Production and detection of an axion dark matter echo

Electromagnetic radiation with angular frequency equal to half the axion mass stimulates the decay of cold dark matter axions and produces an echo, i.e. faint electromagnetic radiation traveling in the opposite direction. We propose to search for axion dark matter by sending out to space a powerful beam of microwave radiation and listening for its echo. We find that this is a promising approach to axion detection in the $2 \times 10^{-7}$ to $3 \times 10^{-4}$ eV mass range.

The identity of dark matter remains one of the central questions in science today [1].One of the leading candidates is the QCD axion.This hypothetical particle was originally postulated as a solution [2] to the Strong CP Problem of the Standard Model of particle physics, i.e. the puzzle why the strong interactions conserve P and CP.The properties of the QCD axion are given almost entirely in terms of a single parameter f a , called the axion decay constant.In particular the mass of the axion m = 0.6 × 10 −5 eV 10 12 GeV f a , and its electromagnetic coupling where φ(x) is the axion field and g γ is a model-dependent dimensionless coupling that is generically of order one.In the KSVZ model [3], g γ = −0.97.In the DFSZ model [4], and in all grand-unified axion models, g γ = 0.36.
Early laboratory limits and stellar evolution constraints require f a > 10 9 GeV [5] in which case the axion is so extremely weakly coupled that it was once dubbed "invisible".However, cosmology came to the rescue.Axions are overproduced during the QCD phase transition in the simplest cosmological scenarios unless f a < ∼ 10 12 GeV [6].The precise limit depends on whether inflation occurs before or after the phase transition during which Peccei-Quinn symmetry is spontaneously broken, and other considerations such axion production by topological defects, the precise temperature dependence of the axion mass, and the amount of entropy production associated with the QCD phase transition.In any case, the axions produced during the QCD phase transition are a form of cold dark matter [7] and therefore a candidate for the constituent particle of galactic halos.The topic is reviewed in refs.[8].
Axions and axion-like particles (ALPs) are by-products of variously motivated proposals for physics beyond the Standard Model, including its supersymmetric extensions [9] and string theory [10].ALPs [11] are light pseudoscalar particles like QCD axions but without the definite relationship between mass and coupling implied by Eqs. ( 1) and (2).Several methods to test experimentally the axion hypothesis have been proposed and some have produced useful limits.For dark matter axion searches, the different approaches include the cavity technique [12,13], wire [14] and dielectric plate [15] detectors, magnetic resonance methods [16,17], the LC circuit approach [18], and atomic transitions [19].Solar axions are searched for by their conversion to x-rays in a laboratory magnetic field [12,20] and in crystals [21], and through the axio-electric effect [22].Finally, "shining light through wall" experiments attempt to convert photons to axions on one side of a wall followed by back conversion on the other side [23].
The purpose of our paper is to propose a new approach to axion dark matter detection.The effect we exploit is the stimulated decay of cold dark matter axions by a powerful beam of microwave radiation.Refs.[24] discuss the stimulated decay of axion dark matter in astrophysical contexts.We first describe the effect in the rest frame of a perfectly cold axion fluid, and then generalize to the case where the observer is moving with respect to the axion fluid and to the case where the axion fluid has velocity dispersion.
Let A 0 ( x, t) be the vector potential of the initial outgoing radiation.In the presence of axions, A 0 is itself a source of electromagnetic radiation A 1 ( x, t).Since axions are very weakly coupled, we have in radiation gauge where φ(t) = A sin(mt) is the axion field, and g ≡ g γ α π 1 fa is the overall coupling that appears in Eq. ( 2).The axion density is ρ = 1 2 A 2 m 2 .Let the outgoing radiation A 0 be stationary, linearly polarized and with angular frequencies ω at and near m/2.The retarded A 1 is in that case identical to A 0 except that 1) it flows exactly backwards because, up to a constant factor, its spatial Fourier transform is the same as that of A 0 whereas its angular frequency is opposite, 2) it is reduced relative to A 0 by a time-dependent factor proportional to gA, and 3) it is linearly polarized at a 90 • angle relative to A 0 .If A 0 is circularly polarized, A 1 has the same circular polariza-tion as A 0 .We call A 1 the echo wave.
