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تسجيل مستخدم جديدOn 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 $D$
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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.
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