Projected bispectrum in spherical harmonics and its application to angular galaxy catalogues
1 Re x] denotes the real part of the complex number x.
2 The number of independent D is 18 as defined in with ℓi=ℓ<40,i=1,2,3. The presence of the 3J symbol for equilateral configurations requires ℓ to be even, ℓ=2 is discarded because it would be contaminated by the galaxy quadrupole.
3 The procedure of subdivision to increase the S/N appears counter-intuitive. In fact, nothing more is gained by this than by relaxing the precise shape of the triangles. It is easiest to demonstrate this in Fourier space in 3D. MVH97 considered the bispectrum B(k1,k2,k3) and demonstrated that the S/N increases ∝N1/2, where N is the number of subvolumes. Alternatively, in increasing the volume by a factor N, one has more triangles to analyse. MVH97’s analysis includes the density of k1 states; relaxing the triangle shape configuration increases the number of k2 states by the ratio of the density of states (i.e. N). k3 is fixed at −k1−k2, so the number of triangles for given k1 is ∝N, giving the same increase of S/N (see ). In practice, correlations may modify the details. This argument is a variant of that in .
Abstract
We present a theoretical and exact analysis of the bispectrum of projected galaxy catalogues. The result can be generalized to evaluate the projection in spherical harmonics of any 3D bispectrum and therefore has applications to cosmic microwave background and gravitational lensing studies.
By expanding the 2D distribution of galaxies on the sky in spherical harmonics, we show how the three-point function of the coefficients can be used in principle to determine the bias parameter of the galaxy sample. If this can be achieved, it would allow a lifting of the degeneracy between the bias and the matter density parameter of the Universe, which occurs in linear analysis of 3D galaxy catalogues. In previous papers, we have shown how a similar analysis can be done in three dimensions, and we show here through an error analysis and by implementing the method on a simulated projected catalogue that ongoing three-dimensional galaxy redshift surveys (even with all the additional uncertainties introduced by partial sky coverage, redshift-space distortions and smaller numbers) will do far better than all-sky projected catalogues with similar selection function.
