Showing papers on "Scalar curvature published in 2019"
TL;DR: In this article, the Ruppeiner scalar curvature of a van der Waals fluid was studied for higher dimensional charged AdS black holes, and it was shown that the curvature is always negative for the van derWaals fluid, indicating that attractive interactions dominate among the fluid microstructures.
Abstract: We present a novel approach for probing the microstructure of a thermodynamic system that combines thermodynamic phase transitions with the Ruppeiner scalar curvature. Originally considered for van der Waals fluids and charged black holes [Phys. Rev. Lett. 123, 071103 (2019)], we extend and generalize our approach to higher-dimensional charged AdS black holes. Beginning with thermodynamic fluctuations, we construct the line element of the Ruppeiner geometry and obtain a universal formula for the scalar curvature $R$. We first review the thermodynamics of a van der Waals fluid and calculate the coexistence and spinodal curves. From this we are able to clearly display the phase diagram. Notwithstanding the invalidity of the equation of state in the coexistence phase regions, we find that the scalar curvature is always negative for the van der Waals fluid, indicating that attractive interactions dominate among the fluid microstructures. Along the coexistence curve, the scalar curvature $R$ decreases with temperature, and goes to negative infinity at a critical temperature. We then numerically study the critical phenomena associated with the scalar curvature, and find that the critical exponent is 2, and that $R(1\ensuremath{-}\stackrel{\texttildelow{}}{T}{)}^{2}{C}_{v}\ensuremath{\approx}1/8$, where $\stackrel{\texttildelow{}}{T}$ and ${C}_{v}$ are the respective reduced temperature and heat capacity. We next consider four-dimensional charged AdS black holes. Vanishing of the heat capacity at constant volume yields a divergent scalar curvature. In order to extract the corresponding information, we define a new scalar curvature that has behaviour similar to that of a van der Waals fluid. We analytically confirm that at the critical point of the small/large black hole phase transition, the scalar curvature has a critical exponent 2, and $R(1\ensuremath{-}\stackrel{\texttildelow{}}{T}{)}^{2}{C}_{v}=1/8$, the same as that of a van der Waals fluid. However we also find a significant distinction: the scalar curvature can be positive for the small charged AdS black hole, implying that repulsive interactions dominate among the black hole microstructures. We then generalize our study to higher-dimensional charged AdS black holes, and investigate the influence of the dimensionality on the black hole microstructures and the scalar curvature. Our novel approach provides a universal way for probing the microstructure of charged AdS black holes from a geometric construction.
122 citations
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TL;DR: In this article, the authors overview main topics and ideas in spaces with their scalar curvatures bounded from below, and present a more detailed exposition of several known and some new geometric constraints on Riemannian spaces implied by the lower bounds on their curvatures.
Abstract: We overview main topics and ideas in spaces with their scalar curvatures bounded from below, and present a more detailed exposition of several known and some new geometric constraints on Riemannian spaces implied by the lower bounds on their scalar curvatures
86 citations
TL;DR: In this article, it was shown that the Palatini curvature scalar is equivalent to Einstein gravity with a nonzero cosmological constant but with no massless scalar field when the Weyl symmetry is spontaneously broken.
Abstract: Pure ${R}^{2}$ gravity has been shown to be equivalent to Einstein gravity with nonzero cosmological constant and a massless scalar field when restricted Weyl symmetry is spontaneously broken. We show that the Palatini formulation of pure ${R}^{2}$ gravity is equivalent to Einstein gravity with a nonzero cosmological constant as before but with no massless scalar field when the Weyl symmetry is spontaneously broken. This is an important new development because the massless scalar field is not readily identifiable with any known particle in nature or unknown particles like cold dark matter which are expected to be massive. We then include a nonminimally coupled Higgs field as well as fermions to discuss how the rest of the standard model fields fit into this paradigm. With Higgs field, Weyl invariance is maintained by using a hybrid formalism that includes both the Palatini curvature scalar $\mathcal{R}$ and the usual Ricci scalar $R$.
70 citations
TL;DR: In this paper, the authors study the cosmological constraints on the variation of the Newton's constant and on post-Newtonian parameters for simple models of scalar-tensor theory of gravity beyond the extended Jordan-Brans-Dicke theory.
