We derive the equations of cosmological evolution from an anti-de Sitter–Schwarzschild black hole via holographic renormalization with appropriate boundary conditions.

We study the cosmological evolution on a brane with induced gravity within a bulk with arbitrary matter content. We consider a Friedmann-Robertson-Walker brane, invariantly characterized by a six-dimensional group of isometries. We derive the effective Friedmann and Raychaudhuri equations. We show that the Hubble expansion rate on the brane depends on the covariantly defined integrated mass in the bulk, which determines the energy density of the generalized dark radiation. The Friedmann equation has two branches, distinguished by the two possible values of the parameter $ϵ=\pm 1$. The branch with $ϵ=1$ is characterized by an effective cosmological constant and accelerated expansion for low energy densities. Another remarkable feature is that the contribution from the generalized dark radiation appears with a negative sign. As a result, the presence of the bulk corresponds to an effective negative energy density on the brane, without violation of the weak energy condition. The transition from a period of domination of the matter energy density by nonrelativistic brane matter to domination by the generalized dark radiation corresponds to a crossing of the phantom divide $w=-1$.

We discuss features of the brane cosmological evolution that arise through the presence of matter in the bulk. As these deviations from the conventional evolution are not associated with some observable matter component on the brane, we characterize them as mirage effects. We review an example of expansion that can be attributed to mirage nonrelativistic matter (mirage cold dark matter) on the brane. The real source of the evolution is an anisotropic bulk fluid with negative pressure along the extra dimension. We also study the general problem of exchange of real nonrelativistic matter between the brane and the bulk, and discuss the related mirage effects. Finally, we derive the brane cosmological evolution within a bulk that contains a global monopole (hedgehog) configuration. This background induces a mirage curvature term in the effective Friedmann equation, which can cause a brane universe with positive spatial curvature to expand forever.

Using a fully covariant treatment for the description of the bulk geometry, we study the brane cosmological evolution in the presence of a smooth bulk matter distribution. We focus on the case of a Friedmann-Robertson-Walker brane, invariantly characterized by the existence of a six-dimensional group of isometries acting on 3D spacelike orbits. With a Friedmann-Robertson-Walker brane, the bulk geometry can be regarded as the 5D generalization of the inhomogeneous orthogonal family of locally rotationally symmetric spacetimes. We show that, for any bulk matter configuration, the expansion rate on the brane depends only on the covariantly defined comoving mass $\mathcal{M}$ of the bulk fluid within a radius equal to the average length scale of the 3D spacelike hypersurfaces of constant curvature. This unique contribution incorporates the effects of the 5D Weyl tensor and the projected tensor related to the bulk matter, and gives a transparent physical picture that includes an effective conservation equation between the brane and the bulk matter.

The existence of self-similar solutions is discussed in symmetric teleparallel $f(Q)$ theory for a Friedmann-Lemaître-Robertson-Walker background geometry with zero and nonzero spatial curvature. For the four distinct families of connections that describe the specific cosmology in symmetric teleparallel gravity, the functional form of $f(Q)$ is reconstructed. Finally, to see if the analogy with GR holds, we discuss the relation of the self-similar solutions with the asymptotic behavior of more general $f(Q)$ functions.