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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 Schrödinger 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 ak/2m , −1 implies an r cusp for r

2 spectrum at k =4.5h Mpc−1 might solve this problem. 0.8 k ∼ . −1 Thus, the FCDM cutoﬀ at 4 5Mpc , produced (a) rJ /L=0.44 if m ∼ 10−22eV is chosen to remove kpc scale cusps, may solve the low mass halo abundance problem as well. 0.6

Whether the required masses actually coincide in detail (offset)

g acceleration can only be addressed by simulations. FCDM Numerical simulations of CDM with a smooth cutoff CDM in the initial power spectrum qualitatively confirm the 0.4 analytic estimates but suggest that a somewhat larger −1 scale may be necessary: half power at k =2h Mpc z 10 h−1 M reduces the = 3 abundance of 10 halos by a ρ 11 −1 0.2 factor of ∼ 5 and the abundance of 10 h M halos byafactorof∼ 3 [16]. Note, however, that our model density produces a much sharper cutoff in the power spectrum than in the model tested. Furthermore, astrophysical 0 influences such as feedback or photoionization may have 0.8 prevented dwarf galaxy halos from accumulating much (b) rJ /L=0.044 gas or stars [17].

First Objects and Reionization.— At very high redshift, 0.6 much of the star formation in a CDM model is pre- (offset) acceleration dicted to occur in low-mass halos which are not present g intheFCDMmodel.InaCDMmodelthefirstround of star formation is thought to occur in objects of mass 0.4 5 ∼ 10 M (see [18] and references therein) due to molecular hydrogen cooling. The consequent destruction of molecular hydrogen [19,20] implies that it is larger mass ρ 0.2 > 8 objects ∼ 10 M , where atomic cooling is possible, that are responsible for reionization. In our scenario, if the density cutoff scale in eqn. (9) were set to reduce the abundance < 9 0 of M ∼ 10 M halos, reionization could be delayed and 0 0.2 0.4 0.6 0.8 1 the number of detectable galaxies prior to reionization x/L reduced by a factor of ∼ 5 [21]. FIG. 1. One dimensional simulations (a) large Jeans scale Live Halos.— Precisely what effect the Jeans (de Broglie) rJ /L =0.44 (b) small Jeans scale rJ /L =0.044. Two snap- t/t scale has on the structure and abundance of low mass shots, dyn = 99 (solid) and 100 (dotted) are shown for the density profile (units of 15ρ0) and gravitational acceleration halos is best answered through simulations. To provide 2 (units of 3L/2t ,offsetforclarity). some insight on these issues we conclude with simulations dyn of the effects in one dimension.

3 Three dimensional numerical simulations are required preprint, astro-ph/9911372 to determine whether our proposal works in detail. Our [6] B. Moore et al., preprint (astro-ph/9903164) one dimensional simulations suggest that the small-scale [7] V. Sahni & L. Wang, astro-ph/9910097 also considered cutoff appears at r ∼ rJ and that the density profile mass effects on halo cusps but assumed stability only on m−1 on these scales not only is not universal but also evolves the much smaller Compton wavelength . continuously on the dynamical time scale (or faster) due [8] S.-J. Sin, Phys. Rev. D., 50, 3650 (1994) considered halos to be stationary energy states. to interference effects. The observable rotation curves [9] W.H. Press, B.S. Ryden, & D.N. Spergel, Phys. Rev. are smoother than the density profile so that this pre- Lett., 64, 1084 (1990) considered m ∼ 10−28eV and a diction, while testable with high-resolution data, is not Jeans scale of ∼ 100 Mpc. obviously in conflict with the data today. Likewise, the [10] M.Yu. Khlopov, B.A. Malomed, & Ya. B. Zel’dovich, time-variation of the potential is smaller than that of Mon. Not. Roy. Astron. Soc. 215, 575 (1985) the density but can in principle transfer energy from the [11] J.F. Navarro, C.S. Frenk, & S.D.M. White, Astrophys. FCDM to the baryons in the halo. This could puff up J. 490, 493 (1996) the baryons in dwarf galaxies while bringing the FCDM [12] W. Hu, Astrophys. J. 506, 485 (1998) closer to a stationary ground state, but the precise evo- [13] If isocurvature perturbations from inflation are negligi- lution requires detailed calculations. ble, gravity rapidly generates adiabatic conditions from curvature perturbations; c.f. D. Seckel & M.S. Turner, Our Jeans scale is a weak function of density, rJ ∝ Phys. Rev. D, 33, 889 (1986) ρ−1/4. This has two testable consequences. The first is a [14] A. Klypin, A.V. Kravtsov, O. Valenzuela, & F. Prada, k ∼ . m1/2 −1 sharp cutoff at 4 5 22 Mpc in the linear power. Astrophys. J., 522, 82 (1999) Quantities related to the abundance of low mass halos, [15] B. Moore, et al., Astrophys. J., 524, L19 (1999) e.g., dwarf galaxies in the Local Group, the first objects, [16] M. White & R.A.C. Croft, preprint (astro-ph/0001247) faint galaxies at very high redshifts, and reionization can [17] G. Efstathiou, Mon. Not. Roy. Astron. Soc., 256, 43 be seriously affected by the cutoff. Counterintuitively, (1992); A.A. Thoul & D.H. Weinberg, Astrophys. J, 465, quantities related to the non-linear power spectrum of 608 (1996); J.S. Bullock, A.V. Kravtsov & D.H. Wein- the dark matter are only weakly affected due to the grav- berg, preprint, astro-ph/0002214 itational regeneration of small-scale power [16]. The sec- [18] Z. Haiman, A.A. Thoul & A. Loeb, Astrophys. J, 464, 523 (1996) ond consequence is that choosing the mass to set the core [19] T.P. Stecher & D.A. Williams, Astrophys. J, 149, L29 radius in one class of dark matter dominated objects sets (1967); Z. Haiman, M. Rees & A. Loeb, Astrophys. J., the core radius in another set, given the ratio of charac- 476, 458 (1996), erratum, ibid, 484, 985 teristic densities. This relation can in principle be tested [20] Quasars might have catalyzed molecular hydrogen for- by comparing the local dwarf spheroidals [24] to higher mation leading to continued star formation in CDM mass systems. Z. Haiman, T. Abel, & M. Rees, preprint, astro- While the detailed implications remain to be worked ph/9903336; in FCDM, it would still be suppressed. out, fuzzy cold dark matter provides an interesting means [21] R. Barkana & A. Loeb, Astrophys. J., in press (astro- of suppressing the excess small-scale power that plagues ph/0001326) the cold dark matter scenario. [22] S.M. Carroll, Phys. Rev. Lett, 81, 3067 (1998) [23] B. Ratra & P.J.E. Peebles, Phys. Rev. D 37, 3406 (1988); Acknowledgements.— We thank J.P. Ostriker, P.J.E. K. Coble, S. Dodelson & J.A. Frieman, Phys. Rev. D 55, Peebles, D.N. Spergel, M. Srednicki, and M. White for 1851 (1997); R.R. Caldwell, R. Dave & P.J. Steinhardt, useful discussions. WH and AG are supported by the Phys. Rev. Lett. 80, 1582 (1998) Keck Foundation and NSF-9513835; RB by Institute [24] M. Mateo, Ann. Rev. Astron. Astrophs. 36, 435 (1998) Funds and the NSF under Grant No. PHY94-07194.

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