Neutrinos from WIMP annihilation in the Sun : Implications of a self-consistent model of the Milky Way's dark matter halo

Abstract

Upper limits on the spin-independent (SI) as well as spin-dependent (SD)
elastic scattering cross sections of WIMPs with protons, imposed by the
Super-Kamiokande (S-K) upper limit on the neutrino flux from WIMP annihilation
in the Sun, and their compatibility with the "DAMA-compatible" regions of the
WIMP parameter space within which the annual modulation signal observed by the
DAMA/LIBRA experiment is compatible with the null results of other direct
detection experiments, are studied within the frame work of a self-consistent
model of the finite-size dark matter (DM) halo of the Galaxy, the parameters of
which are determined by a fit to the rotation curve data of the Galaxy. We find
that the S-K implied upper limits on the WIMP-proton elastic cross section as a
function of WIMP mass impose stringent restrictions on the branching fractions
of the various WIMP annihilation channels. For SI interaction, while the S-K
upper limits are consistent with the DAMA-compatible region of the WIMP
parameter space if the WIMPs annihilate dominantly to $\bbarb$\ and/or
$\cbarc$, portions of the DAMA-compatible region can be excluded if WIMP
annihilations to $\tautau$ and $\nu\anu$ occur at larger than ~ 10% and 0.1%
levels, respectively. For SD interaction, the restrictions on the possible
annihilation channels are much more stringent, essentially ruling out the
DAMA-compatible region of the WIMP parameter space if the relatively low-mass
($\sim$ 2 -- 20 GeV) WIMPs under consideration annihilate predominantly to any
mixture of $\bbarb$, \ $\cbarc$, \ $\tautau$, \ and $\nu\anu$ final states.

Full text

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arXiv:1106.5711v1 [astro-ph.GA] 28 Jun 2011
Neutrinos from WIMP annihilation in the Sun : Implications of a self-consistent
model of the Milky Way’s dark matter halo
Susmita Kundu∗and Pijushpani Bhattacharjee†
AstroParticle Physics & Cosmology Division and Centre for AstroParticle Physics,
Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata 700064. India
Upper limits on the spin-independent (SI) as well as spin-dependent (SD) elastic scattering cross
sections of WIMPs (Weakly Interacting Massive Particles — the dark matter candidate) with pro-
tons, imposed by the upper limit on the neutrino flux from WIMP annihilation in the Sun given
by the Super-Kamiokande (S-K) experiment, and their compatibility with the “DAMA-compatible”
regions of the WIMP parameter space — the regions of the WIMP mass versus cross section param-
eter space within which the annual modulation signal observed by the DAMA/LIBRA experiment is
compatible with the null results of other direct detection experiments — are studied within the frame
work of a self-consistent model of the finite-size dark matter (DM) halo of the Galaxy. The halo
model includes the gravitational influence of the observed visible matter of the Galaxy on the phase
space distribution function of the WIMPs constituting the Galaxy’s DM halo in a self-consistent
manner. Unlike in the so-called “Standard Halo Model” (SHM) which, as we argue, has several
undesirable properties, the velocity distribution of the WIMPs in our model is non-Maxwellian,
with a high-velocity cutoff determined self-consistently by the model itself. The parameters of the
model are determined from a fit to the rotation curve data of the Galaxy. We find that the require-
ment of consistency of the S-K implied upper limits on the WIMP-proton elastic cross section as
a function of WIMP mass imposes stringent restrictions on the branching fractions of the various
WIMP annihilation channels. For SI interaction, while the S-K upper limits do not place addi-
tional restrictions on the DAMA-compatible region of the WIMP parameter space if the WIMPs
annihilate dominantly to¯bb and/or ¯ cc, portions of the DAMA-compatible region can be excluded
if WIMP annihilations to τ+τ−and ν¯ ν occur at larger than 10% and 10−3levels, respectively. For
SD interaction, on the other hand, the restrictions on the possible annihilation channels are much
more stringent, essentially ruling out the DAMA-compatible region of the WIMP parameter space
if the relatively low-mass (∼ 2 – 20 GeV) WIMPs under consideration annihilate predominantly to
any mixture of¯bb, ¯ cc, τ+τ−, and ν¯ ν final states. The upper limits on the branching fractions of
the various annihilation channels obtained here are about a factor of 10 more restrictive than those
obtained earlier within the context of the SHM.