Abstract: We study the cosmological constraints on the variation of the Newton's constant and on post-Newtonian parameters for simple models of scalar-tensor theory of gravity beyond the extended Jordan-Brans-Dicke theory. We restrict ourselves to an effectively massless scalar field with a potential $V \propto F^2$, where $F(\sigma)=N_{pl}^2+\xi\sigma^2$ is the coupling to the Ricci scalar considered. We derive the theoretical predictions for cosmic microwave background (CMB) anisotropies and matter power spectra by requiring that the effective gravitational strength at present is compatible with the one measured in a Cavendish-like experiment and by assuming adiabatic initial condition for scalar fluctuations. When comparing these models with $Planck$ 2015 and a compilation of baryonic acoustic oscilation (BAO) data, all these models accomodate a marginalized value for $H_0$ higher than in $\Lambda$CDM. We find no evidence for a statistically significant deviation from Einstein's general relativity. We find $\xi 0$ (for $\xi 0$ (for $\xi < 0$). For the particular case of the conformal coupling, i.e. $\xi=-1/6$, we find constraints on the post-Newtonian parameters of similar precision to those within the Solar System.
69 citations
TL;DR: In this paper, the authors consider the scalarisation of a black hole surrounded by (non-conformally invariant) classical matter (Einstein-Maxwell-dilaton black holes).
Abstract: Electro-vacuum black holes are scale-invariant; their energy-momentum tensor is traceless. Quantum corrections of various sorts, however, can often produce a trace anomaly and a breakdown of scale-invariance. The (quantum-corrected) black hole solutions of the corresponding gravitational effective field theory (EFT) have a non-vanishing Ricci scalar. Then, the presence of a scalar field with the standard non-minimal coupling $\xi \phi^2 R$ naturally triggers a spontaneous scalarisation of the corresponding black holes. This scalarisation phenomenon occurs for an (infinite) discrete set of $\xi$. We illustrate the occurrence of this phenomenon for two examples of static, spherically symmetric, asymptotically flat black hole solution of EFTs. In one example the trace anomaly comes from the matter sector -- a novel, closed form, generalisation of the Reissner-Nordstrom solution with an $F^4$ correction -- whereas in the other example it comes from the geometry sector -- a noncommutative geometry generalization of the Schwarzschild black hole. For comparison, we also consider the scalarisation of a black hole surrounded by (non-conformally invariant) classical matter (Einstein-Maxwell-dilaton black holes). We find that the scalarised solutions are, generically, entropically favoured.
58 citations
TL;DR: In this article, the authors considered the Weyl quadratic gravity action with spontaneous breaking in the presence of matter fields and showed that it can be recovered with a positive cosmological constant and with spontaneous electroweak symmetry breaking.
Abstract: We consider the (gauged) Weyl gravity action, quadratic in the scalar curvature (
$$ \tilde{R} $$
) and in the Weyl tensor (
$$ {\tilde{C}}_{\mu
u \rho \sigma} $$
) of the Weyl conformal geometry. In the absence of matter fields, this action has spontaneous breaking in which the Weyl gauge field ωμ becomes massive (mass mω ∼ Planck scale) after “eating” the dilaton in the $$ \tilde{R} $$
2 term, in a Stueckelberg mechanism. As a result, one recovers the Einstein-Hilbert action with a positive cosmological constant and the Proca action for the massive Weyl gauge field ωμ. Below mω this field decouples and Weyl geometry becomes Riemannian. The Einstein-Hilbert action is then just a “low-energy” limit of Weyl quadratic gravity which thus avoids its previous, long-held criticisms. In the presence of matter scalar field ϕ1 (Higgs-like), with couplings allowed by Weyl gauge symmetry, after its spontaneous breaking one obtains in addition, at low scales, a Higgs potential with spontaneous electroweak symmetry breaking. This is induced by the non-minimal coupling $$ {\xi}_1{\phi}_1^2\tilde{R} $$
to Weyl geometry, with Higgs mass ∝ ξ1/ξ0 (ξ0 is the coefficient of the $$ \tilde{R} $$
2 term). In realistic models ξ1 must be classically tuned ξ1 ≪ ξ0. We comment on the quantum stability of this value.
