Gauss-Bonnet Curvature Research Articles

Non-linear tides and Gauss-Bonnet scalarization

In linear perturbation theory, a static perturber in the vicinity of a Schwarzschild black hole (BH) enhances [suppresses] the Gauss-Bonnet (GB) curvature invariant, RGB, in the high [low] tide regions. By analysing exact solutions of the vacuum Einstein field equations describing one or two BHs immersed in a multipolar gravitational field, which is locally free of pathologies, including conical singularities, we study the corresponding non-linear tides on a fiducial BH, in full General Relativity (GR). We show that the tidal field due to a far away, or close by, static BH creates high/low tides that can deviate not only quantitatively but also qualitatively from the weak field/Newtonian pattern. Remarkably, the suppression in low tide regions never makes RGB negative on the BH, even though the horizon Gaussian curvature may become negative; but RGB can vanish in a measure zero set, a feature qualitatively recovered in a Newtonian analogue model. Thus, purely gravitational, static, tidal interactions in GR, no matter how strong, cannot induce GB− scalarization. We also show that a close by BH produces noticeable asymmetric tides on another (fiducial) BH.

From inflation to reheating and their dynamical stability analysis in Gauss–Bonnet gravity

We investigate the inflation and reheating phenomenology in scalar-Einstein-Gauss–Bonnet theory of gravity where a scalar field non-minimally couples with the Gauss-Bonnet (GB) curvature term. Regarding the inflationary phenomenology, we find – (1) the inflation starts with a quasi de-Sitter phase and has an exit at a finite e-fold, which is indeed consistent with the resolution of horizon and flatness problems, (2) the scalar and tensor perturbations prove to be ghost free and do not suffer from gradient instability, (3) the curvature perturbation amplitude as well as its tilt and the tensor-to-scalar ratio turn out to be simultaneously compatible with the recent Planck data for suitable values of the parameters which also results to the inflationary energy scale ∼1012GeV. After the inflation ends, the scalar field starts to decay to radiation with a constant decay width. For our considered scalar potential and the GB coupling function, the model results to an analytic power law solution of the Hubble parameter and a logarithmic solution of the scalar field during the reheating era, where the exponent of the Hubble parameter determines the effective EoS parameter (weff) during the same. The stability of such reheating dynamics is examined by dynamical analysis which ensures that weff can go beyond unity and reach up-to the maximum value of max(weff)=1.56. The scenario with weff>1 proves to be purely due to the presence of the GB coupling function, which in turn may have important consequences on enhancing the primordial gravitational waves’ amplitude observed today. The inflationary e-fold number (Nf) gets further constrained by the input of the reheating stage, and we critically examine the constraint of Nf coming from both the inflation and reheating phenomenology. We finally construct the complete forms of scalar potential (V(ϕ)) and the GB coupling function (ξ(ϕ)) that smoothly transits from inflation to reheating, and numerically solve the Hubble parameter and the scalar field for such complete forms of V(ϕ) and ξ(ϕ). Such numerical solutions match with analytic ones in the respective regime. After the reheating ends, the GB term gets decouple from the theory and the radiation energy density dominates the energy budget of the universe, which in turn produces the standard radiation era of the universe.

Curvature induced scalarization of Kerr-Newman black hole spacetimes

With certain nonminimal coupling between a massless scalar field and the Gauss-Bonnet curvature invariant in the extended scalar-tensor-Gauss-Bonnet (ESTGB) field theory, tachyonic instability of the Kerr-Newman (KN) black hole is promoted. Critical spin and charge for the onset of the spontaneous scalarization phenomenon for the KN black hole are calculated in the infinitely large coupling limit. Then the analytical formula for the coupling-strength-dependent critical existence line is obtained for the black-hole-scalar-field configuration in a finite large coupling regime. Moreover, numerical methods are used to perform threshold curves that are boundaries between bald KN black holes and their hairy counterparts. Intriguingly, we get to know the KN black hole can be scalarized in the vanishing spin limit in ESTGB theory.

