Enhancement of dark matter relic density from the late time dark matter conversions
aa r X i v : . [ h e p - ph ] S e p Enhancement of dark matter relic density fromthe late time dark matter conversions
Ze-Peng Liu ∗ , Yue-Liang Wu † , and Yu-Feng Zhou ‡ Kavli Institute for Theoretical Physics China,Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing, 100190, P.R. China
Abstract
We demonstrate that if the dark matter (DM) in the Universe contains multiplecomponents, the possible interactions between the DM components may convertthe heavier DM components into the lighter ones. It is then possible that thelightest DM component with an annihilation cross section significantly larger thanthat of the typical weakly interacting massive particle (WIMP) can obtain a relicdensity in agreement with the cosmological observations, due to an enhancementof number density from the DM conversion process at late time after the ther-mal decoupling, which may provide an alternative source of boost factor relevantto the positron and electron excesses reported by the recent DM indirect searchexperiments. ∗ Email: [email protected] † Email: [email protected] ‡ Email: [email protected] Introduction
In the recent years, a number of experiments such as PAMELA [1], ATIC [2], Fermi-LAT[3] and HESS [4] etc. have reported excesses in the high energy spectrum of cosmic-raypositrons and electrons over the backgrounds estimated from the traditional astrophysics.Besides plausible astrophysical explanations [5–7], the dark matter (DM) annihilation ordecay provides exciting alternative explanations from particle physics.If the DM particle is a thermal relic such as the weakly interacting massive particle(WIMP), the thermally averaged product of its annihilation cross section with the rela-tive velocity at the time of thermal freeze out is typically h σv i F ≃ × − cm s − . Thepositron or electron flux produced by the DM annihilation can be parametrized asΦ e = BN e ρ h σv i F m D , (1)where ρ ≃ . · cm − is the smooth local DM energy density estimated fromastrophysics, N e is the averaged electron number produced per DM annihilation whichdepends on DM models and propagation parameters, and m D is the mass of the DMparticle. The boost factor B is defined as B ≡ ( ρ/ρ ) h σv i / h σv i F with ρ the truelocal DM density and h σv i the DM annihilation cross section multiplied by the relativevelocity and averaged over the DM velocity distribution today. Both the PAMELA andFermi-LAT results indicate that a large boost factor is needed [8, 9]. For a typical DMmass of ∼ B is ∼ µ + µ − and ρ fixed to ρ [9].A large boost factor may arise from the non-uniformity of the DM distribution in theDM halo. The N-body simulations show however that the local cumps of dark matterdensity are unlikely to contribute to a large enough ρ/ρ [10, 11]. An other possibilityof increasing the boost factor is that the DM annihilation cross section may be velocity-dependent which grows at low velocity. The DM annihilation cross section today may bemuch larger than that at the time of thermal freeze out, and thus is not constrained bythe DM relic density. Some enhancement mechanisms have been proposed along this line,such as the Sommerfeld enhancement [12–20] and the resonance enhancement [21–23] etc.In some non-thermal DM scenarios, the number density of the DM particle can beenhanced by the out of equilibrium decay of some heavier unstable particles if the DMparticle is among the decay products of the decaying particle [24, 25]. The decay of theunstable particle must take place at very late time. Otherwise the DM particles with theenhanced number density will annihilate into the Standard Model (SM) particles again,which washes out the effect of the enhancement. This requires that the decay width ofthe unstable particle must be extremely small, typically 10 − GeV for the mass of the1ecaying particle around TeV [24], which is much smaller than that of the typical weakinteraction.In this work, we consider an alternative possibility for generating a boost factor,which does not require the velocity-dependent annihilation cross section or the decay ofunstable particles. We show that in the scenarios of interacting Milt-component DM,the interactions among the DM components may convert the heavier DM componentsinto the lighter ones, which is not sensitive to the details of the conversion interaction.If the interactions are strong enough and the DM components are nearly degenerate inmass, the conversion can enhance the number density of the lighter DM components atlate time after the thermal decoupling. Eventually, the whole DM today in the