The GAPS programme at TNG XXXIV. Activity-rotation, flux-flux relationships, and active-region evolution through stellar age

Context. Active-region evolution plays an important role in the generation and variability of magnetic fields on the surface of lower main sequence stars. However, determining the lifetime of active-region growth and decay as well as their evolution is a complex task. Most previous studies of this phenomenon are based on optical light curves, while little is known about the chromosphere and the transition region. Aims. We aim to test whether or not the lifetime of active-region evolution shows any dependency on stellar parameters, particularly stellar age. Methods. We identified a sample of stars with well-defined ages via their kinematics and membership to young stellar associations and moving groups. We made use of high-resolution echelle spectra from HARPS at La Silla 3.6m-telescope and HARPS-N at TNG to compute rotational velocities, activity levels, and emission excesses. We used these data to revisit the activity-rotation-age relationship. The time-series of the main optical activity indicators, namely Ca II H and K, Balmer lines, Na I D1, D2, and He I D3, were analysed together with the available photometry using state-of-the-art Gaussian processes to model the stellar activity of these stars. Autocorrelation functions of the available photometry were also analysed. We used the derived lifetimes of active-region evolution to search for correlations with stellar age, spectral type, and activity level. We also used the pooled variance technique to characterise the activity behaviour of our targets. Results. Our analysis confirms the decline of activity and rotation as a star ages. We also confirm that the rotation rate decays with age more slowly for cooler stars and that, for a given age, cooler stars show higher levels of activity. We show that F- and G-type young stars also depart from the inactive stars in the flux-flux relationship. The Gaussian process analysis of the different activity indicators does not seem to provide any useful information on the lifetime and evolution of active regions. On the other hand, the lifetimes of active regions derived from the light-curve analysis might correlate with stellar age and temperature. Conclusions. Although we suggest caution because of small number statistics, our results suggest that active regions seem to live longer on younger, cooler, and more active stars.