6 FIG. 4: An ordered list of the Eigenvalues for our cluster covari- ance matrix. evidence for an alternative explanation for the observed cosmic acceleration such as modified gravity. In this pa- per, we provide a first demonstration of our new param- eterization using clusters of galaxies from the SDSS. In detail, we attempt to model the “squashing” of the 2- D correlation function of the C4 cluster sample seen in Figure 5 using the formalism presented herein. We do however caution the reader that we expect the limited size of the DR3 sample to leads to large statistical er- rors, due to a significant shot–noise contribution because of their low number density. However, future cluster and galaxy samples (e.g., LRGs) should provide stronger con- straints and provide a more robust test of these high bulk flow measurements in the literature. In Figure 5, we provide the best fit parameters b and v$_{p}$ for the C4 correlation function presented in Figure 3 and there is as expected a clear anti-correlation be- tween these two parameters because the anisotropic am- plitude is generated by cross-correlations in the density and peculiar velocity fields. The best fit value from Fig- ure 5 is v$_{p}$ = 270 +433 km/s (at the 1 σ level marginalised with other parameters including b ) and is consistent with v$_{p}$ = 0. We do not quote the negative bound of the error on v$_{p}$ as it is below zero and thus has no physical mean- ing. Instead, we quote the upper bound on v$_{p}$ and note that our result is consistent with zero. Our measurement of v$_{p}$ is close to the predicted value of 203 km/s for a WMAP5–normalised ΛCDM model. We propose above that v$_{p}$ is a complementary param- eter for reporting such peculiar velocity measurements. The parameter g Θ , which is equivalent to fσ 8 , is not de- termined precisely without the prior information of A$_{S}$ . But when we report our measurement with v$_{p}$ , there is no uncertainty due to other cosmological parameters which are not determined accurately, as it is equivalent to g ∗ Θ determined statistically from redshift space distortion. The observed value v$_{p}$ at a given redshift is not only in- dependent of bias but also independent of normalisation. Δ C$_{ij}$ = √ 2 N$_{jk}$ (14) C$_{ij}$ = U † $_{ik}$D$_{kl}$ V$_{lj}$, (15) and therefore, if N$_{jk}$ is small, or there are degeneracies within C$_{ij}$ , the inversion will be affected. This problem can be eliminated by performing a Single Value Decom- position (SVD) of the matrix, Before we invert C$_{ij}$ in Eqn. 12, we note that the values of C$_{ij}$ are estimated to limited resolution, tribute to the χ 2 by requiring that λ 2 > √ 2 /N$_{jk}$ . In Figure 4, we rank the eigenvalues ( λ$_{i}$ ) for the in- creasing eigenmodes and see a “kink” in the distribu- tion which we interpret as indicating a transition in the signal–to–noise of the eigenmodes, i.e., only the first ten modes contain most of the signal, while higher-ordered modes are dominated by noise. We therefore remove eigenmodes beyond this kink (with λ$_{i}$ < 0 . 01) where the eigenvalues start to flatten out. where U and V are orthogonal matrices that span the range and the null space of C$_{ij}$ and D$_{kl}$ = λ $^{2}$δ$_{kl}$ , a diag- onal matrix with singular values along the diagonal. In doing the SVD, we select the dominant modes to con- We convert v$_{p}$ measurement into g Θ using A$_{s}$ from WMAP5 ( g Θ : coherent growth factor of peculiar veloc- ity, and it is equivalent to fσ 8 in other parameteriza- tions). With the evolution of g Θ known, dynamics of per- turbations are reconstructed to provide the history of Ψ through the Euler equation. In most theoretical models, the time variation of v$_{p}$ is minimal at these low redshifts discussed here for the C4 sample ( z ≃ 0 . 1), which allows us to ignore the time-derivative part in Eq. 9. Therefore, it is straightforward to transform the coherent evolution of Θ into the coherent evolution of Ψ. If we assume no anisotropic stress, then it is easy to convert to the coher- ent evolution of Φ, g Φ . We are able to determine matter density fluctuations through the Poisson equation. We calculate the coher- ent growth of δ$_{m}$ , g$_{δ}$ = 0 . 7, which is related to g Φ as As discussed in the Introduction, there is recent evi- dence for excessive bulk flow motions compared to the WMAP5-normalised ΛCDM model [18] and therefore, it is important to confirm these results as it may indicate B. Reconstruction of matter density field from v p A. Statistical determination of large scale flow