Fermilab arXiv:astro-ph/0105232 14 May 2001 Ultra-high energy cosmic rays from annihilation of ...

12 Pages·2001·0.28 MB·English

Fermilab
FERMILAB-Pub-01/059-A May 2001
Ultra-high energy cosmic rays from
annihilation of superheavy dark matter
a;1 b;2 a;c;3
Pasquale Blasi , Rainer Dick , and Edward W. Kolb
a
NASA/Fermilab Astrophysics Center, Fermi National Accelerator Laboratory,
1
0 Batavia, Illinois 60510-0500
0 b
Department of Physics and EngineeringPhysics, University of Saskatchewan,
2
116 Science Place, Saskatoon, SK S7N 5E2, Canada
y
c
a Department of Astronomy and Astrophysics, Enrico Fermi Institute,
M
The University of Chicago, Chicago, Illinois 60637-1433

4
1

Abstract
2
3
2 We consider the possibility that ultra-high energy cosmic rays originate from the annihi-
5
lation of relic superheavy dark-matter particles. We (cid:12)nd that a cross section of h(cid:27)Avi (cid:24)
0 (cid:0)26 2 12 3=2
1 10 cm (MX=10 GeV) is required to account for the observed rate of super-GZK events if
0 the superheavy dark matter follows a Navarro{Frenk{White density pro(cid:12)le. We also calculate
/
h the possible signature from annihilation in sub-galactic clumps of dark matterand (cid:12)nd that the
p signal from sub-galactic structures may dominate. Finally, we discuss the expected anisotropy
-
o in the arrival directions of the cosmic rays, which is a characteristic signature of this scenario.
r
t
s
a Key words: Dark Matter, Ultra-High Energy Cosmic Rays
:
v PACS: 98.70.Sa,98.70.-f, 95.35.+d, 14.80.-j
i
X
r
a 1 Introduction ucts of wimpzillas.
The dark matter puzzle results from
The nature of the dark matter and the the observation that visible matter can
origin of the ultra-high energy (UHE) cos- account for only a very small fraction of
micraysaretwoofthemostpressingissues matter bound in large-scale structures.
in contemporary particle astrophysics. In The evidence for dark matter is supported
this paper we explore the possibility that by mass estimates from gravitational lens-
there is a common resolution of the two ing [1], by the peculiar velocities of large
issues: dark matter is a relic superheavy scale structures [2], by measurements of
wimpzilla
dark-matterparticle( ),andthe CMB anisotropy [3] and by measurements
UHEcosmicraysaretheannihilationprod- of the recession velocity of high-redshift
supernovae [4]. Constraints from big-bang
1
nucleosynthesis imply that the bulk of the
[email protected]
2
[email protected] dark matter cannot be baryonic, and most
3
[email protected] of the matter density of the universe must
Preprint submitted to Elsevier Science 14 May 2001
arise from particles not accounted for by present abundance would be too large if
wimps
the Standard Model of particle physics [5]. the are thermal relics. Thus, one
The existenceof UHE cosmicrays of en- predicts a maximum mass for a thermal
wimp
ergies above the Greisen-Zatsepin-Kuzmin ,whichturnsouttobeafewhundred
19
cuto(cid:11) [6], EGZK 5 10 eV, is a ma- TeV. While a thermal origin for wimps is
’ (cid:2)
jor puzzle because the cosmic microwave themostcommonassumption,itisnot the
background constitutes an e(cid:14)cient obsta- simplest possibility. It has been recently
cle for protons or nuclei of ultra-high en- pointed out that dark particles mighthave
ergies to travel farther than a few dozen never experienced local chemical equilib-
Mpc [6,7]. This suggests that the observed rium during the evolution of the universe,
extremely energetic cosmic rays with E > and that their mass may be in the range
12 19
EGZK shouldoriginate inour cosmicneigh- 10 to 10 GeV, much larger than the
wimps
borhood. Furthermore, the approximately massofthermal [13{16].Sincethese
wimps
isotropic distribution of arrival directions would be much more massive than
wimps
makesitdi(cid:14)culttoimaginethatnearbyas- thermal ,suchsuperheavydarkpar-
wimpzillas
trophysical sources are the accelerators of ticles have been called [16].
