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8

Remnants of dark matter clumps

Veniamin Berezinsky,1, 2, ∗ Vyacheslav Dokuchaev,2, † and Yury Eroshenko2, ‡

1INFN, Laboratori Nazionali del Gran Sasso, I-67010 Assergi (AQ), Italy2Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Russia

(Dated: February 13, 2013)

What happened to the central cores of tidally destructed dark matter clumps in the Galactic halo?We calculate the probability of surviving of the remnants of dark matter clumps in the Galaxy bymodelling the tidal destruction of the small-scale clumps. It is demonstrated that a substantialfraction of clump remnants may survive through the tidal destruction during the lifetime of theGalaxy if the radius of a core is rather small. The resulting mass spectrum of survived clumps isextended down to the mass of the core of the cosmologically produced clumps with a minimal mass.Since the annihilation signal is dominated by the dense part of the core, destruction of the outerpart of the clump affects the annihilation rate relatively weakly and the survived dense remnantsof tidally destructed clumps provide a large contribution to the annihilation signal in the Galaxy.The uncertainties in minimal clump mass resulting from the uncertainties in neutralino models arediscussed.

PACS numbers: 12.60.Jv, 95.35.+d, 98.35.Gi

I. INTRODUCTION

According to current observations, about 30% of themass of the Universe is in a form of cold dark matter(DM). The nature of DM particles is still unknown. Thecold DM component is gravitationally unstable and is ex-pected to form the gravitationally bounded clumpy struc-tures from the scale of the superclusters of galaxies anddown to very small clumps of DM. The large-scale DMstructures are observed as the galactic halos and clus-ters of galaxies. They are also seen in numerical simu-lations. Theoretical study of DM clumps are importantfor understanding the properties of DM particles becauseannihilation of DM particles in small dense clumps mayresult in visible signal. The DM clumps in the Galaxycan produce the bright spots in the sky in the gamma orX-bands [1]. A local annihilation rate is proportional tothe square of the DM particle number density. Thus, theannihilation signal from small clumps can dominate overdiffuse component of DM in the halo.

The cosmological formation and evolution of small-scale DM clumps have been studied in numerous works[2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. The minimum massof clumps (the cutoff of the mass spectrum), Mmin is de-termined by the collisional and collisionless damping pro-cesses (see e. g. [6] and references therein). Recent calcu-lations [10] show that the cutoff mass is related to the fric-tion between DM particles and cosmic plasma similar tothe Silk damping. In the case of the Harrison-Zeldovichspectrum of primordial fluctuations with CMB normal-ization, the first small-scale DM clumps are formed atredshift z ∼ 60 (for 2σ fluctuations) with a mean density

∗Electronic address: berezinsky@lngs.infn.it†Electronic address: dokuchaev@lngs.infn.it‡Electronic address: erosh@inr.npd.ac.ru

7 × 10−22 g cm−3, virial radius 6× 10−3 pc and internalvelocity dispersion 80 cm s−1 respectively. Only a verysmall fraction of these clumps survives the early stageof tidal destruction during the hierarchial clustering [4].Nevertheless, these survived clumps may provide the ma-jor contribution to the annihilation signal in the Galaxy[4, 7, 14, 15, 16]. At a high redshift, neutralinos, con-sidered as DM particles, may cause the efficient heatingof the diffuse gas [17] due to annihilation in the denseclumps.

One of the unresolved problem of DM clumps is a valueof the central density or core radius. Numerical simula-tions give a nearly power density profile of DM clumps.Both the Navarro-Frenk-White and Moore profiles giveformally a divergent density in the clump center. A the-oretical modelling of the clump formation [18] predicts apower-law profile of the internal density of clumps

ρint(r) =3 − β

3ρ

( r

R

)−β

, (1)

where ρ and R are the mean internal density and a radiusof clump, respectively, β ≃ 1.8−2 and ρint(r) = 0 at r >R. A near isothermal power-law profile (1) with β ≃ 2has been recently obtained in numerical simulations ofsmall-scale clump formation [19].

It must be noted that density profiles of small-scaleDM clumps and large-scale DM haloes may be differ-ent. The galactic halos are well approximated by theNavarro-Frenk-White profile outside of the central corewhere dynamical resolution of numerical simulations be-comes insufficient. Different physical mechanisms are en-gaged for formation of a central core during the formationand evolution of clumps. A theoretical estimation of therelative core radius of a DM clump xc = Rc/R was ob-tained in [18] from energy criterion, xc ≡ Rc/R ≃ δ3eq,where δeq is a value of density fluctuation at the begin-ning of a matter-dominated stage. A similar estimatefor DM clumps with the minimal mass ∼ 10−6M⊙ orig-inated from 2σ fluctuation peaks gives δeq ≃ 0.013 and

2

Rc/R ≃ 1.8 × 10−5, respectively. In [4], the core radiusxc ≃ 0.3ν−2 has been obtained, where ν is a relativeheight of the fluctuation density peak in units of dis-persion at the time of energy-matter equality (see alsoSection V). This value is a result of the influence of tidalforces on the motion of DM particles in the clump at thestage of formation. This estimate may be considered asan upper limit for the core radius or as the break-scalein the density profile, e.g., a characteristic scale in theNavarro-Frenk-White profile. It could be that a real coreradius, where the density ceases to grow, is determinedby the relaxation of small-scale perturbations inside theforming clump [20]. Another mechanism for core forma-tion arises in the “meta-cold dark matter model” due tolate decay of cold thermal relics into lighter nonrelativis-tic particles with low phase-space density [21, 22].

Nowadays, numerical simulations have a rather lowspace resolution in the central region of clumps to de-termine the core radius. The only example with someindication to presence of a core with radius xc ≃ 10−2 isnumerical simulation of small-scale clump formation [19].The special numerical simulations with a sufficiently highspace resolution to reveal the real core radius are very re-quested.

In this work, we consider the relative core radiusxc = Rc/R of DM clumps as a free parameter in therange 0.001 − 0.1. We investigate the dependence of theprobability of clump survival in the Galaxy on this pa-rameter under the action of tidal forces from galactic diskand stars. As a preferred, value we consider xc ≃ 10−2 inspite of the numerical simulations [19]. The correspond-ing annihilation rate of DM particles is proportional totheir squared number density, and thus is very sensitiveto the value of the core radius.

In our earlier works [7, 12], we used a simplified cri-terium for a tidal destruction of clump. Namely, we pos-tulated that the clump is destructed if a total tidal energygain

∑

(∆E)j after several disk crossings (or collisionswith stars) becomes of order of initial binding energy ofa clump |E|, i.e.

∑

j

(∆E)j ∼ |E|, (2)

where summation goes over the successive disk crossings(or encounters with stars). This criterium is justifiedin the cosmological context of the DM clump formationbecause both the formation of the density profile of theclump and its tidal heating proceed during the same timeof nonlinear evolution of density perturbation. For theGalaxy case, a more detailed consideration is needed todescribe a tidal destruction of DM clumps by stars. Animproved approach includes a gradual mass loss of sys-tems [23, 24, 25], in particular, by small-scale DM clumps[5, 26].

