Gauss-Bonnet Coupling Function Research Articles

Revisiting Einstein-Gauss-Bonnet theories after GW170817

In this work we study the inflationary framework of Einstein-Gauss-Bonnet theories which produce a primordial gravitational wave speed that respects the constraint imposed by the GW170817 event, namely |cT2−1|<6×10−15 in natural units. For these theories in general, the scalar Gauss-Bonnet coupling function ξ(ϕ) is arbitrary and unrelated to the scalar potential. We develop the inflationary formalism of this theory, and present analytic forms for the slow-roll indices, the observational indices and the e-foldings number, assuming solely a slow-roll era and also that the constraints imposed by the GW170817 event hold true. We present in detail an interesting class of models that produces a viable inflationary era. Finally, we investigate the behavior of the Einstein-Gauss-Bonnet coupling function during the reheating era, in which case the evolution of the Hubble rate satisfies a constant-roll-like relation H˙=δH2. As we show, the behavior of the scalar Gauss-Bonnet coupling during the reheating is different from that of the inflationary era. This is due to the fact that the evolution governing the reheating is different compared to the inflationary era.

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.

GW170817 Implementation on Einstein-Gauss-Bonnet Theory with Non Minimal and Non Minimal Derivative Coupling

The GW170817 event manifests that gravitational wave velocity is close to thespeed of light. As a result, several theories of gravity are no longer applicable, including EinsteinGauss-Bonnet (EGB) inflation. However, a constraint equation could be applied so that thetheory could produce a viable result. In this study, the EGB inflation is being extended byadding a non-minimal coupling (NMC) and a non-minimal derivative coupling (NMDC). Freeparameters values were evaluated to obtained viability with observational indices. We use powerlaw and exponential Gauss-Bonnet coupling functions. Each model provides observational valuesof ns and r that are compatible with the observations and has its characteristic. It specifiesthe free parameter that controls the alteration of ns and r values. The power-law model iscontrolled by the power m of the Gauss-Bonnet coupling function and the potential integrationconstant, V2. While the exponential model is controlled by the potential integration constant cand the power m of the exponential function. Some approximations do not hold true so that themodels need to be rectified. Apparently, the rectified power-law model is violating null energycondition (NEC), so we also provide the non-violating NEC power-law model.

Dynamical descalarization with a jump during a black hole merger

Black holes in scalar-Gauss-Bonnet gravity are prone to scalarization, that is a spontaneous development of scalar hair for strong enough spacetime curvature while the weak field regime of the theory coincides with general relativity. Since large spacetime curvature is associated with smaller black hole masses, the merging of black holes can lead to dynamical descalarization. This is a spontaneous release of the scalar hair of the newly formed black hole in case its mass is above the scalarization threshold. Depending on the exact form of the Gauss-Bonnet coupling function, the stable scalarized solutions can be either continuously connected to the Schwarzschild black hole, or the transitions between the two can happen with a jump. By performing simulations of black hole head-on collisions in scalar-Gauss-Bonnet gravity prone to dynamical descalization, we have demonstrated that such a jump can be clearly observed in the accumulated gravitational wave data of multiple merger events with different masses. The distinct signature in the gravitational wave signal will share similarities with the effects expected from first order matter phase transitions happening during neutron star binary mergers.

Reheating era in Gauss-Bonnet theories of gravity compatible with the GW170817 event

In the present article we showcase how the reheating era can be described properly in the context of Einstein-Gauss-Bonnet gravity assuming that the primordial gravitational waves propagate with the velocity of light. The equations of the duration of reheating along with the reheating temperature are derived and as demonstrated, their expressions are quite similar to the case of a canonical scalar field where now the second derivative of the Gauss-Bonnet scalar coupling function appears and effectively alters the numerical value of the scalar potential. The appearance of such term is reminiscing of a λR model of gravity where λ is now dynamical. We consider two viable inflationary models of interest, the former involves an error function as scalar Gauss-Bonnet coupling function and the latter a Woods-Saxon scalar potential. It is shown that for both models the aforementioned quantities can be in agreement with theoretical expectations. The only constraint that is needed is the assumption that the second time derivative of the Gauss-Bonnet scalar coupling function is actually lesser than the Planck mass squared, that is ξ¨<MPl28 in order to obtain a viable description. We find that a free parameter of the theory and specifically the potential amplitude for the Woods-Saxon model during the inflationary era, dictates the effective equations of state and therefore the reheating epoch can be described either by a type of stiff matter with EoS parameter equal to unity or by an EoS parameter close to that of the radiation.

