De las categorías centrales

As described above, inflationary models predict the initial density fluctuations to be Gaussian distributed. If this is the case, then the statistical behavior of the initial seed perturbations is completely determined by its two point statistics or accordingly the matter power-spectrum. Considering the density contrastδi ≡δ(xi) defined at the co-

moving positionxi to be a stochastic variable we can define the joint probability distribution of the initial density

field as a multivariate Gaussian:

P({δi}:i=1, ...,N)= 1 √ det(2πS)e −1 2 PN i,j=1δiS−1i jδj, _(3.48)

withN being an arbitrary positive integer andSi j =hδiδjiis the according covariance matrix (see e.g.Lahav &

Suto 2004). As can be seen from equation (3.48), the Gaussian distribution is completely specified by the two point correlation functionSi j =ξ(xi,xj). However, although the two point statistics are of undoubtable importance,

they do not provide any information on the phases of the initial density fluctuations. Hence, there exists no phase correlations in a Gaussian random field. Also note, that the Gaussian assumption has to break down at later epochs of structure formation since it predicts density amplitudes to be symmetrically distributed in the range−∞< δ_i<∞

but weak and strong energy conditions requireδi≥ −1. The Gaussian assumption is therefore strictly speaking only

valid in the limit of infinitesimal small density fluctuations|δ_i|<<1. Equation (3.48) is most conveniently written in Fourier space, where for an homogeneous and isotropic universe the Fourier space covariance reduces to a diagonal form:

hδ(k)δ∗(k0)i=(2π)3δD(k−k0)Pδ(k), (3.49) withδ(k) = R d3_{xδ(x) exp(−ikx) being the Fourier transform of the density contrast and P}

δ(k) is the according density power-spectrum. Due to the diagonal structure of the Fourier space covariance matrix, in a homogeneous, isotropic universe different Fourier modes are uncorrelated and their probability distribution is given by:

P(δ(k))= √ 1 2πPδ(k)e −1 2 |δ(k)|2 Pδ(k) . _(3.50)

Therefore, if the initial density fluctuations in a homogeneous and isotropic universe were Gaussian distributed then the matter power-spectrum, completely characterizes their statistical properties. Also note, that the assumption of homogeneity and isotropy of the Universe can, in principle, be tested by measuring the off-diagonal elements of

hδ(k)δ∗(k0)i. As already discussed above, the Gaussian assumption only applies to density amplitudes|δi| << 1.

However, gravitational amplification will dramatically increase the amplitudes of the density field and will also introduce mode coupling as well as phase correlations. Hence, the evolved density field in the non-linear regimes will not be isotropic and homogeneous, due to mode coupling, and will also not be Gaussian distributed, due to phase correlations. The statistical properties of the evolved density field will be discussed below.

3.3.2.1. The matter power-spectrum

P

(k)

If the initial density fluctuations in a homogeneous and isotropic universe were Gaussian or close to Gaussian, then the detailed functional form of the matter power-spectrumPδ(k) is of obvious relevance to cosmology. The primordial shape ofPδ(k) can be estimated with the aid of linear perturbation theory as described in3.2.1.4. In linear perturbation theory gravity acts homogeneously on the density field causing each individual Fourier mode to

3.3.2 Statistical description of the large scale structure: Gaussian random fields

evolve independently:

δ(x,a)=D₊(a)δ(x)−→δ(k,a)=D₊(a)δ(k), (3.51) as long as its wavelength is small compared to the horizon sizedH=c/(aH(a)) where the Newtonian treatment is

applicable. HeredHis defined as the distance a photon could have traveled since the Big Bang. A Fourier mode

δ(k) crosses the horizon at the epochastartif its wavelengthλ=2π/kis equal to theλ =dH(astart) horizon size at

that time.