In this work, we will describe a gradual mass loss ofsmall-scale DM clumps assuming that only the outer lay-ers of clumps are involved and influenced by the tidalstripping. Additionally, we assume that inner layers of a

clump are not affected by tidal forces. In this approx-imation, we calculate a continuous diminishing of theclump mass and radius during the successive Galacticdisk crossings and encounters with the stars. We acceptnow for criterium of clump destruction the diminishingof the radius of tidally stripped clump down to the coreradius. An effective time of mass loss for the DM clumpremains nearly the same as in our previous calculations[12]. However, the clump destruction time has now quitedifferent physical meaning: it provides now a character-istic time-scale for the diminishing of the clump massand size instead of the total clump destruction. Thismeans that small remnants of clumps may survive in theGalaxy. Respectively, these remnants would be an ad-ditional source of amplification of the DM annihilationsignal in the Galaxy.

II. TIDAL DESTRUCTION OF CLUMPS BY

DISK

The kinetic energy gain of a DM particle with respectto the center of a clump after one crossing of the Galacticdisk is [27]

δE =4g2m(∆z)2m

v2z,c

A(a), (3)

where m is a constituent DM particle mass, ∆z is a ver-tical distance (orthogonal to the disk plane) of a DMparticle with respect to the center of the clump, vz,c isa vertical velocity of the clump with respect to the diskplane at the moment of disk crossing, and A(a) is theadiabatic correction factor. A gravitational accelerationnear the disk plane is

gm(r) = 2πGσs(r), (4)

where we use an exponential model for a surface densityof disk

σs(r) =Md

2πr20e−r/r0 (5)

with Md = 8 × 1010M⊙, r0 = 4.5 kpc.The factor A(a) in (3) describes the adiabatic protec-

tion from slow tidal effects [28]. This adiabatic correc-tion, further on referred to as the Weinberg correction, isdefined as an additional factor A(a) to the values of en-ergy gain in the momentum approximation. This factorsatisfies the following asymptotic conditions: A(a) = 1for a≪ 1 and A(a) ≪ 1 for a≫ 1. In [23], the followingfitting formula was proposed:

A(a) = (1 + a2)−3/2. (6)

Here the adiabatic parameter a = ωτd, where ω is an or-bital frequency of DM particle in the clump, τd ≃ Hd/vz,cis an effective duration of gravitational tidal shock pro-duced by the disk with a half-thickness Hd. For tidal

3

interactions of clumps with stars in the bulge and thehalo the duration of the gravitational shock can be es-timated as τs ∼ l/vrel, where l is an impact parameterand vrel is a relative velocity of a clump with respect toa star.

As a representative example, we consider the isother-mal internal density profile of a DM clump

ρint(r) =1

4π

v2rot

Gr2(7)

with a cutoff at the virial radius R: ρ(r) = 0 atr > R. A corresponding mass profile of a clump isM(r) = Mi(r/R), where Mi is an initial mass of a clumpat the epoch of the Galaxy formation. With this massdistribution, a circular velocity inside a clump is indepen-dent of the radius, vrot = (GM(r)/r)1/2 = (GMi/R)1/2.A gravitational potential corresponding to the densityprofile (7) is φ(r) = v2

rot[log(r/R)−1]. Let us define a di-mensionless energy of the DM particle ε = E/(mv2

rot) andgravitational potential ψ(r) = φ(r)/v2

rot = ln(r/R) − 1.An internal density profile ρint(r) and the distributionfunction of DM particles in the clump fcl(ε) are relatedby the integral relation [29]

ρint(r) = 25/2π

0∫

ψ(r)

√

ε− ψ(r) fcl(ε) dε. (8)

The corresponding isothermal distribution function is

fcl(ε) ≃v2rot

4π5/2e2GR2e−2ε. (9)

Note what this distribution function provides only anapproximate representation of (7), far from the cutoffradius R. Nevertheless, this approximation is enough forour estimates of tidal destruction of the DM clumps.

By using the hypothesis of a tidal stripping of outerlayers of a DM clump, we see that a tidal energy gainδε causes the stripping of particles with energies in therange −δε < ε < 0. A corresponding variation of densityat radius r is

δρ(r) = 25/2π

0∫

−δε

√

ε− ψ(r) fcl(ε) dε. (10)

In this equation, the tidal energy gain (3) by different DMparticles is averaged over angles, so as 〈(∆z)2〉 = r2/3.A resulting total mass loss by a DM clump during onecrossing of the Galactic disk is

δM = −4π

∫ R

0

r2δρ(r) dr. (11)

Let us specify the dimensionless quantities

Qd =g2m

2πv2z,cGρi

, Sd =4π

3Gρiτ

2d , (12)

where ρi = 3Mi/(4πR3) is a initial mean density of

clump. For the most parts of clumps Qd ≪ 1 with atypical value Qd ∼ 0.03. In the limiting case Qd ≪ 1and in the absence of the adiabatic correction, Sd = 0,the integrals (10) can be calculated analytically. In ageneral case, the fitting formula for the mass loss of aclump during one passage through the Galactic disk is

(

δM

M

)

d

≃ −0.13Qd exp(

−1.58S1/2d

)

. (13)

Now, we calculate the tidal mass loss by clumps usinga realistic distribution of their orbits in the the halo.The method of calculation is similar to the one used in[12], but instead of the rough energetic criterium for atidal clump destruction (2) we will assume now a gradualdecreasing of clump mass and size.

Let us choose some particular clump moving in thespherical halo with an orbital “inclination” angle γ be-tween the normal vectors of the disk plane and orbitplane. The orbit angular velocity at a distance r fromthe Galactic center is dφ/dt = J/(mr2), where J is anorbital angular momentum of a clump. A vertical veloc-ity of a clump crossing the disk is

vz,c =J

mrssin γ, (14)

where rs is a radial distance of a crossing point fromthe Galaxy center. There are two crossing points (withdifferent values of rs) during an orbital period.

The standard Navarro-Frenk-White profile of the DMGalactic halo is

ρH(r) =ρ0

(r/L) (1 + r/L)2 , (15)

where L = 45 kpc, ρ0 = 5 × 106M⊙ kpc−3. It useful tointroduce the dimensionless variables:

x =r

L, ρH(x) =

ρH(r)

ρ0, y =

J2

8πGρ0L4M2, (16)

ε =Eorb/M − Φ0

4πGρ0L2, ψ =

Φ − Φ0

4πGρ0L2, (17)

where Φ0 = −4πGρ0L2, Eorb is a total orbital energy of

a clump. With these variables, the density profile of thehalo (15) is written as

ρH(x) =1

x(1 + x)2. (18)

A gravitational potential ψ(x), corresponding to densityprofile (18) is

ψ(x) = 1 − log(1 + x)

x. (19)

An equation for orbital turning points, r2 = 0, for DMclumps in the potential (19) is

1 − log(1 + x)

x+

y

x2= ε. (20)

4

From (20), one can find numerically the minimum xmin

and maximum xmax radial distance of a clump from theGalactic center as a function of orbital energy ε andsquare of angular momentum y. Denoting p = cos θ,where θ is an angle between the radius-vector ~r and theorbital velocity ~v, we have y = (1 − p2)x2[ε− ψ(x)]. Aswe assumed above, the unit vectors ~v/v are distributedisotropically at each point x, and, therefore, p has a uni-form distribution in the interval [−1, 1].

