Gravitational redshift and other redshift-space distortions of the imaginary part of the power spectrum

Understand

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.