∗susmita.kundu@saha.ac.in
†pijush.bhattacharjee@saha.ac.in

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I.INTRODUCTION
Weakly Interacting Massive Particles (WIMPs) (hereafter generically denoted by χ) with masses mχin the range of
few GeV to few TeV are a natural candidate for the dark matter (DM) in the Universe; See e.g., Refs. [1–5] for reviews.
Several experiments are currently engaged in efforts to directly detect such WIMPs by observing nuclear recoils due to
scattering of WIMPs off nuclei in suitably chosen detector materials in underground laboratories. Recent results from
some of these direct detection (DD) experiments, in particular the annual modulation of the nuclear recoil event rates
reported by the DAMA/LIBRA collaboration [6] and the excess of low energy recoil events reported by the CoGeNT
collaboration [7] have raised the interesting possibility [8, 9] that these events could be due to WIMPs of relatively
low mass, approximately in the range ∼ 5–10 GeV, interacting with nuclei with a WIMP-nucleon spin-independent
elastic cross section in the region of few ×10−4pb, without conflicting with the null results from other experiments
such as XENON10 [10], XENON100 [11] and CDMS-II-Si [12]. Earlier analyses (before the CoGeNT results [7]) had
also found similar compatibility of the DAMA/LIBRA annual modulation signal with the null results from other DD
experiments; see, e.g., Refs. [13–15].1
Scattering of WIMPs off nuclei can also lead to capture of the WIMPs by massive astrophysical bodies such as the Sun
or the Earth if, after scattering off a nucleus inside the body, the velocity of the WIMP falls below the escape velocity
of the body. The WIMPs so captured over the lifetime of the capturing body would gradually settle down to the core
of the body where they would annihilate and produce standard model particles, e.g., W+W−, Z0Z0, τ+τ−, t¯t, b¯b, c¯ c,
etc. Decays of these particles would then produce neutrinos, gamma rays, electrons-positrons, protons-antiprotons,
etc. For astrophysical objects like the Sun or the Earth, only the neutrinos would be able to escape. Detection
of these neutrinos by large neutrino detectors can, albeit indirectly, provide a signature of WIMPs. Although no
detection has yet been reported, the Super-Kamiokande (S-K) detector, for example, has provided upper limits on the
possible neutrino flux from WIMP annihilation in the Sun as a function of the WIMP mass [18]. Similarly, the γ-rays
produced in the annihilation of the WIMPs in suitable astrophysical environments with enhanced DM density but
low optical depth to gamma rays, such as in the central region of our Galaxy, in dark matter dominated objects such
as dwarf galaxies, and in clusters of galaxies, can offer a complimentary avenue of indirect detection (ID) of WIMPs.
Although no unambiguous gamma ray signals of dark matter origin have been reported, a recent analysis [19] of
the spectral and morphological features of the gamma ray emission from the inner Galactic Center region (within a
Galactocentric radius of ∼ 175pc) measured by the Fermi Gamma-ray Space Telescope (FGST) seems to suggest the
presence of a gamma ray emission component which is difficult to explain in terms of known sources and/or process
of gamma ray production, but is consistent with that expected from annihilations of WIMPs of mass in the 7–9 GeV
range (annihilating primarily to tau leptons) with a suitably chosen density and distribution of the dark matter in
the Galactic Center region; see, however, Ref. [20] for a different view.
In this paper we focus on the neutrinos produced by annihilations of WIMPs in the core of the Sun, and study the
constraints imposed on the WIMP mass vs. WIMP-nucleon cross section, for low-mass (<∼20GeV) WIMPs, from
non-detection of such neutrinos. This is done within the context of a self-consistent model of the finite-size dark halo of
the Galaxy [15, 21] that includes the gravitational effect of the observed visible matter on the DM in a self-consistent
manner, with the parameters of the model determined from fits to the rotation curve data of the Galaxy [22, 23].
