Turismo globalizado

The increasing volume of the new large galaxy redshift surveys requires accurate models of the LSS observations in order to extract the maximum amount of information from the data without introducing systematic effects. In Section 2.2, I introduced the basic concepts of renormalised perturbation theory and how it can improve over standard PT in describing the dark matter power spectrum at non-linear scales. However, in order to describe the galaxy clustering, it is necessary to include into the modelling the distortions due to bias and redshift space effects.

As a first step towards a phenomenological model of the power spectrum, I introduce two approximations in the two terms in the right hand side of equation (2.25). First I

4.2 Modelling the full shape of the power spectrum 33

assume that the propagator is described by the growth factor times a Gaussian damping

G(k, z) =D1(z) exp " − k √ 2k? 2# , (4.1)

where the scale k? is given by equation (2.24). Secondly I reduce the full mode coupling

P∞

n=2PnMC(k, τ) to its lowest order contribution: P22 from equation (2.23a). Equation

(4.1) is almost exact in the large-k limit and is a good approximation for the propagator for small values of k.

Crocce & Scoccimarro (2008) proposed a model for the large scale correlation function motivated by the RPT formalism. In this ansatz the correlation function is given by

ξNL(r) = b2 h ξL(r)⊗e−(k?r) 2 +AMCξL0(r)ξ (1) L (r) i , (4.2)

whereb,k? and AMC are treated as free parameters, and the symbol ⊗ denotes a convolu-

tion. The factorbencodes the change in the large scale amplitude caused by the linear bias b1 and the Kaiser factor Slin; AMC quantifies the relative amplitude of the mode coupling

term with respect to the linear one. ξ_L0 is the derivative of the linear correlation function and ξ_L(1)(r) is defined by ξ_L(1)(r)≡ˆr· ∇−1_ξ L(r) = 1 2π2 Z dk kPL(k)j1(kr), (4.3)

with j1(y) denoting the spherical Bessel function of order one. The two terms in equation

(4.2) are, respectively, the linear correlation function convolved with the Fourier trans- form of the approximated propagator of equation (4.1) and the leading order contribution to ξMC from the lowest order approximation of the mode coupling power spectrum of

equation (2.23a). S´anchez et al. (2008) compared this model with the results of N-body simulations and found that it is able to give an accurate description of the full shape of the correlation function, including the effects of bias and redshift space distortions, for volumes up to two orders of magnitude larger than present day datasets. S´anchez et al. (2009) successfully used this model to obtain constraints on cosmological parameters from the correlation function of a sample of luminous red galaxies drawn from the data release 6 of the SDSS as measured by Cabr´e & Gazta˜naga (2009a).

In this analysis, I follow the same approach and model the non-linear power spectrum as

P(k, z) =b2he−(k/k?)2Plin(k, z) +AMCP22(k, z)

i

, (4.4)

and treat b, k? and AMC as free parameters. In the next section I show that this model

allows to obtain unbiased constraints on the dark energy equation of state parameter by accurately describing the full shape of the power spectrum measured in real and redshift space.

Panel a of Figure 4.2 shows the linear theory power spectrum (solid line) and P22(k)

(dashed line) computed assuming the cosmological parameters of the L-BASICC II simu- lations. Panel b showsPL(k) divided by a reference power spectrum without BAO (Eisen-

Figure 4.2: Panel a: Linear the- ory (solid line) and lowest order non-linear (dashed line) power spectra. Panel b: ratio between the power spectra of panel a

and reference ones without oscil- lations. P22(k) shows small oscil-

lations out of phase with respect to Plin(k), generating a net shift

of the BAO peaks when summed.

(2.23a) to the smooth power spectrum. Both power spectra show oscillations, although in P22(k) they have a smaller amplitudes and are out of phase with respect to the ones

in PL(k). When these two terms are summed as in equation (4.4), the BAOs are shifted

towards smaller scales with respect to the ones in the linear power spectrum, in agreement with the findings of Crocce & Scoccimarro (2008), S´anchez et al. (2008) and Smith et al. (2008).

P22(k) does not have the same shape asP2 MC(k): the latter in fact decreases faster than

the former and at k ∼0.15−0.2hMpc−1 _{it is roughly 1.5−2 times smaller. But at those}

scales the amplitude of the three mode coupling, P3 MC(k), is already around 1/4−1/2 of

P2 MC(k) and should be included in the model, as can be seen comparing Figures 4.2 and

2.2. P22(k) is thus somewhat larger than P2 MC(k) +P3 MC(k) and the difference becomes

more and more important with increasing wave number. I will come back to this again in section 4.3, where I discuss the range of scales in which the model of equation (4.4) can be applied to a measurement of the power spectrum.

It can be expected that this model is less efficient at describing the shape of the power spectrum in redshift-space than in real-space. In fact, in equation (4.4) I do not include redshift space distortions explicitly and I let the free parameters compensate some of their effects. The lack of a model for the redshift space distortions will be particularly visible for the dark matter, since the associated scale dependence is stronger than for the haloes (Scoccimarro, 2004; Angulo et al., 2008).