Draft version November 16, 2018
Preprint typeset using LATEX style emulateapj v. 11/26/04
DARK MATTER CAUSTICS AND THE ENHANCEMENT OF SELF-ANNIHILATION FLUX
Roya C. Mohayaee
, Sergei Shandarin
, Joseph Silk
Draft version November 16, 2018
ABSTRACT
Cold dark matter haloes are populated by caustics, which are yet to be resolved in N-body
simulations or observed in the Universe. Secondary infall model provides a paradigm for the
study of caustics in typical haloes assuming that they have had no major mergers and have
grown only by smooth accretion. This is a particular characteristic of the smallest dark
matter haloes of about 10−5 M⊙, which although atypical contain no substructures and
could have survived until now with no major mergers. Thus using this model as the first
guidline, we evaluate the neutralino self-annihilation flux for these haloes. Our results show
that caustics could leave a distinct sawteeth signature on the differential and cumulative
fluxes coming from the outer regions of these haloes. The total annihilation signal from the
regions away from the centre can be boosted by about forty percents.
Subject headings: dark matter haloes, caustics, dark matter detection
1. INTRODUCTION
Evidence from the rotation curves of galaxies,
gravitational lensing, microwave background radi-
ation, peculiar velocity fields, and many other ob-
servations indicate that the visible mass, in the form
of stars and hot gas, is only a small fraction of the
total content of the Universe. The nature of the
missing mass, the dark matter, remains unknown
but is widely presumed to be Weakly Interacting
Massive Particles (WIMPs), such as the lightest su-
persymmetric particles, which are yet to be detected
in particle accelerators (Jungman, Kamionkowski,
& Griest 1996 ; Bertone, Hooper & Silk 2004).
Accelerator searches are complemented by the
vast experimental efforts to detect these particles in
our galaxy and in nearby galaxies which are believed
to be embedded in dark matter haloes (Ostriker
& Peebles 1973). Such complementary techniques
presently involve direct detection in low background
laboratory detectors (Goodman & Witten 1985)
and indirect detection through observation of en-
ergetic neutrinos, gamma rays and other products
of self-annihilation of dark matter particles (Silk &
Srednicki 1984) .
The event rate for self-annihilation depends
quadratically on the local dark matter density,
which falls off with distance from the center of the
halo. The averaged halo density profile obtained in
various numerical simulations diverges at the cen-
tre but is otherwise smooth and is often fit with a(n
asymptotically) double power-law (Navarro, Frenk
& White 1996, Moore et al 1998). However, a con-
sensus on the precise values of the power exponents,
the size of the central core and the resolution of fine
high-density structures are yet to be achieved. The
fine structures, the caustics, are inevitable outcomes
1 Institut d’astrophysique de Paris, 98 bis boulevard
Arago, France
2 Department of Physics and Astronomy, University of
Kansas, KS 66045, U.S.A.
3 University of Oxford, Astrophysics, Keble Road, Oxford
OX1 3RH, U.K.
of the evolution of a collisionless self-gravitating sys-
tem described by the Jeans-Vlassov-Poisson equa-
tion (for a one-dimensional numerical result see
Alard & Colombi 2005). Formally, in three dimen-
sions, the most common caustics are surfaces of zero
thicknesses over which the density diverges. 4 How-
ever, a maximum cut-off to their density is set by
the finite non-negligible velocity dispersion of dark
matter particles. Their density however remains
very high and hence they can be significant for dark
matter search experiments (Sikivie & Ipser 1992,
Sikivie et al 1997, Natarajan 2007). The effect of
velocity dispersion in the smearing of the caustics is
expected to dominate over other effects such as par-
ticle discreteness which would also smooth the caus-
tics but to a far lesser degree. Mergers of haloes can
also smear out the caustics substantially and due to
this fact we restrict our study to haloes that have
grown by slow and smooth accretion. Nevertheless,
caustics are robust, in that while they may break
up into micro-caustics, they remain in the fine-scale
halo substructure and thereby contribute to the gen-
eral clumpiness boost of any annihilation signal.
