Fundamental limits on constraining primordial non-Gaussianity

We study the cosmic variance limit on constraining primordial non-Gaussianity for a variety of theory-motivated shapes. We consider general arguments for 2D and 3D surveys, with a particular emphasis on the CMB. A scale-invariant $N$-point correlator can be measured with a signal-to-noise that naively scales with the square root of the number of observed modes. This intuition generally fails for two reasons. First, the signal-to-noise scaling is reduced due to the blurring of the last scattering surface at short distances. This blurring is caused by the combination of projection and damping, but the loss of signal is not due to exponential decay, as both signal and noise are equally damped. Second, the behavior of the $N$-point correlator in the squeezed and collapsed (for $N>3$) limits can enhance the scaling of the signal-to-noise with the resolution, even with a reduced range of momenta probing these limits. We provide analytic estimates for all $N$-point correlators. We show that blurring affects equilateral-like shapes much more than squeezed ones. We discuss under what conditions the optimistic scalings in the collapsed limit can be exploited. Lastly, we confirm our analytical estimates with numerical calculations of the signal-to-noise for local, orthogonal and equilateral bispectra, and local trispectra. We also show that adding polarization to intensity data enhances the scaling for equilateral-like spectra.