Qun-Ying Xie and colleagues investigated thick branes generated by a scalar field in mimetic gravity theory, which is inspired by considering the conformal symmetry under the conformal transformation of an auxiliary metric. They obtained a series of analytical thick brane solutions and found that, perturbations of the brane system are stable and the effective potentials for the tensor and scalar perturbations are dual to each other. Findings of this study recently appeared in the *Journal Symmetry*.

Mimetic gravity is a Weyl-symmetric extension of General Relativity, related to the latter by a singular disformal transformation, wherein the appearance of a dust-like perfect fluid can mimic cold dark matter at a cosmological level. Within this framework, it is possible to provide an unified geometrical explanation for dark matter, the late-time acceleration, and inflation, making it a very attractive theory. This theory was also extended to Horava-like theory and applied to galactic rotation curves. It was also applied to other gravity theories such as f(R) gravity, Horndeski gravity and Gauss–Bonnet gravity.

On the other hand, Lisa Randall and Raman Sundrum proposed that our four-dimensional world could be a brane embedded in five-dimensional space-time, in order to solve gauge hierarchy problem and the cosmological constant problem. With the warped extra dimension, it was further found that the size of extra dimension can be infinitely large without conflicting with Newtonian gravitational law. This charming idea has attracted substantial researches in particle physics, cosmology, gravity theory, and other related fields. In the RS model, the brane is geometrically thin, therefore the space-time is singular at the brane. Although in the thin brane approximation many interesting results have been obtained, in some situations the effects of the brane thickness cannot be neglected.

“In five-dimensional problems, the thin brane approximation is valid as long as the brane thickness cannot be resolved, in other words, if the energy scale of the brane thickness is much higher than those in the bulk and on the brane. In contrast, when thickness becomes as large as the scale of interest, its effect is no longer negligible.”

Now, Qun-Ying Xie and colleagues investigated the super-potential method with which the second-order equations can be reduced to the first-order ones for thick brane models in modified gravity with Lagrange multiplier. The main step of this method is to introduce a pair of auxiliary super-potentials, i.e., W(φ) and Q(φ). With these two super-potentials, the field equations are rewritten as Equations (1)–(5).

Then, they used this method to find a series of analytical thick brane solutions via some polynomial super-potentials, period super-potentials, and mixed super-potentials.

Finally, they analyzed the tensor and scalar perturbations of the brane system. It was shown that both equations of motion of the perturbations can be transformed into Schrodinger-like equations. They added that, both perturbations are stable and the effective potentials for the tensor and scalar perturbations are dual to each other. Moreover, the tensor zero mode can be localized on the brane while the scalar zero mode cannot. Thus, the four-dimensional Newtonian potential can be recovered on the brane and there is no additional fifth force contradicting with the experiments.

**Featured image:*** The effective potential VS(z(y)) for solution I. The parameter is set as n = 1 (red solid thick lines), n = 3 (blue dashed lines), and n = 5 (black solid thin lines). © Xie et al. *

**Reference***: Xie, Q.-Y.; Fu, Q.-M.; Sui, T.-T.; Zhao, L.; Zhong, Y. First-Order Formalism and Thick Branes in Mimetic Gravity. Symmetry 2021, 13, https://doi.org/10.3390/sym13081345*

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