Momentum and Matter Matter for Axion Dark Matter Matters on Earth

The Gist

Imagine the universe is filled with a ghostly, invisible fog called Axion Dark Matter. For decades, scientists have been trying to catch this fog using giant, sensitive detectors on Earth. They've been looking for two things: how much "fog" is there (the field value) and how fast it's "sloshing" or changing direction (the gradient).

This paper asks a simple but profound question: Does the Earth itself mess with this fog as it passes through?

Think of the Earth not just as a rock we stand on, but as a giant, dense sponge sitting in a river. When the axion "river" flows past the sponge, does the sponge change the water's speed or height?

Here is the breakdown of what the authors found, using everyday analogies:

1. The "Sponge" Effect (Matter Effects)

In a vacuum (empty space), axions behave like a calm, steady wave. But when they hit Earth, they interact with the atoms inside our planet. The authors describe this as the Earth acting like a medium that changes the "refractive index" for these waves, similar to how a straw looks bent when you put it in a glass of water.

• The Twist: The paper argues that previous studies assumed the axions were perfectly still (zero momentum) as they hit Earth. The authors say, "Wait, Earth is moving through the galaxy!" Because the axions have momentum (they are moving), the interaction is more complex and less extreme than previously thought.

2. The Two Main Surprises

The paper identifies two distinct ways Earth changes the axion signal, depending on how heavy or light the axions are:

A. The "Squashed" Fog (Reduced Field Value)

For certain types of axions (specifically lighter ones with stronger interactions), the Earth acts like a giant sponge that sucks the fog out.

• The Analogy: Imagine trying to measure the height of a wave hitting a beach. If the sand is very sticky, the wave might flatten out before it even reaches the shore.
• The Result: In these cases, the amount of axion "stuff" right on Earth's surface is lower than it is in deep space.
• Why it matters: Experiments that rely on detecting the amount of axions might see a weaker signal than expected. They might think the axions don't exist, when really, Earth just hid them.

B. The "Spiky" Slosh (Enhanced Gradient)

Here is the counter-intuitive part. While the amount of fog might drop, the movement or "slope" of the fog can get much steeper.

• The Analogy: Imagine a calm river flowing into a narrow canyon. The water level might drop, but the current becomes incredibly fast and turbulent.
• The Result: The paper finds that the radial gradient (how fast the axion field changes as you move from the ground up into the sky) can be massively enhanced—sometimes by thousands of times—compared to empty space.
• Why it matters: Some experiments don't care about the "amount" of axions; they care about the "slope" or the force the axions exert on spinning particles (like neutrons). For these experiments, Earth might actually act as a magnifying glass, making the signal much easier to detect.

3. The "Sweet Spot" (Resonances)

The authors also found that under very specific conditions, the Earth can act like a musical instrument. If the axion's "wavelength" matches the size of the Earth perfectly, the signal can bounce around and create a resonance (like a singer hitting a note that shatters a glass).

• However, because axions have a spread of speeds (they aren't all moving at the exact same speed), these "shattering glass" moments are rare and usually smoothed out. The main effect is the general squashing or spiking mentioned above.

4. What This Means for Experiments

The paper draws a map (Figure 1 in the text) showing where these effects happen:

• The "Safe Zone": For the most famous type of axion (the "Canonical QCD Axion"), Earth's effects are negligible. The experiments looking for these are safe; they don't need to worry about the Earth hiding the signal.
• The "Danger Zone": For lighter axions with stronger interactions, Earth changes the game.
  • If you are looking for the amount of axions, you might be looking in the wrong place (the signal is suppressed).
  • If you are looking for the force/slope of axions, you might be sitting on a goldmine (the signal is enhanced).

Summary

The paper essentially says: "Don't ignore the Earth."

When hunting for axion dark matter, scientists have been treating Earth as a transparent window. This paper shows that for certain types of axions, Earth is actually a filter. It can hide the "volume" of the signal while simultaneously turning up the "volume" of the signal's movement.

