Friedmann Equations - Rescaled Friedmann Equation

Rescaled Friedmann Equation

Set $\tilde{a}=\frac{a}{a_0}, \;\rho_c=\frac{3H_0^2}{8\pi G},\; \Omega=\frac{\rho}{\rho_c},\; t=\frac{\tilde{t}}{H_0},\; \Omega_c=-\frac{kc^2}{H_0^2 a_0^2}\;$ , where and are separately the scale factor and the Hubble parameter today. Then we can have

where . For any form of the effective potential, there is an equation of state that will produce it.

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