Issue |
A&A
Volume 510, February 2010
|
|
---|---|---|
Article Number | A90 | |
Number of page(s) | 6 | |
Section | Astrophysical processes | |
DOI | https://doi.org/10.1051/0004-6361/200913353 | |
Published online | 17 February 2010 |
Gamma rays from annihilations at the galactic center in a physical dark matter distribution
A. Lapi1,2 - A. Paggi1 - A. Cavaliere1 - A. Lionetto1,3 - A. Morselli1,3 - V. Vitale1
1 - Dip. Fisica, Univ. ``Tor Vergata'', via Ricerca Scientifica 1, 00133 Roma, Italy
2 -
SISSA/ISAS, via Beirut 2-4, 34151 Trieste, Italy
3 -
INFN-Sezione di Roma2, via Ricerca Scientifica 1, 00133 Roma, Italy
Received 25 September 2009 / Accepted 28 November 2009
Abstract
We discuss the -ray signal to be expected
from dark matter (DM) annihilations at the Galactic center. We
describe the DM distribution in the Galactic halo, based on the
Jeans equation for self-gravitating, anisotropic equilibria. In
solving the Jeans equation, we adopted the specific correlation
between the density
and the velocity dispersion
expressed by the powerlaw behavior of the DM
``entropy''
with
.
Indicated (among others) by several
recent N-body simulations, this correlation is privileged by
the form of the radial pressure term in the Jeans equation, and
it yields a main-body profile consistent with the classic
self-similar development of DM halos. In addition, we required
the Jeans solutions to satisfy regular boundary conditions both
at the center (finite pressure, round gravitational potential)
and on the outskirts (finite overall mass). With these building
blocks, we derived physical solutions, dubbed
``
-profiles''. We find the one with
,
suitable for the Galaxy halo, to be intrinsically flatter at
the center than the empirical NFW formula, yet steeper than the
empirical Einasto profile. On scales of 10-1 deg it yields
annihilation fluxes lower by a factor 5 than the
former, yet higher by a factor 10 than the latter.
Such fluxes will eventually fall within the reach of the
Fermi satellite. We show the effectiveness of the
-profile in relieving the astrophysical uncertainties
related to the macroscopic DM distribution, and discuss its
expected performance as a tool instrumental in interpreting the
upcoming
-ray data in terms of DM annihilation.
Key words: cosmology: dark matter - galaxies: evolution - Galaxy: halo - methods: analytical
1 Introduction
Several astrophysical and cosmological probes (for a
review see Bertone et al. 2005) have firmly established that baryons
- which stars, planets, and (known) living creatures are made
of - constitute only some
of the total matter
content in the Universe (adding to the dominant dark-energy
component). The rest is in the form of ``cold dark matter'' (DM),
i.e., massive particles that were nonrelativistic at
decoupling, do not emit/absorb radiations, and basically do not
interact with themselves and with the baryons except via
long-range gravitational forces.
However, no ``direct'' detection of the DM has been made so far, other than Bernabei et al. (2008). Thus the microscopic nature of the DM largely remains a mystery, where several clues suggest as a promising candidate or component the lightest supersymmetric particle, the ``neutralino'' (for a review see Bertone 2009). Given that the latter's mass, depending on the specific supersymmetric model, ranges from several GeVs to tens of TeVs, its laboratory production requires an accelerator at least as powerful as the newly-born Large Hadron Collider (see Baer & Tata 2009). The discovery of supersymmetry and specifically of the neutralino is one of the main aims for the current experiments in high-energy physics.
Meanwhile, evidence of the DM can be looked for ``indirectly'' in
the sky. In fact, the basic aims of the recently launched
Fermi satellite include the search for -ray
signals due to the annihilation of DM particles at the Galactic
center (GC) and in nearby galaxies (see discussion in Sect. 4).
The former provides a favorable target because it is closest to
us, with the DM density expected to increase in moving toward
the inner regions of a galaxy. However, the GC is also a
crowded region, and it remains a challenging task to separate
the DM signal from the contributions of other astrophysical
sources and backgrounds whose energy spectrum and angular
distribution are poorly known.
In principle, if one can predict the strength and angular
distribution of the annihilation signal itself, then the
-ray observations would elicit, or put ``indirect''
constraints on, the (combined) microscopic properties of
the DM particles like mass, annihilation cross section, and
channels. This approach has been pursued extensively
(e.g., Serpico & Hooper 2009; Bertone et al. 2009; Fornengo et al. 2004; Strigari 2007; Bergström et al. 1998) but still suffers from large
uncertainties (see Cesarini et al. 2004), mainly related to
the poor knowledge of the macroscopic DM distribution
throughout the Galaxy. Since the annihilation rate scales like
,
such uncertainties are maximized near the center