wimpzillas
the observedUHEcosmicrays (but for the Since are extremely mas-
opposite point of view, see [8]). sive,the challenge lies in creating very few
wimpzilla
An interesting possibility is that UHE ofthem.Several scenarioshave
cosmic rays are the products of the decay been developed involving production dur-
of some superheavy particle. This possi- ing di(cid:11)erent stages of the evolution of the
bility has been proposed by Berezinsky, universe.
wimpzillas
Kachelrie(cid:25) and Vilenkin and by Kuzmin may be created during
and Rubakov [9,10], see also [11] for a bubble collisions if in(cid:13)ation is completed
discussion of this in the framework of through a (cid:12)rst-order phase transition
string/M-theory, and [12] for a discussion [17,18]; at the preheating stage after the
in the framework of topological defects. end of in(cid:13)ation with masses easily up to
15
Thesuperheavyparticlesmusthavemasses the Grand Uni(cid:12)ed scale of 10 GeV [19]
12
MX 10 GeV. Although this proposal or even up to the Planck scale [20{22]; or
(cid:21)
circumvents some of the astronomical during the reheating stage after in(cid:13)ation
problems, there are two issues to address: [15] with masses which may be as large as
3
Some cosmological mechanism must be 2 10 times the reheat temperature.
(cid:2)
wimpzillas
found for producing particles of such large may also be generated in
mass in the necessary abundance, and the the transition between an in(cid:13)ationary
lifetimeof this very massive state must be and a matter-dominated (or radiation-
20
in excess of 10 yr, if we want these par- dominated) universe due to the \nonadia-
ticles to be both dark matter candidates batic" expansion of the background space-
and sources of UHE cosmic rays. time acting on the vacuum quantum (cid:13)uc-
Thesimpleassumptionthatdarkmatter tuations. This mechanism was studied in
isathermalreliclimitsthemaximummass detailsin Refs.[13,23,24]. The distinguish-
ofthedarkmatterparticle.Thelargestan- ing feature of this mechanism is the ca-
nihilationcrosssectionintheearlyuniverse pability of generating particles with mass
(cid:0)2
is expected to be roughly MX . This im- of the order of the in(cid:13)aton mass (usually
wimps
plies that verymassive have such a much larger than the reheat temperature)
small annihilation cross section that their even when the particles only interact ex-
2
tremely weakly (or not at all) with other the possibility that the observed UHE cos-
particles,anddonotcoupletothein(cid:13)aton. mic rays result from annihilation of relic
The long lifetime required in the decay superheavy dark matter.
scenario for UHE cosmic rays is problem- Annihilation of dark matter in the halo
atic if the UHE cosmic rays originate in has been a subject of much interest, with
single-particle decays. The lifetime prob- particular emphasis on possible neutralino
lem can be illustrated in the decays of signaturesinthecosmicray(cid:13)ux.Areason-
stringorKaluza{Kleindilatons.Thesepar- able annihilation scenario for UHE cosmic
ticles can be decoupled from fermions [25] raysistheproductionoftwojetseachofen-
and therefore decay through dimension- ergyMX,whichthenfragmentintoa(very)
(cid:12)ve couplings to gauge (cid:12)elds, which in many-particle(cid:12)nalstate,includingleading
leading order are particles of mass comparable to MX.
Assumingthattherelicsuperheavydark
(cid:8) (cid:22)(cid:23) a
I(cid:8) = F aF(cid:22)(cid:23) : (1) matter follows a Navarro{Frenk{White
L (cid:0)4f
(NFW) density pro(cid:12)le [28], in Sec. 2 we
calculate the expected spectrum and the
Here, f is a mass scale characterizing the
annihilation cross section required to ac-
strength of the coupling, and in Kaluza{
count for the observed super-GZK events.
