The lightest supersymmetric particle, most likely the lightest neutralino, is
one of the most prominent particle candidates for cold dark matter (CDM). We
show that the primordial spectrum of density fluctuations in neutralino CDM has
a sharp cut-off, induced by two different damping mechanisms. During the
kinetic decoupling of neutralinos, non-equilibrium processes constitute
viscosity effects, which damp or even absorb density perturbations in CDM.
After the last scattering of neutralinos, free streaming induces neutralino
flows from overdense to underdense regions of space. Both damping mechanisms
together define a minimal mass scale for perturbations in neutralino CDM,
before the inhomogeneities enter the nonlinear epoch of structure formation. We
find that the very first gravitationally bound neutralino clouds ought to have
masses above 10^{-6} solar masses, which is six orders of magnitude above the
mass of possible axion miniclusters.Comment: 7 pages, 3 figures, to appear in proceedings of "IDM 2002, 4th
  International Workshop on the Identification of Dark Matter

Hofmann, S.

Schwarz, D. J.

Stocker, H.

English

arXiv

The lightest supersymmetric particle, most likely the lightest neutralino, is one of the most prominent particle candidates for cold dark matter (CDM). We show that the primordial spectrum of density fluctuations in neutralino CDM has a sharp cut-off, induced by two different damping mechanisms. During the kinetic decoupling of neutralinos, non-equilibrium processes constitute viscosity effects, which damp or even absorb density perturbations in CDM. After the last scattering of neutralinos, free streaming induces neutralino flows from overdense to underdense regions of space. Both damping mechanisms together define a minimal mass scale for perturbations in neutralino CDM, before the inhomogeneities enter the nonlinear epoch of structure formation. We find that the very first gravitationally bound neutralino clouds ought to have masses above 10^{-6} solar masses, which is six orders of magnitude above the mass of possible axion miniclusters

Hofmann, Stefan

Schwarz, D.J.