right where detection is favored. Similar if milder
uncertainties affect the source function of the electrons
originating in DM annihilations by production or cascading;
these diffuse outwards and interact with the Galactic magnetic
field and with the interstellar light to produce synchrotron
emission observed in the radio band (see Bertone et al. 2009),
and inverse Compton radiation observable in
rays
(see Papucci & Strumia 2009).
Traditionally, the density profile of an equilibrium DM
structure, or ``halo'', is rendered in terms of different
empirical formulas that fit the results of N-body simulations
and, to some extent, the stellar observations. Perhaps the most
popular one is the Navarro, Frenk & White (hereafter
NFW; see Navarro et al. 1997) profile, which has an asymptotic inner
slope
,
goes over to a powerlaw behavior
in the halo's middle, and declines as
in the outer regions. Despite its
widespread use in the literature, clearly this expression
cannot account for the actual DM distribution in the inner
regions of a galaxy halo where it would imply a centrally
angled gravitational potential well and an infinite pressure,
nor in the halo outskirts where it would yield a diverging
overall mass.
Other empirical density profiles have been proposed, but they
suffer of similarly unphysical features; e.g., the Moore
profile (see Diemand et al. 2005) goes like
and implies a gravitational force diverging towards
the center, while the Einasto profile (see Graham et al. 2006)
behaves like
,
so it yields a
vanishing pressure there. We stress that the differences
in the predicted annihilation signals under these DM
distributions turn out to be quite considerable; e.g., the
ratio of the NFW to the Einasto squared density averaged over
1 degree (about 150 pc) comes to a factor 10 when
normalized at the Sun's location (see also Sect. 4).
Here our stand is that the macroscopic uncertainties yielding
such differences can and ought to be relieved. To this purpose,
in Sect. 2 we present the physical density distributions
that we dub -profiles. These are solutions of
the Jeans equation that satisfy regular inner and outer
boundary conditions. In Sect. 3 we use the
-profile
suitable for the Galaxy halo as the macroscopic benchmark for
evaluating the DM annihilation signal expected from the GC. We
base the microscopic sector on a standard model for the mass,
cross section, and annihilation channel of the DM particles,
because the extension to more complex microphysics is
straightforward. Finally, our findings are summarized and
discussed in Sect. 4.
Throughout this work we adopt a standard, flat cosmology
(see Dunkley et al. 2009) with normalized matter density
,
and Hubble constant H0 = 72 km s-1 Mpc-1.
2 Development and structure of DM halos
Galaxies are widely held to form under the drive of the gravitational instability that acts on initial perturbations modulating the cosmic density of the dominant cold DM component. At first the instability is kept in check by the cosmic expansion, but when the local gravity prevails, collapse sets in, to form a DM halo in equilibrium under self-gravity. The amplitude of more massive perturbations is smaller, so the formation is progressive in time and hierarchical in mass, with the largest structures typically forming later (see Peebles et al. 1983, for a review).
2.1 Two-stage evolution
Such a formation history has been resolved to a considerable
detail by many N-body simulations (e.g., Springel et al. 2006; White 1986); recently, a novel viewpoint emerged. First, the
halo growth has been recognized (see Diemand et al. 2007; Zhao et al. 2003; Wechsler et al. 2006; Hoffman et al. 2007) to comprise two stages:
an early fast collapse including a few violent major mergers,
that builds up the halo main ``body'' with the structure set by
dynamical relaxation; and a later, quasi-equilibrium stage when
the body is nearly unaffected, while the outskirts develop from
the inside-out by minor mergers and smooth accretion
(see Salvador-Solé et al. 2007). The transition is provided
by the time when a DM gravitational potential attains its
maximal depth; i.e., the radial peak of the circular velocity
attains its maximal height, along a given
growth history (see Li et al. 2007).
Second, generic features of the ensuing equilibrium structures
have been sought (see Schmidt et al. 2008; Hansen 2004; Dehnen & McLaughlin 2005) among powerlaw correlations of the form
;
this
involves the density
and the velocity dispersion
,
with
anisotropy inserted via the standard Binney (1978)
parameter
and
modulated by the index D (see Hansen 2007). It is
matter of debate which of these correlations best apply, see
Schmidt et al. (2008) and Navarro et al. (2010); the former
authors, in particular, find that the structure of different
simulated halos may be described by different values of D,
with linearly related values of
and
(see
their Eqs. (4) and (5)).
Here we focus on the specific correlation
![]() |
(1) |
that solely involves the squared radial dispersion






