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), τ 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 proportional to R-α at fixed local density it factorises exactly as τ/τ sph=S α S-α w, with S(Ω) the ellipsoidal shape function. The LMC lies only 0 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 Gaia we find τ LMC/τ 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 Gaia DR3 rotation curve, truncation near 25 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, where the logarithmic slope is steep and rising, amplifying the shape sensitivity: the 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 plus or minus15 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.