Analytic studies of the formation of haloes and
caustics have been carried out mainly under vari-
ous simplifying assumptions, such as spherical sym-
metry, self-similarity, and cold and smooth accre-
tion (Gott 1975, Gunn 1977, Fillmore & Goldreich
1984, Bertschinger 1985). In an Einstein-de Sit-
ter Universe a spherical overdensity expands and
then turns around to collapse. After collapse and
at late times, the fluid motion becomes self-similar:
its form remains unchanged when its length is re-
scaled in terms of the radius, rta, of the shell that
is currently at turn-around and is falling onto the
4 The general theory of singularities (Arnol’d, Shandarin&
Zel’dovich 1982) also predicts singularities on lines and at
points. Despite the greater concentration of mass in these
singularities they probably play a less important role in the
total annihilation rate because they contain a considerably
smaller amount of mass. However, this has not been studied
in detail.
http://arxiv.org/abs/0704.1999v1
galaxy for the first time. Physically, self-similarity
arises because gravity is scale-free and because mass
shells outside the initial overdensity are also bound
and turn around at successively later times. Self-
similar solutions give power-law density profiles on
the scale of the halo which provides an explanation
for the flattening of the rotation curves of galaxies.
However, on smaller scales the density profile con-
tains many spikes (i.e. caustics) of infinite density.
The position and time of formation of these caus-
tics are among the many properties that have been
established in the framework of the self-similar in-
fall model (Fillmore & Goldreich 1984, Bertschinger
1985).
In reality, dark matter has a small velocity dis-
persion and haloes do suffer from major mergers
and non-sphericity. However, until numerical simu-
lations achieve sufficient resolution, the self-similar
accretion model provides a useful guideline to haloes
which have not undergone major mergers.
Here, we use the self-similar model of halo for-
mation and a further elaboration which includes
the velocity dispersion of dark matter (Mohayaee
& Shandarin 2006) as a first guidline to describe
the evolution of the smallest haloes which have sur-
vived major merger and disruption until now and
have grown only by slow accretion.
The application of self-similar model to such
haloes can be viewed from two contradictory angles.
One might assume that minihaloes are expected to
be well-represented by this model, since they con-
tain no substructures, have not undergone merger
and grow very slowly only by smooth mass accre-
tion. On the other hand, minihaloes are not typical
haloes and selfsimilar accretion model is formulated
to describe the evolution of a characteristic halo.
Keeping both of these issues in mind, we use self-
similar model only as a first guidline for the evolu-
tion of minihaloes. A large number of them have
been found in simulations (Diemand et al 2005).
The simulations estimate the size of these haloes
to be of about 0.01 pc (half mass radius) and their
mass of 10−6M⊙ at z = 26. Due to resolution prob-
lems, these simulations are stopped at this redshift
and typical evolution of galactic scale haloes is ex-
tended to minihaloes and the conclusion is drawn
that about 1015 of these haloes could exist in the
halo of MilkWay today. We assume that at least
a fraction of these haloes have evolved by slow ac-
cretion model from z = 26 until now and use the
selfsimilar model to evaluate their radius and mass
at z = 0, which are respectively 1 pc and 10−5
M⊙. For these haloes and working self-consistently
within our model including the contribution from
the caustics, we demonstrate that in the outer re-
gions of these haloes caustics can boost the annihi-
lation signal by about 40%.
2. SECONDARY-INFALL MODEL WITH VELOCITY
DISPERSION
The haloes considered in this work grow by
smooth and slow accretion. A good example are
the earth-mass haloes which were recently resolved
(at z=26) in numerical simulations (Diemand et al
Fig. 1.— A surface-contour plot of the caustic density.
In the self-similar model, caustics form concentric shells of
increasing density and decreasing thicknesses and separations
as we approach the center of the halo.
2005) and which although small are expected to
have clean spherical caustics. We expect that at
least a fraction of these haloes have survive disrup-
tion and major merger and grow by self-similar ac-
cretion model to a virial radius of about 1 pc and a
mass of about 10−5 M⊙.
To comply with the requirement of slow accretion,
we fix the value of the parameter ǫ in the initial den-
sity perturbation δ ∼ M−ǫi , where Mi is the initial
mass, to unity. We emphasis that the self-similar
model aims at describing the evolution of a typi-
cal halo. Typical haloes have mass variance (σ(M)
which varies as M−(n+3)/6, which sets ǫ = (n+3)/6,
where n is the power spectrum index. A typical
σ(M) fluctuation grows as t2/(3ǫ). Minihaloes cor-
respond to the limit n → −3 part of the spectrum.
For this part of the spectrum, there are mass fluc-
tuations of comparable amplitude on all scales and
consequently adiabatic invariance does not apply for
such fluctuations.
Hence, we use the self-similar model only as a
first guidline for the growth of minihaloes. We as-
sume that minihaloes of mass 10−6M⊙ have grown
by very small accretion from z = 26 to z = 0. Once
again, slow accretion corresponds to the case of
ǫ = 1 in the work of Fillmore and Goldreich (1984),
hence we shall adopt this value for ǫ.
