Primordial non-Gaussianity and analytical formula for Minkowski functionals of the cosmic microwave background and large-scale structure

We derive analytical formulae for the Minkowski functionals of the cosmic microwave background (CMB) and large-scale structure (LSS) from primordial non-Gaussianity. These formulae enable us to estimate a nonlinear coupling parameter, f(NL), directly from the CMB and LSS data without relying on numerical simulations of non-Gaussian primordial fluctuations. One can use these formulae to estimate statistical errors on fNL from Gaussian realizations, which are much faster to generate than non-Gaussian ones, fully taking into account the cosmic/sampling variance, beam smearing, survey mask, etc. We show that the CMB data from the Wilkinson Microwave Anisotropy Probe should be sensitive to vertical bar f(NL)vertical bar similar or equal to 40 at the 68% confidence level. The Planck data should be sensitive to vertical bar f(NL)vertical bar similar or equal to 20. As for the LSS data, the late-time non-Gaussianity arising from gravitational instability and galaxy biasing makes it more challenging to detect primordial non-Gaussianity at low redshifts. The late-time effects obscure the primordial signals at small spatial scales. High-redshift galaxy surveys at z > 2 covering similar to 10 Gpc(3) volume would be required for the LSS data to detect vertical bar f(NL)vertical bar similar or equal to 100. Minkowski functionals are nicely complementary to the bispectrum because the Minkowski functionals are defined in real space and the bispectrum is defined in Fourier space. This property makes the Minkowski fianctionals a useful tool in the presence of real-world issues such as anisotropic noise, foreground, and survey masks. Our formalism can be easily extended to scale-dependent f(NL).