The relation between the density profile ρH(x) and thedistribution function is given by the same equation (8)with an obvious substitution fcl ⇒ F (ε), where the dis-tribution function F (ε) for a halo profile (18) can be fit-ted as [30]

F (ε) = F1(1 − ε)3/2ε−5/2

[

− ln(1 − ε)

ε

]q

eP . (21)

Here F1 = 9.1968×10−2, q = −2.7419, P =∑

i

pi(1−ε)i,(p1, p2, p3, p4) = (0.3620,−0.5639,−0.0859,−0.4912).An interval of time for motion from xmin to xmax andback is

Tc(x, ε, p) =1√

2πGρ0

xmax∫

xmin

ds√

ε− ψ(s) − y/s2. (22)

An angle of orbital precession during the time Tc/2 is

φ = y1/2

xmax∫

xmin

ds

s2√

ε− ψ(s) − y/s2− π < 0. (23)

Therefore, an orbital period is longer than Tc and is givenby

Tt = Tc

(

1 + φ/π)−1

. (24)

Choosing a time interval ∆T much longer than a clumporbital period Tt, but much shorter than the age of theGalaxy t0, i.e., Tt ≪ ∆T ≪ t0, we define an averagedrate of mass loss by a selected clump under influence oftidal shocks in successive disk crossings

1

M

(

dM

dt

)

d

≃ 1

∆T

∑

(

δM

M

)

d

, (25)

where (δM/M)d is given by (13) and summation goesover all successive crossing points (odd and even) of aclump orbit with the Galactic disk during the time inter-val ∆T . According to (4) and (14) the gm and vz,c bothdepend on the radius x = r/L. One simplification in cal-culation of (25) follows from the fact that a velocity oforbit precession is constant. For this reason the points ofsuccessive odd crossings are separated by the same an-gles φ from (23). The same is also true for successiveeven crossings. Using this simplification, we transformthe summation in (25) to integration:

1

∆T

∑

(

δM

M

)

d

≃ 2

Tt|φ|

xmax∫

xmin

(

δM

M

)

d

dφ

dxdx,

where

dφ

dx=

y1/2

x2√

ε− ψ(x) − y/x2(26)

is an equation for the clump orbit in the halo. Themethod described will be used in the Section IV for thefinal calculations.

III. TIDAL DESTRUCTION OF CLUMPS BY

STARS

Now, we calculate the diminishing of a clump massdue to a tidal heating by stars in the Galaxy by usingthe same hypothesis of the the preferable stripping of theouter clump layers. During a single close encounter of aDM clump with a star, the energy gain of a constituentDM particle in the clump with respect to clump centeris [7]

δE =2G2m2

sm∆z2

v2rell

4, (27)

where m∗ is a star mass, l is an impact parameter, vrelis a relative star velocity with respect to a clump, ∆z =r cosψ, r is a radial distance of a DM particle from theclump center, and ψ is an angle between the directionsfrom the clump center to the DM particle and to thepoint of the closest approach of a star. Using the samemethod as in Sec. II, we calculate a relative mass lossby the clump (δM/M)s during a single encounter with astar and obtain the same fitting formula as (13) but withsubstituting the dimensionless parameters, Qd ⇒ Qs andSd ⇒ Ss, where

Qs =Gm2

∗

2πv2rell

4ρi, Ss =

4π

3Gρiτ

2s , (28)

where τs ≃ l/vrel.A DM clump acquires the maximum energy gain dur-

ing a single encounter with a star when the impact pa-rameter l ∼ R. Using the relation

dt =1

2√

2πGρ0

dx√

ε− ψ(x) − y/x2, (29)

and integrating over all impact parameters l > R, wecalculate an averaged rate of mass loss by a clump duringsuccessive encounters with stars

1

M

(

dM

dt

)

s

≃ (30)

1

2Tt√

2πGρ0

∞∫

R

2πl dl

xmax∫

xmin

ds n∗(s)vrel√

ε− ψ(s) − y/s2

(

δM

M

)

s

,

where n∗(r) is a radial number density distribution ofstars in the bulge and halo. A DM clump moves through

5

the medium with a varying value of n∗ along the clumporbit. In contrast to the case of the disk crossing, theprecession of the clump orbit during an orbital perioddoes not influence the mass loss due to encounters withstars. Additionally, the mass loss due to encounters withstars is independent of the inclination of clump orbits inthe case of a spherically symmetric distribution of starsin the bulge and halo.

Using the results of [31], we approximate the radialnumber density distribution of stars in the bulge in theradial range r = 1 − 3 kpc as

nb,∗(r) = (ρb/m∗) exp[

−(r/rb)1.6

]

, (31)

where ρb = 8M⊙/pc3 and rb = 1 kpc. A correspondingnumber density distribution of the halo stars at r > 3 kpcoutside the Galactic plane can be approximated as

nh,∗(r) = (ρh/m∗)(r⊙/r)3, (32)

where m∗ = 0.4M⊙ and r⊙ = 8.5 kpc. According to[32], in the region between r = 1 and 40 kpc a total massof stars is 4 × 108M⊙ with a star density profile ∝ r−3.These data correspond to ρh = 1.4 × 10−5 M⊙/pc3 in(32). We neglect in our calculations the oblationes of thestellar halo [32].

IV. SURVIVING FRACTION OF CLUMPS

From Eq. (3) it is seen that the tidal forces influencemainly the outer part of the clump (where ∆z is ratherlarge). Further, we will use our basic assumption thatonly outer layers of a clump undergo the tidal stripping,while the inner parts of a clump are unaffected by tidalforces. Thus, we assume that a clump mass M = M(t)and radius R = R(t) are both diminishing in time due tothe tidal stripping of outer layers, but its internal densityprofile remains the same as given by Eq. (7), e.g., forthe isothermal density profile M(t) ∝ R(t) and ρ(t) ∝M(t)−2. Combining together the rates of mass loss (25)and (30) due to the tidal stripping of a clump by the diskand stars respectively, we obtain the evolution equationfor a clump mass:

dM

dt=

(

dM

dt

)

d

+

(

dM

dt

)

s

. (33)

In the following, we solve this equation numerically start-ing from the time of Galaxy formation at t0 − tG up tothe present moment t0. In numerical calculations, it isconvenient to use the dimensionless variables: t/t0 fortime and M/Mi for a clump mass, where Mi is an initialclump mass. The adiabatic correction provides generallyonly a small effect. In the absence of adiabatic correctionor, equivalently, at Sd = Ss = 0 the evolution equation(33) has a simple form

dµ

dt= − µ

ts− µ3

td, (34)

where µ = M(t)/Mi and parameters td and ts are inde-pendent of µ. The solution of this equation

µ2(t0) =2td

(2td + ts) exp(2t0/ts) − ts(35)

represents a good approximation to numerical solution of(33) with the adiabatic correction taken into account.

