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Year:2010
Impactofdarkmattermicrohalosonsignaturesfordirectandindirect
detection
Schneider,A;Krauss,L;Moore,B
Abstract:DetectingdarkmatterasitstreamsthroughdetectorsonEarthreliesonknowledgeofits
phase space density on a scalecomparable to the size of ourSolar System.Numerical simulations
predictthatourgalactichalocontainsanenormoushierarchyofsubstructures,streamsandcaustics,
theremnantsof themerging hierarchythat beganwithtinyEarth-massmicrohalos.Ifthesebound
orcoherentstructurespersistuntilthepresenttime, theycoulddramaticallyaltersignaturesforthe
detectionofweaklyinteractingelementaryparticledarkmatter.Usingnumericalsimulationsthatfollow
thecoarsegrainedtidaldisruptionwithintheGalacticpotentialandnegrainedheatingfromstellar
encounters,wendthatmicrohalos,streams,andcausticshaveanegligiblelikelihoodofimpactingdirect
detectionsignaturesimplyingthatdarkmatterconstraintsderivedusingsimplesmoothhalomodelsare
relativelyrobust.Wealsondthatmanydensecentralcuspssurvive,yieldingasmallenhancementin
thesignalforindirectdetectionexperiments.
DOI:https://doi.org/10.1103/PhysRevD.82.063525
PostedattheZurichOpenRepositoryandArchive,UniversityofZurich
ZORAURL:https://doi.org/10.5167/uzh-41501
JournalArticle
AcceptedVersion
Originallypublishedat:
Schneider,A;Krauss,L;Moore,B(2010).Impactofdarkmattermicrohalosonsignaturesfordirect
andindirectdetection.PhysicalReviewD,82(6):063525.
DOI:https://doi.org/10.1103/PhysRevD.82.063525
Impact of Dark Matter Microhalos on Signatures for Direct and Indirect Detection
Aurel Schneider
1
, Lawrence Krauss
1,2
and Ben Moore
1
1
Institute for Theoretical Physics, University of Zurich, Zurich,
Switzerland
2
School of Earth and Space Exploration and Department of Phy s i cs ,
Arizona State University, PO Box 871404, Tempe, AZ 85287;
(Dated: August 16, 2010)
Detecting dark matter as it streams through detectors on Earth relies on knowledge of its phase
space density on a scale comparable to the size of our solar system. Numerical simulations predict
that our Galactic halo contains an enormous hierarchy of subst ru c tu res , streams and caustics, the
remnants of the merging hierarchy that began with tiny Earth mass microhalos. If these bound or
coherent structures persist until the present time, they could dramatically a lt er signatures for the
detection of weakly interacting elementary particle dark m a tt er (WIMP). Using numerical simula-
tions tha t follow the coarse grained tidal disruption within the Galactic potential and fine grained
heating from stellar encounters, we find that microhalos, streams and caustics have a n eg l ig i b le
likelihood of impacting direct detection signatures implying that dark matter constraints derived
using simple smo o t h halo models are relatively robust. We also find that many dense central cusp s
survive, yielding a small enhancement in the signal for indirect detection experiments.
PACS numbers: 98.80
Introduction
In a ΛCDM dominated universe, structure forms by the
hierarchical clustering and merging of small density per-
turbations [1]. Numerical simulations that follow these
processes predict that our Galactic halo should contain a
vast hierarchy of surviving substructures - th e remnants
of the entire halo merger tree [2]. The number density
of substructures of a given mass M goes as n ∝ M
−1
subs
and they span over 15 decades in mass [3]. The small-
est, oldest and most abundant are Earth-mass microha-
los with a half mass radius of 10
−2
pc that formed at
z ≃ 80 − 30 [4–8]. This minimum mass is modulated by
the free streaming velocity which is related to the mass
of the neutralino [9, 10].
Simulations of relatively large subhalos suggest that
their gravitational interactions with a disk potential, can
lead to a destruction of subhalos at distances closer than
30 kpc [11]. Smaller subhalos form earlier, however, with
denser cores, and are therefore the most probable dark
matter structures to survi ve gravitational interactions.
