Modelling stellar variability in archival HARPS data: I -- Rotation and activity properties with multi-dimensional Gaussian Processes

Although instruments for measuring the radial velocities (RVs) of stars now routinely reach sub-meter per second accuracy, the detection of low-mass planets is still very challenging. The rotational modulation and evolution of spots and/or faculae can induce variations in the RVs at the level of a few m/s in Sun-like stars. To overcome this, a multi-dimensional Gaussian Process framework has been developed to model the stellar activity signal using spectroscopic activity indicators together with the RVs. A recently published computationally efficient implementation of this framework, S+LEAF 2, enables the rapid analysis of large samples of targets with sizeable data sets. In this work, we apply this framework to HARPS observations of 268 well-observed targets with precisely determined stellar parameters. Our long-term goal is to quantify the effectiveness of this framework to model and mitigate activity signals for stars of different spectral types and activity levels. In this first paper in the series, we initially focus on the activity indicators (S-index and Bisector Inverse Slope), and use them to a) measure rotation periods for 49 slow rotators in our sample, b) explore the impact of these results on the spin-down of middle-aged late F, G & K stars, and c) explore indirectly how the spot to facular ratio varies across our sample. Our results should provide valuable clues for planning future RV planet surveys such as the Terra Hunting Experiment or the PLATO ground-based follow-up observations program, and help fine-tune current stellar structure and evolution models.