Inflaton is the scalar field that causes inflation. Why it is assumed that it is a scalar field with very weak interaction?
You may wish to look at S Shankaranarayanan papers on mass dimension one fermions based on a complete set of eigenspinors of the charge conjugation operator (ELKO) and their role in inflation.
A VEV of a fermion will break Lorentz invariance, which is a good
space time symmetry. When you minimize inflaton potential it will
naturally pick up nonzero VEV due to couplings with doublet Higgs.
Here I assume that inflaton is a singlet scalar.
Otherwise if inflaton transforms under extra gauge groups, then to
break them you will require a VEV of inflaton.
Unless I am mistaken Shankaranarayanan considers a condensate of mass dimension one fermions.
It should also be noted that unless we insist on breaking C invariance, constancy of speed light for all inertial observers only requires Cohen-Glashow symmetries (see their 2006 PRL on Very Special Relativity, VSR) and not the full Lorentz symmetry. The assumption of P symmetry takes from the VSR to SR.
arXiv.org > hep-ph > arXiv:hep-ph/0601236
Very Special Relativity
Andrew G. Cohen, Sheldon L. Glashow
This paper is discussing the effects of giving up full
Poincare symmetry and formulating QFTs based on
a subgroup of Lorentz group + space time translations.
They have suggested experimental tests of their theory.
From citations to their paper, I found a talk by Ralf Lehnert
arXiv:1404.3670 which states,
"At the present time, no convincing experimental evidence for
Lorentz or CPT breakdown exists. Nevertheless,
various approaches to more fundamental physics (e.g., quan
tum-gravity models) contain mechanisms for generat-
ing feeble violations of Lorentz, CPT, and translational in
variance."
Indeed true, and yet there is no strong _empirical_ evidence that dark sector must be Lorentz covariant particularly when dropping P covariance allows one to go to SIM(2) avatar of VSR and provides mass dimension one fermions with two aspects: (a) the interaction of these fermions with SM particles is suppressed by at least one power of unification/Planck scale with one exception, that of Higgs, (b) these new fermions carry unsuppressed quartic interaction. So without compromising with the constancy of speed of light one obtains a natural dark matter candidate with several interesting cosmological consequences (including those for inflation). There is quite a bit of literature on the subject and I"ll be happy to give these references if you wish. Shankaranarayanan's work is among several others.
I will be happy to get a reference, if possible a short talk which
describes qualitative aspects of Shankaranarayanan's work.
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