Large Scale Structure Formation

The real Universe is not exactly homogeneous and isotropic. At small scales, clustering effects are non-negligible: for example, near a black hole, the more appropriate metric is Schwarzschild metric. First-order perturbation theory can be used to find approximate solutions to Einstein equations. These pertur- bations are the seeds of the Large Scale Structure (LSS) we see and observe ev- ery day. They have been originated by quantum fluctuations blown up during the Inflation. They are uncorrelated and the distribution of their amplitudes is Gaussian. The growth of their amplitudes has been caused by gravitational instability, and, since the relative density contrast is much smaller than unity, these deviations from isotropy and homogeneity can be described using linear perturbation theory. These perturbations can be visualised in CMB anisotropy map, since these latter are a direct tracer of density fluctuations.

There are three kind of perturbations that evolve independently: scalar, such as the density perturbations, vector, and tensor, which are the gravita- tional waves. In order to describe the behaviour of an ensemble of interacting particles, one can apply Boltzmann’s equation for the phase space distribution function

D fi

Dt = C[ fj], (2.25)

where on the left-hand side we have the total time derivative of this distribu- tion function for particle species i, while on the right-hand side we have the effect of all the possible interactions, which can possibly depend on the distri- bution function of all other species j. Hence, perturbations of matter and ra- diation can be computed by numerically solving a set of second-order partial differential equations in a certain gauge, that is the correspondancy between physical space-time and FRW one. Working in a certain gauge can produce dif- ferent results from the ones obtained from another gauge. For example, work- ing in a fixed gauge could create some fake modes, called gauge modes, that are non-physical, since they disappear if we move on another gauge. Another choice consists in working with gauge-invariant objects which are linear com- binations of perturbations. Interested readers can find a useful reference in Bartolo et al. (2007). Codes like CAMB1_{can solve numerically this set of differ-}

ential equations, and are used by the majority of the scientific community. Linear Growth Function

We are mainly interested in studying scalar perturbations, i.e. matter den- sity perturbations. We define the density contrast:

δ(x, z) = ρ(x, z) ¯

ρ(z) − 1, (2.26)

where ¯ρ(a) = ρ0a−3is the average cosmic density. The overdensity always sat-

isfies the inequality −1 ≤ δ ≥ ∞. Applying relativistic and non-relativistic per- turbation theory, we can see that perturbations with δ  1 evolve like an−2_,

with n = 4 before Equivalence and n = 3 after that time. In fact, the result of the second-order differential equation generally has decaying and growing solutions, and the growing ones will dominate over time. The key source term ensures them to grow for gravitational instability, identifying the source of per- turbations evolution. The first derivative term in perturbation acts as a friction term, which causes fluctuations to grow slower than they would in a static uni- verse. So perturbations do not evolve exponentially, but they rather grow as a power-law. Not all scales are allowed to grow in this way, but only modes larger than a characteristic scale, called Jeans scale, can grow. This Jeans length cor- responds to the minimum mass that a structure can have to gravitationally collapse and be formed. This mass is MJ ∝ c3sρ¯−1/2, where cs is the speed of

sound. So the Jeans scale λJacts as a sound horizon for matter density pertur-

bations. For example before Recombination, photons and baryons are tightly coupled, causing the speed of sound to be close to that of radiation, namely cs ' c/

√

3. In this regime the Jeans mass is way larger than that of any other known object in the Universe, of order 1019_M

. Only CDM can collapse, while

baryonic structures can not be formed by gravitational collapse. After Recom- bination, this mass is of order 105_M

, and recombination radiation pressure is

not able to block the formation of cosmologically interesting structures. It can be proved that during RDE, perturbations grow extremely slowly (log- arithmically). During MDE, at redshifts z  zeq, where the Universe hasΩm ,

1 or Ω_{Λ , 0, density perturbations evolve proportionally to the scale factor,} namely δ(t) ∝ a(t). During DE era, which will presumably happen in the future, structures will not grow, allowing for static solutions. In general, when a  aeq

withΩm , 1 or ΩΛ , 0, perturbations evolve according to

δ(a) = δ0a

g0(a)

g0₍₁₎ = δ0ag(a), (2.27)

where δ0 is the density contrast extrapolated to the present epoch and g(a) is

the so-called growth function (Carroll et al., 1992; Van Waerbeke & Mellier, 2003) which is a function of redshift (or scale factor), namely

g(z)= 5 2ΩmH(z) Z a 0 da0 a03_H3_(a0₎. (2.28)

