Cosmologists need to check their work by using different coordinates. The Friedmann coordinates conventionally used are rife with fictitious potentials which interact nonlinearly with real sources of gravity. It is Cartesian coordinates alone that have no such effects. And hyperspherical coordinates are still orthogonal everywhere, so they have no off-diagonal fictitious potentials.
The whole point of coordinate systems other than Cartesian is to pretend that they are Cartesian themselves. To maintain this fiction without error, fictitious forces and their potentials must be inserted. The fictitious potentials then seem real and not different from ordinary gravity. They affect sources of gravity in the same way as real potentials.
When the correct fictitious potentials are inserted into the boundary conditions, then the Bianchi identities can be integrated on. This can be a graphical process when the correct tensor ranks are used.
Michael J. Burns