Towards a smooth unification from an ekpyrotic bounce to the dark energy era

In the context of a ghost free f ( R , G ) model, we present a non-singular cosmological scenario in which the universe initially contracts through an ekpyrotic phase having a bouncing like behaviour, and following the bounce, it smoothly transits to a matter or radiation like deceleration era which is further smoothly connected to the dark energy era at present epoch. The ghost free character of the model is ensured by the presence of a Lagrange multiplier, and we consider the Gauss–Bonnet (GB) coupling function in such a way that it gets compatible with the event GW170817. Using suitable reconstruction technique, we obtain the non-trivial scalar field potential as well as the GB coupling function. Such scalar potential and GB coupling function source a smooth unified scenario from an ekpyrotic bounce to the dark energy era with an intermediate deceleration era. The occurrence of ekpyrotic contraction phase justifies the resolution of the anisotropic problem (also known as BKL instability) in the background evolution. Consequently we determine the background Hubble parameter and the corresponding effective equation of state parameter, and discussed several qualitative features of the model. The Hubble radius shows an asymmetric behaviour around the bounce, in particular, the evolution of the Hubble radius leads to the generation era of the primordial perturbation modes far before the bounce in the deep sub-Hubble regime. Accordingly we perform the scalar and tensor perturbation evolution in the present context, and as a result, the scalar power spectrum at large scale modes is found to be problematic. Thus an extended scenario is proposed where we consider a pre-ekpyrotic phase having the equation of state parameter is less than unity, and re-examine the scalar and tensor power spectra, on large scales that cross the Hubble radius during the pre-ekpyrotic stage. In this regard, the GB coupling function shows considerable effects in reducing the tensor to scalar ratio compared to the case where the GB coupling is absent. Furthermore the dark energy epoch is consistent with the Planck + SNe + BAO data.

Canonical scalar field inflation with string and [formula omitted]-corrections

Assuming that a scalar field controls the inflationary era, we examine the combined effects of string and f(R) gravity corrections on the inflationary dynamics of canonical scalar field inflation, imposing the constraint that the speed of the primordial gravitational waves is equal to that of light’s. Particularly, we study the inflationary dynamics of an Einstein–Gauss–Bonnet gravity in the presence of αR2 corrections, where α is a free coupling parameter. As it was the case in the pure Einstein–Gauss–Bonnet gravity, the realization that the gravitational waves propagate through spacetime with the velocity of light, imposes the constraint that the Gauss–Bonnet coupling function ξ(ϕ) obeys the differential equation ξ̈=Hξ̇, where H is the Hubble rate. Subsequently, a relation for the time derivative of the scalar field is extracted which implies that the scalar functions of the model, which are the Gauss–Bonnet coupling and the scalar potential, are interconnected and simply designating one of them specifies the other immediately. In this framework, it is useful to freely designate ξ(ϕ) and extract the corresponding scalar potential from the equations of motion but the opposite is still feasible. We demonstrate that the model can produce a viable inflationary phenomenology and for a wide range of the free parameters. Also, a mentionable issue is that when the coupling parameter α of the R2 correction term is α<10−3 in Planck Units, the R2 term is practically negligible and one obtains the same equations of motion as in the pure Einstein–Gauss–Bonnet theory, however the dynamics still change, since now the time derivative of ∂f∂R is nonzero. We study in detail the dynamics of inflation assuming that the slow-roll conditions hold true and also we briefly address the constant-roll case dynamics, and by using several illustrative examples, we compare the dynamics of the pure and R2-corrected Einstein–Gauss–Bonnet gravity. Finally the ghosts issue is briefly addressed.

