Faint companions to hot stars. Spectroscopic detection sensitivity and implications for survey design

Searches for black holes in binaries must distinguish genuinely dark companions from faint luminous stars. This is particularly challenging for massive hot stars. Their masses can exceed those of typical stellar black holes, while their spectra contain only a few broad lines. We developed a Fourier-domain method that converts isolated, broadened spectral lines into calibrated constraints on faint companions. In Fourier space, Doppler shifts are encoded in the phase, while the line profile defines a smooth amplitude envelope. A faint secondary introduces an oscillatory perturbation to the otherwise smooth log-amplitude. Expanding this perturbation to first order in the secondary-to-primary line-depth ratio, α, allows it to be profiled against the smooth baseline in a likelihood-ratio test with the single-star model. Because the model is analytic and nearly linear, its sensitivity can be evaluated transparently for individual spectra and at survey scale. It can be applied to single or multiple epochs without requiring full spectral disentangling. At small velocity separation v, the companion signal becomes degenerate with the smooth single-star profile. For the Fourier log-amplitude, the fractional uncertainty scales as Δα/α ∝ v^ -4 . If the primary velocity is known, the Fourier phase provides a more favorable detection channel whose sensitivity scales as v^-1. We demonstrated the method on archival FEROS spectra of the massive binary CPD-41^̧irc7742. We detected the known secondary in He,I 6678 Å and obtained a stringent nondetection in He,II 5411 Å, where its contribution is expected to be negligible. The same framework yields local detection efficiencies as functions of line-depth ratio, velocity separation, noise, and instrumental setup. These can be combined across lines, epochs, and targets to construct survey selection functions for spectroscopic black-hole searches, including those enabled by SDSS-V and Gaia.