The power in the echo wave is where dP0 dν is the spectral density of the outgoing wave at angular frequency ω = 2πν = m/2, and t is the time since the outgoing wave was first established.If the outgoing wave is emitted as a parallel beam of finite cross-section, it will spread as a result of its transverse wavevector components.The echo wave retraces the outgoing wave backward in time, returning to the location of emisssion of the outgoing wave with the latter's original transverse size.If the outgoing power P 0 is turned on for a time t and then turned off, the echo power P − given by Eq. ( 4) lasts forever in the future under the assumption that the perfectly cold axion fluid has infinite spatial extent.
Next, let us consider the case where the perfectly cold axion fluid is moving with velocity v with respect to the outgoing power source.Nothing changes in the axion fluid rest frame compared to the above discussion except that each increment dE 0 = P 0 dt of outgoing energy is emitted from a different location.The incremental echo power dP − , given by the RHS of Eq. ( 4) with t replaced by dt, returns forever to the location in the axion fuid rest frame from which the increment dE 0 of outgoing energy was emitted.In the frame of its source, the frequency at which the outgoing power stimulates axion decay is (c = 1) where k is the unit vector in the direction of the outgoing power.The frequency of the echo is The echo of the power emitted a time t e ago arrives displaced from the point of emission by d = v ⊥ t e where v ⊥ is the component of v perpendicular to k.To detect as much echo power as possible at or near the place of emission of the outgoing power, the observer wants v ⊥ as small as possible, i.e. k parallel or anti-parallel to v.
If the axion fluid has velocity dispersion, its density can be viewed as an integral over cold flows Everything said before holds true for each infinitesimal cold flow increment.The echo frequency has a spread δω − = m 2 δv where δv is the spread of axion velocities in the k direction.The echo of power emitted a time t e ago is spread over a transverse size δ d = δ v ⊥ t e where δ v ⊥ is the spread of axion velocities perpendicular to k.
It is clear from the above that the amount of echo power that the observer may easily collect depends sharply on the velocity distribution of axion dark matter on Earth as well as its total local density.We will consider two contrasting models of the Milky Way halo, the isothermal model [25] and the caustic ring model [26].In the isothermal model the local dark matter has density 300 MeV/cm 3 and velocity dispersion 270 km/s.In the caustic ring model, the local dark matter velocity distribution is dominated by a single flow, called the Big Flow, because of our proximity to the 5th caustic ring of dark matter in the Milky Way halo.An upper limit of order 70 m/s on the velocity dispersion of the Big Flow has been derived [27].The properties of the Big Flow depend sharply on our distance and position relative to the mid-plane cusp of the nearby caustic ring.We therefore discuss in some detail the observational evidence that constrains our position relative to the 5th caustic ring.
A triangular feature in the IRAS map of the galactic plane in the direction of galactic coordinates (l, b) = (80 • , 0 • ) is consistent with the expected imprint of the 5th caustic ring on gas and dust when viewed from a direction tangent to the ring [28].Moreover, a rise in the Milky Way rotation curve occurs at galactocentric radii consistent within measurement errors with the position of the IRAS triangle [28].The rise in the rotation curve starts with an upward kink and ends with a downward kink as expected from the gravitational pull of a caustic ring.The GAIA map of the galactic plane shows a clear triangle in the direction (l, b) = (−90 • , 0 • ) as well as the triangle seen originally in the IRAS map [29].The two triangles fix the tangent directions to the 5th caustic ring.The velocity vector of the Big Flow is then determined with good precision, so that the expected magnitude of v ⊥ is of order 5 km/s.This is discussed in detail below.
