Measuring the distance to stars in relation to us is fundamental for understanding the universe we live in, however, it is also one of the most challenging tasks in astronomy. Accurately estimating distances would allow us to understand the distribution of stars within our galaxy, the distribution of galaxies within the universe and how the universe has evolved over time, allowing us to set constraints on the theory governing gravitational interactions. The objective of this thesis is to enhance our understanding of the methods used to measure distances and the physics of the stars involved.

Classical Cepheids are the primary standard candles used for determining galactic and extragalactic distances. They are a fundamental part of one of the most precise methods known for measuring the rate of expansion of the universe H0. Cepheids calibrate the absolute luminosity of Type Ia supernovae, which, in turn, are used as standard candles for more distant galaxies where Cepheids can no longer be detected. By combining this calibration with redshift measurements of galaxies that host these supernovae, it is possible to estimate H0. Alternatively, H0 can be derived from fitting the equations of the standard cosmological model, LCDM, to observations of the Cosmic Microwave Background. If LCDM is correct, both measurements should agree, but they differ by more than five standard deviations, possibly indicating measurement errors or new physics.

Improving our understanding of distance estimation could help identify the source of the discrepancy. Chapter 1 introduces the basic concepts used in this thesis, including the definitions of standard candles and highlights the role of clusters in studying the population of variable stars.

Chapter 2 examines Cepheid. We show that by using cluster parallaxes, it is possible to reduce the uncertainties associated with the parallax of Cepheids by a factor of 3. This allowed us to achieve one of the most precise calibrations of their absolute luminosity, reaching uncertainties of ~1% for Cepheids with pulsation periods of 10 days. As a result, the uncertainty in the measurement of H0 was reduced, increasing the statistical significance of the Hubble tension.

Chapters 3 and 4 focus on identifying RR Lyrae and Population II Cepheids in GCs, in the future, these samples could help verify the distances obtained with Cepheids to galaxies in the Local Group. We found that around 25% of horizontal branch stars showed no signs of variability in the Gaia photometry. This observation could have major implications for RRL models, as they predict that all stars in the instability strip should be photometrically variable.

Chapter 5 examines the geometry of GCs. We found that the on-sky geometry of these clusters is elliptical. Our results show that rotation is present in multiple GCs, and the projected rotation axis aligns with the minor axis of the ellipse. This provides evidence that rotation influences their geometry.

Chapter 6 presents unpublished results on long-period variable stars in GCs. We observe that these stars appear to be split into two types, with some indications that blending may have affected the photometry of one type, but we cannot conclude that this has impacted their classification. The thesis concludes with Chapter 7, which summarizes the main contributions in greater detail and explores how future data releases from Gaia could enhance our understanding of standard candles and clusters.