Gravitational lensing outside and inside of a marginally unstable photon sphere at a $Z_2$-symmetric wormhole throat in strong deflection limits

Technical Summary: Gravitational Lensing at a $Z_2$-Symmetric Wormhole Throat with a Marginally Unstable Photon Sphere

Problem Statement
Gravitational lensing by compact objects, particularly those with photon spheres (unstable circular light orbits), is a critical tool for distinguishing black holes from exotic alternatives like wormholes. In standard strong deflection limits near a photon sphere, the deflection angle $\alpha$ of a light ray typically diverges logarithmically as the impact parameter $b$ approaches the critical value $b_m$ ($b \to b_m \pm 0$). This behavior is well-established for Schwarzschild black holes and many other spacetimes.

However, specific exotic compact objects, such as $Z_2$-symmetric wormholes, may possess a marginally unstable photon sphere located exactly at the throat. In such cases, the standard logarithmic divergence does not apply. Previous studies have suggested that the deflection angle might diverge as a power law, but the specific coefficients and universal properties for $Z_2$-symmetric wormholes required rigorous investigation. Furthermore, existing semi-analytic methods for calculating these coefficients in wormhole spacetimes (specifically Damour-Solodukhin and Simpson-Visser models) have yielded conflicting or potentially incorrect results for the leading-order coefficients.

Methodology
The author extends the semi-analytic and numerical methods originally developed by Eiroa, Romero, and Torres for standard photon spheres to the specific case of a marginally unstable photon sphere located at the throat of a $Z_2$-symmetric wormhole.

1. Theoretical Framework: The study considers a general, static, spherically symmetric, asymptotically flat wormhole spacetime with $Z_2$ symmetry about the throat ($r=r_{th}$). The author defines a marginally unstable photon sphere at the throat where the effective potential $V(r, b)$ and its first three derivatives vanish, while the fourth derivative is non-zero.
2. Deflection Angle Analysis:
   • Outside the Throat ($b > b_m$): Rays are reflected just outside the throat. The deflection angle is calculated by integrating the trajectory from infinity to the closest approach $r_0$ and back.
   • Inside the Throat ($b < b_m$): Rays pass through the throat to the other asymptotic region. The deflection angle is calculated by integrating from the throat ($r_{th}$) to infinity.
3. Power-Law Divergence: The author hypothesizes and numerically verifies that in the strong deflection limit, the deflection angle diverges as a power law rather than logarithmically:
   $$\alpha(b) = \frac{\bar{c}_{\pm}}{|b/b_m - 1|^{1/4}} + \bar{d}_{\pm}$$
   where the exponent is $1/4$ (distinct from the $1/6$ exponent found in non-throat marginally unstable spheres).
4. Numerical Implementation: The author develops a numerical method to compute the coefficients $\bar{c}_{\pm}$ and $\bar{d}_{\pm}$ by evaluating limits of the deflection angle and its derivatives as $b \to b_m$.
5. Application to Specific Models: The method is applied to three specific spacetimes:
   • A commonly-used simple wormhole (often cited in Event Horizon Telescope studies).
   • The Damour-Solodukhin wormhole.
   • The Simpson-Visser black-bounce spacetime.
6. Lens Equation: The author solves the Ohanian lens equation to determine observable quantities such as Einstein ring radii, angular separations, and magnifications.

Key Contributions and Results

• Power-Law Divergence at the Throat: The paper demonstrates that for $Z_2$-symmetric wormholes with a marginally unstable photon sphere at the throat, the deflection angle diverges as $|b/b_m - 1|^{-1/4}$. This contrasts with the logarithmic divergence of standard black holes and the $|b/b_m - 1|^{-1/6}$ divergence found in marginally unstable spheres off the throat.
• Universal Relations: Numerical results across the three tested spacetimes suggest a universal relationship between the coefficients for rays passing inside ($-$) and outside ($+$) the throat:
  $$\bar{c}_{-} = \sqrt{2}\bar{c}_{+} \quad \text{and} \quad \bar{d}_{-} = \bar{d}_{+}$$
  This differs from the $\sqrt{3}$ ratio found in non-throat marginally unstable spheres.
• Correction of Previous Coefficients:
  • Damour-Solodukhin Wormhole: The author finds that the coefficient $\bar{c}_{+}$ calculated in previous semi-analytic works (Tsukamoto [145]; Zhang and Xie [110]) is invalid ($2.5558$ vs. the correct $3.3507$), while the constant term $\bar{d}_{+}$ was approximately correct.
  • Simpson-Visser Wormhole: The author identifies that both the coefficient $\bar{c}_{+}$ and the term $\bar{d}_{+}$ in a previous study (Tsukamoto [146]) are incorrect. The new numerical values are $\bar{c}_{+} \approx 4.1037$ and $\bar{d}_{+} \approx -3.7709$.
• Observables: The paper provides calculated values for the critical impact parameter, Einstein ring diameters, and magnification ratios for the three models, assuming parameters relevant to the supermassive black hole candidate Sgr A* ($M \approx 4 \times 10^6 M_\odot$).

Significance and Claims
The paper claims that the $Z_2$-symmetric wormhole, often considered a "simple" or "common" example in the literature, possesses a "special property" due to the location of its marginally unstable photon sphere at the throat. This property fundamentally alters the strong deflection limit behavior, rendering standard logarithmic analysis inapplicable.

The author asserts that their numerical method successfully corrects previous errors in the semi-analytic determination of deflection angle coefficients for these specific spacetimes. They emphasize that the universal relations derived ($\bar{c}_{-} = \sqrt{2}\bar{c}_{+}$) are specific to the throat geometry and differ from those of marginally unstable spheres located away from the throat.

The study concludes that while the Event Horizon Telescope has ruled out this specific $Z_2$-symmetric wormhole model based on current observations, the detailed investigation of the deflection angle's power-law divergence is necessary to exclude other black hole mimickers with photon spheres in future high-precision space observations. The author notes that a full analytic confirmation of the power-law forms (Eqs. 2.30 and 2.40) remains a task for future work, as the current study relies on numerical verification of forms expected from earlier literature.