1.16) Introducing a (cid:12)nal new variable ∆ = Y (cid:0)Y , leads to [2] eq ( ) d∆ dY s ∆′ = = (cid:0) eq (cid:0)(a+6b=x) ∆ ∆+2Y2 dx dx √Hx eq (1.17) ( ) = (cid:0)Y′ (cid:0)(a+6b=x) (cid:25)g(cid:3)mMPl∆ ∆+2Y2 ; eq 45 x2 eq where M is the Planck mass. Pl Wecansolvethisequationanalyticallyinthecaselongafterfreeze-out, x ≫ x (i.e. F the situation today), since in that case Y ≫ Yeq and thus√also ∆ = Y (cid:0)Yeq (cid:25) Y ≫ Yeq andequation1.17canbeapproximatedas∆′ = (a+6b=x) (cid:25)g(cid:3)mMPl∆2. Ifweintegrate 45 x2 this last equation between xF and 1 and simplify using ∆1 ≪ ∆x we arrive at: p f Y(cid:0)1 = 1 (cid:25)g(cid:3)mMPl; (1.18) 1 (a+6b=x) 45 x F and, since the present density of a relic (cid:31) is given by (cid:26)(cid:31) = m(cid:31)n(cid:31) = m(cid:31)s0Y1, we can write for the relic density [2]: 1:07(cid:2)109 GeV(cid:0)1 x 1 Ω h2 = (cid:26) =(cid:26) (cid:25) pF (cid:31) (cid:31) c MPl g(cid:3)a+3b=xF (1.19) 3(cid:2)10(cid:0)27cm3s(cid:0)1 (cid:25) ⟨(cid:27)v⟩ where in the last step we employed an order of magnitude approximation [13]. Equation 1.19, with the current value for the relic density Ω h2 (cid:25) 0:1 [1] means X that ⟨(cid:27)v⟩ (cid:25) 3(cid:2)10(cid:0)26cm3s(cid:0)1. This result is a strong argument in favor for the WIMP as a DM candidate. For a new particle with a weak-scale interaction, we can estimate its annihilation cross-section to be [13] ⟨(cid:27)v⟩ (cid:24) (cid:11)2(100 GeV)(cid:0)2 (cid:24) 10(cid:0)25cm3s(cid:0)1, where the (cid:12)ne-structure constant (cid:11) (cid:24) 10(cid:0)2. This is remarkably close to the cross-section derived from the relic abundance of DM if comprised of neutralinos. Itshouldbenotedthatinequation1.8,wehaveignoredtheeffectsofcoannihilations between neutralinos and heavier SUSY particles. Including such reactions yields instead ∑N ( ) dn = (cid:0)3Hn(cid:0) ⟨(cid:27) v ⟩ n n (cid:0)neqneq ; (1.20) dt ij ij i j i j i;j=1 which, demanding that the current relic density of neutralinos be the same as in the case without coannihilations, requires a much smaller ⟨(cid:27)v⟩ [5]. { 9 {