The most important astrophysical manifestation ofDM clumps is a possible annihilation of constituent DMparticles. The crucial point is a dominance of the cen-tral core of a clump in annihilation signal if clumps havea steep enough density profile. Namely, annihilation ofDM particles in a clump core will prevail in a total anni-hilation rate in a single clump with a power-law densityprofile (1) if β > 3/2 and xc = Rc/R ≪ 1. More specif-

ically, the quantity N ∝∫ r

r04πr′2dr′ρ2

int(r′) practically

does not depend on r, if r ≫ r0. As a result, the anni-hilation luminosity of a DM clump with approximatelyan isothermal density profile (β ≃ 2) will be nearly con-stant under influence of tidal stripping until a clump ra-dius diminishes to its core radius. In other words, inthe nowadays Galaxy the remnants of tidally strippedclumps with xc < µ(t0) ≪ 1, where µ(t) = M(t)/Mi

and t0 ≃ 1010 yrs is the Galaxy age, obeys the evolutionequation (33) and have the same annihilation luminosityas their progenitors with µ = 1.

By using the evolution equation (33), we now calculatethe probability P of the survival of clump remnant duringthe lifetime of the Galaxy. Let us choose some arbitrarypoint in the halo with a radius-vector ~r and an angle αwith a polar axis of the Galactic disk. Only the clumporbits with an inclination angle π/2 − α < γ < π/2 passthrough this point. A survival probability for clumps canbe written now in the following form

P (x, α) =4π

√2

ρ(x) sinα

1∫

0

dp

sinα∫

0

d cos γ (36)

×1

∫

ψ(x)

dε [ε− ψ(x)]1/2F (ε)Θ[µ(t0) − xc].

In this equation, ρ(x) is a density profile of the halo from(18), p = cos θ, θ is an angle between the radius-vector~r and the orbital velocity of clump, Θ is the Heavisidefunction, ψ(x) is the halo gravitational potential from(18), F (ε) is a distribution function of clumps in the halofrom (21), µ(t0) depends on all variables of the integra-tion, and xc = Rc/R is an initial value of the clump core.The function µ(t0) is calculated from numerical solutionof evolution equation (33). If µ(t0) > xc, the clump rem-nant is survived through the tidal destruction by boththe disk and stars. The annihilation rate in this remnantwould be the same as in the initial clump. On the con-trary, in the opposite case, when µ(tG) < xc, the clumpis totally destructed because (i) the core is not a dynam-ically separated system and composed of particles with

6

1

2

3

4

r, kpc0.4

0.6

0.8

1

Ρ

0

0.25

0.5

0.75

1

PHrL

1

2

3

4

r, kpc

FIG. 1: The survival probability P (r, ρ) plotted as a functionof distance from the Galactic center r and a mean internalclump density ρ in the case xc = 0.1. It gives the normalizedfraction of DM clumps in the halo P calculated from (36),which survives the tidal destruction by the stellar disk andthe halo stars.

1

2

3

4

r, kpc0.4

0.6

0.8

1

Ρ

0

0.25

0.5

0.75

1

PHrL

1

2

3

4

r, kpc

FIG. 2: The same as Fig. 1, but for the case xc = 0.05.

extended orbits, and because (ii) a nearly homogeneouscore is destructed easier than a similar object with thesame mass but with a near isothermal density profile.

We consider the small-scale DM clumps in the initialmass interval Mi = [10−6M⊙, 1M⊙] originated from the2σ (i.e. ν = 2) peaks in the Harrison-Zeldovich pertur-bation spectrum. A reason is as follows. The DM clumpsoriginated from initial density perturbations with ν < 1were almost completely destructed by tidal interactionsduring the early stage of hierarchical clustering as it canbe seen from the distribution function of clumps (44) (seebelow). On the contrary, the most dense DM clumpswith ν > 3 are mostly survived the stage of hierarchicalclustering, but their number according to (44) is expo-nentially falls with ν and is small. For this reason, wewill use the following approximation: the DM clumpswere originated on average from ν ≃ 2 peaks, and forany given mass M we do not consider the distribution ofclumps over their densities. In this approximation, the

initial radius of the clump Ri depends only on the one pa-rameter — the initial clump density, which also dependsonly on the initial clump mass Mi.

The crucial result of numerical calculation of a sur-vival probability (36) for clumps with xc ≪ 0.05 is thatP (x, α) ∼ 1 everywhere. Even inside the bulge thereare clumps which flying through the bulge from externalregions. These means that clump remnants are mostlysurvived through the tidal destruction in the Galaxy. Anoticeable diminishing P (x, α) < 1 near the center ofGalaxy becomes apparent for clumps with xc > 0.05. Itis understandable because with xc → 1, we return to theprevious criterium of tidal destruction of clumps (2) andto a corresponding results for survival probability [7, 12].The survival probability P (r, α) numerically calculatedfrom (36) for the cases xc ∼ 0.05 and 0.1 is shown inthe Figs. 1 and 2. The dependence on α (an angle be-tween a radius-vector ~r and a polar axis of the Galacticdisk) is very weak as it was shown in [12]. For this rea-son, we present the results only for an intermediate valueα = π/4. The density of clumps is normalized to thedensity 7.3 × 10−23 g cm−3 valid for clumps with massM = 10−6M⊙ originated from 2σ density peaks in thecase of power-law index of primordial spectrum of per-turbations np = 1.

It is worth to note that a tidal radius of a clump in thebulge is [33]

r3t =GM(rt)

ω2p − d2φ/dl2

, (37)

where ωp ≃ [GMb(l)/l3]1/2 is an angular velocity at the

pericenter (we consider here a circular orbit for simplic-ity), Mb(l) is a mass profile of the bulge and φ(l) is agravitational potential of the bulge. For the consideredsmall-scale clumps rt ≥ 0.2Ri, and, therefore, the tidalradius is not a crucial factor for destruction of clumps.

V. COSMOLOGICAL DISTRIBUTION

FUNCTION OF CLUMPS

In this section, we provide calculations of a mass func-tion for the small-scale clumps in the Galactic halo bymore transparent method than in our previous works[4, 7].

The first gravitationally bound objects in the Universeare the DM clumps of minimum mass Mmin. A numeri-cal value of Mmin depends strongly on the nature of DMparticle. Even in the case of a particular DM particle,e.g., neutralino, the calculated value of Mmin can differby many orders of magnitude for different sets of param-eters in mSUGRA model, see the Appendix A1. Thelarger scale clumps are formed later. The larger scaleclumps host the smaller ones and are hosted themselvesby next larger clumps. Major parts of small-scale clumpsare destroyed by the tidal gravitational fields of their hostclumps. At small-mass scales, the hierarchial clustering is

7

a fast and complicated nonlinear process. The formationof new clumps and their capturing by the larger ones arenearly simultaneous processes because at small-scales aneffective index of the density perturbation power spec-trum is very close to a critical value, n → −3. TheDM clumps are not totally virialized when they are cap-tured by hosts. The adiabatic invariants cannot preventthe survival of cores at this stage because there are notenough time for the formation of the singular density pro-files in clumps (an internal dynamical time of clump is ofthe same order as its capture time by a host). We use asimplified model to take into account the most importantfeatures of hierarchial clustering.