Universecan be dominated by the lightest DM component with enhanced number density, whichcorresponds to a large boost factor. The scenarios of multi-component DM have beendiscuss previously in Refs. [26–35]. Note however that the models with simply mixednon-interacting multi-component DM cannot generate large boost factors.This paper is organized as follows: in section 2, we first discuss the thermal evolutionof the DM number densities in generic multi-component DM models. We then giveapproximate analytic expressions as well as precise numerical calculations of the boostfactor in a generic two-component DM model. In section 3, we consider a concrete modelcontaining two fermionic DM particles with extra U (1) gauge interactions in the hiddensector. The conclusions are given in section 4. Let us consider a generic model in which the whole cold DM contains N components χ i ( i = 1 , . . . , N ), with masses m i and internal degrees of freedom g i respectively. TheDM components are labeled such that m i < m j for i < j , thus χ is the lightest DMparticle. We are interested in the case that χ i are nearly degenerate in mass, namelythe relative mass differences between χ i and χ satisfy ε i ≡ ( m i − m ) /m ≪
1. Inthis case, we shall show that the interactions between the DM components lead to theDM conversion. The situation is analogous to the neutral meson mixing and neutrinooscillations in particle physics. They all occur at small mass differences. The thermalevolution of the DM number density normalized to the entropy density Y i ≡ n i /s withrespect to the rescaled temperature x ≡ m /T is govern by the following Boltzmann2quation dY i ( x ) dx = − λx " h σ i v i ( Y i − Y ieq ) − X j h σ ij v i ( Y i − r ij Y j ) , (2)where λ ≡ xs/H ( T ) is a combination of x , the entropy density s and the Hubble param-eter H ( T ) as a function of temperature T . Y ieq ≃ ( g i /s )[ m i T / (2 π )] / exp( − ε i x ) is theequilibrium number density normalized to entropy density for non-relativistic particles. h σ i v i are the thermally averaged cross sections multiplied by the DM relative velocityfor the process χ i χ i → XX ′ with XX ′ standing for the light SM particles which are inthermal equilibrium, and h σ ij v i are the ones for the DM conversion process χ i χ i → χ j χ j .The quantity r ij ( x ) ≡ Y ieq ( x ) Y jeq ( x ) = (cid:18) g i g j (cid:19) (cid:18) m i m j (cid:19) / exp[ − ( ǫ i − ǫ j ) x ] (3)is the ratio between the two equilibrium number density functions, In writing down Eq.(2) we have assumed kinetic equilibrium. The first term in the r.h.s. of Eq.(2) describesthe change of number density of χ i due to the annihilation into the SM particles, whilethe second term describes the change due to the conversion to other DM particles.In the case that the cross section of the conversion process h σ ij v i is large enough, theDM particle χ i can be kept in thermal equilibrium with χ j for a long time after both χ i and χ j have decoupled from the thermal equilibrium with the SM particles. In this case,the number densities of χ i,j satisfy a simple relation Y i ( x ) Y j ( x ) ≈ Y ieq ( x ) Y jeq ( x ) = r ij ( x ) . (4)We emphasize that even when χ i is in equilibrium with χ j the ratio of the numberdensity Y i ( x ) /Y j ( x ) can be quite different from unity and can vary with temperature.For instance, if g i ≫ g j and 0 < ( ǫ i − ǫ j ) ≪
1, from Eq. (3) and (4) one obtains Y i ( x ) ≫ Y j ( x ) at the early time when ( ǫ i − ǫ j ) x ≪
1. However, at the late time when( ǫ i − ǫ j ) x ≫
1, one gets Y i ( x ) ≪ Y j ( x ), which is due to the Boltzmann suppression factorexp[ − ( ǫ i − ǫ j ) x ] in the expression of r ij . Thus the heavier particles can be graduallyconverted into lighter ones through this temperature-dependent equilibrium between χ i and χ j .Since all the DM components χ i are stable, in general the co-annihilation process χ i χ j → XX ′ are not allowed as the crossing process χ i → χ j XX ′ corresponds to the de-cay of χ i . Furthermore, unlike the case of co-annihilation, χ i and χ j may not necessarilyshare the same quantum numbers.An interesting limit to consider is that the rates of DM conversion are large comparedwith that of the individual DM annihilation into the SM particles, i.e. h σ ij v i & h σ i v i .3n this limit, after both the DM particles have decoupled from the thermal equilibriumwith the SM particles, which take place at a typical temperature x = x dec ≈
