  Certain covariant theories of the modified Newtonian dynamics paradigm seem
to require an additional hot dark matter (HDM) component - in the form of
either heavy ordinary neutrinos or more recently light sterile neutrinos (SNs)
with a mass around 11eV - to be relieved of problems ranging from cosmological
scales down to intermediate ones relevant for galaxy clusters. Here we suggest
using gravitational lensing by galaxy clusters to test such a marriage of
neutrino HDM and modified gravity, adopting the framework of
tensor-vector-scalar theory (TeVeS). Unlike conventional cold dark matter
(CDM), such HDM is subject to strong phase-space constraints, which allows one
to check cluster lens models inferred within the modified framework for
consistency. Since the considered HDM particles cannot collapse into
arbitrarily dense clumps and only form structures well above the galactic
scale, systems which indicate the need for dark substructure are of particular
interest. As a first example, we study the cluster lens Abell 2390 and its
impressive straight arc with the help of numerical simulations. Based on our
results, we outline a general and systematic approach to model cluster lenses
in TeVeS which significantly reduces the calculation complexity. We further
consider a simple bimodal lens configuration, capable of producing the straight
arc, to demonstrate our approach. We find that such a model is marginally
consistent with the hypothesis of 11eV SNs. Future work including more detailed
and realistic lens models may further constrain the necessary SN distribution
and help to conclusively assess this point. Cluster lenses could therefore
provide an interesting discriminator between CDM and such modified gravity
scenarios supplemented by SNs or other choices of HDM.
