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With photometric redshift estimates for individual sources, one can be more ambitious, and treat the data as a very noisy 3D shear field, which is sampled at a number of discrete locations, and for whom the locations are somewhat imprecisely known. It makes some sense, therefore, to deal with the data one has, and to compare the statistics of the discrete 3D field with theoretical predictions. This was the approach of Heavens (2003), Castro et al., (2006), Heavens et al., (2006). It should yield smaller statistical errors than tomography, as it avoids the binning process which loses information. In common with many other methods, one has to make a decision whether to analyse the data in configuration space or in the spectral domain. The former, usually studied via correlation functions, is advantageous for complex survey geometries, where the convolution with a complex window function implicit in spectral methods is avoided. However, the more readily computed correlation properties of a spectral analysis are a definite advantage for Bayesian parameter estimation, and we follow that approach here. The natural expansion of a 3D scalar field which is derived from a potential is in terms of products of spherical harmonics and spherical Bessel functions, . Such products, characterised by 3 spectral parameters, {k,l,m}, are eigenfunctions of the Laplace operator, thus making it very easy to relate the expansion coefficients of the density field to that of the potential (essentially via k squared from the del squared operator). Similarly, the 3D expansion of the lensing potential,
where the prefactor and the factor of k are introduced for convenience. The expansion of the complex shear field is most
naturally made in terms of spinweight 2 spherical harmonics and spherical Bessel functions, since
where the derivatives here are Newman and Penrose's edth derivative (which are somewhat complicated functions of theta and phi, but tend to in the flatsky limit).
The choice of the expansion becomes clear when we see that the coefficients of the shear field are related very simply to those of
the lensing potential:
The relation of the coefficients of phi to the expansion of the density field is readily computed, but more complicated as the lensing potential is a weighted integral of the gravitational potential. The details will not be given here, but relevant effects such as photometric redshift errors, nonlinear evolution of the power spectrum, and the discreteness of the sampling are easily included. The reader is referred to the original papers for details (Heavens, 2003, Castro et al., 2006, Heavens et al., 2006). In this way the correlation properties of the shear field coefficients can be related to an integral over the power spectrum, involving the z(r) relation, so cosmological parameters can be estimated via standard Bayesian methods from the coefficients. Clearly, this method probes the dark energy effect on both the growth rate and the z(r) relation.
The technique outlined here has been applied to the small COMBO17 survey, with a view to constraining the equation of state parameter of Dark Energy. Kitching et al. (2007) found a conditional error of w = 1.3 +/ 0.6. This is evidently not yet competitive with other probes, but comes from less than one square degree of data, and forecasts based on future surveys are very encouraging.
Castro P.G., Heavens A.F., Kitching T.D., (2006) Phys Rev D72, 023516
Heavens A.F. (2003), MNRAS, 343, 1327
Heavens A.F., Kitching T.D., Taylor A.N. (2006), MNRAS, 373, 105
Kitching T.D. et al. (2007), MNRAS, 376, 771