Shot noise in multi-tracer constraints on $f_\text{NL}$ and relativistic
  projections: Power Spectrum
  Multiple tracers of the same surveyed volume can enhance the signal-to-noise
on a measurement of local primordial non-Gaussianity and the relativistic
projections. Increasing the number of tracers comparably increases the number
of shot noise terms required to describe the stochasticity of the data.
Although the shot noise is white on large scales, it is desirable to
investigate the extent to which it can degrade constraints on the parameters of
interest. In a multi-tracer analysis of the power spectrum, a marginalization
over shot noise does not degrade the constraints on $f_\text{NL}$ by more than
$\sim 30$% so long as halos of mass $M\lesssim 10^{12}M_\odot$ are resolved.
However, ignoring cross shot noise terms induces large systematics on a
measurement of $f_\text{NL}$ at redshift $z<1$ when small mass halos are
resolved. These effects are less severe for the relativistic projections,
especially for the dipole term. In the case of a low and high mass tracer, the
optimal sample division maximizes the signal-to-noise on $f_\text{NL}$ and the
projection effects simultaneously, reducing the errors to the level of $\sim
10$ consecutive mass bins of equal number density. We also emphasize that the
non-Poissonian noise corrections that arise from small-scale clustering effects
cannot be measured with random dilutions of the data. Therefore, they must
either be properly modeled or marginalized over.
