  The light we observe from distant astrophysical objects including supernovae
and quasars allows us to determine large distances in terms of a cosmological
model. Despite the success of the standard cosmological model in fitting the
data, there remains no underlying explanation for the accelerated expansion and
dark matter. Furthermore, there is a current tension between early- and
late-universe determinations of the Hubble constant. New techniques may offer
the possibility of measuring out to larger distances, provide complementary
information, or be able to side-step current limitations. After reviewing in
detail the fundamentals of standard cosmology and gravitational lensing,
including a derivation of the cosmological lens equation, this thesis
investigates a novel method of cosmography based on combining the techniques of
strong gravitational lensing time delay measurements and quasar reverberation
mapping. The motivation for this method was the possibility of avoiding lens
modelling challenges, such as the mass-sheet degeneracy, typically associated
with time delay cosmography. It suggested that differential time delays
originating from spatially separated signals in the Broad Line Region of a
quasar could be distinguished and measured from the spectroscopy of the images,
and utilised to provide a ratio of cosmological distances independent of the
lensing potential. An analytic description of the effect of the differential
lensing on the emission line spectral flux for axisymmetric Broad Line Region
geometries is given, with the inclined ring or disk, spherical shell, and
double cone as examples. This critical examination shows that the proposed
method is unable to recover cosmological information, as the observed time
delay and inferred line-of-sight velocity do not uniquely map to the
three-dimensional position within the quasar.
