Data reconstruction of the dynamical connection function in f(Q) cosmology

We employ Hubble data and Gaussian Processes in order to reconstruct the dynamical connection function in f(Q) cosmology beyond the coincident gauge. In particular, there exist three branches of connections that satisfy the torsionless and curvatureless conditions, parametrized by a new dynamical function gamma. We express the redshift dependence of gamma in terms of the H(z) function and the f(Q) form and parameters, and then we reconstruct it using 55 H(z) observation data. First, we investigate the case where ordinary conservation law holds, and we reconstruct the f(Q) function, which is very well described by a quadratic correction on top of symmetric teleparallel equivalent of general relativity. Proceeding to the general case, we consider two of the most studied f(Q) models of the literature, namely the square-root and the exponential one. In both cases we reconstruct gamma (z), and we show that according to Akaike Information Criterion and Bayesian Information Criterion information criteria its inclusion is favoured compared to both Lambda cold dark matter paradigm, as well as to the same f(Q) models under the coincident gauge. This feature acts as an indication that f(Q) cosmology should be studied beyond the coincident gauge.