  We use deep spectroscopy from the SAMI Galaxy Survey to explore the precision
of the fundamental plane of early-type galaxies (FP) as a distance indicator
for future single-fibre spectroscopy surveys. We study the optimal trade-off
between sample size and signal-to-noise ratio (SNR), and investigate which
additional observables can be used to construct hyperplanes with smaller
intrinsic scatter than the FP. We add increasing levels of random noise
(parametrised as effective exposure time) to the SAMI spectra to study the
effect of increasing measurement uncertainties on the FP-and
hyperplane-inferred distances. We find that, using direct-fit methods, the
values of the FP and hyperplane best-fit coefficients depend on the spectral
SNR, and reach asymptotic values for a mean SNR=40 {\AA}$^{-1}$. As additional
variables for the FP we consider three stellar-population observables:
light-weighted age, stellar mass-to-light ratio and a novel combination of Lick
indices (I$_{\rm age}$). For a SNR=45 {\AA}$^{-1}$ (equivalent to 1-hour
exposure on a 4-m telescope), all three hyperplanes outperform the FP as
distance indicators. Being an empirical spectral index, I$_{\rm age}$ avoids
the model-dependent uncertainties and bias underlying age and mass-to-light
ratio measurements, yet yields a 10 per cent reduction of the median distance
uncertainty compared to the FP. We also find that, as a by-product, the Iage
hyperplane removes most of the reported environment bias of the FP. After
accounting for the different signal-to-noise ratio, these conclusions also
apply to a 50 times larger sample from SDSS-III. However, in this case, only
age removes the environment bias.
