Unitarity Bounds and the Cuspy Halo Problem

Home , Cuspy halo problem

arXiv:astro-ph/0102349v3 30 May 2002 rs-eto of cross-section 10 atr where latter, tnad neatcsatrn rs-eto of cross-section scattering physics elastic particle an by interaction of standard: [7]. level [6] by high collisional (SADM) by require matter of Both dark (SIDM) annihilating proposals matter strongly the of dark and on self-interacting here strongly these focus or of We some on constraints models. non-trivial impose chanics therein). summary those and [5] from (see different CDM properties that conventional proposals have of but several might [3], encouraged matter ( has observations dark This to pro- [4]). compared inner also an steep see have too or is predict core that density models file high CDM the a is that have attention that namely of halos , lot problem a attracted halo uncovered. has cuspy are that scales one of cluster main The to properties galactic and observed from the models, structures with formation discrepancies structure apparent (CDM) Recent Cold Matter of predictions [1]). Dark the (e.g. sharpen [2] structure simulations galactic numerical from properties ter xeiet ratohsclcnieain ..[8]. from e.g. the considerations constraints of astrophysical while complement or nature done results experiments the is Our about This interactions. assumptions constraints. minimal physical as making other bounds as mass of as- particle well unitarity reasonable interesting few the imposes a that with sumptions, show together We matrix, scattering as neutralino. the matter such or candidates dark axion usual proposed from the different The quite therefore approach. is of velocity ative nteohrhn,gnrlpicpe fqatmme- quantum of principles general hand, other the On mat- dark constrain to efforts of history long a is There − 24 ( m X / e)cm GeV) 88.q 88.s 86.q 18.t 11.55Bq 11.80.Et; 98.62.Gq; 98.80.Es; 98.89.Cq; predictio making when discussed. pr account are super-elastic into searches taken involves experimental be likely se must cross scenario change a this ergy with that clusters, show has to We scenario galaxies collisional dwarf the elastic from of constraints significant variant of a Recently, presence masses. the implies scenario second e and GeV oas olsoa akmte rpsdb pre Stein scatt & Efficient Spergel Turner. & by Knox Kaplinghat, proposed by proposed matter matter dark unitarity, be Sev collisional particular have in matter posals: mechanics, observations. dark quantum with the of conflict of principles properties apparent fundamental in modifying are which halos, ovninlCl akMte omlgclmdl predict models cosmological Matter Dark Cold Conventional m X σ eateto hsc,Clmi nvriy 3 et120th West 538 University, Columbia Physics, of Department .INTRODUCTION I. ann stepril as and mass, particle the is . 2 v m rel nttt o dacdSuy colo aua cecs P Sciences, Natural of School Study, Advanced for Institute o h omrada annihilation an and former the for < ∼ ∼ ntrt onsadteCsyHl Problem Halo Cuspy the and Bounds Unitarity 5GVrsetvl.Tesm ruet hwta h strong the that show arguments same The respectively. GeV 25 10 − 28 ( m X / e)cm GeV) v rel [email protected] sterel- the is 2 o the for σ el . a Hui Lam ∼ 1 h ento ftesatrn amplitude scattering the of definition the where where eet opeesto ttswt measure with states of set complete a resents uin otecsyhl problem. halo cuspy the to lutions h ntrt ftesatrn matrix scattering the of [12]. unitarity of the treatment theory density. field relic of the thermal closely be estimating follow might we for here Here given e.g. derivation interest How- and wider purpose. results our un- for the easily adequate ever, probably most is (e.g. be which mechanics [11]), can quantum non-relativistic and using textbooks, derstood many in found lsi pia hoe 1] esmaieorfindings our the summarize using We states in [10]. final theorem arbitrary optical for classic derivation general a hc mle (1 implies which eis suig2bd nlsae.In states. final thermal of 2-body density assuming freeze-out the relics, to related bounds mass h pia hoe 1]i oeflcneuneof consequence powerful a is [10] theorem optical The be can bounds unitarity the of versions Different Z retadKmokwk 9 rvosydrvdsimilar derived previously [9] Kamionkowski and Griest § I,dsusecpin oorbud,adohrso- other and bounds, our to exceptions discuss III, I EIIGTEUIAIYBOUNDS UNITARITY THE DERIVING II. dγ ml neetn osrit ntopro- two on constraints interesting imply enavctdt aif simultaneously satisfy to advocated been cteig atclryfrlreenough large for particularly scattering, p to htsae nesl ihvelocity. with inversely scales that ction α Z cse,adteascae iei en- kinetic associated the and ocesses, h β h β β ad,adsrnl niiaigdark annihilating strongly and hardt, and dγ and | rn eurdi ohimplies both in required ering ml cl tutrs uha cuspy as such structures, scale small | npooe.W hwta general that show We proposed. en 1 1 rlatraieseaisbsdon based scenarios alternative eral s xetosadipiain for implications and Exceptions ns. (2 − − β tet e ok Y10027 NY York, New Street, p π S S α ) ersn w pcfidsae and states specified two represent | | 4 γ ictn J08540 NJ rinceton, α − r h oa ormmna n obtains one four-momenta, total the are δ ih − ≡ i 4 S ( γ p ) | α † 1 (1 − − i (2 − S p niiaini the in annihilation γ π † S ) | ) α | (1 = ) 4 A δ i γα 4 Re 2 = ( p | 2 β − Im 2 = − m S p h S † α < β ∼ (1 + ) ) i.e. , | § A 1 12 I eprovide we II, A A − βα βα αα S dγ − S | α † S Using . i S ,or ), γ 1, = rep- (2) (3) (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. 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, weP obtain [12]: that kℓhkˆ|ℓℓzi is a polynomial function of k, we expect 2π the ℓ partial wave contribution to Aβα (eq. 5) to scale ℓ 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 hℓsn|1 − S|ℓsnij,Etot. range from 10 to 1000 km/s ≪ c), the ℓ = 0 or s-wave Xℓ,s contribution dominates. Setting ℓ = 0 in eq. (10), (11):

2 2 2 < σtot. ≤ 16π/(mX vrel) , σinel.vrel ≤ 4π/(mX vrel) (12) 1000km/s, gives us a bound of mX ∼ 25 GeV for strongly annihilating dark matter. 2 2 v v 2 2 2 where k1 = mX | 2 − 1| /4 = mX vrel/4 is used. The 3. For SADM, efficient annihilation (a form of inelastic second inequality agrees with [9]. Hence, scattering) inevitably implies some elastic scattering as vel 2 −1 2 well. From eq. (15), and using v ∼ 1000 km/s as −17 2 GeV 10kms σtot. ≤ 1.76 × 10 cm (13) before, we have mX vrel. −22 2 2 2 −1 σel. ≥ 4 × 10 cm [ GeV/mX ] (18) −22 2 GeV 10kms σinel.vrel. ≤ 1.5 × 10 cm (14) −5 3 2 mX vrel. [1 − 1 − 7 × 10 (mX / GeV) ] p Furthermore, if σinel. is bounded from below, say σinel. ≥ Two simple limiting cases: when mX is close to the up- > −22 2 σann., one can derive a lower bound on σel. using eq. per bound of 25 GeV, σel. ∼ 4 × 10 cm ; when mX −31 2 4 (8) & (9), and setting ℓ = 0. Defining hXiJ ≡ [(2s1 + is small, σel. >∼ 5 × 10 cm (mX / GeV) . Hence, elas- −1 1)(2s2 + 1)] j,ℓ,s(2j + 1)X, using S at the moment to tic scattering is inevitable in this scenario, but can be denote hℓsn|S|Pℓsni, and noting that h1iJ = 1 for ℓ = 0, it reduced by having a sufficiently small mass. 2 2 2 2 4. can be shown (π/k1)(1 − h|S|iJ ) ≥ (π/k1)(1 − h|S| iJ ) ≥ Recent simulations suggest that the simplest version 2 σinel. ≥ σann., which implies h|S|iJ ≤ 1 − k1σann./π. of SIDM fails to match simultaneously the observed halo 2 2 2 2 Also, σel. ≥ (π/k1)h|1 − S| iJ ≥ (π/k1p)h(1 − |S|) iJ ≥ properties from dwarf galaxies to clusters [13,5] (see also 2 2 (π/k1)(1 − h|S|iJ ) . Combining, we have [14]), which have vrel. ranging over 3 orders of magnitude. It was suggested that an elastic scattering cross-section 2 2 2 rel σel. ≥ (π/k1)[ 1 − 1 − k1σann./π ] (15) of σ ∝ 1/v . might solve the problem. But as shown q in eq. (16), elastic scattering generally implies σ → con- This tells us that the elastic scattering cross-section