New insights into cosmological tensions

From the tiny anisotropies of the cosmic microwave background (CMB) to the distribution of galaxies in the large-scale structure (LSS), observations across multiple wavelengths and redshift ranges have converged on a simple yet powerful theoretical framework: the $\Lambda$CDM model. However, as measurement precision has improved, significant tensions have emerged between parameters inferred from early- and late-Universe probes, including CMB, Type Ia supernovae (SNIa) and baryon acoustic oscillations (BAO). The most prominent is the \emph{Hubble tension}, a $\gtrsim 5\sigma$ discrepancy between the value of the Hubble parameter, $H_{0}$, inferred from the CMB under $\Lambda$CDM and that measured independently via the local distance ladder. Alongside other inconsistencies, such as those in the growth of structure, this challenges the completeness of the standard cosmological model and raises the pressing question of whether these tensions arise from unaccounted systematics or new physics. This thesis addresses these issues from a \emph{model-independent} perspective, a timely and essential approach to refining the statistical interpretation of current cosmological tensions. By combining complementary cosmological probes, we reassess the calibration of the cosmic distance ladder, test fundamental relations underpinning the $\Lambda$CDM model and its extensions, and evaluate the robustness and mutual consistency of several key cosmological observables. Leveraging data from BAO, SNIa, and cosmic chronometers, we estimate the spatial curvature of the Universe and refine the distance ladder calibration independently of the main drivers of the current tensions. This methodology is further extended to calibrate high-$z$ probes -- Gamma-Ray Bursts and Quasars -- yielding a Hubble diagram up to $z \sim 7$ which provides an independent avenue for probing the expansion history beyond the reach of current SNIa and BAO data. In parallel, this thesis examines possible late-time solutions to the Hubble tension by investigating the shape of the Hubble function, the calibration of SNIa, and the impact of different BAO datasets. A new diagnostic is indeed introduced to test the consistency of different distance measurements, which allows us to reveal an unexpectedly strong mismatch between anisotropic and angular BAO data, underscoring the need for independent verification and methodology for future surveys. We further contribute to studies of LSS by advocating alternative LSS parameterizations that reduce biases in evaluating the growth of structure arising from the dependence on $H_0$. This approach provides a more physically consistent framework for assessing the impact of new physics on structure formation and the Hubble tension. This thesis provides novel, data-driven insights into key challenges in modern cosmology, critically assessing the $\Lambda$CDM framework and paving the way for future high-precision surveys to uncover the physical origin of current cosmological tensions.