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Unitarity Bounds and the Cuspy Halo Problem Unitarity Bounds and the Cuspy Halo Problem 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 pro- posals: 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 en- ergy change must be taken into account when making predictions. Exceptions and 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]. arXiv:astro-ph/0102349v3 30 May 2002 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 = 2Re 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| = 2Im 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 ′ 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 (n′ ′ ′ ′ ′ ′ 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- ′ ′ 2 |hℓ s n|1 − S|ℓsni tot. | section for scattering from a two-body initial state to all j,E ℓ,s,ℓX′,s′ 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 ,ℓ,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, ℓ 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 − |hℓsn|S|ℓsni|2 − |hℓ′s′n|1 − S|ℓsni|2] 1 1 ′ ′ 2 ′ ′ ′ 2 2 Xℓ,s ℓ 6=Xℓ,s 6=s Aβα=4i(2π) [E1 + E2/|k1|] [E1 + E2/|k1|] (5) ′ ′ ′ hℓ s n |1 − S|ℓsnij,Etot. From eq. (6) & (9), we can derive two bounds: Xj,jz ℓ′,sX′,ℓ,s 2 −1 tot k1 1 2 ′ z′ z′ z′ z z′ z′ ˆ′ ′ z′ σ . ≤ 4π[| | (2s + 1)(2s + 1)] 2j + 1 (10) ′ ′ ′ ′ k1 hs s |s1 s2 is1,s2 hjj |ℓ s iℓ ,s h |ℓ ℓ i Xj Xℓ,s ℓzX′,sz ′ 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 |ℓ s iℓ,sh |ℓℓ i Xj Xℓ,s ℓXz,sz The first inequality uses |hℓsn|S|ℓsni|2 ≤ 1, ob- where the crucial assumption is that S is rotationally tained from dγhℓsn|S†|γihγ|S|ℓsni ≥ |hℓsn|S|ℓsni|2 invariant and so j and jz are conserved, in addition to † ′ ′ ′ and S S = 1.R A similar bound can be derived for σel. energy conserving. The notation hℓ s n |1 − S|ℓsnij,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 |ℓ s iℓ,s give ˆ z ˆ ear and angular momentum. No assumptions are made the Clebsch-Gordon coefficients, and hk1|ℓℓ i = Yℓℓz (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 Yℓℓz (k1) = δℓz,0 2ℓ +1/(4π). The index ′ ′ ′ ′ ′ particles in the final states. To obtain useful limits from β denotes a 2-body final state p|k1,s1; k2,s2; n i.
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