Unitarity Bounds and the Cuspy Halo Problem

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Lam Hui Department of Physics, Columbia University, 538 West 120th Street, New York, NY 10027 Institute for Advanced Study, School of Natural Sciences, Princeton, NJ 08540 [email protected]

Conventional Cold Dark Matter cosmological models predict small scale structures, such as cuspy halos, which are in apparent conflict with observations. Several alternative scenarios based on modifying fundamental properties of the dark matter have been proposed. We show that general principles of quantum mechanics, in particular unitarity, imply interesting constraints on two proposals: collisional dark matter proposed by Spergel & Steinhardt, and strongly annihilating dark matter proposed by Kaplinghat, Knox & Turner. Efficient scattering required in both implies m <∼ 12 GeV and m <∼ 25 GeV respectively. The same arguments show that the strong annihilation in the second scenario implies the presence of significant elastic scattering, particularly for large enough masses. Recently, a variant of the collisional scenario has been advocated to satisfy simultaneously constraints from dwarf galaxies to clusters, with a cross section that scales inversely with velocity. We show that this scenario likely involves super-elastic processes, and the associated kinetic energy change must be taken into account when making predictions. Implications for experimental searches are discussed.

98.89.Cq; 98.80.Es; 98.62.Gq; 11.80.Et; 11.55Bq

I. INTRODUCTION Griest and Kamionkowski [9] previously derived similar mass bounds related to the freeze-out density of thermal There is a long history of efforts to constrain dark mat- relics, assuming 2-body final states. In §II, we provide ter properties from galactic structure (e.g. [1]). Recent a general derivation for arbitrary final states using the numerical simulations [2] sharpen the predictions of Cold classic optical theorem [10]. We summarize our findings Dark Matter (CDM) structure formation models, and in §III, discuss exceptions to our bounds, and other so- apparent discrepancies with the observed properties of lutions to the cuspy halo problem. structures from galactic to cluster scales are uncovered. The main one that has attracted a lot of attention is the II. DERIVING THE UNITARITY BOUNDS cuspy halo problem , namely that CDM models predict halos that have a high density core or have an inner pro- file that is too steep compared to observations ( [3], but Different versions of the unitarity bounds can be see also [4]). This has encouraged several proposals that found in many textbooks, and can be most easily un- dark matter might have properties different from those derstood using non-relativistic quantum mechanics (e.g. of conventional CDM (see [5] and summary therein). [11]), which is probably adequate for our purpose. How- On the other hand, general principles of quantum me- ever, the results and derivation given here might be of chanics impose non-trivial constraints on some of these wider interest e.g. for estimating thermal relic density. models. We focus here on the proposals of collisional Here we follow closely the field theory treatment of [12]. or strongly self-interacting dark matter (SIDM) by [6] The optical theorem [10] is a powerful consequence of † and of strongly annihilating dark matter (SADM) by [7]. the unitarity of the scattering matrix S, i.e. S S = 1, † † Both require high level of interaction by particle physics which implies (1 − S) (1 − S) = (1 − S ) + (1 − S), or standard: an elastic scattering cross-section of σel. ∼ −24 2 † 10 (mX / GeV) cm for the former and an annihilation dγhβ|1 − S|γihγ|1 − S |αi = 2 Re hβ|1 − S|αi (1) −28 2 Z cross-section of σann.vrel ∼ 10 (mX / GeV) cm for the latter, where mX is the particle mass, and vrel is the rel- where α and β represent two specified states and γ rep- ative velocity of approach. The proposed dark matter resents a complete set of states