The nine rings of the galaxy LEDA 1313424

Technical Summary: The Nine Rings of the Galaxy LEDA 1313424

Problem Statement
The galaxy LEDA 1313424, recently discovered by I. Pasha et al. (2025), exhibits nine luminous rings, a unique feature as no previously known galaxy possessed more than three. The discovery paper interprets these rings as "collisional ring galaxies" (CRGs), resulting from a head-on collision where a smaller "impactor" galaxy passed through the center of LEDA approximately 56 Myr ago. This interpretation relies on an analytic theory of CRGs (Appleton & Struck-Marcell 1996; Struck 2010) which predicts a specific pattern of ring radii. However, the authors of this paper identify significant physical inconsistencies in the CRG interpretation when applied to LEDA, specifically regarding the kinematics of the rings and the validity of the underlying analytic assumptions.

Methodology and Critique of the Collisional Ring Interpretation
The authors first re-evaluate the CRG interpretation using the analytic theory of collisional ring galaxies. This theory posits that ring radii $r_i$ scale as $1/(2i-1)$, where $i$ is the ring index. While the observed ratios of ring radii in LEDA roughly match this prediction, the authors identify a critical kinematic discrepancy:

• Velocity Mismatch: Based on the analytic theory and LEDA's rotation speed ($v_{rot} \approx 370$ km/s), the outermost observed ring ($i=2$, radius $\approx 70$ kpc) should expand at approximately 55 km/s. However, to reach a radius of 70 kpc in the inferred time of 56 Myr, the ring would need to travel at 1,225 km/s. This discrepancy exceeds a factor of 20. Even assuming the collision occurred earlier (up to the 200 Myr limit for ring visibility), the required speed remains six times higher than the theoretical prediction.
• Ring Widths: The analytic theory predicts that ring widths ($\Delta r_i$) increase with radius. For the parameters required to form the inner rings, the theory predicts that the outer rings ($i=7, 8, 9, 10$) should overlap and blur. In LEDA, however, all nine rings are observed as distinct.
• Violation of Assumptions: The analytic theory assumes a "sudden impulse" approximation and collisionless particles. The authors demonstrate that the condition for a sudden impulse ($2\sqrt{2}v_{rot} \ll v_{imp}$) is violated, as the required impact speed ($v_{imp} \approx 705$ km/s) is not sufficiently larger than the rotational term ($\approx 1,046$ km/s). Furthermore, the assumption of collisionless particles fails for the gas component in galactic disks, where the mean free path is short, causing gas to deviate from the predicted caustic behavior.

Alternative Interpretation: Dark Matter Caustic Rings
As an alternative, the authors propose that the rings are imprints of caustic rings of dark matter on the baryonic matter within the galaxy's disk. This hypothesis rests on the following theoretical framework:

• Dark Matter Nature: The proposal assumes dark matter consists of axions that thermalize via gravitational self-interactions and form a Bose-Einstein condensate (BEC).
• Formation Mechanism: In galactic halos, cold dark matter with large-scale vorticity falls in and out of the gravitational potential well, creating caustics (singularities in the density distribution) where the phase-space sheet folds. These caustics form circular tubes (rings) in the galactic disk.
• Predicted Radii: The radii of these caustic rings are predicted by the formula:
  $$r_n \simeq 40 \text{ kpc} \frac{n}{\left(\frac{v_{rot}}{220 \text{ km/s}}\right) \left(\frac{j_{max}}{0.18}\right)}$$
  where $n$ is an integer index, $v_{rot}$ is the rotation speed, and $j_{max}$ is a dimensionless angular momentum parameter.

Results and Quantitative Analysis
The authors perform a $\chi^2$ fit of the observed ring radii in LEDA to the caustic ring prediction (Eq. 3), treating $j_{max}$ as the sole fitting parameter.

• Fit Quality: Without accounting for theoretical errors, the fit yields $j_{max} = 0.19$ with $1/8\chi^2 = 1.94$. When a theoretical error term ($\delta r_{th} = \alpha r_n$) is included, a value of $\alpha = 0.077$ results in $1/8\chi^2 = 1.0$. This indicates the model is accurate to within approximately 7.7%.
• Statistical Significance: The authors note that the probability of the eight observed ring ratios agreeing with the theoretical prediction within a 7.7% margin by random chance is extremely low ($\approx 1.2 \times 10^{-9}$).
• Kinematic Consistency: Unlike the CRG model, caustic rings move slowly. The outward speed of the $n$-th ring is predicted to be $\dot{r}_n \simeq \frac{4}{3} \frac{r_n}{t_0}$. For LEDA, the $n=2$ and $n=3$ rings move at 3.7 km/s and 2.6 km/s, respectively. Over a star formation burst duration of 200–400 Myr, this results in a radial spread of $\sim 1$ kpc, which is consistent with the observed distinctness of the rings.

Claims of Significance
The paper claims that the CRG interpretation of LEDA's nine rings is physically untenable due to the implausible expansion speeds required and the violation of the theory's underlying assumptions. Conversely, the caustic ring interpretation, grounded in the axion dark matter hypothesis, provides a consistent explanation for:

1. The existence of multiple rings without requiring a rare, specific collision event.
2. The observed pattern of ring radii.
3. The distinctness of the rings (due to slow expansion speeds).
4. The slight offsets of the rings from the galactic center (attributed to the complex flow of dark matter).

The authors modestly conclude that while not all galactic rings are necessarily caustic rings (many may still be collisional), the axion dark matter hypothesis implies that all isolated disk galaxies possess caustic rings. The visibility of these rings depends on specific conditions regarding the interaction between the enhanced dark matter density and baryonic matter, which remains an area for further investigation.