This represents the growth relative to that in an Einstein-de Sitter Universe. The growth function has to satisfy, for any DE equation of state ω(z), the di- mensionless form of the following second order ordinary differential equation

2d 2 g(a) d ln a2 + [5 − 3ω(a)ΩDE(a)] dg(a) d ln a + 3 [1 − ω(a)] Ω(a)g(a) = 0. (2.29) We conclude this section demonstrating how DM can be argumented also using the growth of linear perturbations. CMB shows temperature fluctuations of order 10−5_{that are linked to density fluctuations of the same order of magni-}

tude by Sachs-Wolfe effect. CMB originated at redshift well after the Universe became matter-dominated, where its fluctuations should reach a level of 10−2_,

that CMB displays only baryonic matter component fluctuations, but δ(a) is made of baryonic and Dark Matter components. We know that this additional matter component interacts only through weak gravitational interactions with the rest of known physics. We already realised that DM component will de- couple from cosmic plasma well before baryonic matter, and their fluctuations have all the time to reach the amplitudes observed today. These fluctuations, which will evolve indipendently in Fourier space, do not collapse because of a growth suppression experienced during the RDE: the expansion scale, domi- nated by radiation, will be be smaller than the typical DM collapse scale. This suppression is restricted to scales that are smaller than the horizon, i.e. the size of causally connected regions in the Universe. A scale will enter the horizon if k = d−1_h (a). Large-scales perturbations remain unaffected preventing dark mat- ter fluctuations from collapsing. This suppression factor is f = (k0/k)2, where

k = 1/λ is the independent Fourier mode and k0 = 1/dh

 aeq



is the horizon size scale at Equality. This is roughly 0.083(Ωmh2)Mpc−1 ' 10−2Mpc−1. Remem-

ber that the horizon distance is different from the Hubble radius c/H0 and a

scale will enter the Horizon when its length is comparable to the Hubble length 1/H. So, if the perturbation is smaller that the horizon at Equality, it enters the sound horizon at zenter < zeqduring RDE. This prevents perturbations from

growing further, while perturbations bigger than the horizon at the Equality will keep growing.

The Primordial Power Spectrum of Density Fluctuations

As previously stated, in order to solve some BBT inconsistency and to ex- plain why CMB is so isotropic and presents super-horizon correlations, we need an early period called Inflation. We will not give a full review for it (which can be found in Baumann (2009) for example), but we will only mention its consequences and how these are linked to LSS formation. Inflation is basically a period of accelarated expansion (¨a < 0) driven by a slow-rolling scalar field, in which the early Universe occupied a very small volume. This blown up so largely and so quickly that any inhomogeneities or curvature in such volume are smoothed out, diluting the density of non-relativistic particles. During this period, the horizon scale is much larger than the Hubble length at that time (where H was constant), and small-scale quantum fluctuations are blown up very quickly. Once they are larger than the apparent horizon set by the Hubble length, they are frozen in and behave as completely classical fluctuations. So, when a fluctuation re-enters the Horizon, overdensities begin to collapse.

One of the most powerful predictions coming from Inflation theory, other than the creation of a potentially detectable primordial gravitational waves background, is that primordial density perturbations can be modeled as Gaus- sian Random Fields (GRF) having scale-invariant power spectrum. This has also been confirmed with CMB measurements, since the temperature anisotropy power spectrum is proportional to the matter density fluctuation one. We give a review of GRF statistics in Appendix A. The Fourier transform of density fluc-

tuations will be

δ(k) =Z d3xδ(x) e−ik·x_{, (2.30)}

and their phases are random and these modes are uncorrelated with each other. All the statistical properties of a GRF are described by its variance, i.e. its cor- relation function at the comoving position x, hδ(x)δ(x0_)i = ξ (|x − x0_{|). The}

Fourier transform of the correlation function is the Power Spectrum P(k), whose definition is

hδ(k)δ?_(k0

)i = (2π)3δD(k − k0)Pδ(k), (2.31)

where k = |k|. We can safely apply Fourier decomposition because perturba- tions scales are much smaller than the curvature radius of the Universe. The scale-invariant power spectrum of primordial density fluctuations is called Harrison- Zel’dovich-Peebles spectrum and scales as Pδ(k) ∝ k. Usually this is written

modifying it with a scale-dependent transfer function T (k),

Pδ(k, z)= Pδ(k)g2(z)T2(k), (2.32)

where g(z) is the growth function. The transfer function encodes modes-growth information when the Universe is radiation or matter dominated. It has to be constant for k < k0and T (k) ∝ k−2for k > k0, with k0the Horizon scale at equal-

ity. So, growth perturbation modes whose wavelength was small enough to have entered the horizon during RDE could not grow until the Universe be- came matter dominated. Longer wavelengths, which entered the horizon dur- ing MDE, have not been suppressed. Because of this, the power spectrum pic-