Klein or string theory, it is of order of the
(cid:0)1=2 We(cid:12)ndthatthenecessarycrosssectionex-
reducedPlanck massmPl = (8(cid:25)GN)
18 ’ ceedstheunitaritybound.Wethendiscuss
2:4 10 GeV. In heterotic string theory,
(cid:2) 18 possibilities for circumventingthe bound.
e.g., f = mPl=p2 1:7 10 GeV [26]. If
’ (cid:2) In Sec. 3 we consider the contribution
thenumberofthevector(cid:12)eldsisdG,we(cid:12)nd
fromannihilationinsub-galacticclumpsof
the lifetime estimated from the dilaton{
darkmatter.We(cid:12)ndthatthisresultisvery
vector coupling of Eq. (1)
sensitive to the assumed density pro(cid:12)le of
2
32(cid:25)f the subclumps.
(cid:28)(cid:8)= 3
In Sec. 4 we discuss the expected
dGm(cid:30)
(cid:0)22 12 3 anisotropy in arrival direction of UHE cos-
1:9 10 s 10 GeV
mic rays if the source is superheavy relic
= (cid:2)
dG m(cid:30) ! particle annihilation.
2
f Sec. 5 contains our conclusions.
18 :
(cid:2) 1:7 10 GeV!
(cid:2)
2 Annihilation in the smooth com-
If there are direct decay channels through
ponent
(cid:12)rst order couplings, superheavy particles
decayextremelyfast,evenifthecouplingis
In calculating the UHE cosmic ray (cid:13)ux
only of gravitational strength and dimen- 4
from a smooth superheavy dark matter
sionallysuppressed.Superheavyrelicparti-
distribution in the galactic halo, we as-
cleswitha su(cid:14)cientlylong lifetimerequire
sumeasuperheavyX-particlehalo density
sub-gravitational couplings or exponential
spherically symmetric about the galactic
suppression of the decay mechanism due
4
to wormhole e(cid:11)ects [9], instantons [10], or We refer to a superheavy dark matter dis-
magic from the brane world [27]. tribution as smooth if it can be described by
Motivated by the attractiveness of the a particle density nX(d) which decreases uni-
decayscenario,inthispaperweinvestigate formlywith distance fromthe galactic center.
3
center, nX(d) = nX(d), where d is the dis- where (cid:0)X is the decay width of the wim-
pzilla
tance from the galactic center. We will as- .
sume nX(d) is given by a NFW pro(cid:12)le [28] There are many discussions about the
extrapolations and approximations to the
N0
fragmentation function d (E;E =
nX(d) = 2: (2) jet
N
d(d+ds)
MX)=dE (for a recent review see [31] and
for a numerical approach, see [32]). In the
Navarro’sestimateforthe(cid:12)ducialradius
results of Fig. 1, we use the fragmentation
ds for the Milky Way is of order 25kpc
functionsthataretheMLLAlimitingspec-
[29]. Dehnen and Binney have examined
trum of ordinary QCD [33]. The salient
a (cid:13)attened NFW pro(cid:12)le as a special case
features of the results can be understood
of a whole class of halo models and give
by making the simple approximation that
a value ds = 21:8kpc [30]. We will use
mostof thecontentof thejetis pions,with
ds = 3d(cid:12) = 24kpc in our numerical esti-
a spectrum in terms of the usual variable
mates, where d(cid:12) 8kpc is the distance of
’ x E=E (of course, 0 x 1),
the solar system from the galactic center. (cid:17) jet (cid:20) (cid:20)
ThedimensionlessparameterN0 maybe
found by requiring that the total mass of d (x) 15 (cid:0)3=2 2
N = x (1 x)
12
the Galaxy is 2 10 M(cid:12), which gives dx 16 (cid:0)
(cid:2) 15 (cid:0)3=2
55 (cid:0)9 3 x (x 1):
8 10 2:8 10 kpc (cid:24) 16 (cid:28)
N0 = (cid:2) = (cid:2) 3 ;
M12 M12 cm
The UHE cosmic rays in this picture are
12 photons resulting from the decay of neu-
where M12 = MX=10 GeV.
