Scattering from compact objects: Debye series and Regge-Debye poles

Abstract

We investigate elastic scattering by a compact, horizonless body in curved spacetime, considering a massless scalar wave incident on a static, spherically symmetric, uniform-density star of radius $R$ and mass $M$ with a Schwarzschild exterior. We introduce an exact Debye-series decomposition of the scattering matrix, in the spirit of Debye expansions in Mie scattering. This decomposition separates direct surface reflection from contributions involving transmission into the interior and subsequent propagation, and admits a natural trajectory interpretation. We then determine the associated Regge-Debye pole spectrum in the complex angular-momentum plane. For neutron-star-like tenuities ($R>3M$), the spectrum exhibits two pole families: a surface-wave branch associated with the surface matching condition and a broad-resonance branch associated with the interior regularity condition. For ultracompact objects ($R<3M$), the surface-wave branch persists, while the interior-resonance sector splits into broad- and narrow-resonance branches. We next reconstruct the scattering amplitude from the Debye partial-wave contributions and find excellent agreement with direct partial-wave calculations. Finally, we develop complex angular-momentum representations order by order in the Debye series, making explicit how the pole families and non-pole sectors contribute to each Debye term. In the neutron-star-like regime, we find a genuine competition between Regge-Debye pole sums and branch-cut contributions, and show that, at high frequency, the rainbow-like enhancement already arises from the first interior-transmission contribution and is dominated by the interior-resonance Regge-Debye poles. By contrast, in the ultracompact regime, the Debye amplitudes are overwhelmingly pole dominated.

Figures (11)

• Figure 1: Regge--Debye pole spectrum for the massless scalar field with $R=6M$. The upper panel corresponds to $2M\omega=2$ and the lower panel to $2M\omega=16$. Blue circles denote the surface-wave branch and red squares denote the broad-resonance branch. For comparison, we also show the Regge poles of a Schwarzschild black hole at the same frequency.
• Figure 2: Regge--Debye pole spectrum for the massless scalar field with $R=2.26M$. The upper panel corresponds to $2M\omega=2$ and the lower panel to $2M\omega=6$. Blue circles denote the surface-wave branch, red squares denote the broad-resonance branch, and purple triangles denote the narrow-resonance branch. For comparison, we also show the Regge poles of a Schwarzschild black hole at the same frequency.
• Figure 3: Comparison between the Regge--Debye poles and the Regge poles. Top panel: neutron-star-like configuration with $R=6M$ at frequency $2M\omega=16$. The Regge--Debye spectrum, obtained from the conditions $\alpha^{\rm in}=0$ and $c^{\rm in}=0$ for the surface-wave and broad-resonance branches, respectively, is compared with the Regge poles obtained from the condition $A^{(-)}=0$ in Ref. OuldElHadj:2019kji. Two families are present in this regime: a surface-wave branch and a broad-resonance branch. Bottom panel: ultracompact configuration with $R=2.26M$ at frequency $2M\omega=6$. In this case, the Regge--Debye surface-wave branch is again generated by the condition $\alpha^{\rm in}=0$, whereas the broad- and narrow-resonance branches are generated by the condition $c^{\rm in}=0$. The Regge poles, by contrast, are obtained in all cases from the single condition $A^{(-)}=0$.
• Figure 4: Differential scattering cross section for a neutron-star-like compact body with $R=6M$. Upper panels: exact result (black curve) and the first Debye-order contributions $p=0,\ldots,4$ for $2M\omega=2$ (left) and $2M\omega=16$ (right). Lower panels: Debye partial sums compared with the exact cross section. At $2M\omega=2$ (left) only a few Debye orders are required, whereas at $2M\omega=16$ (right) higher orders become increasingly important, especially at large scattering angles.
• Figure 5: Differential scattering cross section for an ultracompact object with $R=2.26M$. Upper panels: exact result (black curve) and the first Debye-order contributions $p=0,\ldots,4$ for $2M\omega=2$ (left) and $2M\omega=6$ (right). Lower panels: Debye partial sums compared with the exact cross section. For this ultracompact configuration, a small number of Debye orders already yields an accurate approximation over a broad angular range at both frequencies.