58 citations
TL;DR: In this paper, an exponential form for the dependence of the f(R, T) gravity model on T was proposed, and the authors derived the field equations as well as the non-continuity equation and solved those to wormhole metric and energymomentum tensor.
Abstract: Alternative gravity is nowadays an extremely important tool to address some persistent observational issues, such as the dark sector of the universe. They can also be applied to stellar astrophysics, leading to outcomes one step ahead of those obtained through General Relativity. In the present article we test a novel f(R, T) gravity model within the physics and geometry of wormholes. The f(R, T) gravity is a reputed alternative gravity theory in which the Ricci scalar R in the Einstein-Hilbert gravitational lagrangian is replaced by a general function of R and T, namely f(R, T), with T representing the trace of the energy-momentum tensor. We propose, for the first time in the literature, an exponential form for the dependence of the theory on T. We derive the field equations as well as the non-continuity equation and solve those to wormhole metric and energy-momentum tensor. The importance of applying alternative gravity to wormholes is that through these theories it might be possible to obtain wormhole solutions satisfying the energy conditions, departing from General Relativity well-known outcomes. In this article, we indeed show that it is possible to obtain wormhole solutions satisfying the energy conditions in the exponential f(R, T) gravity. Naturally, there is still a lot to do with this model, as cosmological, galactical and stellar astrophysics applications, and the reader is strongly encouraged to do so, but, anyhow, one can see the present outcomes as a good indicative for the theory.
52 citations
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TL;DR: In this article, the authors extend the Kronheimer-Mrowka characterization of the Thurston norm in terms of the Euler characteristic of the level sets to any closed, oriented manifold with no nonseparating spheres.
Abstract: For a harmonic map $u:M^3\to S^1$ on a closed, oriented $3$--manifold, we establish the identity $$2\pi \int_{\theta\in S^1}\chi(\Sigma_{\theta})\geq \frac{1}{2}\int_{\theta\in S^1}\int_{\Sigma_{\theta}}(|du|^{-2}|Hess(u)|^2+R_M)$$ relating the scalar curvature $R_M$ of $M$ to the average Euler characteristic of the level sets $\Sigma_{\theta}=u^{-1}\{\theta\}$. As our primary application, we extend the Kronheimer--Mrowka characterization of the Thurston norm on $H_2(M;\mathbb{Z})$ in terms of $\|R_M^-\|_{L^2}$ and the harmonic norm to any closed $3$--manifold containing no nonseparating spheres. Additional corollaries include the Bray--Brendle--Neves rigidity theorem for the systolic inequality $(\min R_M)sys_2(M)\leq 8\pi$, and the well--known result of Schoen and Yau that $T^3$ admits no metric of positive scalar curvature.
47 citations
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TL;DR: In this article, a lower bound for the mass of an asymptotically flat Riemannian 3-manifold is given in terms of linear growth harmonic functions and scalar curvature.
Abstract: An explicit lower bound for the mass of an asymptotically flat Riemannian 3-manifold is given in terms of linear growth harmonic functions and scalar curvature. As a consequence, a new proof of the positive mass theorem is achieved in dimension three. The proof has parallels with both the Schoen-Yau minimal hypersurface technique and Witten's spinorial approach. In particular, the role of harmonic spinors and the Lichnerowicz formula in Witten's argument is replaced by that of harmonic functions and a formula introduced by the fourth named author in recent work, while the level sets of harmonic functions take on a role similar to that of the Schoen-Yau minimal hypersurfaces.
43 citations
TL;DR: In this article, the authors investigated some bouncing models in the framework of an extended gravity theory where the usual Ricci scalar in the gravitational action is replaced by the sum of the Ricci scalear and a term proportional to the trace of the energy momentum tensor.
Abstract: We have investigated some bouncing models in the framework of an extended gravity theory where the usual Ricci scalar in the gravitational action is replaced by the sum of the Ricci scalar and a term proportional to the trace of the energy momentum tensor. The dynamical parameters of the models are derived in a most general manner. We considered two bouncing scenarios described by an exponential and a power law scale factors. The non-singular bouncing models also favour a late time cosmic speed up phenomenon. The dynamical behaviour of the equation of state parameter is studied for the models. It is observed that, near the bounce, the dynamics is substantially affected by the coupling parameter of the modified gravity theory and is least affected by the parameter of the bouncing scale factors.
