Slide 70
Slide 70 text
flip . tanedo 81
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@ PRESENTING SCIENCE
70
Equations
ius
rN
that maximizes the radial number density
nN
(
r
)
r2
of target nucleus
alues, the contact interaction limit fails for
mA
0
. 3 MeV. Rather than
mentum term altogether, a slightly more sophisticated approach would be
tution p2
(1 cos
✓CM
) !
µ2
N w2
. In this work, however, we keep the full p
propagator and evaluate the capture rate numerically so that our results
ut parameter space. We have confirmed that our results reproduce those
n the corners of parameter space where simplifying assumptions are valid.
match Ref. [48] in the large-
mA
0
, point-like cross section limit.
apture rates, it is convenient to re-express the di↵erential cross section
ecoil energy
ER
=
µ2
N w2
(1 cos
✓CM
)
/mN
in the lab frame. In the non-
he expression simplifies to [49]
d N
dER
⇡ 8
⇡"2
↵X↵Z2
N
mN
w2
(2
mN ER
+
m2
A
0
)2
|
FN
|2
.
(9)
the Helm form factor [50],
|
FN
(
ER
)|2 = exp [
ER/EN
]
,
(10)
y Gould [13, 14, 16] where
v
(
r
) is the escape velocity at radius
r
and
u
is the dar
elocity asymptotically far from the Earth.
The total capture rate is obtained by integrating Eq. (11) over the region of p
pace where the final state dark matter particle has energy less than
mXv2
(
r
)
/
hus gravitationally captured. The escape velocity
v
(
r
) and number densities
n
etermined straightforwardly from the density data enumerated in the Preliminary R
arth Model [51]. Following Edsj¨
o and Lundberg [20], the target number den
modeled by dividing the Earth into two layers, the core and the mantle, with
ensities and elemental compositions given in Table I. The capture rate is then
P
N CN
cap
, where the rate on target
N
is
CN
cap
=
nX
Z
R
0
dr
4
⇡r2
nN
(
r
)
Z 1
0
dw
4
⇡w3
f
(
w, r
)
Z
Emax
Emin
dER
d N
dER
⇥(
E
)
.
ere ⇥(
E
) = ⇥(
Emax Emin
) imposes the constraint that capture is kinematically
y enforcing that the minimum energy transfer,
Emin
, to gravitationally capture
6
radius
r
, which is distorted from the free-space Maxwell–Boltzmann distribution,
f
(
u
), by
the Earth’s motion and gravitational potential. We follow the velocity notation introduced
by Gould [13, 14, 16] where
v
(
r
) is the escape velocity at radius
r
and
u
is the dark matter
velocity asymptotically far from the Earth.
The total capture rate is obtained by integrating Eq. (11) over the region of parameter
space where the final state dark matter particle has energy less than
mXv2
(
r
)
/
2 and is
thus gravitationally captured. The escape velocity
v
(
r
) and number densities
nN
(
r
) are
determined straightforwardly from the density data enumerated in the Preliminary Reference
Earth Model [51]. Following Edsj¨
o and Lundberg [20], the target number densities are
modeled by dividing the Earth into two layers, the core and the mantle, with constant
densities and elemental compositions given in Table I. The capture rate is then
Ccap
=
P
N CN
cap
, where the rate on target
N
is
CN
cap
=
nX
Z
R
0
dr
4
⇡r2
nN
(
r
)
Z 1
0
dw
4
⇡w3
f
(
w, r
)
Z
Emax
Emin
dER
d N
dER
⇥(
E
)
.
(12)
Here ⇥(
E
) = ⇥(
Emax Emin
) imposes the constraint that capture is kinematically possible
by enforcing that the minimum energy transfer,
Emin
, to gravitationally capture the dark
6
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