Ultra-high energy cosmic rays from annihilation of superheavy dark matter

13 Pages·2008·0.23 MB·English

Ultra-high energy cosmic rays from
annihilation of superheavy dark matter
Pasquale Blasia,b,1, Rainer Dickc,2, and Edward W. Kolbb,d,3
aOsservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125 Firenze, Italy
1
0 bNASA/Fermilab Astrophysics Center, Fermi National Accelerator Laboratory,
0 Batavia, Illinois 60510-0500
2
cDepartment of Physics and Engineering Physics, University of Saskatchewan,
v
o 116 Science Place, Saskatoon, SK S7N 5E2, Canada
N
dDepartment of Astronomy and Astrophysics, Enrico Fermi Institute,

9 The University of Chicago, Chicago, Illinois 60637-1433
1

3
v
Abstract
2
3
2 We consider the possibility that ultra-high energy cosmic rays originate from the annihilation of
5 relicsuperheavydarkmatter.Wefindthatacrosssectionof σ v 10−26cm2(M /1012GeV)3/2
0 A X
h i ∼
1 is required to account for the observed rate of super-GZK events if the superheavy dark matter
0 followsaNavarro–Frenk–Whitedensityprofile.Thiswouldrequireextremelylarge-lcontributions
/
h to the annihilation cross section. We also calculate the possible signature from annihilation in
p
sub-galactic clumpsofdark matter andfindthatthesignal fromsub-clumpsdominates andmay
-
o explain the observed flux with a much smaller cross section than if the superheavy dark matter
r
t is smoothly distributed. Finally, we discuss the expected anisotropy in the arrival directions of
s
a the cosmic rays, which is a characteristic signature of this scenario.
:
v
i
X Key words: Dark Matter, Ultra-High Energy Cosmic Rays
r PACS: 98.70.Sa, 98.70.-f, 95.35.+d, 14.80.-j
a
1 Introduction between these issues: dark matter is a very
massiverelicparticlespecies(wimpzilla),
The nature of the dark matter and the and wimpzilla annihilation products are
origin of the ultra-high energy (UHE) cos- the ultra-high energy cosmic rays.
micraysaretwoofthemostpressingissues The dark matter puzzle results from
in contemporary particle astrophysics. In the observation that visible matter can
thispaperweexploreapossibleconnection account only for a small fraction of the
matter bound in large-scale structures.
1 [email protected] The evidence for dark matter is supported
2 [email protected] by mass estimates from gravitational lens-
3 [email protected], ing [1], by the peculiar velocities of large
[email protected]
Preprint FNAL-PUB-01/059-A submitted to Elsevier Science 1 February 2008
scale structures [2], by measurements of Thesimpleassumptionthatdarkmatter
CMB anisotropy [3] and by measurements isathermalreliclimitsthemaximummass
of the recession velocity of high-redshift ofthedarkmatterparticle.Thelargestan-
supernovae [4]. Constraints from big-bang nihilationcrosssectionintheearlyuniverse
nucleosynthesis imply that the bulk of the is expected to be roughly M−2. This im-
X
dark matter cannot be baryonic, and most plies that very massive wimps have such a
of the matter density of the universe must small annihilation cross section that their
arise from particles not accounted for by present abundance would be too large if
the Standard Model of particle physics [5]. the wimps are thermal relics. Thus, one
The existence of UHE cosmic rays of en- predicts a maximum mass for a thermal
ergies above the Greisen-Zatsepin-Kuzmin wimp,whichturnsouttobeafewhundred
cutoff [6], E 5 1019eV, is a ma- TeV. While a thermal origin for wimps is
GZK
≃ ×
jor puzzle because the cosmic microwave the most commonassumption, it isnot the
background constitutes an efficient obsta- simplest possibility. It has been recently
cle for protons or nuclei of ultra-high en- pointed out that dark particles might have
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
E shouldoriginateinour cosmicneigh- 1012 to 1019 GeV, much larger than the
GZK
borhood. Furthermore, the approximately massofthermalwimps[13–16].Sincethese
isotropic distribution of arrival directions wimps would be much more massive than
makesitdifficulttoimaginethatnearbyas- thermalwimps,suchsuperheavydarkpar-
trophysical sources are the accelerators of ticles have been called wimpzillas [16].