can- stant in the small velocity limit. Hence, σ ∝ 1/vrel. likely not be arbitrarily small given a non-vanishing inelastic requires inelastic processes. As eq. (17) shows, processes ′ cross-section, e.g. via annihilation. in which the net kinetic energy increases (|k1| > |k1| The above 3 bounds are the main results of this section. in c.o.m. frame) can give such a velocity dependence. Two more results will be useful for our later discussions. SADM provides an example. More generally, the net For two-body to two-body processes, recall that the ℓ, ℓ′ kinetic energy increase (super-elasticity) must be taken ′ ℓ ′ ℓ contribution to Aβα scales as |k1| |k1| . Using dσ/dΩ= into account when considering the viability of a model 2 ′ 2 2 |Aβα| (|k1|/|k1|)/[64π (E1 + E2) ] (obtained from eq. 7 with σ ∝ 1/vrel. e.g. it may delay core collapse and make by integrating over β except for solid angle Ω), it can be the core larger. Note, however, the general considerations ′ seen that for elastic scattering, where |k1| = |k1|, in the last section does not forbid an elastic cross-section that increases as vrel. decreases e.g. the O(vrel.) term in dσ/dΩ → const.[1 + O(vrel.)] (16) eq. (16) can have a negative coefficient. A 1/vrel. power- law may approximate such a cross-section, but likely only as |k1| → 0. For inelastic scattering where the sys- for a limited range of vrel.. An example is the neutron- tem gains kinetic energy by losing rest mass (e.g. de- neutron scattering cross-section, which approaches a con- excitation of a composite particle or annihilation), since −2 stant for |k1| < 10 GeV, and scales as 1/vrel only for k′ k ∼ . | 1| approaches a non-zero value as | 1| → 0, we have −2 k −2 10 <∼ | 1| <∼ 5 × 10 GeV [17]. It is helpful to mention here possible exceptions to the dσ/dΩ → ( const./vrel.)[1 + O(vrel.)] (17) above limits. Our bounds are obtained from eq. (13) & instead in the low velocity limit. The opposite case where (14), which are the ℓ = 0 (s-wave) versions of eq. (10) the particle gains mass is discussed in [12]. & (11). The argument for putting ℓ = 0 in the small velocity limit assumes the analyticity of Aβα at k1 = 0. The latter breaks down if the interaction is long-ranged, III. DISCUSSION 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 −24 −23 2 1. 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 (appropriate nance seems unlikely. Finally, the most likely situation in for clusters), eq. (13) tells us mX <∼ 12 GeV for SIDM. which the bounds break down is if the particle has a large 2. The annihilation cross-section from [7], σann.vrel. ∼ enough size, or the interaction has a large enough effec- −28 2 10 cm (mX / GeV), together with eq. (14) and vrel. ∼ tive range, R, such that |k1|R> 1 (e.g. see [16]). In such

3 cases, higher partial waves in addition to s-wave gener- 2 ally contribute, 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 > −9 appropriate for dwarf galaxies: R ∼ 10 cm(GeV/mX ). [1] S. Tremaine & J. E. Gunn, Phys. Rev. Lett., 42, 407 One can compare this with R for neutron-neutron scat- (1979) −13 tering ∼ 10 cm [17]. [2] e.g. J. F. Navarro, C. S. Frenk & S. D. M. White, ApJ, It is intriguing that halo structure might be telling us 490, 493 (1997); A. V. Kravtsov, A. A. Klypin, J. S. the elementary properties, in particular the mass, of dark Bullock & J. R. Primack, ApJ, 502, 48 (1998); B. Moore, matter. It is interesting that several proposals to ad- T. Quinn, F. Governato, J. Stadel & G. Lake, MNRAS, dress the cuspy halo problem, such as Warm Dark Matter 310, 1147 (1999); Y. P. Jing & Y. Suto, ApJL, 529, 69 [18] and Fuzzy Dark Matter [19] make explicit assump- (2000) [3] e.g. R. Flores & J. Primack, ApJ, 427, L1 (1994); B. tions about the mass of the particles – mX ∼ 1 keV and −22 Moore, Nature, 370, 629 (1994) mX ∼ 10 eV respectively. For SIDM and SADM, as- [4] F. C. Van Den Bosch, B. E. Robertson & J. J. Dalcanton, trophysical considerations