# Correlation functions in cosmology

I'm reading an article about Non-Gaussianity of Large-Scale Cosmic Microwave Background (link) and the authors write that the n-point correlation function of $$e^{\varphi(x)}$$ where $$\varphi(x)$$ is a random Gaussian field can be written as $$\begin{equation} \langle e^{\varphi(x_1)}\ldots e^{\varphi(x_N)}\rangle=e^{\frac{1}{2}\int dx dyJ(x)\langle\varphi(x)\varphi(y)\rangle J(y)} \end{equation}$$ where $$J(x)=\sum_i^N \delta(x-x_i)$$. I can't understand how to get this relation and what is its physical meaning.

It's perhaps more illuminating to check out the finite version. Let $$X\in\mathbb R^N$$ be a random Gaussian vector of dimension $$N$$, $$0$$ mean, and covariance matrix $$C$$ ie $$\langle X_i X_j\rangle = C_{ij}$$ with $$C$$ positive definite. All the statistics is captured by $$C$$. Let $$J \in\mathbb R$$, you can calculate: $$\langle e^{J^TX}\rangle = \int e^{J^T X}\frac{e^{-X^TC^{-1}X/2}}{\sqrt{(2\pi)^N\det(C)}}d^N X \\ = e^{\frac{1}{2}J^TCJ}\int \frac{e^{-\frac{1}{2}(X-CJ)^TC^{-1}(X-CJ)}}{\sqrt{(2\pi)^N\det(C)}}d^N X \\ = e^{\frac{1}{2}J^TCJ}$$ (completed the square and integration variable translation)
To see the analogy better, you can rewrite: $$\langle e^{\phi(x_1)}...e^{\phi(x_n)} \rangle = \langle e^{\int dx J(x)\phi(x)} \rangle$$ so you just need to formally set the dimension to the continuum and replace sums in the the dot product by integrals.
The importance of this is that generally you write $$\mathcal Z(J) = \langle e^{\int dx J(x)\phi(x)} \rangle$$ the generating functional as it gives you by successive differentiation at $$0$$ the Green's functions which correspond to correlators.