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 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 halos and show that the shape dependence is carried entirely by the projection of the line of sight onto the halo principal axes, and we derive an exact factorization for scale free halos. The LMC lies only 0.78 degrees 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. We find that the ratio between optical depths for the halo shapes favoured by GAIA and for spherical halos to be 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 kpc is offset by the larger local density it implies and the optical depth falls by only 7 - 15 per cent. 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 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 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.