the observed UHE cosmic rays (but for the Since wimpzillas are extremely mas-
opposite point of view, see [8]). sive, the challenge lies in creating very few
An interesting possibility is that UHE ofthem.Severalwimpzillascenarioshave
cosmic rays arise from the decay of some been developed involving production dur-
superheavy particle. This possibility has ing different stages of the evolution of the
been proposed by Berezinsky, Kachelrieß universe.
andVilenkin andby KuzminandRubakov wimpzillas may be created during
[9,10], see also [11] for a discussion of this bubble collisions if inflation is completed
in the framework of string/M-theory, and through a first-order phase transition
[12] for a discussion in the framework of [17,18]; at the preheating stage after the
topological defects. The superheavy par- end of inflation with masses easily up to
ticles must have masses M 1012GeV. the Grand Unified scale of 1015GeV [19]
X
≥
Althoughthisproposalcircumvents theas- or even up to the Planck scale [20–22]; or
tronomical problems, there are two issues during the reheating stage after inflation
to address: Some cosmological mechanism [15] with masses which may be as large as
must be found for producing particles of 2 103 times the reheat temperature.
×
such large mass in the necessary abun- wimpzillas may also be generated in
dance,andthelifetimeofthisverymassive the transition between an inflationary
state must be in excess of 1020yr, if we and a matter-dominated (or radiation-
want these particles to be both dark mat- dominated) universe due to the “nonadia-
ter candidates and sources of UHE cosmic batic” expansion of the background space-
rays. time acting on the vacuum quantum fluc-
2
tuations. This mechanism was studied in ticleswithasufficientlylonglifetimethere-
details in Refs. [13,23,24]. The distinguish- fore require sub-gravitational couplings or
ing feature of this mechanism is the ca- exponentialsuppressionofthedecaymech-
pability of generating particles with mass anism due to wormhole effects [9], instan-
of the order of the inflaton mass (usually tons [10], or magic from the brane world
much larger than the reheat temperature) [27].
even when the particles only interact ex- Motivated by the attractiveness of the
tremely weakly (or not at all) with other decay scenario, we investigate the possibil-
particles,anddonotcoupletotheinflaton. ity that UHE cosmic rays may result from
The long lifetime required in the decay annihilation of relic superheavy dark mat-
scenario for UHE cosmic rays is problem- ter.
atic if the UHE cosmic rays originate in Annihilation of dark matter in the halo
single-particle decays. The lifetime prob- has been a subject of much interest, with
lem can be illustrated in the decays of particular emphasis on possible neutralino
string or Kaluza–Klein dilatons. These signatures in the cosmic ray flux. A rea-
particles can be decoupled from fermions sonable annihilation scenario for UHE cos-
[25], and therefore decay in leading order mic rays is the production of two jets, each
through their dimension-five couplings to of energy M , which then fragment into a
X
gauge fields many-particle final state including leading
particlesofenergiescomparabletoasignif-
Φ
IΦ = FµνaFµνa. (1) icant fraction of MX.
L −4f
Assumingthattherelicsuperheavydark
matter follows a Navarro–Frenk–White
Here, f is a mass scale characterizing the
(NFW) density profile [28], in Sec. 2 we
strength of the coupling, and in Kaluza–
calculate the expected spectrum and the
Klein or string theory, it is of order of the
reduced Planck mass m = (8πG )−1/2 annihilation cross section required to ac-
Pl N
≃ count for the observed super-GZK events.
2.4 1018GeV. In heterotic string theory,
× We find that the necessary cross section
e.g., f = m /√2 1.7 1018GeV [26].
Pl
≃ × exceeds the unitarity bound unless the
If the number of the vector fields is d , we
G
annihilation involves very large angular
findthelifetimeestimatefromthedilaton–
momentum contributions or non-standard
vector coupling of Eq. (1)
mechanisms, like, e.g., in monopolonium
32πf2 decay.
