2 FIG. 1: Δ ξ ∗ l ( r ) for various CMB experiments priors. In the upper panel, we show the change in ξ ∗ l ( r ) for variations in ω$_{m}$ . In the lower panel, we show the change in ξ ∗ l ( r ) as a function of n$_{S}$ . The thick black curves are based on WMAP priors, while the thin blue curves are for Planck prior. On the y-axis, we focus on the range of scales probed by recent and planned reshift–space distortion mea- surements. decomposition of the correlation function into two vec- tors; one parallel ( π ) to the line–of–sight and the other perpendicular ( σ ) to the line–of–sight. On small scales, any incoherent velocities of galaxies within a single dark matter halo (or cluster) will just add to the cosmological Hubble flow thus causing the famous “Fingers-of-God” (FoG) effect which stretches the 2-D correlation function preferentially in line-of-sight ( π ) direction. These dis- tortions depend on the inner dynamics and structure of halos and therefore, any cosmological information is dif- ficult to distinguish from the halo properties. However, on large scales (outside individual dark matter halos), the peculiar velocities become coherent and follow the linear motion of the matter thus providing crucial information on the formation of large-scale structure [22]. In this paper, we compare predictions for ξ$_{s}$ ( π, σ ) to observations based on the C4 cluster catalogue [23] from the Sloan Digital Sky Survey (SDSS) [24]. Us- ing the Kaiser formulation [22], a theoretical model for ξ$_{s}$ ( π, σ ) is fit to the measured 2-D correlation function in configuration–space, with the ξ$_{s}$ parameterised by a shape dependent part and a coherent evolution compo- nent. We also propose that the 1-D linear velocity disper- sion ( v$_{p}$ ) is a interesting quantity to report when measur- ing redshift–space distortions, and complementary to tra- ditional quantities like β , f or f σ 8 discussed recently [11– 13], as it is independent of both bias and the normalisa- tion method. Therefore, the measured v$_{p}$ provides an unbiased tracer of the evolution of structure formation. tities [31]. This consistency test was developed further to show that, even in the Kaiser limit, the description of ξ$_{s}$ ( σ, π ) in linear theory can be modified due to the correlation between the ”squashing” (in the σ direction) and dispersion effects (in the π direction) [32]. With the assumption of a Gaussian pair-wise velocity distribu- tion function, the dispersion effect smears into the Kaiser limit description of ξ$_{s}$ ( σ, π ) at around the percent level which for our present work can be ignored. Thus we adopt the Kaiser limit for the description of ξ$_{s}$ ( σ, π ) in linear regime while considering dispersion effect as a sys- tematic uncertainty. We introduce below a new param- eterisation of ξ$_{s}$ ( σ, π ) in terms of the cosmological pa- rameters and construct a method to measure the mean velocity dispersion v$_{p}$ in a model independent way. II. STATISTICAL DETERMINATION OF PECULIAR VELOCITY A. Model independent parameterisation of power spectra The redshift–space two-point correlation function of mass tracers ( ξ$_{s}$ ( σ, π )) is an anisotropic function [22]. On small scales, it is elongated in the π -direction by the “Fingers-of-God” effect, while on large scales, the gravi- tational infall into overdense regions preferentially com- presses the correlation function in the σ direction. There- fore, peculiar velocities can be statistically measured by analyzing the observed anisotropic pattern of ξ$_{s}$ ( σ, π ) in both the linear and non-linear regimes. ξ$_{s}$ ( σ, π ) is derived from the convolution of ξ ( r ) with a probability distribution function of peculiar velocities along the line of sight, which is usually called the stream- ing model [25]. Even with the simplest form of a Gaussian probability distribution, the streaming model describes the suppression effect on ξ$_{s}$ ( σ, π ) on small scales. In the linear regime, the density fluctuations and pe- culiar velocity are coherently evolved through the conti- nuity equation, which is known as the Kaiser limit. Thus the known correlation function of ξ ( r ) from the linear perturbation theory developed by gravitational instabil- ity is uniquely transformed into ξ$_{s}$ ( σ, π ) [26–30]. The large scale limit of the streaming model is con- sistent with the Kaiser limit [22], when both the density and peculiar velocity fields are treated as statistical quan- The discovery of cosmic acceleration has prompted rapid progress in theoretical cosmological research and prompted many authors to propose modification to the law of gravity beyond our solar system. For example, some theories based upon General Relativity can be mod- ified by screening (or anti-screening) the mass of gravi- tationally bound objects [3], while others include a non- trivial dark energy component (e.g. interacting dark en- ergy [33, 34], or clumping dark energy [35]) thus break- ing the dynamical relations between density fluctuations