The Big Flow density is less well constrained.It was estimated to be 1 GeV/cm 3 [26] when only the IRAS triangle was known (before the GAIA data appeared) under the following three assumptions: 1) that the 5th caustic ring is axially symmetric, 2) that it is centered on the galactic center, and 3) that the dark matter falls from afar onto the galaxy equally from all directions.The positions of the GAIA and IRAS triangles imply that the caustic ring center is not at the galactic center but displaced from it by approximately 700 pc in the direction of decreasing galactic longitude [29].This places the Earth much closer to the mid-plane cusp of the 5th caustic ring than under assumption 2).As a result the Big Flow density is boosted by a factor of approximately nine.A second consideration leads to a further upward revision of the Big Flow density; namely the caustic ring model was modified to include the presence of a Big Vortex along the symmetry axis of the Milky Way [30].A Big Vortex is expected to result from the merger of many mutually attractive small vortices.The Big Vortex explains the average size of the rises caused by caustic rings in the Milky Way rotation curve.It implies that the dark matter does not fall isotropically onto the galaxy but preferentially along the galactic plane, boosting the density of the Big Flow by a factor of approximately five [30] compared to its estimate under assumption 3).In view of the uncertainty on the Big Flow density, we will give sensitivity estimates for our approach to axion dark matter detection in the caustic ring model for ρ = 1, 9 and 45 GeV/cm 3 , as well as in the isothermal model.
Let us first consider the case where the local axion dark matter density is dominated by a single cold flow, as in the caustic ring model.As was discussed above, the largest amount of echo power is available for detection near the source of outgoing power when the outgoing power has direction k parallel or anti-parallel to the velocity vector v of the axion fluid with respect to the observer.That velocity vector is a sum where v a is the velocity of the axion fluid with respect to a non-rotating coordinate system attached to the Milky Way galaxy, v LSR is the velocity of the Local Standard of Rest (LSR) in that same coordinate system, v ⊙ is the velocity of the Sun with respect to the LSR, and v ⊗ is the velocity of the observer with respect to the Sun as a result of the orbital and rotational motions of the Earth.We are particularly interested in the extent to which the uncertainties in the several terms on the RHS of Eq. ( 8) affect our ability to minimize v ⊥ .v ⊗ (t) is known with great precision.The components of v ⊙ are known with a precision of order one or two km/s.v LSR is in the direction of galactic rotation by definition.Its magnitude (often quoted to be 220 km/s) is known with an uncertainty of order 20 km/s.The magnitude of v a for the Big Flow is approximately 520 km/s [26].Its direction is fixed by the positions of the IRAS and GAIA triangles on the sky with a precision of order 0.01 radians.So we expect that it is not possible to reduce v ⊥ to less than of order 5 km/s, the nominal value we use below.Because the Big Flow is almost in the direction of galactic rotation (within approximately 12 • ), the uncertainty on the magnitude of v LSR is less important.Consider a dish (e.g. a radiotelescope) of radius R collecting echo power at the location of the outgoing power source.Because the echo from outgoing power emitted a time t e ago is displaced by d = v ⊥ t e , the amount of echo power collected by the dish is given by Eq (4) with t of order R/| v ⊥ |, i.e.
where C is a number of order one which depends on the configuration of the source relative to the dish detector.
As an example, C = 0.42 in case the outgoing power is emitted uniformly by the receiver dish itself.This may not be the best option, however.It is probably better to place several source dishes around the receiver dish.It is straightforward to calculate C for each configuration.