wimpzilla tralpions.Theneutralpionsareaboutone-
annihilation produces two
third of the total number of pions in the
jets, each of energy MX, while decay of 0
wimpzilla jet, and each (cid:25) decay produces two pions.
a produces two jets, each of
Using this fragmentation approxima-
energy MX=2. The energy spectrum of
tion, the scaling of the (cid:13)ux with MX can
observed UHE cosmic ray events from an-
be found to be
nihilation is
1=2 (cid:0)2
d (E;Ejet = MX) MX (cid:27)A MX (annihilation)
=2 N (cid:27)Av F / 1=2h i (cid:0)1
F dE h i MX (cid:0)XMX (decay): (4)
2 /
3 nX(d)
dd 2: (3) 1=2
(cid:2)Z 4(cid:25) d d(cid:12) The factor of MX is from the fragme(cid:0)n2-
j (cid:0) j
tation function, and the factors of MX
(cid:0)1 2
Here, d (E;Ejet)=dE is the fragmenta- or MX arise from nX or nX, respectively.
N
tionspectrumresultingfromajetofenergy Therefore, for a given , the necessary
F3=2
Ejet.Forcomparison,theenergyspectrum cross section scales as MX in the anni-
of observed UHE cosmic ray events from hilation scenario and the necessary decay
wimpzilla 1=2
decay is width scalesas MX in thedecayscenario.
Calculating the resulting UHE cosmic
d (E;E = MX=2)
jet ray (cid:13)ux in the annihilation model, and
=2 N (cid:0)X
F dE comparing it to the similar calculation in
3 nX(d) the decay scenario, we obtain the results
dd 2;
(cid:2)Z 4(cid:25) d d(cid:12) shown in Fig. 1.
j (cid:0) j
4
We have already discussed the fact that
alifetimeinthisrangeforaparticlesomas-
sive is rather unexpected (but, of course,
not impossible).Therequiredcross section
in the annihilation scenario is also orders
of magnitude larger than expected.
The necessary annihilation cross section
is well in excess of the unitarity bound to
the l-wave reaction cross section [34{36]:
(cid:0)47
4(cid:25) 1:5 10
(cid:27)lv 2 (2l+1) (cid:2) 2
(cid:20) M v ’ M12
(cid:0)1
100km s 2
(2l+1) cm :
(cid:2) v
However,as emphasized by Hui [36], there
are several ways to evade the bound. The
most probable evasion in our scenario is
wim-
possible (cid:12)nite size e(cid:11)ects of the
pzilla
. The relevant limit in that case
2
is (cid:27)A . 64(cid:25)R , where R is the \size" of
either the particle or the range of the in-
Fig. 1. UHE cosmic ray spectra from su-
teraction. For the type of cross sections
perheavy particle annihilation (upper panel)
discussed above, we would require
or decay (lower panel). For both (cid:12)gures the
12
solid lines are for MX = 10 GeV and 1=2
14 (cid:0)13 (cid:27)Av
the dashed lines are for MX = 10 GeV. R&4 10 h(cid:0)26 i 2
(cid:2) 10 cm !
For annihilation, the solid curve is for
(cid:0)27 2 (cid:0)1 1=2
h(cid:27)Avi = 6(cid:2)10 cm and the dashed curve 100km s
(cid:0)25 2
corresponds to h(cid:27)Avi= 10 cm . In the de- cm:
20 (cid:2) v !
cay case, the solid curve is for (cid:0)X = 10 yr
20
and the dashed curve is for (cid:0)X = 8(cid:2)10 yr.
Thissizeisstilluncomfortablylarge,butis
oforderofthelengthscaleofthestrong in-
The shape of the spectrum is deter-
teractions.Inthenextsectionwewillshow
wimpzilla
mined by the mass of the and
that the signal froma clumpedcomponent
the overall normalization can be scaled by
of dark mattermaydominateand the nec-
adjusting (cid:27)Av or (cid:0)X. Clearly, in order
h i essarycrosssectionmaybesmallerandthe
to produce UHE cosmic rays in excess of
20 requirementson R not as severe.