τ =
Φ d m3 In Sec. 3 we consider the contribution
G φ
from annihilation in sub-galactic clumps
1.9 10−22s 1012GeV 3
of dark matter. We find that annihilation
= ×
dG mφ ! in isothermal sub-galactic clumps may ex-
f 2 plain the UHE cosmic rays without violat-
.
ing the unitarity bound.
× 1.7 1018GeV!
× In Sec. 4 we discuss the expected
If there are direct decay channels through anisotropy in arrival direction of UHE
first order couplings, superheavy particles cosmic rays if the primary source is su-
decayextremelyfast,evenifthecouplingis perheavy relic particle annihilation in sub-
only of gravitational strength and dimen- galactic clumps.
sionally suppressed. Superheavy relic par- Sec. 5 contains our conclusions.
3
2 Annihilation in the smooth com- d (E,E = M )
jet X
=2 N σ v
ponent A
F dE h i
n2 (d)
d3d X . (3)
In calculating the UHE cosmic ray flux × 4π d d 2
Z ⊙
from a smooth4 superheavy dark matter | − |
distribution in the galactic halo, we as- Here, d (E,E )/dE is the fragmenta-
jet
N
sume asuperheavy X-particlehalodensity tionspectrumresultingfromajetofenergy
spherically symmetric about the galactic E .Forcomparison,theenergy spectrum
jet
center, nX(d) = nX(d), where d is the of observed UHE cosmic ray events from
distance from the galactic center. We will wimpzilla decay is
assume n (d) is given by an NFW profile
X
[28] d (E,E = M /2)
jet X
=2 N Γ
X
F dE
N
0 n (d)
nX(d) = d(d+ds)2. (2) × d3d 4π dX d 2,
Z ⊙
| − |
Navarro’sestimateforthefiducialradius where Γ is the decay width of the wim-
X
ds for the Milky Way is of order 25kpc pzilla.
[29]. Dehnen and Binney have examined
Extrapolations and approximations to
a flattened NFW profile as a special case
thefragmentationfunctiond (E,E )/dE
jet
of a whole class of halo models and give N
have been reviewed in [31]. Birkel and
a value ds = 21.8kpc [30]. We will use Sarkar applied the Herwig Monte Carlo
d = 3d = 24kpc in our numerical esti-
s ⊙ program to calculate the spectrum from
mates, where d⊙ 8kpc is the distance of wimpzilla decay [32], and Fodor and
≃
the solar system from the galactic center.
Katz employed a numerical integration of
Thedimensionless parameterN maybe
0 the DGLAP evolution equations [33]. The
found by requiring that the total mass of
resulting spectrum in the interesting range
the Galaxy is 2 1012M⊙, which gives between 1019GeV and 1020GeV is similar
×
to the spectrum from the modified leading
8 1055 2.8 10−9kpc3
logarithmic approximation (MLLA) [34],
N = × = × ,
0 M12 M12 cm3 which was employed by Berezinsky et al.
in their proposal of UHE cosmic rays from
where M = M /1012GeV.
12 X decay of superheavy darkmatterdecay [9].
Forsimplicity, we willassume thatwim-
WealsousetheMLLAlimitingspectrum
pzilla annihilation produces two leading
in theresults of Fig.1.The salient features
jets, each of energy M , while decay of a
X oftheresultscanbeunderstoodbymaking
wimpzilla produces two jets, each of en-
the simple approximation that most of the
ergy M /2. The energy spectrum of ob-
X content of the jet is pions, with a spectrum
served UHE cosmic ray events from anni-
in terms of the usual variable x E/E
jet
hilation is ≡
(of course, 0 x 1),
≤ ≤
4 We refer to a superheavy dark matter dis- d (x) 15
N = x−3/2(1 x)2
tribution as smooth if it can be described by dx 16 −
a particle density n (d) which decreases uni- 15
X x−3/2 (x 1).
formly with distance from thegalactic center.