38 citations
TL;DR: In this article, the authors considered the problem of constructing solutions to the fractional Yamabe problem which are singular at a given smooth submanifold, for which they established the classical gluing method of Mazzeo and Pacard (J. Differential Geom., 1996) for the scalar curvature in fractional setting.
Abstract: We consider the problem of constructing solutions to the fractional Yamabe problem which are singular at a given smooth submanifold, for which we establish the classical gluing method of Mazzeo and Pacard (J. Differential Geom., 1996) for the scalar curvature in the fractional setting. This proof is based on the analysis of the model linearized operator, which amounts to the study of a fractional-order ordinary differential equation (ODE). Thus, our main contribution here is the development of new methods coming from conformal geometry and scattering theory for the study of nonlocal ODEs. Note, however, that no traditional phase-plane analysis is available here. Instead, we first provide a rigorous construction of radial fast-decaying solutions by a blowup argument and a bifurcation method. Then, second, we use conformal geometry to rewrite this nonlocal ODE, giving a hint of what a nonlocal phase-plane analysis should be. Third, for the linear theory, we use complex analysis and some non-Euclidean harmonic analysis to examine a fractional Schrodinger equation with a Hardy-type critical potential. We construct its Green’s function, deduce Fredholm properties, and analyze its asymptotics at the singular points in the spirit of Frobenius method. Surprisingly enough, a fractional linear ODE may still have a 2-dimensional kernel as in the second-order case.
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TL;DR: In this paper, the Ricci scalar was modeled as a scalar field and the Higgs field was used to play the role of the scalar fields for the purpose of gravitational reheating.
Abstract: We present a model for viable gravitational reheating involving a scalar field directly coupled to the Ricci curvature scalar. Crucial to the model is a period of kination after inflation, which causes the Ricci scalar to change sign thus inducing a tachyonic effective mass $m^{2} \propto -H^2$ for the scalar field. The resulting tachyonic growth of the scalar field provides the energy for reheating, allowing for temperatures high enough for thermal leptogenesis. Additionally, the required period of kination necessarily leads to a blue-tilted primordial gravitational wave spectrum with the potential to be detected by future experiments. We find that for reheating temperatures $T_{\rm RH} \lesssim 1$ GeV, the possibility exists for the Higgs field to play the role of the scalar field.
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TL;DR: Bamler and Kleiner as mentioned in this paper showed that the space of metrics of positive scalar curvature on any 3-manifold is either empty or contractible, and showed that diffeomorphism group of every 3-dimensional spherical space form deformation retracts to its isometry group.
Abstract: Author(s): Bamler, Richard H; Kleiner, Bruce | Abstract: We show that the space of metrics of positive scalar curvature on any 3-manifold is either empty or contractible. Second, we show that the diffeomorphism group of every 3-dimensional spherical space form deformation retracts to its isometry group. This proves the Generalized Smale Conjecture. Our argument is independent of Hatcher's theorem in the $S^3$ case and in particular it gives a new proof of the $S^3$ case.
TL;DR: In this article, the authors proved a reductiveness result of the reduced Lie algebra of holomorphic vector fields for conformally Einstein-Maxwell Kahler manifolds, extending the Lichnerowicz-Matsushima Theorem for constant scalar curvature.
Abstract: Let (M, g) be a compact Kahler manifold and f a positive smooth function such that its Hamiltonian vector field $$K = J\mathrm {grad}_g f$$
for the Kahler form $$\omega _g$$
is a holomorphic Killing vector field. We say that the pair (g, f) is conformally Einstein–Maxwell Kahler metric if the conformal metric $$\tilde{g} = f^{-2}g$$
has constant scalar curvature. In this paper we prove a reductiveness result of the reduced Lie algebra of holomorphic vector fields for conformally Einstein–Maxwell Kahler manifolds, extending the Lichnerowicz–Matsushima Theorem for constant scalar curvature Kahler manifolds. More generally we consider extensions of Calabi functional and extremal Kahler metrics, and prove an extension of Calabi’s theorem on the structure of the Lie algebra of holomorphic vector fields for extremal Kahler manifolds. The proof uses a Hessian formula for the Calabi functional under the set up of Donaldson-Fujiki picture.
