Academic literature on the topic 'Phi-Weyl'

We aim to derive the most general f(R) gravity theory, including thematter, so that it be Weyl invariant. Making use of the mathematicalequivalence of these theories with an type of scalar-tensor theory, and byimposing the Weyl invariance for the pure gravity as well as for the mattersector, we obtain the fundamental equation that restricts the form of V (phi) (and, accordingly, of f(R)) so that the resulting action to be Weylinvariant in the Jordan frame. We show that this action is not otherthan the so-called dilaton gravity action with one scalar eld,, whicheective mass is R and Phi dependent. In the Einstein frame, the actionbecomes the Einstein-Hilbert action with the Ricci scalar being constantdue to that the eective mass of scalar eld in this frame vanish. So,we can assume that the Ricci scalar, in the Einstein frame, is the trueCosmological Constant. Therefore, is not preposterous to guess that, atleast mathematically, all Weyl invariant metric f(R) theory in the Jordanframe is equivalent, at classical level, to the Einstein gravity, in theEinstein frame, with a constant Ricci scalar. At quantum level, as it isknown, both theories are not equivalent due to the presence of anomaliesin one of the frames.

In this paper, we determine the components of the Weyl tensor of almost contact metric (ACR-) manifold of class $C_{12}$ on associated G-structure (AG-structure) space. As an application, we prove that the conformally flat ACR-manifold of class $C_{12}$ with $n>2$ is an $\eta$-Einstein manifold and conclude that it is an Einstein manifold such that the scalar curvature $r$ has provided. Also, the case when $n=2$ is discussed explicitly. Moreover, the relationships among conformally flat, conformally symmetric, $\xi$-conformally flat and $\Phi$-invariant Ricci tensor have been widely considered here and consequently we determine the value of scalar curvature $r$ explicitly with other applications. Finally, we define new classes with identities analogously to Gray identities and discuss their connections with class $C_{12}$ of ACR-manifold.

AbstractWe study quadratic gravity $$R^2+R_{[\mu \nu ]}^2$$ R 2 + R [ μ ν ] 2 in the Palatini formalism where the connection and the metric are independent. This action has a gauged scale symmetry (also known as Weyl gauge symmetry) of Weyl gauge field $$v_\mu = (\tilde{\Gamma }_\mu -\Gamma _\mu )/2$$ v μ = ( Γ ~ μ - Γ μ ) / 2 , with $$\tilde{\Gamma }_\mu $$ Γ ~ μ ($$\Gamma _\mu $$ Γ μ ) the trace of the Palatini (Levi-Civita) connection, respectively. The underlying geometry is non-metric due to the $$R_{[\mu \nu ]}^2$$ R [ μ ν ] 2 term acting as a gauge kinetic term for $$v_\mu $$ v μ . We show that this theory has an elegant spontaneous breaking of gauged scale symmetry and mass generation in the absence of matter, where the necessary scalar field ($$\phi $$ ϕ ) is not added ad-hoc to this purpose but is “extracted” from the $$R^2$$ R 2 term. The gauge field becomes massive by absorbing the derivative term $$\partial _\mu \ln \phi $$ ∂ μ ln ϕ of the Stueckelberg field (“dilaton”). In the broken phase one finds the Einstein–Proca action of $$v_\mu $$ v μ of mass proportional to the Planck scale $$M\sim \langle \phi \rangle $$ M ∼ ⟨ ϕ ⟩ , and a positive cosmological constant. Below this scale $$v_\mu $$ v μ decouples, the connection becomes Levi-Civita and metricity and Einstein gravity are recovered. These results remain valid in the presence of non-minimally coupled scalar field (Higgs-like) with Palatini connection and the potential is computed. In this case the theory gives successful inflation and a specific prediction for the tensor-to-scalar ratio $$0.007\le r\le 0.01$$ 0.007 ≤ r ≤ 0.01 for current spectral index $$n_s$$ n s (at $$95\%$$ 95 % CL) and $$N=60$$ N = 60 efolds. This value of r is mildly larger than in inflation in Weyl quadratic gravity of similar symmetry, due to different non-metricity. This establishes a connection between non-metricity and inflation predictions and enables us to test such theories by future CMB experiments.