To independently probe the matter, Lapi & Cavaliere (2009a) performed
a semianalytical study of the two-stage halo development, and
derived (consistent with the simulations) that
is set
at the transition time via scale-free stratification of the
particle orbits throughout the halo body, and thereafter
remains closely constant and uniform at a value within the
narrow range 1.25-1.3. Moreover, they find that on
average the values of
weakly depend on the mass of
the halo, such that
applies to galaxy
clusters, while
applies to Milky Way sized
galaxies.
2.2 The DM
-profiles
The halo physical profiles may be derived from the radial Jeans
equation, with the radial pressure
and anisotropies described by the
standard Binney (1978) parameter
.
Thus the Jeans
equation simply writes as
![]() |
(2) |
in terms of the logarithmic density slope





To set the context for the Milky Way DM distribution, we recall
that the space of solutions for Eq. (2) spans the range
:
the one for the upper bound and
the behaviors of others ones have been analytically
investigated by Austin et al. (2005) and Dehnen & McLaughlin (2005). In
Lapi & Cavaliere (2009a), we explicitly derived the Jeans solutions
with
(meaning isotropy) for the full range
subjected to regular
boundary conditions both at the center and in the outskirts,
i.e., a round minimum of the potential with a
finite pressure (or energy density) and a finite
(hence definite) overall mass, respectively. These we dubbed
``
-profiles''.
The corresponding density runs steepen monotonically
outwards and are summarized by the pivotal slopes
![]() |
(3) |
These start from the central (





![]() |
Figure 1:
Density and mass profiles in the Milky Way. The dashed
and solid lines illustrate the |
Open with DEXTER |
For a density profile, a relevant parameter is the
concentration
,
defined in terms of the
virial radius Rv and of the radius r-2 where
.
In the context of
-profiles c may be
viewed as a measure either of central condensation (small
r-2) or of outskirts' extension (large Rv). The
concentration constitutes an indicator of the halo age; in
fact, numerical experiments (see Diemand et al. 2007; Zhao et al. 2003; Wechsler et al. 2006; Bullock et al. 2001) show that
holds at the end of the fast collapse stage, to grow as
during the slow accretion
stage after the transition at
.
Current values
apply for a galaxy like the Milky Way
that had its transition at
.
The density and mass distribution in the Milky Way are
illustrated in Fig. 1 for the isotropic -profiles with
,
for the NFW formula, and for the Einasto
profile. All densities have been normalized to the local
density 0.3 GeV cm-3 at the Sun's location
kpc within the Galaxy. We further adopt
r-2 = 20 kpc (consistent with c = 10). Note in Fig. 1
that the Einasto and NFW profiles differ substantially
at the center as for the density and in the outskirts as for
the mass, while the
-profile strikes an
intermediate course between the two.
2.3 Anisotropy
It is clear from Eq. (2) that anisotropy will steepen
the density run for positive
meaning radial velocity
dominance, as expected in the outskirts from infalling
cold matter. On the other hand, tangential components
(corresponding to
)
must develop toward the
center, as expected from the increasing strength of angular
momentum effects. This view is supported by numerical
simulations (see Hansen & Moore 2006; Austin et al. 2005; Dehnen & McLaughlin 2005),
which in detail suggest the effective linear approximation
![]() |
(4) |
to hold with



