 The curvature of surfaces \(\Sigma(t)\) of constant cosmic time is given by \(\frac{k}{R^2(t)}\), where \(k = \{-1,0,1\}\) for negative, flat, and positive curvature (respectively). The assumed symmetries force the matter to be described as a perfect  fluid[8]  with energy density \(\rho\) and pressure \(p\), which obey the energy conservation equation 