non-Riemannian Volume Form Research Articles

Broken Scale Invariance, Gravity Mass, and Dark Energy inModified Einstein Gravity with Two Measure Finsler like Variables

We study new classes of generic off-diagonal and diagonal cosmological solutions for effective Einstein equations in modified gravity theories (MGTs), with modified dispersion relations (MDRs), and encoding possible violations of (local) Lorentz invariance (LIVs). Such MGTs are constructed for actions and Lagrange densities with two non-Riemannian volume forms (similar to two measure theories (TMTs)) and associated bimetric and/or biconnection geometric structures. For conventional nonholonomic 2 + 2 splitting, we can always describe such models in Finsler-like variables, which is important for elaborating geometric methods of constructing exact and parametric solutions. Examples of such Finsler two-measure formulations of general relativity (GR) and MGTs are considered for Lorentz manifolds and their (co) tangent bundles and abbreviated as FTMT. Generic off-diagonal metrics solving gravitational field equations in FTMTs are determined by generating functions, effective sources and integration constants, and characterized by nonholonomic frame torsion effects. By restricting the class of integration functions, we can extract torsionless and/or diagonal configurations and model emergent cosmological theories with square scalar curvature, R2, when the global Weyl-scale symmetry is broken via nonlinear dynamical interactions with nonholonomic constraints. In the physical Einstein–Finsler frame, the constructions involve: (i) nonlinear re-parametrization symmetries of the generating functions and effective sources; (ii) effective potentials for the scalar field with possible two flat regions, which allows for a unified description of locally anisotropic and/or isotropic early universe inflation related to acceleration cosmology and dark energy; (iii) there are “emergent universes” described by off-diagonal and diagonal solutions for certain nonholonomic phases and parametric cosmological evolution resulting in various inflationary phases; (iv) we can reproduce massive gravity effects in two-measure theories. Finally, we study a reconstructing procedure for reproducing off-diagonal FTMT and massive gravity cosmological models as effective Einstein gravity or Einstein–Finsler theories.

Dynamically generated inflationary two-field potential via non-Riemannian volume forms

We consider a simple model of modified gravity interacting with a single scalar field φ with weakly coupled exponential potential within the framework of non-Riemannian spacetime volume-form formalism. The specific form of the action is fixed by the requirement of invariance under global Weyl-scale symmetry. Upon passing to the physical Einstein frame we show how the non-Riemannian volume elements create a second canonical scalar field u and dynamically generate a non-trivial two-scalar-field potential Ueff(u,φ) with two remarkable features: (i) it possesses a large flat region for large u describing a slow-roll inflation; (ii) it has a stable low-lying minimum w.r.t. (u,φ) representing the dark energy density in the “late universe”. We study the corresponding two-field slow-roll inflation and show that the pertinent slow-roll inflationary curve φ=φ(u) in the two-field space (u,φ) has a very small curvature, i.e., φ changes very little during the inflationary evolution of u on the flat region of Ueff(u,φ). Explicit expressions are found for the slow-roll parameters which differ from those in the single-field inflationary counterpart. Numerical solutions for the scalar spectral index and the tensor-to-scalar ratio are derived agreeing with the observational 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 and the tensor-to-scalar ratio r approx 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.

Two scalar fields inflation from scale-invariant gravity with modified measure

A new class of gravity-matter model defined with an independent non-Riemannian volume form is studied, in the second order formalism. The action has a global scale invariance symmetry, which can be broken by the equation of motion of the measure fields. By a conformal transformation the theory transforms into a theory which governed by two independent scalar fields and a specific potential. When the scale invariance symmetry is not broken also in the equations of motion, only one field appear in the effective potential. This effective potential has a flat region which is responsible for the inflation, and it ends with a minimum, which is responsible for the present vacuum energy. In the general case, with a spontaneous symmetry breaking of the scale symmetry, one scalar field is responsible for the inflation and the other could be responsible for the particle creation. The first field (inflaton) can slowly roll, while the second field (curvaton) is more strongly coupled to the potential. The number of e-folds for both models is also dissuaded and could be constraint in the future.

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.

Unified dark energy and dust dark matter dual to quadratic purely kinetic K-essence

We consider a modified gravity plus single scalar-field model, where the scalar Lagrangian couples symmetrically both to the standard Riemannian volume-form (spacetime integration measure density) given by the square root of the determinant of the Riemannian metric, as well as to another non-Riemannian volume-form in terms of an auxiliary maximal-rank antisymmetric tensor gauge field. As shown in a previous paper, the pertinent scalar-field dynamics provides an exact unified description of both dark energy via dynamical generation of a cosmological constant, and dark matter as a “dust” fluid with geodesic flow as a result of a hidden Noether symmetry. Here we extend the discussion by considering a non-trivial modification of the purely gravitational action in the form of $$f(R) = R - \alpha R^2$$ generalized gravity. Upon deriving the corresponding “Einstein-frame” effective action of the latter modified gravity-scalar-field theory we find explicit duality (in the sense of weak versus strong coupling) between the original model of unified dynamical dark energy and dust fluid dark matter, on one hand, and a specific quadratic purely kinetic “k-essence” gravity–matter model with special dependence of its coupling constants on only two independent parameters, on the other hand. The canonical Hamiltonian treatment and Wheeler–DeWitt quantization of the dual purely kinetic “k-essence” gravity–matter model is also briefly discussed.