The expected flux of neutrinos from the Sun due to WIMP annihilations depends on the rate at which WIMPs are
captured by the Sun. The capture rate depends on the density as well as the velocity distribution of the WIMPs in
the solar neighborhood as the Sun goes around the Galaxy. The density and velocity distribution of the WIMPs in the
Galaxy are a priori unknown. Most earlier studies of neutrinos from WIMP capture and annihilation in the Sun have
been done within the context of the so-called “Standard Halo Model” (SHM) in which the DM halo of the Galaxy is
described by a single component isothermal sphere [24] with a Maxwellian velocity distribution of the DM particles in
the Galactic rest frame [1, 25, 26]). The velocity distribution is isotropic, and is usually truncated at a chosen value
of the escape speed of the Galaxy. The density of DM in the solar neighborhood is typically taken to be in the range
ρDM,⊙∼ 0.3 ± 0.1GeV/cm3[27–30]. The velocity dispersion, ?v2?1/2, the parameter characterizing the Maxwellian
velocity distribution, is typically taken to be ∼ 270kms−1. This follows from the relation [24], ?v2?1/2=
?
3
2vc,∞,
1The question of compatibility of the DAMA/LIBRA and CoGeNT results with the null results of other experiments, however, remains
controversial; see, e.g., the results of a recent reanalysis of the CDMS-II Germanium data with a lowered recoil-energy threshold of 2
keV [16], as well as the recent results from the XENON100 collaboration [17], both of which claim to disfavor such a compatibility.

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between the velocity dispersion of the particles constituting a single-component self-gravitating isothermal sphere and
the asymptotic value of the circular rotation speed, vc,∞, of a test particle in the gravitational field of the isothermal
sphere and assuming vc,∞≈ vc,⊙≈ 220kms−1, where vc,⊙is the measured value of the circular rotation velocity of
the Galaxy in the solar neighborhood.2Neutrino flux from DM annihilation in the Sun for low mass WIMPs and
the resulting constraints on WIMP properties from the Super-Kamiokande upper limits on such neutrinos [18] have
been studied within the context of the SHM in Ref. [32].
Whereas the SHM serves as a useful benchmark model, there are a number of reasons why the SHM does not provide
a satisfactory description of the dynamics of the Galaxy. First, it does not take into account the modification of
the phase space structure of the DM halo due to the significant gravitational effect of the observed visible matter
on the DM particles inside and up to the solar circle. Second, the isothermal sphere model of the halo is infinite in
extent and has a formally divergent mass, with mass inside a radius r, M(r) ∝ r, as r → ∞, and is thus unsuitable
for representing a halo of finite size. Third, the procedure of truncating the Maxwellian speed distribution at a
chosen value of the local (solar neighborhood) escape speed is not a self-consistent one because the resulting speed
distribution is not in general a self-consistent solution of the steady-state collisionless Boltzmann equation describing
a finite system of collisionless DM particles. In addition, since the rotation curve for such a truncated Maxwellian
is, in general, not asymptotically flat, the relation ?v2?1/2=
Maxwellian speed distribution of the isothermal sphere, as done in the SHM, is not valid in general. Finally, recent
numerical simulations [30] seem to find that the velocity distribution of the Dark Matter particles deviates significantly
from the usual Maxwellian form. These issues are further discussed in detail in Ref. [15], where we have also discussed
the constraints on WIMP properties from the results of the direct detection (DD) experiments within the context of
our self-consistent halo model mentioned above. The present paper extends this study to the case of indirect detection
(ID) of WIMPs via neutrinos from WIMP annihilations in the Sun.
?
3
2vc,∞used to determine the value of ?v2?1/2in the
Our model of the phase space structure of the finite-size DM halo of the Galaxy is based on the so-called “lowered”
(or truncated) isothermal models (often called “King models”) [24] of the phase-space distribution function (DF)
of collisionless particles. These models are proper self-consistent solutions of the collisionless Boltzmann equation
representing nearly isothermal systems of finite physical size and mass. There are two important features of these
models: First, at every location within the system a DM particle can have speeds up to a maximum speed which is
self-consistently determined by the model itself. A particle of maximum velocity at any location within the system
can just reach its outer boundary, fixed by the truncation radius, a parameter of the model, where the DM density
by construction vanishes. Second, the speed distribution of the particles constituting the system is non-Maxwellian.
To include the gravitational effect of the observed visible matter on the DM particles, we modify the “pure” King
model DF by replacing the gravitational potential appearing in the King model DF by the total gravitational potential
consisting of the sum of those due to DM and the observed visible matter. This interaction with the visible matter
influences both the density profile and the velocity distribution of the dark matter particles as compared to those for
a “pure” King model. In particular, the dark matter is pulled in by the visible matter, thereby increasing its central
density significantly. When the visible matter density is set to zero and the truncation radius is set to infinity, our
halo model becomes identical to that of a single-component isothermal sphere used in the SHM. For further discussion
of the model, see [15, 21].
