  The quest for understanding the late-time acceleration is haunted by an
immense freedom in the analysis of dynamical models for dark energy in extended
parameter spaces. Often-times having no prior knowledge at our disposal,
arbitrary choices are implemented to reduce the degeneracies between
parameters. We also encounter this issue in the case of quintessence fields,
where a scalar degree of freedom drives the late-time acceleration. In this
study, we implement a more physical prescription, the \textit{flow} condition,
to fine-tune the quintessence evolution for several field potentials. We find
that this prescription agrees well with the most recent catalogue of data,
namely supernovae type Ia, baryon acoustic oscillations, cosmic clocks and
distance to last scattering surface, and it enables us to infer the initial
conditions for the field, both potential and cosmological parameters. At
$2\sigma$ we find stricter bounds on the potential parameters $f/m_{pl}>0.26$
and $n<0.15$ for the PNGB and IPL potentials, respectively, while constraints
on cosmological parameters remain extremely consistent across all assumed
potentials. By implementing information criteria to assess their ability to fit
the data, we do not find any evidence against thawing models, which in fact are
statistically equivalent to $\Lambda$CDM, and the freezing ones are moderately
disfavoured. Through our analysis we place upper bounds on the slope of
quintessence potentials, consequently revealing a strong tension with the
recently proposed swampland criterion, finding the 2$\sigma$ upper bound of
$\lambda \sim 0.31$ for the exponential potential.

Cosmology and Nongalactic Astrophysics