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PHYSICAL REVIEW D, VOLUME 64, 021302͑R͒
Kinetic decoupling of neutralino dark matter
Xuelei Chen* Department of Physics, The Ohio State University, 174 West 18th Avenue, Columbus, Ohio 43210
Marc Kamionkowski† California Institute of Technology, Mail Code 130-33, Pasadena, California 91225
Xinmin Zhang‡ Institute of High Energy Physics, Chinese Academy of Sciences, P.O. Box 918-4, Beijing 100039, People’s Republic of China ͑Received 27 March 2001; published 25 June 2001͒ After neutralinos cease annihilating in the early Universe, they may still scatter elastically from other particles in the primordial plasma. At some point in time, however, they will eventually stop scattering. We calculate the cross sections for neutralino elastic scattering from standard-model particles to determine the time at which this kinetic decoupling occurs. We show that kinetic decoupling occurs above a temperature T ϳMeV. Thereafter, neutralinos act as collisionless cold dark matter.
DOI: 10.1103/PhysRevD.64.021302 PACS number͑s͒: 98.80.Cq, 13.15.ϩg, 12.60.Jv, 95.35.ϩd
I. INTRODUCTION matter candidates, Gunn et al. ͓6͔ showed that they would fall out of kinetic equilibrium shortly after their annihilation Neutralinos provide perhaps the most promising candidate freezes out. Later, Schmid et al. ͓7͔ estimated the decoupling for the mysterious dark matter in galactic halos ͑see, e.g., temperature of neutralino-neutrino interaction. In this paper Refs. ͓1,2͔ for reviews͒. Such particles would have existed in we consider the process of kinetic decoupling of neutralinos thermal equilibrium in the early Universe when the tempera- in more detail. We calculate the cross sections for elastic ture exceeded the mass of the particle. However, as the tem- scattering of neutrinos and photons from neutralinos, and for perature drops below the mass, the rate of the production and completeness, we consider the elastic scattering of neutrali- destruction reactions that maintain the equilibrium density nos from other particles as well. We show that the estimates become smaller than the expansion rate. At this point, anni- for neutrino-neutralino and photon-neutralino scattering used hilations cease, and a relic population of neutralinos remains. by Bœhm et al. neglect important kinematic effects, and thus The calculation of this relic density is straightforward, and to hugely overestimate these cross sections. We show that neu- a first approximation, the relic density is fixed by the cross tralinos are more likely to undergo kinetic freezeout much section for neutralino annihilation to lighter standard-model earlier, at temperatures between a MeV and a GeV, rather particles. The cross section required for the critical density is than a keV. Thus, although Bœhm et al.’s mechanism for ϳ10Ϫ37 cm2, coincidentally within a few orders of magni- collisional damping may be applicable to more general dark- tude of the neutralino annihilation cross section in the mini- matter candidates, it is unlikely to be relevant to the most mal supersymmetric extension of the standard model. promising supersymmetric dark-matter candidates. After freezeout ͑i.e., after they fall out of chemical equi- In the next section we briefly review the cross sections librium͒ neutralinos may still continue scattering elastically required for the Bœhm et al. collisional damping. In Sec. III, from other particles in the primordial plasma, and thus re- we calculate the cross section for neutralino-photon scatter- main in chemical equilibrium. In an interesting recent paper ing. In Sec. IV, we move on to the neutralino-neutrino cross ͓3͔, Bœhm et al. pointed out that if these particles remain in section. We consider elastic scattering of neutralinos from kinetic equilibrium until a temperature Tϳ0.1 keV, at other species in Sec. V, and we conclude in Sec. VI. which point the mass enclosed in a Hubble volume is com- parable to that of a dwarf galaxy, then collisional damping II. REQUIRED CROSS SECTIONS due to elastic scattering of neutrinos and/or photons from neutralinos could