∼ 16 ≪
4
Using this fragmentation approxima-
tion, the scaling of the flux with M can
X
be found to be
M1/2 σ M−2 (annihilation)
F ∝ X h Ai X
M1/2Γ M−1 (decay). (4)
∝ X X X
1/2
The factor of M is from the fragmen-
X
tation function, and the factors of M−2
X
or M−1 arise from n2 or n , respectively.
X X X
Therefore, for a given , the necessary
F3/2
cross section scales as M in the anni-
X
hilation scenario and the necessary decay
1/2
width scales as M in the decay scenario.
X
Calculating the resulting UHE cosmic
ray flux in the annihilation model, and
comparing it to the similar calculation in
the decay scenario, we obtain the results
shown in Fig. 1.
The shape of the spectrum is deter-
mined by the mass of the wimpzilla and
the overall normalization can be scaled
Fig. 1. UHE cosmic ray spectra from su-
by adjusting σ v or Γ . Clearly, in or-
h A i X perheavy particle annihilation (upper panel)
der to produce UHE cosmic rays in excess
or decay (lower panel). For both figures the
of 1020eV, MX cannot be much smaller solid lines are for M = 1012GeV and
X
than 1012GeV. In order to provide enough the dashed lines are for M = 1014GeV.
X
events to explain the observed UHE cos- For annihilation, the solid curve is for
1m0i−c25racmys2, thoσA10v−i27hcams 2t.o be in the range hcσorArveisp=on6d×s t1o0−σ27cvm=2 a1n0d−2t5hcemd2a.sIhnedthceudrvee-
A
This is well in excess of the unitarity cay case, the sohlid ciurve is for Γ−1 = 1020yr
X
bound to the l-wave reaction cross section and the dashed curve is for Γ−1 = 8 1020yr.
X ×
[35–37]:
than M−1. One example of a fundamental
X
4π 1.5 10−47 length scale might be the physical size of
σ v (2l+1) ×
l ≤ M2v ≃ M2 the particle. In this case, the scale for the
12
100km s−1 annihilation cross section could be related
(2l+1) cm2. to the size of the particle. Another possi-
× v
bility for the scale of the annihilation cross
The unitarity bound essentially states section might be the range of the interac-
that the annihilation cross section must tion. If the interaction has a range λ, then
be smaller than M−2. However, as empha- one might imagine that the scale for the
X
sized by Hui [37], there are several ways to cross section is λ2.
evade the bound. The annihilation cross Arelatedissueisthetypicalenergyofthe
section may be larger if there are funda- annihilation products. In this paper we as-
mental length scales in the problem larger sume that annihilation produces two jets,
5
each with energy approximately M . It is tion in the halo [41].
X
easy to imagine that with the finite-size ef- The monopolonium example is an ex-
fects discussed above, there is the possibil- istence proof that the annihilation cross
itythatannihilationwillproducemanysoft section may be many orders of magnitude
particles, rather thanessentially two parti- larger than M−2, while annihilation prod-
X
cles each of energy M . uctscanhaveenergyoforderM .Whileit
X X
An example that suits our needs is the isanexistence proof,italsoillustratesthat
annihilation of a monopole-antimonopole thecombinedconditionsofσ M−2 and
A ≫ X
pair. Assume the monopole mass is M . E M probablyrequiressomeunusual
M X
h i ∼
In the early universe, the annihilation interactions of the dark matter. Of course
cross section is roughly M−2 [38]. It is we know essentially nothing about the in-
M
this cross section that would determine teractions of dark matter, in particular its
the present monopole abundance. How- self-interactions, which may not be com-
ever, monopole-antimonopole annihilation municated to the visible sector.
in the present universe is another matter We regard the requisite size of the an-
[39]. In the late universe, monopoles and nihilation cross section to be the most
antimonopoles can capture with a large unattractive feature of our proposal.
cross section and form monopolonium, a
monopole–antimonopole Rydberg state. It
3 Annihilation in the clumped com-
may take as long as the age of the uni-
ponent
verse for the state to lose energy by radi-
ation, becoming smaller and more tightly
So far we have assumed that the galac-
bound. Eventually the state will have a
tic dark matter is smoothly distributed. In
size of the inner core of the monopole (for
this section we will consider the contribu-
a GUT monopole, about α/M ). Then
M tion from inhomogeneities in the galactic
the monopole-antimonopole annihilates in
distribution.
a final burst of radiation.