• For experiments measuring the field strength: Earth might be making them less sensitive than they think.
• For experiments measuring the field's slope (spin-magnetic experiments): Earth might be making them more sensitive, potentially allowing them to find axions that were previously thought to be too weak to detect.

The authors conclude that while the "standard" axion models are likely unaffected, the search for lighter, more interactive axions needs to be recalibrated to account for the fact that we are standing on a giant, signal-altering planet.

Technical Summary

Technical Summary: Momentum and Matter Effects on Axion Dark Matter Searches on Earth

Problem Statement
Axion dark matter (DM) is a leading candidate for solving the strong CP problem and accounting for the universe's missing mass. Standard experimental searches for axion DM typically assume the axion field behaves as a free, oscillating wave in a vacuum, with an amplitude determined by the local DM density ($\rho_{DM} \sim 0.4 \text{ GeV/cm}^3$). However, axions possess non-derivative quadratic couplings to nucleons, which modify the axion's effective mass in the presence of matter.

Recent analyses (Refs. [43, 44]) suggested that in the zero-momentum limit, the axion field amplitude near Earth could diverge for specific values of the axion decay constant ($f_a$), implying a "runaway" behavior. This paper investigates whether these effects persist when the finite momentum of the axion DM field is properly accounted for. The authors aim to determine how Earth's matter density alters both the axion field value ($a$) and its spatial gradient ($\nabla a$) at the surface and within the Earth, and how these modifications impact current and planned experimental sensitivities.

Methodology
The authors model Earth as a sphere of constant density ($\rho_\oplus \approx 5.5 \text{ g/cm}^3$) and radius $R_\oplus$. They solve the axion Equation of Motion (EOM) for a field with finite asymptotic momentum $k$, rather than the zero-momentum limit used in previous studies.

1. Effective Mass Shift: The quadratic coupling to nucleons induces a shift in the axion mass-squared inside Earth: $\delta m^2 \propto \sigma_N n / f_a^2$, where $\sigma_N$ is the nucleon sigma term and $n$ is the nucleon density. This leads to an effective in-medium mass $m_{eff}^2 = m_a^2 - \delta m^2$.
2. Scattering Formalism: The problem is treated as the scattering of a plane wave (representing virialized DM) off a spherical potential well. The solution is expanded in spherical harmonics (partial wave expansion) with angular momentum modes $\ell$.
3. Boundary Conditions: The solution matches the standard virialized DM solution at spatial infinity ($r \to \infty$) with the interior solution at the Earth's surface ($r=R_\oplus$). The authors explicitly avoid the zero-momentum limit ($k \to 0$) to regulate potential divergences.
4. Virial Averaging: Recognizing that DM velocity is stochastic (modeled by a Maxwell-Boltzmann distribution in the Standard Halo Model), the authors integrate over the velocity distribution to compute observable quantities like the power spectral density, rather than relying on a single monochromatic momentum.

