Dynamically Generated Inflationary ΛCDM

Our primary objective is to construct a plausible, unified model of inflation, dark energy and dark matter from a fundamental Lagrangian action first principle, wherein all fundamental ingredients are systematically dynamically generated starting from a very simple model of modified gravity interacting with a single scalar field employing the formalism of non-Riemannian spacetime volume-elements. The non-Riemannian volume element in the initial scalar field action leads to a hidden, nonlinear Noether symmetry which produces an energy-momentum tensor identified as the sum of a dynamically generated cosmological constant and dust-like dark matter. The non-Riemannian volume-element in the initial Einstein–Hilbert action upon passage to the physical Einstein-frame creates, dynamically, a second scalar field with a non-trivial inflationary potential and with an additional interaction with the dynamically generated dark matter. The resulting Einstein-frame action describes a fully dynamically generated inflationary model coupled to dark matter. Numerical results for observables such as the scalar power spectral index and the tensor-to-scalar ratio conform to the latest 2018 PLANCK data.

Dynamically Generated Inflationary Lambda-CDM

Our primary objective is to construct a plausible unified model of inflation, dark energy and dark matter from a fundamental Lagrangian action first principle, where all fundamental ingredients are systematically dynamically generated starting from a very simple model of modified gravity interacting with a single scalar field employing the formalism of non-Riemannian spacetime volume-elements. The non-Riemannian volume element in the initial scalar field action leads to a hidden nonlinear Noether symmetry which produces energy-momentum tensor identified as a sum of a dynamically generated cosmological constant and a dust-like dark matter. The non-Riemannian volume-element in the initial Einstein-Hilbert action upon passage to the physical Einstein-frame creates dynamically a second scalar field with a non-trivial inflationary potential and with an additional interaction with the dynamically generated dark matter. The resulting Einstein-frame action describes a fully dynamically generated inflationary model coupled to dark matter. Numerical results for observables such as the scalar power spectral index and the tensor-to-scalar ratio conform to the latest 2018 PLANCK data.

Cosmological Solutions from a Multi-Measure Model with Inflaton Field

In a recent work, we demonstrated that a modified gravity model in which a scalar “darkon” field is coupled to both the standard Riemannian metric and to another non-Riemannian volume form is compatible with observational data from Supernovae Type Ia. Here, we investigate a more complicated model with an additional “inflaton” scalar field. We demonstrate numerically that the model can qualitatively reproduce the Universe inflation epoch, matter dominated epoch, and present accelerating expansion in a seamless way. We show that such solutions occur only when the model parameters are within a very particular range. The main numerical problem we are faced with is reproducing the extremely small time of the inflation epoch. Here, we present how the variation of some parameters affects this time.

Dynamically generated inflation from non-Riemannian volume forms

We propose a simple modified gravity model without any initial matter fields in terms of several alternative non-Riemannian spacetime volume elements within the metric (second order) formalism. We show how the non-Riemannian volume-elements, when passing to the physical Einstein frame, create a canonical scalar field and produce dynamically a non-trivial inflationary-type potential for the latter with a large flat region and a stable low-lying minimum. We study the evolution of the cosmological solutions from the point of view of theory of dynamical systems. The theory predicts the spectral index $$n_s \approx 0.96$$ns≈0.96 and the tensor-to-scalar ratio $$r \approx 0.002$$r≈0.002 for 60 e-folds, which is in accordance with the observational data. In the future Euclid and SPHEREx missions or the BICEP3 experiment are expected to provide experimental evidence to test those predictions.

Cosmological aspects of a unified dark energy and dust dark matter model

Recently, a model of modified gravity plus single scalar field was proposed, in which the scalar couples both to the standard Riemannian volume form given by the square root of the determinant of the Riemannian metric, as well as to another non-Riemannian volume form given in terms of an auxiliary maximal rank antisymmetric tensor gauge field. This model provides an exact unified description of both dark energy (via dynamically generated cosmological constant) and dark matter (as a “dust” fluid due to a hidden nonlinear Noether symmetry). In this paper, we test the model against Supernovae type Ia experimental data and investigate the future Universe evolution which follows from it. Our results show that this model has very interesting features allowing various scenarios of Universe evolution and in the same time perfectly fits contemporary observational data. It can describe exponentially expanding or finite expanding Universe and moreover, a Universe with phase transition of first kind. The phase transition occurs to a new, emerging at some time ground state with lower energy density, which affects significantly the Universe evolution.