25, thestrong interactions of conversion will maintain an equilibrium between χ i and χ j for along time until the rate of the conversion cannot compete with the expansion rate of theUniverse. Making use of Eq. (4), the evolution of the total density Y ( x ) ≡ P Ni =1 Y i ( x )can be written as dYdx = − λx h σ eff v i (cid:0) Y − Y eq (cid:1) , (5)where h σ eff v i is the effective thermally averaged product of DM annihilation cross sectionand the relative velocity which can be written as h σ eff v i = P Ni =1 w i g i (1 + ε i ) exp( − ε i x ) g eff h σ v i , (6)where w i ≡ h σ i v i / h σ v i is the annihilation cross section relative to that of the lightestone. The total equilibrium number density can be written as Y eq ≡ N X i =1 Y ieq ( x ) ≈ g eff (cid:18) m T π (cid:19) / exp( − x ) , (7)with effective degrees of freedom g eff = P i g i (1 + ε i ) / exp( − ε i x ). Note that the con-version terms do not show up explicitly in Eq. (5). Through the conversion processes χ i χ i → χ j χ j the slightly heavier components will be converted into the lighter ones,because the factor r ij ( x ) is proportional to exp[ − ( m i − m j ) /T ] which suppresses thedensity of the heavier components at lower temperature. If the conversion cross sectionis large enough, most of the DM components will be converted into the lightest χ beforethe interaction of conversion decouples, which may result in a large enhancement of therelic density of χ and leads to a large boost factor.As an example, let us consider a generic DM model with only two components. Forrelatively large conversion cross section u ≡ h σ v i / h σ v i &
1, The effective total crosssection is given by h σ eff v i = 1 + wg exp( − εx )[1 + g exp( − εx )] h σ v i , (8)where w ≡ w , g ≡ g /g and ε ≡ ε . Because of the x -dependence in h σ eff v i , thethermal evolution of Y ( x ) differs significantly from that of the standard WIMP. In thecase that χ has large degrees of freedom but a small annihilation cross section, namely g ≫ w ≪ wg ≪
1, the thermal evolution of the total density Y can beapproximated by dYdx ≈ − λx g exp( − εx )] h σ v i (cid:0) Y − Y eq (cid:1) , (9)4he thermal evolution of the total number density can be roughly divided into four stages:i) At high temperature region where 3 . x ≪ x dec , both the DM components arein thermal equilibrium with the SM particles. Y i ( x ) must closely track Y ieq ( x ) whichdecrease exponentially as x increases. However, since g ≫ ǫ ≪
1, the number den-sity of χ is much higher than that of χ , i.e. Y ( x ) ≫ Y ( x ). ii) When the temperaturegoes down and x is close to the decoupling point x dec , both the DM components start todecouple from the thermal equilibrium. In the region x dec . x ≪ /ε , h σ eff v i is nearlya constant and h σ eff v i ≈ h σ v i / (1 + g ) ≪ h σ v i , the total density Y ( x ) behaves justlike that of an ordinary WIMP which converges quickly to Y ( x ) ≈ x dec / ( λ h σ v i ). iii)As x continues growing, the suppression factor exp( − εx ) in h σ eff v i becomes relevant.The value of h σ eff v i grows rapidly especially after x reaches the point εx ≈ O (1), whichleads to the further reduction of Y ( x ). In this stage, although both χ , have decoupledfrom the thermal equilibrium with the SM particles. The strong conversion interaction χ χ ↔ χ χ maintains an equilibrium between the two DM components. Accordingto Eq. (4), the relative number density Y ( x ) /Y ( x ) decreases with x increasing, whichcorresponds to the conversion from the heavier DM component into the lighter one. Atthe point x c = (1 /ε ) ln g one has Y ( x ) ≈ Y ( x ). For the region x > x dec and x is notclose to x c , because of Y eq ( x ) ≪ Y ( x ) and g exp( − εx ) ≫
1, the Eq. (9) can be analyt-ically integrated out, using the expression I ( x ) = R x − exp( x ) dx = Ei( x ) − exp( x ) /x where Ei( x ) is the exponential integral function. The integral has an asymptotic formof I ( x ) ≈ exp( x ) /x for x ≫
1. Thus Y ( x ) in this region can be approximated by Y ( x ) ≈ g x dec λ h σ v i (cid:20) (cid:16) x dec x (cid:17) exp(2 εx )2 εx (cid:21) − . (10)iv) When x becomes very large εx ≫ O (1) , h σ eff v i quickly approaches h σ v i , andbecomes independent of x again. The evolution of Y ( x ) in this region can be obtainedby a simple integration as it was done in the stage ii). The solution of Y ( x ) shows asecond decoupling. Finally when the conversion rate cannot compete with the expansionrate of the Universe at some point x F corresponding to sY h σ v i /H ≈
1, both Y ( x )and Y ( x ) remain unchanged as relics. The whole DM can be dominated by χ if theconversion is efficient enough.By matching the analytic solutions of Y ( x ) in different regions near the points x dec and x c , and requiring that the final total relic density is equivalent to the observedΩ CDM h ≈ .