Rectifying Einstein-Gauss-Bonnet inflation in view of GW170817

In this work we introduce a new theoretical framework for Einstein-Gauss-Bonnet theories of gravity, which results to particularly elegant, functionally simple and transparent gravitational equations of motion, slow-roll indices and the corresponding observational indices. The main requirement is that the Einstein-Gauss-Bonnet theory has to be compatible with the GW170817 event, so the gravitational wave speed cT2 is required to be cT2≃1 in natural units. This assumption was also made in a previous work of ours, but in this work we express all the related quantities as functions of the scalar field. The constraint cT2≃1 restricts the functional form of the scalar Gauss-Bonnet coupling function ξ(ϕ) and of the scalar potential V(ϕ), which must satisfy a differential equation. However, by also assuming that the slow-roll conditions hold true, the resulting equations of motion and the slow-roll indices acquire particularly simple forms, and also the relation that yields the e-foldings number is N=∫ϕiϕfξ″/ξ′dϕ, a fact that enables us to perform particularly simple calculations in order to study the inflationary phenomenological implications of several models. As it proves, the models we presented are compatible with the observational data, and also satisfy all the assumptions made during the process of extracting the gravitational equations of motion. More interestingly, we also investigated the phenomenological implications of an additional condition ξ′/ξ″≪1, which is motivated by the slow-roll conditions that are imposed on the scalar field evolution and on the Hubble rate. As we shall show, the resulting constraint differential equation that constrains the functional form of the scalar Gauss-Bonnet coupling function ξ(ϕ) and of the scalar potential V(ϕ), is simpler in this case, and in effect the whole study becomes somewhat easier. As we also show, compatibility with the observational data can also be achieved in this case too, in a much simpler and less constrained way. Our approach opens a new window in viable Einstein-Gauss-Bonnet theories of gravity.

Swampland implications of GW170817-compatible Einstein-Gauss-Bonnet gravity

We revisit the Einstein-Gauss-Bonnet theory in view of the GW170817 event, which compels that the gravitational wave speed is equal to cT2=1 in natural units. We use an alternative approach compared to one previous work of ours, which enables us to express all the slow-roll indices and the observational indices as functions of the scalar field. Using our formalism we investigate if the Swampland criteria are satisfied for the Einstein-Gauss-Bonnet theory and as we demonstrate, the Swampland criteria are satisfied for quite general forms of the potential and the Gauss-Bonnet coupling function ξ(ϕ), if the slow-roll conditions are assumed to hold true.

Self-interactions and spontaneous black hole scalarization

It has recently been shown that nontrivial couplings between a scalar and the Gauss-Bonnet invariant can give rise to black hole spontaneous scalarization. Theories that exhibit this phenomenon are among the leading candidates for testing gravity with upcoming black hole observations. All models considered so far have focused on specific forms for the coupling, neglecting scalar self-interactions. In this work, we take the first steps towards placing this phenomenon on a more robust theoretical footing by considering the leading-order scalar self-interactions as well as the scalar-Gauss-Bonnet coupling. Our approach is consistent with the principles of effective field theory and yields the simplest and most natural model. We find that a mass term for the scalar alters the threshold for the onset of scalarization, and we study the mass range over which scalarized black hole solutions exist. We also demonstrate that the quartic self-coupling is sufficient to produce scalarized solutions that are stable against radial perturbations, without the need to resort to higher-order terms in the Gauss-Bonnet coupling function. Our model therefore represents a canonical model that can be studied further, with the ultimate aim of developing falsifiable tests of black hole scalarization.

