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Register a new userOn Taking the $Dto 4$ limit of Gauss-Bonnet Gravity: Theory and Solutions
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We comment on the recently introduced Gauss-Bonnet gravity in four dimensions. We argue that it does not make sense to consider this theory to be defined by a set of $Dto 4$ solutions of the higher-dimensional Gauss-Bonnet gravity. We show that a well-defined $Dto 4$ limit of Gauss-Bonnet Gravity is obtained generalizing a method employed by Mann and Ross to obtain a limit of the Einstein gravity in $D=2$ dimensions. This is a scalar-tensor theory of the Horndeski type obtained by a dimensional reduction methods. By considering simple spacetimes beyond spherical symmetry (Taub-NUT spaces) we show that the naive limit of the higher-dimensional theory to four dimensions is not well defined and contrast the resultant metrics with the actual solutions of the new theory.
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We investigate the $Drightarrow 4$ limit of the $D$-dimensional Einstein-Gauss-Bonnet gravity, where the limit is taken with $tilde{alpha}=(D-4), alpha$ kept fixed and $alpha$ is the original Gauss-Bonnet coupling. Using the ADM decomposition in $D$ dimensions, we clarify that the limit is rather subtle and ambiguous (if not ill-defined) and depends on the way how to regularize the Hamiltonian or/and the equations of motion. To find a consistent theory in $4$ dimensions that is different from general relativity, the regularization needs to either break (a part of) the diffeomorphism invariance or lead to an extra degree of freedom, in agreement with the Lovelock theorem. We then propose a consistent theory of $Drightarrow 4$ Einstein-Gauss-Bonnet gravity with two dynamical degrees of freedom by breaking the temporal diffeomorphism invariance and argue that, under a number of reasonable assumptions, the theory is unique up to a choice of a constraint that stems from a temporal gauge condition.
We study the slow-roll single field inflation in the context of the consistent $Dto4$ Einstein-Gauss-Bonnet gravity that was recently proposed in cite{Aoki:2020lig}. In addition to the standard attractor regime, we find a new attractor regime which we call the Gauss-Bonnet attractor as the dominant contribution comes from the Gauss-Bonnet term. Around this attractor solution, we find power spectra and spectral tilts for the curvature perturbations and gravitational waves (GWs) and also a model-independent consistency relation among observable quantities. The Gauss-Bonnet term provides a nonlinear $k^4$ term to the GWs dispersion relation which has the same order as the standard linear $k^2$ term at the time of horizon crossing around the Gauss-Bonnet attractor. The Gauss-Bonnet attractor regime thus provides a new scenario for the primordial GWs which can be tested by observations. Finally, we study non-Gaussianity of GWs in this model and estimate the nonlinear parameters $f^{s_1s_2s_3}_{rm NL,;sq}$ and $f^{s_1s_2s_3}_{rm NL,;eq}$ by fitting the computed GWs bispectra with the local-type and equilateral-type templates respectively at the squeezed limit and at the equilateral shape. For helicities $(+++)$ and $( -- )$, $f^{s_1s_2s_3}_{rm NL,;sq}$ is larger while $f^{s_1s_2s_3}_{rm NL,;eq}$ is larger for helicities $(++-)$ and $(--+)$.
In a very recent paper [1], we have proposed a novel $4$-dimensional gravitational theory with two dynamical degrees of freedom, which serves as a consistent realization of $Dto4$ Einstein-Gauss-Bonnet gravity with the rescaled Gauss-Bonnet coupling constant $tilde{alpha}$. This has been made possible by breaking a part of diffeomorphism invariance, and thus is consistent with the Lovelock theorem. In the present paper, we study cosmological implications of the theory in the presence of a perfect fluid and clarify the similarities and differences between the results obtained from the consistent $4$-dimensional theory and those from the previously considered, naive (and inconsistent) $Drightarrow 4$ limit. Studying the linear perturbations, we explicitly show that the theory only has tensorial gravitational degrees of freedom (besides the matter degree) and that for $tilde{alpha}>0$ and $dot{H}<0$, perturbations are free of any pathologies so that we can implement the setup to construct early and/or late time cosmological models. Interestingly, a $k^4$ term appears in the dispersion relation of tensor modes which plays significant roles at small scales and makes the theory different than not only general relativity but also many other modified gravity theories as well as the naive (and inconsistent) $Dto 4$ limit. Taking into account the $k^4$ term, the observational constraint on the propagation of gravitational waves yields the bound $tilde{alpha} lesssim (10,{rm meV})^{-2}$. This is the first bound on the only parameter (besides the Newtons constant and the choice of a constraint that stems from a temporal gauge fixing) in the consistent theory of $Dto 4$ Einstein-Gauss-Bonnet gravity.
To ensure the existence of a well defined linearized gravitational wave equation, we show that the spacetimes in the so-called Einstein-Gauss-Bonnet gravity in four dimension have to be locally conformally flat.
Exact solutions with torsion in Einstein-Gauss-Bonnet gravity are derived. These solutions have a cross product structure of two constant curvature manifolds. The equations of motion give a relation for the coupling constants of the theory in order to have solutions with nontrivial torsion. This relation is not the Chern-Simons combination. One of the solutions has a $AdS_2times S^3$ structure and is so the purely gravitational analogue of the Bertotti-Robinson space-time where the torsion can be seen as the dual of the covariantly constant electromagnetic field.
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