Key Contributions and Results

• Resolution of Divergences: The authors demonstrate that the apparent divergence of the axion field amplitude found in zero-momentum analyses (Refs. [43, 44]) is an artifact of the approximation. When finite momentum is included, the solution remains finite. Instead of a divergence, the system exhibits resonant behavior (Sommerfeld enhancement) at specific discrete values of $f_a$ where the axion wavelength matches the Earth's size, but these resonances are regulated by the finite width of the DM velocity distribution.
• Field Amplitude Suppression (Blue Region): For axions with sufficiently small decay constants ($f_a \lesssim 3 \times 10^{13} \text{ GeV}$) and masses such that the momentum shift is significant, the axion field amplitude at Earth's surface is generally suppressed compared to the vacuum expectation. This suppression scales roughly as $f_a / (f_a)_c^\oplus$, where $(f_a)_c^\oplus$ is a critical coupling dependent on $m_a$. This implies that experiments targeting canonical QCD axion solutions (where $m_a f_a \approx m_\pi f_\pi$) are largely unaffected, but those targeting lighter axions with stronger couplings may experience reduced sensitivity due to a lower field amplitude.
• Gradient Enhancement (Green Region): A crucial finding is that while the field amplitude may be suppressed, the radial component of the axion gradient ($\hat{r} \cdot \nabla a$) is significantly enhanced. In the regime where $k R_\oplus < 1$ (low mass or large coupling), the radial gradient can be enhanced by a factor of $\sim 1/(k R_\oplus)$ or even $1/(k R_\oplus)^2$ on resonance. Conversely, gradients perpendicular to the radial direction are suppressed.
• Parameter Space Mapping: The authors map the parameter space ($m_a$ vs. $1/f_a$) into distinct regions:
  • Blue Region: Where the field amplitude is significantly modified (suppressed).
  • Green Region: Where the radial gradient is significantly enhanced.
  • Red Region: Where the mass shift is so large ($\delta m^2 > m_a^2$) that the axion vacuum inside Earth shifts from $a=0$ to $a=\pi f_a$, leading to "sourcing" effects that dominate over the DM signal.
• Resonances: The paper identifies that resonances occur when the axion wavefunction overlaps with bound states of the Earth's potential well. However, for a virialized DM distribution, these resonances are narrow and their contribution is regulated, preventing the infinite amplitudes predicted in static limits.

Experimental Implications
The authors analyze how these modifications affect two primary classes of axion experiments:

1. Axion Wind Experiments (Spin-Magnetic Coupling): These experiments (e.g., CASPEr, nEDM searches) are sensitive to the axion gradient coupling to fermion spins. The authors argue that if these experiments are sensitive to the radial gradient, their sensitivity could be dramatically improved (by up to nine orders of magnitude for very light axions) due to the enhancement of $\hat{r} \cdot \nabla a$. However, they caution that the parameter space where this enhancement is largest often overlaps with the "Earth Bound" (where the vacuum shifts), potentially excluding these models via other constraints (e.g., from White Dwarfs or Neutron Stars) before the gradient enhancement can be utilized.
2. Axion-Photon Coupling Experiments:
   • DC/Magneto-Quasistatic (MQS) Experiments: In the presence of a background magnetic field, the induced electric field ($E_1$) is proportional to $\nabla a$, while the magnetic field ($B_1$) is proportional to $a$. In the matter-affected regime, the ratio $E_1/B_1$ can be significantly enhanced, and the magnetic signal may become independent of $f_a$ in certain limits, altering how experimental constraints on $g_{a\gamma\gamma}$ should be interpreted.
   • AC/Heterodyne Experiments: Similar enhancements to the electric field signal are expected if the experiment is sensitive to the gradient term. However, no existing AC experiments currently target the specific parameter space where these effects are dominant.

Significance and Claims
The paper claims that the finite momentum of axion DM is a crucial factor in accurately modeling Earth's impact on axion searches. By moving beyond the zero-momentum limit, the authors:

• Correct the prediction of divergent field amplitudes, replacing them with regulated resonances.
• Identify a regime where the axion gradient is enhanced, potentially boosting the reach of gradient-sensitive experiments.
• Highlight a regime where the field amplitude is suppressed, potentially reducing the sensitivity of amplitude-sensitive experiments (like traditional haloscopes) for non-canonical axion models.
• Emphasize that for canonical QCD axions (yellow band in their figures), Earth's matter effects are negligible, but for lighter axions with stronger couplings, these effects are profound.

The authors conclude that while their findings offer a pathway to enhanced sensitivity for specific gradient-based searches, the parameter space where these enhancements are maximal is often constrained by astrophysical bounds (Earth sourcing, White Dwarf stability). They call for dedicated analyses of proposed experiments to determine if they can access the un-constrained regions of this modified parameter space.