  We consider general linear Higgs-sigma models as ultra-violet completions of
the Higgs inflation. We introduce general higher curvature terms beyond
Einstein gravity and recast them into a class of linear Higgs-sigma models in
the scalar-dual formulation where conformal symmetry is manifest. Integrating
out the sigma field in this class of linear sigma models, we obtain the same
Higgs inflation Lagrangian of non-linear sigma model type in the effective
theory. We show that the successful inflation for sigma field singles out the
sigma-field potential derived from the $R^2$ term and the tracker solution for
dark energy at late times can be realized for the $R^{p+1}$ term with $-1<p<0$.
We also discuss the implications of Higgs-sigma interactions for the inflation
and the vacuum stability in the Standard Model.