with measure dγ. Using is therefore quite different from usual candidates such as the definition of the scattering amplitude Aβα the axion or neutralino. We show that the unitarity of 4 4 the scattering matrix, together with a few reasonable as- hβ|1 − S|αi ≡ −i(2π) δ (pβ − pα)Aβα (2) sumptions, imposes interesting particle mass bounds as well as other physical constraints. This is done while where pβ and pα are the total four-momenta, one obtains making minimal assumptions about the nature of the 4 4 2 interactions. Our results complement constraints from dγ(2π) δ (pα − pγ)|Aγα| = 2 Im Aαα (3) experiments or astrophysical considerations e.g. [8]. Z

1 if β = α in eq. (1). We are interested in the case where α This gives the total spin-averaged cross-section for scat- represents a 2-body state of X + X or X + X¯ approach- tering from X + X or X + X¯ to all possible final states. ing each other. The final state γ, on the other hand, For X +X¯ annihilation, we exclude from the above the is completely general, and the integration over γ covers contribution due to elastic scattering (where type and the entire spectrum of possible final states. To be more mass of particles do not change i.e. X + X¯ → X + X¯, z z z z 0 precise, suppose |αi = |k1, s1; k2, s2; ni where s1 and s2 implying |k1| = |k1|) [9]. To do so, we need the following represent the spin-states of the two incoming particles expression for 2-body to 2-body scattering cross-section: with spins s1 and s2 (in our particular case, s1 = s2) dσ |A |2 while k1 and k2 are their respective 4-momenta, and n βα 4 4 dβ = k (2π) δ (pβ − pα)dβ , (7) labels the particle types (e.g. mass, etc.). Recalling that dβ 4(E1 + E2)| 1| 2 dσ/dγ ∝ |Aγα| , eq. (3) gives in the center of mass frame We average over initial spin states and integrate over out- k k (adopted hereafter i.e. 1 + 2 = 0): going momenta, but focus on the elastic contribution (n0 0 0 0 0 0 dσ Im A in |βi = |k1, s1; k2, s2; n i is set to n in |αi) [12]: dγ (α → γ) = αα (4) Z dγ 2(E1 + E2)|k1| π σel. = 2 (2j + 1) (8) |k1| (2s1 + 1)(2s2 + 1) where the left hand side is exactly the total cross-section. Xj This is the optical theorem. It states that the total cross- 0 0 2 |h` s n|1 − S|`sni tot. | section for scattering from a two-body initial state to all j,E `,s,`X0,s0 possible final states equals the imaginary part of the two- body to two-body forward scattering amplitude. The above is the total cross-section for elastic scattering To use this theorem, we expand the scattering am- (note: the same expression also describes X+X → X+X plitude in terms of partial waves i.e. states labeled as elastic scattering) that has to be subtracted from σtot. to z |ktot., Etot., j, j , `, s, ni where ktot. is the total linear mo- yield the total inelastic scattering cross-section, which is mentum (= 0 in the center of mass frame), Etot. is the relevant for annihilation into all possible final states: total energy , j is the total angular momentum, jz is its π z-component, ` is the orbital momentum and s is the to- σinel. = 2 (2j + 1) (9) k1(2s1 + 1)(2s2 + 1) tal spin. Inserting appropriate complete sets of partial Xj wave outer products into eq. (2), we obtain [1 − |h`sn|S|`sni|2 − |h`0s0n|1 − S|`sni|2] 1 1 0 0 2 0 0 0 2 2 X`,s ` 6=X`,s 6=s Aβα= 4i(2π) [E1 + E2/|k1|] [E1 + E2/|k1|] (5) 0 0 0 h` s n |1 − S|`snij,Etot. From eq. (6) & (9), we can derive two bounds: Xj,jz `0,sX0,`,s 2 −1 tot k1 1 2 0 z0 z0 z0 z z0 z0 ˆ0 0 z0 σ . ≤ 4π[| | (2s + 1)(2s + 1)] 2j + 1 (10) 0 0 0 0 k1 hs s |s1 s2 is1,s2 hjj |` s i` ,s h |` ` i Xj X`,s `zX0,sz 0 2 −1 σinel ≤ π[|k1| (2s1 + 1)(2s2 + 1)] 2j + 1 (11) z z z ∗ z z z ∗ ˆ z ∗ . k1 hss |s1s2is1,s2 hjj |` s i`,sh |`` i Xj X`,s `Xz,sz The first inequality uses |h`sn|S|`sni|2 ≤ 1, ob- where the crucial assumption is that S is rotationally tained from dγh`sn|S†|γihγ|S|`sni ≥ |h`sn|S|`sni|2 invariant and so j and jz are conserved, in addition to † 0 0 0 and S S = 1.R A similar bound can be derived for σel. energy