  The time delay between the arrival of photons of multiple images of time
variable sources can be used to constrain absolute distances in the Universe
(Refsdal 1964), and in turn obtain a direct estimate of the Hubble constant and
other cosmological parameters. To convert the time delay into distances, it is
well known that the gravitational potential of the main deflector and the
contribution of the matter along the line-of-sight need to be known to a
sufficient level of precision. In this paper, we discuss a new astrometric
requirement that is becoming important as time-delay cosmography improves in
precision and accuracy with larger samples, and better data and modelling
techniques. We derive an analytic expression for the propagation of astrometric
uncertainties on the multiple image positions into the inference of the Hubble
constant and derive requirements depending on image separation and relative
time delay. We note that this requirement applies equally to the image position
measurements and to the accuracy of the model in reproducing them. To
illustrate the requirement, we discuss some example lensing configurations and
highlight that, especially for time delays of order 10 days or shorter, the
relative astrometric requirement is of order milli-arcseconds, setting a tight
requirement on both measurements and models. With current optical infrared
technology, astrometric uncertainties may be the dominant limitation for strong
lensing cosmography in the small image-separation regime when high-precision
time-delays become accessible.
