  We study some possible astrophysical implications of a very weakly coupled
ultralight dilaton-type scalar field. Such a field may develop an
(approximately stable) network of domain walls. The domain wall thickness is
assumed to be comparable with the thickness of the luminous part of the spiral
galaxies. The walls provide trapping for galactic matter. This is used to
motivate the very existence of the spiral galaxies. A zero mode existing on the
domain wall is a massless scalar particle confined to 1+2 dimensions. At
distances much larger than the galaxy/wall thickness, the zero-mode exchange
generates a logarithmic potential, acting as an additional term with respect to
Newton's gravity. The logarithmic term naturally leads to constant rotational
velocities at the periphery. We estimate the scalar field coupling to the
matter energy-momentum tensor needed to fit the observable flat rotational
curves of the spiral galaxies. The value of this coupling turns out to be
reasonable -- we find no contradiction with the existing data.
