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(dR/dt)^{2} + kc^{2} = 8GR^{2} / 3 + R^{2} c^{2} / 3 ---------- (20a)where is the cosmological constant. It can be expressed in term of the corresponding density _{} by the formula = 8G_{} / c^{2}. Assuming a flat universe, current observations of distance supernovae, the cosmic microwave background radiation, and the dynamics of galaxies together favor a value of _{} = 0.7 _{c}, the numeric value for is about 1.3x10^{-56} cm^{-2}. Past attempts to identify the cosmological constant with the vacuum energy of the various quantum fields was not very successful. See "Vacuum Energy Density" for a 2015 update. The effect of the cosmological constant on the cosmic expansion is summarized in Figure 10h. | |

## Figure 10h Cosmological Constant [view large image] |

The de Sitter universe is devoid of matter energy containing only vacuum energy. It follows that Eq.(20a) is simplified to:

(dR/dt)

For k = 0, the solution is: R(t) = R

where H = (/3)^{1/2}c = (dR/dt)/R is the Hubble constant, and R_{0} is a constant.. This solution shows the weird property of a constant matter-energy density _{}, i.e., vacuum energy is continuously being created to fill up the void in the wake of the expansion. The idea is similar to the steady state universe^{¶}, which requires the continuous creation of matter. Since the event horizon is d_{h} = c / H = 1 / (/3)^{1/2} = constant; therefore, like the earth's horizon, the de Sitter horizon can never be reached - it is always a finite distance away. The de Sitter universe with k = 0 starts at t - from a singularity. It reaches a size of R_{o} at t = 0, and expands to infinity as t +. Alternatively, R_{o} at t = 0 can be taken as the initial condition. The universe then can either grow or decay exponentially. Note that Eq.(20b) is time reversal invariant. For some reason, this universe chose to grow exponentially in the positive time direction. | |

## Figure 10i de Sitter Universe with k = +1 |
For k = +1, R(t) = (c/H) cosh(Ht) (c/2H) e^{H|t|} as t _{}. |

For k = -1, R(t) = (c/H) sinh(Ht) (c/2H) e

All de Sitter universes expand forever exponentially. Such behaviour is similar to the situation at the very beginning and very end of our universe, either when energy matter had not been created or they have been diluted so much at the end.

The anti-de Sitter (AdS) universe is also described by Eq.(20b) except that is now negative.

For k = -1, R(t) = (c/H) sin(Ht).

where H = (-/3)

The simple de Sitter, anti-de Sitter models suggest that for large positive , everything flies apart so quickly that there is no chance for matter to assemble itself into sturctures like galaxies, stars, planets, atoms or even nuclei. On the other hand, with large negative , the expanding universe quickly turns around and terminates the evolution of life before it can arrive at the present state. The estimated value of about 0.7x10

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