The DM distribution in the Galaxy may have significant amount of substructures which may have interesting effects
on the WIMP capture and annihilation rates [33]. However, not much information, based on observational data, is
available about the spatial distribution and internal structures of these substructures. As such, in this paper we shall
be concerned only with the smooth component of the DM distribution in the Galaxy described by our self-consistent
model mentioned above, the parameters of which are determined from the observed rotation curve data for the Galaxy.
We calculate the 90% C.L. upper limits on the WIMP-proton spin-independent (SI) as well as spin-dependent (SD)
elastic cross sections as a function of the WIMP mass, for various WIMP annihilation channels, using the null result
of the S-K experiment’s search for WIMP annihilation neutrinos from the Sun [18], and study the consistency of those
limits with the 90% C.L.“DAMA-compatible” regions — the regions of the WIMP mass versus cross section parameter
space within which the annual modulation signal observed by the DAMA/LIBRA experiment [6] is compatible with
2A somewhat higher value of vc,⊙ ≈ 250kms−1, as suggested by a recent study [31], would imply a correspondingly higher value of
?v2?1/2iso≈ 306kms−1.

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the null results of other DD experiments — determined within the context of our halo model [15]. We find that
the requirement of such consistency imposes stringent restrictions on the branching fractions of the various WIMP
annihilation channels. For example, in the case of spin-independent WIMP-proton interaction, while the S-K upper
limits do not place additional restrictions on the DAMA-compatible region of the WIMP parameter space if the WIMPs
annihilate dominantly to¯bb, ¯ cc, portions of the DAMA-compatible region can be excluded if WIMP annihilations to
τ+τ−and ν¯ ν occur at larger than 10% and 10−3levels, respectively. In the case of spin-dependent interactions, on
the other hand, the restrictions on the branching fractions of various annihilation channels are much more stringent,
essentially ruling out the DAMA-compatible region of the WIMP parameter space if the relatively low-mass WIMPs
under consideration annihilate predominantly to any mixture of¯bb, ¯ cc, τ+τ−, and ν¯ ν final states. In quantitative
terms, the upper limits on the branching fractions of the various annihilation channels obtained here are about a
factor of 10 more restrictive than the result obtained in [32] within the context of the SHM. Our results, based as
they are on a self-consistent model of the Galaxy’s dark matter halo, the parameters of which are determined by a fit
to the rotation curve data of the Galaxy, therefore, strengthen the conclusion of [32]3.
The rest of the paper is organized as follows: In section II we briefly describe the self-consistent model of the DM halo
of the Galaxy. The formalism of calculating the WIMP capture and annihilation rates in the Sun within the context
of our DM halo model, and that for calculating the resulting neutrino flux and event rate in the Super-Kamiokande
detector, are discussed in sections III and IV, respectively. Our results and the constraints on the WIMP properties
implied by these results are described in section V. The paper ends with a Summary in section VI.
II. THE SELF-CONSISTENT TRUNCATED ISOTHERMAL MODEL OF THE MILKY WAY’S DARK
MATTER HALO
The phase space distribution function (DF) of the DM particles constituting a truncated isothermal halo of the Galaxy
can be taken, in the rest frame of the Galaxy, to be of the “King model” form [15, 21, 24],
f(x,v) ≡ f(E) =
?
ρ1(2πσ2)−3/2?
0
eE/σ2− 1
?
for E > 0,
for E ≤ 0,
(1)
where
E(x) ≡ Φ(rt) −
?1
2v2+ Φ(x)
?
≡ Ψ(x) −1
2v2, (2)
is the so-called “relative energy” and Ψ(x) = −Φ(x)+Φ(rt) the “relative potential”, Φ(x) being the total gravitational
potential under which the particles move, with boundary condition Φ(0) = 0. The relative potential and relative
energy, by construction, vanish at |x| = rt, the truncation radius, which represents the outer edge of the system where
the particle density vanishes. At any location x the maximum speed a particle of the system can have is
vmax(x) =
?