conserving. The notation h` s n |1 − S|`snij,Etot. z z as well, which coincides exactly with that for σtot.. emphasizes that S is diagonal in j, j and Etot but the j We pause to note that the above bounds assume only dependence drops out because S commutes with Jx iJy. z z z z z z unitarity and the conservation of total energy and lin- The inner products hss |s1s2is1,s2 and hjj |` s i`,s give ˆ z ˆ ear and angular momentum. No assumptions are made the Clebsch-Gordon coefficients, and hk1|`` i = Y``z (k1) ˆ about the nature of the particles, whether they are com- is the spherical harmonic function. We assume k1 = ˆz ˆ posite or point-like. Nor do we assume the number of in which case Y``z (k1) = δ`z,0 2` + 1/(4π). The index 0 0 0 0 0 particles in the final states. To obtain useful limits from β denotes a 2-body final state p|k1, s1; k2, s2; n i. the bounds, we take the low velocity limit. Assuming the Setting β = α, and averaging over the spin-states (i.e. k1 −1 −1 scattering amplitude Aβα is an analytic function of as (2s1 + 1) (2s2 + 1) z z ) on both sides of eq. (4), s1 ,s2 k1 → 0 (exceptions will be discussed in §III), and noting the optical theorem, wePobtain [12]: that k`hkˆ|``zi is a polynomial function of k, we expect 2π the ` partial wave contribution to Aβα (eq. 5) to scale ` σtot. = 2 (2j + 1) (6) k1 |k1| (2s1 + 1)(2s2 + 1) as | | . This means in the low velocity limit, as is rele- Xj vant for our purpose (typical velocity dispersion in halos

Re h`sn|1 − S|`snij,Etot. range from 10 to 1000 km/s c), the ` = 0 or s-wave X`,s contribution dominates. Setting ` = 0 in eq. (10), (11):

2 2 2 2. σtot. ≤ 16π/(mX vrel) , σinel.vrel ≤ 4π/(mX vrel) (12) The annihilation cross-section from [7], σann.vrel. ∼ −28 2 10 cm (m / GeV), together with eq. (14) and vrel ∼ 2 2 2 2 2 X . where k = m |v2 − v1| /4 = m v /4 is used. The 1 X X rel 1000 km/s, gives us a bound of mX <∼ 25 GeV for strongly second inequality agrees with [9]. Hence, annihilating dark matter. 3. For SADM, efficient annihilation (a form of inelastic 2 −1 2 −17 2 GeV 10 km s scattering) inevitably implies some elastic scattering as σtot. ≤ 1.76 × 10 cm (13) mX vrel. well. From eq. (15), and using vvel ∼ 1000 km/s as 2 −1 before, we have −22 2 GeV 10 km s σinel.vrel. ≤ 1.5 × 10 cm (14) m vrel −22 2 2 X . σel. ≥ 4 × 10 cm [ GeV/mX ] (18) −5 3 2 Furthermore, if σinel. is bounded from below, say σinel. ≥ [1 − 1 − 7 × 10 (mX / GeV) ] σann., one can derive a lower bound on σel. using eq. p 1 Two simple limiting cases: when mX is close to the up- (8) & (9), and setting ` = 0. Defining hXiJ ≡ [(2s + −22 2 −1 per bound of 25 GeV, σel > 4 × 10 cm ; when m 1)(2s2 + 1)] (2j + 1)X, using S at the moment to . ∼ X j,`,s −31 2 4 is small, σel > 5 × 10 cm (m / GeV) . Hence, elas- denote h`sn|S|P`sni, and noting that h1iJ = 1 for ` = 0, it . ∼ X 2 2 2 2 tic scattering is inevitable in this scenario, but can be can be shown (π/k1)(1 − h|S|iJ ) ≥ (π/k1)(1 − h|S| iJ ) ≥ 2 reduced by having a sufficiently small mass. σinel. ≥ σann., which implies h|S|iJ ≤ 1 − k1σann./π. 2 2 2 2 4. Recent simulations suggest that the simplest version Also, σel. ≥ (π/k1)h|1 − S| iJ ≥ (π/k1p)h(1 − |S|) iJ ≥ 2 2 of SIDM fails to match simultaneously the observed halo (π/k1)(1 − h|S|iJ ) . Combining, we have properties from dwarf galaxies to clusters [13,5] (see also 2 2 2 [14]), which have vrel. ranging over 3 orders of magnitude. σel. ≥ (π/k1)[ 1 − 1 − k1σann./π ] (15) q It was suggested that an elastic scattering cross-section of σ ∝ 1/vrel might solve the problem. But as shown This tells us that the elastic scattering cross-section can- . in eq. (16), elastic scattering generally implies σ → con- not be arbitrarily small given a non-vanishing inelastic stant in the small velocity limit. Hence, σ ∝ 1/vrel likely cross-section, e.g. via annihilation. . requires inelastic processes. As eq. (17) shows, processes The above 3 bounds are the main results of this section. 