Stoecker, Horst

CERN Document Server

arXiv:astro-ph/0211325v1  14 Nov 2002January 3, 2014 20:20 WSPC/Trim Size: 9in x 6in for Proceedings idm02v2FORMATION OF SMALL-SCALE STRUCTURE IN SUSYCDMS. HOFMANN♯, D. J. SCHWARZ♭ AND H. STO¨CKER♯♯Institut fu¨r Theoretische Physik, Johann Wolfgang Goethe-Universita¨t,60054 Frankfurt am Main, Germany♭Theory Division, CERN, 1211 Geneva 23, SwitzerlandE-mail: stehof@th.physik.uni-frankfurt.de, dominik.schwarz@cern.ch,stoecker@th.physik.uni-frankfurt.deThe lightest supersymmetric particle, most likely the lightest neutralino, is oneof the most prominent particle candidates for cold dark matter (CDM). We showthat the primordial spectrum of density fluctuations in neutralino CDM has a sharpcut-off, induced by two different damping mechanisms. During the kinetic decou-pling of neutralinos, non-equilibrium processes constitute viscosity effects, whichdamp or even absorb density perturbations in CDM. After the last scattering ofneutralinos, free streaming induces neutralino flows from overdense to underdenseregions of space. Both damping mechanisms together define a minimal mass scalefor perturbations in neutralino CDM, before the inhomogeneities enter the non-linear epoch of structure formation. We find that the very first gravitationallybound neutralino clouds ought to have masses above 10−6M⊙, which is six ordersof magnitude above the mass of possible axion miniclusters.1. IntroductionRecent measurements support the idea that there is a significant amount ofcold dark matter (CDM) in the Universe. An analysis of the temperatureanisotropies in the cosmic microwave background (CMB) gives for the CDMmass density Ωcdmh2 = 0.13+0.03−0.02 (all data with weak h-prior)1, while fromthe same analysis the mass density of the baryons is much smaller Ωbh2 =0.023+0.003−0.003. The latter is consistent with the measurement of the primordialdeuterium abundance in high-redshift hydrogen clouds, which gives themost precise determination of the baryon density2, Ωbh2 = 0.0205±0.0018.Large galaxy redshift surveys support the CMB measurements: from theanalysis of 160,000 galaxies Percival et al3 find (Ωb +Ωcdm)h = 0.20± 0.03and Ωb/(Ωb + Ωcdm) = 0.15± 0.07. Taking various observations together,Turner4 estimates that the non-baryonic mass in the Universe, which is justthe mass of CDM, is given by Ωcdm = 0.29± 0.04.1January 3, 2014 20:20 WSPC/Trim Size: 9in x 6in for Proceedings idm02v22The characteristic feature of CDM is its non-relativistic equation ofstate at the time when the Universe contains enough matter within oneHubble volume to form one galaxy. The two leading particle candidatesfor CDM are the axion, a very light particle with a mass in the rangema ∼ (10−6 − 10−5) eV, which is cold since it has been produced in aBose condensate, and the neutralino, a heavy particle with a mass Mχ˜ ∼(50−500) GeV, which is cold by virtue of its large mass. In the constrainedminimal supersymmetric extension of the standard model (MSSM), thelightest neutralino is most likely the bino.For almost 20 years there has been a strong working hypothesis: axionCDM and neutralino CDM cannot be distinguished by purely cosmologicalobservations because both particle candidates interact only via gravity, asfar as cosmology goes. This fact has made the study of structure formationon large scales > 1 Mpc simple—the microphysics of CDM is irrelevant.However, at the galactic scale and below, various particle candidates mightbe distinguishable. To learn more about the nature of CDM, we study thesmall-scale structure of neutralino CDM with emphasis on the very first,purely gravitationally bound, neutralino clouds (for details, see our recentwork5).2. Chemical and kinetic decouplingThere are two distinct temperature scales for CDM particles that are mas-sive and weakly interacting and obey a thermal history. For temperaturesT > Tcd, binos are kept in chemical equilibrium with all fermions in theheat bath via the annihilation processes χ˜+ χ˜↔ F + F , at the rateΓann ≈ 10−3 Mχ˜M 4χ˜(M 2F˜+M 2χ˜)2 M 4F˜ +M 4χ˜(M 2F˜+M 2χ˜)2 x− 52 e−x . (1)Here, x =Mχ˜/T andMF˜ denotes the universal sfermion mass. The numer-ical prefactor is the effective neutralino–fermion coupling, the second factorshows the dependence on the MSSM parameters and the x-dependent fac-tor gives the temperature dependence in units of the neutralino mass andis proportional to the bino number density. Chemical decoupling of binoshappens at the temperature Tcd = Mχ˜/xcd, when the neutralino annihila-tion rate becomes comparable to the Hubble rate:xcd ≈ ln10−4 MPl(M 4F˜+M 4χ˜)M 3χ˜(M 2F˜+M 2χ˜)4 . (2)January 3, 2014 20:20 WSPC/Trim Size: 9in x 6in for Proceedings idm02v23Scanning the MSSM parameter space, we typically find xcd ≈ 22, see Fig-ure 1. The