# What are the dimensions of the inflation slow-roll parameters?

The inflation slow-roll parameters are:

$$\epsilon = \frac{M_{pl}^2}{2}(\frac{V'}{V})^2$$
$$\eta=M_{pl}^2 \frac{V''}{V}$$

What are the dimensions of $$\epsilon$$ and $$\eta$$? What about $$V$$ and its derivatives? $$M_{pl}$$ clearly has dimensions of mass.

The slow-roll conditions are usually given as $$\epsilon \ll 1$$ and $$\eta \ll 1$$, which suggests that they are dimensionless, but in that case I am not sure what the dimensions of $$V$$ and its derivatives are.

Both parameters are indeed dimensionless. Dimension of $$V$$ (which in units $$c=\hbar=1$$ has the dimension of $$\text{mass}^4$$, the dimension of energy density) is irrelevant since it enters both in numerator and denominator. So, the dimension of $$\frac{V'}{V}$$ is simply the inverse dimension of the scalar field $$\phi$$, (since $$V'\equiv\frac{d V(\phi)}{d\phi}$$) which has the dimension of mass.
Note, that both quantities could also be expressed as a purely geometric dimensionless quantities relating to the evolution of Hubble parameter $$H$$: $$\epsilon = - \frac{\dot H}{H^2} = - \frac{d\ln H}{dN},$$ $$\eta =\epsilon - \frac{1}{2 \epsilon} \frac{d \epsilon}{dN} ,$$ where $$dN=H dt$$ is dimensionless, with $$N$$ measuring the number of $$e$$-foldings of exponential expansion.
• Do you know of any reference that explicitly treats the dimension of $V$, preferably in SI units? If it's in Daniel Bauman's lecture notes, I can't find it. I can't find it in other resources I've looked at either. Commented Jan 19, 2021 at 6:22
• For that matter, why is $m^4$ the dimension of energy density? Energy density should intuitively have dimensions $J/m^3$. $J$ is related to mass by $E = mc^2$, but that still leaves the length cubed in the denominator. Commented Jan 19, 2021 at 6:32
• @Allure: For that matter, why is $m^4$ the dimension of energy density? Energy density should have dimension $\text{Energy}/\text{volume}$ in units $\hbar=c=1$ energy has dimension $m^1$ and volume $m^{-3}$.This is not GR units (which have $G=c=1$), but HEP units! Here length has dimension of inverse mass (as a check you could use SI definition of Compton length $\ell=\hbar/(mc)$). Commented Jan 19, 2021 at 8:07
• If it's in Daniel Bauman's lecture notes, I can't find it Baumann's Appendix D2 is a reference to expressing $\epsilon$ and $\eta$ as a geometric quantities independent of specific unit system used for field equation. Commented Jan 19, 2021 at 8:16