A general reconstruction of the recent expansion history of the universe
Distance measurements are currently the most powerful tool to study the expansion history of the universe without specifying its matter content nor any theory of gravitation. Assuming only an isotropic, homogeneous and flat universe, in this work we introduce a model-independent method to reconstruct directly the deceleration function via a piecewise function. Including a penalty factor, we are able to vary continuously the complexity of the deceleration function from a linear case to an arbitrary $(n+1)$-knots spline interpolation. We carry out a Monte Carlo (MC) analysis to determine the best penalty factor, evaluating the bias-variance trade-off, given the uncertainties of the SDSS-II and SNLS supernova combined sample (JLA), compilations of baryon acoustic oscillation (BAO) and $H(z)$ data. The bias-variance analysis is done for three fiducial models with different features in the deceleration curve. For each fiducial model, we test different reconstructions using, in each case, more than $10^4$ catalogs in a total of about $5\times 10^5$. This investigation proved to be essential in determining the best reconstruction to study these data. We show that, evaluating a single fiducial model, the conclusions about the bias-variance ratio are misleading. We determine the reconstruction method in which the bias represents at most $10\%$ of the total uncertainty. Finally, we apply the Ensemble Sampler Markov Chain Monte Carlo (ESMCMC) method to explore the posterior of the deceleration function up to redshift $1.3$ (using only JLA) and $2.3$ (JLA+BAO+$H(z)$). We obtain that the standard cosmological model agrees within $3\sigma$ level with the reconstructed results in the whole studied redshift intervals. Since our method is calibrated to minimize the bias, the error bars of the reconstructed functions are a good approximation for the total uncertainty.