content_image/355332.jpg content_image/355333.jpg content_image/355334.jpg content_image/355335.jpg content_image/355336.jpg content_image/355337.jpg content_image/355338.jpg content_image/355339.jpg
## 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 

<img src='content_image/355332.jpg'>

The energy flow across the apparent horizon is − dE = dMc 2 + PdV = ( ρc 2 + P ) dV 

<img src='content_image/355333.jpg'>

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 

<img src='content_image/355334.jpg'>

<img src='content_image/355335.jpg'>

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 

<img src='content_image/355336.jpg'>

<img src='content_image/355337.jpg'>

Rearranging one has one form of the standard Friedmann acceleration equation 

<img src='content_image/355338.jpg'>

If one takes for the continuity (conservation) equation for the perfect cosmological fluids, 

<img src='content_image/355339.jpg'>