In the model of spherical collapse (see for example[34]), a formation time for a clump with an internal den-sity ρ is t = (κρeq/ρ)

1/2teq, where κ = 18π2, ρeq =ρ0(1 + zeq)

3 is a cosmological density at the time ofmatter-radiation equality teq, 1 + zeq = 2.35× 104Ωmh

2,and ρ0 = 1.9×10−29Ωmh

2 g cm−3. The index “eq” refersto quantities at the time of matter-radiation equality teq.The DM clumps of mass M can be formed from densityfluctuations of a different peak-height ν = δeq/σeq(M),where σeq(M) is a fluctuation dispersion on a mass-scale M at the time teq. A mean internal density ofthe clump ρ is fixed at the time of the clump formationand according to [34] is ρ = κρeq[νσeq(M)/δc]

3, where

δc = 3(12π)2/3/20 ≃ 1.686.A tidal destruction of clumps is a complicated process

and depends on many factors: the formation history ofclumps, host density profile, the existence of another sub-structures inside the host, orbital parameters of individ-ual clumps in the hosts, etc. Only in numerical simula-tions, all these factors can be taken into account properly.The first such simulation in the small-scale region wasproduced in [19]. We use a simplified analytical approachby parameterizing the energy gains in tidal interactionsby the number of tidal shocks per dynamical time in thehosts. Using the model [35] for tidal heating, we deter-mine the survival time T , i.e. time of tidal destruction,for a chosen small-scale clump due to the tidal heatinginside of a host clump with larger mass. During the dy-namical time tdyn ≃ 0.5(Gρh)

−1/2, where ρh is a meaninternal density of the host, the chosen small-scale clumpmay belong to several successively destructed hosts. Aclump trajectory in the host experiences successive turnsaccompanied by the “tidal shocks” [35]. Similar shockscome from interactions with other substructure, and ingeneral due to any varying gravitational field. For theconsidered small-scale clump with a mass M and radiusR, the corresponding internal energy increase after a sin-gle tidal shock is

∆E ≃ 4π

3γ1GρhMR2, (38)

where a numerical factor γ1 ∼ 1. Let us denote by γ2

the number of tidal shocks per dynamical time tdyn. Thecorresponding rate of internal energy growth for a clumpis E = γ2∆E/tdyn. A clump is destroyed in the host if

its internal energy increase due to tidal shocks exceeds atotal energy |E| ≃ GM2/2R. As a result, for a typicaltime T = T (ρ, ρh) of the tidal destruction of a small-scaleclump with density ρ inside a more massive host with adensity ρh we obtain

T−1(ρ, ρh) =E

|E| ≃ 4γ1γ2G1/2ρ

3/2h ρ−1. (39)

It turns out that a resulting mass function of small-scaleclumps (see this Sec. below) depends rather weakly onthe value of the product γ1γ2.

During its lifetime, a small-scale clump can stay inmany host clumps of larger mass. After tidal disruptionof the first lightest host, a small-scale clump becomesa constituent part of a larger one, etc. The process ofhierarchical transition of a small-scale clump from onehost to another occurs almost continuously in time upto the final host formation, where the tidal interactionbecomes inefficient. The probability of clump survival,determined as a fraction of the clumps with mass M sur-viving the tidal destruction in hierarchical clustering, isgiven by the exponential function e−J with

J ≃∑

h

∆thT (ρ, ρh)

. (40)

Here, ∆th is a difference of formation times th for twosuccessive hosts, and summation goes over all clumpsof intermediate mass-scales, which successively host theconsidered small-scale clump of a mass M . Changing thesummation by integration in (40) we obtain

J(ρ, ρf ) =

tf∫

t1

dthT (ρ, ρh)

≃ γρ1 − ρf

ρ≃ γ

ρ1

ρ≃ γ

t2

t21,

(41)where

γ = 2γ1γ2κ1/2G1/2ρ1/2

eq teq ≃ 14(γ1γ2/3), (42)

and t, t1, tf , ρ, ρ1 and ρf are, respectively, the formationtimes and internal densities of the considered clump andof its first and final hosts. One may see from Eq. (41)that the first host provides a major contribution to thetidal destruction of the considered small-scale clump, es-pecially if the first host density ρ1 is close to ρ, and con-sequently e−J ≪ 1. Therefore, Eq. (41) gives a qualita-tively correct description of the tidal destruction. How-ever, in the more detailed approach the dependence of γon another parameters is possible to take into account.As reasonable estimate, we will use the ansatz given byEq. (41) for further calculation of mass function.

Now we need to track the number of clumps M (orig-inated from the density peak ν) which enter some largerhost during time intervals ∆t1 around each t1 beginningfrom the time t of clump formation. A mass functionof small-scale clumps (i.e., a differential mass fraction of

8

DM in the form of clumps survived in hierarchical clus-tering) can be expressed as

ξdM

Mdν = (43)

dM dνe−ν

2/2

√2π

t0∫

t(νσeq)

dt1

∣

∣

∣

∣

∂2F (M, t1)

∂M ∂t1

∣

∣

∣

∣

e−J(t,t1).

In this expression, t0 is the Universe age and F (M, t) is amass fraction of unconfined clumps (i.e., clumps, not be-longing to more massive hosts) with a mass smaller thanM at time t. According to [34], the mass fraction of un-

confined clumps is F (M, t) = erf(

δc/[√

2σeq(M)D(t)])

,where erf(x) is the error-function and D(t) is the growthfactor normalized by D(teq) = 1. An upper limit of inte-gration t0 in Eq. (43) is not crucial and may be extrapo-lated to infinity because a main contribution to the tidaldestruction of clumps is provided by the early formedhosts at the beginning of the hierarchical clustering.

Two processes are responsible for time evolution of thefraction ∂2F/(∂M∂t) for unconfined clumps in the massinterval dM : (i) the formation of new clumps and (ii)the capture of smaller clumps into the larger ones. Boththese processes are equally efficient at the time when∂2F/(∂M∂t) = 0. To take into account the confinedclumps (i.e., clumps in the hosts) we need only the secondprocess (ii) for the fraction ∂F (M, t)/∂M . Nevertheless,in Eq. (43), which it is used the fraction ∂F (M, t)/∂Mwhich depends on both processes. This is not accurate ata typical formation time of a clump with a massM , whenclump density is comparable with the density of hosts.Fortunately, for this time the exponent in Eq. (43) is verysmall, e−J ≪ 1, as it can be seen from (41) and (42). Re-spectively, an uncertain contribution from the process (i)to the integral (43) is also very small. Meanwhile, onlyprocess (ii) dominates in the integration region wherethe exponent e−J is not small. For this reason, Eq. (43)provides a suitable approximation for the mass fractionof clumps survived in the hierarchical clustering. Thecharacteristic epoch t∗ of the clumps M formation canbe estimated from the equation σeq(M)D(t∗) ≃ δc. Ifone considers, the times t ≫ t∗, then the exponentsexp

−δ2c/[2σ2eqD

2(t)]

can be putted approximately tounity for simplification of integration in (43).

Finally, we transform the distribution function (43) tothe following form:

ξdM

Mdν ≃ ν dν√

2πe−ν

2/2f1(γ)d log σeq(M)

dMdM, (44)

where

f1(γ) =2[Γ(1/3)− Γ(1/3, γ)]

3√

2πγ1/3, (45)

Γ(1/3) and Γ(1/3, γ) are the Euler gamma-function andincomplete gamma-function, respectively. The function(45) is shown in Fig. 3. It is seen in this figure that

10 15 20 25 30 35 40Γ

0.25

0.3

0.35

0.4

f1HΓL

FIG. 3: The function f1(γ) from (45).

f1(γ) varies rather slowly in the interesting interval of14 < γ < 40, and one may use f1(γ) ≃ 0.2 − 0.3.

