4 few percent effect with the projected Planck priors. In the bottom panel of Fig. 1, the dependence of ξ ∗ l ( r ) on n$_{S}$ is given for both WMAP5 and Planck priors. The overall shifting can be re-scaled by adjusting the pivot point to the effective median scale of the survey. With the measured WMAP5 prior of Δ n$_{S}$ = 0 . 015 [36], we ex- pect variations of a few percent on the shape, while for an estimated Planck prior of Δ n$_{S}$ = 0 . 0071 [37], we expect ξ ∗ l ( r ) to be nearly invariant to n$_{S}$ . The shape of ξ ∗ l ( r ) is affected maximally during the intermediate epoch, from horizon crossing to the matter-radiation epoch. The de- cay rate of the inhomogeneities differs by the ratio be- tween matter and radiation energy densities. Once CMB constraints are placed on the shape part of Eqn. 6, the coherent history of structure formation is obtained from the anisotropic moment of ξ$_{s}$ ( σ, π ). Even though both g ∗ b i and g ∗ Θ$_{m}$ i weight the evolution sector simultaneously, their contribution to ξ$_{s}$ ( σ, π ) are differ- ent, which enables us to discriminate g ∗ Θ$_{m}$ i from g ∗ $_{b i}$. In the monopole moment, g ∗ b i is the dominant component since g ∗ b i > g ∗ Θ$_{m}$ i unless their is an excessive bulk flow. Thus the variation of g ∗ b i generates a near isotropic am- plification as illustrated in the top panel of Figure 2. In the quadrupole moment, the cross-correlation between δ$_{m}$ and Θ$_{m}$ is leading order. The reversed sign of the quadrupole moment results in the squashing effect, and it is sensitive to the variation of g ∗ Θ$_{m}$ i as the cross- correlation is the leading order. In the bottom panel of Figure 2 the variation of g ∗ Θ$_{m}$ i mainly contributes to the anisotropic moment. It is this signal which allows both g ∗ b i and g ∗ Θ$_{m}$ i to be probed separately using the anisotropic structure of ξ$_{s}$ ( σ, π ). The contribution from the term having peculiar velocity autocorrelation is not significant if excessive bulk flows are excluded. C. Implication for cosmology from measuring g b i and g ∗ Θ$_{m}$ i A measurement of g ∗ Θ$_{m}$ i is equivalent to the quantity fσ mass 8 [14] and therefore, an excellent test of dark en- ergy models (where f is the logarithmic derivative of the linear growth rate and σ mass 8 is the root-mean-square mass fluctuation in spheres with radius 8 h − $^{1}$Mpc). While cosmological test of g ∗ Θ$_{m}$ i are free from bias, which is notoriously difficult to measure accurately in a model– independent way, the reported value of g ∗ Θ$_{m}$ i does depend on the normalization which is also poorly constrained (i.e., primordial amplitude) or model–dependent (i.e., σ mass 8 ). FIG. 2: The ξ$_{s}$ ( σ, π ) correlation function. We plot for three con- tours of ξ$_{s}$ ( σ, π ) = 1 , 0 . 1 , 0 . 01 (from the inner to outer contour). In the upper panel, the solid curves are for a ΛCDM cosmology, while the dash and dotted curves are for models with g$_{b}$ = 1 . 5 and 2 respectively. In the lower panel, the solid curves are for a ΛCDM cosmology, while the dash and dotted curves are for models with g$_{Θ}$$_{m}$ = 1 . 5 and 2 respectively. where P$_{l}$ ( µ ) is the Legendre polynomial and the spherical harmonic moment ξ ∗ l ( r ) is given by, ξ ∗ l ( r ) = ∫ k $^{2}$dk 2 π 2 D ∗ $_{m}$( k ) j$_{l}$ ( kr ) , (7) where j$_{l}$ is a spherical Bessel function and ∗ denotes scal- ing of the shape factor with A ∗ 2 S . As discussed above, we ignore the effect on ξ$_{s}$ ( σ, π ) of the small-scale velocity dispersions within a single dark matter halo [32] as the effect is only a few percent, and split Eqn. 6 into a shape–dependent part ( ξ ∗ l ( r )), which is determined by the cosmological parameters ( n$_{S}$ and ω$_{m}$ ), and a coherent evolution component, which is pa- rameterised by g ∗ $_{b i}$, g ∗ Θ$_{m}$ i at the targeted redshift z$_{i}$ . The shape part is therefore almost completely determined by CMB priors, while the coherent evolution of structure formation can be determined from fitting ξ$_{s}$ ( σ, π ), in redshift–space, as a function of redshift. In Figure 1, we present the effect of CMB priors on the value of ξ ∗ l ( r ). In the top panel of Fig. 1, we provide the expected variation in ξ ∗ l ( r ) from varying ω$_{m}$ . We see that varying a$_{eq}$ causes greater tilting in the shape of ξ ∗ l ( r ), since larger scale modes can come into the hori- zon earlier. In addition to this contribution, the overall amplitude of ξ ∗ l ( r ) depends on ω$_{m}$ by a weighted trans- formation between D Φ ( k ) and D$_{m}$ ( k ). Considering the marginalisation over CMB priors, we expect a discrep- ancy of ≃ 5% with WMAP5 measurements, and just a Thus, we introduce a more convenient parameterisa- tion of peculiar velocity which is independent of these normalization issues. The measured g ∗ Θ$_{m}$ i (in the redshift bin i ) which can be translated into the one-dimensional (1-D) velocity dispersion in that redshift bin ( v i $_{p}$) by, v i 2 p = g ∗ 2 Θ$_{m}$ i H 2 3 ∫ ∞ 0 dk k D$_{m}$ ( k ) dk. (8)