H0LiCOW-III. Quantifying the effect of mass along the line of sight to the gravitational lens HE 0435-1223 through weighted galaxy counts

Based on spectroscopy and multiband wide-field observations of the gravitationally lensed quasar HE 0435-1223, we determine the probability distribution function of the external convergence K-ext for this system. We measure the under/overdensity of the line of sight towards the lens system and compare it to the average line of sight throughout the Universe, determined by using the CFHTLenS (The Canada France Hawaii Lensing Survey) as a control field. Aiming to constrain K-ext as tightly as possible, we determine under/overdensities using various combinations of relevant informative weighting schemes for the galaxy counts, such as projected distance to the lens, redshift and stellar mass. We then convert the measured under/overdensities into K-ext distribution, using ray-tracing through the Millennium Simulation. We explore several limiting magnitudes and apertures, and account for systematic and statistical uncertainties relevant to the quality of the observational data, which we further test through simulations. Our most robust estimate of K-ext has a median value K-ext(med)= 0.004 and a standard deviation sigma(k)= 0.025. The measured sigma(k) corresponds to 2.5 per cent relative uncertainty on the time delay distance, and hence the Hubble constant H-0 inferred from this system. The median K-ext(med) value varies by similar to 0.005 with the adopted aperture radius, limiting magnitude and weighting scheme, as long as the latter incorporates galaxy number counts, the projected distance to the main lens and a prior on the external shear obtained from mass modelling. This corresponds to just similar to 0.5 per cent systematic impact on H-0. The availability of a well-constrained.ext makes HE 0435-1223 a valuable system for measuring cosmological parameters using strong gravitational lens time delays.