Analysis method for 3D power spectrum of projected tensor field with
  fast estimator and window convolution modelling: an application to intrinsic
  alignments
  Rank-2 tensor fields of large-scale structure, e.g. a tensor field inferred
from shapes of galaxies, open up a window to directly access 2 scalar, 2 vector
and 2 tensor modes, where the scalar fields can be measured independently from
the standard density field that is traced by distribution of galaxies. Here we
develop an estimator of the multipole moments of coordinate-independent power
spectra for the three-dimensional tensor field, taking into account the
projection of the tensor field onto plane perpendicular to the line-of-sight
direction. To do this, we find that a convenient representation of the power
spectrum multipoles can be obtained by the use of the associated Legendre
polynomials in the form which allows for the fast Fourier transform estimations
under the local plane-parallel (LPP) approximation. The formulation also allows
us to obtain the Hankel transforms to connect the two-point statistics in
Fourier and configuration space, which are needed to derive theoretical
templates of the power spectrum including convolution of a survey window. To
validate our estimators, we use the simulation data of the projected tidal
field assuming a survey window that mimics the BOSS-like survey footprint. We
show that the LPP estimators fairly well recover the multipole moments that are
inferred from the global plane-parallel approximation. We find that the survey
window causes a more significant change in the multipole moments of projected
tensor power spectrum at $k\lesssim 0.1\,h{\rm Mpc}^{-1}$ from the input power
spectrum, than in the density power spectrum. Nevertheless, the theoretical
predictions including the survey window effects match the multipole moments
measured from the simulations. The analysis method presented here paves the way
for a cosmological analysis using three-dimensional tensor-type tracers of
large-scale structure for current and future surveys.
