Towards Precision Cosmography with Cosmic Chronometers

The Hubble tension, the growing discrepancy between early- and late-Universe determinations of $H_0$, has renewed interest in late-time, model-independent probes of the cosmic expansion history. Cosmic chronometers (CCs), passively evolving early-type galaxies whose differential ageing directly traces the cosmic expansion rate, $H(z)\propto \text{d}z/\text{d}t$, offer one such probe, entirely decoupled from the distance ladder and from early-Universe physics. This thesis develops and applies a Lick-index-based CC pipeline, combining high signal-to-noise spectral stacking, stellar-population synthesis fitting, and a cosmographic (Taylor-expansion) reading of the resulting age-redshift relations, to two large spectroscopic samples: $90\,396$ SDSS Legacy galaxies at $0.05\lesssim z\lesssim0.35$ and $527\,287$ DESI-DR1 luminous red galaxies at $0.36\lesssim z\lesssim0.80$. Applied to the SDSS sample, a classic, $z_0=0$ cosmographic fit of TMJ-modelled stellar ages yields $H_0=70.0^{+4.1}_{-7.6}\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$, comfortably compatible with both sides of the Hubble tension, together with a $w_0\mathrm{CDM}$ extension fully consistent with a cosmological constant. Applied to the substantially larger and higher-redshift DESI sample, a pivotal-redshift cosmography anchored at $z_0\approx0.57$ delivers a fully covariant $H(z)$ reconstruction across $0.36\lesssim z\lesssim0.80$, together with two independent, differential $H(z)$ measurements at $z\approx0.55$ and $z\approx0.61$; every one of these results is individually consistent with the $\Lambda$CDM prediction of Planck Collaboration et al. (2020), taken today as the typical benchmark cosmology to compare with. As a by-product of the same stacks, age-metallicity-$\alpha$-enhancement scaling relations are derived at $z\approx0$ from SDSS and, indirectly, at $z\approx0.57$ from DESI, both consistent with the downsizing scenario for the assembly of early-type galaxies. Taken together, these results establish Lick-index cosmic chronometers, built on carefully stacked spectra and interpreted through a cosmographic rather than a purely finite-difference framework, as a competitive, genuinely model-independent probe of the low- and intermediate-redshift expansion history of the Universe. The thesis closes by identifying the main systematic limitations of the current pipeline, including an unexplained oscillatory pattern in the index-redshift relations, and by outlining concrete directions for future work on both the astrophysical and the cosmographic sides of the method.