MANUAL DEL OPERADOR PARA LA GESTIÓN DE FALLOS

By applying the peak-background split formalism, we analytically derive the La- grangian linear halo bias associated to the LV mass function of Eq. (2.62), using Eq. (2.36): bLLV(ν) = δc σ2 − 1 σ 6 +S3σ(4ν3−4ν) + 2_ddS_ln3_σ σ ν 6ν+S3σ(ν4−2ν2−1) + _ddS_ln3_σ σ(ν2−1) . (2.66) It follows that the Lagrangian bias associated to the rescaled mass function of Eq. (2.65) is hence ebL_LV(ν) =−1 σ ∂feLV(ν) ∂ν =b L LV(ν) +bLT(ν)−bLPS(ν), (2.67) where bL

LV is given in Eq. (2.66), while bLT and bLPS are derived from the Eulerian

biases of Eqs. (2.39) and (2.37) by subtracting unity.

In the presence of PNG, the halo density perturbations depend not only on the dark matter perturbations δ, but also on the potential ϕ. The latter can then be related back to the density in Fourier space by using the Poisson equation, so that the effective Eulerian bias can be written at a fixed redshift ¯z as

bobs_eff (M, k, fNL) =b(M, fNL) + ∆b(M, k, fNL), (2.68)

where the bias contains implicitly the following two corrections:

(i) a scale-independent correction with respect to the Gaussian case, following from the difference in the mass function, given by

δb(fNL)≡b(M, fNL)−b(M,0) ; (2.69)

(ii) a scale-dependent correction, given by ∆b(M, k, fNL)≡

2fNLδcbL(M, fNL)

α(k,z¯) . (2.70) Fig. 2.6 shows the linear Tinker halo bias bT(M) as a function of halo mass M at

¯

z = 0.2, compared with the PS case, and with the scale-independent part of the bias in the presence of primordial non-Gaussianity described in Eq. (2.68): the cosmology is fixed to the best-fit model for our analysis. As in the Gaussian case, we will then

2.5 Primordial non-Gaussianity 63

Figure 2.6: Mass dependence of the linear halo bias at ¯z= 0.2 for three mass functions: Press-Schechter (cyan dotted), Tinker (black solid) and modified LoVerde mass function in the presence of primordial non-Gaussianity (magenta dot-dashed), withfNL = 400. The cosmological model is fixed at the fiducial model in our analysis.

.

average the bias over the masses in our catalogue, following Eq. (2.40). To include the uncertainty on the assumption of a mass function, we introduce a nuisance pa- rameter B, which rescales the bias as bobs _{= ¯b B.}

The scale-independent correction δb(fNL) is small, easily confused with other

normalisation effects, and relies on the assumed form of the mass function and the peak-background split method. For these reasons, it is worth ensuring that the re- sults do not depend on this contribution. We make sure this happens in our case because any constant rescaling of the bias can be equally explained by either a change in the nuisance parameter B or a change in fNL. But since a model with fNL 6= 0

also predicts the scale-dependent bias, it will be favoured only in case such a feature is indeed observed in the data, otherwise the B 6= 1 model will be assigned a better likelihood. In practice, we impose a Gaussian prior centred on B = 1 (details in Chapter 4), but we have checked that the results on fNL do not depend significantly

Figure 2.7: The effect of PNG on the cluster power spectrum. We compare here the predictions for the best-fit model we obtain withfNL =−46 (red solid) and for two cases

with fNL = −200 (blue dashed) and fNL = 200 (green dot-dashed). The dotted line at kmax= 0.15h Mpc−1 represents the smallest scale we use in the analysis.

Finally, we show in Fig. 2.7 the full power spectrum Peobs_{(k) in the presence of}

PNG for a choice of fNL values. High and positive values of fNL increase evidently

the power spectrum on large scales, while weakly suppressing it on small scales. The scale-dependent bias induced by PNG impacts on large scales, while the smaller scale- independent contribution affects the small scales. The survey window convolution partially suppresses the effect of PNG on the largest scales, which become comparable with the survey volume.

Chapter 3

Observations, data and errors

This Chapter is dedicated to the description of the data sets and error estimates we use in our combined cosmological analysis. First, we describe briefly galaxy cluster observations through the whole electromagnetic spectrum. We list the most impor- tant X-ray, millimetre (SZ), weak lensing and optical surveys of past, present and future times, as well as the cosmological constraints obtained from previous works with cluster catalogue. A particular focus is given to the Sloan Digital Sky Sur- vey maxBCG optical catalogue, which is used in this work, and to the data sets we derived from it. We first describe the cluster counts and their covariance matrix. Sec- ondly, we concentrate on the weak lensing mass estimates, the richness-mass scaling relation and the cluster total masses. Finally, the description of the redshift space cluster power spectrum is provided, together with its covariance error matrix. We end with some considerations on the cross-covariance matrix between counts and power spectrum.

3.1

Multi-wavelength surveys of galaxy clusters

Clusters have been detected across multiple wavelengths with varying degrees of suc- cess. To date, a few hundreds have been observed in the millimetre, few thousands in the X-ray, many tens of thousands in the optical. The efficiency of a survey is due to the combination of the technical properties of the detection instrument (e.g. flux sensitivity and angular resolution) and the physical features of the observed objects (e.g. intrinsic luminosity and redshift). Surveys are thus characterised by complete- ness (namely amount of objects that should have been detected), and purity (i.e. contamination due to spurious detected objects). This section includes a description

of the the various detection methods as well as the main galaxy cluster surveys and the derived catalogues.