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Cold and Fuzzy Dark Matter View Metadata, Citation and Similar Papers at Core.Ac.Uk Brought to You by CORE Cold and Fuzzy Dark Matter View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server Wayne Hu, Rennan Barkana & Andrei Gruzinov Institute for Advanced Study, Princeton, NJ 08540 Revised March 28, 2000 Cold dark matter (CDM) models predict small-scale structure in excess of observations of the cores and abundance of dwarf galaxies. These problems might be solved, and the virtues of CDM models retained, even without postulating ad hoc dark matter particle or field interactions, if the dark matter is composed of ultra-light scalar particles (m ∼ 10−22eV), initially in a (cold) Bose-Einstein condensate, similar to axion dark matter models. The wave properties of the dark matter stabilize gravitational collapse providing halo cores and sharply suppressing small-scale linear power. Introduction.| Recently, the small-scale shortcomings of the Compton wavelength m−1 but much smaller than the otherwise widely successful cold dark matter (CDM) the particle horizon, one can employ a Newtonian ap- models for structure formation have received much at- proximation to the gravitational interaction embedded in tention (see [1–4] and references therein). CDM models the covariant derivatives of the field equation and a non- predict cuspy dark matter halo profiles and an abundance relativistic approximation to the dispersion relation. It is of low mass halos not seen in the rotation curves and lo- then convenient to define the wavefunction ψ ≡ Aeiα,out cal population of dwarf galaxies respectively. Though the of the amplitude and phase of the field φ = A cos(mt−α), significance of the discrepancies is still disputed and so- which obeys lutions involving astrophysical processes in the baryonic 3 a˙ 1 2 gas may still be possible (e.g. [5]), recent attention has i(∂t + )ψ =(− ∇ + mΨ)ψ, (2) mostly focused on solutions involving the dark matter 2 a 2m sector. where Ψ is the Newtonian gravitational potential. For m ∼ In the simplest modification, warm dark matter ( the unperturbed background, the right hand side van- keV) replaces CDM and suppresses small-scale struc- ishes and the energy density in the field, ρ = m2|ψ|2/2, ture by free-streaming out of potential wells [3], but this redshifts like matter ρ ∝ a−3. modification may adversely affect structure at somewhat On time scales short compared with the expansion larger scales. Small-scale power could be suppressed time, the evolution equations become more cleanly in the initial fluctuations, perhaps originat- ing from a kink in the inflaton potential [2], but its regen- 1 2 2 i∂tψ =(− ∇ + mΨ)ψ, ∇ Ψ=4πGδρ . (3) eration through non-linear gravitational collapse would 2m likely still produce halo cusps [6]. Assuming the dark matter also dominates the energy More radical suggestions include strong self- density, we have δρ = m2δ|ψ|2/2. This is simply the non- interactions either between dark matter particles [1] or linear Schr¨odinger equation for a self-gravitating particle in the potential of axion-like scalar field dark matter [4]. in a potential well. In the particle description, ψ is pro- While interesting, these solutions require self-interactions portional to the wavefunction of each particle in the con- wildly in excess of those expected for weakly interacting densate. massive particles or axions respectively. In this Letter, we propose a solution involving free par- Jeans / de Broglie Scale.| The usual Jeans analysis tells ticles only. The catch is that the particles must be ex- us that when gravity dominates there exists a growing −22 eγt γ2 πGρ traordinarily light (m ∼ 10 eV) so that their wave mode where =4 ; however a free field oscillates as e−iEt or γ2 = −(k2/2m)2.Infact,γ2 =4πGρ − nature is manifest on astrophysical scales. Under this 2 2 proposal, dark matter halos are stable on small scales for (k /2m) and therefore there is a Jeans scale the same reason that the hydrogen atom is stable: the 3/4 −1/4 −1/2 rJ =2π/kJ = π (Gρ) m , uncertainty principle in wave mechanics. We call this −1/2 −1/4 2 −1/4 dark matter candidate fuzzy cold dark matter (FCDM). =55m22 (ρ/ρb) (Ωmh ) kpc , (4) Equations of Motion.