chemical decoupling temperature Tcd increases with increas-ing bino mass, since the neutralino number density is suppressed withexp (−Mχ˜/T ). As a consequence, the annihilation rate approaches the Hub-ble rate faster for larger bino masses. For a fixed bino mass, Tcd increases foran increasing sfermion mass, since the interaction range decreases. (Notethat this discussion is not correct on a quantitative level, since we neglecthere important effects like co-annihilation, in order to render an analyticdiscussion possible.)Figure 1. Chemical and kinetic decoupling as a function of the universal sfermion (slep-ton) mass for bino masses Mχ˜ ∈ {50, 100, 200} GeV (from bottom to top).For temperatures Tkd < T < Tcd neutralinos are in local thermal equi-librium (lte) with all fermions of the heat bath. Equilibrium is maintaineddue to elastic scattering processes χ˜+ {F , F} ↔ χ˜+ {F, F} at the rateΓel ≈ 10−2 Mχ˜M 4χ˜(M 2F˜−M2χ˜)2 x−5 . (3)Kinetic decoupling of binos takes place when the relaxation time τrelax inthe bino system exceeds the Hubble time. The relaxation time τrelax =Nτcoll differs from the collision time τcoll = 1/Γel by the number N(T ) ofelastic scatterings needed to keep or establish lte in the bino system. Thisnumber is given by the relative momentum transfer per elastic scattering,N(T ) = (∆Pχ˜/Pχ˜)−1 ≈ Mχ˜/T , with ∆Pχ˜ =√< t > ≈ 3T/√2 denotingJanuary 3, 2014 20:20 WSPC/Trim Size: 9in x 6in for Proceedings idm02v24the rms of the Mandelstam variable t. Kinetic decoupling happens atTkd ≈102 M αχ˜(M 2F˜−M 2χ˜)2MPl13+α, (4)with α = 0 if mistakenly τcoll ≡ τrelax is assumed and α = 1 whenthe number of scatterings N is taken into account. We typically findTkd = (10−100) MeV, see Figure 1. The kinetic decoupling temperature isincreasing with increasing bino mass because the momentum transfer to theheat bath is decreasing. As a consequence the number of elastic scatteringprocesses needed to keep or establish lte is increasing.We find Tcd ≫ Tkd, because Γann/Γel ≈ 10−1x5/2 exp (−x) ≪ 1 forT < Tcd < Mχ˜; the number density of neutralinos is Boltzmann suppressedwith respect to the number densities of the relativistic fermions. This is thereason for the temperature hierarchy Tcd ≫ Tkd > Tls, with Tls denotingthe temperature at which binos scatter for the last time.3. Local transport coefficientsDuring the process of kinetic decoupling, non-equilibrium processes consti-tute themselves as viscosity phenomena in the bino system. We thereforeuse hydrodynamics for the description. For T ≫ Tcd, CDM and the heatbath can be described by a single ideal fluid. For Tcd > T > Tkd the CDMfluid is strongly coupled to the radiation fluid (rad), which keeps CDM inlte. Around Tkd, the CDM fluid decouples from the radiation fluid, in whichlte persists, since Ωcdm = (a/aeq)Ωrad ≪ Ωrad for T ≫ Teq.The resulting non-equilibrium processes in the CDM fluid can be takeninto account by additional Lorentz tensors J(1) and T(1) in the currentdensity Jcdm and the energy momentum tensor Tcdm of the CDM fluid.We fix the ambiguities in the relativistic description of imperfect fluids bydemanding J(1) ≡ 0 and T(1) = ζ h∇ ·U + η W(T) + χ Sym (U⊗Q(T)).U denotes the adiabatic velocity field and h projects on the hypersurfaceperpendicular to it. The first term is the bulk viscosity, describing theflow of U in the hypersurface defined by h. The second term is the shearviscosity and describes the bending of U in the direction perpendicular tothe adiabatic current. The last term is the heat conduction.The strength of the dissipative processes is given by the local transportcoefficients ζ, η and χ. An efficient method for calculating these coefficientswas proposed by Silk6. It has been applied in great detail to the case of aJanuary 3, 2014 20:20 WSPC/Trim Size: 9in x 6in for Proceedings idm02v25relativistic fluid by Weinberg7. A generalisation to an arbitrary equationof state can be found in our recent work5.The main idea is to compare the Lorentz tensors in the hydrodynami-cal description with the Lorentz tensors in the kinetic description. In thekinetic description we make the ansatz F (1)(ω, n, x) = A(ω, x) +B(ω, x) ·n + C(ω, x) · (n⊗ n+ 1/3h) for the CDM phase-space distribution de-scribing the non-equilibrium state of the bino system. Here, ω denotes theprojection of the momentum on the velocity field and n denotes a vectorperpendicular to the adiabatic current. We calculate5 the coefficients inthe expansion of F (1) into irreducible polynoms in n and h. For the tensorstructure we find A ∝ ∇ ·V, B ∝ Q(T) and C ∝W(T). The scalar A gen-erates dissipative processes, which are not perpendicular to the adiabaticcurrent. This contribution deserves special care.Calculating the energy momentum tensor in the kinetic descriptionand