The full-sky relativistic correlation function and power spectrum of galaxy number counts. Part I: theoretical aspects

Abstract

We derive an exact expression for the correlation function in redshift shells including all the relativistic contributions. This expression, which does not rely on the distantobserver or flat-sky approximation, is valid at all scales and includes both local relativistic corrections and integrated contributions, like gravitational lensing. We present two methods to calculate this correlation function, one which makes use of the angular power spectrum C (z 1 , z 2 ) and a second method which evades the costly calculations of the angular power spectra. The correlation function is then used to define the power spectrum as its Fourier transform. In this work theoretical aspects of this procedure are presented, together with quantitative examples. In particular, we show that gravitational lensing modifies the multipoles of the correlation function and of the power spectrum by a few percent at redshift z = 1 and by up to 30% and more at z = 2. We also point out that large-scale relativistic effects and wide-angle corrections generate contributions of the same order of magnitude and have consequently to be treated in conjunction. These corrections are particularly important at small redshift, z = 0.1, where they can reach 10%. This means in particular that a flat-sky treatment of relativistic effects, using for example the power spectrum, is not consistent. 1 We point out that the original derivation of redshift-space distortion from [20] contains a contribution proportional to n · v = vr. This term does contribute to the monopole and quadrupole and it consequently modifies (1.6). It is however neglected in most redshift-space distortion analysis and therefore we do not consider it as 'standard' and we do not include it in (1.6). We include it however in the relativistic corrections, along with the other Doppler corrections, which are of the same order of magnitude (see Eq. (2.5)). Note that, as discussed in more detail in Section 2.2.2, this specific contribution has been studied in detail in [9][10][11]29] and its impact on the correlation function was found to be important at small redshift and large separation.2 Note that these expressions are valid in the linear regime only. Theoretical models accounting for nonlinearities have been developed and are used to extend the constraints to non-linear scales, see e.g. [30].

Galaxy two-point correlation function in general relativity

“…A detailed analysis of the correlation function was performed in [36]. The bottom left panel of fig.…”

The dipole of the galaxy bispectrum

“…We also use the standard plane-parallel approximation, which is reasonable on ultra-large scales. However, we note that wide-angle effects in the power spectrum can be of the same order of magnitude as the Doppler-type effects in certain circumstances (Tansella et al 2018), and these should be incorporated in a more complete treatment.…”

Limited accuracy of linearized gravity

“…Studies of nonlinear gravitational effects have also shown that energy from small-scale gravitational interactions can have a considerable impact on the cosmological properties of a spacetime [3,4]. Even within a standard perturbative cosmological framework, it has been shown that neglected relativistic effects can lead to percent-level or larger corrections when computing observables [5][6][7], providing a means by which we can study general relativity beyond its dominant behavior.…”