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Fuzzy Cold Dark Matter: the Wave Properties of Ultralight Particles

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Fuzzy Cold Dark Matter: the Wave Properties of Ultralight Particles VOLUME 85, NUMBER 6 PHYSICAL REVIEW LETTERS 7AUGUST 2000 Fuzzy Cold Dark Matter: The Wave Properties of Ultralight Particles Wayne Hu, Rennan Barkana, and Andrei Gruzinov Institute for Advanced Study, Princeton, New Jersey 08540 (Received 27 March 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 ultralight scalar particles ͑m ϳ 10222 eV͒, 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. PACS numbers: 95.35.+d, 98.80.Cq Introduction.—Recently, the small-scale shortcomings particle horizon, one can employ a Newtonian approxi- of the otherwise widely successful cold dark matter (CDM) mation to the gravitational interaction embedded in the models for structure formation have received much atten- covariant derivatives of the field equation and a nonrela- tion (see [1–4] and references therein). CDM models pre- tivistic approximation to the dispersion relation. It is then dict cuspy dark matter halo profiles and an abundance of convenient to define the wave function c ϵ Aeia, out of low mass halos not seen in the rotation curves and local the amplitude and phase of the field f ෇ A cos͑mt 2a͒, population of dwarf galaxies, respectively. Although the which obeys µ ∂ µ ∂ significance of the discrepancies is still disputed and so- ᠨ 3 a 1 2 lutions involving astrophysical processes in the baryonic i ≠t 1 c ෇ 2 = 1 mC c , (2) gas may still be possible (e.g., [5]), recent attention has fo- 2 a 2m cused mostly on solutions involving the dark matter sector. where C is the Newtonian gravitational potential. For the ͑ ϳ In the simplest modification, warm dark matter m unperturbed background, the right-hand side vanishes and ͒ keV replaces CDM and suppresses small-scale structure the energy density in the field, r ෇ m2jcj2͞2, redshifts by free streaming out of potential wells [3], but this modi- like matter r~a23. fication may adversely affect structure at somewhat larger On time scales short compared with the expansion time, scales [6]. Small-scale power could be suppressed more the evolution equations become cleanly in the initial fluctuations, perhaps originating from µ ∂ a kink in the inflaton potential [2], but its regeneration 1 i≠ c ෇ 2 =2 1 mC c, =2C ෇ 4pGdr . through nonlinear gravitational collapse would likely still t 2m produce halo cusps [7]. (3) More radical suggestions include strong self-interactions either between dark matter particles [1] or in the potential Assuming the dark matter also dominates the energy of axionlike scalar field dark matter [4]. While interesting, density, we have dr ෇ m2djcj2͞2. This is simply the these solutions require self-interactions wildly in excess of nonlinear Schrödinger equation for a self-gravitating par- those expected for weakly interacting massive particles or ticle in a potential well. In the particle description, c is axions. proportional to the wave function of each particle in the In this Letter, we propose a solution involving free par- condensate. ticles only. The catch is that the particles must be extraor- Jeans͞de Broglie scale.—The usual Jeans analysis tells dinarily light ͑m ϳ 10222 eV͒ so that their wave nature us that when gravity dominates there exists a growing is manifest on astrophysical scales. Under this proposal, mode egt where g2 ෇ 4pGr; however, a free field dark matter halos are stable on small scales for the same oscillates as e2iEt or g2 ෇ 2͑k2͞2m͒2. In fact, g2 ෇ reason that the hydrogen atom is stable: the uncertainty 4pGr2͑k2͞2m͒2 and therefore there is a Jeans scale principle in wave mechanics. We call this dark matter can- ෇ ͞ ෇ 3͞4͑ ͒21͞4 21͞2 didate fuzzy cold dark matter (FCDM). rJ 2p kJ p Gr m , 21͞2 21͞4 2 21͞4 Equations of motion.—If the dark matter is composed of ෇ 55m22 ͑r͞rb͒ ͑Vmh ͒ kpc , (4) ultralight particles m ø 1 eV, the occupation numbers in galactic halos are so high [8] that the dark matter behaves below which perturbations are stable and above which they 222 as a classical field obeying the wave equation behave as ordinary CDM. Here m22 ෇ m͞10 eV and r ෇ 2.8 3 1011V h2M Mpc23 is the background den- ᮀf ෇ m2f b m Ø , (1) sity. The Jeans scale is the geometric mean between the where we have set h¯ ෇ c ෇ 1. On scales much larger than dynamical scale and the Compton scale (cf. [9–11]), as the Compton wavelength m21 but much smaller than the originally shown in a more convoluted manner by [12]. 