erase structure on such scales and thus help In the standard scenario, neutrinos undergo chemical de- resolve the discrepancy between the predicted and observed coupling at a temperature of one MeV and their kinetic de- substructure in Galactic halos ͓4,5͔. They also argued briefly coupling follows shortly thereafter. Now suppose that the that supersymmetric dark-matter candidates may have the re- neutrino has a nonzero elastic-scattering cross section with quired neutralino-photon and/or neutralino-neutrino cross the dark-matter particle . This elastic scattering will keep sections. the dark-matter particles in kinetic equilibrium until the rate When heavy neutrinos were first considered as cold dark at which neutrinos scatter from a given dark-matter particle becomes comparable to the expansion rate. The scattering rate is ⌫ϭnc, where n is the number density of neu- *Electronic address: xuelei@pacific.mps.ohio-state.edu trinos and is the neutrino–dark-matter elastic-scattering †Electronic address: [email protected] cross section. The Hubble expansion rate is H ‡ Electronic address: [email protected] ϭ1.66g1/2T2/m , where g ϭ3.36 is the effective number * Pl * 0556-2821/2001/64͑2͒/021302͑4͒/$20.0064 021302-1 ©2001 The American Physical Society RAPID COMMUNICATIONS
XUELEI CHEN, MARC KAMIONKOWSKI, AND XINMIN ZHANG PHYSICAL REVIEW D 64 021302͑R͒
tralinos are related to these by crossing symmetry, and this led Bœhm et al. to speculate that the elastic-scattering cross sections could also be as large as 10Ϫ38 cm2. This, however, neglects a dramatic suppression of the elastic-scattering cross section that arises from the spinor structure of the amplitude. The amplitude for either process can be written,
2 e FIG. 1. One of the Feynman diagrams for neutralino annihila- Aϭ k k ⑀ ⑀ A¯ , ͑2͒ 42 tion to two photons. There are also other Feynman diagrams for this process. where k and k are the four-momenta of the photons and ⑀ and ⑀ are their polarization four-vectors, and is the of relativistic degrees of freedom at a temperature TϳkeV, Ӎ 19 completely antisymmetric tensor. The reduced amplitude A¯ and mPl 10 GeV is the Planck mass. The temperature at depends on the particle masses and couplings in the diagram which the Hubble volume encloses a baryonic mass M b is loops. Its value for annihilation will differ from that for elas- ϭ 9 ͒Ϫ1/3 ⍀ 2 ͒1/3 ͑ ͒ T 0.15 keV͑M b/10 M ᭪ ͑ bh /0.02 , 1 tic scattering by factors of order unity, but not by many orders of magnitude. For the annihilation process, the four- ⍀ where b is the baryon density and h is the present Hubble momenta will be equal to m , while for elastic scattering, Ϫ Ϫ parameter in units of 100 km sec 1 Mpc 1. Setting ⌫ϭH, they will be equal to the photon energy. Thus, neglecting the at Tϳ0.1 keV, we find that the cross section required to A¯ Ϫ small difference in , we see that the factors of k in the Շ 9 ϳ 40 2 4 erase structure on 10 M ᭪ scales is 10 cm , amplitude will lead to a reduction by a factor (E␥ /m) of which recovers the estimate of Bœhm et al. Similar argu- the elastic-scattering cross section relative to that for annihi- ments for scattering from photons lead Bœhm et al. to esti- lation. For E␥ϳ0.1 keV and a neutralino mass m Ϫ38 2 mate a dark-matter–photon cross section of 10 cm for ϳ100 GeV, this suppression is 10Ϫ36. More precisely, the collisional damping via photon scattering. annihilation cross section is In fact, we believe that Bœhm et al. may have underesti- mated the required cross section by several orders of magni- 2 2 ␣ m ϭ ͉A¯ ͉2 ͑ ͒ tude. The cold-dark-matter particles are non-relativistic, so →␥␥ 3 , 3 even if neutrino and photon scattering keep them in kinetic 16 equilibrium, they could hardly move far enough to smooth out fluctuations in the dark-matter density. For collisional while the elastic-scattering cross section is damping to work, the motion of photons and neutrinos must ␣2 be affected. Indeed, this is how collisional damping works in 4 2 ␥→␥ϭ E␥͉A¯ ͉ . ͑4͒ ͑ ͒ 2 2 the case of baryons Silk damping in which density pertur- 3 m bations are smoothed on scales which