10 eV, MX cannot be too much smaller
12
than 10 GeV. In order to provide enough
events to explain the observed UHE cos- 3 Annihilationin the clumpedcom-
mic rays, (cid:27)Av has to be in the range ponent
(cid:0)25 2 h (cid:0)i27 2
10 cm to 10 cm .
wim-
Similarly,inthedecayscenario,the So far we have assumed that the galac-
20
pzilla
lifetimemustbeintherange10 yr ticdark matteris smoothly distributed.In
21
to 10 yr. this section we will consider the contribu-
5
tion from inhomogeneities in the galactic dividualsubclumps.Letusnowdiscussthe
distribution. We will (cid:12)rst consider the sig- e(cid:11)ect of the density pro(cid:12)le of individual
nal from the sort of sub-galactic clumps subclumps.
(subclumps) predicted by numerical simu-
lations. Then we willconsider the possibil- 3.1.1 Isothermal subclumps
ity that we live within the core radius of First, we assume that the densitypro(cid:12)le
such a subclump. of an individualsubclumpis an isothermal
sphere, with radial pro(cid:12)le
3.1 Individual Subclumps
(cid:0)2
r
nX(r) = nH(d) ;
We have modeled the smooth compo- (cid:18)r0(cid:19)
nent of the dark matter by a NFW pro(cid:12)le,
wherer0 istheradiusoftheindividualsub-
Eq. (2). Inaddition to this smooth compo-
clump, de(cid:12)ned as the radius at which the
nent,the dark mattermayhavea clumped
subclumpdensityequals the densityof the
component, as suggested by N-body sim-
backgroundhaloatthedistancedfromthe
ulations. Using the results of these simu-
galactic center. If the subclump mass is
lations, the number of subclumps of mass
Mcl, we can write
Mcl per unit volumeat distanced fromthe
galactic center can be written in the form: r0
2 (cid:0)2
Mcl = 4(cid:25)MXnH(d) drr (r=r0) ;
(cid:0)(cid:11)
0 Mcl Z0
ncl(d;Mcl)=ncl
(cid:18)MH(cid:19)
2 (cid:0)3=2 from which we obtain
d
1+ ; 1=3
(cid:2)2 Rccl! 3 Mcl
r0 = :
4 5 "4(cid:25)MXnH(d)#
cl
whereRc isthecoreradiusofthesubclump
distribution in the galaxy, typically of or- The rate of annihilationin an individual
der10to20kpc.Themassofthehaloofthe subclump at distance d from the galactic
GalaxyisMH.Thenormalizationconstant center is
0
ncl can be calculated by requiring that the r0
2
mass in the clumped component is a frac-
(Mcl;d)= (cid:27)Av 4(cid:25) drr
tion (cid:24) of the halo mass MH. From simula- R h i RmZin
tions, (cid:24) 10%. The power index, (cid:11), may 2 (cid:0)4
(cid:24) nH(d)(r=r0) (5)
also be found from simulations, with the (cid:2) 2 4
+ (cid:27)Av (4(cid:25)=3)nH(d)r0=Rmin;
result (cid:11) 1:9. It is then easy to (cid:12)nd h i
(cid:24)
0 (2 (cid:11))(cid:24) whereRmin istheminimumradiusofasub-
ncl cl 2(cid:0) 2(cid:0)(cid:11); clump(theinnerradiuswherethepro(cid:12)leis
(cid:25) 4(cid:25)(Rc ) RHMH(cid:17)
(cid:13)attened by e(cid:14)cient annihilations).
