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We discuss models involving two scalar fields coupled to classical gravity that satisfy the general criteria: (i) the theory has no mass input parameters, (ii) classical scale symmetry is broken only through $-frac{1}{12}varsigma phi^2 R$ couplings where $varsigma$ departs from the special conformal value of $1$; (iii) the Planck mass is dynamically generated by the vacuum expectations values (VEVs) of the scalars (iv) there is a stage of viable inflation associated with slow roll in the two--scalar potential; (v) the final vacuum has a small to vanishing cosmological constant and an hierarchically small ratio of the VEVs and the ratio of the scalar masses to the Planck scale. This assumes the paradigm of classical scale symmetry as a custodial symmetry of large hierarchies.
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Scalar fields, $phi_i$ can be coupled non-minimally to curvature and satisfy the general criteria: (i) the theory has no mass input parameters, including the Planck mass; (ii) the $phi_i$ have arbitrary values and gradients, but undergo a general expansion and relaxation to constant values that satisfy a nontrivial constraint, $K(phi_i) =$ constant; (iii) this constraint breaks scale symmetry spontaneously, and the Planck mass is dynamically generated; (iv) there can be adequate inflation associated with slow roll in a scale invariant potential subject to the constraint; (v) the final vacuum can have a small to vanishing cosmological constant (vi) large hierarchies in vacuum expectation values can naturally form; (vii) there is a harmless dilaton which naturally eludes the usual constraints on massless scalars. These models are governed by a global Weyl scale symmetry and its conserved current, $K_mu$ . At the quantum level the Weyl scale symmetry can be maintained by an invariant specification of renormalized quantities.
Weyl (scale) invariant theories of scalars and gravity can generate all mass scales spontaneously. In this paper we study a particularly simple version -- scale invariant $R^2$ gravity -- and show that, during an inflationary period, it leads to fluctuations which, for a particular parameter choice, are almost indistinguishable from normal $R^2$ inflation. Current observations place tight constraints on the primary coupling constant of this theory and predict a tensor to scalar ratio, $0.0033 > r > 0.0026$, which is testable with the next generation of cosmic microwave background experiments.
We study whether the relaxion mechanism solves the Higgs hierarchy problem against a high scale inflation or a high reheating temperature. To accomplish the mechanism, we consider the scenario that the Higgs vacuum expectation value is determined after inflation. We take into account the effects of the Hubble induced mass and thermal one in the dynamics of the relaxion.
We analyse Starobinsky inflation in the presence of the Brout Englert Higgs (BEH) boson with a non-minimal coupling to the Ricci scalar, $R$. The latter induces a coupling of the massive scaleron associated with the $R^2$ term to the BEH boson and this leads to a radiative correction to the BEH mass that must be fine tuned to keep the scalar light. For the case of $R^2$ driven inflation this requires a high level of fine tuning of order 1 part in $10^{8}$; for the case of Higgs inflation it is very much greater. We consider a scale invariant extension of the $R^2$/Higgs model and find that for $R^2$ driven inflation but not for Higgs inflation the required fine tuning is significantly reduced to one part in $10^{3-4}$. We consider the vacuum stability of the fine tuned model and its reheating and dilaton abundance after inflation. We also discuss possible gravitational wave signals associated with the model and the constraint on the mass of scalar or fermion dark matter candidates if they are produced by the gravitational couplings of the scalaron.
No-scale supergravity provides a successful framework for Starobinsky-like inflation models. Two classes of models can be distinguished depending on the identification of the inflaton with the volume modulus, $T$ (C-models), or a matter-like field, $phi$ (WZ-models). When supersymmetry is broken, the inflationary potential may be perturbed, placing restrictions on the form and scale of the supersymmetry breaking sector. We consider both types of inflationary models in the context of high-scale supersymmetry. We further distinguish between models in which the gravitino mass is below and above the inflationary scale. We examine the mass spectra of the inflationary sector. We also consider in detail mechanisms for leptogenesis for each model when a right-handed neutrino sector, used in the seesaw mechanism to generate neutrino masses, is employed. In the case of C-models, reheating occurs via inflaton decay to two Higgs bosons. However, there is a direct decay channel to the lightest right-handed neutrino which leads to non-thermal leptogenesis. In the case of WZ-models, in order to achieve reheating, we associate the matter-like inflaton with one of the right-handed sneutrinos whose decay to the lightest right handed neutrino simultaneously reheats the Universe and generates the baryon asymmetry through leptogenesis.
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