TL;DR: In this article, the authors studied positive scalar curvature on the regular part of Riemannian manifolds with singular, uniformly Euclidean metrics that consolidate Gromov's scalar comparison theory and edge metrics that appear in the study of Einstein manifolds.
Abstract: We study positive scalar curvature on the regular part of Riemannian manifolds with singular, uniformly Euclidean (
$$L^\infty $$
) metrics that consolidate Gromov’s scalar curvature polyhedral comparison theory and edge metrics that appear in the study of Einstein manifolds. We show that, in all dimensions, edge singularities with cone angles $$\le 2\pi $$
along codimension-2 submanifolds do not affect the Yamabe type. In three dimensions, we prove the same for more general singular sets, which are allowed to stratify along 1-skeletons, exhibiting edge singularities (angles $$\le 2\pi $$
) and arbitrary $$L^\infty $$
isolated point singularities. We derive, as an application of our techniques, Positive Mass Theorems for asymptotically flat manifolds with analogous singularities.
TL;DR: In this article, a class of local definitions of weak lower scalar curvature bounds that are well defined for $C^0$ metrics is proposed. But they are not stable under greater-than-second-order perturbation of the metric.
Abstract: In this paper we propose a class of local definitions of weak lower scalar curvature bounds that is well defined for $C^0$ metrics. We show the following: that our definitions are stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starting from $C^0$ initial data which is smooth for positive times, and that the weak lower scalar curvature bounds are preserved under evolution by the Ricci flow from $C^0$ initial data.
TL;DR: In this paper, interior C2 estimates for convex solutions of the scalar curvature equation and the σ2-Hessian equation were established for isometrically immersed hypersurfaces (Mn,g)⊂Rn+1 with positive scalar convex curvature.
Abstract: We establish interior C2 estimates for convex solutions of the scalar curvature equation and the σ2-Hessian equation. We also prove interior curvature estimates for isometrically immersed hypersurfaces (Mn,g)⊂Rn+1 with positive scalar curvature. These estimates are consequences of interior estimates for these equations obtained under a weakened condition.
TL;DR: In this paper, the authors investigated spatially homogeneous and totally anisotropic perfect cosmological model in the presence of an attractive massive scalar field in the background of Bianchi type space-time.
Abstract: In this paper, we are interested to investigate spatially homogeneous and totally anisotropic perfect cosmological model in the presence of an attractive massive scalar field in $f(R,T)$
gravity in the background of Bianchi type-
$\mathit{III}$
space-time. Here $R$
is the Ricci scalar and $T$
is the trace of the energy momentum tensor. In order to solve the field equations, we have used (i) the expansion scalar of the space-time is proportional to the shear scalar which leads to a relationship between metric potentials and (ii) a power law between the scalar field and the average scale factor. We obtain a cosmological model of the universe with variable deceleration parameter. We have computed all the cosmological parameters of the model and discussed their physical importance.
TL;DR: In this article, an analysis of the normalized Yamabe flow starting at and preserving a class of compact Riemannian manifolds with incomplete edge singularities and negative Yamabe invariant is presented.
Abstract: This article presents an analysis of the normalized Yamabe flow starting at and preserving a class of compact Riemannian manifolds with incomplete edge singularities and negative Yamabe invariant. Our main results include uniqueness, long-time existence and convergence of the edge Yamabe flow starting at a metric with everywhere negative scalar curvature. Our methods include novel maximum principle results on the singular edge space without using barrier functions. Moreover, our uniform bounds on solutions are established by a new ansatz without in any way using or redeveloping Krylov–Safonov estimates in the singular setting. As an application we obtain a solution to the Yamabe problem for incomplete edge metrics with negative Yamabe invariant using flow techniques. Our methods lay groundwork for studying other flows like the mean curvature flow as well as the porous medium equation in the singular setting.
TL;DR: In this paper, it was shown that perfect fluid spacetimes with divergence-free projective, concircular, conharmonic or quasi-conformal curvature tensors are generalized Robertson Walker (GRW) spacetimits.