We have modeled the smooth compo-
Theannihilationspectrumofmonopolo-
nent of the dark matter by a NFW profile,
nium contains many particles. There are
Eq. (2). In addition to this smooth compo-
many low-energy photons radiated during
nent, the dark matter may have a clumped
the long decay period, followed by a rapid
component, as suggested by N-body sim-
burst of high-energy jets (E αM ) in
∼ M ulations. Using the results of these simu-
the final death throes of annihilation.
lations, the number of subclumps of mass
The monopolonium example also illus-
M per unit volume at distance d fromthe
trates that the early-universe annihilation cl
galactic center can be written in the form:
cross section may be many orders of mag-
nitude different than the present annihi- M −α
lation cross section. However, for mag- ncl(d,Mcl)=n0cl Mcl
(cid:18) H(cid:19)
netic monopolesthe Parker boundontheir
2 −3/2
abundance imposes a further constraint d
1+ ,
[40]. This arises from the requirement that × Rcl! 
c
magnetic monopoles must not cause too
 
much depletion of the galactic magnetic whereRclisthecoreradiusofthesubclump
c
field and seems to rule out a noticeable distribution in the galaxy, typically of or-
UHE cosmic ray flux from their annihila- der10to20kpc.Themassofthehaloofthe
6
GalaxyisMH.Thenormalizationconstant 1 r0 dr
n0 can be calculated by requiring that the + (7)
cl ×3R r2
mass in the clumped component is a frac- min Z
 Rmin 
tion ξ of the halo mass MH. From simula- 16πn2 (d) σ v r04 ,
tions, ξ 10%. The power index, α, may ≈ 3 H h A iR
∼ min
also be found from simulations, with the
whereR istheminimumradiusofasub-
result α 1.9. It is then easy to find min
∼ clump(theinnerradiuswheretheprofileis
(2 α)ξ flattened by efficient annihilations).
n0 − ,
cl ≈ 4π(Rcl)3ln(R /Rcl)M η2−α OnewaytoestimateRmin wouldassume
c H c H
equality of the annihilation time scale
where η is the ratio of the mass of the
−1
τ =[n(r) σ v ]
largest subclump to the halo mass, ex- A A
h i
pected to be in the range 0.01 to 0.1, and 1 r 2
=
R is the radius of the halo. n (d) σ v r
H H h A i (cid:18) 0(cid:19)
Up to this point, all the discussion is in-
with the radial infall time
dependent of the density profile of the in-
dividual subclumps.
τ =r /v(r )
ff 0 0
Let us now assume that the density pro-
=[4πGM n (d)]−1/2,
fileofanindividualsubclumpisanisother- X H
mal sphere, with radial profile
see e.g. [42], where such an estimate was
applied to the central core of the galaxy.
r −2
n (r) = n (d) , (5) In the present setting this would yield a
X H
r
(cid:18) 0(cid:19) ratio
wherer0 istheradiusoftheindividualsub- R n (d) 1/4
min = σ v 1/2 H (8)
clump, defined as the radius at which the A
subclump density equals the density of the r0 (cid:12)(cid:12)τff=τA h i "4πGMX#
(cid:12)
backgroundhaloatthedistancedfromthe and an(cid:12) annihilation rate in a subclump of
galactic center. mass M at distance d from the galactic
cl
IfthesubclumpmassisM ,wecanwrite center:
cl
4
r0 (M ,d)= M σ v 1/2(4πG)1/4
M = 4πM n (d) drr2(r/r )−2, R cl 3 clh A i
cl X H 0
Z0 nH(d) 3/4
.
