Planck units ($\hslash =c=G=1$)

Planck units use fundamental constants to set natural units of mass, length and time. The displayed convention sets the ordinary Planck mass to one. The related reduced Planck units convention instead normalizes the reduced Planck mass.

Here the reduced Planck mass $M_{\mathrm{Pl}}=(8\pi G{)}^{-1/2}$ is one, so the Friedmann equation for an energy density $\rho$ is $3H^2=\rho$. This differs by a factor of $\sqrt{8\pi }$ from setting the ordinary Planck mass to one; the convention must be stated in inflationary amplitudes.