where (cid:17) is the ratio of the mass of A reasonable upper limitto Rmin can be
the largest subclump to the halo mass, obtained by requiring that the dynamical
(cid:17) = Mcl(max)=MH 0:01 0:1, and RH free-fall timeof dark matter from the edge
(cid:24) (cid:0)
is the radius of the halo. of the subclump is comparable with the
Up to this point, all the discussion is in- annihilation rate at Rmin. The dynamical
dependent of the density pro(cid:12)le of the in- free-fall time is
6
(cid:28)ff =r0=v(r0) Now turning to the location of the so-
=[4(cid:25)GMXnH(d)](cid:0)1=2; lar system in the Milky Way, using Rccl =
15kpc, (cid:17) = 0:1, and (cid:24) = 0:1, we (cid:12)nd
while the annihilation time scale is 1=2
clumped 6 MX
(cid:0)1 F 2 10 12
(cid:28)A=[n(r) (cid:27)Av ] ’ (cid:2) 10 GeV
smooth (cid:18) (cid:19)
h1 i r 2 F (cid:0)26 2 1=2
= : 3 10 cm
nH(d)h(cid:27)Avi (cid:18)r0(cid:19) (cid:2) (cid:2) (cid:27)Av ! :
h i
It is then easy to obtain the ratio
1=4
Rmin 1=2 nH(d)
= (cid:27)Av : (6)
r0 h i "4(cid:25)GMX#
From Eqs. (6) and (6), we then obtain for
the rate of annihilations in a subclump of
mass Mcl at distance d from the galactic
center:
4 1=2 1=4
(Mcl;d)= Mcl (cid:27)Av (4(cid:25)G)
R 3 h i
3=4
nH(d)
: Fig.2. The ratio of eventsfrom the subclump
(cid:2) MX !
component to the smooth component assum-
ing either an isothermal or a NFW pro(cid:12)le for
Nowwecancalculatethecontributionof
the subclumps.
the clumped component in a similar man-
ner as Eq. (3): The ratio of the (cid:13)uxes from the sub-
clump component and the smooth compo-
d (E;Ejet = MX) nent are given in Fig. 2. Comparing Fig. 1
=2 N 12
F dE andFig.2,weseethatforMX = 10 GeV,
3 1 to normalize the (cid:13)ux to the observed
d d 2 (7) 3
(cid:2)Z 4(cid:25) d d(cid:12) events would require a cross section 10
Mmax j (cid:0) j timessmallerthan the isotropic case if the
wimpzilla
dMcl ncl(d;Mcl) (Mcl;d): subclumps have an isothermal
(cid:2) R
MmZin pro(cid:12)le.
As before we will use an NFW pro(cid:12)le for 3.1.2 NFW subclumps
nH(d) with ds = 24kpc. Now, we take the density pro(cid:12)le of the
To understand the scaling of the events subclumps according to a NFW pro(cid:12)le,
from subclumps, we can make the simple
2
approximation that one is at the galactic r0[r0 +rs]
nX(r) = nH(d) 2 ; (8)
center( d(cid:12) = 0).Inthelimitthatthecon- r[r +rs]
j j cl
tributions come from d Rc and d ds,
(cid:28) (cid:28)
we (cid:12)nd where rs is the (cid:12)ducial radius of the sub-
clumpandagain r0 istheradiusofthesub-
1=2 (cid:0)1
(cid:27)Av MX clump where nX(r) = nH(d). We assume
F / h i
7
8
herethattheratior0=rs 1isconstantfor isothermal subclump with mass 10 M(cid:12),
(cid:29) (cid:0)27 2
all subclumps. The mass of the subclump and for (cid:27)Av = 3 10 cm , we have
h i (cid:2) 15
can be written in terms of r0 and rs as r0 1kpc and Rmin 10 cm, so that
(cid:24) (cid:0)21 (cid:24)
r0 Pcore 10 . The probability is quite
(cid:24)
2 in(cid:12)nitesimal,but the possibility has inter-
Mcl=4(cid:25)MX drr nX(r)
esting consequences.
Z0
The cases to consider are again that of
3 r0
4(cid:25)MXnH(d)r0ln : an isothermal subclump and a NFW sub-
’ (cid:18)rs(cid:19)
clump,with the Earth in the core. In both
which provides the core radius r0 for the cases it is relevantto calculate the value of
subclump once we know rs. the minimum radius, Rmin, within which
The rate of annihilations for one sub- the e(cid:14)ciency of annihilations (cid:13)attens the
clump is given by radial pro(cid:12)le of dark matter. The reason
forthisisthatthemaincontributiontothe
r0
2 2 (cid:13)ux of UHE events comes from this region
(Mcl;d)=4(cid:25) (cid:27)Av drr nX(r) if the detector is within the core.
