# The definition of $f_{NL}$ and transfer function

To me there seems to be quite a few different definitions of $f_{NL}$ in cosmology and I would like to know if or how they are equivalent. Let me cite at least 3 such,

• One can see the equation 6.71 on page 100 of the book "The Primordial Density Perturbation" by Liddle and Lyth. The closest one can make 6.71 look like the following two is through equations 25.21 and 25.22 - but it still doesn't really become equal to the following.

• One can see equation 2 (page 3) of this paper. In this equation I am hoping that the quantity $\Phi(x)$ called the "Bardeen's curvature perturbation" really what is called $\xi$ in the Liidle and Lyth book or in Dodelson's.

• One can see the definition at the bottom of page 1 in this very famous paper. Here curiously though the equation looks exactly the same as in the paper above, the quantity $\Phi(x)$ is called the "gravitational potential in the matter era"

I wonder if one can hope to somewhat translate between the last two definitions by using the relation $\xi = \frac{2}{3}\Phi \vert_{post\text{ }inflation}$ - but even then the last two definitions are not really "equal".

Further this "discrepancy" between the definitions of $f_{NL}$ seem to be kind of related to the same problem with the definition of the transfer function" ($T(k)$),

• In the first paper cited above, one sees a definition of $T(k)$ in equation 9 and 10.

• But the closest one can get to the above is if one eliminates $\Phi(\vec{k},a)$ between equation 7.7 and 7.5 in Dodelson and then replaces $\Phi_p$ as $\Phi \vert_{post\text{ }inflation}$ and converts that into $\xi$ through $\xi = \frac{2}{3}\Phi \vert_{post\text{ }inflation}$ (..and hope that $\Phi$ in equation 2 and 9 of of the paper is actually $\xi$..)

But still the two definitions of transfer function don't match!

• And the above two definitions are anyway still very far from the definition of $T(k)$ as in equation 8.52 (page 125) of the above cited book by Liddle and Lyth.

It would be very helpful if someone can help reconcile the above!