Does Higgs Boson influence gravity? Does the Higgs Boson influence gravity? Is there any way that in the expanding universe, Higgs particle has influence on it?
 A: The ubiquitous "nonzero vacuum expectation value" of the Higgs field, by which the Higgs supplies mass to quarks, leptons, and W and Z bosons, should actually be contributing to the vacuum energy of the universe, and therefore to the dark energy, in a way that isn't seen. So there is definitely something we don't understand about how the Higgs field interacts with gravity, e.g. perhaps something unknown cancels out its contribution to the vacuum energy. This is one aspect of what is called the cosmological constant problem. 
A: In addition to the V.E.V. of the Higgs contributing to the vacuum energy (part of the cosmological constant problem; there'a s good review in Rev. Mod. Phys. by Weinberg from the '80s), there have been investigations that the Higgs could be the inflaton field, which drove inflation in the early universe.
However, I think such a model is disfavoured, but I haven't looked into it.
Search `Higgs inflation Shaposhnikov' (the main proponent is M. Shaposhnikov) in Google.
A: Higgs field can act as attractive force carrier among the Standard Model fermions. The force is indeed greater than the gravitational force in the early universe (Hubble constant $H \gg m_{\text{Higgs}}$) and hence it could have induced collapse if there was no electromagnetic interactions.