In Lapi & Cavaliere (2009b) we extended the -profiles to such
anisotropic conditions in the full range
,
inspired by the analysis by Dehnen & McLaughlin (2005) for
the specific case
.
We find the
corresponding
to be somewhat flattened at the
center by a weakly negative
and further
steepened into the outskirts where
grows
substantially positive. Specifically, the following simple
rules turn out to apply: the slope
in Eq. (4) drops
out from the derivatives of the Jeans equation
(see Dehnen & McLaughlin 2005); the upper bound to
now
reads
;
moreover,
is modified
into
,
while
and
retain their form.
The anisotropic -profiles for the Milky Way are shown
in Fig. 1. We note, in particular, that even a limited central
anisotropy (e.g.,
)
causes an appreciable
flattening of the inner density slope, bringing it down
to
for
.
Plainly, this
result produces an even more considerable flattening in the
slope of the squared density, the relevant quantity in our
context of DM annihilations.
2.4 A guide to profile computations
Finally, in the Appendix we provide user-friendly analytic fits
for the density runs of the -profiles in terms of
standard deprojected Sérsic formulas, but with parameters
directly derived from the Jeans equation. We stress that
these physical
-profiles with their analytic fits are
relevant to, and recently tested in several contexts, including
the interpretation of gravitational lensing observations
(see Lapi & Cavaliere 2009b), the physics of the hot diffuse
baryons constituting the intra-cluster plasma
(see Cavaliere et al. 2009), and galaxy kinematics (Lapi &
Cavaliere, in preparation). In the following we focus on the
specific
-profile with
suitable for the
Milky Way halo (see Sect. 2.1) to predict the DM annihilation
signal from the GC.
2.5
-ray signal from DM annihilation at the GC
The -ray flux per solid angle due to DM annihilation
along a direction at an angle
relative to the
line-of-sight toward the GC may be written (under the commonly
assumed spherical symmetry) as
![]() |
(5) |
in units of







For the sake of definiteness we begin by considering a
neutralino DM particle with mass
GeV,
annihilating through the
channel (with
branching ratio). We use the benchmark value for the
annihilation rate
cm3 s-1, corresponding to a thermal relic
with a density close to the cosmological DM abundance
![]() |
(6) |
as measured by WMAP (see Dunkley et al. 2009). To compute

![]() |
(7) |
obtained from extrapolating the results by Fornengo et al. (2004) down to energies



The astrophysical term of Eq. (5) is given by the integral of
the (squared) DM density projected along the line-of-sight
![]() |
(8) |
normalized to