generally only put constraints AJ, 119, 1579 (2000) on the cross-section per unit mass. We have shown here [5] R. Davé, D. N. Spergel, P. J. Steinhardt, B. Wandelt, that unitarity arguments imply a rather modest mass for submitted to ApJ, astro-ph 0006218 (2000) both scenarios as well. It is also worth pointing out that [6] D. N. Spergel & P. J. Steinhardt, Phys. Rev. Lett, 84, our arguments, with suitable modification to take into 3760 (2000) account bose enhancement and multiple incoming parti- [7] M. Kaplinghat, L. Knox & M. S. Turner, submitted to cles, can be extended to cover dark matter in the form of Phys. Rev. Lett, astro-ph 0005210 (2000) a bose condensate, as has been proposed as yet another [8] e.g. B. Wandelt, R. Davè, G. R. Farrar, P. C. McGuire, solution to the cuspy halo problem [20]. They generally D. N. Spergel & P. J. Steinhardt, proceedings of Dark < Matter 00, hep-ph 0006344 (2000); J. Miralda-Escude, require small masses as well ∼ 10 eV. A few issues are worth further investigation. Wan- preprint, astro-ph 0002050 (2000) [9] K. Griest & M. Kamionkowski, Phys. Rev. Lett., 64, 615 delt et al. [8] recently argued a version of SIDM, where (1990) the dark matter interacts strongly also with baryons, is 5 [10] E. Feenberg, Phys. Rev., 40, 40 (1932); N. Bohr, R. E. experimentally viable, but requires mX >∼ 10 GeV, or Peierls & G. Placzek, Nature, 144, 200 (1939) mX <∼ 0.5 GeV. Our bound here is inconsistent with [11] L. D. Landau & E. M. Lifshitz, Quantum Mechanics, the large mass region (but see exceptions above); exper- Pergamon Press (1977) imental constraints on the low mass region will be very [12] S. Weinberg, The Quantum Theory of Fields, Vol. 1, Sec. < −25 2 3.6 - 3.7, Cambridge University Press (1995) interesting (σel. ∼ 10 cm ). It would be useful to find a micro-physics realization of the collisional scenario [21] [13] N. Yoshida, V. Springel, S. D. M. White & G. Tormen, or its variant where σ scales appropriately with velocity ApJL, 544, L87 (2000); C. Firmani, E. D’Onghia, G. Chi- to match observations. The impact of inelastic collisions narini, submitted to MNRAS, astro-ph 00010497 (2000) on halo structures is worth exploring in more detail. It [14] C. Firmani, E. D’Onghia, V. Avila-Reese, G. Chincar- ini & X. Hernández, MNRAS, 315L, 29 (2000); J. A. is also timely to reconsider possible astrophysical solu- Sellwood, ApJL, 540, 1 (2000); A. Burkert, ApJL, 534, tions to the cuspy halo problem, such as the use of mass 143 (2000); C. S. Kochanek & M. White, ApJ, 543, 514 loss mechanisms [22]. We hope to examine some of these (2000) issues in the future. [15] A. Gould, B. T. Draine & R. Romani, Phys. Lett. B, 238, 337 (1990) [16] A. Kusenko & P. J. Steinhardt, in prep. ACKNOWLEDGMENTS [17] J. M. Blatt & V. F. Weisskopf, Theoretical Nuclear Physics, John Wiley & Sons (1952) The author thanks Manoj Kaplinghat, Avi Loeb, Dam [18] C. J. Hogan & J. J. Dalcanton, Phys. Rev. D62, 063511 Son, Paul Steinhardt and Igor Tkachev for useful discus- (2000); see also V. K. Narayanan, D. N. Spergel, R. Davé sions, and Jonathan Feng, Andrei Gruzinov and Yossi Nir & C. P. Ma, preprint, astro-ph 0005095 (2000) [19] W. Hu, R. Barkana & A. Gruzinov, preprint, astro-ph for helpful comments. Thanks are due to David Spergel 0003365 (2000) for pointing out the example of neutron scattering, and [20] P. J. E. Peebles, preprint, astro-ph 0002495 (2000); J. to Kim Griest and Marc Kamionkowski for educating the Goodman, New Astronomy, 5, 103(2000); I. Tkachev & author on elastic scattering, which led to a tightening of A. Riotto, Phys. Lett. B484, 177 (2000). a bound from an earlier version. Support by the Taplin [21] R. N. Mohapatra and V. L. Teplitz, Phys. Rev. D62, Fellowship at the IAS, and by the Outstanding Junior 063506 (2000) Investigator Award from DOE, DE-FG02-92-ER40699, [22] J. F. Navarro, V. R. Eke & C. S. Frenk, MNRAS, 283, is gratefully acknowledged. L72 (1996)