While this analysis is promising, it remains to be tested if non-linear clustering and complicated biasing (which is quite plausible for red galaxies) would not ‘contaminate’ the measurement of the equation of state. Even if the Alcock–Paczynski test turns out to be less accurate than other cosmological tests (e.g., CMB and SN Ia), the effect itself is an interesting and important ingredient in analyzing the clustering pattern of galaxies at high redshifts. We shall now present the formalism for this effect.

Due to a general-relativistic effect through the geometry of the Universe, the observable separations perpendicular and parallel to the line-of-sight direction, and , are mapped differently to the corresponding comoving separations in real space and :

with being the angular diameter distance. The difference between and generates an apparent anisotropy in the clustering statistics, which should be isotropic in the comoving space. Then the power spectrum in cosmological redshift space is related to defined in the comoving redshift space as where the first factor comes from the Jacobian of the volume element , and and are the wavenumber perpendicular and parallel to the line-of-sight direction.Using Equation (131), Equation (156) reduces to

whereFigure 16 shows anisotropic power spectra . As specific examples, we consider SCDM, LCDM, and OCDM models, which have , , and , respectively. Clearly the linear theory predictions (; top panels) are quite different from the results of -body simulations (bottom panels), indicating the importance of the nonlinear velocity effects ( computed according to [58]; middle panels).

Next we decompose the power spectrum into harmonics,

where are the -th order Legendre polynomials. Similarly, the two-point correlation function is decomposed as using the direction cosine between the separation vector and the line-of-sight. The above multipole moments satisfy the following relations: with being spherical Bessel functions. Substituting in Equation (159) yields , and then can be computed from Equation (161).A comparison of the monopoles and quadrupoles from simulations and model predictions exhibits how the results are sensitive to the cosmological parameters, which in turn may put potentially useful constraints on . Figure 17 indicates the feasibility, which interestingly results in a constraint fairly orthogonal to that from the supernovae Ia Hubble diagram.

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