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Appendix: Luminosity fluctuations

Dans le document Relativistic effects in galaxy clustering (Page 124-128)

variables: we replace r in the first two lines and ˜r in the last line with rS. With this, the expression becomes which is identical to the thin shell expression in Eq. (6.17), with an additional integra-tion over the window funcintegra-tion, drSWi(rS).

6.7 Appendix: Luminosity fluctuations

In this appendix we derive in detail the luminosity fluctuation used in expression (6.8).

We use the fact that the fractional fluctuation in the luminosity at fixed flux is given by twice the fractional fluctuation in the luminosity distance,

δLS

S

=2δDL

L

.

We start from the luminosity distance fluctuation derived in [65] and we change the integration variable from the conformal time τ to r=τ0−τ

δDL

where we have neglected the local monopole and dipole terms and we have written the Laplacian in spherical coordinates, ∆=∂r2+2

rr+ 1

r2. We have also used n⋅ ∇ = −∂r. The last term is not present in [65], since there it is assumed that Ψ=Φ, however, it can be found e.g. in Ref. [101]. Considering the total derivative along the geodesic path

d f(τ,x(τ))

dr = −d f(τ,x(τ))

dτ = −f−n⋅ ∇f = −f+∂rf. (6.52) we can rewrite the last integral of (6.51) as

1 Combining all terms together we finally arrive at

δDL

Chapter 7

Cosmological Parameter Estimation with Large Scale Structure Observations

Based on:

[105] E. Di Dio, F. Montanari, R. Durrer and J. Lesgourgues, “Cosmological Parameter Estimation with Large Scale Structure Observations,” JCAP 1401 (2014) 042 [arXiv:1308.6186 [astro-ph.CO]].

Here we include typo corrections to the publish version and to arXiv:1308.6186v2 [astro-ph.CO].

Abstract. We estimate the sensitivity of future galaxy surveys to cosmological pa-rameters, using the redshift dependent angular power spectra of galaxy number counts, C`(z1, z2), calculated with all relativistic corrections at first order in perturbation the-ory. We pay special attention to the redshift dependence of the non-linearity scale and present Fisher matrix forecasts for Euclid-like and DES-like galaxy surveys. We compare the standard P(k) analysis with the new C`(z1, z2) method. We show that for surveys with photometric redshifts the new analysis performs significantly better than the P(k) analysis. For spectroscopic redshifts, however, the large number of redshift bins which would be needed to fully profit from the redshift information, is severely limited by shot noise. We also identify surveys which can measure the lensing contribution and we study the monopole, C0(z1, z2).

7.1 Introduction

Observations and analysis of the cosmic microwave background (CMB) have led to stunning advances in observational cosmology [110, 8]. This is due on the one hand

105

to an observational effort which has led to excellent data, but also to the theoretical simplicity of CMB physics. In a next (long term) step, cosmologists will try to repeat the CMB success story with observations of large scale structures (LSS), i.e. the distribution of galaxies in the Universe.

The advantage of LSS data is the fact that it is three-dimensional, and therefore contains much more information than the two-dimensional CMB. The disadvantage is that the interpretation of the galaxy distribution is much more complicated than that of CMB anisotropies. First of all, our theoretical cosmological models predict the fluctuations of a continuous density field, which we have to relate to the discrete galaxy distribution. Furthermore, on scales smaller than 30h−1Mpc, matter density fluctuations become large and linear perturbation theory is not sufficient to compute them. On these scales, in principle, we rely on costly N-body simulations.

In an accompanying paper [106] we describe a code, CLASSgal, which calculates galaxy number counts, ∆(n, z), as functions of direction n and observed redshift z in linear perturbation theory. In this code all the relativistic effects due to pecu-liar motion, lensing, integrated Sachs Wolfe effect (ISW) and other effects of metric perturbations as described in [64, 80] are fully taken into account. Even if a realis-tic treatment of the problem of biasing mentioned above is still missing, the number counts have the advantage that they are directly observable as opposed to the power spectrum of fluctuations in real space which depends on cosmological parameters. The problem how the galaxy distribution, number counts and distance measurements are affected by the propagation of light in a perturbed geometry has also been investigated in other works; see, e.g. [293, 290, 148, 243, 50,146].

In this paper we use CLASSgal to make forecasts for the ability to measure cos-mological parameters from Euclid-like and DES-like galaxy surveys. This also helps to determine optimal observational specifications for such a survey. The main goal of the paper is to compare the traditional P(k)analysis of large scale structure with the new C`(z1, z2) method. To do this we shall study and compare the figure of merit (FoM) for selected pairs of parameters. As our goal is not a determination of the cosmological parameters but a comparison of methods, we shall not use constraints on the param-eters from the Planck results or other surveys. We just use the Planck best fit values as the fiducial values for our Fisher matrix analysis. We mainly want to analyze the sensitivity of the results to redshift binning, and to the inclusion of cross correlations, i.e., correlations at different redshifts. We shall also study the signal-to-noise of the different contributions to C`(z1, z2) in order to decide whether they are measurable with future surveys.

In the next section we exemplify how the same number counts lead to different 3D power spectra when different cosmological parameters are employed. In Section7.3 we use the Fisher matrix technique to estimate cosmological parameters from the number count spectrum. We pay special attention on the non-linearity scale which enters in a

7.2. NUMBER COUNTS VERSUS THE REAL SPACE POWER SPECTRUM 107

Dans le document Relativistic effects in galaxy clustering (Page 124-128)