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2.3. Standard cosmological solutions

When k = 0 the Friedmann and fluid equations can readily be solved for the equations of state given earlier, leading to the classic cosmological solutions

Equation 8 (8)

Equation 9 (9)

In both cases the density falls as t-2. When k = 0 we have the freedom to rescale a and it is normally chosen to be unity at the present, making physical and comoving scales coincide. The proportionality constants are then fixed by setting the density to be rho0 at time t0, where here and throughout the subscript zero indicates present value.

A more intriguing solution appears for the case of a so-called cosmological constant, which corresponds to an equation of state p = - rho. The fluid equation then gives rhodot = 0 and hence rho = rho0, leading to

Equation 10 (10)

More complicated solutions can also be found for mixtures of components. For example, if there is both matter and radiation the Friedmann equation can be solved be using conformal time tau = integdt / a, while if there is matter and a non-zero curvature term the solution can be given either in parametric form using normal time t, or in closed form with conformal time.