7 to revisit these measurements using larger datasets and different tracers of the density field. As discussed in Section II-C, our measurement of v$_{p}$ is correlated with g$_{b}$ , which is the combination of b and δ$_{m}$ , since the observed anisotropic shape of the 2-D cor- relation function in redshift–space is generated by a cross correlation between the density field and peculiar veloc- ities. However, one of the important implications of our method is that we can measure v$_{p}$ without knowing how to decompose g$_{b}$ , and thus without the uncertainty of determining b . There are however some caveats to our analysis. We do not analyse our data in Fourier space, but in config- uration space. Small scales have been removed from our data analysis ( < 10Mpc), to ensure that the FoG effect will not contaminate our results. Our methodology is in- sensitive to a possible shape dependence at large scales for any exotic reason; scale dependent later time growth (e.g. f(R) gravity models [41]), or scale–dependent bias at large scales. In follow-up studies, we will measure the redshift–space distortions in Fourier space to test the effect of small–scales on our results. In addition, the for- mulation to derive ξ$_{s}$ used in this paper can be slightly biased due to the dispersion effect studied in [32]. This ef- fect is not parameterised properly here, but the reported level of uncertainty is approximately 5% which is much smaller than the statistical errors on our present mea- surements. Therefore, we dismiss this shift here but it is worth revisiting this issue in the future to know how to incorporate this effect in a new parameterisation. Finally, Song and Percival [13] recently proposed a method to re-construct the structure formation observ- ables from Θ measurements. Although it is not yet es- timated precisely in that narrow range of measured val- ues, we apply their methodology in practice. From the observed coherent evolution of Θ at z = 0 . 1 from the DR3 C4 clusters, we re-construct Ψ, Φ and δ$_{m}$ . We then find that bias can be derived from the estimated δ$_{m}$ and the measured g ∗ $_{b}$. Here, for the first time, we estimate bias from peculiar velocity measurements only. The esti- mated values are reasonable at b = g ∗ $_{b}$/g$_{δ}$ = 2 . 9 ± 0 . 8. It is not precise measurement yet, as the time variation is ignored, but we will revisit this in a following paper. The authors would like to thank Nick Kaiser, Kazuya Koyama and Will Percival for helpful conversations, Prina Patel for useful suggestions on the presentation of this article, and the referee for helpful comments. Y-SS, RCN and CGS are grateful for support from STFC . Acknowledgments IV. DISCUSSION We outline in this paper a new theoretical model for ξ$_{s}$ ( π, σ ), the 2-D two–point correlation function in configuration–space, which allows us to constrain the bulk flow motion of matter on large scales. We also propose that the 1-D linear velocity dispersion ( v$_{p}$ ) is a interesting quantity to report when measuring redshift– space distortions. We demonstrate this method using C4 clusters from the SDSS and find a value for v$_{p}$ that is consistent with a WMAP5–normalised ΛCDM cosmol- ogy (within our large statistical errors). Our observed value for these bulk flows is marginally inconsistent with other recent observations in the literature, which find an excess flow compared to a WMAP5–normalised ΛCDM model [18–20]. We do not discuss this further as we plan g$_{δ}$ = ag Φ , if no modified Poisson equation is assumed. Finally, the estimated g$_{δ}$ can be used to derive bias using measured g ∗ $_{b}$. Through fitting to the redshift distortion effect, we extract both g ∗ b and g ∗ $_{Θ}$. The density fluctu- ation evolution g$_{δ}$ is estimated only from g ∗ $_{Θ}$, and the other measurement g ∗ b is not yet used. The combination of the estimated g$_{δ}$ and the measured g ∗ Θ provides bias from b = g ∗ $_{b}$/g$_{δ}$ = 2 . 9 ± 0 . 8, which is fully consistent with our expectations for such massive clusters of galaxies in the C4 sample. FIG. 5: The 2-D contours between b and v$_{p}$ with the DR3 cluster sample. [1] S. Perlmutter et al. (Supernova Cosmology Project), As- trophys. J. 517 , 565 (1999), astro-ph/9812133. [2] A. G. Riess et al. (Supernova Search Team), Astron. J. 116 , 1009 (1998), astro-ph/9805201.