0 in which the net kinetic energy increases (|k1| > |k1| Two more results will be useful for our later discussions. in c.o.m. frame) can give such a velocity dependence. For two-body to two-body processes, recall that the `, `0 0 ` 0 ` SADM provides an example. More generally, the net contribution to Aβα scales as |k1| |k1| . Using dσ/dΩ = 2 0 2 2 kinetic energy increase (super-elasticity) must be taken |A | (|k1|/|k1|)/[64π (E1 + E2) ] (obtained from eq. 7 βα into account when considering the viability of a model by integrating over β except for solid angle Ω), it can be 0 with σ ∝ 1/vrel.. Note, however, the general considera- seen that for elastic scattering, where |k | = |k1|, 1 tions in the last section does not forbid an elastic cross- section that increases as vrel. decreases e.g. the O(vrel.) dσ/dΩ → const.[1 + O(vrel.)] (16) term in eq. (16) can have a negative coefficient. A as |k1| → 0. For inelastic scattering where the sys- 1/vrel. power-law may approximate such a cross-section, tem gains kinetic energy by losing rest mass (e.g. de- but likely only for a limited range of vrel.. An example is excitation of a composite particle or annihilation), since the neutron-neutron scattering cross-section, which ap- −2 0 k1 < |k1| approaches a non-zero value as |k1| → 0, we have proaches a constant for | | ∼ 10 GeV, and scales as −2 −2 < k1 < 1/vrel. only for 10 ∼ | | ∼ 5 × 10 GeV [16]. dσ/dΩ → ( const./vrel.)[1 + O(vrel.)] (17) It is helpful to mention here possible exceptions to the above limits. Our bounds are obtained from eq. (13) & instead in the low velocity limit. The opposite case where (14), which are the ` = 0 (s-wave) versions of eq. (10) the particle gains mass is discussed in [12]. & (11). The argument for putting ` = 0 in the small velocity limit assumes the analyticity of Aβα at k1 = 0. III. DISCUSSION The latter breaks down if the interaction is long-ranged, e.g. Coulomb scattering. This is unlikely to be relevant, because there are strong constraints on dark matter We can derive the following four constraints for with such long ranged interaction [15]. Our argument for strongly self-interacting dark matter (SIDM) [6] and the dominance of s-wave scattering can also be invalid if strongly annihilating dark matter (SADM) [7]. there is a resonance. However, given that the scattering 1. −24 −23 2 The range σel. ∼ 10 − 10 cm (mX / GeV) is cross section should vary smoothly over three orders of given by [5] for SIDM to yield the desired halo properties. magnitude in velocities from dwarfs to clusters, a reso- Using the lower σel., and vrel. ∼ 1000 km/s as appropriate nance seems unlikely. Finally, the most likely situation for clusters, we obtain from eq. (13) mX <∼ 12 GeV for in which the bounds break down is if the particle has a collisional dark matter. large enough size, or the interaction has a large enough

3 effective range, R, such that |k1|R > 1. In such cases, higher partial waves in addition to s-wave generally con- 2 tribute, and σtot. <∼ 64πR and our arguments turn into a limit on R [9]. The condition |k1|R > 1 gives the most stringent constraint on R for vrel. = 10 km/s, as appro- −9 [1] S. Tremaine & J. E. Gunn, Phys. Rev. Lett., 42, 407 priate for dwarf galaxies: R >∼ 10 cm(GeV/mX ). One (1979) can compare this with R for neutron-neutron scattering [2] e.g. J. F. Navarro, C. S. Frenk & S. D. M. White, ApJ, −13 ∼ 10 cm [16]. 490, 493 (1997); A. V. Kravtsov, A. A. Klypin, J. S. It is intriguing that halo structure might be telling us Bullock & J. R. Primack, ApJ, 502, 48 (1998); B. Moore, the elementary properties, in particular the mass, of dark T. Quinn, F. Governato, J. Stadel & G. Lake, MNRAS, matter. It is interesting that several proposals to ad- 310, 1147 (1999); Y. P. Jing & Y. Suto, ApJL, 529, 69 dress the cuspy halo problem, such as Warm Dark Matter (2000) [17] and Fuzzy Dark Matter [18] make explicit assump- [3] e.g. R. Flores & J. Primack, ApJ, 427, L1 (1994); B. Moore, Nature, 370, 629 (1994) tions about the mass of the particles – mX ∼ 1 keV