  We constrain the expansion history of the Universe and the cosmological
matter density fraction in a model-independent way by exclusively making use of
the relationship between background and perturbations under a minimal set of
assumptions. We do so by employing a Gaussian process to model the expansion
history of the Universe from present time to the recombination era. The
expansion history and the cosmological matter density are then constrained
using recent measurements from cosmic chronometers, Type-Ia supernovae, baryon
acoustic oscillations, and redshift-space distortion data. Our results show
that the evolution in the reconstructed expansion history is compatible with
the \textit{Planck} 2018 prediction at all redshifts. The current data
considered in this study can constrain a Gaussian process on $H(z)$ to an
average $9.4 \%$ precision across redshift. We find $\Omega_m = 0.224 \pm
0.066$, lower but statistically compatible with the \textit{Planck} 2018
cosmology. Finally, the combination of future DESI measurements with the CMB
measurement considered in this work holds the promise of $8 \%$ average
constraints on a model-independent expansion history as well as a five-fold
tighter $\Omega_m$ constraint using the methodology developed in this work.
