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

TL;DR

This paper extends linear theory to include gravitational redshift effects in galaxy clustering, predicting a small but potentially measurable imaginary component in the cross-power spectrum of different galaxy types, dominated by shot noise.

Contribution

It introduces a new theoretical framework for gravitational redshift effects in redshift-space clustering, highlighting the imaginary part of the cross-power spectrum and its detectability.

Findings

Imaginary part of cross-power spectrum is proportional to galaxy bias differences.

Signal-to-noise converges at scales around 0.05 h/Mpc.

Shot noise dominates measurement noise, not sample variance.

Abstract

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.