The self-similar density profile is given by
(Bertschinger 1985)
(−1)jexp
where ρ̄ is the critical density and
is the dimensionless radius and r is the physical ra-
dius and ξi = ln(t/tta) is the dimensionless time
given in terms of the turnaround time, tta, of the
particle that is at the jth point where λ = λ(ξ) (see
Bertschinger 1985 for further explanation).
The density (1) is evaluated numerically and plot-
ted in Fig. 2 after an appropriate cut-off of the caus-
tics which shall be discussed now. In principle the
0.06 0.1 0.2
1e+02
1e+03
1e+04
1e+05
1e+06
2nd caustic 1st caustic
Axion
Neutralinoρ
λ2 λ1
Fig. 2.— The plot is made by numerically solving (1) and
cutting the caustics using (3). The density of halos (divided
by the critical density ρ̄) can be enhanced significantly at the
caustics. This enhancement is far larger for axions (dotted
violet spikes) than for neutralinos (continuous black line).
The number of the streams increases to the center of the halo
which explains the rapid growth of the smooth component of
the density profile and the caustic contribution to the halo
density diminishes. The dashed-dotted red line marked ρss is
the approximate self-similar density profile given by (6). The
inset shows a magnified view of the second and third caustics
for neutralinos.
density at the caustics diverges if the velocity dis-
persion of dark matter is zero. In the presence of a
small velocity dispersion the maximum density and
thickness of the caustic shells and their density pro-
files have been evaluated (Mohayaee & Shandarin
2006). The maximum density at the caustics and
their profile are given by
ρcaustic,k =
|∆λk|
ρ̄ λk − |∆λk| < λ < λk
ρ̄ λ < λk − |∆λk|
where λk is the non-dimensional radius of the kth
caustic counted inwards and
−2λ′′k
e−2ξk/3
and the thickness of the caustic shell is given by
∆λk =
(3π)2/3e5ξk/9Λk
t σ(t)
, (5)
where t is the age of the Universe, σ is the present-
day velocity dispersion of dark matter particles
which is that at decoupling re-scaled with the ex-
pansion factor. The values of these parameters vary
from one caustic to another (see Table 1 of Mo-
hayaee & Shandarin 2006 for the first ten caustics).
The profile (1) together with appropriate cut-off
given by (3) is plotted in Fig. 2.
The peaked density profile given by (1) and shown
in Fig. 2 has to be evaluated numerically. However,
as is evident from Fig. 2 a “self-similar” profile5 is
5 This profile can also be well-fitted by a power-law and
an exponential cut-off
reached which we fit with
ρss =
2.8λ−9/4
(1 + λ3/4)2
ρ̄ , (6)
as shown in Fig. 2 by the dashed-dotted red line,
marked ρss. The turnaround radius, rta can now be
evaluated by considering that at the virial radius the
density is about 200 times the background density,
and is given by rta ∼ 4rvir, which corresponds to
the density profile given by (6). This approximate
profile has been shown to be a good fit also to the
mass profile (see Mohayaee & Shandarin 2006). In
the next section we shall show that using this profile
which ignores the caustics would yield an under-
estimated value for the flux.
Both the extrapolated numerical and the approx-
imate density profiles shown in Fig. 2 formally di-
verge at the centre. However, due to finite dark
matter velocity dispersion, haloes can develop cen-
tral cores. Dark matter haloes are expected to have
central cores due to the dark matter velocity disper-
sion, self-annihilations at the centre, angular mo-
mentum, tidal and various other effects. The core
could be very small and the minimum scale asso-
ciated with a generic dark matter merging history
would conserve traces of the original cores in the
initial substructure. These should be of order the
free-streaming mass as for example computed in
Bertschinger (2006).
In principle for small core sizes the total flux from
the whole of the halo is dominated by the annihi-
lation in the centre of the halo and the boost due
to caustics is negligible. However, we shall show in
the next section that the differential (similarly cu-
mulative) flux would be distinctly marked by the
caustics and shall have a sawteeth pattern and the
contribution to the total flux from the outer region
of these haloes by the caustics is significant and can
yield a boost factor of about 40%.
3. THE FLUX DUE TO SELF-ANNIHILATION
INCLUDING THE EFFECT OF CAUSTICS
Caustics if detected would be clear evidence of
the existence of dark matter and could rule out al-
ternative models of gravity. Two major methods
for their detection are through gravitational lensing
(see e.g. Gavazzi et al 2006) and the flux of dark
matter annihilation product which is expected to
be significantly enhanced by the caustics. Here we
shall discuss the second method.
The flux of the self-annihilation product (e.g. γ-
rays) is given by
F lux ∼
ρ2(4πr2) dr , (7)
where the proportionality coefficient is a function of
dark matter particle mass, interaction cross section
and the number of photons produced per annihila-
tion.
