The existence of envelopes is studied for systems of differential equations in connection with the method of asymptotic splittings which allows one to dete

We construct a regular five-dimensional brane-world with localised gravity on a flat 3-brane. The matter content in the bulk is parametrised by an analog of a non-linear fluid with equation of state $$p=\gamma \rho ^\lambda $$p=γρλ between the ‘pressure’pand the ‘density’ $$\rho $$ρ dependent on the 5th dimension. For $$\gamma $$γ negative and $$\lambda >1$$λ>1 , the null energy condition is satisfied and the geometry is free of singularities within finite distance from the brane, while the induced four-dimensional Planck mass is finite.

We discuss qualitative features of the conformal relation between certain classes of gravity theories and general relativity, common to different themes such as $f\left(R\right)$, Brans-Dicke-type, and string theories. We focus primarily on the frame relations of the fields involved, slice energy, traceless and Palatini extensions, and selected cosmological applications.

We review studies on the singularity structure and asymptotic analysis of a 3-brane (flat or curved) embedded in a five-dimensional bulk filled with a ‘perfect fluid’ with an equation of state p = γ ρ , where p is the ‘pressure’ and ρ is the ‘density’ of the fluid, depending on the fifth space coordinate. Regular solutions satisfying positive energy conditions in the bulk exist only in the cases of a flat brane for γ = − 1 or of AdS branes for γ ∈ [ − 1 , − 1 / 2 ) . More cases can be found by gluing two regular brunches of solutions at the position of the brane. However, only a flat brane for γ = − 1 leads to finite Planck mass on the brane and thus localizes gravity. In a more recent work, we showed that a way to rectify the previous findings and obtain a solution for a flat brane and a range of γ , which is both free from finite-distance singularities and compatible with the physical conditions of energy and finiteness of four-dimensional Planck mass, is by introducing a bulk fluid component that satisfies a nonlinear equation of state of the form p = γ ρ λ with γ < 0 and λ > 1 . This article is part of the theme issue ‘The future of mathematical cosmology, Volume 2’.