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Corresponding author.

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The Palatini

As the most successful gravitational theory, general relativity (GR) is able to explain the gravitational phenomena in the scale ranging from submillimeter to Solar System scales

Recently, based on the pure metric Born-Infeld theory of gravity

It was later found that, by introducing a new definition,

Inspired by the recent interesting research papers on EIBI branes

In this case, it is crucial to confirm whether it is available to construct the braneworld scenario in

This paper is organized as follows. In Sec.

We start with the action of the

In Ref.

In this paper, we mainly focus on the five-dimensional

Now, for the sake of simplicity, we replace the equations of motion

In this section, we mainly focus on the construction of thick branes in

Next, we discuss the behavior of the scalar potential. By using Eq.

The shapes of the thick brane solution. (a) The warp factor

To check whether the background solution corresponds to a thick braneworld model, we should discuss the energy density of the brane. As we know, in most braneworld models

After the analysis of the localization of the thick brane, we should further investigate the effects of the thick brane on the geometry of five-dimensional spacetime. Using the above thick brane solution and the definition of the physical curvature scalar,

In this subsection, we investigate the metric perturbations and the localization of the massless graviton. With great interest in the tensor perturbation, we use the transverse-traceless (TT) gauge to decouple the vector perturbations and the scalar perturbations from the tensor perturbation in the spacetime metric, as done in Ref.

In order to recover the four-dimensional Newtonian potential, we investigate the localization of the massless graviton. The corresponding analytical function of the massless graviton is given as

The shapes of the effective potential, the massless graviton, and the function

In this subsection, we investigate the low-energy effective theory of a specific family of theories, i.e.,

To give a prediction on the correction to the effective Newtonian gravitational potential in this

The relation between the parameters

As is shown in Fig.

The constraints on (a) the parameter

We finally conclude that, though the contributions from the massive gravitons have not been seen on the recent gravitational test, our model is still not excluded yet. From the theoretical side, this deviation is probably apparent with the separation at the length scale of a nanometer at most. Therefore, we are still hopeful that the signal from the extra dimension can be obtained using the next generation of the gravitational test.

In summary, in this paper, we investigated the thick

We also investigated the tensor perturbations of the two metrics and showed that the choice of TT gauge on the spacetime metric perturbations would lead to the TT gauge condition on the auxiliary metric perturbations. It is important to note that, unlike what we have assumed before the calculation of the linear perturbation equations, the tensor perturbations of the two metrics were found to be connected with each other and to be independent of

The localization of the fermion fields and the gravitons is a conspicuous problem in the braneworld model. Many efforts have been paid in this area, and it is now well known that the localization problem could be addressed well in some thick braneworld models

We are thankful to the referees’ suggestions and criticisms. We also thank Bao-Min Gu, Wen-Di Guo, Hao Yu, and Zi-Qi Chen for helpful discussions. This work was supported by the National Natural Science Foundation of China (Grants No. 11522541, No. 11375075, and No. 11705070); the Fundamental Research Funds for the Central Universities (Grants No. lzujbky-2018-k11 and No. lzujbky-2017-it68); and the Strategic Priority Research Program on Space Science, the Chinese Academy of Sciences (Grant No. XDA15020701).