R h i
RmZin
For an isothermal subclump,the (cid:13)ux is
3 2
+(4(cid:25)=3)Rmin (cid:27)Av nH(Rmin); proportional to
h i
4 11=12 (cid:0)1=2
which becomes = nH(d(cid:12)) (cid:27)Av
F 3 h i
6 1=3 5=12 3=4
4(cid:25) r0 2 Mcl (4(cid:25)MX) G ;
(Mcl;d) (cid:27)Av 3nH(d): (cid:2)
R ’ 3 h irs
To calculate for NFW subclumps, we where Mcl is the subclump mass and
follow a procedFure similar to Eq. 7, but nH(d(cid:12)) is the typical density at the dis-
with (Mcl;d)givenabove.Againexpress- tance of the solar system from the galac-
R
ing the (cid:13)ux in terms of the (cid:13)ux from the tic center. This (cid:13)ux should be compared
smooth component, we (cid:12)nd with the total (cid:13)ux from a distribution of
isothermalsubclumpsinthehalo, givenby
clumped 3 Eq. (7).
F 2 10 ;
’ (cid:2) One has to calculate Rmin in the NFW
smooth
F
case as well. As before, the way to do this
independent of MX or (cid:27)Av . Again, this is to require that the free-fall time scale
h i
result is represented in Fig. 2. equals the time scale for annihilation at
Rmin. We assume that the external radius
3.2 If we live in a subclump of the subclump and its core radius,rs, are
related by r0=rs = (cid:14) 1. Then we can
(cid:29)
Now consider the possibility that we write:
live within the dense core of a subclump.
The probability to be in the center of a 9=2 (cid:0)1=2 5=2
Rmin=(cid:14) (GMcl) rs
subclump is given by the probability of
nH(d(cid:12)) (cid:27)Av :
being in a subclump, multiplied by the (cid:2) h i
probability to be within the minimum ra-
3
dius, so Pcore 0:1(Rmin=r0) . For an Notealsothatrs isrelatedtothetotalsub-
(cid:24)
8
clump mass by
1=3
Mcl
rs = 3 ;
"4(cid:25)nH(d(cid:12))(cid:14) MXf((cid:14))#
wheref((cid:14)) = ln(1+(cid:14)) (cid:14)=(1+(cid:14)).The(cid:13)ux
(cid:0)
in this case is proportional to
(cid:0)2
Rmin 2
Rmin (cid:27)Av nH;0:
F / rs h i
(cid:18) (cid:19)
3
Here, nH;0 (cid:14) nH(d(cid:12)). This number
(cid:25) Fig. 3. The angular dependence of events
should be compared with the total (cid:13)ux
originating from annihilation or decay of the
from the distribution of subclumps with
smoothcomponentofdarkmatterthatfollows
NFW pro(cid:12)les in the Galaxy.
a NFW pro(cid:12)le. In this (cid:12)gure, cos(cid:18) = 1 corre-
Note that the (cid:13)uxes are proportional
(cid:0)1=2 (cid:0)1 sponds to the direction of the galactic center.
to (cid:27)Av ( (cid:27)Av ) for the isothermal
h i h i
(NFW) case. This means that decreasing
matter subclumps within the galaxy, then
the cross section actually helps increasing
the density in the subclumps would be
the (cid:13)ux. This is true provided the mini-
larger than the ambient background dark-
mum radius, Rmin, reaches the radius of
matterdensity,andeventsoriginatingfrom
(say)theatmosphere.Thisimposesalower
subclumps will dominate the observed sig-
limit on the cross section, which is still
nal.
above the unitarity limit.