11, we obtain the following approximate expression of the boost factor B ≈ g (cid:20) (cid:18) x dec x c (cid:19) (cid:18) exp(2 εx c )2 εx c + g (cid:19)(cid:21) − . (11)As expected, the enhancement essentially comes from the conversion of the degrees offreedom. Thus the maximum enhancement is g . The two terms in the r.h.s of the above5quation correspond to the reduction of Y ( x ) during the late time conversion stages. Forlarge enough g , the boost factor can be approximated by B ≈ g / (1 + εg x dec / ln g ). Inorder to have a large boost factor, a small ε ≪ ln g/ ( g x dec ) is also required. As shownin Eq. (11) the boost factor is not sensitive to the exact values of the cross sections aslong as the conditions w ≪ u ≫ Y i ( x ) and the boost factor withoutusing approximations for a generic two-component DM model. The results for w = 10 − , u = 10 and ε = 2 × − is shown in Fig. 1. The value of h σ v i is adjusted such thatthe final total DM relic abundance is always equal to the observed value Ω CDM h . Themass of the light DM particle is set to m = 1 TeV. For an illustration the ratio betweenthe internal degrees of freedom is set to be large g = 60. From the figure, the four stagesof the thermal evolution of Y ( x ) as well as the crossing point can be clearly seen. Thecrossing point at x = x c ≈ × − indicates the time when the number density of χ start to surpass that of χ and eventually dominant the whole DM relic density. In thisparameter set a large boost factor B ≈ h σ v i / h σv i F ≈
585 is obtained which is in aremarkable agreement with Eq. (11) with error less than ∼ B varies with the mass difference ε fordifferent relative internal degrees of freedom g . In general, B becomes larger for smaller ε and larger g . For ε = 10 − and g = 60, the boost factor can reach B ∼ . For amuch smaller g = 20 and a larger ε = 8 × − , the boost factor can still reach O (100).The dependence of B on the cross sections u and w is shown in Fig. 2 (right). A small w and large u lead to the increasing of B . However, for very small w . − and varylarge u & B becomes insensitive to the exact values of w and u , whichis also in agreement with the approximate solution given in Eq. (11). For models with multiple DM components, it is nature that there exists interactionsamong the DM components which may lead to the conversions among them. In this workwe consider a simple interacting two-component DM model by adding to the standardmodel (SM) with two SM gauge singlet fermionic DM particles χ , . The particles χ , are charged under a local U (1) symmetry which is broken spontaneously by thevacuum expectation value (VEV) of a scalar field φ through the Higgs mechanism. Thecorresponding massive gauge boson is denoted by A which may cause the interaction¯ χ χ ↔ ¯ χ χ . The stability of χ , is protected by two different global U (1) numbersymmetries. An SM gauge singlet pseudo-scalar η is introduced as a messenger field6 ( x ) i Y -17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 Y Y +Y Y (no conversion) Y (no conversion) Y Figure 1: Thermal evolution of the number densities Y ( x ) (red solid) and Y ( x ) (bluesolid) with respect to x . The solid (dashed) curves correspond to the case with (without)DM conversions. The green dotted curve corresponds to the sum of Y and Y , forparameters g = 60, m = 1TeV, ε = 2 × − , w = 10 − and u = 10 respectively.which couples to both the dark sector and the SM sector. In order to have the leptophilicnature of DM annihilation, we also introduce an SM SU (2) L triplet field ∆ with the SMquantum number (1 , ,
