The MSSM invisible Higgs in the light of dark matter and g-2

Giving up the assumption of the gaugino mass unification at the GUT scale, the latest LEP and Tevatron data still allow the lightest supersymmetric Higgs to have a large branching fraction into invisible neutralinos. Such a Higgs may be difficult to discover at the LHC and is practically unreachable at the Tevatron. We argue that, for some of these models to be compatible with the relic density, light sleptons with masses not far above the current limits are needed. There are however models that allow for larger sleptons masses without being in conflict with the relic density constraint. This is possible because these neutralinos can annihilate efficiently through a Z-pole. We also find that many of these models can nicely account, at the 2\sigma level, for the discrepancy in the latest g-2 measurement. However, requiring consistency with the g-2 at the 1\sigma level, excludes models that lead to the largest Higgs branching fraction into LSP's. In all cases one expects that even though the Higgs might escape detection, one would have a rich SUSY phenomenology even at the Tevatron, through the production of charginos and neutralinos