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].

Future galaxy clustering surveys will probe small scales where non-linearities become important. Since the number of modes accessible on intermediate to small scales is very high, having a precise model at these scales is important especially in the context of discriminating alternative cosmological models from the standard one. In the mildly nonlinear regime, such models typically differ from each other, and galaxy clustering data will become very precise on these scales in the near future. As the observable quantity is the angular power spectrum in redshift space, it is important to study the effects of non-linear density and redshift space distortion (RSD) in the angular power spectrum. We compute non-linear contributions to the angular power spectrum using a flat-sky approximation that we introduce in this work, and compare the results of different perturbative approaches with N -body simulations. We find that the TNS perturbative approach is significantly closer to the N -body result than Eulerian or Lagrangian 1-loop approximations, effective field theory of large scale structure or a halofit-inspired model. However, none of these prescriptions is accurate enough to model the angular power spectrum well into the non-linear regime. In addition, for narrow redshift bins, ∆z 0.01, the angular power spectrum acquires nonlinear contributions on all scales, right down to = 2, and is hence not a reliable tool at this time. To overcome this problem, we need to model non-linear RSD terms, for example as TNS does, but for a matter power spectrum that remains reasonably accurate well into the deeply non-linear regime, such as halofit.

The subject of this paper is to build a physical model describing shape and size correlations of galaxies due to weak gravitational lensing and due to direct tidal interaction of elliptical galaxies with gravitational fields sourced by the cosmic large-scale structure. Setting up a linear intrinsic alignment model for elliptical galaxies which parameterises the reaction of the galaxy to an external tidal shear field is controlled by the velocity dispersion, we predict intrinsic correlations and cross-correlations with weak lensing for both shapes and sizes, juxtaposing both types of spectra with lensing. We quantify the observability of the intrinsic shape and size correlations and estimate with the Fisher-formalism how well the alignment parameter can be determined from the Euclid weak lensing survey. Specifically, we find a contamination of the weak lensing convergence spectra with an intrinsic size correlation amounting to up to 10% over a wide multipole range ℓ = 100…300, with a corresponding cross-correlation exhibiting a sign change, similar to the cross-correlation between weak lensing shear and intrinsic shapes. A determination of the alignment parameter yields a precision of a few percent forecasted for Euclid, and we show that all shape and many size correlations should be measurable with Euclid.

We investigate the corrections which relativistic light-cone computations induce on the correlation of the tangential shear with galaxy number counts, also known as galaxygalaxy lensing. The standard-approach to galaxy-galaxy lensing treats the number density of sources in a foreground bin as observable, whereas it is in reality unobservable due to the presence of relativistic corrections. We find that already in the redshift range covered by the DES first year data, these currently neglected relativistic terms lead to a systematic correction of up to 50% in the density-shear correlation function for the highest redshift bins. This correction is dominated by the the fact that a redshift bin of number counts does not only lens sources in a background bin, but is itself again lensed by all masses between the observer and the counted source population. Relativistic corrections are currently ignored in the standard galaxy-galaxy analyses, and the additional lensing of a counted source populations is only included in the error budget (via the covariance matrix). At increasingly higher redshifts and larger scales, these relativistic and lensing corrections become however increasingly more important, and we here argue that it is then more efficient, and also cleaner, to account for these corrections in the density-shear correlations.

Even though we know that physical observations are frame independent, the frame dependence of cosmological perturbations is relatively subtle and has led to confusion in the past. In this paper we show that while the (unobservable) matter power spectrum is frame dependent, the observable number counts are not. We shall also determine how the frame dependence of the power spectrum depends on scale.