Physically, the first factor ν in (44) corresponds to amore effective survival of high-density clumps (i.e., withlarge values of ν) with respect to the low-density ones(with small values of ν). Integrating Eq. (44) over ν, weobtain

ξintdM

M≃ 0.02(n+ 3)

dM

M. (46)

An effective power-law index n in Eq. (46) is determinedas n = −3(1 + 2∂ log σeq(M)/∂ logM) and depends veryweakly on M . Equation (46) implies that for suitablevalues of n only a small fraction of clumps, about 0.1 −0.5 %, survives the stage of hierarchical tidal destructionin the each logarithmic mass interval ∆ logM ∼ 1. Itmust be stressed that a physical meaning of the survivedclump distribution function ξ dM/M is different from thesimilar one for the unconfined clumps, given by the Press-Schechter mass function ∂F/∂M .

The simple M−1 shape of the mass function (46) is invery good agreement with the corresponding numericalsimulations [19], but our normalization factor is a fewtimes smaller. One also can see a reasonable agreementbetween the extrapolation of our calculations and thecorresponding numerical simulations of the large-scaleclumps with M ≥ 106M⊙ (for a comparison see [7]).The obtained mass function (46) is further transformedin the process of tidal destructions of clumps by stars inthe Galaxy (see previous sections).

VI. MODIFIED DISTRIBUTION FUNCTION

OF CLUMPS

In this section we calculate the modified mass functionfor the small-scale clumps in the Galaxy taking into ac-count clump mass loss instead of the clump destructionconsidered in [4, 12].

According to theoretical model [4] and numerical sim-ulations [19], a differential number density of small-scale clumps in the comoving frame in the Universe is

9

10-8 10-6 10-4

MM

108

1010

1012

1014

dnHMLdlogHML,Mpc-3

8.5 kpc

3 kpc

FIG. 4: Numerically calculated modified mass function ofclump remnants for galactocentric distances 3 and 8.5 kpc.The solid curve shows the initial mass function.

n(M) dM ∝ dM/M2. This distribution is shown in Fig. 4by the solid line. The damping of small-scale pertur-bations with M < Mmin provides an additional factorexp[−(M/Mmin)

2/3] responsible for the fading of distri-bution at small M . The result of the numerical simu-lations [19] can be expressed in the form of a differen-tial mass fraction of the DM clumps in the Galactic halof(M) dM ≃ κ(dM/M), where κ ≃ 8.3 × 10−3. The ana-lytical estimation (46) gives approximately κ ≃ 4× 10−3

for the mass interval 10−6M⊙ < M < 1M⊙. The dis-crepancy by the factor ≃ 2 may be attributed to the ap-proximate nature of our approach as well as to the wellknown additional factor 2 in the original Press-Schechterderivation of the mass function. In the later case onemust simply multiply equation (43) by factor 2. To clar-ify this discrepancy the more sophisticated calculationsare necessary.

As it was described earlier, we consider DM clumpsoriginated from 2σ density peaks. Therefore, in our ap-proximation the density of clumps and their distributiondepends only on one parameterM . In general, the distri-bution of DM clumps depends on the pair of parameters,e.g. mass and radius, mass and velocity dispersion ormass and peak-high ν as in the distribution (44). Mean-while the authors of numerical simulations do not presenta general distribution of clumps over two parameters.The general distribution of clumps can be in principleextracted from simulations and is very requested for fur-ther investigations of DM clumpiness.

By using the formalism of Sec. V, we derive the massdistribution of the clump remnants in dependence of theinitial masses Mi of clumps. To do this, we calculate nu-merically the value of the mass µ of the clump remnant independence of the initial mass Mi for separate elements∆p∆γ∆ε in the parameter space in (36). Then for fixedintervals ∆µ of values of µ, we provide the summationof the weights of distribution function, which is given byEq. (36) without symbols of integration and Θ-function.By using the derived µ-distribution, we transform the ini-tial (cosmological) mass function of clumps to the final

(nowadays) mass function in the halo at the present mo-ment. This final mass function is shown in Fig. 4 for twodistances from the Galactic center. We supposed in nu-merical calculations that a core radius is very small andall masses of remnants are admissible. With a finite coresize, the final mass function has a cutoff near the coresmass of clump with a minimal mass Mmin. The adiabaticcorrection leads to the accumulation of remnants of somemass corresponding to violation of momentum approxi-mation. One can see from Fig. 4 that clump remnantsexist below the Mmin. Deep in the bulge (very near to theGalactic center) the clump remnants are more numerousbecause of intensive destructions of clumps in the densestellar environment in comparison with the rarefied onein the halo. The main contribution to the low-mass tailof the mass function of remnants comes from the clumpswith the near-disk orbits where the destructions are moreefficient.

The another important point is an efficient destructionof clumps with orbits confined inside the stellar bulge.Nevertheless, a number density of clumps inside the bulgeis nonzero because a major part of clumps have orbitsextending far beyond the bulge, These “transit” clumpsspend only a small part of their orbital time traversingthe bulge and survive the tidal destruction.

VII. AMPLIFICATION OF ANNIHILATION

SIGNAL

A local annihilation rate is proportional to the squareof the DM particle number density. A number density ofDM particles in clump is much large than a correspond-ing number density of the diffuse (not clumped) compo-nent of DM. For this reason, an annihilation signal fromeven a small fraction of DM clumps can dominate overan annihilation signal from the diffuse component of DMin the halo. In this section, we calculate the amplifi-cation (or “boosting”) of an annihilation signal due tothe presence of the survived DM clump remnants in theGalactic halo. We consider here the Harrison-Zeldovichinitial perturbation spectrum with power index np = 1 asa representative example. The value of np is not exactlyfixed by the current observations of cosmic microwavebackground (CMB) anisotropy. In the case of np < 1,the DM clumps are less dense, and a corresponding am-plification of annihilation signal would be rather small[4].

The gamma-ray flux from the annihilation of the dif-fuse distribution (15) of DM in the halo is proportionalto

IH =

rmax(ζ)∫

0

ρ2H(ξ) dx, (47)

where the integration is over r goes along the line of sight,ξ(ζ, r) = (r2 + r2⊙ − 2rr⊙ cos ζ)1/2 is the distance to the

Galactic center, rmax(ζ) = (R2H − r2⊙ sin2 ζ)1/2 + r⊙ cos ζ

10

0 25 50 75 100 125 150 175Ζ, degrees

0

10

20

30

40

50

Iann

FIG. 5: The annihilation signal (48) (upper curve) as a func-tion of the angle ζ between the line of observation and thedirection to the Galactic center. For comparison the annihi-lation signal is also shown (by the bottom curve) from theGalactic halo without DM clumps (47). The values of bothintegrals (48) and (47) are multiplied by a factor of 1048.

is a distance to the external halo border, ζ is an anglebetween the line of observation and the direction to theGalactic center, RH is a virial radius of the Galactic halo,r⊙ = 8.5 kpc is the distance between the Sun and theGalactic center. The corresponding signal from annihi-lations of DM in clumps is proportional to the quantity[4]

Icl = S

rmax(ζ)∫

0

dx

∫

Mmin

f(M) dMρρH(ξ)P (ξ, ρ), (48)

where ρ(M) is the mean density of the clump. The func-tion S depends on the clump density profile and coreradius of the clump [4], and we use S ≃ 14.5 as a rep-resentative example. The observed amplification of theannihilation signal is defined as η(ζ) = (Icl + IH)/IH isshown in Fig. 6 for the case xc = 0.1. It tends to unityat ζ → 0 because of the divergent form of the halo pro-file (15). The annihilation of diffuse DM prevails oversignal from clumps at the the Galactic center. The η(ζ)very slightly depends on xc, and corresponding graphsfor xc < 0.1 are almost indistinguishable from the one inFig. 5. This is because the observed signal is obtainedby integration along the line of sight and the effect of theclump’s destruction at the Galactic center is masked bythe signal from another regions of the halo.