2Ψ(x), (3)
at which the relative energy E and, as a consequence, the DF (1), vanish. The model has three parameters, namely,
ρ1, σ and rt. Note that the parameter σ in the King model is not same as the usual velocity dispersion parameter of
the isothermal phase space DF [24]. Also, in our calculations below, we shall use the parameter ρDM,⊙, the value of
the DM density at the location of the Sun, in place of the parameter ρ1.
Integration of f(x,v) over all velocities gives the DM density at the position x:
ρDM(x) =
ρ1
(2πσ2)3/2
?
?√
2Ψ(x)
0
dv 4πv2
?
exp
?Ψ(x) − v2/2
?
πσ2
σ2
?
− 1
?
(4)
= ρ1
exp
?Ψ(x)
σ2
?
erf
?√Ψ(x)
σ
?
−
4Ψ(x)
?
1 +2Ψ(x)
3σ2
??
, (5)
3In the present paper, the CoGeNT results [7] are not included in the analysis. Preliminary results of the analysis [34] to find the
“CoGeNT-compatible” region in the WIMP mass vs. cross section plane within the context of our halo model indicates that its inclusion
will not significantly change the above constraints on the branching fractions for the various annihilation channels.

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which satisfies the Poisson equation
∇2ΦDM(x) = 4πGρDM(x), (6)
where ΦDMis the contribution of the DM component to the total gravitational potential,
Φ(x) = ΦDM(x) + ΦVM(x), (7)
in presence of the visible matter (VM) whose gravitational potential, ΦVM, satisfies its own Poisson equation, namely,
∇2ΦVM(x) = 4πGρVM(x). (8)
We choose the boundary conditions
ΦDM(0) = ΦVM(0) = 0, and (∇ΦDM)|x|=0= (∇ΦVM)|x|=0= 0. (9)
The mass of the system, defined as the total mass contained within rt, is given by GM(rt)/rt = [Φ(∞) − Φ(rt)].
Note that, because of the chosen boundary condition Φ(0) = 0, Φ(∞) is a non-zero positive constant.
Since the visible matter distribution ρVM(x), and hence the potential ΦVM(x), are known from observations and
modeling, the solutions of equation (6) together with equations (5), (7) and the boundary conditions (9), give us a
three-parameter family of self-consistent pairs of ρDM(x) and ΦDM(x) for chosen values of the parameters (ρ1,σ,rt).
The values of these parameters for the Galaxy can be determined by comparing the theoretically calculated rotation
curve, vc(R), given by
v2
c(R) = R∂
∂R
?
Φ(R,z = 0)
?
= R∂
∂R
?
ΦDM(R,z = 0) + ΦVM(R,z = 0)
?
, (10)
with the observed rotation curve data of the Galaxy. (Here R is Galactocentric distance on the equatorial plane and
z is the distance normal to the equatorial plane.) This procedure was described in detail in Refs. [15, 21] where,
for the visible matter density distribution described there, we determined the values of the parameters rt and σ
that gave reasonably good fit to the rotation curve data of the Galaxy [22, 23] for each of the three chosen values
of the parameter ρDM,⊙= 0.2, 0.3 and 0.4 GeV/cm3. These models are summarized in Table I, which we use for
our calculations in this paper. The density profiles, mass profiles, velocity distributions of the DM particles and the
ModelρDM,⊙
(GeV/cm3) (kpc) (kms−1)
rt
σ
M1
M2
M3
0.2
0.3
0.4
120.0
80.0
80.0
300.0
400.0
300.0
TABLE I: Parameters of our self-consistent model of the Milky Way’s Dark Matter halo that give good fits to the Galaxy’s
rotation curve data, for the three chosen values of the DM density at the solar neighborhood
resulting rotation curves in each of these models are discussed in detail in Ref. [15].
With our halo model specified, we now briefly review the basic formalism of calculating the WIMP capture and
annihilation rates within the context of our halo model.
III. CAPTURE AND ANNIHILATION RATES
The capture rate per unit volume at radius r inside the Sun can be written as [35, 36]
dC
dV(r) =
?
d3u
˜f(u)
u
wΩ−(w), (11)
where˜f(u) is the WIMP velocity distribution, as measured in the Sun’s rest frame, in the neighborhood of the
Sun’s location in the Galaxy, and w(r) =
?u2+ wesc(r) is the WIMP’s speed at the radius r inside the Sun,