𝒫𝒯 and anti-𝒫𝒯 symmetries for astrophysical waves

\nContext. Discrete symmetries have found numerous applications in photonics and quantum mechanics, but remain little studied in fluid mechanics, particularly in astrophysics.\nAims. We aim to show how 𝒫𝒯 and anti-𝒫𝒯 symmetries determine the behaviour of linear perturbations in a wide class of astrophysical problems. They set the location of ‘exceptional points’ in the parameter space and the associated transitions to instability, and are associated with the conservation of quadratic quantities that can be determined explicitly.\nMethods. We study several classical local problems: the gravitational instability of isothermal spheres and thin discs, the Schwarzschild instability, the Rayleigh-Bénard instability and acoustic waves in dust–gas mixtures. We calculate the locations and the order of the exceptional points using the resultant of two univariate polynomials, as well as the conserved quantities in the different regions of the parameter space using Krein theory.\nResults. All problems studied here exhibit discrete symmetries, even though Hermiticity is broken by different physical processes (self-gravity, buoyancy, diffusion, and drag). This analysis provides genuine explanations for certain instabilities, and for the existence of regions in the parameter space where waves do not propagate. Those two aspects correspond to regions where 𝒫𝒯 and anti-𝒫𝒯 symmetries are broken respectively. Not all instabilities are associated to symmetry breaking (e.g. the Rayleigh-Benard instability).\n