This amplification of an annihilation signal is oftencalled a “boost-factor”. A boost-factor of the order of10 is required for interpretation of the observed EGRETgamma-ray excess as a possible signature of DM neu-tralino annihilation [36].

VIII. CONCLUSION

In [5] it was found that almost all small-scale clumpsin the Galaxy are destructed by tidal interactions with

0 25 50 75 100 125 150 175Ζ, degrees

0

50

100

150

200

250

300

HIcl+IHLIH

FIG. 6: The amplification of the annihilation signal (Icl +IH)/IH as function of the angle between the line of observationand the direction to the Galactic center, where fluxes are givenby (47) and (48).

stars and transformed into “ministreams” of DM. Theproperties of these ministreams may be important forthe direct detection of DM particles because DM parti-cles in streams arrive anisotropically from several discretedirections. In this work, we demonstrate that the cores ofclumps (or clump remnants) survive in general during thetidal destruction by stars in the Galaxy. Although theirouter shells are stripped and produce the ministreamsof DM, the central cores are protected by the adiabaticinvariant and survived as the sources of annihilation sig-nals. This conclusion depends crucially on the unknownsizes of the cores: the smaller cores are more protectedbecause DM particles there have higher orbital frequen-cies and therefore the larger the adiabatic parameter.

Despite the small survival probability of clumps dur-ing early stage of hierarchial clustering, they provide themajor contribution to the annihilation signal (in compar-ison with the unclumpy DM). The amplification (boost-factor) can reach 102 or even 103 depending on the ini-tial perturbation spectrum and minimum mass of clumps.This boost-factor must be included in calculations of theannihilation signals. Some promising interpretations ofobservations and calculations of annihilation signal fromthe Galactic halo require this boost-factor (see, e.g., [36]).The discussed dense remnants survive the tidal destruc-tion and provide the enhancement of DM annihilationin the Galaxy. These remnants of DM clumps form thelow-mass tail in the standard mass distribution of small-scale clumps extended much below Mmin of the standarddistribution. It does not mean of course the increasing ofannihilation signal in comparison with the case withoutclump destruction. It only indicates that galactic clumpdestruction does not diminishes strongly the annihilationsignal.

The principle simplifying assumption of this work isthat only the outer layers of clumps are subjected by thetidal stripping. The main difficulties in considering thefull problem with the mass loss from inner layers are inthe complicated dynamical reconstruction of clumps just

11

after tidal shocks. We believe that our approach providesa rather good result by the two reasons. First, the in-fluence of tidal forces depends on the system size, and,therefore, the outer layers are greatly subjected to tidalforces. Second, the adiabatic protection is more efficientin the inner part of the clump because of higher orbitalfrequencies here. In reality, we expect some expansion ofclump and diminishing of its central density due to en-ergy deposit from tidal forces. It is very interesting taskto clarify this process in future works.

The numerical estimate of the boost-factor for DM par-ticle annihilation inside clumps is very model-dependent.It depends on nature of DM particles and on their in-teraction with ambient plasma. The important physicalparameters, which affects the annihilation rate in clumps,are decoupling temperature Td and minimal mass Mmin

in the clump mass distribution. The boost-factor in-creases strongly for small Mmin. The minimal mass instandard calculations is determined by the escape of DMparticles from a growing fluctuation due to, e.g., diffu-sion, free streaming or Silk effect. Uncertainties in thecalculated values of Td and Mmin are discussed in Ap-pendix A 1. For the lightest neutralino as a DM par-ticle, assuming it to be the pure bino B, one can seefrom Table III a huge difference in Mmin caused bythe variation of supersymmetry (SUSY) parameters mχ

and m. For these parameters, we use cosmologically al-lowed values from the benchmark scenarios of the work[37]. Moreover, inclusion of other neutralino composi-tions, e.g., mixed bino-Higgsino, the other allowed bench-mark scenarios with co-annihilation and focus-point re-gions, and some other modifications, may very consid-erably increase the allowed region of Mmin values up to(3 × 10−12 − 7 × 10−4)M⊙ [38]. Inclusion of the otherparticle candidates extends further this region.

Another parameter variation which affects strongly theboost-factor is the spectral index of density perturba-tion np (see [4]). We conclude thus that the annihila-tion boost-factor (enhancement) even for neutralino haslarge uncertainties due to the difference in SUSY pa-pameters and spectral index np. It can reach the factor104 and even more, The largest values of boost-factor canbe already excluded by observations of indirect signal,since mSUGRA parameters can be fixed for this largestvalue. On the contrary, a tidal destruction of clumpsin the Galaxy affects the annihilation boost-factor muchweaker.

Acknowledgments

We thank K. P. Zybin and V. N. Lukash for helpfuldiscussions. This work has been supported in part bythe Russian Foundation for Basic Research grants 06-02-16029 and 06-02-16342, the Russian Ministry of Sciencegrants LSS 4407.2006.2 and LSS 5573.2006.2.

APPENDIX A: MINIMUM CLUMP MASS

1. Uncertainties in minimal clump mass and

decoupling temperature

The low-mass cutoff of the clump mass-spectrum ac-companies the process of decoupling. It starts when DMparticles coupled strongly with surrounding plasma inthe growing density fluctuations. The smearing of thesmall-scale fluctuations is due to the collision dampingoccurring just before decoupling, in analogy with the Silkdamping [39]. It occurs due to the diffusion of DM parti-cles from a growing fluctuation, and only the small-scalefluctuations can be destroyed by this process. The corre-sponding diffusive cutoff Mdiff

min is very small. As couplingbecomes weaker, the larger fluctuations are destroyedand Mmin increases. One may expect that the largestvalue of Mmin is related to a free-streaming regime. How-ever, as recent calculations show [10], the largest Mmin

is related to some friction between DM particles andcosmic plasma similar to the Silk damping. The pre-dicted minimal clump masses range from very low values,Mmin ∼ 10−12M⊙ [40], produced by diffusive escape ofDM particles, up to Mmin ∼ 10−4M⊙, caused by acous-tics oscillations [42] and quasi-free-streaming with limitedfriction [10].

The calculations of minimal clump mass Mmin and de-coupling temperature Td are determined by elastic scat-tering of DM particles off leptons l = (νL, eL, eR) incosmic plasma. The uncertainties in cross-section verystrongly influence the resulting values of Mmin and Td.In all works cited above, the lightest neutralino (χ) in

the form of pure bino (B) is assumed as a DM parti-cle, and χl-scattering occurs due to exchange by sleptonsl = (νL, eL, eR).