AbstractScalar fields in curved backgrounds are assumed to be composite objects. As an example realizing such a possibility we consider a model of the massless tensor field $$l_{\mu \nu }(x)$$ l μ ν ( x ) in a 4-dim. background $$g_{\mu \nu }(x)$$ g μ ν ( x ) with spontaneously broken Weyl and scale symmetries. It is shown that the potential of $$l_{\mu \nu }$$ l μ ν , represented by a scalar quartic polynomial, has the degenerate extremal described by the composite Nambu–Goldstone scalar boson $$\phi (x):=g^{\mu \nu }l_{\mu \nu }$$ ϕ ( x ) : = g μ ν l μ ν . Removal of the degeneracy shows that $$\phi $$ ϕ acquires a non-zero vev $$\langle \phi \rangle _{0}=\mu $$ ⟨ ϕ ⟩ 0 = μ which, together with the free parameters of the potential, defines the cosmological constant. The latter is zero for a certain choice of the parameters.

Let $\Phi$ be an irreducible crystallographic root system with Weyl group $W$, coroot lattice $\check{Q}$ and Coxeter number $h$. Recently the second named author defined a uniform $W$-isomorphism $\zeta$ between the finite torus $\check{Q}/(mh+1)\check{Q}$ and the set of non-nesting parking functions $\operatorname{Park}^{(m)}(\Phi)$. If $\Phi$ is of type $A_{n-1}$ and $m=1$ this map is equivalent to a map defined on labelled Dyck paths that arises in the study of the Hilbert series of the space of diagonal harmonics. In this paper we investigate the case $m=1$ for the other infinite families of root systems ($B_n$, $C_n$ and $D_n$). In each type we define models for the finite torus and for the set of non-nesting parking functions in terms of labelled lattice paths. The map $\zeta$ can then be viewed as a map between these combinatorial objects. Our work entails new bijections between (square) lattice paths and ballot paths.

AbstractWe present a comparative study of inflation in two theories of quadratic gravity with gauged scale symmetry: (1) the original Weyl quadratic gravity and (2) the theory defined by a similar action but in the Palatini approach obtained by replacing the Weyl connection by its Palatini counterpart. These theories have different vectorial non-metricity induced by the gauge field ($$w_\mu $$ w μ ) of this symmetry. Both theories have a novel spontaneous breaking of gauged scale symmetry, in the absence of matter, where the necessary scalar field is not added ad-hoc to this purpose but is of geometric origin and part of the quadratic action. The Einstein-Proca action (of $$w_\mu $$ w μ ), Planck scale and metricity emerge in the broken phase after $$w_\mu $$ w μ acquires mass (Stueckelberg mechanism), then decouples. In the presence of matter ($$\phi _1$$ ϕ 1 ), non-minimally coupled, the scalar potential is similar in both theories up to couplings and field rescaling. For small field values the potential is Higgs-like while for large fields inflation is possible. Due to their $$R^2$$ R 2 term, both theories have a small tensor-to-scalar ratio ($$r\sim 10^{-3}$$ r ∼ 10 - 3 ), larger in Palatini case. For a fixed spectral index $$n_s$$ n s , reducing the non-minimal coupling ($$\xi _1$$ ξ 1 ) increases r which in Weyl theory is bounded from above by that of Starobinsky inflation. For a small enough $$\xi _1\le 10^{-3}$$ ξ 1 ≤ 10 - 3 , unlike the Palatini version, Weyl theory gives a dependence $$r(n_s)$$ r ( n s ) similar to that in Starobinsky inflation, while also protecting r against higher dimensional operators corrections.

