  The dark matter enclosed in a density perturbation with a large initial
amplitude (delta-rho/rho > 1e-3) collapses shortly after recombination and
forms an ultracompact minihalo (UCMH). Their high central densities make UCMHs
especially suitable for detection via astrometric microlensing: as the UCMH
moves, it changes the apparent position of background stars. A UCMH with a mass
larger than a few solar masses can produce a distinctive astrometric
microlensing signal that is detectable by the space astrometry mission Gaia. If
Gaia does not detect gravitational lensing by any UCMHs, then it establishes an
upper limit on their abundance and constrains the amplitude of the primordial
power spectrum for k~2700 Mpc^{-1}. These constraints complement the upper
bound on the amplitude of the primordial power spectrum derived from limits on
gamma-ray emission from UCMHs because the astrometric microlensing signal
produced by an UCMH is maximized if the dark-matter annihilation rate is too
low to affect the UCMH's density profile. If dark matter annihilation within
UCMHs is not detectable, a search for UCMHs by Gaia could constrain the
amplitude of the primordial power spectrum to be less than 1e-5; this bound is
three orders of magnitude stronger than the bound derived from the absence of
primordial black holes.
