Modifications of class

We now describe how to implement the fNL parameter (and its running nNG) for local primordial non-Gaussianity (PNG). In Class, it is convenient to use the Poisson equation Φ= − (3Ω_0mH₀²/2ak²)δco in Equation (8.20), to obtain:

8.8. APPENDIX: PRIMORDIAL NG IN CLASS 157 for more details about the derivation of this result see [93]. Here fNL, nNG, k∗ and δec are constants describing the PNG parameter, its running, its pivot scale and the critical density for ellipsoidal collapse, respectively. The linear and scale-independent galaxy bias in absence of PNG is given by the functionbG(z). The variableR = −3Φ_p/2 (here Φ_p is the primordial Bardeen potential), is the primordial potential of curvature perturbations. The linear growth factor g(z) = (1+z)D(z), where D(z) is the ampli-tude of the growing mode, appears since Class, by normalizing all source functions with respect to R (e.g., δ(z, k)/R(k)), would otherwise evaluate Φ of Equation (8.19) at decoupling, while in LSS convention it is evaluated at z=0. In other words, we take into consideration the following relation between LSS and CMB conventions [289, 77]:

f_NL^LSS= g(zdec)

g(0) f_NL^CMB, (8.32)

and we use fNL ≡f_NL^LSS.

We refer to Class v2.3.2², where a constant galaxy bias is implemented in the transfer.c module as a rescaling of the density transfer function. As a first step, we remove this constant bG from thetransfer.c module. As described in [106], the code computes the density perturbation in the comoving gauge, related to the longitudinal Newtonian gauge by:

δco=δlong+3aH

k² θlong , (8.33)

where θlong =kVlong is the divergence of the velocity in the longitudinal gauge. Fur-thermore, δco is a gauge-invariant variable and at first order in perturbation theory it coincides with the density perturbation in the synchronous-gauge comoving with dark matter used in [93]. Then, we modify the perturbations.c module to rescale the matter density source functionδco(z, k)/R(k)by a function bG(z)specified within the same module. Finally, we add the quantity ∆b⋅δco/R from Equation (8.31) to the rescaled density contrast bG(z)δco/R.

With this modifications, and allowing also for a time-dependent magnification bias s(z), the transfer functions given in Equation (8.2) become the terms given in Equa-tion (8.29).

2http://class-code.net/

Chapter 9

Measuring the Lensing Potential with Galaxy Number Counts

Based on:

[201] F. Montanari and R. Durrer, “Measuring the lensing potential with galaxy clus-tering,” arXiv:1506.01369[astro-ph.CO].

We include additional updates to arXiv:1506.01369v2.

Abstract. We investigate how the lensing potential can be measured tomographi-cally with future galaxy surveys using their number counts. Such a measurement is an independent test of the standard ΛCDM framework and can be used to discern modified theories of gravity. We perform a Fisher matrix forecast based on galaxy angular-redshift power spectra, assuming specifications consistent with future photo-metric Euclid-like surveys and spectroscopic SKA-like surveys. For the Euclid-like survey we derive a fitting formula for the magnification bias. Our analysis suggests that the cross correlation between different redshift bins is very sensitive to the lensing potential such that the survey can measure the amplitude of the lensing potential at the same level of precision as other standard ΛCDM cosmological parameters.

9.1 Introduction

Cosmology has become a data driven science. After the amazing success story of the cosmic microwave background (CMB), see [6, 12, 10, 112, 110], we now also want to profit in an optimal way from present and future galaxy catalogs. Contrary to the CMB which comes from the two dimensional surface of last scattering, galaxy catalogs are three dimensional and therefore contain potentially more, richer information.

159

In the large galaxy surveys planned at present one distinguishes spectroscopic sur-veys, which determine the redshift of galaxies very precisely and are used to determine the large scale structure (LSS), i.e., the clustering properties of galaxies, and photo-metric surveys which determine the redshift with less precision but which are optimized for galaxy shape measurements. From shape measurements one can then statistically infer the shear γ and from it the lensing potential. In this paper we study to which extent the lensing potential can be obtained by simply looking at the angular corre-lation function of galaxies at different redshift without invoking shape measurements which are plagued by difficult systematics and intrinsic alignment [249].

The lensing potential is especially interesting since its relation to the matter density is determined by Einstein’s equations. The lensing potential ψ is given by (see, e.g., [110])

ψ(n, z) = − ∫

r(z)

0 d˜rr(z) −r˜

r(z)˜r (Φ+Ψ)(˜rn, τ0−r˜) (9.1) wherer(z)is the comoving distance of the source andτ0 is the present conformal time.

We neglect spatial curvature, K =0, and we define the lensing potential as a function of redshift z and observer direction n. The variables Φ and Ψ are the well known Bardeen potentials [28, 110]. By measuring both, the density fluctuation δ(n, z) and the lensing potential we can in principle test General Relativity on cosmological scales.

When observing galaxies we measure their redshift and angular position. We can then determine the number of galaxies per solid angle and per redshift bin. The correlation function of these number counts within linear perturbation theory has been determined in Refs. [64, 80]. Apart from the galaxy number density and velocity it also depends on the convergence

κ= −1

2∆_Ωψ , (9.2)

which affects the volume corresponding to a given observed solid angle and redshift bin. Here ∆_Ω is the angular Laplacian.

Number counts have also been studied in [293, 148, 243, 50] and relativistic ex-pressions to second order have been derived in [48, 49, 46, 296, 103]. Their po-tential for cosmological parameter constraints has been analyzed in several papers [105,228,219, 294, 292, 295, 224, 16]. In Ref [106] a code (Classgal) for fast compu-tation is presented and forecasts for DES and Euclid-like catalogs are compared with the traditional LSS analysis. In Ref. [77,74] the capacity of SKA (the square kilometer array [185]) to determine primordial non-Gaussianity is analyzed using number counts.

Furthermore, relativistic contributions to the number counts can be isolated using spe-cial observational techniques [66, 63, 75]. In most of this paper we are concerned with the well known convergence term κ which strictly speaking is also a “relativistic con-tribution”. However, due to the Laplacian, on small and intermediate scales, this term

9.2. DETERMINING THE LENSING POTENTIAL WITH NUMBER COUNTS161