A second source of fine grained structure to survive are
the numerous caustic sheets and folds that form due to
the very high initial phase space density of the cold dark
matter particles [12]. These are wrapped in a complex
way within all the subsequent structures that form, how-
ever in th e absence of a heating term, the fine grained
phase density would be preserved.
With low int e rna l velocity dispersion and high mean
density, both the event rate and the characteristic spec-
trum of energy deposited by dark matter in direct de-
tection experiments [13] could be affected by any fea-
tures survivin g in the phase space distribution of CDM
particles. Direct detecti on experiments are sensitive to
the density and velocity distributions of WIMPs on a
scale of ≈ 10
13
m, the distanc e the Earth travels over a
year. In order to make predictions and exclusion lim-
its, these experiments assume that the dark matter i s
completely smooth on these scales, with a well mixed
Maxwell-Boltzmann velocity distribution [14, 15]. Fur-
thermore, if any such small high-density clumps actually
dominate the dark matter distribution in the solar neigh-
borhood, the indirect detection signal due to dark matter
annihilation in the galaxy might also be affected.
Since existing N-body simulations of galaxy formation
do not have a resolution that goes down to objects with
mass as small as 10
−6
M
⊙
, in order to address the ques-
tion of the survival and impact of such microhalos we
need to combine analytical esti mat e s with the results of
smaller scale simulations that can resolve such objects.
Previous studies have made analytic [16] and numerical
estimates [17] of the d is ru pt ion timescale of microhaloes
as they orbit through the stellar field. Zhao et. al. [16]
argued that most of the microhaloes should be completely
destroyed by encounters with stars, whilst Goerdt et. al.
[17] show that indeed, whilst most of the mass is unbound
the dense central cusp may survive intact.
We extend this work by numerical calculating the tidal
disruption of microhaloes as they actually orbit through
a field of stars, as we l l as se lf -c ons i st ently including the
Galactic halo potential. We calculate t h e survival statis-
tics of microhaloes using realistic orbital distributions
within the disk, allowing us to follow the dynamical struc-
ture of the dark matter streams and thus to estimate the
fine grained phase space distribution function of WIMPs
on scales relevant to dark matter detection experiments.
Microhalo parameters and disruption processes
The dominant processes that can affect microhalos in-
volve gravitational interactions with baryons in the stel-
lar field during the crossing of the disk and also tidal
effects of the disk potential during the orbit of the micro-
arXiv:1004.5432v2 [astro-ph.GA] 13 Aug 2010
2
halo. Unfortunately we cannot account for both effects
simultaneously in our simulations as it would require set-
ting up a self-consistent disk with billions of stars. In-
stead we look separately at the effects of stellar disruption
and tidal streaming and we t h en estimate the combined
behavior.
As a substructure halo crosses t hrou gh a stellar field,
high-speed interactions with single stars will heat up the
halo distribution, causing it to increase its velocity dis-
persion and hence its scale size will grow. This process
is analogous to galaxy harassment that occurs in clus-
ters [18] and basically has a timescale p roportional to
the relaxation time of the stellar disk and the time e ach
microhalo spends within the fluctuating potential fiel d of
the stars. For an analytical estimation of t h i s process we
follow the Goerdt et. al. paper [17].
In th e ’dis t ant-tide’ app roximation [ 19] the internal en-
ergy increase of the microhalo due to a single encounter
with a fixed star is given by
δE(b) =
1
2
2GM
∗
b
2
V
mh
2
2
3
hr
2
i, (1)
where b is the impact parameter and V
mh
is the velocity
of the microhalo. Since halos with an early formation
time have a low concentration we can set hri ≈ 0.5r
v ir
.