![]() |
(9) |
with

Table 1:
Values of the astrophysical factor .
Table 2:
Values of the -ray flux (in m-2 s-1)
for
MeV.
![]() |
Figure 2:
The astrophysical factor |
Open with DEXTER |
We computed and report in Table 1 the values of
at
angular resolutions
sr and 10-5sr for the
-profile with
in the isotropic
and anisotropic cases, for the NFW formula, and for the Einasto
profile. In Table 2 we list the corresponding values of the
-ray flux for energies
MeV. These
outcomes are illustrated in Fig. 2. It is seen that, relative
to the NFW distribution, the fluxes predicted from the
isotropic and anisotropic
-profile are lower by factors
from a few to several. Such fluxes are still within the reach
of the Fermi satellite; in fact, on the basis of the
simulations performed by Baltz et al. (2008),
Striani (2009), and Vitale et al. (2009a), we expect the
annihilation signal to be probed at a 3-
confidence
level over a few years.
The above values may be compared with the current upper bound
to the integrated flux of
m-2 s-1 based on Fermi measurements at
MeV during a 8-month observation of the GC
over a solid angle
sr
(see Vitale et al. 2009b; Atwood et al. 2009; Abdo et al. 2009); this bound
will decrease as
with the observation time t.
However, the flux observed currently includes contributions
from diffuse or not-yet-resolved Galactic sources, which are
being progressively removed (see Vitale et al. 2009b; Striani 2009; Goodenough & Hooper 2009). Next stages of such a process will
take longer observations aimed at determining the spectrum of
individual resolved sources and a careful likelihood analysis
of the backgrounds (see discussion by Cesarini et al. 2004).
4 Discussion and conclusions
We have presented our -profile with
for
the equilibrium density and mass distributions in a galactic DM
halo, and specifically in the Milky Way. We have shown that
this profile constitutes the robust solution of the
equilibrium Jeans equation with physical inner
and outer boundary conditions, i.e., finite pressure and round
potential minimum at the center and finite overall mass. The
corresponding density profile
is intrinsically
flatter at the center and intrinsically steeper
in the outskirts, relative to the empirical NFW formula. These
features are still sharpened in halos with anisotropic
random velocities. We also provided the reader with a precise
and user-friendly analytic fit to the
-profile (see
Appendix for details).
Then we focused on the role of this -profile as a
benchmark for computing the DM annihilation signal
expected from the GC. In fact, we computed the ``astrophysical
factor''
(angular distribution, independent of
microphysics) entering the expression of the annihilation flux.
As a definite example, we also computed the
-ray flux
on adopting a simple, fiducial microscopic model. This we find
consistent with current Fermi observations, which may
include contributions from still unresolved point sources.
Given the physical -profile and the corresponding
factor
,
the extension to more complex microscopic
scenarios like mSUGRA (started
by Barbieri et al. 1982; Hall et al. 1983; Chamseddine et al. 1982; Ohta et al. 1983) will
be easily made in terms of annihilations channels, cross
sections and particle masses. In this context our
-profile relieves astrophysical uncertainties
related to the macroscopic DM distribution. We stress that
constraints on particle cross sections and masses inferred from
radio and
-ray observations of the GC have been to now
more sensitive to the assumed DM distribution than to specific
annihilation channels (different from leptonic
),
see Figs. 3 and 4 in Bertone et al. (2009). In fact, the latter
show that a DM distribution with an inner slope like our
-profile is required to allow cross sections
cm3 s-1 with
masses
GeV for the non-leptonic channels,
which are widely considered on grounds of theoretical
microphysics.
Concerning small scales
a few tens of pcs around the
GC, we touch upon a number of possible deviations of the very
inner DM density distribution from our benchmark
-profile (Fig. 1, top panel). For example, the process
of galaxy formation could lead either to flattening or to some
steepening of the inner DM distribution. The former may occur
either owing to transfer of energy and/or angular momentum from
the baryons to the DM (see Tonini et al. 2006; El-Zant et al. 2001) or
owing to quick mass removal following the energy feedback from
stars or active galactic nuclei (see discussion
by Lauer et al. 2007; Kormendy et al. 2009). On the other hand, steepening
might be induced by the `adiabatic' contraction of the baryons
into the disk (see Blumenthal et al. 1986; Mo et al. 1998); but even
in extreme cases (see discussion by Abadi et al. 2009) such a
contraction would yield an inner DM density profile
,
still flatter than
1 though somewhat steeper than the original
.
Finally, at the very center of the Galaxy any
accretion of DM (e.g., Bertone et al. 2002; Gondolo et al. 1999) onto
the nuclear supermassive black hole might enhance the DM
distribution on tiny scales
r< 10-1 pc.
Summing up, we stress that all such alterations of the inner
slope would occur on scales smaller than some 10 pcs.
Although significant at levels of a few percent to account for
the central stellar light, their import is far
smaller where the annihilation signal is concerned, and on the
average over 10-1 deg the flux is altered by less than
.
In fact, these corrections are currently at or below
the resolution limit and the prospective sensitivity of
Fermi.
Other possible targets include the dwarf spheroidal galaxies in
the Local Group. On the one hand, these constitute cleaner
environments than the GC owing to the dearth of stellar
sources; on the other, their distance, if modest on
intergalactic scales, already makes detecting and resolving the
related annihilation signal a real challenge for Fermi
(e.g., Pieri et al. 2009). In addition, the shallow
gravitational potential wells of these systems make them
particularly prone to energy feedback events (see above), which
may flatten the inner DM distribution to flat slopes
(consistent with kinematical observations), with the
effect of further lowering the annihilation signals. Upper
limits more stringent than the current value
cm3 s-1 at a mass
GeV will require delicate stacking over an
ensemble of dwarfs.
Concerning particle cross section and masses, we recall that
the PAMELA satellite recently observed an excess of
the positron fraction
in the cosmic ray
spectra relative to the expected astrophysical
background above 10 GeV (see Adriani et al. 2009). This
excess can simply be explained in terms of a single or a few
sources like pulsars, which are expected to produce a powerlaw
spectrum of
pairs with a cutoff at several TeVs
(see Bertone 2009). On the other hand, the signal
may also be interpreted in terms of DM annihilations occurring
throughout the Galactic halo (e.g., Bertone et al. 2009).
If this is to be the case, however, the flux measured by
PAMELA mandates for very large effective annihilation
cross sections
cm3s-1, well above the natural value suggested by the
cosmological DM abundance (see Sect. 3). From a microphysical
point of view, this is still conceivable in scenarios with
Sommerfeld enhancements (see discussion
by Arkani-Hamed et al. 2009). The cross section may be enhanced by a
factor
for velocities
.
On the
other hand, such a large Sommerfeld effects would also yield a
strong
-ray annihilation signal towards the GC; for
this, little room is allowed on the basis of the current upper
limit provided by Fermi (see Sect. 3), unless the DM
particle mass substantially exceeds 50 GeV.
Another possibility is to invoke a large boost factor of the effective cross section due to clumpiness in the Galactic halo, i.e., a crowd of dense subhalos; however, state-of-the-art numerical simulations suggest such boosts are not realistic in the Galaxy, even less at the GC (see Springel et al. 2008; and discussion by Lattanzi & Silk 2009).
To sum up, we have discussed why the -profile with
(see Fig. 1) constitutes a reliable DM
distribution in the Galaxy. We argued that it will provide a
benchmark to gauge the
rays from the GC to be
detected with Fermi in terms of DM annihilation (see
Fig. 2). Such an
-profile will be instrumental in
deriving reliable information concerning the microscopic nature
of the DM particles.
This work was supported by Agenzia Spaziale Italiana (ASI), Istituto Nazionale di Astrofisica (INAF), and Istituto Nazionale di Fisica Nucleare (INFN). We acknowledge our referee S. H. Hansen for keen and helpful comments. We are indebted to L. Ciotti, A. Cirelli, P. Salucci, and M. Tavani for useful discussions. We thank F. Vagnetti for the critical reading. A. Lapi thanks the SISSA/ISAS and INAF-OATS for warm hospitality.
Appendix A: Analytic fit to the
-profile
To complement the analytical details extensively dealt with by
Lapi & Cavaliere (2009a) and to enable a straightforward comparison
with the classic NFW and Einasto density runs, we provide here
a handy analytic fit to the -profiles in terms of the
deprojected Sérsic formula substantiated with parameters
directly derived from the Jeans equation. We base it on
the expression (see Prugniel & Simien 1997)
![]() |
(A.1) |
where










