Ah, this makes a lot of sense, indeed. I was expecting Line 8 to be correct, since the rest of the proof follows through with this value, but couldn't find the error. Thanks a lot, @TSny, for spotting the error before that.TSny said:
Box 5.3 in the MCP (Modern Cosmology and Physics) book focuses on the derivation of key equations related to the dynamics of cosmological models, particularly in the context of general relativity and the expanding universe.
The key assumptions include the isotropy and homogeneity of the universe, the use of the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, and the consideration of a perfect fluid as the source of energy-momentum in the Einstein field equations.
The proof in Box 5.3 derives the Friedmann equations, which describe the expansion of the universe. It shows how the energy density and pressure of the cosmological fluid influence the rate of expansion and the curvature of space-time.
The results imply that the dynamics of the universe's expansion are governed by the interplay between matter, radiation, and dark energy. They provide insights into the evolution of the universe, including the potential for accelerated expansion.
Many readers find the transition from the Einstein field equations to the Friedmann equations challenging, particularly in understanding how to properly apply the conservation laws and the equation of state for the perfect fluid. It is crucial to carefully follow the metric and energy-momentum tensor definitions throughout the derivation.