One encounter can totally disrupt the microhalo if δE
exceeds the bi n di n g energy E
b
. Since E
b
≈ 0.4v
2
v ir
with
v
2
v ir
= Gm
v ir
/r
v ir
[20], the minimal encounter parameter
that does not entirely disrupt the microhalo is found to
be
b
min
≈ 0.8
GM
∗
r
v ir
V
mh
v
v ir
1/2
. (2)
We can now define the disruption probability of a micro-
halo in a stellar field
p =
1
E
b
Z
δEdN =
Z
b
min
0
dN +
1
E
b
Z
∞
b
min
δEdN, (3)
where dN = 2πnbdbV
mh
dt. Here we have used δE =
E
b
for b < b
min
. Performing the int egr at ion leads to a
disruption probability of
p ≈ 4GM
∗
n
∗
t
r
v ir
v
v ir
=
2GM
∗
n
∗
t
5H
0
Ω
1/2
m0
(1 + z)
−3/2
, (4)
where we have used the definition of the virial radius
M
v ir
=
4π
3
r
3
v ir
200ρ
c
with ρ
c
=
3H
2
8π G
. The microhalo is
completely destroyed at p = 1. Therefore we get the
average disruption time
t = 250
0.04M
⊙
pc
−3
M
∗
n
∗
1 + z
61
3/2
Myr. (5)
A microhalo with a formation redshift z ∼ 60 should
therefore survive about 250 Myr in a stellar field wi th
a density simil ar to the one in the solar neighbourhod.
However this is only true on average, since one very close
encounter can immediately lead to total disruption.
The above estimate does not take into account the in-
ternal structure of the microhalos and should therefore
only give a very rough estimation of t he disrupt i on time.
Also it does not allow one to follow the mass l oss during
the disruption process. We therefore perform a simula-
tion where a microhalo is crossing a periodic box of stars.
The box has a length of 50 pc and is filled with randomly
distributed stars with th e density ρ = 0.04M
⊙
pc
−3
and
the velocity dispersion σ = 50 km/s. This constellation
corresponds to the stellar field in the disk at the solar
radius [19]. For simplicity all the stars have the average
mass of 0.7 M
⊙
. The microhalo which is crossing the box
at 200 km/s is set up by the halogen-code of Zemp et.
al. [21] . Corresponding to the results in [7] it has a mass
of 10
−6
M
⊙
and the density profile
ρ(r) ∝
1
r
r
s
γ
1 +
r
r
s
α
β−γ
α
(6)
with α = 1, β = 3 and γ = 1.2, as well as a concentration
parameter of c = r
200
/r
s
= 1.6. The virial radius r
200
is
defined with respect to the background density at z = 60,
the average formation redshift of a microhalo [4].
In our simulation we find that 50% of the microhalo
mass is unboun d after 80 Myr of box-crossing ( wh ich
corresponds to about 40 perpendicular disk passages).
After 160 Myr (80 disk passages) even the central core
starts to disappear and more than 90% of the microhalo
is complet e l y disrupt ed (see pictures in Table I). At latest
after 200 Myr (100 disk passsages) no bound structure is
left (see Fig 1). Our simulation gives therefore a slightly
shorter disruption time than the simp li fi ed analytical es-
timation of equation (5).
In order to determine what fraction of microh alos sur-
vive until the present day, we have to calculate orbital
statistics and the distribution of disk crossing times. This
can be established by tracing back the orbits of particles
in a galactic potential. We use the standard Milky Way
model with disk and halo part icles set up by the G al ac-
tICS code [22] and we select a sample of halo particles in
a small box around the position of the sun. The orbits
of the se particles are followed backwards in time and we
find that the average number of disk crossings for these
particles is
c = 80 with a standard deviation of σ
c
= 43.
The average crossing rad i us is (not surprisingly) R = 8
kpc with σ
R
= 4 kpc. The spread of disk crossing events
for different particles follows a Maxwell-Boltzmann dis-
tribution.
We use this disk crossing distribution combined with
the rate of mass loss determin ed from ou r numerical
study to calculate the surv ival statistics of microhalos
in the vicinity of the s un . Since the timescale for com-
plete disruption in our simulation is equivalent to the
average time a microhalo spends in the stellar disk, we
conclude that the average microhalo in the vicinity of the
sun is just about to be entirely destroyed at the present
3
TABLE I: Microhalo density map at t = 0, 20, 40, 60, 80, 100,
120, 140, 160 Myr (from the upper left to the lower right).
