Galaxy bias and redshift-space distortions

Define matter contrast as density minus its mean divided by that mean. On sufficiently large scales, a deterministic linear bias model sets galaxy contrast equal to b times matter contrast. This is an effective approximation for a selected tracer population, not a fundamental law identifying galaxies with all mass. Bias can depend on redshift, scale, and sample selection, and stochastic contributions need not follow exactly the simplest Poisson prescription.

Observed redshift coordinates differ from real comoving coordinates by peculiar velocity divided by a H along the line of sight. Number conservation contributes a Jacobian between those coordinates. Linear continuity connects coherent velocity divergence to density growth, producing an angular dependence in Fourier space. The familiar compact formula assumes a common line of sight, small perturbations, an appropriate growing mode, and negligible velocity bias. It cannot be applied unchanged inside a virialized galaxy cluster.

A transverse mode has wavevector perpendicular to the line of sight, so the coherent velocity contribution vanishes in this approximation. A radial mode receives the full growth term. Their ratio depends on beta, the growth rate divided by bias, rather than on the matter normalization separately. Angular multipoles summarize this anisotropy; integrating the squared angular factor produces the monopole coefficients below. The angle in this formula is a Fourier-mode direction, not the angular separation of a galaxy pair.

The overall amplitude mixes bias and the matter fluctuation normalization at the bin’s redshift. Redshift-space analyses therefore often constrain $f(z)\sigma_8(z)$ and $b(z)\sigma_8(z)$. Here $\sigma_8(z)$ is the linear matter root-mean-square contrast at redshift z in spheres of radius eight inverse-h megaparsecs; h is H zero divided by one hundred km per second per megaparsec. For scale-independent growth, $\sigma_8(z)=[D(z)/D(0)]\sigma_{8,0}$, with $\sigma_{8,0}$ the present-day value. Converting a fitted bin amplitude to that value requires an assumed growth history and power-spectrum shape.

Random virial velocities smear radial positions on small scales and generate fingers of God, which differ from the coherent linear compression. Nonlinear bias, velocity dispersion, wide-angle geometry, and relativistic projection corrections require additional modeling in their relevant regimes. A wrong fiducial distance-redshift relation also creates anisotropy through the Alcock–Paczynski effect. Separating geometric anisotropy from velocities is part of the inference, rather than something guaranteed by measuring an anisotropic correlation function.

A finite survey multiplies the density field by its angular and radial window, convolving Fourier modes and correlating estimated powers. The measured mean density imposes an integral constraint, while incompleteness and redshift failures alter the effective selection. Random catalogs and forward-convolved models address these effects. More tracer density reduces shot noise, but the number of accessible large-scale modes is limited by volume and geometry. Multiple tracers can exploit shared modes under controlled bias and noise assumptions, without creating an unlimited number of independent universes.