× MX !
from which we obtain
Nowwecancalculatethecontributionof
1/3
Mcl the clumped component in a similar man-
r = . (6)
0
"4πMXnH(d)# ner as Eq. (3):
The rate of annihilation in an individual = dN(E,MX) d3d 1
subclump at distance d from the galactic F dE 2π d d 2
Z ⊙
| − |
center is Mmax
dM n (d,M ) (d,M ). (9)
cl cl cl cl
(M ,d)=4πn2 (d) σ v r4 × Z R
R cl H h A i 0 Mmin
7
As before we will use an NFW profile for
n (d) with d = 24kpc.
H s
UsingRcl = 15kpc,η = 0.1,andξ = 0.1, which becomes
c
we find
4π r6
(M ,d) σ v 0n2 (d).
Fclumped 2 106 MX 1/2 R cl ≃ 3 h A irs3 H
≃ × 1012GeV
Fsmooth (cid:18) (cid:19) To calculate for NFW subclumps, we
3 10−26cm2 1/2 follow a procedFure similar to Eq. 9, but
× .
× hσAvi ! withR(Mcl,d)givenabove.Againexpress-
ing the flux in terms of the flux from the
To understand the scaling of the events smooth component, we find
from subclumps, we can make the simple
approximation that one is at the galactic clumped
F 2 103,
center( d⊙ = 0).Inthelimitthatthecon- smooth ≃ ×
| | F
tributions come from d Rcl and d d ,
≪ c ≪ s
we find independent of M or σ v . Again, this
X A
h i
result is represented in Fig. 2.
σ v 1/2M−1.
F ∝ h A i X
Now, we take the density profile of the
subclumps according to a NFW profile,
2
r [r +r ]
0 0 s
n (r) = n (d) , (10)
X H 2
r[r +r ]
s
where r is the fiducial radius of the sub-
s
clumpandagainr istheradiusofthesub-
0
clump where n (r) = n (d). We assume
X H
herethattheratior /r 1isconstantfor
0 s
≫
all subclumps. The mass of the subclump
Fig. 2. Theratio of events from the subclump
can be written in terms of r and r as
0 s component to the smooth component assum-
ing either an isothermal or a NFW profile for
r0
M =4πM drr2n (r) the subclumps. This assumes the ratio (8)
cl X X
Z0 The ratio of the fluxes from the sub-
r
4πM n (d)r3ln 0 . clump component and the smooth compo-
≃ X H 0 r
(cid:18) s(cid:19) nent are given in Fig. 2. Comparing Fig. 1
andFig.2,weseethatforM = 1012GeV,
which provides the core radius r for the X
0
tonormalizethefluxtotheobservedevents
subclump once we know r .
s
would require a cross section 103 times
The rate of annihilations for one sub-
smaller than the isotropic case if the wim-
clump is given by
pzilla subclumps have an NFW profile.
r0 For NFW subclumps the calculation of
R(Mcl,d)=4πhσAvi drr2n2X(r) the flux has a natural cutoff at small r.
RmZin This is not the case for isothermal sub-
+(4π/3)R3 σ v n2 (R ), clumps,andtheresultingfluxcouldbevery
minh A i H min
8
different with a different choice for R . much smaller than what found in Eq. (8).
min
For example, one might estimate R by With this estimate the rate of annihilation
min
requiring detailed hydrodynamic balance in an individual subclump at distance d
between the infalling and the annihilating from the galactic center is
matter in the central core of the dark mat-
1/2
ter clumps. In general we can write for the 2πGn (d)
H
(d,M ) 80M , (14)
change in density of a radially symmetric R cl ≈ cl" MX #
dark matter clump
and the flux is
∂ ∂
n (r,t)= (n (r,t)v(r,t)) (11)
X X d (E,M )
∂t ∂r N X 4.2m−2s−1
clumped
2 F ≈ dE ×
+ n (r,t)v(r,t)
r X ξ
.
