Scale factor vs. time for various cosmological models. Models are all chosen to have R = 1, now, and a Hubble constant = 65 km/s/Mpc (so they all have the same slope now). In terms of the density parameter (defined in equation 20.45), the models are (from top to bottom):

(nM = o, nA = i),(flM = o, nA = 0) (nM = i, nA = 0)

(HM = 2, Oa = 0). [© Edward L.Wright, used with permission]

like those with a mass density, pEFF = —A/8-n-G, constant in space and time. A positive value of A behaves like a negative effective mass density (repulsion), and a negative value of A behaves like a positive effective mass density (attraction). So, for positive A, we would expect the expansion to accelerate, and for negative A we would expect the expansion to stop and reverse.

Note that for positive A, this corresponds to an exponential growth in the expansion rate. The flat model with zero density and a non-zero A is sometimes called the deSitter model.

For positive curvature (k = +1), there are

solutions only for A > 0, R(t) > ( — ) , in which case

where t1 is any convenient constant by which to scale the result. If we choose it to be t0, the current age of the universe, taking the constant in front to be equal to unity makes R = 1 now. So, the flat model which describes our universe would be cosh

where t is zero when R(t) has its minimum value. For negative curvature (k = —1)

sin h

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