and [4] F. C. Van Den Bosch, B. E. Robertson & J. J. Dalcanton, m ∼ 10−22 eV respectively. For SIDM and SADM, as- X AJ, 119, 1579 (2000) trophysical considerations generally only put constraints [5] R. Davé, D. N. Spergel, P. J. Steinhardt, B. Wandelt, on the cross-section per unit mass. We have shown here submitted to ApJ, astro-ph 0006218 (2000) that unitarity arguments imply a rather modest mass for [6] D. N. Spergel & P. J. Steinhardt, Phys. Rev. Lett, 84, both scenarios as well. It is also worth pointing out that 3760 (2000) our arguments, with suitable modification to take into [7] M. Kaplinghat, L. Knox & M. S. Turner, submitted to account bose enhancement and multiple incoming parti- Phys. Rev. Lett, astro-ph 0005210 (2000) cles, can be extended to cover dark matter in the form of [8] e.g. B. Wandelt, R. Davè, G. R. Farrar, P. C. McGuire, a bose condensate, as has been proposed as yet another D. N. Spergel & P. J. Steinhardt, proceedings of Dark solution to the cuspy halo problem [19]. They generally Matter 00, hep-ph 0006344 (2000); J. Miralda-Escude, require small masses as well < 10 eV. preprint, astro-ph 0002050 (2000) ∼ [9] K. Griest & M. Kamionkowski, Phys. Rev. Lett., 64, 615 A few issues are worth further investigation. Wan- (1990) delt et al. [8] recently argued a version of SIDM, where [10] E. Feenberg, Phys. Rev., 40, 40 (1932); N. Bohr, R. E. the dark matter interacts strongly also with baryons, is 5 Peierls & G. Placzek, Nature, 144, 200 (1939) experimentally viable, but requires mX >∼ 10 GeV, or [11] L. D. Landau & E. M. Lifshitz, Quantum Mechanics, mX <∼ 0.5 GeV. Our bound here is inconsistent with Pergamon Press (1977) the large mass region (but see exception above); exper- [12] S. Weinberg, The Quantum Theory of Fields, Vol. 1, Sec. imental constraints on the low mass region will be very 3.6 - 3.7, Cambridge University Press (1995) −25 2 [13] N. Yoshida, V. Springel & S. D. M. White, submit- interesting (σel. <∼ 10 cm ). It would be useful to find a micro-physics realization of the collisional scenario or ted to ApJL, astro-ph 0006134 (2000); C. Firmani, E. its variant where σ scales appropriately with velocity to D’Onghia, G. Chinarini, submitted to MNRAS, astro-ph match observations. The impact of inelastic collisions on 00010497 (2000) [14] C. Firmani, E. D’Onghia, V. Avila-Reese, G. Chincarini halo structures is worth exploring in more detail. It is & X. Hernández, MNRAS, 315L, 29 (2000); J. A. Sell- also timely to reconsider possible astrophysical solutions wood, ApJL, 540, 1 (2000); A. Burkert, submitted to to the cuspy halo problem, such as the use of mass loss ApJL, astro-ph 0002409 (2000); C. S. Kochanek & M. mechanisms [20]. We hope to examine some of these is- White, ApJ, 543, 514 (2000) sues in the future. [15] A. Gould, B. T. Draine & R. Romani, Phys. Lett. B, 238, 337 (1990) [16] J. M. Blatt & V. F. Weisskopf, Theoretical Nuclear ACKNOWLEDGMENTS Physics, John Wiley & Sons (1952) [17] C. J. Hogan & J. J. Dalcanton, Phys. Rev. D62, 063511 The author thanks Manoj Kaplinghat, Avi Loeb, Dam (2000); see also V. K. Narayanan, D. N. Spergel, R. Davé Son, Paul Steinhardt and Igor Tkachev for useful discus- & C. P. Ma, preprint, astro-ph 0005095 (2000) [18] W. Hu, R. Barkana & A. Gruzinov, preprint, astro-ph sions, and Jonathan Feng, Andrei Gruzinov and Yossi Nir 0003365 (2000) for helpful comments. Thanks are due to David Spergel [19] P. J. E. Peebles, preprint, astro-ph 0002495 (2000); J. for pointing out the example of neutron scattering, and Goodman, preprint, astro-ph 0003018 (2000); I. Tkachev to Kim Griest and Marc Kamionkowski for educating the & A. Riotto, preprint, astro-ph 0003388 (2000). author on elastic scattering, which led to a tightening of [20] J. F. Navarro, V. R. Eke & C. S. Frenk, MNRAS, 283, a bound from an earlier version. Support by the Taplin L72 (1996) Fellowship at the IAS, and by the Outstanding Junior Investigator Award from DOE, DE-FG02-92-ER40699, is gratefully acknowledged.