Gravitationally trapped axions on the Earth

We advocate for the idea that there is a fundamentally new mechanism for axion production on the Earth, as recently suggested in [H. Fischer, X. Liang, Y. Semertzidis, A. Zhitnitsky, and K. Zioutas, Phys. Rev. D 98, 043013 (2018); X. Liang and A. Zhitnitsky, Phys. Rev. D 99, 023015 (2019)]. We specifically focus on production of axions within the Earth, with low velocities such that they will be trapped in the gravitational field. Our computations are based on the so-called axion quark nugget (AQN) dark matter model, which was originally invented to explain the similarity of the dark and visible cosmological matter densities. This occurs in the model irrespective of the axion mass $m_{\mathrm{a}}$ or initial misalignment angle ${\theta }_0$. Annihilation of antimatter AQNs with visible matter inevitably produces axions when AQNs hit the Earth. The emission rate of axions with velocities below escape velocity is very tiny compared to the overall emission; however these axions will be accumulated over the 4.5 billion year lifetime of the Earth, which greatly enhances the discovery potential. We perform numerical simulations with a realistically modeled incoming AQN velocity and mass distribution, and we explore how AQNs interact as they travel through the interior of the Earth. We use this to estimate the axion flux on the surface of the Earth, the velocity-spectral features of trapped axions, the typical annihilation pattern of AQNs, and the density profile of the axion halo around the Earth. Knowledge of these properties is necessary to make predictions for the observability of trapped axions using CAST, ADMX, MADMAX, CULTASK, ORPHEUS, ARIADNE, CASPEr, ABRACADABRA, QUAX, ORGAN, TOORAD, and DM Radio.

Figure 1

The coordinate system used in the calculation of the AQN flux when we isotropize the incoming wind direction. The solid-angle element $\mathrm{d}\mathrm{\Omega }=\sin \theta \mathrm{d}\theta \mathrm{d}\varphi$ is integrated out over $4\pi$ to obtain an isotropic average.

Figure 2

Probability density for the heat-emission profile of axions within the Earth: $q(r,v_{\mathrm{AQN}})$ for $\alpha =(1.2,2.5)$, $B_{\min }=3\times {10}^{24}$. We show the radius, $r$, from the center at which axions are emitted and the velocity of the AQN, $v_{\mathrm{AQN}}$, responsible for generating those axions. The color intensity corresponds to probability density. The dashed lines denote radii of the different layers of the Earth, where the abrupt changes in density translate to abrupt changes in the heat of axion emission. This figure demonstrates that most axion heat is emitted close to the surface of the Earth, but that there is a significant tail to low depths. There is a particular spike in axion emission at the boundary between the mantle and core, at $r\simeq 0.55R_{\oplus }$, where the density jumps by more than a factor of 2. In total, $10^6$ AQN samples were used to generate this figure.

Figure 3

The axion-density profile around the Earth, ${\rho }_{\mathrm{a}}(r)$, corresponding to an incident AQN distribution with $\alpha =(1.2,2.5)$, $B_{\min }=3\times {10}^{24}$. The simulation (blue) is compared with the analytical fit (orange). In total, $2\times {10}^5$ axion samples were used in the simulation.

Figure 4

The axion velocity-distribution profile on the surface of the Earth $F(v_{\mathrm{a}})$ vs $v_{\mathrm{a}}(R_{\oplus })$: $\alpha =(1.2,2.5)$, $B_{\min }=3\times {10}^{24}$. The simulation (blue) is compared with the analytical estimation from our simple toy model $F^{(\text{est})}(v_{\mathrm{a}})$ (orange) with $v_{\mathrm{esc}}(r_0)=13.5\mathrm{km}{\mathrm{s}}^{-1}$. The distribution peaks at $\simeq 7\mathrm{km}{\mathrm{s}}^{-1}$ and there is (obviously) a sharp cutoff at the escape speed $\simeq 11\mathrm{km}{\mathrm{s}}^{-1}$. In total, $2\times {10}^5$ axion samples were used for the simulation.

Figure 5

Definition of the $\eta$ parameter.

Figure 6

Coordinate system used in the calculation of AQN flux (fixed wind).

Figure 7

The flux distribution (fixed wind) profile obtained by Monte Carlo simulation ($10^6$ samples). Here $\mu =220\mathrm{km}{\mathrm{s}}^{-1}$, $\sigma =110\mathrm{km}{\mathrm{s}}^{-1}$.

Figure 8

Summary of simulations ($B_{\min }=3\times {10}^{24}$). The AQN heat emission profiles $q(r,v_{\mathrm{AQN}})$ are presented in the first row ($10^6$ samples), followed by the axion density distribution ${\rho }_a(r)$ and the velocity spectrum $F(v_{\mathrm{a}})$ on the Earth’s surface [with $v_{\mathrm{esc}}(r_0)=13.5\mathrm{km}{\mathrm{s}}^{-1}$] respectively (blue, $2\times {10}^5$ samples). The analytical fits (orange) are also presented as in the main text. In general, all choices of parameters in Table 3 share similar features in the left column. The moderate exception is the case $\epsilon =3$ as presented in the right column here.