Bayesian Linear Regression for estimation of the Fractal Dimension of the Cerebral Cortex

The cerebral cortex presents self-similarity in a proper interval of spatial scales, a property typical of natural objects exhibiting fractal geometry. Its complexity therefore can be characterized by the value of its fractal dimension (FD). In the computation of this metric, it has usually been employed a frequentist approach to probability, with point estimator methods yielding only the optimal values of the FD. In our study, we aimed at retrieving a more complete evaluation of the FD by utilizing a Bayesian model for the linear regression analysis of the box-counting algorithm. We used T1-weighted MRI data of 86 healthy subjects (age 44.2 ± 17.1 years, mean ± standard deviation, 48% males) in order to gain insights into the confidence of our measure and investigate the relationship between mean Bayesian FD and age. Our approach yielded a stronger and significant (P < .001) correlation between mean Bayesian FD and age as compared to the previous implementation. Thus, our results make us suppose that the Bayesian FD is a more truthful estimation for the fractal dimension of the cerebral cortex compared to the frequentist FD.

Gravitational waves as dark sirens: an astrophysical and cosmological analysis

Despite the success of the ΛCDM model in describing the Universe, a possible tension between early- and late-Universe cosmological measurements is calling for new independent cosmological probes. Amongst the most promising ones, gravitational waves (GWs) can provide a self-calibrated measurement of the luminosity distance. However, to obtain cosmological constraints, additional information is needed to break the degeneracy between parameters in the gravitational waveform. In this thesis, we exploit the latest LIGO-Virgo-KAGRA Gravitational Wave Transient Catalog (GWTC-3) of GW sources to constrain the background cosmological parameters together with the astrophysical properties of Binary Black Holes (BBHs), using information from their mass distribution. We expand the public code MGCosmoPop, previously used for the application of this technique, by implementing a state-of-the-art model for the mass distribution, needed to account for the presence of non-trivial features, i.e. a truncated power law with two additional Gaussian peaks, referred to as Multipeak. We then analyse GWTC-3 comparing this model with simpler and more commonly adopted ones, both in the case of fixed and varying cosmology, and assess their goodness-of-fit with different model selection criteria, and their constraining power on the cosmological and population parameters. We also start to explore different sampling methods, namely Markov Chain Monte Carlo and Nested Sampling, comparing their performances and evaluating the advantages of both. We find concurring evidence that the Multipeak model is favoured by the data, in line with previous results, and show that this conclusion is robust to the variation of the cosmological parameters. We find a constraint on the Hubble constant of H0 = 61.10+38.65−22.43 km/s/Mpc (68% C.L.), which shows the potential of this method in providing independent constraints on cosmological parameters. The results obtained in this work have been included in [1].