The dominance of events from sub-
clumps has two e(cid:11)ects. The (cid:12)rst e(cid:11)ect is
4 Predicted signals that a smaller annihilation cross section is
required to account for the observed UHE
The annihilation rate is very sensitive (cid:13)ux. The second e(cid:11)ect is that there will
to the local density. This implies that if be a very large probability of detecting a
the UHE cosmic rays originate from dark nearby subclump.
matterannihilationanddarkmatterinour In Fig. 4 we present histograms showing
galaxy is clumped, the arrival direction of the number of occurrences of single and
UHE cosmic rays should re(cid:13)ect the dark multipleevents in arrival directions within
matter distribution. a square degree. We generated many real-
The(cid:12)rstpossibilityweconsideredisthat izations of the expected subclump distri-
thedarkmatterdistributionissmooth,and bution, and calculated the (cid:13)ux from the
follows a NFW pro(cid:12)le. In this case, the subclumps that would result in about 100
galactic center should be quite prominent. detected events. We see that the average
In Fig. 3 we illustrate the expected angu- probabilities are quite reasonable, with
lardependenceofthearrivaldirection.The most events in square degree areas with
galactic center is prominent for the decay onlyoneevent,anda fewpairsand triplets
scenariowhere nX [12],andevenmore of events. However if we examine indi-
F /
prominent for the annihilation case where vidual realizations containing about 100
2
nX. events, we see that there is a large prob-
F /
Now if we assume that there are dark ability of a large number of events from
9
eral features can be understood as as-
suming that each elementary process of
quark-antiquark generation or gluon radi-
ation implies an average angular widen-
ing of the initial beam by the amount
(cid:14)(cid:11) (cid:3)QCD=E , where (cid:3)QCD 0:3 GeV
jet
(cid:24) (cid:24)
and E MX. While the jets further
jet
(cid:24)
fragments into other quarks and gluons,
its opening angle is expected to make a
random walk for a number of steps ap-
proximately given by E =(cid:3)QCD. There-
jet
fore, the ultimate opening angle should
1=2
be (cid:11) ((cid:3)QCD=E ) . We stress that
jet jet
(cid:24)
this has to be intended as a rough esti-
mateofthequantity(cid:11) ,whose realvalue
jet
is determined by a variety of elementary
particle physics processes.
If we adopt the expression obtained
above for (cid:11) , the size of the hadronic
jet
shower at distance d from the production
region can be estimated as:
Fig. 4. The probability of (cid:12)nding a certain 8 (cid:0)1=2
l = (cid:11) d 5 10 M12 d15cm;
number of events in a one square degree area jet
(cid:24) (cid:2)
of the sky. The upper (cid:12)gure is for isothermal
15
subclumps and thelower(cid:12)gure forNFWsub- where d15 = d=10 cm. For simplicity, let
clumps us assume that the Earth is sitting in the
center of a clump with an isothermal pro-
single nearby subclumps. For instance, in
(cid:12)le. In this case, most of the contribution
the single realizations shown in Fig. 4, the
tothelocal(cid:13)uxof UHEcosmicrays,as de-
isothermal subclumps result in a square
scribedinthepreviousesection,isprovided
degree bin with 14 events and a square
byparticlesannihilatingwithinRmin.From
degree bin with 33 events. In the typical
Eq. (6) we obtain for the minimumradius
NFW distribution, there is one bin with
16 events and another bin with 43 events.
14 (cid:27)Av
Therefore, if more than 100 events are ob-
Rmin 7 10 h(cid:0)26 i 2
servedwithfullskycoverage,thesignature (cid:25) (cid:2) 10 cm
1=3
of subclumps will be unmistakable. (cid:0)1=2 Mcl
M12 8 cm;
Finally,weremarkon thecharacteristics (cid:2) 10 M(cid:12)!
of cosmic ray events originating from an-
wim-
nihilation (or decay) of very nearby where Mcl;8 is the clump mass in units of
pzillas 8
, as might be expected if we live 10 M(cid:12).Substitutingintotheexpressionfor
within the core radius of a subclump. l we obtain
The process of jet production after
wimpzilla
the annihilation or decay is 8 (cid:27)Av
extremely complicated, but some gen- l<3:5 10 h(cid:0)26 i 2
(cid:2) 10 cm
10