2021-USAPhO_plus-02
The Dark Forest
USAPhO · 2021 · Elektromagnetisme
The Dark Forest
Dark matter could be made of hypothetical, extremely light particles called axions. Because individual axions are so light, experiments do not search for individual axions, but rather for the
classical axion field formed by a large collection of axions, which oscillates as
a(t) = a0 sin(t).
This is analogous to how a large collection of photons can form a classical electromagnetic field. In
the presence of a magnetic field B and an axion field a, the axion field produces an effective current
J = g B
where we define = da/dt. The effective current produces electromagnetic fields in exactly the same
way as ordinary current, though it does not come from the motion of actual charges. Experiments
can search for axion dark matter using systems which are resonantly driven by this current.
You may use fundamental constants in your answers, such as
c = 3.00
~ = 1.055 10 s
e = 1.602 10
G = 6.67 10 m2 /kg2
= 4 10
kB = 1.38 10.
You do not have to provide numeric answers unless asked. When asked to “estimate”, you may drop
constants of order one. The numeric values provided below are from standard references where ~,
c, , and are set to one; to get correct numeric results, you must restore these factors yourself.
1. First, we will describe some physical properties of the axion field.
(a) Consider a single axion at rest, with mass m. Find its associated angular frequency . This
will be the angular frequency of the corresponding classical field, when there are many axions.
(b) Suppose dark matter is distributed spherically symmetrically in the galaxy with uniform
density . The solar system is a distance r from the center of the galaxy and orbits around
it with period T . Neglecting everything besides dark matter, find the dark matter density .
(c) The energy density of the axion field is m2 a20 /(2~3 c). Find the axion field amplitude a0 .
(d) The radius and period of the Sun’s orbit, as well as a typical axion mass, are
r = 2.5 ,
T = 7.1 ,
m = 1.0 10.
Numerically compute the axion field amplitude a0 .
(e) In this problem, we treat the axion field as spatially uniform within a terrestrial laboratory.
To verify that this assumption is reasonable, numerically estimate the axion field’s wavelength
, assuming the axions have the same galactic speed as the Sun.
(f) In part (a), you found by neglecting the axion’s speed. In reality, the axion’s finite speed
changes the frequency to + , in a frame at rest with respect to the galactic center.
Numerically estimate / to show that it is reasonable to neglect this effect.
Copyright ©2021 American Association of Physics Teachers
2021 USAPhO+
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The ABRACADABRA1 experiment, currently taking data at MIT, is a toroidal solenoid with
inner and outer radius Rin and Rout and height h. You may assume h
Rout for simplicity. A
superconducting wire carrying current I wraps N times around the toroid, where N is high enough
to neglect the discreteness of the wires. A circular pickup loop with radius slightly less than Rin is
placed at the center of the toroid.
2. Now, we will find the axion signal generated in the ABRACADABRA apparatus.
(a) Find the magnetic field B(r) inside the toroid due to the superconducting current.
(b) The superconducting wires lose their superconductivity when exposed to a magnetic field
greater than Bmax . Find the maximum possible current Imax that can be used, and assume
this current is used in later parts.
(c) Assuming that is small, find the magnetic flux B (t) through the pickup loop due to the
axion field in terms of a0 , g, , Bmax , and the dimensions of the apparatus. (You may ignore
any currents induced on the surfaces of the superconducting wires. Accounting for them
makes the problem much harder, but does not substantially affect the final result.)
(d) If is too large, the result above breaks down due to radiation effects. Estimate the frequency
c where this happens.
(e) Using the design values
Rin = ,
Rout = ,
h =
estimate the numerical value of /c .
(f) Let be the amplitude of the time-varying axion flux. Using the typical values
Bmax = ,
g = 1.0 1016 GeV1
and your previous results, compute the numerical value of .
1
aka, A Broadband/Resonant Approach to Cosmic Axion Detection with an Amplifying B-field Ring Apparatus.
Copyright ©2021 American Association of Physics Teachers
2021 USAPhO+
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The pickup loop has inductance L and is attached to a capacitor, forming a circuit with resonant
frequency equal to the axion frequency . The circuit also has a small internal resistance R in series,
and is at temperature T . The axion signal can be detected by monitoring the current in the circuit.
The main source of noise is thermal noise, which causes fluctuations in the current.
3. We will now estimate the sensitivity of ABRACADABRA to axions.
(a) The axion produces a current which oscillates sinusoidally. Find the signal current amplitude
Is in terms of , , and the circuit parameters.
(b) Find the average value of the current squared hI 2 i in the circuit due to thermal noise.
(c) At anypmoment in time, the noise current is oscillating sinusoidally with typical amplitude
In = hI 2 i, which is much larger than Is . However, the phase of the noise current also
fluctuates randomly, so that after a typical time tc , its phase will be roughly independent of
the phase it had before. Find an estimate for tc in terms of and the circuit parameters.
(Hint: at any given moment, the thermal noise current is simultaneously being produced by
the random motion of electrons in the circuit, and damped by the resistor.)
(d) Suppose the experiment runs for a total time te
tc . Roughly estimate the average amplitude
of the noise current over this period of time.
(e) The axion is detectable if the signal current amplitude is larger than the averaged noise
current amplitude, and the circuit parameters are
L = 1 mH,
R = ,
T = .
Roughly numerically estimate the time needed to potentially detect the axion. (Hint: if your
answer seems strange, note that in reality, the axion’s phase also fluctuates over time, because
of the effect of part 1(f). In addition, we don’t know ahead of time, so the experiment
needs to be run many times. We ignored these effects here to keep things simple.)
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2021 USAPhO+
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Question 3
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