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

Question

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.

Answer 1

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:

• Baumann, Daniel. TASI lectures on inflation. (2009), arXiv:0907.5424.