AbstractWe generalise Einstein’s formulation of the traceless Einstein equations to f(R) gravity theories. In the case of the vacuum traceless Einstein equations, we show that a non-constant Weyl tensor leads via a conformal transformation to a dimensionally homogeneous (‘no-scale’) theory in the conformal frame with a scalar field source that has an exponential potential. We then formulate the traceless version of f(R) gravity, and we find that a conformal transformation leads to a no-scale theory conformally equivalent to general relativity and a scalar field $$\phi $$ ϕ with a potential given by the scale-invariant form: $$V(\phi )=\frac{D-2}{4D}Re^{-\phi }$$ V ( ϕ ) = D - 2 4 D R e - ϕ , where $$\phi =[2/(D-2)]\ln f^{\prime }(R)$$ ϕ = [ 2 / ( D - 2 ) ] ln f ′ ( R ) . In this theory, the cosmological constant is a mere integration constant, statistically distributed in a multiverse of independent causal domains, the vacuum energy is another unrelated arbitrary constant, and the same is true of the height of the inflationary plateau present in a huge variety of potentials. Unlike in the conformal equivalent of full general relativity, flat potentials are found to be possible in all spacetime dimensions for polynomial lagrangians of all orders. Hence, we are led to a novel interpretation of the cosmological constant vacuum energy problem and have accelerated inflationary expansion in the very early universe with a very small cosmological constant at late times for a wide range of no-scale theories. Fine-tunings required in traceless general relativity or standard non-traceless f(R) theories of gravity are avoided. We show that the predictions of the scale-invariant conformal potential are completely consistent with microwave background observational data concerning the primordial tilt and the tensor-to-scalar ratio.

A brief review of the essentials of Asymptotic Safety and the Renormalization Group (RG) improvement of the Schwarzschild Black Hole that removes the r = 0 singularity is presented. It is followed with a RG-improvement of the Kantowski-Sachs metric associated with a Schwarzschild black hole interior and such that there is no singularity at t = 0 due to the running Newtonian coupling G(t) (vanishing at t = 0). Two temporal horizons at t _- \simeq t_P and t_+ \simeq t_H are found. For times below the Planck scale t < t_P, and above the Hubble time t > t_H, the components of the Kantowski-Sachs metric exhibit a key sign change, so the roles of the spatial z and temporal t coordinates are exchanged, and one recovers a repulsive inflationary de Sitter-like core around z = 0, and a Schwarzschild-like metric in the exterior region z > R_H = 2G_o M. The inclusion of a running cosmological constant \Lambda (t) follows. We proceed with the study of a dilaton-gravity (scalar-tensor theory) system within the context of Weyl's geometry that permits to single out the expression for the classical potential V (\phi ) = \kappa\phi^4, instead of being introduced by hand, and find a family of metric solutions which are conformally equivalent to the (Anti) de Sitter metric. To conclude, an ansatz for the truncated effective average action of ordinary dilaton-gravity in Riemannian geometry is introduced, and a RG-improved Cosmology based on the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric is explored.

The aim of this thesis is to study the geometry of a Riemannian manifold M, with a special structure, called Einstein-type structure, depending on 3 real parameters, a smooth map phi into a target Riemannian manifold N, and a smooth function, called potential function, on M itself. We will occasionally let some of the parameters be smooth functions. The setting generalizes various previously studied situations:, Ricci solitons, almost Ricci-solitons, Ricci-harmonic solitons, quasi-Einstein manifolds and so on. By taking a constant potential function those structures reduces to harmonic-Einstein manifolds, that are a generalization of Einstein manifolds. The main ingredient of our analysis is the study of certain modified curvature tensors on M related to the map phi, called phi-curvatures, obtaining, for instance, their transformation laws under a conformal change of metric, and to develop a series of results for harmonic-Einstein manifolds that parallel those obtained for Einstein manifolds some times ago and also in the very recent literature. Einstein-type structures may be obtained, for some special values of the parameters involved, by a conformal deformation of a harmonic-Einstein manifold or even as the base of a warped product harmonic-Einstein manifold. The latter fact applies not only in the Riemannian but also in the Lorentzian setting and thus some Einstein-type structures are connected with solutions of the Einstein field equations, which are of particular interest in General Relativity. The main result of the thesis is the locally characterization, via a couple of integrability conditions and mild assumptions on the potential function, of Einstein-type structures with vanishing phi-Bach curvature (in the direction of the potential) as a warped product with harmonic-Einstein base and with an open real interval as fibre, extending in a very non trivial way a recent result for Bach flat Ricci solitons. Moreover the map phi depends only on the base of the warped product and not on the fibre . We also consider rigidity, triviality and non-existence results, both in the compact and non-compact cases. This is done via integral formulas and, in the non-compact case, via analytical tools, like the weak maximum principle and the classical results of Obata, Tashiro, Kanai.