In the context of cosmology, what is meant by "..arbitrary quadratic non-Gaussianity i.e non-Gaussianity that is described to leading order by a 3-point function.."? (.."quadratic non-Gaussianity" sounds like an oxymoron and isn't 3-point function always the lowest Green's function that will see the non-Gaussianity?..) And why is this "non-local non-Gaussianity"?

One defines the "small-scale density field", $\delta_s (x)$, as local fluctuations around the coarse-grained field $\delta_L(x)$ as $\delta_s (x) = \delta(x) - \delta_L(x)$ and let $\sigma_s^2(x) = \langle\delta_s ^2\rangle$. Then define a parameter $y(x) = \frac{1}{2}(\frac{\delta_s ^2 (x)}{\sigma _s ^2} - 1 )$

Given the definitions above, apparently it's a "famous" relation that,

$$\langle\delta _s ^2(x)\rangle\vert _ {\xi(x)} = \sigma_s^2[1+4f_{NL}\xi(x)] + O(f_{NL}^2)$$

where $\xi(x)$ is the "Bardeen potential" which I guess is the same as the conserved curvature perturbation. ($x$ here always refers to positions on the spatial slice)

I would like to know of a derivation of the above relationship. I haven't been able to trace it anywhere. Is it there in the 2009 book by Liddle and Lyth? (..may be garbed in some different notation?..)


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