 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 $|\Delta \lambda |/\lambda \simeq 6\times 10^{-10}$ 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 Loeb's 1998 proposal.

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).