are small compared with the photon diffusion length ͓8͔. If it is in fact the neu- We have calculated this cross section for a number of super- trinos ͑or photons͒ that are responsible for collisional damp- symmetric models using a modified version of the code ing with neutralino-neutrino scattering, then the appropriate NEUTDRIVER ͓1͔. For a range of typical supersymmetric pa- reaction rate to equate to the Hubble length is the rate at rameters and a photon energy of 100 eV, we find which a given neutrino scatters from a neutralino, rather than 10Ϫ79 cm2ϽϽ10Ϫ73 cm2. We thus conclude that elastic vice versa. In this case, the required neutrino–dark-matter neutralino-photon scattering will play no role in maintaining cross section must be larger by the ratio of the neutrino num- kinetic equilibrium of neutralinos, nor of scattering photons ber density to the dark-matter number density, and this factor from neutralinos. 7 is 5ϫ10 (m /GeV), where m is the dark-matter-particle mass. IV. NEUTRALINO-NEUTRINO SCATTERING
III. NEUTRALINO-PHOTON SCATTERING The neutrino-neutralino differential scattering cross sec- tion is given by The cross section for annihilation of two neutralinos to two photons has been calculated by a number of authors d ͉M͉2 ͑e.g., Refs. ͓9–14͔͒. Annihilation can proceed, e.g., through a ͩ ͪ ϭ , ͑5͒ ⍀ 2 2 Feynman diagram such as that shown in Fig. 1, but there are d 64 m also many other diagrams that can contribute. The value of the annihilation cross section depends on a number of super- where symmetric couplings and sparticle masses, and it may span a MϭM ϩM ϩM ͑ ͒ range of several orders of magnitude. In some regions of the s t u 6 plausible supersymmetric parameter space, it can indeed be as large as 10Ϫ38 cm2 ͓13͔. is the scattering amplitude. The Feynman diagrams are The diagrams for elastic scattering of photons from neu- shown in Fig. 2 ͑we have omitted Higgs-boson–exchange
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KINETIC DECOUPLING OF NEUTRALINO DARK MATTER PHYSICAL REVIEW D 64 021302͑R͒
2 ͉ Ј ͉2 g Xij0 1 M M*ϭϪ ͓2OЉ* ͑p p ͒͑p p ͒ s t 2 Ϫ 2 Ϫ 2 00L 1 2 3 4 cos w s m˜ t m˜ j j Ϫ Љ 2͑ ͔͒ ͑ ͒ O00*Rm p2p4 , 12
FIG. 2. Feynman diagrams for elastic neutralino-neutrino scat- ͉ Ј ͉4 Xij0 tering. M M*ϭϪ m2͑p p ͒, ͑13͒ s u Ϫ 2 ͒ Ϫ 2 ͒ 2 4 ͑s m˜ ͑u m˜ j j diagrams since the Yukawa coupling of the Higgs boson is zero in the limit of zero neutrino mass͒. These are given by1 2 ͉ Ј ͉2 g Xij0 M M*ϭϪ ͓2OЉ ͑p p ͒ t u 2 ͑ Ϫ 2 ͒͑ Ϫ 2 ͒ 00R 1 4 cos w t mZ u m˜ ͉ Ј ͉2 j Xij0 M ϭi ¯u P Cu¯ TuTCϪ1P u , s Ϫ 2 4 R 3 1 L 2 ϫ͑p p ͒ϪOЉ m2͑p p ͔͒. ͑14͒ s m˜ 2 3 00L 2 4 j In the limit mӷE , we have 2 g 1 Ӎ 2 ӍϪ ͑ ͒ Ӎ 2 ͑ ͒ M ϭi ¯u ␥ P u ¯u ␥͑OЉ P s m , t 2 p2p4 , u m , 15 t 2 Ϫ 2 4 L 2 3 00L L 2 cos w t mZ and ϩ Љ ͒ O00RPR u1 , ͒Ӎ ͒Ӎ ͑p1p2 ͑p2p3 mE , ͉ Ј ͉2 Xij0 ͑p p ͒Ӎ͑p p ͒ӍmE , ͑16͒ M ϭϪi ¯u P u ¯u P u , ͑7͒ 1 4 3 4 u Ϫ 2 4 R 1 3 L 2 u m˜ j ͒Ӎ 2 Ϫ ͒ ͑p2p4 E͑1 cos . where i, j are the flavor indices of the neutrino and sneutrino, We thus see that for mӷE , the square of the matrix ele- ϭ Ϫ␥ ϭ ϩ␥ Ј respectively, PL (1 5)/2 and PR (1 5)/2, Xij0 is ment, and thus the cross section, are proportional to 2 the neutralino-sneutrino-neutrino coupling ͑given in Ref. (E /m) , and so they are enormously suppressed as the ͓ ͔͒ Љ Љ 0 ͑ neutrinos cool. The cross section is thus, schematically, 1 , and O00R and O00L are neutralino-Z couplings also given in Ref. ͓1͔͒. The superscript T denotes the transpose of ϳ͑ ͒2͑ 2 4 ͒ ͑ ͒ the matrix, and C is the charge-conjugation matrix. Then, el couplings E /m , 17 and should be at most ͑for couplings of order unity͒ ͉M͉2ϭ͉M ͉2ϩ͉M ͉2ϩ͉M ͉2ϩ ͑M M ϩM M Ϫ46 2 s t u 2R s t* s u* 10 cm . A survey of the supersymmetric parameter space shows that when realistic couplings are included, the typical ϩM M*͒. ͑8͒ Ϫ t u cross section is more like 10 53 cm2. The cross section can be enhanced if the sneutrino mass is nearly degenerate with The various terms are given the neutralino mass, in which case there can be an s-or u-channel resonance. However, the sneutrino would have to ͉ Ј ͉4 be extraordinarily close in mass to the neutralino, and there 2 Xij0 ͉M ͉2ϭ ͑p p ͒͑p p ͒, ͑9͒ is no reason why this should be so in supersymmetric mod- s Ϫ 2 ͒2 1 2 3 4 ͑s m˜ els. j