The differential and cumulative flux (i.e. the in-
tegrand in expression (7) and the integral evaluated
from rta inwards) for neutralino (σ = 0.03 cm/s)
and a minihalo of rta = 3.24 pc (which corresponds
to a virial radius of about 0.8 pc) is shown in Fig.
0.3 0.5 1
r (pc)
1e+02
1e+04
1e+06
approximate  [" eq. (6)]
numerical [" eq. (1)]
0.14 0.29 0.57 1.1
r (pc)
1e+03
1e+04
1e+05
1e+06
1e+07
for neutralino minihaloes of  r
~1 pc ; M~ 10
σ=0.03 cm/s
σ=0.0003 cm/s
Fig. 3.— Cumulative flux is obtained by summing the
flux inwards: i.e. from the first outer caustic towards the
most inner (i.e. the integral (7) evaluated inwards). The flux
is shown for two different values of the velocity dispersion.
The red dashed line shows the cumulative flux obtained by
using our approximate analytic expression for the density (6)
which neglects the contribution from the caustics and can
considerably underestimate the annihilation flux and ignore
the distinct sawteeth characteristic of the caustics. The inset
shows the differential flux (integrand of expression (7)) using
the full density profile (1) as shown by the solid black spiky
line and the approximate profile (6) as shown by the dashed-
dotted red line. The sawteeth pattern is once again neglected
in using the later profile.
3. The fluctuations, due to caustics, become less
prominent as we go towards the centre. Decreasing
the velocity dispersion would increase both the am-
plitude of the peaks in the density profile and the
fluctuations in the flux, as shown in Figs. 2 and 3.
Using our numerical solution to (1) and approx-
imation (6), we can now determine the flux from
the neutralino minihaloes (Diemand et al 2005) and
its enhancement due to the first twenty caustics.
Clearly the total flux from the whole halo is dom-
inated by the emission from the centre, where the
density of the caustics reaches the background den-
sity (see also . However in the outer regions where
the first twenty caustics dominate, as shown in Fig.
2 the ratio of the flux using the self-similar density
profile given by (6) and the complete density profile
(1) gives a boost factor of about
Boost = 1.4 (8)
Thus, not only we expect a distinct signature on
the cumulative and (similarly differential) flux due
to caustics as highlighted schematically in Fig. 1 and
shown numerically in Fig. 3, we also expect that the
total flux from the outer halo region including the
first twenty caustics to be boosted by about %40.
Quantitative works on the gamma-ray flux is not
carried out here, as it requires more realistic model
than the self-similar model which can at best ex-
plain the growth of a typical halo. Minihaloes are
atypical in the sense that they evolve in isolation,
accreting almost no mass.
In conclusion, we have modeled dark matter
haloes by an extended version of secondary infall
model to include non-vanishing velocity dispersion.
We have shown that the differential and cumula-
tive fluxes would have distinct sawteeth pattern due
to caustics. We have demonstrated that caustics
can boost the total annihilation flux by about 40%
percents in the outer regions of smallest haloes of
about 10−5 M⊙. As for the prospect of detecting
caustics, the nearest minihaloes could be detectable
in gamma rays by proper motions observed with
GLAST (Koushiappas 2006), and should display a
caustic-like substructure. One would expect to find
a series of caustics, detectable as arclets. The pre-
dicted spacings could be used as a template to dig
more deeply into the noisy background.
Acknowledgment: S.S. acknowledges the support
from LANL T8, CITA and IAP during sabbatical
year 2005-06 when most of this study was done. We
thank M. Kuhlen and J. Diemand for discussions.
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Arnol’d V.I., Shandarin S., Zel’dovich Ya.-B 1982,
Geophysical and Astrophysical Fluid Dynamics, 20, 111
Arnol’d V.I. 1990, Singularities of caustics and wave
fronts, Kluwer Academic publishers, Mathematics and its
applications (Soviet Series) volume 62.
Bertone G., Hooper D., Silk J. 2004, Phys. Rep. 405, 279
Bertschinger E. 1985, ApJ 58, 39
Bertschinger E. 2006, Phys. Rev. D74, 3509
Diemand J., Moore B., Stadel J. 2005, Nature 433, 389
Fillmore J.A., Goldreich P. 1984, ApJ 281, 1
Gavazzi R., Mohayaee R. & Fort B. 2006, A&A 445, 43 ; and
erratum: Gavazzi R., Mohayaee R. & Fort B. 2006, A&A
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Jungman G., Kamionkowski M., Griest K. 1996, Phys. Rep.