1) and flavor contents ∆ = ( δ ++ , δ + , δ ). The triplet carries thequantum number B − L =2 such that it can couple to the SM left-handed leptons ℓ L through Yukawa interactions ¯ ℓ cL ∆ ℓ L , but cannot couple to quarks directly. The VEV ofthe triplet has to be very small around eV scale, which is required by the smallness ofthe neutrino masses. As a consequence, the couplings between one triplet and two SMgauge bosons such as δ ±± W ∓ W ∓ , δ ± W ∓ Z and δ Z Z are strongly suppressed as theyare all proportional to the VEV of the triplet, which makes it difficult for the triplet todecay even indirectly into quarks through SM gauge bosons [36, 37]. If η has a strongercoupling to ∆ than that to the SM Higgs boson H and φ then the annihilation productsof the dark matter particles χ , will be mostly leptons.The full Lagrangian of the model can be written as L = L SM + L The new in-teractions in L which are relevant to the DM annihilation and conversion are given7 -4 -3 B g=80g=60g=40g=20 u -1
10 1 10 B -5 w=10 -4 w=10 -4 · w=5 -3 w=10 Figure 2: Left) boost factor B as a function of the relative mass difference ε for differentrelative degrees of freedom g =80 (solid), 60 (dashed), 40 (dotted) and 20 (dot-dashed)respectively, for w = 10 − and u = 10 ; Right) boost factor as a function of the relativeconversion cross section u . Four curves correspond to w = 10 − (solid), 10 − (dashed),5 × − (dotted) and 10 − (dot-dashed) respectively, for parameters g = 60, m = 1TeV and ε = 1 × − respectively.by L ⊃ ¯ χ i ( i /D − m i ) χ i + ( D µ φ ) † ( D µ φ ) − m φ φ † φ + 12 ∂ µ η∂ µ η − m η η − y i ¯ χ i iγ ηχ i − y ℓ ¯ ℓ cL ∆ ℓ L + h.c − ( µη + ξη ) (cid:2) Tr(∆ † ∆) + κ ( H † H ) + ζ ( φ † φ ) (cid:3) , ( i = 1 ,
2) (12)with D µ = ∂ µ + ig A A µ and g A standing for the gauge coupling constant. Note that φ and η do not directly couple to the SM fermions. After the spontaneous symmetry breakingin V ( φ ), the scalar φ obtains a nonzero VEV h φ i = v φ / √ m A = g A v φ . At the tree level, the three components of the triplet δ ++ , δ + and δ are degenerate in mass, i.e. m δ ++ = m δ + = m δ + ≡ m ∆ .After the spontaneous symmetry breaking in the scalar sectors, the fields ∆, H and φ obtain nonzero VEVs, which also generates a linear term in η through the last term of Eq.(12). The linear term in η in turn leads to a nonzero VEV of η , i.e., h η i = v η = 0, whichwill give corrections to the masses of χ i and may enlarge the mass difference between χ and χ . This problem can be avoided by using the above mentioned assumption that η has a much stronger coupling to ∆ than that to H and φ , which requires that κ, ζ ≪ η is proportional to the ratio between the linear and quadratic terms in η ,and can be estimated as v η ≈ − µ ( κv H + ζ v φ ) / (2( m η + κv H + ζ v φ )). Since the VEV ofthe triplet ∆ is extremely small and v H ≈ O (10 )GeV, if µ , m η , and v φ are all aroundTeV scale, for κ . O (10 − ) and ζ . O (10 − ) the VEV of η is v η . O (10 − )TeV whichis small enough to avoid breaking the degeneracy in the masses of χ , .We assume that χ has large internal degrees of freedom relative to that of χ , i.e., g ≫ g , which can be realized if χ belongs to a multiplet of the product of some globalnonabelian groups. For instance g = 4˜ g with ˜ g =16, 8, and 4 if it belongs to thespinor representation of a single group of SO (8), SO (6) and SO (4) respectively. When χ belongs to a representation of the product of these groups, its internal degrees offreedom can be very large.At the early time when the temperature of the Universe is high enough, the triplet∆ can be kept in thermal equilibrium with SM particles through the SM gauge interac-tions. The DM particles χ i can reach thermal equilibrium by annihilating into the tripletthrough the intermediate particle η . The annihilation ¯ χ χ → η ∗ → δ ±± δ ∓∓ , δ ± δ ∓ , δ δ ∗ is an s -wave process which is dominant contribution . The cross section before averagingover the relative velocity v is given by σ i v = N f y i µ πg i ( s − m η ) r − m s , (13)where N f = 3 is the number of final states, m η is the mass of η and s is the squareof the total energy in the center of mass frame. For s -wave annihilation we use theapproximation that the thermally averaged cross section is the same as the one beforethe average, i.e., h σv i ≃ σv . From the above equation the ratio of the two annihilationcross sections is w = ( y /y ) ( g /g ). It is easy to get a very small w provided that y ≪ y and g ≪ g . In order to have a large enough h σ v i ≫ h σv i F the product ofthe coupling constants y µ must be large enough, or the squared mass of η is close to s .The cross section of the conversion process ¯ χ χ → A ∗ → ¯ χ χ is given by σ v = 3 g A m π ( s − m A ) (cid:18) g g (cid:19) r − m s . (14)The cross section is suppress by g /g and also the phase space factor p − m /s when s is close to 4 m at the vary late time of the thermal evolution. However, the crosssection be greatly enhanced if m A is close to a resonance when the relation s ≃ m A issatisfied. In the numerical calculations, we find that for the following selected parameters: m = 1TeV, ǫ = 1 × − , g = 1, g = 60, m ∆ = 500 GeV, m η = 1 . m A = 2 . y = 3, y = 0 . µ/m = 3, and g A = 2 .
5, the following ratio of the cross sectioncan be obtained w ≃ × − , u ≃ . , and h σ v i / h σv i F ≃ .
9n this parameter set the relative mass difference between m A and 2 m is around 1%.From Fig. 2, one can see that the corresponding boost factor is B ∼ The mechanism proposed here does not require velocity-dependent annihilation crosssections which is essential to the Sommerfeld enhancement. There exists stringent con-straints from astrophysical observations if the DM annihilation cross section scales withvelocity as 1 /v or 1 /v and saturates at very low velocity. Those constraints involvesthe bound on the µ -type distortion of CMB spectrum [38–40] and the bounds on diffusegamma-rays from the cold structures which have lower velocity dispersion than that inthe solar neighborhood in which v ∼ − . For instance, in the subhalos the averagevelocity can be as low as v ∼ − [41], and the DM velocity in the protohalos canbe even lower v ∼ − [42] . If the enhancement is insensitive to the velocity, thoseapstrophysical bounds can be relaxed significantly. Furthermore, unlike the Sommerfeldenhancement, no attractive long-range force between the DM particles is involved. Theexistence of such a long-range force can change the halo shape and is constrained byobservations [43–45]. The boost factor from DM conversion is free from this type ofconstraint as well.In summary, We have considered an alternative mechanism for obtaining boost fac-tors from DM conversions which does not require the velocity-dependent annihilationcross section or the decay of unstable particles. We have shown that if the whole DMis composed of multiple components, the relic density of each DM component may notnecessarily be inversely proportional to its own annihilation cross section. We demon-strate the possibility that the number density of the lightest DM component with anannihilation cross section much larger than h σv i F can get enhanced in late time throughDM conversation processes, and finally dominates the whole relic abundance, which cor-responds to a boost factor needed to explain the excesses in cosmic-ray positron andelectrons reported by the recent experiments. Acknowledgments
This work is supported in part by the National Basic Research Program of China (973Program) under Grants No. 2010CB833000; the National Nature Science Foundation ofChina (NSFC) under Grants No. 10975170, No. 10821504 and No. 10905084; and theProject of Knowledge Innovation Program (PKIP) of the Chinese Academy of Science.10 eferences [1]
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