Calibrations and laminations

A calibration of degree k ∈ ℕ on a Riemannian manifold M is a closed differential k-form θ such that the integral of θ over every k-dimensional, oriented submanifold N is smaller or equal to the Riemannian volume of N. A calibration θ is said to calibrate N if θ restricts to the oriented volume form of N. We investigate conditions on a calibration θ that ensure the existence of submanifolds calibrated by θ. The cases k = 1 and k > 1 turn out to be essentially different. Our main result says that, on a compact manifold M, a calibration θ calibrates a lamination if θ is simple, of class C1, and if θ has minimal comass norm in its cohomology class.

Vacuum structure and gravitational bags produced by metric-independent space–time volume-form dynamics

We propose a new class of gravity-matter theories, describing [Formula: see text] gravity interacting with a nonstandard nonlinear gauge field system and a scalar “dilaton,” formulated in terms of two different non-Riemannian volume-forms (generally covariant integration measure densities) on the underlying space–time manifold, which are independent of the Riemannian metric. The nonlinear gauge field system contains a square-root [Formula: see text] of the standard Maxwell Lagrangian which is known to describe charge confinement in flat space–time. The initial new gravity-matter model is invariant under global Weyl-scale symmetry which undergoes a spontaneous breakdown upon integration of the non-Riemannian volume-form degrees of freedom. In the physical Einstein frame we obtain an effective matter-gauge-field Lagrangian of “k-essence” type with quadratic dependence on the scalar “dilaton” field kinetic term [Formula: see text], with a remarkable effective scalar potential possessing two infinitely large flat regions as well as with nontrivial effective gauge coupling constants running with the “dilaton” [Formula: see text]. Corresponding to each of the two flat regions we find “vacuum” configurations of the following types: (i) [Formula: see text] and a nonzero gauge field vacuum [Formula: see text], which corresponds to a charge confining phase; (ii) [Formula: see text] (“kinetic vacuum”) and ordinary gauge field vacuum [Formula: see text] which supports confinement-free charge dynamics. In one of the flat regions of the effective scalar potential we also find: (iii) [Formula: see text] (“kinetic vacuum”) and a nonzero gauge field vacuum [Formula: see text], which again corresponds to a charge confining phase. In all three cases, the space–time metric is de Sitter or Schwarzschild–de Sitter. Both “kinetic vacuums” (ii) and (iii) can exist only within a finite-volume space region below a de Sitter horizon. Extension to the whole space requires matching the latter with the exterior region with a nonstandard Reissner–Nordström–de Sitter geometry carrying an additional constant radial background electric field. As a result, we obtain two classes of gravitational bag-like configurations with properties, which on one hand partially parallel some of the properties of the solitonic “constituent quark” model and, on the other hand, partially mimic some of the properties of MIT bags in QCD phenomenology.

OBSERVATIONS ON GAUSSIAN UPPER BOUNDS FOR NEUMANN HEAT KERNELS

Given a domain ${\rm\Omega}$ of a complete Riemannian manifold ${\mathcal{M}}$, define ${\mathcal{A}}$ to be the Laplacian with Neumann boundary condition on ${\rm\Omega}$. We prove that, under appropriate conditions, the corresponding heat kernel satisfies the Gaussian upper bound $$\begin{eqnarray}h(t,x,y)\leq \frac{C}{[V_{{\rm\Omega}}(x,\sqrt{t})V_{{\rm\Omega}}(y,\sqrt{t})]^{1/2}}\biggl(1+\frac{d^{2}(x,y)}{4t}\biggr)^{{\it\delta}}e^{-d^{2}(x,y)/4t}\quad \text{for}~t>0,~x,y\in {\rm\Omega}.\end{eqnarray}$$ Here $d$ is the geodesic distance on ${\mathcal{M}}$, $V_{{\rm\Omega}}(x,r)$ is the Riemannian volume of $B(x,r)\cap {\rm\Omega}$, where $B(x,r)$ is the geodesic ball of centre $x$ and radius $r$, and ${\it\delta}$ is a constant related to the doubling property of ${\rm\Omega}$. As a consequence we obtain analyticity of the semigroup $e^{-t{\mathcal{A}}}$ on $L^{p}({\rm\Omega})$ for all $p\in [1,\infty )$ as well as a spectral multiplier result.