I wrote an undergrad paper on this decades ago now so I'm not sure I can explain that very well any more.
We know the Higgs exists now so electroweak symmetry breaking is on pretty good terms. That means at a high enough energy the W and Z bosons will lose their mass and kind of reverse fuse with the Higgs and photon and you'll get SU(2) Yang-Mills theory. In order to get our universe with broken SU(2)xU(1) symmetry and a massive Higgs and W, Z bosons then you really do need the symmetry breaking mechanism.
I cannot explain why that dumps energy into space-time and why that in particular causes the inflation and expansion of the Universe. I think its dependent upon the exact shape of the Higgs "sombrero" potential and we don't know exactly what that looks like. To have a Higgs mechanism breaking the electroweak symmetry without dumping energy into space time though I think is considered not likely. Once you dump the energy from the Higgs mechanism into space-time I think you're fairly guaranteed to get an inflating universe, but here is where I'm just entirely trusting what I've read in words, with no personal connection to math at all (although I do trust that the math exists for this)
Then on the flip side you can use inflation to explain the large scale mass structure in the universe and the globby strands of matter become quantum fluctuations in the pre-inflation universe. Of course I think they're still off by orders and orders of magnitude between theory and reality still(?) but that does lend some plausibility to it all.
The electroweak symmetry breaking is on much more firm ground since we can point at the Higgs particle and it doesn't make a lot of sense except as a broken symmetry. Since we can produce lots and lots of Higgs particles in the LHC now it means that the plausibility of at least some kind of simple inflationary cosmology is pretty high.
There is a real possibility that there are multiple different regions of the universe (and since the universe is presumably infinite that means presumably infinite numbers of regions) where the Higgs potential broke differently and physics is very different there. We find ourselves in a pretty uniform area of the universe, though, which is probably pretty good, and which is likely also explained by inflation. That might be wrong, but at least it needs to be considered seriously. And someone would need to come up with an explanation of why the infinite Universe would freeze uniformly to having exactly one way of breaking the primordial Yang-Mills symmetry across all of it -- it seems more plausible there would be different regions with fundamentally different electroweak symmetry. We find ourselves in one of the regions conducive to complex life because you need things like stellar nucleosynthesis to have anything interesting to talk about. That is all more of a philosophical story though -- except that we know there was one symmetry breaking and it just seems weirder to have only one and not N.
Well... if the Higgs symmetry breaking did drive inflation, then it happened before inflation. So it may well be the same throughout the universe.
A bit more about why I question the symmetry-breaking driving inflation, though: Let's say that when the symmetry breaks, it releases a lot of energy. Fine. Where does it release the energy? Where certain particles are, or throughout all space? I strongly suspect that it's where the particles are. That leads to those particles having more energy. But inflation isn't an explosion of the matter in the universe; it's an explosion of spacetime itself. Adding a bunch of energy to the matter may make the matter move more energetically; that shouldn't have any effect on the rate of expansion of spacetime.
Open to being proved wrong (preferably in ways I can follow...)
matter is energy and energy is matter. the false vacuum state is where the energy comes from (which is at every point in space-time). even in the universe right now the vacuum expectation value of any field is nonzero since virtual particles are being created and destroyed spontaneously at every point. the false vacuum is similar. when the symmetry is broken that energy is released everywhere.
since energy is matter i assume that means that the energy really is "stored" in a quantum field, so a particle of some sort. the vacuum expectation value of that particle must be large, so that when symmetry is broken those virtual particles decay and create the shower of matter and energy which drives inflation.
iirc, when symmetry breaking occurred, the universe found a new low-energy state, under the floor of the unified field. relative to the new low energy floor, the high energy bits look like particles. so a bunch of particles were produced, each of which had collosal kinetic energy, obeying new physical laws.
We know the Higgs exists now so electroweak symmetry breaking is on pretty good terms. That means at a high enough energy the W and Z bosons will lose their mass and kind of reverse fuse with the Higgs and photon and you'll get SU(2) Yang-Mills theory. In order to get our universe with broken SU(2)xU(1) symmetry and a massive Higgs and W, Z bosons then you really do need the symmetry breaking mechanism.
I cannot explain why that dumps energy into space-time and why that in particular causes the inflation and expansion of the Universe. I think its dependent upon the exact shape of the Higgs "sombrero" potential and we don't know exactly what that looks like. To have a Higgs mechanism breaking the electroweak symmetry without dumping energy into space time though I think is considered not likely. Once you dump the energy from the Higgs mechanism into space-time I think you're fairly guaranteed to get an inflating universe, but here is where I'm just entirely trusting what I've read in words, with no personal connection to math at all (although I do trust that the math exists for this)
Then on the flip side you can use inflation to explain the large scale mass structure in the universe and the globby strands of matter become quantum fluctuations in the pre-inflation universe. Of course I think they're still off by orders and orders of magnitude between theory and reality still(?) but that does lend some plausibility to it all.
The electroweak symmetry breaking is on much more firm ground since we can point at the Higgs particle and it doesn't make a lot of sense except as a broken symmetry. Since we can produce lots and lots of Higgs particles in the LHC now it means that the plausibility of at least some kind of simple inflationary cosmology is pretty high.
There is a real possibility that there are multiple different regions of the universe (and since the universe is presumably infinite that means presumably infinite numbers of regions) where the Higgs potential broke differently and physics is very different there. We find ourselves in a pretty uniform area of the universe, though, which is probably pretty good, and which is likely also explained by inflation. That might be wrong, but at least it needs to be considered seriously. And someone would need to come up with an explanation of why the infinite Universe would freeze uniformly to having exactly one way of breaking the primordial Yang-Mills symmetry across all of it -- it seems more plausible there would be different regions with fundamentally different electroweak symmetry. We find ourselves in one of the regions conducive to complex life because you need things like stellar nucleosynthesis to have anything interesting to talk about. That is all more of a philosophical story though -- except that we know there was one symmetry breaking and it just seems weirder to have only one and not N.