= Short-scale variance modulation by local non-Gaussianity
For a long potential approximately constant over the short-scale region, <local-type primordial non-Gaussianity> gives $\varphi_s=(1+2f_{\rm NL}\varphi_{G,L})\varphi_{G,s}$, to first order in the long mode and ignoring short-mode squares. A linear <cosmological Poisson equation> therefore gives
$$
\sigma_s^2=\sigma_{G,s}^2(1+4f_{\rm NL}\varphi_{G,L})
$$
to the same order. The <variance> is conditional on the long mode.
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