Scalarized black holes in the presence of the coupling to Gauss-Bonnet gravity

In this paper, we study static and spherically symmetric black hole (BH) solutions in the scalar-tensor theories with the coupling of the scalar field to the Gauss-Bonnet (GB) term $\xi (\phi) R_{\rm GB}$, where $R_{\rm GB}:=R^2-4R^{\alpha\beta}R_{\alpha\beta}+R^{\alpha\beta\mu\nu}R_{\alpha\beta\mu\nu}$ is the GB invariant and $\xi(\phi)$ is a function of the scalar field $\phi$. Recently, it was shown that in these theories scalarized static and spherically symmetric BH solutions which are different from the Schwarzschild solution and possess the nontrivial profiles of the scalar field can be realized for certain choices of the coupling functions and parameters. These scalarized BH solutions are classified in terms of the number of nodes of the scalar field. It was then pointed out that in the case of the pure quadratic order coupling to the GB term, $\xi(\phi)=\eta \phi^2/8$, scalarized BH solutions with any number of nodes are unstable against the radial perturbation. In order to see how a higher order power of $\phi$ in the coupling function $\xi(\phi)$ affects the properties of the scalarized BHs and their stability, we investigate scalarized BH solutions in the presence of the quartic order term in the GB coupling function, $\xi(\phi)=\eta \phi^2 (1+\alpha \phi^2)/8$. We clarify that the existence of the higher order term in the coupling function can realize scalarized BHs with zero nodes of the scalar field which are stable against the radial perturbation.

Inflation with Gauss-Bonnet coupling

We consider inflationary models with the inflaton coupled to the Gauss-Bonnet term assuming a special relation $\delta_1=2\lambda\epsilon_1$ between the two slow-roll parameters $\delta_1$ and $\epsilon_1$. For the slow-roll inflation, the assumed relation leads to the reciprocal relation between the Gauss-Bonnet coupling function $\xi(\phi)$ and the potential $V(\phi)$, and it leads to the relation $r=16(1-\lambda)\epsilon_1$ that reduces the tensor-to-scalar ratio $r$ by a factor of $1-\lambda$. For the constant-roll inflation, we derive the analytical expressions for the scalar and tensor power spectra, the scalar and tensor spectral tilts, and the tensor-to-scalar ratio to the first order of $\epsilon_1$ by using the method of Bessel function approximation. The tensor-to-scalar ratio is reduced by a factor of $1-\lambda+\lambda\tilde \eta$. Comparing the derived $n_s$-$r$ with the observations, we obtain the constraints on the model parameters $\tilde\eta$ and $\lambda$.

Perturbation, non-Gaussianity, and reheating in a Gauss-Bonnet α -attractor model

Motivated by $\alpha$-attractor models, in this paper we consider a Gauss-Bonnet inflation with E-model type of potential. We consider the Gauss-Bonnet coupling function to be the same as the E-model potential. In the small $\alpha$ limit we obtain an attractor at $r=0$ as expected, and in the large $\alpha$ limit we recover the Gauss-Bonnet model with potential and coupling function of the form $\phi^{2n}$. We study perturbations and non-Gaussianity in this setup and we find some constraints on the model's parameters in comparison with PLANCK datasets. We study also the reheating epoch after inflation in this setup. For this purpose, we seek the number of e-folds and temperature during reheating epoch. These quantities depend on the model's parameter and the effective equation of state of the dominating energy density in the reheating era. We find some observational constraints on these parameters.

Constraining scalar-Gauss-Bonnet inflation by reheating, unitarity, and Planck data