However, at early times, during the era of radiation domination, structure growth on small scales is greatly sup- pressed due to the radiation driven expansion of the Universe. This can be seen by comparing the expansion time scaletHubbleto the collapse time scaletdmof dark matter:

tHubble∝ 1 √ Gρr ≤ √1 Gρm ∝tdm, (3.52)

withρr > ρmat the epoch of radiation domination. This radiation driven expansion effectively suppresses structure

growth from astart until the epoch of matter radiation aeq. Therefore, structures first start growing at a redshift

z ' 24500 whenρr(aeq) = ρm(aeq). Then, according to equation (3.37), all fluctuations withλ < dH(aeq) are

suppressed by (astart/aeq)2and do not grow before matter radiation equality. The epochsastartat which the different

modes enter the horizon can now be expressed as a function of wavelength byλ=dH(astart). In the early Einstein-

de Sitter phase,dH(a) can be approximated as being proportional to the scale factor bya, yielding a suppression

proportional toλ2 on scales smaller than the horizon size at the epoch of matter-radiation equality. The transition from the mode suppression- to the mode growing region therefore happens at a wavelength corresponding to the horizon size at matter radiation equality, the numerical value of which is 0.025/(Ωmh) Hubble radii.

To estimate the shape of the primordial power-spectrum one commonly assumes the initial power-spectrum to be scale invariant on large scales,Pδ(k)∝kns _withn

s '1 (Harrison 1970,Peebles & Yu 1970,Zeldovich 1972).

With this assumption and the suppression of Fourier modes∝λ2 _=(2π/k)2 _∝k−2_{, the asymptotic behavior of the}

power-spectrum on small scales can be estimated to be Pδ(k) ∝ kns−4 = _k−3_{. As already pointed out before, the}

transition point between the large scale and small scale regime is related to a horizon size of roughly 0.025/(Ωmh)

Hubble radii. From this simple arguments it is clear that this point of turn over in the cosmic power-spectrum carries some information about the dark matter content of the Universe.

Nevertheless, much more accurate fitting formulas, which link these two asymptotic regimes in a smooth way have been obtained by large-volumen-body simulations. Especially the fitting function provided byBardeen et al.

(1986) has been proven to be very accurate:

Pδ(k)∝kns·T2(k), (3.53) where the transfer function is given by

T(q)= ln(1+2.34q) 2.34q h 1+3.89q+(16.1q)2+(5.46q)3+(6.71q)4i− 1 4 . (3.54) Here,qis the wave vectorkdivided by the shape parameterΓ, as introduced byEfstathiou et al.(1992) for CDM models and extended to models withΩ_,1 bySugiyama(1995):

q= k/Mpc −1_h Γ withΓ = Ωmhexp      −Ωb·      1+ √ 2h Ωm            . (3.55)

The power-spectrumPδ(k) is normalized such that the variance of the density fluctuationsδon scales ofR=8 Mpc is given by the parameter σ8. The variance of the density field on an arbitrary scaleR is usually measured by

applying a top-hat shaped filter function to the density field. This yields:

σ2 R = 1 2π2 Z ∞ 0 dk k2Wˆ2(kR)P(k), (3.56)

Figure 3.4.:Three different power-spectraPδ(k) of the overdensityδ(x): The power-spectrum for an adiabatic

cold dark matter model calculated according to eq. (3.54) (dash-dotted line), a linear power-spectrum for a universe with baryonic matter calculated according to the fitting formula provided byEisenstein & Hu(1998) andEisenstein & Hu(1999) (solid line) and the nonlinear fitting function ofSmith et al.(2003) (dotted line).

withW(x) being the top-hat filter. Its Fourier-transform is given by: ˆ

W(y)= 3

y3

_sin(y₎₋y_cos(y₎= 3

yJ1(y). (3.57)

In Fig.3.4three different power-spectraPδ(k) of the overdensity fieldδ(x) are plotted, for the case of a pure dark

matterΛCDM model, aΛCDM model with baryonic matter, and the fully evolved nonlinear power-spectrum. Here, the set of parameters (σ8 =0.9,ns=1) was assumed.