  We study the decay of 125 GeV Higgs boson to light LSP neutralino in the
phenomenological minimal supersymmetric standard model in the context of
collider searches and astrophysical experiments. We consider the parameter
space for light neutralinos that can be probed via the invisible Higgs decays
and higgsino searches at the ILC. We consider the cases where the light
neutralino is compatible with the observed relic density or where the thermal
relic is over-abundant, pointing to non-standard cosmology. In the former case,
when the neutralino properties give rise to under-abundant relic density, the
correct amount of relic abundance is assumed to be guaranteed by either
additional DM particles or by non-thermal cosmology. We contrast these
different cases. We assess what astrophysical measurements can be made, in
addition to the measurements made at the ILC, which can provide a clue to the
nature of the light neutralino. We find that a number of experiments, including
Xenon-nT, PICO-250, LZ in conjunction with measurements made at the ILC on
invisible Higgs width can pin down the nature of this neutralino, along with
its cosmological implications. Additionally, we also point out potential LHC
signatures that could be complementary in this region of parameter space.
