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.