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Moore B., Governato F., Quinn T., Stadel J., Lake G. 1998,
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http://arxiv.org/abs/astro-ph/0606208
http://arxiv.org/abs/astro-ph/0703704

  Cold dark matter haloes are populated by caustics, which are yet to be
resolved in N-body simulations or observed in the Universe. Secondary infall
model provides a paradigm for the study of caustics in "typical" haloes
assuming that they have had no major mergers and have grown only by smooth
accretion. This is a particular characteristic of the smallest dark matter
haloes of about 10^{-5} Mo, which although "atypical" contain no substructures
and could have survived until now with no major mergers. Thus using this model
as the first guidline, we evaluate the neutralino self-annihilation flux for
these haloes. Our results show that caustics could leave a distinct sawteeth
signature on the differential and cumulative fluxes coming from the outer
regions of these haloes. The total annihilation signal from the regions away
from the centre can be boosted by about forty percents.

Draft version November 16, 2018
Preprint typeset using LATEX style emulateapj v. 11/26/04
DARK MATTER CAUSTICS AND THE ENHANCEMENT OF SELF-ANNIHILATION FLUX
Roya C. Mohayaee
, Sergei Shandarin
, Joseph Silk
Draft version November 16, 2018
ABSTRACT
Cold dark matter haloes are populated by caustics, which are yet to be resolved in N-body
simulations or observed in the Universe. Secondary infall model provides a paradigm for the
study of caustics in typical haloes assuming that they have had no major mergers and have
grown only by smooth accretion. This is a particular characteristic of the smallest dark
matter haloes of about 10−5 M⊙, which although atypical contain no substructures and
could have survived until now with no major mergers. Thus using this model as the first
guidline, we evaluate the neutralino self-annihilation flux for these haloes. Our results show
that caustics could leave a distinct sawteeth signature on the differential and cumulative
fluxes coming from the outer regions of these haloes. The total annihilation signal from the
regions away from the centre can be boosted by about forty percents.
Subject headings: dark matter haloes, caustics, dark matter detection
1. INTRODUCTION
Evidence from the rotation curves of galaxies,
gravitational lensing, microwave background radi-
ation, peculiar velocity fields, and many other ob-
servations indicate that the visible mass, in the form
of stars and hot gas, is only a small fraction of the
total content of the Universe. The nature of the
missing mass, the dark matter, remains unknown
but is widely presumed to be Weakly Interacting
Massive Particles (WIMPs), such as the lightest su-
persymmetric particles, which are yet to be detected
in particle accelerators (Jungman, Kamionkowski,
& Griest 1996 ; Bertone, Hooper & Silk 2004).
Accelerator searches are complemented by the
vast experimental efforts to detect these particles in
our galaxy and in nearby galaxies which are believed
to be embedded in dark matter haloes (Ostriker
& Peebles 1973). Such complementary techniques
presently involve direct detection in low background
laboratory detectors (Goodman & Witten 1985)
and indirect detection through observation of en-
ergetic neutrinos, gamma rays and other products
of self-annihilation of dark matter particles (Silk &
Srednicki 1984) .
The event rate for self-annihilation depends
quadratically on the local dark matter density,
which falls off with distance from the center of the
halo. The averaged halo density profile obtained in
various numerical simulations diverges at the cen-
tre but is otherwise smooth and is often fit with a(n
asymptotically) double power-law (Navarro, Frenk
& White 1996, Moore et al 1998). However, a con-
sensus on the precise values of the power exponents,
the size of the central core and the resolution of fine
high-density structures are yet to be achieved. The
fine structures, the caustics, are inevitable outcomes
1 Institut d’astrophysique de Paris, 98 bis boulevard
Arago, France
2 Department of Physics and Astronomy, University of
Kansas, KS 66045, U.S.A.
3 University of Oxford, Astrophysics, Keble Road, Oxford
OX1 3RH, U.K.
of the evolution of a collisionless self-gravitating sys-
tem described by the Jeans-Vlassov-Poisson equa-
tion (for a one-dimensional numerical result see
Alard & Colombi 2005). Formally, in three dimen-
sions, the most common caustics are surfaces of zero
thicknesses over which the density diverges. 4 How-
ever, a maximum cut-off to their density is set by
the finite non-negligible velocity dispersion of dark
matter particles. Their density however remains
very high and hence they can be significant for dark
matter search experiments (Sikivie & Ipser 1992,
Sikivie et al 1997, Natarajan 2007). The effect of
velocity dispersion in the smearing of the caustics is
expected to dominate over other effects such as par-
ticle discreteness which would also smooth the caus-
tics but to a far lesser degree. Mergers of haloes can
also smear out the caustics substantially and due to
this fact we restrict our study to haloes that have
grown by slow and smooth accretion. Nevertheless,
caustics are robust, in that while they may break
up into micro-caustics, they remain in the fine-scale
halo substructure and thereby contribute to the gen-
eral clumpiness boost of any annihilation signal.