V. KINETIC DECOUPLING MORE GENERALLY 2g4 ͉M ͉2ϭ ͓͉OЉ ͉2͑p p ͒͑p p ͒ t 4 ͑ Ϫ 2 ͒2 00L 1 2 3 4 More generically, the annihilation cross section should be cos w t mZ ϳ͑ ͒2 2 ͑ ͒ ϩ͉ Љ ͉2͑ ͒͑ ͒Ϫ Љ Љ 2͑ ͔͒ ann couplings /m , 18 O00R p1p4 p2p3 O00LO00Rm p2p4 , ͑10͒ and this must be ϳ10Ϫ38 cm2 if the neutralino is to be the dark matter. This refers to the cross section for annihilation ͉ Ј ͉4 to light quarks, leptons, and neutrinos. The elastic-scattering 2 Xij0 ͉M ͉2ϭ ͑p p ͒͑p p ͒, ͑11͒ cross section will be related to this by crossing symmetry, u Ϫ 2 ͒2 1 4 2 3 ͑u m˜ but with the correct kinematic factor, we obtain an elastic j cross section like Eq. ͑17͒͑but with E replaced more gen- erally by the light-fermion energy͒. The rate at which a given ⌫ neutralino will scatter from these light particles will be el 1 ϳ ϳ 3 For Feynman rules of Majorana fermions, see, e.g., Ref. ͓15͔. n f el , where n f T is the light-particle density, T is the
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XUELEI CHEN, MARC KAMIONKOWSKI, AND XINMIN ZHANG PHYSICAL REVIEW D 64 021302͑R͒ temperature, and at this temperature, the elastic-scattering galaxy masses. Moreover, the cross sections required for col- ϳ Ϫ37 2 2 cross section will be el 10 cm (T/m) . Equating lisional damping may be ten orders of magnitude bigger, in this elastic-scattering rate to the expansion rate, H which case neutrino and photon elastic scattering are that ϳ 2 T /mPl , we find that neutralinos should undergo kinetic much further from being able to provide collisional damping decoupling at on these scales. Therefore, elastic scattering of neutralinos from neutrinos or photons cannot erase structure on sub- ϳ ͑ ͒2/3 ͑ ͒ Tkin dec MeV m /GeV . 19 galactic scales. We have moreover argued that neutralinos should generi- Thus, after the neutralino freezes out at a temperature T ϳ cally decouple kinetically from the primordial plasma shortly m/20, it will remain in kinetic equilibrium with the after they cease annihilating, above temperatures TϳMeV plasma and then undergo kinetic decoupling shortly thereaf- ͑ Շ Շ at which time a Hubble volume encloses no more than a ter. For neutralino masses 10 GeV m 1000 GeV, this solar mass of baryons͒. Neutralinos will thereafter act as col- takes place well before neutrino decoupling and electron- lisionless cold dark matter and thus be incapable of altering positron annihilation. Շ the power spectrum on scales relevant for galaxy and large- At temperatures T 100 MeV, the baryons will be in the scale-structure formation. form of neutrons and protons, and shortly thereafter, 25% of Note added in proof. Shortly after the publication of our the mass will be bound up in helium nuclei. The neutralino- paper, another paper ͓17͔ which also discussed kinetic de- nucleon cross section required to maintain neutralino kinetic coupling of cold dark matter came out. equilibrium at a temperature TϳMeV is roughly 10Ϫ35 cm2. This cross section is extremely high for supersymmetric Ϫ43 2 models ͑typical cross sections are Շ10 cm ), and is in ACKNOWLEDGMENTS fact ruled out by null searches for energetic neutrinos from weakly interacting massive particle ͑WIMP͒ annihilation in X.C. acknowledges the hospitality of the TAPIR group at the Sun and Earth ͓16͔. Caltech where part of this work was completed. X.C. was supported in part by the U.S. Department of Energy under VI. CONCLUSIONS grant No. DE-FG02-91ER40690. M.K. was supported by the DOE under DE-FG03-92-ER40701, the NSF under AST- We have shown that the cross sections for elastic scatter- 0096023, and NASA under NAG5-8506. X.Z. was supported ing of neutrinos and photons from neutralinos are far too in part by the National Natural Science Foundation of China small to keep neutralinos in kinetic equilibrium until T and the Ministry of Science and Technology of China under ϳ10– 100 eV, when the Hubble length encloses dwarf- grant No. NKBRSF G19990754.
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