Does the Higgs field gravitate?

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Ive heard both that the Higgs field has a constant energy throughout space, but I also know that in GR all energy gravitates. Does the energy of non-ground state fields move around? If so, what effect would slightly more or less energy in the field have?
 

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The vacuum energy of the Higgs field contributes to the total vacuum energy. In theory it should gravitate by contributing to the cosmological constant. However, the experimental value of the cosmological constant is much smaller than the vacuum energy of the Higgs field, or any other similar contribution. It is not known why the cosmological constant is much smaller than the value given by the theory.
 
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Cool, does that imply that there is likely more energy in the Higgs field in intra vs intergalactic space or on the surface of a neutron star? Does that have any effect on the mass there? Mass comes from its interaction with the field, does that interaction change?

Does that also imply for deep time cosmology that the Higgs field will have less and less energy in vacuums as gravitationally bound system expand away from each other? Would the lower energy Higgs field eventually jump down to its ground state and obliterate the universe at c?
 

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