Abstract: We show that $n$-dimensional perfect fluid spacetimes with diver\-gen\-ce-free conformal curvature tensor and constant scalar curvature are generalized Robertson Walker (GRW) spacetimes; as a consequence a perfect fluid Yang pure space is a GRW spacetime. We also prove that perfect fluid spacetimes with harmonic generalized curvature tensor are, under certain conditions, GRW spacetimes. As particular cases, perfect fluids with divergence-free projective, concircular, conharmonic or quasi-conformal curvature tensor are GRW spacetimes. Finally, we explore some physical consequences of such results.
TL;DR: In this article, a metric with constant weighted scalar curvature on a compact K-ahler manifold was introduced, and the Mabuchi energy was extended to the weighted setting.
Abstract: We introduce a notion of a K\\\"ahler metric with constant weighted scalar curvature on a compact K\\\"ahler manifold $X$, depending on a fixed real torus $\\mathbb{T}$ in the reduced group of automorphisms of $X$, and two smooth (weight) functions $\\mathrm{v}>0$ and $\\mathrm{w}$, defined on the momentum image (with respect to a given K\\\"ahler class $\\alpha$ on $X$) of $X$ in the dual Lie algebra of $\\mathbb{T}$. A number of natural problems in K\\\"ahler geometry, such as the existence of extremal K\\\"ahler metrics and conformally K\\\"ahler, Einstein--Maxwell metrics, or prescribing the scalar curvature on a compact toric manifold reduce to the search of K\\\"ahler metrics with constant weighted scalar curvature in a given K\\\"ahler class $\\alpha$, for special choices of the weight functions $\\mathrm{v}$ and $\\mathrm{w}$.
We show that a number of known results obstructing the existence of constant scalar curvature K\\\"ahler (cscK) metrics can be extended to the weighted setting. In particular, we introduce a functional $\\mathcal M_{\\mathrm{v}, \\mathrm{w}}$ on the space of $\\mathbb{T}$-invariant K\\\"ahler metrics in $\\alpha$, extending the Mabuchi energy in the cscK case, and show (following the arguments of Li and Sano--Tipler in the cscK and extremal cases) that if $\\alpha$ is Hodge, then constant weighted scalar curvature metrics in $\\alpha$ are minima of $\\mathcal M_{\\mathrm{v},\\mathrm{w}}$. Motivated by the recent work of Dervan--Ross and Dyrefelt in the cscK and extremal cases, we define a $(\\mathrm{v},\\mathrm{w})$-weighted Futaki invariant of a $\\mathbb{T}$-compatible smooth K\\\"ahler test configuration associated to $(X, \\alpha, \\mathbb{T})$, and show that the boundedness from below of the $(\\mathrm{v},\\mathrm{w})$-weighted Mabuchi functional $\\mathcal M_{\\mathrm{v}, \\mathrm{w}}$ implies a suitable notion of a $(\\mathrm{v},\\mathrm{w})$-weighted K-semistability.
TL;DR: In this article, the authors investigate the correspondence between the morphon field and fundamental scalar field dark matter models, in order to describe dark matter as an effective phenomenon arising from kinematical backreaction and the averaged spatial curvature of the inhomogeneous universe.
Abstract: In inhomogeneous cosmology, restricting attention to an irrotational dust matter model, backreaction arises in terms of the deviation of the averaged spatial scalar curvature from a constant-curvature model on some averaging domain , , and the kinematical backreaction . These backreaction variables can be modeled as an effective scalar field, called the ‘morphon field’. The general cosmological equations still need a closure condition to be solved. A simple example is the class of scaling solutions where and are assumed to follow a power law of the volume scale factor . But while they can describe models of quintessence, these and other models still assume the existence of dark matter in addition to the known sources. Going beyond scaling solutions by using a model for structure formation that we argue is reasonably generic, we investigate the correspondence between the morphon field and fundamental scalar field dark matter models, in order to describe dark matter as an effective phenomenon arising from kinematical backreaction and the averaged spatial curvature of the inhomogeneous Universe. While we find significant differences with those fundamental models, our main result is that the energy budget on typical collapsing domains is provided by curvature and matter in equal parts already around the turn-around time, leading to curvature dominance thereafter and increasing to a curvature contribution of 3/4 of the energy budget at the onset of virialization. Kinematical backreaction is subdominant at early stages, but its importance rises quickly after turn-around and dominates the curvature contribution in the final phase of the collapse. We conclude that backreaction can indeed mimic dark matter (in the energy budget) during the collapse phase of megaparsec-scale structures.