| It is well known that if the dark matter is composed of ultra-light scalar particles m below which perturbations are stable and above which m m/ −22 1eV, the occupation numbers in galactic halos are so high they behave as ordinary CDM. Here 22 = 10 eV ρ . × 11 h2M −3 that the dark matter behaves as a classical field obeying and b =28 10 Ωm Mpc is the background the wave equation density. The Jeans scale is the geometric mean between the dynamical scale and the Compton scale (c.f. [7–9]) as 2φ = m2φ, (1) originally shown in a more convoluted manner by [10]. The existence of the Jeans scale has a natural interpre- where we have set ¯h = c = 1. On scales much larger than tation: it is the de Broglie wavelength of the ground state 1 of a particle in the potential well. To see this, note that Euler equations of a relativistic imperfect fluid. First, the velocity scales as v ∼ (Gρ)1/2r so that the de Broglie note that in the Newtonian approximation, the current wavelength λ ∼ (mv)−1 ∼ m−1(Gρ)−1/2r−1. Setting density j ∝ ψ∗∇ψ − ψ∇ψ∗ plays the role of momentum rJ = λ = r, returns the Jeans scale. Stability below the density so that “probability conservation” becomes the Jeans scale is thus guaranteed by the uncertainty prin- continuity equation. The dynamical aspect of equation ciple: an increase in momentum opposes any attempt to (2) then becomes the Euler equation for a fluid with an 2 2 2 2 confine the particle further. effective sound speed ceff = k /4a m ,wherek is the The physical scale depends weakly on the density, but comoving wavenumber. in a dark matter halo ρ will be much larger than the This Newtonian relation breaks down below the Comp- background density ρb. Consider the density profile of a ton scale which for any mode will occur when a<k/2m. 3 halo of mass M [≡ (4πrv/3)200ρb, in terms of the virial In this regime, the scalar field is slowly-rolling in its po- radius rv] found in CDM simulations [11] tential rather than oscillating and it behaves like a fluid with an effective sound speed c2 = 1 [12]. For our pur- fρ eff ρ r, M ∼ 200 b , poses, it suffices to simply join these asymptotic solutions ( ) 2 (5) 3 (cr/rv )(1 + cr/rv) and treat the FCDM as GDM with 3 where f(c)=c /[ln(1 + c) − c/(1 + c)] and the concen- 2 1 ,a≤ k/2m , ceff = 2 2 2 (7) tration parameter c depends weakly on mass. This profile k /4a m ,a>k/2m , −1 implies an r cusp for r<rv/c which will be altered with no anisotropic stresses in linear theory. We have by the presence of the Jeans scale. Solving for the Jeans verified that the details of this matching have a negligi- scale in the halo rJh as a function of its mass using the ble effect on the results. Since the underlying treatment enclosed mean density yields is relativistic, this prescription yields a consistent, covari- 1/3 −2/3 −1/9 2 −2/9 ant treatment of the dark matter inside and outside the rJh ∼ 3.4(c10/f10) m M (Ωmh ) kpc , (6) 22 10 horizon. The linear theory equations including radiation where we have scaled the mass dependent factors to the and baryons are then solved in the usual way but with regime of interest c10 = c/10, f10 = f(c)/f(10), and initial curvature perturbations in the radiation and no 10 M10 = M/10 M . For estimation purposes, we have perturbations in the FCDM [13]. assumed rJh rv/c which is technically violated for The qualitative features of the solutions are easily un- < −22 M10 ∼ 1andm =10 eV, but with only a mild ef- derstood. The comoving Jeans wavenumber scales with 1/4 1/4 fect. In the smallest halos, the Jeans scale is above the the expansion as kJ ∝ aρb (a)or∝ a during matter turnover radius rv/c, and there is no region where the domination and constant during radiation domination. density scales as r−1. The maximum circular velocity Because the comoving Jeans scale is nearly constant, per- will then be lower than that implied by eqn. (5). More turbation growth above this scale generates a sharp break massive halos will have their cuspy r−1 behavior extend in the spectrum. More precisely, the critical scale is kJ 1/2 −1 from r = rv/c down to rJh. at matter-radiation equality kJeq =9m22 Mpc . Nu- These simple scalings show that the wave nature of merically, we find that the linear density power spectrum dark matter can prevent the formation of the kpc scale of FCDM is suppressed relative to the CDM case by cusps and substructure in dark matter halos if m ∼ x3 10−22eV. However, alone they do not determine what P k T 2 k P k ,T k ≈ cos , FCDM( )= F( ) CDM( ) F( ) x8 (8) does form instead. To answer this question, cosmologi- 1+ cal simulations will be required and this lies beyond the 1/18 where x =1.61 m k/kJeq. The power drops by a fac- scope of the present work. Instead we provide here the 22 tor of 2 at tools necessary to perform such a study, a discussion of possible astrophysical implications and illustrative one 1 −1/18 4/9 −1 k1/2 ≈ kJeqm22 =4.5m22 Mpc .
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