Non-linear evolution

On small scales, Fourier modes evolve in the highly non-linear regime at low redshifts. These non-linear effects lead to an increase in the power spectrum on small scalesk>0.2h Mpc−1 with

2.4 Distortions in the linear clustering pattern 21

respect to the amplitude expected from linear theory. Moreover, non-linearities can introduce shifts and distortions in the location and shape in the signal of BAO, which suggests that a good understanding of non-linear effects is imperative if the such feature is to be used as standard ruler in the LSS analysis.

A numerically-based attempt to model non-linear effects using two-point statistics is given by Hamilton et al. (1991) by means of the stable clustering hypotheses (Peebles, 1980). With the advent of large N-body simulations, it has been possible to describe the transition from linear to non-linear regimes by designing fitting algorithms. Among these procedures are the work developed by Peacock and Dodds (1994) and the Halo-Fit of Smith et al. (2003). Phenomenological parameterizations of the non-linear evolution of the galaxy power spectrum has been also constructed based onN₋body simulations and tested against observations, such as theQ₋model of Cole et al. (2005). In Chapter 5 we will implement this phenomenological model to describe the shape of the cluster power spectrum.

On the theoretical side, perturbation theory (e.g. Jain and Bertschinger, 1994; Scoccimarro and Frieman, 1996) provides a theoretical framework based on first principles (Euler equation and continuity equations) capable of accurately reproducing non-linear effects at intermediate redshifts z > 1 and has been successfully tested against numerical simulations (Jeong and Komatsu, 2009). The theory is based on a perturbative analysis of the complete set of conservation equations for matter, the matter distribution and its peculiar velocity fields. Under the assumption of a CDM model of the Universe, the evolution equations for the matter perturbations allow for power-law solutions, leading to a power spectrum written as

P(k,z)=g2(z)P(k,0)+P22+P13· · ·, where the quantitiesPi j correspond to terms of order O(δ)i+j in the density fluctuation, arising due to mode coupling. PT, being a perturbative approach, fails at low redshifts where the perturbative parameter, i.e,δ, goes above unity.

In the renormalized perturbation theory (Wyld, 1964; Crocce and Scoccimarro, 2006) a re-summation of the different non-linearities leads to an expansion whose perturbative parameter is not the perturbation amplitude. Instead, the re-summation is done such that when the expansion is truncated at a given scale, all non-linearities are already taken into account. This model has succeeded in reproducing the non-linear matter power spectrum measured from N-body simulations (e.g. Crocce and Scoccimarro, 2008; Smith et al., 2008), without the need for free parameters. The full-shape modeling of the measured correlation function from the LRG sample of the SDSS based on renormalized perturbation theory has led to tighter constraints on cosmological parameters (S´anchez et al., 2009). Non-linear effects must be taken into account in order to construct theoretical predictions capable of modeling the high-precision data expected from the next generation of cosmological probes. As will be discussed in Chapter 5, the power spectrum that we measured from a galaxy cluster catalogue will display non-linear evolution. Nevertheless, given the moderate volume probed by the survey we will explore (REFLEX, see Chapter 4), it will be sufficient to model non-linearities with a simple model such as theQ₋model.

In Fig. 2.3 the matter power spectrum at redshift z = 0 is shown. The linear power spectrum is taken from the fitting formulae of (Eisenstein and Hu, 1998), while the non- linear power spectrum is calculated with the fitting formulae of Smith et al. (2003). In order to see how the shape of the power spectrum changes with the underlying cosmological parameters, we chose three sets: the concordance set which describes a flat universe with

Ωmat=0.237,n=0.98,σ8=0.77and h=0.7. The second model refers to of a matter dominated

universe ΩΛ = 0 and the third contains Ωmat = 0.5. Only the matter content (and thus the dark energy content) is changed. In addition, we have introduced redshift distortions,

22 2. Cosmological model and structure formation

first through the Kaiser boost factor using β = f (z;Ωi) and second introducing small scale distortions implementing a velocity dispersion of σ =500km/s with a Gaussian distribution function. By varying the cosmological parameters, the shape and the amplitude of the matter power spectrum are both modified, as was already discussed. Interestingly, note how in a matter dominated universe the linear power spectrum has higher amplitudes in a small range of scales than the non-linear contribution. Notice also that when translated to redshift space, the non-linearities can be suppressed by the ”fingers-of-god” effect such that the resulting matter non-linear power spectrum in redshift space could naively been reproduced by linear theory. This is clearly seen in the last plot of Fig. 3.1. This situation can be also translated to the galaxy clustering, where, together with the aforementioned distortions, galaxy bias together with an implementation of a statistical model for the halo occupation distribution of galaxies can generate variations in the shape of the galaxy power spectrum in the non linear regimes.