Bouncing Cosmology in Modified Gravity with Higher-Order Gauss–Bonnet Curvature Term

In this paper, we studied the bouncing behavior of the cosmological models formulated in the background of the Hubble function in the F(R,G) theory of gravity, where R and G, respectively, denote the Ricci scalar and Gauss–Bonnet invariant. The actions of the bouncing cosmology are studied with a consideration of the different viable models that can resolve the difficulty of singularity in standard Big Bang cosmology. Both models show bouncing behavior and satisfy the bouncing cosmological properties. Models based on dynamical, deceleration, and energy conditions indicate the accelerating behavior at the late evolution time. The phantom at the bounce epoch is analogous to quintessence behavior. Finally, we formulate the perturbed evolution equations and investigate the stability of the two bouncing solutions.

Stability of neutron stars in Horndeski theories with Gauss-Bonnet couplings

In Horndeski theories containing a scalar coupling with the Gauss-Bonnet (GB) curvature invariant ${R}_{\mathrm{GB}}^{2}$, we study the existence and linear stability of neutron star (NS) solutions on a static and spherically symmetric background. For a scalar-GB coupling of the form $\ensuremath{\alpha}\ensuremath{\xi}(\ensuremath{\phi}){R}_{\mathrm{GB}}^{2}$, where $\ensuremath{\xi}$ is a function of the scalar field $\ensuremath{\phi}$, the existence of linearly stable stars with a nontrivial scalar profile without instabilities puts an upper bound on the strength of the dimensionless coupling constant $|\ensuremath{\alpha}|$. To realize maximum masses of NSs for a linear (or dilatonic) GB coupling ${\ensuremath{\alpha}}_{\mathrm{GB}}\ensuremath{\phi}{R}_{\mathrm{GB}}^{2}$ with typical nuclear equations of state, we obtain the theoretical upper limit $\sqrt{|{\ensuremath{\alpha}}_{\mathrm{GB}}|}<0.7\text{ }\text{ }\mathrm{km}$. This is tighter than those obtained by the observations of gravitational waves emitted from binaries containing NSs. We also incorporate cubic-order scalar derivative interactions, quartic derivative couplings with nonminimal couplings to a Ricci scalar besides the scalar-GB coupling, and show that NS solutions with a nontrivial scalar profile satisfying all the linear stability conditions are present for certain ranges of the coupling constants. In regularized four-dimensional Einstein-GB gravity obtained from a Kaluza-Klein reduction with an appropriate rescaling of the GB coupling constant, we find that NSs in this theory suffer from a strong coupling problem as well as Laplacian instability of even-parity perturbations. We also study NS solutions with a nontrivial scalar profile in power-law $F({R}_{\mathrm{GB}}^{2})$ models, and show that they are pathological in the interior of stars and plagued by ghost instability together with the asymptotic strong coupling problem in the exterior of stars.

Spin-induced scalarization and magnetic fields

In the presence of certain non-minimal couplings between a scalar field and the Gauss-Bonnet curvature invariant, Kerr black holes can scalarize, as long as they are spinning fast enough. This provides a distinctive violation of the Kerr hypothesis, occurring only for some high spin range. In this paper we assess if strong magnetic fields, that may exist in the vicinity of astrophysical black holes, could facilitate this distinctive effect, by bringing down the spin threshold for scalarization. This inquiry is motivated by the fact that self-gravitating magnetic fields, by themselves, can also promote “spin-induced” scalarization. Nonetheless, we show that in the vicinity of the horizon the effect of the magnetic field B on a BH of mass M, up to BM≲1, works against spin-induced scalarization, requiring a larger dimensionless spin j from the black hole. A geometric interpretation for this result is suggested, in terms of the effects of rotation vs. magnetic fields on the horizon geometry.

