The influence of weak lensing on measurements of the Hubble constant
  with quad-image gravitational lenses
  We investigate the influence of matter along the line of sight and in the
strong lens vicinity on the properties of quad image configurations and on the
measurements of the Hubble constant (H0). We use simulations of light
propagation in a nonuniform universe model with the distribution of matter in
space based on the data from Millennium Simulation. For a given strong lens and
haloes in its environment we model the matter distribution along the line of
sight many times, using different combinations of precomputed deflection maps
representing subsequent layers of matter on the path of rays. We fit the
simulated quad image configurations with time delays using nonsingular
isothermal ellipsoids (NSIE) with external shear as lens models, treating the
Hubble constant as a free parameter. We get a large artificial catalog of
lenses with derived values of the Hubble constant, Hfit. The average and median
of Hfit differ from the true value used in simulations by < 0.5 km/s/Mpc which
includes the influence of matter along the line of sight and in the lens
vicinity, and uncertainty in lens parameters, except the slope of the matter
distribution, which is fixed. The characteristic uncertainty of Hfit is ~3
km/s/Mpc. Substituting the lens shear parameters with values estimated from the
simulations reduces the uncertainty to ~2 km/s/Mpc.