1158 0031-9007͞00͞85(6)͞1158(4)$15.00 © 2000 The American Physical Society VOLUME 85, NUMBER 6 PHYSICAL REVIEW LETTERS 7AUGUST 2000 The existence of the Jeans scale has a natural interpre- scalar field in Eq. (1) onto the continuity and Euler equa- tation: it is the de Broglie wavelength of the ground state tions of a relativistic imperfect fluid. Note that in the New- c 2=ءof a particle in the potential well. To see this, note that the tonian approximation the current density j ~c plays the role of momentum density so that ءvelocity scales as y ϳ͑Gr͒1͞2r so that the de Broglie c=c wavelength l ϳ͑my͒21 ϳ m21͑Gr͒21͞2r21. Setting “probability conservation” becomes the continuity equa- rJ ෇ l ෇ r returns the Jeans scale. Stability below the tion. The dynamical aspect of Eq. (2) then becomes the Jeans scale is thus guaranteed by the uncertainty principle: Euler equation for a fluid with an effective sound speed 2 2 2 2 an increase in momentum opposes any attempt to confine ceff ෇ k ͞4a m , where k is the comoving wave number. the particle further. This Newtonian relation breaks down below the Comp- The physical scale depends weakly on the density, but ton scale which for any mode will occur when a , k͞2m. in a dark matter halo r will be much larger than the back- In this regime, the scalar field is slowly rolling in its po- ground density rb. Consider the density profile of a halo tential rather than oscillating, and it behaves like a fluid ϵ͑ 3͞ ͒ 2 of mass M [ 4pry 3 200rb in terms of the virial radius with an effective sound speed ceff ෇ 1 [15]. For our pur- ry] found in CDM simulations [13], poses, it suffices to simply join these asymptotic solutions and treat the FCDM as GDM with 200 fr Ω ͑ ͒ϳ b ͞ r r, M 2 , (5) 2 1, a # k 2m , 3 ͑cr͞ry͒͑1 1 cr͞ry͒ c ෇ (7) eff k2͞4a2m2, a . k͞2m , ͑ ͒ ෇ 3͓͞ ͑ ͒ ͑͞ ͔͒ where f c c ln 1 1 c 2 c 1 1 c and the con- with no anisotropic stresses in linear theory. We have veri- centration parameter c depends weakly on mass. This pro- fied that the details of this matching have a negligible 21 ͞ file implies an r cusp for r , ry c which will be altered effect on the results. Since the underlying treatment is by the presence of the Jeans scale. Solving for the Jeans relativistic, this prescription yields a consistent, covariant scale in the halo rJh as a function of its mass using the treatment of the dark matter inside and outside the hori- enclosed mean density yields zon. The linear theory equations including radiation and 22͞3 21͞9 r ϳ 3.4͑c ͞f ͒1͞3m M ͑V h2͒22͞9 kpc , (6) baryons are then solved in the usual way but with initial Jh 10 10 22 10 m curvature perturbations in the radiation and no perturba- where we have scaled the mass dependent factors to the tions in the FCDM [16]. regime of interest c10 ෇ c͞10, f10 ෇ f͑c͒͞f͑10͒, and The qualitative features of the solutions are easily un- ෇ ͞ 10 derstood. The comoving Jeans wave number scales with M10 M 10 MØ. For estimation purposes, we have ͞ 1 4 1͞4 assumed rJh ø ry͞c which is technically violated for the expansion as kJ ~ arb ͑a͒ or ~ a during matter 222 M10 & 1 and m ෇ 10 eV, but with only a mild effect. domination and is constant during radiation domination. In the smallest halos, the Jeans scale is above the turnover Because the comoving Jeans scale is nearly constant, per- radius ry͞c, and there is no region where the density turbation growth above this scale generates a sharp break 21 scales as r . The maximum circular velocity will then in the spectrum. More precisely, the critical scale is kJ at ͞ 1 2 21 be lower than that implied by Eq. (5). More massive halos matter-radiation equality kJeq ෇ 9m22 Mpc . Numeri- 21 ෇ ͞ will have their cuspy r behavior extend from r ry c cally, we find that the linear density power spectrum of down to rJh. FCDM is suppressed relative to the CDM case by These simple scalings show that the wave nature of dark cosx3 matter can prevent the formation of the kpc scale cusps ͑ ͒ ෇ 2͑ ͒ ͑ ͒ ͑ ͒ഠ PFCDM k TF k PCDM k , TF k 8 , and substructure in dark matter halos if m ϳ 10222 eV. 1 1 x However, alone they do not determine what does form (8) instead. To answer this question, cosmological simulations 1͞18 where x ෇ 1.61m22 k͞kJeq. The power drops by a factor will be required [14], and this lies beyond the scope of of 2 at this paper.
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