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In the standard cosmological model, where space expands according to the FLRW metric under the influence of $\Lambda$CDM (dark energy, $\Lambda$, and cold dark matter).

In the standard model of particle physics the masses of the fundamental particles are set by the vacuum expectation value (VEV) of the Higgs field and the strength of the particle's interaction with that field.

So, the question is this: how do we disentangle the effect of an expanding universe from the effect of a Higgs field VEV that is changing with time? Specifically, an increasing Higgs VEV shrinks the physical size of atoms, increasing the frequency of their transitions, making existing photons lower energy, by comparison.

I'm especially interested in observable differences that have been observed or looked for because both a changeable Higgs VEV and dynamic space-time are predicted by separate, well supported, theories (the standard model of particle physics and general relativity, respectively). For instance, is there a difference in how the cosmic microwave background would appear? Or would the VEV changing alter the proton's mass in a way that keeps $m_e / m_p$ constant (affecting hyperfine transitions)? The latter is not a trivial question because most of the proton's mass comes from gluons at a scale set by color confinement and that, as far as I know, has not yet been connected to the Higgs VEV.

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One way that we can rule out a cosmology with a purely changing VEV in a static universe is the observed nearly perfect sameness in every direction, that is the isotropy, of the cosmic microwave background (CMB). The cosmology that can be ruled out 'starts' as a static space-time filled with a heat bath of particles and a low Higgs VEV. The Higgs field then quantum tunnels out of a local minimum somewhere, and this zone of Higher Higgs VEV spreads through the universe, expanding at the speed of light. The odds that we would be near the center of such an expanding bubble of increasing Higgs VEV, as required to see an isotropic CMB, are essentially nil compared to the odds of being far from the center and seeing a huge anisotropy.

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