The boxlength of the images is 0.3 8 pc.
time (see also [23]). At most five percent of its initial
mass is still in a bound core. However the spread i n the
number of disc crossings is relatively wide an d a signi f-
icant fraction of microhalos should still have survivi ng
cores. Mass loss is nevertheless important: microhalos
maintaining more than 50% of their initial mass should
be rare. Figure 1 illustrates the mass loss, where the red
curve shows the d is ru p t ion of a typical microhalo with
80 disk crossings in 10 Gyr at the radius of the sun.
However, disk crossing is not the onl y source of dynam-
ical disruption. W h il e orbit in g the galaxy, a microhalo
is under the constant influence of the global Galactic po-
tential, and tidal forces will act so that the microhalo’s
structure b ec om es elongated and unbound particles will
form le adi ng and trailing tidal streams. The detailed
impact of tidal streaming depends on the orbit of the mi-
crohalo and on the shape of the host potential. In our
simulations we use a disk potential that emerges from a
density distribution of the form
ρ(R, z) ∝ exp(−R/R
d
)sech
2
(z/z
d
). (7)
Here R and z are the disk radius and the hei ght respec-
tively, which we set to be R
d
= 2.8 kpc, z
d
= 0.4 kpc.
The disk mass i s M
d
= 4.5 · 10
10
M
⊙
. In al l our simula-
tions the orbit of the microhalo is chosen to be roughly
spherical with a distance of 7.9 kpc from the galactic
center.
We cannot model both heating due to stellar inter-
actions and tidal elongation at the same time since this
would require following the motion of 50 bill ion disk stars.
We therefore performed orbit al simulations for three dif-
0
2
4
6
8
10
12
0.0
0.2
0.4
0.6
0.8
1.0
t @GyrD
N
bound
N
tot
FIG. 1: Ratio between bound and total mass of a microhalo
crossing a stellar field (red) and orbiting in a Mi lky Wa y po-
tential (black, grey). The red curve is stretched in order to
simulate the effect of disk crossing on an average microhalo
with about 80 disk crossings. The black and the grey curve cor-
respond to a microhalo that spent 0 Myr respectively 80 Myr in
the stellar field before orbiting around the galactic potential.
The dominant mass loss is coming from stellar interaction
until complete disruption after about 12 Gyr.
ferent cases: an initially completely un d is t ur bed micro-
halo, a microhalo that first crossed the stellar field for
80 Myr and has lost about 60 percent of its mass, and
a completely disrupted microhalo that spent more than
160 Myr in the stellar field.
The length of the tidal streams l due to the orbit-
ing process can be crudely estimated with the relati on
l(t) ∼ σ
mh
t, where σ
mh
is the velocity dispersion of t h e
initial microhalo. For the initially unperturbed microhalo
σ
mh
∼ 10
−3
km/s, causing a stream length of roughly
l ∼ 10 pc after one Hubble time. For the initially c om-
pletely disrupted microhalo σ
mh
∼ 10
−2
km/s, and the
stream length is about l ∼ 100 pc after a Hub b le time.
These length scales agr ee well with our simulations (i.e.
see Table II).
Orbiting in the galactic potential significantly reduces
the mass of the microhalo (see black and grey lines i n Fig
1). However, the rate of tidal mass loss is s up pr e sse d as
the ti d al radius is steadily red u ced . The ce ntral cusp of
each dark matter microhalo has a very deep potential, as
a consequence there is always a bound core remaining,
even for a microhalo that has been heated in t h e stellar
field before orbiting.
Comparing the curves in Fig 1 leads to the conclusion
that disk crossing is the domin ant disruption process and
the only one that can lead to complete distruc t i on of
the microhalo. The step-like d ec r ease of the curve is an
indication of very close encounters that pl ay a mayor role
in the disruption process. Tidal stri pp i ng on the other
hand can also significantly reduc e the mass but it never
completely di s tr oys the microhalo because of its tightly
bound inner core.
4
TABLE II: Streaming microhalo after 10 Gyr on a roughly
circular orbit around a Milky Way potential: The two images
on the top show the sheet-like streams from t he top and from
the side (boxlength: 30 pc). The third image is a zoom in at
the ce ntre where the still bound core is visible (boxleng th: 2
pc). For these pictures we have used an initially non disr u p ted
microhalo with formation redshift 60.