Curvaton reheating mechanism in a scale invariant two measures theory

The curvaton reheating mechanism in a Scale Invariant Two Measures Theory defined in terms of two independent non-Riemannian volume forms (alternative generally covariant integration measure densities) on the space-time manifold which are metric independent is studied. The model involves two scalar matter fields, a dilaton, that transforms under scale transformations and it will be used also as the inflaton of the model and another scalar, which does not transform under scale transformations and which will play the role of a curvaton field. Potentials of appropriate form so that the pertinent action is invariant under global Weyl-scale symmetry are introduced. Scale invariance is spontaneously broken upon integration of the equations of motion. After performing transition to the physical Einstein frame we obtain: (i) For given value of the curvaton field an effective potential for the scalar field with two flat regions for the dilaton which allows for a unified description of both early universe inflation as well as of present dark energy epoch;(iii) In the phase corresponding to the early universe, the curvaton has a constant mass and can oscillate decoupled from the dilaton and that can be responsible for both reheating and perturbations in the theory. In this framework, we obtain some interesting constraints on different parameters that appear in our model; (iii) For a definite parameter range the model possesses a non-singular "emergent universe" solution which describes an initial phase of evolution that precedes the inflationary phase. Finally we discuss generalizations of the model, through the effect of higher curvature terms, where inflaton and curvaton can have coupled oscillations.

Stable emergent Universe – a creation without Big‐Bang

AbstractBased on an earlier introduced new class of generalized gravity‐matter models defined in terms of two independent nonRiemannian volume forms (alternative generally covariant integration measure densities) on the space‐time manifold, we derive an effective “Einstein‐frame” theory featuring the following remarkable properties: (i) We obtain effective potential for the cosmological scalar field possessing two infinitely large flat regions which allows for a unified description of both early Universe inflation as well as of present dark energy epoch; (ii) for a specific parameter range the model possesses a non‐singular stable “emergent Universe” solution which describes an initial phase of evolution that precedes the inflationary phase. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Dark energy and dark matter from hidden symmetry of gravity model with a non-Riemannian volume form

We show that dark energy and dark matter can be described simultaneously by ordinary Einstein gravity interacting with a single scalar field provided the scalar field Lagrangian couples in a symmetric fashion to two different spacetime volume forms (covariant integration measure densities) on the spacetime manifold – one standard Riemannian given by $$\sqrt{-g}$$ (square root of the determinant of the pertinent Riemannian metric) and another non-Riemannian volume form independent of the Riemannian metric, defined in terms of an auxiliary antisymmetric tensor gauge field of maximal rank. Integration of the equations of motion of the latter auxiliary gauge field produce an a priori arbitrary integration constant that plays the role of a dynamically generated cosmological constant or dark energy. Moreover, the above modified scalar field action turns out to possess a hidden Noether symmetry whose associated conserved current describes a pressureless “dust” fluid which we can identify with the dark matter completely decoupled from the dark energy. The form of both the dark energy and dark matter that results from the above class of models is insensitive to the specific form of the scalar field Lagrangian. By adding an appropriate perturbation, which breaks the above hidden symmetry and along with this couples dark matter and dark energy, we also suggest a way to obtain growing dark energy in the present universe’s epoch without evolution pathologies.

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.

Emergent cosmology, inflation and dark energy

A new class of gravity-matter models defined in terms of two independent non-Riemannian volume forms (alternative generally covariant integration measure densities) on the space-time manifold are studied in some detail. These models involve an additional $R^2$ (square of the scalar curvature) term as well as scalar matter field potentials of appropriate form so that the pertinent action is invariant under global Weyl-scale symmetry. Scale invariance is spontaneously broken upon integration of the equations of motion for the auxiliary volume-form degrees of freedom. After performing transition to the physical Einstein frame we obtain: (i) An effective potential for the scalar field with two flat regions which allows for a unified description of both early universe inflation as well as of present dark energy epoch; (ii) For a definite parameter range the model possesses a non-singular "emergent universe" solution which describes an initial phase of evolution that precedes the inflationary phase; (iii) For a reasonable choice of the parameters the present model conforms to the Planck Collaboration data.