Role of Gauss-Bonnet corrections in a DGP brane gravitational collapse

An Oppenheimer-Snyder (OS)-type collapse is considered for a Dvali-Gabadadze-Porrati (DGP) brane, whereas a Gauss-Bonnet (GB) term is provided for the bulk. We study the combined effect of the DGP induced gravity plus the GB curvature, regarding any modification of the general relativistic OS dynamics. Our paper has a twofold objective. On the one hand, we investigate the nature of singularities that may arise at the collapse end state. It is shown that all dynamical scenarios for the contracting brane would end in one of the following cases, depending on conditions imposed: either a central shell-focusing singularity or what we designate as a "sudden collapse singularity." On the other hand, we also study the deviations of the exterior spacetime from the standard Schwarzschild geometry, which emerges in our modified OS scenario. Our purpose is to investigate whether a black hole always forms regarding this brane world model. We find situations where a naked singularity emerges instead.

Nonequatorial scalar rings supported by magnetized Schwarzschild-Melvin black holes

It has recently been demonstrated that magnetized black holes in composed Einstein-Maxwell-scalar-Gauss-Bonnet field theories with a nonminimal negative coupling of the scalar field to the Gauss-Bonnet curvature invariant may support spatially regular scalar hairy configurations. In particular, it has been revealed that, for Schwarzschild-Melvin black-hole spacetimes, the onset of the near-horizon spontaneous scalarization phenomenon is marked by the numerically computed dimensionless critical relation $(BM{)}_{\mathrm{crit}}\ensuremath{\simeq}0.971$, where ${M,B}$ are respectively the mass and the magnetic field of the spacetime. In the present paper we prove, using analytical techniques, that the boundary between bald Schwarzschild-Melvin black-hole spacetimes and hairy (scalarized) black-hole solutions of the composed Einstein-Maxwell-scalar-Gauss-Bonnet theory is characterized by the exact dimensionless relation $(BM{)}_{\mathrm{crit}}=\sqrt{\frac{\sqrt{6}\ensuremath{-}2}{2\sqrt{6}}+\sqrt{\frac{\sqrt{6}\ensuremath{-}1}{2}}}$ for the critical magnetic strength. Intriguingly, we prove that the critical dimensionless magnetic parameter $(BM{)}_{\mathrm{crit}}$ corresponds to magnetized black holes that support a pair of linearized nonminimally coupled thin scalar rings that are characterized by the nonequatorial polar angular relation $({\mathrm{sin}}^{2}\ensuremath{\theta}{)}_{\mathrm{scalar}\ensuremath{-}\mathrm{ring}}=\frac{690\ensuremath{-}72\sqrt{6}+4\sqrt{3258\sqrt{6}\ensuremath{-}7158}}{789}<1$. It is also proved that the classically allowed angular region for the negative-coupling near-horizon spontaneous scalarization phenomenon of magnetized Schwarzschild-Melvin spacetimes is restricted to the black-hole poles, ${\mathrm{sin}}^{2}{\ensuremath{\theta}}_{\text{scalar}}\ensuremath{\rightarrow}0$, in the asymptotic large-strength magnetic regime $BM\ensuremath{\gg}1$.

Nontrivial spatial behavior of the Gauss-Bonnet curvature invariant of rapidly-rotating Kerr black holes

