AI 中文总结
本文计算三轴暗物质晕的微引力透镜光学深度，发现其形状依赖由视线投影决定，对银河系晕形状导致约±15%的系统误差，影响MACHO比例推断。
AI 中文摘要
朝向大麦哲伦云（LMC）的微引力透镜光学深度 $\tau_{\rm LMC}$ 约束了银河系暗物质晕中致密天体所占的比例，但该光学深度几乎总是在球对称晕的假设下进行评估。我们针对三轴晕计算了该光学深度，并表明其形状依赖性完全由视线在晕主轴上的投影所决定：对于无标度晕 $\rho\propto R^{-\alpha}$，在固定局部密度下，该光学深度精确分解为 $\tau/\tau_{\rm sph}=S_\odot^{\alpha}\langle S^{-\alpha}\rangle_w$，其中 $S(\Omega)$ 为椭球形状函数。LMC 仅位于太阳-银心连线垂直方向 $0.78^\circ$ 处，使其成为面内伸长率的近最优探针，而对垂直扁平度的探测能力较差。对于 \textit{Gaia} 所偏好的晕形状，我们发现 $\tau_{\rm LMC}/\tau_{\rm sph}=0.95$--$1.15$（固定局部密度），对 Law-Majewski 对数势为 $1.57$，而当主轴方位角自由时则为 $0.67$--$1.54$。对于 \textit{Gaia} DR3 下降的旋转曲线所要求的 Einasto 晕，在 $25\kpc$ 附近的截断被其隐含的更大局部密度所抵消，$\tau$ 仅下降 $7$--$15\\%$；但透镜距离的中位数从 $13.4\kpc$ 向内移至 $8.5\kpc$，此处对数斜率陡峭且上升，从而放大了形状敏感性：\textit{Gaia} RR~Lyrae 增强从 $15\\%$ 增至 $23\\%$，且无方向依赖的范围扩大至 $0.62$--$1.62$。对推断的 MACHO 或原初黑洞比例的系统性误差是双向的，对于实测形状约为 $\pm15\\%$，在完全方向自由下因子为 $2.6$，超过了 EROS-2 和 OGLE 约束的统计误差。
英文摘要
The microlensing optical depth towards the Large Magellanic Cloud (LMC), $τ_{\rm LMC}$, bounds the compact-object fraction of the Galactic dark matter halo, but is almost always evaluated for a spherical halo. We compute it for triaxial haloes and show that the shape dependence is carried entirely by the projection of the line of sight onto the halo principal axes: for a scale-free halo $ρ\propto R^{-α}$ at fixed local density it factorises exactly as $τ/τ_{\rm sph}=S_\odot^α\langle S^{-α}\rangle_w$, with $S(Ω)$ the ellipsoidal shape function. The LMC lies only $0\fdg78$ from perpendicular to the Sun-Galactic Centre line, making it a near-optimal probe of in-plane elongation and a poor probe of vertical flattening. For the halo shapes favoured by \textit{Gaia} we find $τ_{\rm LMC}/τ_{\rm sph}=0.95$--$1.15$ at fixed local density, $1.57$ for the Law-Majewski logarithmic potential, and $0.67$--$1.54$ when the major-axis azimuth is left free. For the Einasto halo required by the declining \textit{Gaia} DR3 rotation curve, truncation near $25\kpc$ is offset by the larger local density it implies and $τ$ falls by only $7$--$15$ per cent; but the median lens distance moves inwards from $13.4$ to $8.5\kpc$, where the logarithmic slope is steep and rising, amplifying the shape sensitivity: the \textit{Gaia} RR~Lyrae enhancement grows from $15$ to $23$ per cent and the orientation-free range widens to $0.62$--$1.62$. The systematic on the inferred MACHO or primordial-black-hole fraction is two-sided, of order $\pm15$ per cent for the measured shapes and a factor $2.6$ under full orientation freedom, exceeding the statistical errors of the EROS-2 and OGLE bounds.
Comments9 pages, 5 figures
