= Solution
The <Lyman-alpha forest> supplies many relatively narrow absorption features along each bright distant <quasar> sightline. These arise in intervening intergalactic <hydrogen> rather than in the <quasar>'s rapidly varying emission region. A repeat high-resolution spectrum can be fitted against the same absorption template, and the information from many lines and many sightlines can be combined to estimate a common <spectroscopic velocity drift>. Diffuse intergalactic absorbers also generally have smaller peculiar accelerations than gas close to galactic nuclei, although their peculiar accelerations and changing line profiles still require control.
The preceding calculation requires a relative <wavelength> displacement $|\Delta\lambda|/\lambda\simeq6\times10^{-10}$ over a decade. This is much smaller than one instrumental resolution element. Accurate line centroids can nevertheless be measured to a fraction of a resolution element when the signal-to-noise ratio and template information are sufficient. The real obstacles are long-term <wavelength> calibration, instrumental drift, sufficient <photon> counts and corrections for observer motion, along with astrophysical variability. Statistical averaging reduces independent errors; a common calibration error remains common to all lines. The <Sandage-Loeb test> therefore needs large collecting area, stable high-resolution spectroscopy and a long time baseline. The original absorption-forest strategy is described in https://arxiv.org/abs/astro-ph/9802122[Loeb's 1998 proposal].
\b[The measurement is feasible in principle but demands exceptional long-term stability.] It directly probes the change of <redshift> with observer time, giving $H_0(1+z)-H(z)$ rather than a distance inferred from a <standard candle>. Measuring its sign and <redshift> dependence would distinguish the expansion histories in part (ii).
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