The elastic cross-sections for lLχ and lRχ scatteringhave been calculated in [4] as

(

dσ

dΩ

)

lLχ

=α2

em

8 cos4 θW

ω2(1 + cos θcm)

(m2L −m2

χ)2(A1)

and(

dσ

dΩ

)

lRχ

= 16

(

dσ

dΩ

)

lLχ

, if mL = mR, (A2)

where ω ≫ ml is a c.m.-energy of l, θcm is a scatteringangle of l in c.m.-system, mχ is a neutralino (bino) mass,mL and mR are, respectively a mass of the left and rightsfermions, and θW is the Weinberg angle.

The values of Td and Mmin as cited in [4, 6, 41, 42]differ very much from each other, but a very big contri-bution to this difference comes from the differences in theused values for mχ and m. To see the difference, whichmust be attributed to the different damping mechanismsused in these works, we recalculated Td and Mmin withthe same values of mχ and m, for which we used 100and 200 GeV respectively. The results are presented in

12

Reference: [41]1) [4]1) [6]2) [42]3) [10]4)

Td, MeV 28 26 25 20 22.6

Mmin/M⊙ 2.5× 10−7 1.7 × 10−7 1.5× 10−6 1.3 × 10−5 8.4× 10−6

TABLE I: The values of decoupling temperature Td and min-imal clump mass Mmin with mχ = 100 GeV and m =200 GeV for different damping mechanisms: 1)free-streaming,2)collision damping, 3)acoustic oscillations, 4)quasi-free-streaming with friction.

scenario χ eL eR νe, νµ ντ

B’ 95 188 117 167 167

E’ 112 1543 1534 1539 1532

M’ 794 1660 1312 1648 1492

TABLE II: Selected benchmark scenarios from [37]. Themasses of particles are given in GeV.

Table I. We did not include there the work [40] becausea pure diffusive damping results in too low a value ofMmin. From the Table I one can see a reasonableagreement in values of Td and Mmin, and a successiveincreasing of Mmin from 2.5×10−7M⊙ for free-streamingto ∼ 10−5M⊙ for oscillation damping and quasi-free-streaming with friction.

2. Uncertainties in SUSY parameters

We shall consider now the range of predictions for dif-ferent values of SUSY parameters allowed in cosmology.For this aim, we shall use the SUSY benchmark scenar-ios from the work [37], which agree with the WilkinsonMicrowave Anisotropy Probe (WMAP) and other cos-mological data. These benchmark scenarios are obtainedwithin mSUGRA model with universal parameters at thegrand unified theory scale: m0 (the universal scalar softbreaking mass), m1/2 (the universal gaugino soft break-ing mass), A0 (the universal cubic soft breaking terms)and tanβ (the ratio of two Higgs v.e.v.’s). The LEP andb→ sγ constraints are imposed. The resulting relic den-sity of neutralinos from these scenarios is in agreementwith the WMAP data or can be obtained with smallchanges of m0 and m1/2. In Table II we display threebenchmark scenarios from [37]. The scenario B’ givesthe lower value mχ ≈ 100 GeV and m close to 200 GeV,which we discussed above. Scenario M’ gives the highestvalue mχ ≈ 800 GeV and m ≈ 1600 GeV. Respectively,scenario E’ gives the intermediate value mχ ≈ 110 GeVand m ≈ 1500 GeV, similar to those we used in [4].

To illustrate the uncertainties in Td and Mmin due touncertainties inmχ and m (in the simplifying assumptionthat mν = meL

= meR) we choose the calculations of

Bertschinger [10] in quasi-free-streaming scenario withfriction, which seem to be at present the most detailed

mχ m Td Mmin

100 GeV 200 GeV 22.6 MeV 8.4× 10−6M⊙

100 GeV 1500 GeV 196 MeV 1.3× 10−8M⊙

800 GeV 1600 GeV 305 MeV 3.5× 10−9M⊙

TABLE III: Values of Td and Mmin for the Bertschinger [10]damping scenario and three benchmark scenarios [37] whichclose to scenarios B’, E’ and M’ shown in the Table II.

ones. We use the Bertschinger formulaes

Td = 7.65C−1/4g1/8∗

( mχ

100 GeV

)5/4

MeV, (A3)

Mmin = 7.59 × 10−3C3/4

(

mχ√g∗

100 GeV

)−15/4

M⊙, (A4)

with a dimensionless constant

C = 256 (GFm2W )2

(

m2

m2χ

− 1

)−2∑

L

(b4L + c4L), (A5)

where GF is the Fermi coupling constant, bL and cL areleft and right chiral vertices, and mW , m, and mχ are, re-spectively, the masses of the W boson GFm

2W = 0.0754,

the slepton, the neutralino and the number of freedom atthe decoupling epoch g∗ = 43/4. (Our own calculationsin [4] of C, which is related with the square of the matrixelement for l+ χ→ l+ χ scattering, differ from (A4) bya factor 1.6.) As a result we obtain for the benchmarkscenarios which approximately coincide with model B’(minimum mχ and m), E’ (minimum mχ and large m)and M’ (very large mχ and m) the values of Td and Mmin

listed in Table III. The predicted range of parameters forMmin from this Table: 3.5×10−9−8.4×10−6M⊙ is not ro-bust at all. It is obtained within mSUGRA assumptionsabout possible universality of SUSY parametersm0, m1/2

and A0. Lifting the universality restriction, the mass ofthe neutralino can increase up to the TeV range scale(though mχ > 200 GeV needs a fine-tuning less than 1%in SUSY [43] or decreased down to a few GeV [44]).

In the numerical predictions above, we limited our-selves by rather restrictive assumptions on the mSUGRAmodel. The most important of them are assumptionthat the neutralino is a pure bino state and a choiceof cosmologically allowed benchmark scenario. The de-tailed analysis made in [38] showed that allowed parame-ters of mSUGRA result in much wider possibilities, e.g.,neutralino as mixed bino-Higgsino and the other bench-mark scenarios. These possibilities are considered un-der WMAP cosmological constraints and a conditionof producing the corresponding DM density for eachset of mSUGRA parameters. The considered modifica-tions allow new channels of neutralino interactions withordinary particles, e.g., the exchange by Z-boson, co-annihilation and resonances in neutralino-fermion scat-tering. It results in a wide range (many orders of mag-nitude) of scattering cross-sections, and, respectively, in

13

a wide range of decoupling temperature from 5 MeV to3 GeV. The corresponding range of Mmin is given by(3 × 10−12 − 7 × 10−4) M⊙. The authors consider alsothe Kaluza-Klein particle as a DM candidate.

The small-scale mass of Mmin results in the large den-sity of DM clumps, and thus in a much stronger annihi-lation signal from the Galactic halo. For typical valuesof power-index of perturbation spectrum (from CMB ob-

servations) the small-scale mass of Mmin results in thelarge density of DM clumps, and thus in a much strongerannihilation signal from the Galactic halo. However, thedependance of a mean clump density on the clump massis rather weak due to the nearly flat form of the pertur-bation spectrum at small scales. The crucial factor forthe amplification of the annihilation signal by clumps isthe value of the perturbation power-law index np.

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