Analytic studies of the formation of haloes and
caustics have been carried out mainly under vari-
ous simplifying assumptions, such as spherical sym-
metry, self-similarity, and cold and smooth accre-
tion (Gott 1975, Gunn 1977, Fillmore & Goldreich
1984, Bertschinger 1985). In an Einstein-de Sit-
ter Universe a spherical overdensity expands and
then turns around to collapse. After collapse and
at late times, the fluid motion becomes self-similar:
its form remains unchanged when its length is re-
scaled in terms of the radius, rta, of the shell that
is currently at turn-around and is falling onto the
4 The general theory of singularities (Arnol’d, Shandarin&
Zel’dovich 1982) also predicts singularities on lines and at
points. Despite the greater concentration of mass in these
singularities they probably play a less important role in the
total annihilation rate because they contain a considerably
smaller amount of mass. However, this has not been studied
in detail.
http://arxiv.org/abs/0704.1999v1
galaxy for the first time. Physically, self-similarity
arises because gravity is scale-free and because mass
shells outside the initial overdensity are also bound
and turn around at successively later times. Self-
similar solutions give power-law density profiles on
the scale of the halo which provides an explanation
for the flattening of the rotation curves of galaxies.
However, on smaller scales the density profile con-
tains many spikes (i.e. caustics) of infinite density.
The position and time of formation of these caus-
tics are among the many properties that have been
established in the framework of the self-similar in-
fall model (Fillmore & Goldreich 1984, Bertschinger
1985).
In reality, dark matter has a small velocity dis-
persion and haloes do suffer from major mergers
and non-sphericity. However, until numerical simu-
lations achieve sufficient resolution, the self-similar
accretion model provides a useful guideline to haloes
which have not undergone major mergers.
Here, we use the self-similar model of halo for-
mation and a further elaboration which includes
the velocity dispersion of dark matter (Mohayaee
& Shandarin 2006) as a first guidline to describe
the evolution of the smallest haloes which have sur-
vived major merger and disruption until now and
have grown only by slow accretion.
The application of self-similar model to such
haloes can be viewed from two contradictory angles.
One might assume that minihaloes are expected to
be well-represented by this model, since they con-
tain no substructures, have not undergone merger
and grow very slowly only by smooth mass accre-
tion. On the other hand, minihaloes are not typical
haloes and selfsimilar accretion model is formulated
to describe the evolution of a characteristic halo.
Keeping both of these issues in mind, we use self-
similar model only as a first guidline for the evolu-
tion of minihaloes. A large number of them have
been found in simulations (Diemand et al 2005).
The simulations estimate the size of these haloes
to be of about 0.01 pc (half mass radius) and their
mass of 10−6M⊙ at z = 26. Due to resolution prob-
lems, these simulations are stopped at this redshift
and typical evolution of galactic scale haloes is ex-
tended to minihaloes and the conclusion is drawn
that about 1015 of these haloes could exist in the
halo of MilkWay today. We assume that at least
a fraction of these haloes have evolved by slow ac-
cretion model from z = 26 until now and use the
selfsimilar model to evaluate their radius and mass
at z = 0, which are respectively 1 pc and 10−5
M⊙. For these haloes and working self-consistently
within our model including the contribution from
the caustics, we demonstrate that in the outer re-
gions of these haloes caustics can boost the annihi-
lation signal by about 40%.
2. SECONDARY-INFALL MODEL WITH VELOCITY
DISPERSION
The haloes considered in this work grow by
smooth and slow accretion. A good example are
the earth-mass haloes which were recently resolved
(at z=26) in numerical simulations (Diemand et al
Fig. 1.— A surface-contour plot of the caustic density.
In the self-similar model, caustics form concentric shells of
increasing density and decreasing thicknesses and separations
as we approach the center of the halo.
2005) and which although small are expected to
have clean spherical caustics. We expect that at
least a fraction of these haloes have survive disrup-
tion and major merger and grow by self-similar ac-
cretion model to a virial radius of about 1 pc and a
mass of about 10−5 M⊙.