Implications for Dark Matter Detection
In direct detection experiments the differential inter-
action rate is sensitive t o the fine grained density and
the velocity distribution of dark matter particles on A.U.
scales [24, 25]. Substructures like microhalos can affect
the interaction rate if they are abundant enough to h ave
a substantial likelihood of existing in the solar neigh-
bourho od and if their density is at least the same order
of magnitude as the background dark matter density in
this region, ρ
bg
∼ 10
7
M
⊙
kpc
−3
(see for example [26]).
Equally important, the phase space for the energy de-
posits associated with dark matter events will not be that
appropriate for an isothermal halo if a single microhalo
were to dominate the density distribution i n t h e solar
neighbourhood [13, 27].
Our results above suggest that none of these condi-
tions are generally achieved. In Figure 2 we plot the
stream densities of microhalos that crossed the stellar
field for 0 Myr (red), 80 Myr (gre y) and 160 Myr (blue),
before orbiting in the galactic potential for 10 Gyr. The
tidal streams of the initially unperturbed halo (red) have
an average density of ρ ∼ 10
4
M
⊙
kpc
−3
, which is al-
ready negligibly low compared to the background. Only
the very tiny core still maintains its initial density of
ρ ≈ 10
11
M
⊙
kpc
−3
. The ini ti al l y disrupted microhalo
(blue) has no more bound core. Its stream density is
only at about ρ ∼ 10
2
− 10
3
M
⊙
kpc
−3
. The approach
of first measuring the stellar disruption and then looking
at tidal effects underestimates the stream density, since
the microhalos get most of their heat energy r ight at the
beginning. Therefore, the actual str eam density of an
average microhalo should be somewhe re between the red
and the blue line in Figure 2.
Since the stream densities are far below the value of
the local galactic density, only a surviving core existing
in the region of the earth would any effect upon direct
10
-7
10
-5
0.001
0.1
10
-6
0.001
1
1000
10
6
10
9
10
12
r @kpcD
Ρ @Mo kpc
-3
D
FIG. 2: Stream densities of microh a lo s after an orbital time of
ten Gyr. Before orbiting the micro h al os have spent a time of
0 Myr (red), 80 Myr (grey) and 160 Myr (blue) in the stellar
field. There is still a visible bound core in the red and the grey
profile but no more in the blue one. Th e black dots represent
the density profile of a completely undisrupted microhalo that
spend no time on an orbit. The straight blue line cor responds
to t he average dark matter density around the sun.
detection. However, only about half of the microh al os
still have bound cores because of disk crossing, and tidal
effects further reduce the mass of the cores to less than
ten percent of their original value. We note that any sub-
structures orbiting primarily within the disk plan e woul d
be qu i ckly destroyed by stellar encounters.
The chance of being in such an overde ns e region can
then be optimistically estimated: An extrapolation of
the subhalo mass function leads to a microhalo number
density n
mh
of about 500 pc
−3
at the solar radius [7].
This number can be divided by two due to the disruption
processes stated above and again by two since microhalos
orbiting in the disk plane are completely disrupted. We
then end up with the approximation of n
mh
∼ 100 pc
−3
at the solar radiu s. Each microhalo has a volume of about
V
mh
∼ 10
−9
pc
3
and therefore there is a chance of about
0.0001% of being in such an overdense region.
The streams of particles stripped from microhalos are
coherent and long, thus it is appropriate t o calculate
their volume filling factor. Since the stream den si ty is
ρ ∼ 10
2
− 10
4
M
⊙
kpc
−3
we ex pect that our solar sys-
tem is criss -c ros sed with f
b
×(10
3
−10
5
) streams, where
f
b
≈ 0.1 is the frac ti on of the local Galactic halo den-
sity that forms from substructure s up to a solar mass.
Larger substructures may be completely disrupt ed at the
Sun’s position in the Galaxy due to global disk shocking
and tides [11]. The velocity dispersion within an average
stream due to heating by disk stars is σ ∼ 10
−2
km/s.
Thus, the local density is determined by the superposi -
tion of a lar ge number of in de pendent streams, and the
overall velocity distribution at the solar radius should
be essentially Maxwellian, isotropic and smooth with no
spiky structure, as we would assume for a s mooth halo
model with no substructures. The signatures of streams