The Gauss-Bonnet curvature invariant has attracted the attention of physicists and mathematicians over the years. In particular, it has recently been proved that black holes can support external matter configurations that are non-minimally coupled to the Gauss-Bonnet invariant of the curved spacetime. Motivated by this physically interesting behavior of black holes in Einstein-Gauss-Bonnet theories, we present a detailed {\it analytical} study of the physical and mathematical properties of the Gauss-Bonnet curvature invariant ${\cal G}_{\text{Kerr}}(r,\cos\theta;a/M)$ of spinning Kerr black holes in the spacetime region outside the horizon. Interestingly, we prove that, for all spinning Kerr spacetimes in the physically allowed regime $a/M\in[0,1]$, the spin-dependent maximum curvature of the Gauss-Bonnet invariant is attained at the equator of the black-hole surface. Intriguingly, we reveal that the location of the global minimum of the Gauss-Bonnet invariant has a highly non-trivial functional dependence on the black-hole rotation parameter: (i) For Kerr black holes in the dimensionless slow-rotation $a/M<(a/M)^{-}_{\text{crit}}=1/2$ regime, the Gauss-Bonnet curvature invariant attains its global minimum asymptotically at spatial infinity, (ii) for black holes in the intermediate spin regime $1/2=(a/M)^{-}_{\text{crit}}\leq a/M\leq(a/M)^{+}_{\text{crit}}= \sqrt{\Big\{{{7+\sqrt{7}\cos\Big[3^{-1}\arctan\big(3\sqrt{3}\big)\Big]- \sqrt{21}\sin\Big[3^{-1}\arctan\big(3\sqrt{3}\big)\Big]\Big\}}/12}}$, the global minima are located at the black-hole poles, and (iii) Kerr black holes in the super-critical regime $a/M>(a/M)^{+}_{\text{crit}}$ are characterized by a non-trivial functional behavior of the Gauss-Bonnet curvature invariant along the black-hole horizon with a spin-dependent polar angle for the global minimum point.

Spin-induced black hole spontaneous scalarization: Analytic treatment in the large-coupling regime

It has recently been demonstrated numerically that spinning black holes whose dimensionless rotation parameter is larger than the critical value $(J/{M}^{2}{)}_{\mathrm{crit}}\ensuremath{\equiv}{\overline{a}}_{\mathrm{crit}}=\frac{1}{2}$ can support spatially regular hairy scalar configurations which are characterized by a nonminimal negative coupling to the Gauss-Bonnet curvature invariant. Motivated by this physically important observation, we explore, using analytical techniques, the physical and mathematical properties of the composed spinning-Kerr-black-hole-nonminimally-coupled-linearized-scalar-field configurations in the large-coupling $\ensuremath{-}\overline{\ensuremath{\eta}}\ensuremath{\gg}1$ regime (here $\overline{\ensuremath{\eta}}$ is the dimensionless coupling parameter of the composed black-hole-scalar-field theory). In particular, we derive a remarkably compact analytical formula for the spin-dependent critical existence line ${\overline{\ensuremath{\eta}}}_{\mathrm{crit}}={\overline{\ensuremath{\eta}}}_{\mathrm{crit}}(\overline{a})$ of the composed black-hole-field system. The physical significance of this critical line stems from the fact that, for a given value of the black-hole dimensionless rotation parameter $\overline{a}$, it marks the boundary between bald (scalarless) Kerr black holes and nonlinearly coupled hairy spinning-black-hole-scalar-field configurations.

Gravastars in modified Gauss–Bonnet gravity

This paper aims to discuss the viable and analytic solution of the spherically symmetric gravastar model under the influence of modification of Gauss–Bonnet gravity, i.e., f(G) gravity, where G is the Gauss–Bonnet curvature term. For this purpose, we evaluate the field equations in corresponding theory and conservation equation with the help of an effective energy–momentum tensor. A mathematical formalism of the gravastar’s three regions, i.e., interior de-Sitter region, thin shell, and the exterior Schwarzschild vacuum region have been discussed. We, then analyze different realistic features, in particular energy, entropy, and length of the shell. The viability of these physical features is then examined through the graphical representations separately. Within the framework of an alternative theory, we have obtained the exact and singularity free model of gravastar.

Object picture of scalar field perturbation on Kerr black hole in scalar-Einstein-Gauss-Bonnet theory

Scalar perturbations around the Kerr black hole in scalar-Einstein-Gauss-Bonnet (sEGB) theory are studied in the time domain. To overcome the "outer boundary problem" that usually encountered in traditional numerical calculations, we apply the hyperboloidal compactification technique to perform a $(2+1)$-dimensional simulation aiming to obtain a precise object picture of the wave propagation under the scalar field perturbation. We find that the big enough coupling constant between the scalar field and the Gauss-Bonnet curvature is responsible to destroy the original Kerr black hole. The breakdown of the Kerr spacetime happens earlier and the instability becomes more violent when the coupling becomes stronger. We further present object confirmations on the special case for the negative coupling where there exists a minimum rotation and below which the instability can never happen no matter how strong the coupling is. We also illustrate the fine structure property in the quasinormal ringing frequency once there is the coupling, and present the characteristic imprint of the sEGB theory. We expect that such a fine structure can be detected in the future gravitational wave observation to test the sEGB theory.