To comply with the requirement of slow accretion,
we fix the value of the parameter ǫ in the initial den-
sity perturbation δ ∼ M−ǫi , where Mi is the initial
mass, to unity. We emphasis that the self-similar
model aims at describing the evolution of a typi-
cal halo. Typical haloes have mass variance (σ(M)
which varies as M−(n+3)/6, which sets ǫ = (n+3)/6,
where n is the power spectrum index. A typical
σ(M) fluctuation grows as t2/(3ǫ). Minihaloes cor-
respond to the limit n → −3 part of the spectrum.
For this part of the spectrum, there are mass fluc-
tuations of comparable amplitude on all scales and
consequently adiabatic invariance does not apply for
such fluctuations.
Hence, we use the self-similar model only as a
first guidline for the growth of minihaloes. We as-
sume that minihaloes of mass 10−6M⊙ have grown
by very small accretion from z = 26 to z = 0. Once
again, slow accretion corresponds to the case of
ǫ = 1 in the work of Fillmore and Goldreich (1984),
hence we shall adopt this value for ǫ.
The self-similar density profile is given by
(Bertschinger 1985)
(−1)jexp
where ρ̄ is the critical density and
is the dimensionless radius and r is the physical ra-
dius and ξi = ln(t/tta) is the dimensionless time
given in terms of the turnaround time, tta, of the
particle that is at the jth point where λ = λ(ξ) (see
Bertschinger 1985 for further explanation).
The density (1) is evaluated numerically and plot-
ted in Fig. 2 after an appropriate cut-off of the caus-
tics which shall be discussed now. In principle the
0.06 0.1 0.2
1e+02
1e+03
1e+04
1e+05
1e+06
2nd caustic 1st caustic
Axion
Neutralinoρ
λ2 λ1
Fig. 2.— The plot is made by numerically solving (1) and
cutting the caustics using (3). The density of halos (divided
by the critical density ρ̄) can be enhanced significantly at the
caustics. This enhancement is far larger for axions (dotted
violet spikes) than for neutralinos (continuous black line).
The number of the streams increases to the center of the halo
which explains the rapid growth of the smooth component of
the density profile and the caustic contribution to the halo
density diminishes. The dashed-dotted red line marked ρss is
the approximate self-similar density profile given by (6). The
inset shows a magnified view of the second and third caustics
for neutralinos.
density at the caustics diverges if the velocity dis-
persion of dark matter is zero. In the presence of a
small velocity dispersion the maximum density and
thickness of the caustic shells and their density pro-
files have been evaluated (Mohayaee & Shandarin
2006). The maximum density at the caustics and
their profile are given by
ρcaustic,k =
|∆λk|
ρ̄ λk − |∆λk| < λ < λk
ρ̄ λ < λk − |∆λk|
where λk is the non-dimensional radius of the kth
caustic counted inwards and
−2λ′′k
e−2ξk/3
and the thickness of the caustic shell is given by
∆λk =
(3π)2/3e5ξk/9Λk
t σ(t)
, (5)
where t is the age of the Universe, σ is the present-
day velocity dispersion of dark matter particles
which is that at decoupling re-scaled with the ex-
pansion factor. The values of these parameters vary
from one caustic to another (see Table 1 of Mo-
hayaee & Shandarin 2006 for the first ten caustics).
The profile (1) together with appropriate cut-off
given by (3) is plotted in Fig. 2.
The peaked density profile given by (1) and shown
in Fig. 2 has to be evaluated numerically. However,
as is evident from Fig. 2 a “self-similar” profile5 is
5 This profile can also be well-fitted by a power-law and
an exponential cut-off
reached which we fit with
ρss =
2.8λ−9/4
(1 + λ3/4)2
ρ̄ , (6)
as shown in Fig. 2 by the dashed-dotted red line,
marked ρss. The turnaround radius, rta can now be
evaluated by considering that at the virial radius the
density is about 200 times the background density,
and is given by rta ∼ 4rvir, which corresponds to
the density profile given by (6). This approximate
profile has been shown to be a good fit also to the
mass profile (see Mohayaee & Shandarin 2006). In
the next section we shall show that using this profile
which ignores the caustics would yield an under-
estimated value for the flux.
Both the extrapolated numerical and the approx-
imate density profiles shown in Fig. 2 formally di-
verge at the centre. However, due to finite dark
matter velocity dispersion, haloes can develop cen-
tral cores. Dark matter haloes are expected to have
central cores due to the dark matter velocity disper-
sion, self-annihilations at the centre, angular mo-
mentum, tidal and various other effects. The core
could be very small and the minimum scale asso-
ciated with a generic dark matter merging history
would conserve traces of the original cores in the
initial substructure. These should be of order the
free-streaming mass as for example computed in
Bertschinger (2006).
