Faint Companions to Hot Stars. Spectroscopic Detection Sensitivity and Implications for Survey Design

Abstract

Searches for black holes in binaries must distinguish dark companions from faint luminous stars. This is challenging for hot stars, whose spectra contain few broad lines. We develop a Fourier-domain method that converts isolated, broadened lines into calibrated constraints on faint companions. Doppler shifts are encoded in the Fourier phase, while the line profile defines a smooth amplitude envelope. A faint secondary introduces an oscillatory perturbation to the log-amplitude. Expanding to first order in the secondary-to-primary line-depth ratio, $α$, yields a nearly linear model for a likelihood-ratio test against the single-star model. The method applies to single or multiple epochs without full spectral disentangling. At small velocity separation $v$, the companion signal becomes degenerate with the primary line profile, and the fractional uncertainty from the log-amplitude scales as $Δα/α\propto v^{-4}$. If the primary velocity is known, the Fourier phase provides a more favorable detection channel whose sensitivity scales as $v^{-1}$. We demonstrate the method on archival FEROS spectra of CPD$-41^\circ$7742, detecting the known secondary in He I 6678 Angstrom and obtaining a stringent non-detection in He II 5411 Angstrom, where its contribution is expected to be negligible. The framework yields 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 selection functions for spectroscopic black-hole searches, including those enabled by SDSS-V and Gaia.