Linear Redshift Distortions: A Review

TL;DR

This review comprehensively surveys linear redshift distortions as a probe of cosmological parameters, focusing on the parameter $β=f(Ω_0)/b$ and its relation to the matter density and galaxy bias. It develops the theory via the linear redshift-distortion operator ${\bf S}$, contrasting plane-parallel and radial forms, and discusses the Local Group frame and selection-function issues that affect redshift-space statistics. Three principal measurement methods are surveyed—ratioing redshift to real-space power, quadrupole-to-monopole ratios, and maximum-likelihood analyses—illustrated with an explicit example comparing optical and IRAS galaxy samples and highlighting the impact of nonlinearities. The paper compiles measurements up to 1997, revealing a distinction between optical and IRAS-based $β$ values (approximately $β_{optical} \approx 0.52\pm0.26$ and $β_{IRAS} \approx 0.77\pm0.22$), which translates into a relative bias $b_{optical}/b_{IRAS} \approx 1.5$, and thus constrains $Ω_0$ under reasonable bias assumptions. It also discusses non-linear corrections (e.g., fingers-of-God), translinear modeling with Zel'dovich theory, and the potential for cosmological redshift distortions to inform geometry through Alcock–Paczynski-type tests at higher redshift, though with challenges from bias evolution and degeneracies with peculiar velocities.

Abstract

Redshift maps of galaxies in the Universe are distorted by the peculiar velocities of galaxies along the line of sight. The amplitude of the distortions on large, linear scales yields a measurement of the linear redshift distortion parameter, which is $β\approx Ω_0^{0.6}/b$ in standard cosmology with cosmological density $Ω_0$ and light-to-mass bias $b$. All measurements of $β$ from linear redshift distortions published up to mid 1997 are reviewed. The average and standard deviation of the reported values is $β_{optical} = 0.52 \pm 0.26$ for optically selected galaxies, and $β_{IRAS} = 0.77 \pm 0.22$ for IRAS selected galaxies. The implied relative bias is $b_{optical}/b_{IRAS} \approx 1.5$. If optical galaxies are unbiased, then $Ω_0 = 0.33^{+0.32}_{-0.22}$, while if IRAS galaxies are unbiased, then $Ω_0 = 0.63^{+0.35}_{-0.27}$.

Figures (8)

• Figure 1: A spherical overdensity appears distorted by peculiar velocities when observed in redshift space. On large (linear) scales the overdensity appears squashed along the line of sight, while on small (nonlinear) scales fingers-of-god appear. At left, the overdensity is far from the observer (who is looking upward from somewhere way below the bottom of the diagram), and the distortions are effectively plane-parallel. At right, the overdensity is near the observer (large dot), and the large scale distortions appear kidney-shaped, while the finger-of-god is sharpened on the end pointing at the observer. The observer shares the infall motion towards the overdensity. A similar diagram appears in Kaiser (1987).
• Figure 2: Detail of how peculiar velocities lead to the redshift distortions illustrated in Figure \ref{['fogs']}. The dots are 'galaxies' undergoing infall towards a spherical overdensity, and the arrows represent their peculiar velocities. At large scales, the peculiar velocity of an infalling shell is small compared to its radius, and the shell appears squashed. At smaller scales, not only is the radius of a shell smaller, but also its peculiar infall velocity tends to be larger. The shell that is just at turnaround, its peculiar velocity just cancelling the general Hubble expansion, appears collapsed to a single velocity in redshift space. At yet smaller scales, shells that are collapsing in proper coordinates appear inside out in redshift space. The combination of collapsing shells with previously collapsed, virialized shells, gives rise to fingers-of-god.
• Figure 3: Contour plots of the redshift space two-point correlation function $\xi^{s}$ as a function of separations $s_{\hbox{$/$ $/$}}$ and $s_\perp$ parallel and perpendicular to the line of sight in: (left) the IRAS QDOT and 1.2 Jy redshift surveys, merged over the angular region of the sky common to both surveys; and (right) the optical Stromlo-APM survey. In each case the region within $25 h^{-1} {{\rm Mpc}}$ of the Milky Way has been excluded, so as to eliminate bias from the local overdensity. A near minimum variance pair weighting has been applied. The thick contour signifies $\xi^{s} = 0$, and other contours are logarithmically spaced at intervals of 0.5 dex above $10^{-1.5}$ (the left panel also shows one negative contour, at $-10^{-1.5}$). Shading is graduated at intervals of 0.1 dex above $10^{-1.5}$. The correlation function here has been smoothed over pair separation $s = ( s_\perp^2 + s_{\hbox{$/$ $/$}}^2 )^{1/2}$ with a tophat window of width $0.2$ dex, and over angles $\theta = \tan^{-1} ( s_\perp / s_{\hbox{$/$ $/$}} )$ to the line of sight with a Gaussian window with a 1$\sigma$ width of $10^\circ$.
• Figure 4: Problem: From the linearized continuity equation (\ref{['cty']}) show that, in the linear regime, the peculiar velocity ${\hbox{\boldmath$v$}}$ of a particle (galaxy) at the present time is related to its comoving displacement $\Delta {\hbox{\boldmath$r$}}$ (measured in velocity units) since the Big Bang by ${\hbox{\boldmath$v$}} = \beta \Delta {\hbox{\boldmath$r$}} \ .$ In deriving this result you will need to assume that the density field is initially uniform, and that the velocity is the gradient of a potential. Amongst other things, you should find that in the linear regime particles move in straight lines.
• Figure 5: A wave of amplitude $\delta({\hbox{\boldmath$k$}})$ in real space (thin line) appears as a wave with enhanced amplitude $\delta^{s}({\hbox{\boldmath$k$}})$ in redshift space (thick line) because of peculiar velocities (arrows). If the wavevector ${\hbox{\boldmath$k$}}$ is along the line of sight, then the amplification factor is $1 + \beta$. More generally, a wave that is angled to the line of sight appears amplified in redshift space by a factor $1 + \beta \mu_{\hbox{\boldmath$k$}}^2$, equation (\ref{['dsk']}), where $\mu_{\hbox{\boldmath$k$}} \equiv \hat{{\hbox{\boldmath$z$}}} . \hat{{\hbox{\boldmath$k$}}}$ is the cosine of the angle between the wavevector ${\hbox{\boldmath$k$}}$ and the line of sight ${\hbox{\boldmath$z$}}$.