  We investigate the momentum distribution of weakly interacting 1D Bose gases
at thermal equilibrium both experimentally and theoretically. Momentum
distribution of single 1D Bose gases is measured using a focusing technique,
whose resolution we improve via a guiding scheme. The momentum distribution
compares very well with quantum Monte Carlo calculations for the Lieb-Liniger
model at finite temperature, allowing for an accurate thermometry of the gas
that agrees with (and improves upon) the thermometry based on in situ density
fluctuation measurements. The quasi-condensation crossover is investigated via
two different experimental parameter sets, corresponding to the two different
sides of the crossover. Classical field theory is expected to correctly
describe the quasi-condensation crossover of weakly interacting gases. We
derive the condition of validity of the classical field theory, and find that,
in typical experiments, interactions are too strong for this theory to be
accurate. This is confirmed by a comparison between the classical field
predictions and the numerically exact quantum Monte Carlo calculations.
