Triaxiality of the Milky Way dark matter halo and the microlensing optical depth towards the Large Magellanic Cloud

The microlensing optical depth towards the Large Magellanic Cloud (LMC), ${\tau }_{\mathrm{L}\mathrm{M}\mathrm{C}}$, 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 $\rho \propto R^{-\alpha }$ at fixed local density it factorises exactly as $\tau \mathrm{/}{\tau }_{\mathrm{s}\mathrm{p}\mathrm{h}}=S_{\odot }^{\alpha }⟨S^{-\alpha }{⟩}_w$, with $S(\mathrm{\Omega })$ the ellipsoidal shape function. The LMC lies only $0\text{fdg}78$ 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 $\mathit{\text{Gaia}}$ we find ${\tau }_{\mathrm{L}\mathrm{M}\mathrm{C}}\mathrm{/}{\tau }_{\mathrm{s}\mathrm{p}\mathrm{h}}=0.95$–$1.15$ 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 $\mathit{\text{Gaia}}$ DR3 rotation curve, truncation near $25\text{kpc}$ is offset by the larger local density it implies and $\tau$ falls by only $7$–$15$ per cent; but the median lens distance moves inwards from $13.4$ to $8.5\text{kpc}$, where the logarithmic slope is steep and rising, amplifying the shape sensitivity: the $\mathit{\text{Gaia}}$ 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 $\pm 15$ 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.