The mass corresponding to the density distribution of Eq. (A1)
writes as
![]() |
(A.2) |
where
![$\Gamma[a,x]\equiv \int_0^x{\rm
d}t~t^{a-1}~{\rm e}^{-t}\big/\int_0^\infty{\rm
d}t~t^{a-1}~{\rm e}^{-t}$](/articles/aa/full_html/2010/02/aa13353-09/img144.png)

Table A.1:
Values of the fitting parameters of Eq. (A.1) in the
isotropic case, where
applies for the Galactic
halo.
Table A.2:
Values of the fitting parameters of Eq. (A.1) in the
anisotropic case, where
applies for the Galactic
halo.
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Footnotes
- ... light
- A flat slope is known to describe the very central light distribution in luminous ellipticals, which relates to complex small-scale dynamics (see Lauer et al. 2007; Kormendy et al. 2009).
All Tables
Table 1:
Values of the astrophysical factor .
Table 2:
Values of the -ray flux (in m-2 s-1)
for
MeV.
Table A.1:
Values of the fitting parameters of Eq. (A.1) in the
isotropic case, where
applies for the Galactic
halo.
Table A.2:
Values of the fitting parameters of Eq. (A.1) in the
anisotropic case, where
applies for the Galactic
halo.
All Figures
![]() |
Figure 1:
Density and mass profiles in the Milky Way. The dashed
and solid lines illustrate the |
Open with DEXTER | |
In the text |
![]() |
Figure 2:
The astrophysical factor |
Open with DEXTER | |
In the text |
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