LOS CAMBIOS QUÍMICOS EN LA MATERIA Objetivos de la unidad

The two-point correlation function, as we have mentioned in previous section (eq. 2.82), is one of the main quantity to study the clustering of galaxies. Although we have already provided the statistical denition of the two-point correlation function, when considering a generic redshift survey the two-point function is given by counting the pairs of given distance in the sample, with a certain selection function, with respect to the galaxies in a random sample with the same selection function of the survey sample:

ˆ

ξ(r) = DD(r) − 2DR(r) + RR(r)

RR(r) − 1 (3.1)

where DD(r)∆r is the number of observed pairs with separation in the range

r ± ∆r/2, RR(r)∆r is the expected number of pairs in a random sample and

DR(r)is the number of cross-pair between the real and random samples in

the same range. Eq. 3.1 is the Landy and Szalay estimator (Landy et al., 1998) and it is built to minimize the variance of the two-point correlation function. The number of objects in the random catalog is expected to be larger than the size of the observed sample, because we want that the shot- noise is as low as possible with respect to the observed sample.

As we have done with the two-point correlation function, we can consider an estimator for the counterpart in Fourier space, the power spectrum, in the simple case of periodic cubic box of side L; using the reality condition 2.30 we have: ˆ Ptot(k) = k3 f Nk X q∈k δqδ−q , (3.2)

where kf = 2π/L, Nkare all the modes within each shell, the sum runs over

all the modes within the shell and the density contrast is dened as:

δk= 1 k_f3 1 Nk X i eik·xi _, for k 6= 0 (3.3)

where the sum runs over all the galaxies in the box.

In this case the total power spectrum, including the shot-noise contribu- tion, is given by:

ˆ

Ptot = h ˆP (k)i = ˆP (k) + ˆPSN , (3.4)

where ˆP_SNis the shot-noise estimator, that in the more simple case is given

by the Poissonian noise, 1/N, with N the total number of galaxy in the box. In order to recover the cosmological power spectrum we have to subtract the shot-noise contribution.

As we have already mentioned in the initial part of this section the real surveys are characterized by a selection function.

3.1 Measurements of galaxy clustering 43 One possibility is to enclose the survey into a box, (Baumgart and Fry, 1991), and dene an auxiliary eld (Feldman et al., 1994) given by:

F (x) = wk(x)θ(x)[n(x) − αnr(x)] , (3.5)

where n(x) = ¯n(1+δ(x)) is the observed density, nr(x)is the density from a

random catalog with no correlation; as we have already stressed, the density of the random catalog is higher than the observed one, because we want that the shot-noise in the random catalogs is as small as possible compared to

the observed sample; so, we dene α as the ratio ¯n/¯nr; θ(x) is the window

function with value 1 or 0 for points inside or outside the survey; wk(x)is a

weight dened as:

wk(x) =

1

1 + (2π)3_{n(x)P (k)¯} . (3.6)

wk(x) is obtained imposing that the fractional variance of the power spec-

trum σ2

P(k)/P2(k)is minimized. As it it clear from eq. 3.6, the denition of

the weight requires a preliminary estimation of the power spectrum we want

to measure. Usually we can assume P (k) ≡ P (k∗) ≡ P∗, with k∗ the scale

of interest for our analysis; for simplicity ¯n is assumed to be constant. The Fourier transform of eq. 3.5 is:

Fk= k3f

X

q

W (q − k)δq , (3.7)

where W (k) = wkθk and kf is the fundamental frequency of the box given

by 2π/L with L the dimension of the box. The Fourier transform gives the convolution between the weighted window function and the galaxy density eld. The power spectrum of this auxiliary eld is then:

PF(k) = k6f

X

p

|W (k − p)|2_P

tot(p) , (3.8)

where Ptot is the power spectrum of the box including the shot-noise contri-

bution. For a top-hat window function if k  1/L we can approximate the auxiliary power spectrum as:

PF(k) ' P (k)kf6 X p |W (k − p)|2+ k_f3X p |W (k − p)|2 1 N . (3.9)

The estimator for the power spectrum is given by subtracting the shot-noise contribution and dividing by the selection function. As we can see from the equation above, the presence of the window function modies the power spectrum, so that the quantity we have to estimate is both the convolution between the power spectrum and the window function itself, and the shot- noise term.

44 Probes of the galaxy distribution

In all the relations we have described, we did not consider the redshift dependence. In the introduction of this section, we pointed out that cosmo- logical surveys are carried out in redshift space and that peculiar velocities distort the spatial distribution of cosmological objects (see section 3.2.2). This distortion makes the galaxy distribution anisotropic and we have to include this eect in the power spectrum estimator. In the following we describe the estimator for the redshift power spectrum.

The general expression for the power spectrum in redshift space can be given in terms of Legendre polynomials (Taylor and Hamilton, 1996):

P (k, z) = P (k, µ, z) = X

l=0,2,4,...

Pl(k, z)Ll(µ)(2l + 1) , (3.10)

where Ll are the Legendre polynomials, µ is the the directional cosine be-

tween the line of sight direction and k. For l = 0 we dene the monopole

P0(k, z) as the angular averaged power spectrum; for l = 2 we have the

quadrupole P2(k, z) quantifying the leading anisotropy in the power spec-

trum due to the redshift-space distortion. The quadrupole is an important quantity that can be used to improve the constraints on the bias and cos- mological parameters (Yamamoto et al., 2005).