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(dR/dt)2 + kc2 = 8GR2 / 3 + R2 c2 / 3 where R(t) is the scale factor related to the size of the universe, the cosmological constant. It can be expressed in term of the corresponding density by the formula = 8G / c2. 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] |
where H = (/3)1/2c = (dR/dt)/R is the Hubble constant, and R0 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 dh = 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 Ro at t = 0, and expands to infinity as t +. Alternatively, Ro 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) = R0 cosh(Ht) R0 eH|t| as t . |