Gravitational redshift and other redshift-space distortions of the
  imaginary part of the power spectrum
  I extend the usual linear-theory formula for large-scale clustering in
redshift-space to include gravitational redshift. The extra contribution to the
standard galaxy power spectrum is suppressed by k_c^{-2}, where k_c=c k/a H (k
is the wavevector, a the expansion factor, and H=\dot{a}/a), and is thus
effectively limited to the few largest-scale modes and very difficult to
detect; however, a correlation, \propto k_c^{-1}, is generated between the real
and imaginary parts of the Fourier space density fields of two different types
of galaxy, which would otherwise be zero, i.e., the cross-power spectrum has an
imaginary part: P_{ab}(k,\mu)/P(k)=(b_a+f\mu^2)(b_b+f\mu^2)
-i(3\Omega_m/2)(\mu/k_c)(b_a-b_b)+\mathcal{O}(k_c^{-2}), where P(k) is the
real-space mass-density power spectrum, b_i are the galaxy biases, \mu is the
cosine of the angle between the wavevector and line of sight, and f=dlnD/dlna
(D is the linear growth factor). The total signal-to-noise of measurements of
this effect is not dominated by the largest scales -- it converges at k~0.05
h/Mpc. This gravitational redshift result is pedagogically interesting, but
naive in that it is gauge dependent and there are other effects of similar form
and size, related to the transformation between observable and proper
coordinates. I include these effects, which add other contributions to the
coefficient of \mu/k_c, and add a \mu^3/k_c term, but don't qualitatively
change the picture. The leading source of noise in the measurement is galaxy
shot-noise, not sample variance, so developments that allow higher S/N surveys
can make this measurement powerful, although it would otherwise be only
marginally detectable in a JDEM-scale survey.
