12 Appendix: The Friedmann Equations from the 1st Law In this Appendix, we generalize our discussion to include the possibility that Ω = 1. We include a curvature term and use the apparent event horizon r$_{A}$ as our preferred screen rather than the Hubble horizon r$_{H}$ –these are equivalent for the case of vanishing curvature. We have The energy flow across the apparent horizon is − dE = dMc 2 + PdV = ( ρc 2 + P ) dV where A$_{A}$ = 4 πr 2 $_{A}$. Assuming that the apparent horizon has an associate entropy S and approximate temperature T given by we can use the first law of thermodynamics − dE = TdS to find Dividing by c $^{2}$dt and using T = ℏ 2 πk$_{B}$ c r$_{A}$ one finds and then substituting in for ˙ r$_{A}$ , Dividing through by Hr$_{A}$/G and using A = 4 πr 2 A we have Rearranging one has one form of the standard Friedmann acceleration equation If one takes for the continuity (conservation) equation for the perfect cosmological fluids, r$_{A}$ = c $^{√}$H 2 + k/a 2 and ˙ r$_{A}$ = − Hr 3 A c 2 ( ˙ H − k a 2 ) (A.1) − dE = ( ρ + P/c $^{2}$) c $^{2}$dV = ( ρ + P/c $^{2}$) c $^{2}$A$_{A}$ vdt = ( ρ + P/c $^{2}$) c $^{2}$A$_{A}$ Hr$_{A}$dt (A.2) S = k$_{B}$c 3 $_{ℏ}$G A 4 , T = $_{ℏ}$H 2 πk$_{B}$ = ℏ 2 πk$_{B}$ c r$_{A}$ , (A.3) − dE = A ( ρ + P/c $^{2}$) c $^{2}$Hr$_{A}$ dt = TdS = T k$_{B}$c 3 4$_{ℏ}$ G dA$_{A}$ dt dt = T k$_{B}$c 3 4$_{ℏ}$ G 2 × 4 πr$_{A}$ ˙ r$_{A}$dt (A.4) A ( ρ + P/c $^{2}$) Hr$_{A}$ = c 2 G ˙ r$_{A}$ = − 1 G Hr 3 A ( ˙ H − k a 2 ) (A.5) 4 πG ( ρ + P/c $^{2}$) = −$^{(}$ ˙ H − k a 2 ) (A.6) ˙ H − k a 2 = − 4 πG ( ρ + P/c $^{2}$) (A.7) ˙ ρ + 3 H ( ρ + P/c $^{2}$) = 0 (A.8)