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Register a new userConformal inflation in the metric-affine geometry
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Systematic understanding for classes of inflationary models is investigated from the viewpoint of the local conformal symmetry and the slightly broken global symmetry in the framework of the metric-affine geometry. In the metric-affine geometry, which is a generalisation of the Riemannian one adopted in the ordinary General Relativity, the affine connection is an independent variable of the metric rather than given e.g. by the Levi-Civita connection as its function. Thanks to this independency, the metric-affine geometry can preserve the local conformal symmetry in each term of the Lagrangian contrary to the Riemannian geometry, and then the local conformal invariance can be compatible with much more kinds of global symmetries. As simple examples, we consider the two-scalar models with the broken $mathrm{SO}(1,1)$ or $mathrm{O}(2)$, leading to the well-known $alpha$-attractor or natural inflation, respectively. The inflaton can be understood as their pseudo Nambu-Goldstone boson.
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We motivate a minimal realization of slow-roll $k$-inflation by incorporating the local conformal symmetry and the broken global $mathrm{SO}(1,1)$ symmetry in the metric-affine geometry. With use of the metric-affine geometry where both the metric and the affine connection are treated as independent variables, the local conformal symmetry can be preserved in each term of the Lagrangian and thus higher derivatives of scalar fields can be easily added in a conformally invariant way. Predictions of this minimal slow-roll $k$-inflation, $n_mathrm{s}sim 0.96$, $rsim 0.005$, and $c_mathrm{s}sim 0.03$, are not only consistent with current observational data but also have a prospect to be tested by forthcoming observations.
We study the Standard Model (SM) in Weyl conformal geometry. This embedding is truly minimal, {it with no new fields} beyond the SM spectrum and Weyl geometry. The action inherits a gauged scale symmetry $D(1)$ (known as Weyl gauge symmetry) from the underlying geometry. The associated Weyl quadratic gravity undergoes spontaneous breaking of $D(1)$ by a geometric Stueckelberg mechanism in which the Weyl gauge field ($omega_mu$) acquires mass by absorbing the spin-zero mode of the $tilde R^2$ term in the action. This mode also generates the Planck scale. The Einstein-Hilbert action emerges in the broken phase. In the presence of the SM, this mechanism receives corrections (from the Higgs) and it can induce electroweak (EW) symmetry breaking. The Higgs field has direct couplings to the Weyl gauge field while the SM fermions only acquire such couplings following the kinetic mixing of the gauge fields of $D(1)times U(1)_Y$. One consequence is that part of the mass of $Z$ boson is not due to the usual Higgs mechanism, but to its mixing with massive $omega_mu$. Precision measurements of $Z$ mass set lower bounds on the mass of $omega_mu$ which can be light (few TeV), depending on the mixing angle and Weyl gauge coupling. The Higgs mass and the EW scale are proportional to the vev of the Stueckelberg field. In the early Universe the Higgs field can have a geometric origin, by Weyl vector fusion, and the Higgs potential can drive inflation. The dependence of the tensor-to-scalar ratio $r$ on the spectral index $n_s$ is similar to that in Starobinsky inflation but mildly shifted to lower $r$ by the Higgs non-minimal coupling to Weyl geometry.
We classify the metric-affine theories of gravitation, in which the metric and the connections are treated as independent variables, by use of several constraints on the connections. Assuming the Einstein-Hilbert action, we find that the equations for the distortion tensor (torsion and non-metricity) become algebraic, which means that those variables are not dynamical. As a result, we can rewrite the basic equations in the form of Riemannian geometry. Although all classified models recover the Einstein gravity in the Palatini formalism (in which we assume there is no coupling between matter and the connections), but when matter field couples to the connections, the effective Einstein equations include an additional hyper energy-momentum tensor obtained from the distortion tensor. Assuming a simple extension of a minimally coupled scalar field in metric-affine gravity, we analyze an inflationary scenario. Even if we adopt a chaotic inflation potential, certain parameters could satisfy observational constraints. Furthermore, we find that a simple form of Galileon scalar field in metric-affine could cause G-inflation.
Open inflation scenario is attracting a renewed interest in the context of string landscape. Since there are a large number of metastable de Sitter vacua in string landscape, tunneling transitions to lower metastable vacua through the bubble nucleation occur quite naturally. Although the deviation of Omega_0 from unity is small by the observational bound, we argue that the effect of this small deviation on the large angle CMB anisotropies can be significant for tensor-type perturbation in open inflation scenario. We consider the situation in which there is a large hierarchy between the energy scale of the quantum tunneling and that of the slow-roll inflation in the nucleated bubble. If the potential just after tunneling is steep enough, a rapid-roll phase appears before the slow-roll inflation. In this case the power spectrum is basically determined by the Hubble rate during the slow-roll inflation. If such rapid-roll phase is absent, the power spectrum keeps the memory of the high energy density there in the large angular components. The amplitude of large angular components can be enhanced due to the effects of the wall fluctuation mode if the bubble wall tension is small. Therefore, one can construct some models in which the deviation of Omega_0 from unity is large enough to produce measurable effects. We also consider a more general class of models, where the false vacuum decay may occur due to Hawking-Moss tunneling, as well as the models involving more than one scalar field. We discuss scalar perturbations in these models and point out that a large set of such models is already ruled out by observational data, unless there was a very long stage of slow-roll inflation after the tunneling. These results show that observational data allow us to test various assumptions concerning the structure of the string theory potentials and the duration of the last stage of inflation.
We point out that the successful generation of the electroweak scale via gravitational instanton configurations in certain scalar-tensor theories can be viewed as the aftermath of a simple requirement: the existence of a quadratic pole with a sufficiently small residue in the Einstein-frame kinetic term for the Higgs field. In some cases, the inflationary dynamics may also be controlled by this residue and therefore related to the Fermi-to-Planck mass ratio, up to possible uncertainties associated with the instanton regularization. We present here a unified framework for this hierarchy generation mechanism, showing that the aforementioned residue can be associated with the curvature of the Einstein-frame target manifold in models displaying spontaneous breaking of dilatations. Our findings are illustrated through examples previously considered in the literature.
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