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Twinkle twinkle dark star: Oscillating profiles from dark matter scalar solitons

Real scalar fields, e.g. the axion, cannot condensate into stationary solitonic configurations to form star-like structures, eventually either dispersing or collapsing. However, by relaxing the stationarity condition on the metric, it has been shown that oscillatory solitonic solutions -- known as oscillatons -- exist. Oscillatons share several properties with boson stars, including comparable compactness and mass ranges. However, their time-dependent nature can lead to potentially discriminating observable signatures. In this work, we explore the observational properties of oscillatons. We find that stable oscillatory circular orbits exist, extending down to the center of the configuration, supporting the possibility of accretion disk structures within the star. We compute the deflection of light rays and verify that it is largely insensitive to the time dependence of the metric. Despite this, the oscillatory behavior of the redshift factor has a strong effect on the observed intensity profiles from accretion disks, producing a breathing-like image whose frequency depends on the mass of the scalar field. In fact, their oscillation period may lie within the observational windows of the Event Horizon Telescope for Sgr A$^{*}$ and M87$^{*}$, suggesting the possibility that this ``twinkling" behavior could be tested via near-horizon imaging of these objects.