The stability and localization of the gravitational perturbations for a special brane system in Eddingtoninspired Born-Infeld gravity were studied in Liu et al. [Phys. Rev. D 85, 124053 (2012)]. In this paper, we show that the gravitational perturbations for a general brane system are stable, the four-dimensional graviton (massless KK graviton) can be localized on the brane, and the mass spectra of massive KK gravitons are gapless and continuous. Two models are constructed as examples. In the first model, which is a generalization of Liu et al. [Phys. Rev. D 85, 124053 (2012)], the brane has no inner structure and there is no gravitational resonance (quasilocalized KK gravitons). In the second one, the background scalar field is a double kink when the parameter in the model approaches its critical value. Correspondingly, the brane has inner structure and some gravitational resonances appear.

In this paper, we study the thick brane system in the so-called f(Q) gravity, where the gravitational interaction was encoded by the nonmetricity Q like scalar curvature R in general relativity. With a special choice of $$f(Q)=Q-b Q^n$$ f ( Q ) = Q - b Q n , we find that the thick brane system can be solved analytically with the first-order formalism, where the complicated second-order differential equation is transformed to several first-order differential equations. Moreover, the stability of the thick brane system under tensor perturbation is also investigated. It is shown that the tachyonic states are absent and the graviton zero mode can be localized on the brane. Thus, the four-dimensional Newtonian potential can be recovered at low energy. Besides, the corrections of the massive graviton Kaluza–Klein modes to the Newtonian potential are also analyzed briefly.

Horndeski theory is the most general scalar-tensor theory retaining second-order field equations, although the action includes higher-order terms. This is achieved by a special choice of coupling constants. In this paper, we investigate thick brane system in reduced Horndeski theory, especially the effect of the non-minimal derivative coupling on thick brane. First, the equations of motion are presented and a set of analytic background solutions are obtained. Then, to investigate the stability of the background scalar profile, we present a novel canonically normalized method, and show that although the original background scalar field is unstable, the canonical one is stable.The stability of the thick brane under tensor perturbation is also considered. It is shown that the tachyon is absent and the graviton zero mode can be localized on the brane. The localized graviton zero mode recovers the four-dimensional Newtonian potential and the presence of the non-minimal derivative coupling results in a splitting of its wave function. The correction of the massive graviton KK modes to the Newtonian potential is also analyzed briefly.

It is known that the metric and Palatini formalisms of gravity theories have their own interesting features but also suffer from some different drawbacks. Recently, a novel gravity theory called hybrid metric-Palatini gravity was put forward to cure or improve their individual deficiencies. The action of this gravity theory is a hybrid combination of the usual Einstein-Hilbert action and a $f(\mathcal{R})$ term constructed by the Palatini formalism. Interestingly, it seems that the existence of a light and long-range scalar field in this gravity may modify the cosmological and galactic dynamics without conflicting with the laboratory and Solar System tests. In this paper we focus on the tensor perturbation of thick branes in this novel gravity theory. We consider two models as examples, namely, the thick branes constructed by a background scalar field and by pure gravity. The thick branes in both models have no inner structure. However, the graviton zero mode in the first model has inner structure when the parameter in this model is larger than its critical value. We find that the effective four-dimensional gravity can be reproduced on the brane for both models. Moreover, the stability of both brane systems against the tensor perturbation can also be ensured.Comment: 7 pages, 4 figure