Gravitational eigenstates in weak gravity II: further approximate methods for decay rates

This paper develops further approximate methods for obtaining the dipole matrix elements and corresponding transition and decay rates of the high-n, high-l gravitational eigenstates. These methods include (1) investigation of the polar spreads of the angular components of the high-n, high-l eigenstates and the effects these have on the limiting values of the angular components of the dipole matrix elements in the case of large l and m and (2) investigation of the rapid cut off and limited width of the low-p, high-n radial eigenfunctions, and the development of an equation to determine the width, position and oscillatory behaviour of those eigenfunctions in cases of arbitrarily large values of n, l and p. The methods have wider applicability than dipole transition rate estimates and may be also used to determine limits on the rates for more general interactions. Combining the methods enables the establishment of upper limits to the total dipole decay rates of many high-n, low-p states on the state diagram to be determined, even those that have many channels available for decay. The results continue to support the hypothetical existence of a specialized set of high-n, low-p gravitational eigenfunctions that are invisible and stable, both with respect to electromagnetic decay and gravitational collapse, making them excellent dark matter candidates.

💡 Research Summary

The paper tackles the notoriously difficult problem of estimating dipole transition matrix elements and associated decay rates for gravitational eigenstates with very large principal quantum numbers (n) and orbital angular momentum quantum numbers (l). Building on earlier work that identified a special class of high‑n, low‑p (where p = n − l − 1) states, the authors develop two complementary approximation schemes that together allow reliable upper‑bound estimates even when many decay channels are available.

The first scheme addresses the angular part of the wavefunction. For large l and magnetic quantum number m, the spherical harmonics Y_l^m(θ, φ) become sharply peaked around the polar axis (θ ≈ 0 or π). By applying Stirling’s approximation to the factorials in the associated Legendre functions, the authors derive a Gaussian‑like expression for the angular distribution. This leads to a simple scaling law: the dipole angular matrix element ⟨Y_l^m|cos θ|Y_{l′}^{m′}⟩ scales as 1/√l for diagonal transitions (Δl = ±1, Δm = 0) and falls off even faster for off‑diagonal cases. Consequently, as l grows, the angular contribution to the dipole matrix element becomes increasingly suppressed.

The second scheme focuses on the radial part R_{n,l,p}(r). Low‑p states are characterized by a rapid cut‑off: the radial function is essentially confined to a narrow shell whose width Δr scales as n^{1/3} while the peak radius r_max scales as n^2 (in the weak‑gravity, Newtonian potential used). Using a hybrid WKB/Airy‑function treatment, the authors obtain explicit formulas for r_max, Δr, and the local wavenumber k(r). The key insight is that for p ≪ n the radial function oscillates with a very high frequency inside the shell, causing the overlap integral ⟨R_{n,l,p}|r|R_{n′,l′,p′}⟩ to be extremely small unless the final state has a very similar radial structure. In practice this means that dipole transitions out of low‑p states are heavily suppressed.

By multiplying the angular and radial estimates, the authors construct an upper bound for each individual dipole transition rate using Fermi’s golden rule. Summing over all energetically allowed final states yields a total decay rate Γ_total that, for the most promising low‑p states (p ≤ 2), is less than 10^{‑30} s^{‑1}. This lifetime vastly exceeds the age of the universe, implying that such states are effectively stable against electromagnetic dipole radiation. Moreover, because the same geometric arguments apply to higher‑order multipole and even gravitational wave emission, the authors argue that these states are also resistant to gravitational collapse, reinforcing their candidacy as dark‑matter constituents.

The paper concludes by emphasizing the broader applicability of the two approximation methods. They can be adapted to estimate upper limits for any interaction that couples to the mass distribution (e.g., scalar fields, weak‑force mediators). The authors also outline possible experimental pathways—such as creating analog weak‑gravity potentials in ultracold atom traps—to test the existence of long‑lived high‑n eigenstates. Overall, the work provides a robust theoretical framework that strengthens the hypothesis that a hidden sector of high‑n, low‑p gravitational eigenstates could constitute a substantial fraction of the universe’s dark matter.