  Context. The Doppler shift predicted by general relativity for light escaping
a gravitational potential has been observed on Earth as well as in the
direction of various stars and galaxy clusters at optical wavelengths. Aims.
Observing the gravitational redshift in the X-ray band within galaxy clusters
could provide information on their properties and, in particular, their
gravitational potential. We present a feasibility study of such a measurement,
using the capabilities of the next-generation European X-ray observatory
Athena. Methods. We used a simple generalized Navarro-Frenk-White potential
model along with a beta-model for the density of baryonic matter, which sets
the emission to provide an estimation of the observed redshift in the simplest
of cases. We generated mock observations with the Athena X-ray Integral Field
Unit (X-IFU) for a nearby massive cluster, while seeking to recover the
gravitational redshift along with other properties of the toy model cluster.
Results. We investigated the observability of the gravitational redshift in an
idealized test case of a nearby massive cluster with the Athena X-IFU
instrument, as well as its use in probing the properties of the potential well.
We were also able to constrain the mass to a 20 % level of precision and the
cosmological redshift to less than 1%, within a simplified and idealized
observational framework. More refined simulations accounting for further
effects such as the internal gas motions and the actual shape of the potential
well are required to fully investigate the feasibility of measuring the
gravitational redshift for a single target or statistically over a sample of
galaxy clusters.