Let us assume that a pulse of outgoing power P 0 , with frequency ν 0 and uniform spectral density dP0 dν = P0 ∆ν0 over bandwidth ∆ν 0 , is emitted during a time t m .Provided that the echo power P c = 2.33 × 10 −31 P 0 10 kHz ∆ν 0 g γ 0.36 is received over the same time interval t m .Since the magnitude of the velocity of the Big Flow relative to us v ≃ 520 km/s -220 km/s = 300 km/s, the frequency of the echo power is red-or blue-shifted from ν 0 by ∆ν ≃ 2 × 10 −3 ν 0 .The echo power has bandwidth B = 2δv ν < 5 × 10 −7 ν since the velocity dispersion of the Big Flow is less than 70 m/s [30].The frequency range of interest is approximately 30 MHz to 30 GHz because the Earth's atmosphere is mostly transparent at those frequencies.It corresponds to the mass range 2.5 × 10 −7 < m < 2.5×10 −4 eV, which happens to be prime hunting ground for QCD axions.The cosmic microwave background and radio emission by astrophysical sources are irreducible sources of noise.In addition there will be instrumental noise to contend with.We use a nominal value of 10 K for the total noise temperature.The signal to noise ratio with which the echo power is detected when ω 0 falls within the angular frequency range of the emitted power is given by Dicke's radiometer equation Combining Eqs. ( 11) and ( 12) and setting B = 5×10 −7 ν, the total outgoing energy per logarithmic frequency interval necessary to detect the axion echo with a given signal to noise ratio is found to be: We used Eq. ( 1) and m = 4πν.Fig. 1 shows the sensitivity to g γ of an axion echo search that consumes 10  13) and (15).
MWyear of outgoing energy for each octave (factor of 2) in axion mass covered, for ρ = 1, 9 and 45 GeV/cm 3 and the nominal values of all other experimental parameters used in Eq. ( 13).
In the isothermal model, v a = 0 on average.In a nonrotating galactic reference frame the velocity distribution is Gaussian with dispersion √ 3σ ≡ v • v ≃ 270 km/s.In the LSR, the axion fluid moves with speed 220 km/s in the direction opposite to that of galactic rotation.Assuming the direction k of the outgoing power is parallel (anti-parallel) to the direction of galactic rotation the echo power is blue (red)-shifted in frequency by a fractional amount whose average is ∆ν ν ≃ 440 km/s = 1.5 × 10 −3 and whose rms deviation is δν ν = 2σ ≃ 1.04 × 10 −3 .The echo from outgoing energy that was emitted a time t e ago is spread in space over a fuzzy circular region whose radius is Gaussiandistributed with average value σt e .Eq. ( 9) holds with In view of Eq. ( 10) we now require t m > 2 × 10 −4 sec R 50 m .Using Eq. ( 12) with B = 4σν = 2.1 × 10 −3 ν and setting ρ = 0.3 GeV/cm 3 , we find The sensitivity to g γ in the isothermal model is shown in Fig. 1 as well.13) and (15).
The echo method appears an attractive approach to axion dark matter detection because it uses relatively old technology and because it is applicable over a wide a range of axion masses.The method works better in the caustic ring model than in the isothermal model for three reasons: 1) the density is higher, 2) the echo has less spread in frequency, and 3) the echo has less spread in physical space.Higher density helps the cavity method equally.In Fig. 2 we show the sensitivity of the echo method to |g γ | √ ρ as well as the published limits obtained by searches using the cavity method.

FIG. 1 :
FIG. 1: Sensitivity to |gγ| as function of mass of an axion echo search consuming 10 MWyear of outgoing energy for each factor two in axion mass range covered, in the caustic ring model with Big Flow densities ρ = 1, 9, and 45 GeV/cm 3 , and in the isothermal model with density ρ = 0.30 GeV/cm 3 , assuming all other experimental parameters have the nominal values shown in Eqs.(13) and(15).

FIG. 2 :
FIG. 2: Sensitivity to |gγ | √ ρ as a function of mass of an axion echo search consuming 10 MWyear of outgoing energy for each factor two in axion mass range searched, in the caustic ring model and in the isothermal model, assuming the nominal values of the experimental parameters shown in Eqs.(13) and(15).