Constraining primordial non-Gaussianity with cosmological weak lensing:
  shear and flexion
  We examine the cosmological constraining power of future large-scale weak
lensing surveys on the model of \emph{Euclid}, with particular reference to
primordial non-Gaussianity. Our analysis considers several different estimators
of the projected matter power spectrum, based on both shear and flexion, for
which we review the covariances and Fisher matrices. The bounds provided by
cosmic shear alone for the local bispectrum shape, marginalized over
$\sigma_8$, are at the level of $\Delta f_\mathrm{NL} \sim 100$. We consider
three additional bispectrum shapes, for which the cosmic shear constraints
range from $\Delta f_\mathrm{NL}\sim 340$ (equilateral shape) up to $\Delta
f_\mathrm{NL}\sim 500$ (orthogonal shape). The competitiveness of cosmic
flexion constraints against cosmic shear ones depends on the galaxy intrinsic
flexion noise, that is still virtually unconstrained. Adopting the very high
value that has been occasionally used in the literature results in the flexion
contribution being basically negligible with respect to the shear one, and for
realistic configurations the former does not improve significantly the
constraining power of the latter. Since the flexion noise decreases with
decreasing scale, by extending the analysis up to $\ell_\mathrm{max} = 20,000$
cosmic flexion, while being still subdominant, improves the shear constraints
by $\sim 10%$ when added. However on such small scales the highly non-linear
clustering of matter and the impact of baryonic physics make any error
estimation uncertain. By considering lower, and possibly more realistic, values
of the flexion intrinsic shape noise results in flexion constraining power
being a factor of $\sim 2$ better than that of shear, and the bounds on
$\sigma_8$ and $f_\mathrm{NL}$ being improved by a factor of $\sim 3$ upon
their combination. (abridged)
