angular power spectra it is also possible to obtain the correlation function as
ξ(θ)ij = 1 4π
(2`+1)C`ijP`(cosθ)W`, (4.6) were P`(cosθ) are Legendre polynomials, and W` is a smoothing function on large multipoles taking into account that in practice the sum depends on a finite `max< ∞. Besides the fact that all relativistic terms can be easily included in this formalism, it is also worth noticing that simple analytic estimates of the covariance matrices are available. The accuracy of the calculation is similar to the one of the original Classcode, i.e., overall about 0.1% when using default precision, or up to 0.01% with boosted accuracy settings. In particular, we find that Limber approximation is quite not accurate enough, especially for redshift bins cross-correlations. Whenever possible, we checked thatClassgal agrees well with the similar output from Cambsources.5 As a first application, in  constraints on the matter density parameter Ωm are also considered assuming specifications consistent with the spectroscopic DES survey. The constraining power increases with the non-linear cut-off `max and with the number of bins Nbin. However, given that the total redshift extension of the survey is fixed 0.45<z<0.65, increasing Nbin also implies that the width of the shells decreases. As a consequence shot-noise starts dominating at about Nbin ≃50–100. The conclusions depends on the survey specification assumed, so it is important to study different cases.
Moreover, another important question is how the constraints obtained with angular correlations compare to those obtained from correlations in comoving space.
4.2 Cosmological parameters forecasts with rela-tivistic correlations
In reference , reproduced in chapter 7, we study how to make optimal use of the angular power spectra to constrain cosmological parameters. We assume specifications consistent with the DES spectroscopic catalog in the range 0.45 < z < 0.65, and the Euclid spectroscopic and photometric surveys in the range 0.7<z <2. The photomet-ric catalog contains about 100 more galaxies then the spectroscopic catalog, hence it is less affected by shot-noise. Photometric errors on redshift determination are taken into account by assuming uncertainties of the same order of the bin width. Then, Tophat photometric redshift bins roughly correspond to Gaussian bin in terms of the true redshifts, with root mean square determined by the redshift error. The overlap between different bins is consistently taken into account in the covariance matrix. The
only assumption is that shot-noise is diagonal in redshift, i.e., that the observed spec-tra are related to the fiducial ones as C`obsij = C`ij +δij/Ni, where Ni is the number of galaxies per steradian in the bin around zi, and δij is the Kronecker delta. Hence, we neglect the fact that bin overlap may also imply a shot-noise contribution com-ing from i ≠ j. Nevertheless, the covariance of bin i–j cross-correlation only assume that it is block-diagonal in multipole ` space, but not in redshift. While shot-noise is important on small scales, cosmic variance dominates large scales. Non-linear scales are carefully excluded from the analysis by fixing a larger multipole `max(z). To op-timally include information from linear scales, `max is larger for higher redshifts, as a fixed comoving scale subtends a smaller angle (hence a larger multipole). For bin auto-correlations around redshift ¯z, `max =2πd(z¯)/λmin can be used, where λmin cor-responds to the comoving non-linear scale. For redshift bins separated by large radial separations, `max is fixed by the survey angular resolution. For intermediate bin i–j cross-correlation, more elaborated algorithms can be used, especially when overlapping due to photometric redshift is present.
First, we compare a traditional power spectrumP(k)analysis in terms of comoving (not observable) wave-numbers k, to the directly observable angular power spectra C`(z1, z2). The methodology is to compute for both cases error ellipses of cosmological parameters through a Fisher matrix analysis, assuming the same non-linear scale. The redshift range is divided into Nbin redshift shells. In the case of spectroscopic surveys, assuming that P(k) measurements inside each bin are independent so that the total Fisher matrix is the sum of those computed for every bin, the power spectrum within each bin can be estimated as the one given at the mean redshift of the shell P(k,z¯).
Instead, angular spectra C` strongly depends on the width and shape of the bin. If few bins are considered, the P(k)method brings significantly better constraints. This is due to the fact that to obtain the comoving wave-numbers k we need to know the comoving separation of each galaxy pair. In particular we need to convert observable angles and redshifts into comoving position within each shell. On the other hand, when considering angular C`ij the radial information of galaxies within a given bin is lost.
Hence, the two approaches give similar constraints when the size of the bins is of order the non-linear scale. This can be in principle achieved with spectroscopic surveys, but in practice the estimation of spectra and their covariances from data for Nbin∼ O (100) is computationally challenging.
In the case of photometric surveys, the redshift estimation is affected by errors of the size of the redshift bins. This limits the constraining power of P(k), as the argument illustrated above does not longer apply. The comoving radial information is not reconstructed better than in the case of tomographic C`’s. Angular power spectra in this case bring better constraints than P(k) already for Nbin∼ 5. Besides from redshift uncertainties, the largest Nbin is also limited by the scale where shot-noise dominates. As the bin width shrinks, the signal of the angular spectra increases
4.2. FORECASTS WITH RELATIVISTIC CORRELATIONS 49 because it is smoothed over a smaller radial extension. On the other hand, less galaxies are included in each bin, increasing shot-noise. The limit for DES is about Nbin∼50, and Nbin∼200 for Euclid.