Spontaneous holographic scalarization of black holes in Einstein-scalar-Gauss-Bonnet theories

We holographically investigate the scalarization in the Einstein-Scalar-Gauss-Bonnet gravity with a negative cosmological constant. We find that instability exists for both Schwarzschild-AdS and Reissner-Nordstrom-AdS black holes with planar horizons when we have proper interactions between the scalar field and the Gauss-Bonnet curvature corrections. We relate such instability to possible holographic scalarization and construct the corresponding hairy black hole solutions in the presence of the cosmological constant. Employing the holographic principle we expect that such bulk scalarization corresponds to the boundary description of the scalar hair condensation without breaking any symmetry, and we calculate the related holographic entanglement entropy of the system. Moreover, we compare the mechanisms of the holographic scalarizations caused by the effect of the coupling of the scalar field to the Gauss-Bonnet term and holographic superconductor effect in the presence of an electromagnetic field, and unveil their differences in the effective mass of the scalar field, the temperature dependent property and the optical conductivity.

Locally constrained curvature flows and geometric inequalities in hyperbolic space

In this paper, we first study the locally constrained curvature flow of hypersurfaces in hyperbolic space, which was introduced by Brendle, Guan and Li [7]. This flow preserves the $m$th quermassintegral and decreases $(m+1)$th quermassintegral, so the convergence of the flow yields sharp Alexandrov-Fenchel type inequalities in hyperbolic space. Some special cases have been studied in [7]. In the first part of this paper, we show that h-convexity of the hypersurface is preserved along the flow and then the smooth convergence of the flow for h-convex hypersurfaces follows. We then apply this result to establish some new sharp geometric inequalities comparing the integral of $k$th Gauss-Bonnet curvature of a smooth h-convex hypersurface to its $m$th quermassintegral (for $0\leq m\leq 2k+1\leq n$), and comparing the weighted integral of $k$th mean curvature to its $m$th quermassintegral (for $0\leq m\leq k\leq n$). In particular, we give an affirmative answer to a conjecture proposed by Ge, Wang and Wu in 2015. In the second part of this paper, we introduce a new locally constrained curvature flow using the shifted principal curvatures. This is natural in the context of h-convexity. We prove the smooth convergence to a geodesic sphere of the flow for h-convex hypersurfaces, and provide a new proof of the geometric inequalities proved by Andrews, Chen and the third author of this paper in 2018. We also prove a family of new sharp inequalities involving the weighted integral of $k$th shifted mean curvature for h-convex hypersurfaces, which as application implies a higher order analogue of Brendle, Hung and Wang's [8] inequality.

Inflation with a quartic potential in the framework of Einstein-Gauss-Bonnet gravity

We investigate inflationary dynamics in the framework of the Einstein-Gauss-Bonnet gravity. In the model under consideration, the inflaton field is non-minimally coupled to the Gauss-Bonnet curvature invariant, so that the latter appears to be dynamically important. We consider a quartic potential for the inflaton field, in particular the one asymptotically connected to the Higgs inflation, and a wider class of coupling functions not considered in the earlier work. Keeping in mind the observational bounds on the parameters - the amplitude of scalar perturbations $A_s$, spectral index $n_s$ and tensor-to-scalar ratio $r$, we demonstrate that the model a quartic potential and the proposed coupling function is in agreement with observation.

