  Large-scale Fourier modes of the cosmic density field are of great value for
learning about cosmology because of their well-understood relationship to
fluctuations in the early universe. However, cosmic variance generally limits
the statistical precision that can be achieved when constraining model
parameters using these modes as measured in galaxy surveys, and moreover, these
modes are sometimes inaccessible due to observational systematics or
foregrounds. For some applications, both limitations can be circumvented by
reconstructing large-scale modes using the correlations they induce between
smaller-scale modes of an observed tracer (such as galaxy positions). In this
paper, we further develop a formalism for this reconstruction, using a
quadratic estimator similar to the one used for lensing of the cosmic microwave
background. We incorporate nonlinearities from gravity, nonlinear biasing, and
local-type primordial non-Gaussianity, and verify that the estimator gives the
expected results when applied to N-body simulations. We then carry out
forecasts for several upcoming surveys, demonstrating that, when reconstructed
modes are included alongside directly-observed tracer density modes,
constraints on local primordial non-Gaussianity are generically tightened by
tens of percents compared to standard single-tracer analyses. In certain cases,
these improvements arise from cosmic variance cancellation, with reconstructed
modes taking the place of modes of a separate tracer, thus enabling an
effective "multitracer" approach with single-tracer observations.