It is worth to stress that the P(k) analysis is performed in the plane-parallel ap-proximation. As seen in section 3.2, this assume galaxy velocities to be parallel and, for the Kaiser redshift-distortion term, that their angular separation vanishes. In fact it replaces the terms x⋅v, where x is the direction of observation and v the peculiar velocity, by ˆz⋅v, where ˆz is a radial versor pointing to the center of the galaxies. This approximation is of course not defined for angular correlations C`’s. Furthermore, to compare the results to the P(k), we only consider the Kaiser term besides intrinsic clustering. As a next step, we estimate the signal-to-noise of additional terms con-tributing to eq. (2.55) for number counts. These effects are easily taken into account in the tomographic C` analysis. In this analysis we only consider a single redshift bin auto-correlation. For narrow redshift bins, redshift-space distortions in the Kaiser ap-proximation are the only relevant term together with the intrinsic clustering. However, RSD contribution decreases as the width of the bin increases. Interestingly, lensing convergence is more and more relevant especially on small angular scales. As a toy model, we consider a catalog for which the radial position of galaxies is not known, except for their redshift extension 0.5<z <2.5. With such a large redshift bin, lensing convergence becomes the second more important effect besides the density contribu-tion, whereas RSD are negligible. This is a promising indication of the detectability of the lensing potential in this kind of analysis.
Another point worth noticing about the tomographic analysis is the fact that, while the monopole C0ij and dipole C1ij are not observable because of the non-linear gravitational potential and velocity at the observer, combinations like
M (z, δz¯ ) = 1
2[C0(z−, z−) +C0(z+, z+)] −C0(z−, z+), (4.7) wherez±=z¯±δz/2, are independent of the observer position. Indeed, the contributions at the observer are equal for all redshifts. Adding the information from several weekly correlated redshifts it is possible to obtain a good signal-to-noise. Such a measurement would allow to extract cosmological information from the monopole and dipole terms, so far neglected in cosmological analysis.
In reference , reproduced in chapter 8, we investigate how much neglecting relativistic terms can affect cosmological parameter constraints. We define a standard
“Newtonian” reference case where only redshift bin auto-correlations are taken into account, and in which only the Kaiser term is considered. As explained above, since here we compute angular spectra C`, we include terms beyond the usual plane-parallel approximation. As we consider relatively large redshifts, these terms are not expected to bring a substantial contribution . The “Newtonian” case is compared to the
“full-relativistic”, which includes all the terms contributing to number counts as well
as all bin cross-correlations. In both cases, we include realistic observational specifi-cations, such as redshift dependent galaxy bias, magnification bias and evolution bias.
We assume Euclid photometric and SKA spectroscopic catalogs, spanning a similar redshift range 0 < z < 2 divided into five redshift shells, and containing about 109 sources. The question we want to answer is if neglecting relativistic terms may bias cosmological parameter constraints. In the context of Fisher matrices, given a system-atic error it is possible to compute the expected shift on the best-fit parameters that would be caused by neglecting it . In our case the systematic error would be given by C`syst=C`full−C`Newt. However, the formalism assumes a small systematic error (in particular, it expects that changes in contours size are negligible), whereas in our case the lensing convergence is a dominant contribution in redshift bin cross-correlations and C`syst can be much larger than C`Newt. As confirmed by interesting independent analysis , this translates into a shift on the best-fit values that can be≳10σ, while the Fisher formalism can only be applied around the fiducial cosmology. So the analy-sis should in this case be interpreted as a qualitative indication of a possible significant bias. Another method to account for a systematic error in the context of Fisher matri-ces is to marginalize over nuisance parameters. Given the several non-standard terms contributing to the fully relativistic number counts, this analysis is quite computation-ally expensive and it would not give a quantitative indication anyway. We then opt for a third complementary indication to investigate qualitatively the importance of the systematic error introduced in the standard Newtonian analysis. We compare error contours obtained by assuming C`Newt and C`full as fiducial models for the Fisher fore-cast, respectively. We stress that this corresponds to compare two different universes.
A reliable analysis from the quantitative point of view should instead consider always the full C`full as fiducial model. Then, the theoretical model to be fitted would be provided in turn by the same C`full or the biased C`Newt, respectively, including cosmic variance and other observational effects. This can be done with a Markov chain Monte Carlo analysis.
Our Fisher comparison assuming C`Newt and C`full as fiducial models shows differ-ences of 20%–40% in the two cases. This is an indication, consistent with independent conclusions from best-fit values shifts , that neglecting some of the relativistic ef-fects can significantly bias parameter constraints. We identify the main relativistic contribution to be those of the well-known lensing convergence, though usually ne-glected in galaxy clustering studies. As discussed below in details, lensing is especially important for bin-cross correlations, where it can dominate the signal. The effect is larger for photometric surveys, due to the fact that photometric bins will cause an in-creased importance of the integrated radial terms coming from lensing with respect to the local intrinsic galaxy clustering and RSD signals. In such a tomographic analysis, other relativistic terms turn out to be subleading. Nevertheless, the conclusion may change when considering, e.g., correlations in comoving space. As already mentioned,
4.2. FORECASTS WITH RELATIVISTIC CORRELATIONS 51 these probes are less sensitive to the redshift binning and local velocity terms that are negligible for the angular C`’s, may become relevant especially on large scales in the context of multi-tracer observations for which cosmic variance is significantly reduced.