−n2X(r,t)hσAvi, × 0.1!
where v(r,t) is the velocity of radial infall. This is by about a factor 1015 too large,
Detailed balance at r = R amounts and can be reconciled with the observed
min
to the requirement UHEcosmicrayspectrum onlyifthewim-
pzillacontributiontoΩ amountsonly
1 dark
v(R ) = R n (R ) σ v . (12) to a small admixture 10−15 of annihilat-
min 3 min X min h A i ing wimpzillas in su∼b-galactic dark mat-
Such a requirement was applied in [43] to ter clumps.
estimate the core radii of neutralino stars. Note that this result is independent of
For a particle of energy E = 0 we find the annihilation cross section σ v , and
A
h i
fromenergyconservationorfromthequasi- therefore also complies with annihilation
stationary Euler equation in [43] crosssectionswellbelowthes-waveunitar-
itylimit.Withtheestimate(12)asmallan-
v2(r) = 2U(r), nihilationcrosssectionimpliesaverysmall
−
denselypackedcoreregionofthedarkmat-
where the potential U(r) in the clump can
ter clumps. The high density increases the
be calculated as a solution of the Poisson
luminosity of the clumps and compensates
equation:
for the small factor σ v in the local an-
A
h i
nihilation rate.
2πGM
X
U(r)= (13)
r
r0
4 Predicted signals
′ ′ ′ ′ ′
dr n (r )r ( r r r r ).
X
× | − |− −
Z0
The annihilation rate is very sensitive
This yields to the local density. This implies that if
the UHE cosmic rays originate from dark
r
U(R ) 4πGM n (d)r2ln 0 matterannihilationanddarkmatterinour
min ≈ − X H 0 R
(cid:18) min(cid:19) galaxy is clumped, the arrival direction of
and amounts to a ratio UHE cosmic rays should reflect the dark
matter distribution.
1/2
R n (d) Thefirstpossibilityweconsideredisthat
min H
0.1 σ v ,
A
r0 (cid:12)balance ≈ h i"72πGMX# thedarkmatterdistributionissmoothand
(cid:12)
(cid:12)
(cid:12)
9
follows a NFW profile. In this case, the
galactic center should be quite prominent.
In Fig. 3 we illustrate the expected angu-
lardependenceofthearrivaldirection.The
galactic center is prominent for the decay
scenariowhere n [12],andevenmore
X
F ∝
prominent for the annihilation case where
n2 .
F ∝ X
Fig. 4. The probability of finding a certain
number of events in a one square degree area
of the sky
subclumps that would result in about 100
detected events. We see that the average
probabilities are quite reasonable, with
most events in square degree areas with
onlyoneevent, andafewpairsandtriplets
ofevents.Howeverifweexamineindividual
Fig. 3. The angular dependence of events
realizations containing about 100 events,
originating from annihilation or decay of the
smoothcomponentofdarkmatterthatfollows we see that there is a large probability of a
a NFW profile. In this figure, cosθ = 1 corre- large number of events from single nearby
sponds to the direction of the galactic center. subclumps. For instance, in the single re-
alizations shown in Fig. 4, the isothermal
Now if we assume that there are isother-
subclumps result in a square degree bin
mal dark matter subclumps within the
with 14 events and a square degree bin
galaxy, then the density in the subclumps
with 33 events. If morethan100 events are
would be larger than the ambient back-
observed with full sky coverage, the signa-
ground dark-matter density, and events
ture of subclumps will be unmistakable.
originating from subclumps will dominate
the observed signal.
The dominance of events from sub- 5 Conclusion
clumps has two effects. The first effect is
that a smaller annihilation cross section is The origin of UHE cosmic rays from an-
required to account for the observed UHE nihilation (or decay) of wimpzillas has
flux. The second effect is that there will the attractive feature of the simplicity of
be a very large probability of detecting a generating ultrahigh energies: simple con-
nearby subclump. version of rest mass energy.
InFig.4wepresent ahistogramshowing UHEcosmicraysfromdecayofasmooth
the number of occurrences of single and relic superheavy dark matter population
multiple events in arrival directions within in the halo requires a lifetime of about
a square degree. We generated many real- 1020years or a correspondingly reduced
izations of the expected subclump distri- contribution of the superheavy relics to
bution, and calculated the flux from the Ω .
dark
10