Joint analyses of 2D CMB lensing and 3D galaxy clustering in the
  spherical Fourier-Bessel basis
  Cross-correlating cosmic microwave background (CMB) lensing and galaxy
clustering has been shown to greatly improve the constraints on the local
primordial non-Gaussianity (PNG) parameter $f_{\rm NL}$ by reducing sample
variance and also parameter degeneracies. To model the full use of the 3D
information of galaxy clustering, we forecast $f_{\rm NL}$ measurements using
the decomposition in the spherical Fourier-Bessel (SFB) basis, which can be
naturally cross-correlated with 2D CMB lensing in spherical harmonics. In the
meantime, such a decomposition would also enable us to constrain the growth
rate of structure, a probe of gravity, through the redshift-space distortion
(RSD). As a comparison, we also consider the tomographic spherical harmonic
(TSH) analysis of galaxy samples with different bin sizes. Assuming galaxy
samples that mimic a few future surveys, we perform Fisher forecasts using
linear modes for $f_{\rm NL}$ and the growth rate exponent $\gamma$,
marginalized over standard $\Lambda$ cold dark matter ($\Lambda$CDM)
cosmological parameters and two nuisance parameters that account for clustering
bias and magnification bias. Compared to TSH analysis using only one bin, SFB
analysis could improve $\sigma(f_{\rm NL})$ by factors 3 to 12 thanks to large
radial modes. With future wide-field and high-redshift photometric surveys like
the LSST, the constraint $\sigma(f_{\rm NL}) < 1$ could be achieved using
linear angular multipoles up to $\ell_{\rm min}\simeq 20$. Compared to using
galaxy auto-power spectra only, joint analyses with CMB lensing could improve
$\sigma(\gamma)$ by factors 2 to 5 by reducing degeneracies with other
parameters, especially the clustering bias. For future spectroscopic surveys
like the DESI or $\textit{Euclid}$, using linear scales, $\gamma$ could be
constrained to $3\,\%$ precision assuming the GR fiducial value.