comparing it with T(1), we find the local transport coefficients ζ =5ρcdm/3Γel, η = ρcdm/Γel and χ ≡ 0 in first order in 1/Γel and 1/x. Allcoefficients are decreasing with an increasing elastic scattering rate, sinceelastic scattering allows the transfer of derivations from lte to the heat bath.Heat conduction is a subdominant process for non-relativistic particles andvanishes in leading order.4. Acoustic absorptionThe dissipative processes presented in the last section transfer energy andmomentum in the direction perpendicular to the adiabatic flow. This pro-vides a damping mechanism for acoustic perturbations in CDM7. Thedamping of density inhomogeneities is given by Im(ω) as calculated in lin-ear perturbation theory for relativistic hydrodynamics. For an acousticwave with wave number k we findδρχ˜ρχ˜∝ exp−32t(Tkd)∫0dtk2Γel = exp[(−MdM)2/3], (5)where Md ≈ 3 · 10−8(GeV2/Mχ˜Tkd)3/2(Ωχ˜h2)M⊙ is the characteristicdamping scale. We find Md ≈ 10−9M⊙, see Figure 2. Thus, only acousticperturbations with masses M > Md, contained in the overdense volume,are not absorbed and enter the free streaming regime.January 3, 2014 20:20 WSPC/Trim Size: 9in x 6in for Proceedings idm02v26Figure 2. Acoustic damping scale as a function of the bino mass for sfermion massesMF˜∈ {150, 200, 300, 400} GeV (from bottom to top).5. Free streamingFor temperatures T < Tls, CDM does freely streaming on geodesics. Asa consequence, CDM propagates from overdense to underdense regions,thus smearing out local inhomogeneities. We find for the induced dampingscale Mfs(a) =Mdln3(a/als). Free streaming becomes the leading dampingmechanism once the universe has doubled its size after last scattering. Itis interesting to calculate the free streaming scale at the time of matter-radiation equality, when CDM density perturbations start to grow linearlywith the expansion factor. We typically find Mfs(aeq) ≈ 10−6M⊙.6. ConclusionsWe have shown that collisional damping and free streaming smear out allpower of primordial density inhomogeneities in bino CDM below 10−6M⊙by the time of matter-radiation equality. This is in striking contrast toclaims in the literature8 that the minimal mass for the first purely gravi-tationally bound neutralino cloud is 10−13 (150 GeV/Mχ˜)3 M⊙. The hugedifference to our result stems from the assumption that chemical and ki-netic decoupling happened simultaneously, which we proved to be wrong.We find instead that the very first bino objects have to have masses above10−6M⊙. This result is very robust with respect to the MSSM parameters.These bino clouds are very different from possible axion mini-clusters withtypical masses9 around 10−12M⊙.According to hierarchical structure formation these very first CDMclouds are supposed to merge and form larger objects. Large scale struc-ture simulations show structure formation on all accessible scales down toJanuary 3, 2014 20:20 WSPC/Trim Size: 9in x 6in for Proceedings idm02v27the resolution of the simulation10. However, the dynamic range of todayssimulations is not sufficient to deal with the very first CDM objects, sothe fate of these objects is an open issue. A cloudy distribution of CDM ingalactic halos would have important implications for direct11 and indirect12searches for dark matter.AcknowledgementsS. H. thanks J. Edsjo¨, G. Fuller, A. Greene, P. Ullio and J. Silk for valuablecomments and suggestions and acknowledges financial support of “Vereini-gung von Freunden und Fo¨rderern der JohannWolfgang Goethe-Universita¨tFrankfurt am Main e.V.” and the Marie Curie fellowship.References1. J. I. Sievers et al, [astro-ph/0205387].2. J. M. O’Meara et al, Astrophys. J. 552, 718 (2001), [astro-ph/0011179].3. W. J. Percival et al, MNRAS 327, 1297 (2001), [astro-ph/0105252].4. M. S. Turner, Astrophys. J. 576, L101 (2002), [astro-ph/0108103]; [astro-ph/0207297].5. S. Hofmann, D. J. Schwarz and H. Sto¨cker, Phys. Rev. D59, 083507 (2001),[astro-ph/0104173].6. J. Silk, Astrophys. J. 151, 459 (1968).7. S. Weinberg, Astrophys. J. 168, 175 (1971).8. A. V. Gurevich, K. P. Zybin and V. A. Sirota, Physics-Uspekhi 40, 869 (1997),[astro-ph/9801314].9. E. Kolb and I. Tkachev, Astrophys. J. 460, L25 (1996), [astro-ph/9510043].10. E. van Kampen, submitted to MNRAS, [astro-ph/0008453].11. A. M. Green, contribution to these proceedings and A. M. Green, to appearin Phys. Rev. D, [astro-ph/0207366].12. L. Bergstro¨m, J. Edsjo¨ and P. Gondolo, Phys. Rev. D58, 083507 (1998),[astro-ph/9804050]; L. Bergstro¨m, J. Edsjo¨ and P. Ullio, Phys. Rev. Lett. 87,251301 (2001), [astro-ph/0105048].

Formation of small-scale structure in SUSY CDM

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Formation of small-scale structure in SUSY CDM

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