Can one determine cosmological parameters from multi-plane strong lens
  systems?
  Strong gravitational lensing of sources with different redshifts has been
used to determine cosmological distance ratios, which in turn depend on the
expansion history. Hence, such systems are viewed as potential tools for
constraining cosmological parameters. Here we show that in lens systems with
two distinct source redshifts, of which the nearest one contributes to the
light deflection towards the more distant one, there exists an invariance
transformation which leaves all strong lensing observables unchanged (except
the product of time delay and Hubble constant), generalizing the well-known
mass-sheet transformation in single plane lens systems. The transformation
preserves the relative distribution of mass and light, so that a
`mass-follows-light' assumption does not fix the MST. All time delays (from
sources on both planes) scale with the same factor -- time-delay ratios are
therefore invariant under the MST. Changing cosmological parameters, and thus
distance ratios, is essentially equivalent to such a mass-sheet transformation.
As an example, we discuss the double source plane system SDSSJ0946+1006, which
has been recently studied by Collett and Auger, and show that variations of
cosmological parameters within reasonable ranges lead to only a small
mass-sheet transformation in both lens planes. Hence, the ability to extract
cosmological information from such systems depends heavily on the ability to
break the mass-sheet degeneracy.
