  We present new and accurate measurements of the cosmic distance-redshift
relation, spanning 0.2 < z < 1, using the topology of large-scale structure as
a cosmological standard ruler. Our results derive from an analysis of the
Minkowski functionals of the density field traced by the WiggleZ Dark Energy
Survey. The Minkowski functionals are a set of statistics which completely
describe the topological nature of each isodensity surface within the field, as
a function of the density value. Given the shape of the underlying matter power
spectrum, measured by fluctuations in the Cosmic Microwave Background
radiation, the expected amplitudes of the Minkowski functionals are specified
as an excursion set of a Gaussian random field, with minimal non-Gaussian
corrections for the smoothing scales > 10 Mpc/h considered in this analysis.
The measured amplitudes then determine the cosmic distance D_V(z), which we
obtain with 3-7% accuracies in six independent redshift slices, with the
standard ruler originating in the known curvature of the model power spectrum
at the smoothing scale. We introduce a new method for correcting the
topological statistics for the sparse-sampling of the density field by the
galaxy tracers, and validate our overall methodology using mock catalogues from
N-body simulations. Our distance measurements are consistent with standard
models which describe the cosmic expansion history, and with previous analyses
of baryon acoustic oscillations (BAOs) detected by the WiggleZ Survey, with the
topological results yielding a higher distance precision by a factor of 2.
However, the full redshift-space power-spectrum shape is required to recover
the topological distances, in contrast to the preferred length scale imprinted
by BAOs, which is determined by simpler physics.
