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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.

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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.

Reference:

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  • $\begingroup$ 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. $\endgroup$
    – Allure
    Jan 19, 2021 at 6:22
  • $\begingroup$ 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. $\endgroup$
    – Allure
    Jan 19, 2021 at 6:32
  • $\begingroup$ @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 $\ell=\hbar/(mc)$). $\endgroup$
    – A.V.S.
    Jan 19, 2021 at 8:07
  • $\begingroup$ 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. $\endgroup$
    – A.V.S.
    Jan 19, 2021 at 8:16

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