TL;DR: In this paper, the authors studied the special class of exact solutions in cosmological models based on generalized scalar-tensor gravity with nonminimal coupling of a scalar field to the Ricci scalar and to the Gauss-Bonnet scalar.
Abstract: We study the special class of exact solutions in cosmological models based on generalized scalar-tensor gravity with nonminimal coupling of a scalar field to the Ricci scalar and to the Gauss-Bonnet scalar in a 4D Friedmann universe corresponding to similar ones in GR. The parameters of cosmological perturbations in such models correspond to the case of Einstein gravity with a high precision. As an example of the proposed approach, we obtain the exact solutions for the power-law and exponential power-law inflation.
TL;DR: On an asymptotically flat manifold M-n with nonnegative scalar curvature, with outer minimizing boundary Sigma, this paper proved a Penrose-like inequality in dimensions n < 8, under suitable assumptions.
Abstract: On an asymptotically flat manifold M-n with nonnegative scalar curvature, with outer minimizing boundary Sigma, we prove a Penrose-like inequality in dimensions n < 8, under suitable assumptions ...
TL;DR: In this article, a sequence of 3-manifolds of nonnegative scalar curvature with outermost minimal boundary, converging in the pointed $C^0$ Cheeger-Gromov sense to an asymptotically flat limit space, is shown to have lower semicontinuous total mass.
Abstract: Given a sequence of asymptotically flat 3-manifolds of nonnegative scalar curvature with outermost minimal boundary, converging in the pointed $C^0$ Cheeger--Gromov sense to an asymptotically flat limit space, we show that the total mass of the limit is bounded above by the liminf of the total masses of the sequence. In other words, total mass is lower semicontinuous under such convergence. In order to prove this, we use Huisken's isoperimetric mass concept, together with a modified weak mean curvature flow argument. We include a brief discussion of Huisken's work before explaining our extension of that work. The results are all specific to three dimensions.
TL;DR: In this paper, the exact solutions of spherically symmetric space-times are explored by using Noether symmetries in f(R,,X) gravity with R the scalar curvature, a scalar field and X the kinetic term of.
Abstract: The exact solutions of spherically symmetric space-times are explored by using Noether symmetries in f(R,,X) gravity with R the scalar curvature, a scalar field and X the kinetic term of . Some of these solutions can represent new black holes solutions in this extended theory of gravity. The classical Noether approach is particularly applied to acquire the Noether symmetry in f(R,,X) gravity. Under the classical Noether theorem, it is shown that the Noether symmetry in f(R,,X) gravity yields the solvable first integral of motion. With the conservation relation obtained from the Noether symmetry, the exact solutions for the field equations can be found. The most important result in this paper is that, without assuming R=constant, we have found new spherically symmetric solutions in different theories such as: power-law f(R)=f0 Rn gravity, non-minimally coupling models between the scalar field and the Ricci scalar f(R,,X)=f0 Rn m+f1 Xq−V(), non-minimally couplings between the scalar field and a kinetic term f(R,,X)=f0 Rn +f1mXq , and also in extended Brans-Dicke gravity f(R,,X)=U(,X)R. It is also demonstrated that the approach with Noether symmetries can be regarded as a selection rule to determine the potential V() for , included in some class of the theories of f(R,,X) gravity.
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TL;DR: In this article, it was shown that a Kahler cone appears as the tangent cone of a complete expanding gradient Kahler-Ricci soliton with quadratic curvature decay with derivatives if and only if it has a smooth canonical model.
