Natural units ($\hslash =c=1$)

Natural units choose units so that selected fundamental constants have numerical value one. In relativistic quantum field theory the standard choice is $\hslash =c=1$: energy, mass and inverse length then have the same units. This is a convention, not a physical change of the constants; other natural-unit systems may also normalize other constants.

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.

Reduced Planck units choose $\hslash =c=8\pi G=1$. With the reduced Planck mass defined by $M_{\mathrm{Pl}}=(8\pi G{)}^{-1/2}$ after setting $\hslash =c=1$, this convention gives $M_{\mathrm{Pl}}=1$ and the Einstein-Hilbert action coefficient $M_{\mathrm{Pl}}^2/2=1/2$.

For a spatially flat Friedmann-Lemaître-Robertson-Walker metric, the Friedmann equation is then $3H^2=\rho$, where $\rho$ is the total energy density, including any vacuum contribution. With spatial-curvature parameter $k$, it is instead $3(H^2+k/a^2)=\rho$. Setting the unreduced Planck mass to one would set $G=1$, leaving the factor $8\pi$ in the gravitational equation. Keeping these conventions distinct prevents spurious factors in gravitational and inflationary calculations.