In principle for small core sizes the total flux from
the whole of the halo is dominated by the annihi-
lation in the centre of the halo and the boost due
to caustics is negligible. However, we shall show in
the next section that the differential (similarly cu-
mulative) flux would be distinctly marked by the
caustics and shall have a sawteeth pattern and the
contribution to the total flux from the outer region
of these haloes by the caustics is significant and can
yield a boost factor of about 40%.
3. THE FLUX DUE TO SELF-ANNIHILATION
INCLUDING THE EFFECT OF CAUSTICS
Caustics if detected would be clear evidence of
the existence of dark matter and could rule out al-
ternative models of gravity. Two major methods
for their detection are through gravitational lensing
(see e.g. Gavazzi et al 2006) and the flux of dark
matter annihilation product which is expected to
be significantly enhanced by the caustics. Here we
shall discuss the second method.
The flux of the self-annihilation product (e.g. γ-
rays) is given by
F lux ∼
ρ2(4πr2) dr , (7)
where the proportionality coefficient is a function of
dark matter particle mass, interaction cross section
and the number of photons produced per annihila-
tion.
The differential and cumulative flux (i.e. the in-
tegrand in expression (7) and the integral evaluated
from rta inwards) for neutralino (σ = 0.03 cm/s)
and a minihalo of rta = 3.24 pc (which corresponds
to a virial radius of about 0.8 pc) is shown in Fig.
0.3 0.5 1
r (pc)
1e+02
1e+04
1e+06
approximate  [" eq. (6)]
numerical [" eq. (1)]
0.14 0.29 0.57 1.1
r (pc)
1e+03
1e+04
1e+05
1e+06
1e+07
for neutralino minihaloes of  r
~1 pc ; M~ 10
σ=0.03 cm/s
σ=0.0003 cm/s
Fig. 3.— Cumulative flux is obtained by summing the
flux inwards: i.e. from the first outer caustic towards the
most inner (i.e. the integral (7) evaluated inwards). The flux
is shown for two different values of the velocity dispersion.
The red dashed line shows the cumulative flux obtained by
using our approximate analytic expression for the density (6)
which neglects the contribution from the caustics and can
considerably underestimate the annihilation flux and ignore
the distinct sawteeth characteristic of the caustics. The inset
shows the differential flux (integrand of expression (7)) using
the full density profile (1) as shown by the solid black spiky
line and the approximate profile (6) as shown by the dashed-
dotted red line. The sawteeth pattern is once again neglected
in using the later profile.
3. The fluctuations, due to caustics, become less
prominent as we go towards the centre. Decreasing
the velocity dispersion would increase both the am-
plitude of the peaks in the density profile and the
fluctuations in the flux, as shown in Figs. 2 and 3.
Using our numerical solution to (1) and approx-
imation (6), we can now determine the flux from
the neutralino minihaloes (Diemand et al 2005) and
its enhancement due to the first twenty caustics.
Clearly the total flux from the whole halo is dom-
inated by the emission from the centre, where the
density of the caustics reaches the background den-
sity (see also . However in the outer regions where
the first twenty caustics dominate, as shown in Fig.
2 the ratio of the flux using the self-similar density
profile given by (6) and the complete density profile
(1) gives a boost factor of about
Boost = 1.4 (8)
Thus, not only we expect a distinct signature on
the cumulative and (similarly differential) flux due
to caustics as highlighted schematically in Fig. 1 and
shown numerically in Fig. 3, we also expect that the
total flux from the outer halo region including the
first twenty caustics to be boosted by about %40.
Quantitative works on the gamma-ray flux is not
carried out here, as it requires more realistic model
than the self-similar model which can at best ex-
plain the growth of a typical halo. Minihaloes are
atypical in the sense that they evolve in isolation,
accreting almost no mass.
In conclusion, we have modeled dark matter
haloes by an extended version of secondary infall
model to include non-vanishing velocity dispersion.
We have shown that the differential and cumula-
tive fluxes would have distinct sawteeth pattern due
to caustics. We have demonstrated that caustics
can boost the total annihilation flux by about 40%
percents in the outer regions of smallest haloes of
about 10−5 M⊙. As for the prospect of detecting
caustics, the nearest minihaloes could be detectable
in gamma rays by proper motions observed with
GLAST (Koushiappas 2006), and should display a
caustic-like substructure. One would expect to find
a series of caustics, detectable as arclets. The pre-
dicted spacings could be used as a template to dig
more deeply into the noisy background.
Acknowledgment: S.S. acknowledges the support
from LANL T8, CITA and IAP during sabbatical
year 2005-06 when most of this study was done. We
thank M. Kuhlen and J. Diemand for discussions.
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