3 and peculiar velocity in the simplest dark energy mod- els. These theoretical ideas motivate us to express various power spectra of the density field in a more convenient way to test such theories. We assume a standard cosmology model for epochs ear- lier than the last scattering surface, and that the coherent evolution of structure formation from the last scattering surface to the present day is undetermined due to new physics relevant to the cosmic acceleration. Thus we di- vide the history of structure formation into two regimes; epochs before matter-radiation equality ( a$_{eq}$ ) and a later epoch of coherent evolution of unknown effect on struc- ture formation from new physics.We can then express various power spectra of the density field splits into these two epochs, with the shape-dependent part determined by knowledge of our standard cosmology, and the coher- ent evolution part only affected by new physics. Mathe- matically, this is written as, The shape factor of the perturbed metric power spectra D Φ ( k ) is defined as D Φ ( k ) = 2 π 2 k 3 9 25 Δ 2 ζ $_{0}$( k ) T 2 $_{Φ}$( k ) (3) which is a dimensionless metric power spectra at a$_{eq}$ , where Δ 2 ζ $_{0}$( k ) is the initial fluctuations in the comov- ing gauge and T Φ ( k ) is transfer function normalized at T Φ ( k → 0) = 1. The primordial shape Δ 2 ζ $_{0}$( k ) depends on n$_{S}$ , as Δ 2 ζ $_{0}$( k ) = A 2 $_{S}$( k/k$_{p}$ ) n$_{S}$ − $^{1}$, where A 2 S is the am- plitude of the initial comoving fluctuations at the pivot scale, k$_{p}$ = 0 . 002Mpc − $^{1}$. The intermediate shape factor T Φ ( k ) depends on ω$_{m}$ . The shape factor for matter fluc- tuations and peculiar velocities D$_{m}$ ( k ) are given by the conversion from D Φ ( k ) of, D$_{m}$ ( k ) ≡ 4 9 k 4 H 4 $_{∗}$ω 2 m D Φ ( k ) , (4) g Φ ( a$_{eq}$ ) = 1 , g$_{δ}$$_{m}$ ( a$_{eq}$ ) = a$_{eq}$g Φ ( a$_{eq}$ ) , g Θ$_{m}$ ( a$_{eq}$ ) = $_{−}$dg$_{δ}$$_{m}$ ( a$_{eq}$ ) d ln a . (5) where H$_{∗}$ = 1 / 2997 Mpc − $^{1}$. Unlike the shape part, the coherent evolution compo- nent, g Φ , g$_{b}$ and g Θ$_{m}$ are not generally parameterized by known standard cosmological parameters. We thus normalize these growth factors at a$_{eq}$ such that, where Φ denotes the curvature perturbation in the New- tonian gauge, P ΦΦ ( k, a ) = D Φ ( k ) g 2 $_{Φ}$( a ) , P$_{bb}$ ( k, a ) = D$_{m}$ ( k ) g 2 $_{b}$( a ) , P Θ$_{m}$Θ$_{m}$ ( k, a ) = D$_{m}$ ( k ) g 2 $_{Θ$_{m}$}$( a ) , (1) ds 2 = − (1 + 2Ψ) dt 2 + a $^{2}$(1 + 2Φ) dx $^{2}$, (2) Instead of determining growth factors using cosmologi- cal parameters, we measure these directly in a model- independent way at the given redshift without referenc- ing to any specific cosmic acceleration model and with the minimal assumption of coherent evolution of modes after a$_{eq}$ . Considering the uncertainty in the determi- nation of A 2 S from the CMB anisotropy, which is degen- erate with the optical depth of re-ionization, we com- bine both A 2 S and g$_{X}$ (where X denotes each compo- nent of Φ, b and Θ$_{m}$) with proper scaling for conve- nience as g ∗ X = g$_{X}$A$_{S}$/A ∗ $_{S}$. Throughout this paper, we use A ∗ 2 S = 2 . 41 × 10 − 9 for the mean A 2 S value from the WMAP5 results. Our result on measuring the bulk flow motion is independent of our choice of an arbitrary con- stant A ∗ 2 S . and Θ$_{m}$ denotes the map of the re-scaled divergence of peculiar velocity θ$_{m}$ as Θ$_{m}$ = θ$_{m}$/aH . These power spectra are then partitioned into a scale–dependent part ( D Φ ( k ) , D$_{m}$ ( k )) and a scale-independent (coherent evolu- tion) component ( g Φ , g$_{b}$ , g Θ$_{m}$ ). We define here g$_{b}$ = b g$_{δ}$$_{m}$ where b is the standard linear bias parameter between galaxy (or cluster) tracers and the underlying dark mat- ter density. The shape of the power spectra is determined be- fore the epoch of matter–radiation equality. Under the paradigm of inflationary theory, initial fluctuations are stretched outside the horizon at different epochs which generates the tilt in the power spectrum. The predicted initial tilting is parameterised as a spectral index ( n$_{S}$ ) which is just the shape dependence due to the initial condition. When the initial fluctuations reach the coher- ent evolution epoch after matter-radiation equality, they experience a scale-dependent shift from the moment they re-enter the horizon to the equality epoch. Gravitational instability is governed by the interplay between radiative pressure resistance and gravitational infall. The different duration of modes during this period results in a sec- ondary shape dependence on the power spectrum. This shape dependence is determined by the ratio between matter and radiation energy densities and sets the loca- tion of the matter-radiation equality in the time coordi- nate. As the radiation energy density is precisely mea- sured by the CMB blackbody spectrum, these secondary shape dependences are parameterised by the matter en- ergy density ω$_{m}$ = Ω$_{m}$ h $^{2}$. Both of these parameters are now well–determined by CMB experiments. In the linear regime of the standard gravitational in- stability theory, the Kaiser effect (the observed squeezing of ξ$_{s}$ ( σ, π ) due to coherent infall around large–scale struc- tures) can be written in configuration space as, B. Correlation function in the configuration space ξ$_{s}$ ( σ, π )( a ) = ( g ∗ 2 b + 1 3 g ∗ $_{b}$g ∗ Θ$_{m}$ + 1 5 g ∗ 2 Θ$_{m}$ ) ξ ∗ $_{0}$( r ) P$_{0}$ ( µ ) − 4 3 g ∗ $_{b}$g ∗ Θ$_{m}$ + 4 7 g ∗ 2 Θ$_{m}$ ) ξ ∗ $_{2}$( r ) P$_{2}$ ( µ ) + 8 35 g ∗ 2 $_{Θ$_{m}$}$ξ ∗ $_{4}$( r ) P$_{4}$ ( µ ) , (6)