Spinning black holes in shift-symmetric Horndeski theory

We construct spinning black holes (BHs) in shift-symmetric Horndeski theory. This is an Einstein-scalar-Gauss-Bonnet model wherein the (real) scalar field couples linearly to the Gauss-Bonnet curvature squared combination. The BH solutions constructed are stationary, axially symmetric and asymptotically flat. They possess a non-trivial scalar field outside their regular event horizon; thus they have scalar hair. The scalar “charge” is not, however, an independent macroscopic degree of freedom. It is proportional to the Hawking temperature, as in the static limit, wherein the BHs reduce to the spherical solutions found by Sotirou and Zhou. The spinning BHs herein are found by solving non-perturbatively the field equations, numerically. We present an overview of the parameter space of the solutions together with a study of their basic geometric and phenomenological properties. These solutions are compared with the spinning BHs in the Einstein-dilaton-Gauss-Bonnet model and the Kerr BH of vacuum General Relativity. As for the former, and in contrast with the latter, there is a minimal BH size and small violations of the Kerr bound. Phenomenological differences with respect to either the former or the latter, however, are small for illustrative observables, being of the order of a few percent, at most.

Cosmological instability of scalar-Gauss-Bonnet theories exhibiting scalarization

In a subclass of scalar-tensor theories, it has been shown that standard general relativity solutions of neutron stars and black holes with trivial scalar field profiles are unstable. Such an instability leads to solutions which are different from those of general relativity and have non-trivial scalar field profiles, in a process called scalarization. In the present work we focus on scalarization due to a non-minimal coupling of the scalar field to the Gauss-Bonnet curvature invariant. The coupling acts as a tachyonic mass for the scalar mode, thus leading to the instability of general relativity solutions. We point out that a similar effect may occur for the scalar modes in a cosmological background, resulting in the instability of cosmological solutions. In particular, we show that a catastrophic instability develops during inflation within a period of time much shorter than the minimum required duration of inflation. As a result, the standard cosmological dynamics is not recovered. This raises the question of the viability of scalar-Gauss-Bonnet theories exhibiting scalarization.

The 6D Gauss–Bonnet Supergravity Invariant

We review the recent construction of the off-shell N = (1, 0) supersymmetrization of the Gauss-Bonnet curvature squared combination in six dimensions.

Multipole analysis for linearizedf(R)gravity with irreducible Cartesian tensors

The field equations of \(f(R,{\mathcal {G}})\) gravity are rewritten in the form of obvious wave equations with the stress–energy pseudotensor of the matter fields and the gravitational field as its source under the de Donder condition. The linearized field equations of \(f(R,{\mathcal {G}})\) gravity are the same as those of linearized f(R) gravity, and thus, their multipole expansions under the de Donder condition are also the same. It is also shown that the Gauss–Bonnet curvature scalar \({\mathcal {G}}\) does not contribute to the effective stress–energy tensor of gravitational waves in linearized \(f(R,{\mathcal {G}})\) gravity, though \({\mathcal {G}}\) plays an important role in the nonlinear effects in general. Further, by applying the 1 / r expansion in the distance to the source to the linearized \(f(R,{\mathcal {G}})\) gravity, the energy, momentum, and angular momentum carried by gravitational waves in linearized \(f(R,{\mathcal {G}})\) gravity are provided, which shows that \({\mathcal {G}}\), unlike the nonlinear term \(R^2\) in the gravitational Lagrangian, does not contribute to them either.

Effects of f(G) gravity on the dynamics of self-gravitating fluids

We study the dynamics of self-gravitating fluids bounded by spherically symmetric surface in the background of f (G) gravity. The link between physical and geometrical variables, such as anisotropy, density inhomogeneity, dissipation, the Weyl tensor, expansion scalar, shear tensor and modified (Gauss-Bonnet) curvature terms, is given. We also investigate some particular fluid models according to various dynamical conditions. It is found that our results are consistent with general relativity for constant f (G) model (regular distribution of dark energy in the universe). Any other choice of the model leads to irregular distribution of dark energy and deviates from general relativity.