The nature of medium-period variables on the extreme horizontal branch I. X-shooter study of variable stars in the globular cluster $\omega$ Cen

A fraction of the extreme horizontal branch stars of globular clusters exhibit a periodic light variability that has been attributed to rotational modulation caused by surface spots. These spots are believed to be connected to inhomogeneous surface distribution of elements. However, the presence of such spots has not been tested against spectroscopic data. We analyzed the phase-resolved ESO X-shooter spectroscopy of three extreme horizontal branch stars that are members of the globular cluster $\omega$ Cen and also display periodic light variations. The aim of our study is to understand the nature of the light variability of these stars and to test whether the spots can reproduce the observed variability. Our spectroscopic analysis of these stars did not detect any phase-locked abundance variations that are able to reproduce the light variability. Instead, we revealed the phase variability of effective temperature and surface gravity. In particular, the stars show the highest temperature around the light maximum. This points to pulsations as a possible cause of the observed spectroscopic and photometric variations. However, such an interpretation is in a strong conflict with Ritter's law, which relates the pulsational period to the mean stellar density. The location of the $\omega$ Cen variable extreme horizontal branch stars in HR diagram corresponds to an extension of PG 1716 stars toward lower temperatures or blue, low-gravity, large-amplitude pulsators toward lower luminosities, albeit with much longer periods. Other models of light variability, namely, related to temperature spots, should also be tested further. The estimated masses of these stars in the range of $0.2-0.3\,M_\odot$ are too low for helium-burning objects.