Abstract: We first show that a Kahler cone appears as the tangent cone of a complete expanding gradient Kahler-Ricci soliton with quadratic curvature decay with derivatives if and only if it has a smooth canonical model (on which the soliton lives). This allows us to classify two-dimensional complete expanding gradient Kahler-Ricci solitons with quadratic curvature decay with derivatives. We then show that any two-dimensional complete shrinking gradient Kahler-Ricci soliton whose scalar curvature tends to zero at infinity is, up to pullback by an element of $GL(2,\,\mathbb{C})$, either the flat Gaussian shrinking soliton on $\mathbb{C}^{2}$ or the $U(2)$-invariant shrinking gradient Kahler-Ricci soliton of Feldman-Ilmanen-Knopf on the blowup of $\mathbb{C}^{2}$ at one point. Finally, we show that up to pullback by an element of $GL(n,\,\mathbb{C})$, the only complete shrinking gradient Kahler-Ricci soliton with bounded Ricci curvature on $\mathbb{C}^{n}$ is the flat Gaussian shrinking soliton and on the total space of $\mathcal{O}(-k)\to\mathbb{P}^{n-1}$ for $0
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TL;DR: In this paper, it was shown that for any Riemannian metric on a spin manifold with scalar curvature bounded below a constant σ > 0, the distance between the boundary components of the manifold is at most σ n/ σ σ 1/σ 2, where σ < 8(1+ σ 2 ) being a universal constant.
Abstract: Let $M$ be a closed connected spin manifold such that its spinor Dirac operator has non-vanishing (Rosenberg) index. We prove that for any Riemannian metric on $V = M \times [-1,1]$ with scalar curvature bounded below by $\sigma > 0$, the distance between the boundary components of $V$ is at most $C_n/\sqrt{\sigma}$, where $C_n = \sqrt{(n-1)/{n}} \cdot C$ with $C < 8(1+\sqrt{2})$ being a universal constant. This verifies a conjecture of Gromov for such manifolds. In particular, our result applies to all high-dimensional closed simply connected manifolds $M$ which do not admit a metric of positive scalar curvature. We also establish a quadratic decay estimate for the scalar curvature of complete metrics on manifolds, such as $M \times \mathbb{R}^2$, which contain $M$ as a codimension two submanifold in a suitable way. Furthermore, we introduce the "$\mathcal{KO}$-width" of a closed manifold and deduce that infinite $\mathcal{KO}$-width is an obstruction to positive scalar curvature.
TL;DR: In this article, it was shown that if the Reeb vector field of a Ricci almost soliton is a contact vector field, then the Ricci operator is invariant and the scalar curvature invariant.
Abstract: First, we prove that if the Reeb vector field $\xi$ of a Kenmotsu manifold $M$ leaves the Ricci operator $Q$ invariant, then $M$ is Einstein. Next, we study Kenmotsu manifold whose metric represents a Ricci soliton and prove that it is expanding. Moreover, the soliton is trivial (Einstein) if either (i) $V$ is a contact vector field, or (ii) the Reeb vector field $\xi$ leaves the scalar curvature invariant. Finally, it is shown that if the metric of a Kenmotsu manifold represents a gradient Ricci almost soliton, then it is $\eta$-Einstein and the soliton is expanding. We also exhibited some examples of Kenmotsu manifold that admit Ricci almost solitons.
TL;DR: The self coupling of the Higgs boson in the Standard Model may show critical behavior, i.e. it may serve as inflaton even if its nonminimal coupling to the curvature scalar is only ${cal O(10)$.
Abstract: The self coupling $\lambda$ of the Higgs boson in the Standard Model may show critical behavior, i.e. the Higgs potential may have a point at an energy scale $\sim 10^{17-18}$ GeV where both the first and second derivatives (almost) vanish. In this case the Higgs boson can serve as inflaton even if its nonminimal coupling to the curvature scalar is only ${\cal O}(10)$, thereby alleviating concerns about the perturbative unitarity of the theory. We find that just before the Higgs as inflaton enters the flat region of the potential the usual slow--roll conditions are violated. This leads to "overshooting" behavior, which in turn strongly enhances scalar curvature perturbations because of the excitation of entropic (non--adiabatic) perturbations. For appropriate choice of the free parameters these large perturbations occur at length scales relevant for the formation of primordial black holes. Even if these perturbations are not quite large enough to trigger copious black hole formation, they source second order tensor perturbations, i.e. primordial gravitational waves; the corresponding energy density can be detected by the proposed space-based gravitational wave detectors DECIGO and BBO.