We revisit the inflationary dynamics in detail for theories with Gauss-Bonnet gravity coupled to scalar functions, in light of the Planck data. Considering the chaotic inflationary scenario, we constrain the parameters of two models involving inflaton-Gauss-Bonnet coupling by current Planck data. For non zero inflaton-Gauss-Bonnet coupling $\beta$, an inflationary analysis provides us a big cosmologically viable region in the space of $(m,\beta)$, where $m$ is the mass of inflaton. However, we study further on constraining $\beta$ arising from reheating considerations and unitarity of tree-level amplitude involving we have studied the constraints on $\beta$ arising from reheating considerations and unitarity of tree level amplitude involving $2$ graviton $\rightarrow 2$ graviton ($h h\rightarrow hh$) scattering. Our analysis, particularly on reheating significantly reduces the parameter space of $(m,\beta)$ for all models. he quadratic Gauss-Bonnet coupling parameter turns out to be more strongly constrained than that of the linear coupling. For the linear Gauss-Bonnet coupling function, we obtain $\beta \lesssim 10^3$, with the condition $\beta (m/M_P)^2 \simeq 10^{-4}$. However, study of the Higgs inflation scenario in the presence of Gauss-Bonnet term turned out to be strongly disfavored.

Anisotopic inflation with a non-abelian gauge field in Gauss-Bonnet gravity

In presence of Gauss-Bonnet corrections, we study anisotropic inflation aided by a massless SU(2) gauge field where both the gauge field and the Gauss-Bonnet term are non-minimally coupled to the inflaton. In this scenario, under slow-roll approximations, the anisotropic inflation is realized as an attractor solution with quadratic forms of inflaton potential and Gauss-Bonnet coupling function. We show that the degree of anisotropy is proportional to the additive combination of two slow-roll parameters of the theory. The anisotropy may become either positive or negative similar to the non-Gauss-Bonnet framework, a feature of the model for anisotropic inflation supported by a non-abelian gauge field but the effect of Gauss-Bonnet term further enhances or suppresses the generated anisotropy.

Tachyon Cosmology with Gauss–Bonnet and Non-Minimal Kinetic Couplings

We consider a tachyonic model of dark energy in which scalar field non-minimally coupled with curvature and kinetic part of its Lagrangian density. Additionally the model contains the Gauss–Bonnet coupling to the scalar field through an arbitrary function. The non-minimal Gauss–Bonnet coupling function and scalar field potential have been obtained for power-law solution and then for a dynamically varying equation of state. We have extracted the required condition for the so-called phantom divide line crossing in the model and represented such a crossing numerically.

Bounce universe from string-inspired Gauss-Bonnet gravity

We explore cosmology with a bounce in Gauss-Bonnet gravity where the Gauss-Bonnet invariant couples to a dynamical scalar field. In particular, the potential and and Gauss-Bonnet coupling function of the scalar field are reconstructed so that the cosmological bounce can be realized in the case that the scale factor has hyperbolic and exponential forms. Furthermore, we examine the relation between the bounce in the string (Jordan) and Einstein frames by using the conformal transformation between these conformal frames. It is shown that in general, the property of the bounce point in the string frame changes after the frame is moved to the Einstein frame. Moreover, it is found that at the point in the Einstein frame corresponding to the point of the cosmological bounce in the string frame, the second derivative of the scale factor has an extreme value. In addition, it is demonstrated that at the time of the cosmological bounce in the Einstein frame, there is the Gauss-Bonnet coupling function of the scalar field, although it does not exist in the string frame.

Slow-roll inflation with the Gauss-Bonnet and Chern-Simons corrections

We study slow-roll inflation with the Gauss-Bonnet and Chern-Simons corrections. We obtain general formulas for the observables: spectral indices, tensor-to-scalar ratio and circular polarization of gravitational waves. The Gauss-Bonnet term violates the consistency relation r = −8nT. Particularly, blue spectrum nT > 0 and scale invariant spectrum |8nT|/r << 1 of tensor modes are possible. These cases require the Gauss-Bonnet coupling function of ξ,ϕ ∼ 108/MPl. We use examples to show new-inflation-type potential with 10MPl symmetry breaking scale and potential with flat region in ϕ≳10MPl lead to observationally consistent blue and scale invariant spectra, respectively. Hence, these interesting cases can actually be realized. The Chern-Simons term produce circularly polarized tensor modes. We show an observation of these signals supports existence of the Chern-Simons coupling function of ω,ϕ ∼ 108/MPl. Thus, with future observations, we can fix or constrain the value of these coupling functions, at the CMB scale.