In reference , reproduced in chapter9, we study more in details the importance of the lensing convergence in galaxy clustering analysis. While magnification has been constrained by correlating foreground galaxies with well separated background quasars (see, e.g., ), usual galaxy clustering analysis neglect this effect. This is justified by the fact that when close redshift bins are considered, local terms like density and RSD dominate the signal. We show that in the case of tomographic analysis based on angular spectra, lensing convergence is instead a relevant contribution if all redshift bin cross-correlations are taken into account. These are dominated by the density-lensing correlation
b(zi) (5s(zj) −2) ⟨D(zi)κ(zj)⟩, (4.8) where b is a linear galaxy bias, s is the magnification bias, and zi < zj. This result holds when the redshift bins are separated by comoving distances larger than ∆r = d(zj) −d(zi) ≳150 Mpc/h, hence ∆z=zj−zi≳150 Mpc/hH(¯z) ≃0.09 for ¯z =1. This roughly corresponds to the zero-crossing of the correlation function ξ(r)in real space, since at larger scales the auto-correlation of the density term is small compared to
We assume again catalogs consistent with Euclid photometric and SKA spectro-scopic specifications, with about 109 sources within 0 < z <2. We perform a Fisher matrix analysis including, beside to standard ΛCDM parameters, also an arbitrary rescaling β of the lensing potential ψ, such that:
ψ(zi,n) =βψΛCDM(zi,n) , (4.9) where in the standard ΛCDM model we have β = 1. Hence a constraint on β repre-sents a consistency test of our cosmological model. Furthermore, it can be employed together with, e.g., modified gravity measurements on the slip parameter η and clus-tering parameter Q defined as
Φ=ηΨ, −k2Φ=4πGa2QD , (4.10)
where a is the scale factor. If for simplicity we consider constant values of η and Q, we have β= 12Q(1+η−1).
In the Fisher analysis we assumeNbin=1,5 and 10. While lensing is the second most important contribution for Nbin=1, cosmological parameters are not well constrained in this case. Instead, Nbin = 5 and 10 show competitive constraints. In particular, we expect to constrain the lensing parameter β at the same level as other ΛCDM parameters. All parameters in general show significant better constraints in the case Nbin = 10, because radial information is better resolved in this case. The lensing
amplitudeβis especially sensitive to the inclusion of redshift cross-correlations. Indeed, even if it is not negligible also for redshift auto-correlations, an important part of its constraining power comes from bins well separated in redshift.
Finally, we stress that, as it is clear from eq. (4.8), the analysis is affected by uncer-tainties on galaxy and magnification biases. We reconstructed galaxy bias according to simple fits to simulations, and magnification bias from the observed luminosity func-tions. However, errors on these parameters are still important. Indeed, constraints on the lensing amplitude are mostly interesting when combined with other standard probes such as shear or magnification measurements from galaxy-quasars correlations.
The joint analysis would be useful to break degeneracies. Lensing constraints from the tomographic analysis here consider are a complementary independent measurement worth considering for large-scale surveys.
4.3 Relativistic effects to second order: bispectrum
In reference , reproduced in chapter10, we determine the number counts to second order in cosmological perturbation theory. Indeed, on intermediate and small scales of the large Scale Structure, non-linearities become important. The analysis of the highly non-linear regime is usually done via full numerical simulations. In the intermediate weakly non-linear regime, higher order perturbation theory can be applied. Because of non-linear gravitational effects, the power spectrum does not include all the statistical information. Non-Gaussian statistic is generated by gravitational dynamics, giving rise to non-trivial higher order correlation functions even if initial condition set by inflation follow a Gaussian statistic. Second order perturbation theory allows us to study consistently the galaxy bispectrum.
The full computation to second order including all relativistic effects is quite in-volved. We used e novel method, based on the recently proposed geodesic light-cone gauge (GLC) . The GLC coordinates consist of a time-like coordinate τ, of a null coordinate w (i.e., such that ∂µw∂µw = 0) and of two angular coordinates ˜θa (a=1,2). The GLC metric depends on a scalar Υ, a two-dimensional “vector”Ua and a symmetric matrix γab rising and lowering the two-dimensional indices:
ds2GLC=Υ2dw2−2Υdwdτ+γab(dθ˜a−Uadw) (dθ˜b−Ubdw) . (4.11) When specifying the metric in the GLC gauge, the coordinates (τ, w,θ˜a) still allow some residual gauge freedom. We denote the angular coordinates ˜θa with a tilde to distinguish these exact screen space angles from the angles in the FLRW metric. The time-like coordinate τ can be identified with the proper time of synchronous gauge.
The condition w = constant defines a null hypersurface (∂µw∂µw =0), corresponding to the past light-cone of a given observer. Interestingly, photons travel at constant values of w and ˜θa, so that the calculation of the redshift particularly simple in this
4.3. RELATIVISTIC BISPECTRUM 53