We consider galaxy number counts as functions of the observational direction and the observed redshift, ∆(n, z). In Ref. [106] we describe how the code CLASSgal calculates the corresponding power spectrumC`(z1, z2). We shall consider these spectra as our basic observables and assume that different`-values are uncorrelated. The truly observed spectra have finite resolution in redshift, and are of the form
C`W(z, z′) = ∫ W(z1, z,∆z)W(z2, z′,∆z)C`(z1, z2)dz1dz2. (7.8) HereW(z1, z,∆z)is a normalized window function centered aroundz with half-width
∆z. We shall use Gaussian and top hat windows (with half-width ∆z). Here C`W are the Legendre coefficients of the smoothed angular correlation function
ξW(θ, z, z′) = ∫ W(z1, z,∆z)W(z2, z′,∆z)ξ(θ, z1, z2)dz1dz2. (7.9)
7.3.1 Fisher matrix forecasts
We perform a Fisher matrix analysis to compare a forecast for future redshift surveys derived from the angular power spectrum C`(z1, z2) with the one derived from the three-dimensional Fourier power spectrum P(k). For a given list of Nbin redshift bins with mean redshiftszi, we denote the auto- and cross-correlation angular power spectra
7.3. THE FISHER MATRIX AND THE NONLINEARITY SCALE 111 by C`ij ≡C`W(zi, zj). Since C`ij =C`ji, there are[Nbin(Nbin+1)/2] power spectra to be considered. Assuming that the fluctuations are statistically homogeneous, isotropic and Gaussian distributed, the covariance matrix between different power spectra can be approximated as in Ref. [23],
Cov[`,`′][(ij),(pq)]=δ`,`′
C`obs,ipC`obs,jq+C`obs,iqC`obs,jp
fsky(2`+1) , (7.10)
where fsky is the sky fraction covered by the survey. For each multipole `, the co-variance matrix is a symmetric matrix, Cov[`,`][(ij),(pq)]=Cov[`,`][(pq),(ij)], of dimension [Nbin(Nbin+1)/2)]2. The definition of the truly observable power spectrum C`obs,ij takes into account the fact that we observe a finite number of galaxies instead of a smooth field. This leads to a shot noise contribution in the auto-correlation spectra,
C`obs,ij=C`ij+ δij
∆N(i), (7.11)
where ∆N(i) is the number of galaxies per steradian in the i-th redshift bin. In principle also instrumental noise has to be added, but we neglect it here, assuming that it is smaller than the shot noise. More details about the Fisher matrix and the definition of ’figure of merit’ (FoM) are given in Appendix 7.7.
7.3.2 The nonlinearity scale
Our code CLASSgal uses linear perturbation theory, which is valid only for small density fluctuations, D = δρm/¯ρm ≪ 1. However, on scales roughly of the order of λ ≲30h−1Mpc, the observed density fluctuations are of order unity and larger at late times. In order to compute their evolution, we have to resort to Newtonian N-body simulations, which is beyond the scope of this work. Also, on non-linear scales, the Gaussian approximation used in our expression for likelihoods and Fisher matrices becomes incorrect. Hence, there exists a maximal wavenumber kmax and a minimal comoving wavelength,
λmin= 2π kmax
beyond which we cannot trust our calculations. There are more involved procedures to deal with non-linearities instead of using a simple cutoff, see e.g. [25]. However, in this work we follow the conservative approach of a cutoff.
For a given power spectrumC`ij this nonlinear cutoff translates into an`-dependent and redshift-dependent distance which can be approximated as follows. The harmonic
mode ` primarily measures fluctuations on an angular scale1 θ(`) ∼ 2π/`. Let us consider two bins with mean redshifts ¯zi≤z¯j and half-widths ∆zi and ∆zj. At a mean redshift ¯z= (¯zi+z¯j)/2 the scaleθ(`)corresponds to a comoving distancer` =r(¯z)θ(`), where r(¯z) denotes the comoving distance to ¯z. Let us define the “bin separation”
δzij =zjinf−zisup, wherezjinf =z¯j−∆zj and zisup=z¯i+∆zi. Henceδzij is positive for well-separated bins, and negative for overlapping bins (or for the case of an auto-correlation spectrum with i=j)2.
When δzij < 0, excluding non-linear scales simply amounts in considering corre-lations only for distances r` > λmin. When δzij > 0, the situation is different. The comoving radial distance corresponding to the bin separation δzij is rz =δzijH−1(¯z).
If rz > λmin, we can consider correlations on arbitrarily small angular scales without ever reaching non-linear wavelengths (in that case, the limiting angular scale is set by the angular resolution of the experiment). Finally, in the intermediate case such that 0<δzijH−1(z¯) <λmin, the smallest wavelength probed by a given angular scale is given by √
r2` +r2z. In summary, the condition to be fulfilled by a given angular scale θ(`) and by the corresponding multipole ` is
λmin ≤
The highest multipole fulfilling these inequalities is
`ijmax=
In the last case, the cut-off is given by the experimental angular resolution, `ijmax = 2π/θexp. Only multipolesC`ij which satisfy the condition`≤`ijmaxare taken into account
1Here we refer to the angular scale separating two consecutive maxima in a harmonic expansion, and not to the scale separating consecutive maxima and minima, given by π/`. Since we want to relate angular scales to Fourier modes, and Fourier wavelengths also refer to the distance between two consecutive maxima, the relationθ(`) =2π/`is the relevant one in this context.
2In the case of a spectroscopic survey, we will use top-hat window functions. In this case there is no ambiguity in the definition of δzij, since (zisup,zjinf) are given by the sharp edges of the top-hats.
In the case of a photometric survey, we will work with Gaussian window functions. Then (zsupi ,zjinf) can only be defined as the redshifts standing at an arbitrary number of standard deviations away from the mean redshifts (¯zi, ¯zj). In the following, the standard deviation in the i-th redshift bin is denoted ∆zi, and we decided to show results for two different definitions of the “bin separation”, corresponding either to 2σ distances, with zinfj =z¯j−2∆zj and zisup=¯zi+2∆zi, or to 3σ distances, withzjinf=z¯j−3∆zj andzisup=z¯i+3∆zi.
7.4. RESULTS 113