Figure 2.2: The processed power spectrum with scale-invariant Harrison-Zel’dovich- Peebles power spectrum Pδ(k) ∝ k.

tured on Figure 2.2, is parametrised as Pδ(k, z)= ( Akns, _{(k  k} 0) Akns−4, _{(k  k} 0) . (2.33)

nsis the spectral index of primordial power spectrum. This is measured, and

0.9655 ± 0.0062. A is the normalisation of the power spectrum. This scaling can be predicted by a making a simple argument. Since the matter density contrast grows as δ ∝ an−2_{, its power spectrum will scale as P}

δ(k) ∝ a2(n−2).

When perturbations enter the horizon scale the total power spectrum is scale- invariant. Since Penter ∝ a2(n−2)_enter Pδ(k) ∝ k−4Pδ(k)for k  k0. So, this implies that

Penter(k) ∝ k−3, while the primordial power spectrum scales as Pδ(k) ∝ k. One

of the biggest confirmations for the validity ofΛCDM model comes from pri- mordial power spectrum measurement, since this is well approximated by a power spectrum produced by CDM density fluctuation in an universe whose expansion is driven by Dark Energy. We can see the detection of this power spectrum from several techniques in Figure 2.3. We have to take into account that observation will detect smoothed fluctuations up to a certain characteris- tic cut scale ks. Such a field will be the convolution of the field with a window

function (it can be Gaussian or Top-Hat), and the resulting power spectrum is P(k, ks)= |W(k, ks)|2P(k).

Figure 2.3: The observed power spectrum of density perturbations, as measured from a variety of techniques. Taken from Tegmark & Zaldarriaga (2002).

This spectrum is sensitive to Dark Matter nature: if DM is made of rela- tivistic particles, the produced fluctuation needs to have a minimum size in order to keep them gravitationally bound. All the perturbations smaller that this size will be smoothed away by free-streaming of particles, and the power spectrum will show an exponential cut-off at large k. From this we can de- fine the Hot Dark Matter (HDM) as particles that smooth small-scale perturba- tions, while the Cold Dark Matter (CDM) particles are slow enough to cause no significant damping. Without taking into account exotic particles such SUSY ones, DM is well modelised by massive and collisionless particles which fell out from equilibrium at very early times, and Weakly Interacting Massive Par- ticles (WIMPs) are among the leading candidates, disfavouring a top-down

HDM scenario driven by relativistic particles like neutrinos. Cosmological ob- servations clearly favor the CDM bottom-up scenario instead, where massive objects form first. Recently there have been attempts trying to implement into simulations an evolution made by Warm DM.

We can note that power spectrum normalisation has to be fixed. It is pos- sible to measure it using several different ways which have different results. We can have it from CMB anisotropies, but the computed normalisation is good only for small scales and CMB is sentitive to the amplitude of scalar and tensor perturbation modes, and not to scalar only, which determine the fluctuation growth. We can measure it from local variance of galaxy counts, since they trace DM fluctuations, but the result will be biased because of its dependance from galaxy formation mechanism and statistical sampling. The normalisation can be also measured from local abundance of galaxy clusters, since the cluster formation will sensitively depend on the amplitude of dark- matter fluctuations. In this case the normalisation can be measured only on scales of order 10h−1_{Mpc. Finally it can be measured from gravitational lens-}

ing by large-scale structures, since this detection is sensitive to scales of order k−1₀ ∼ 12Ω0h2



Mpc. Usually this factor is denoted as σ8, which is the vari-

ance of mass for a sphere with radius 8h−1_{Mpc and with a top hat smoothing}

window such that

σ2_{(R) = hδ}2 Ri= Z d3kP(k)W2(kR) = Z ∆2_(k)" 3 j1(kR) kR #2 d ln k, (2.34)

where∆2_{(k) = k}3_P(k)/2π2 _{is the dimensionless power spectrum usually imple-}

mented in numerical codes, and j1is the type 1 Bessel function. This is the rms

amplitude of mass fluctuations smoothed over a scale R. Planck analysis gives a value σ8 = 0.829 ± 0.014.

There is another effect to consider on primordial power spectrum. At small scales, the growth of density fluctuations begins to depart from linear behaviour, and so fluctuations of different sizes begin to interact. In this case the com- putation of P(k, z) becomes complicated and numerical methods, such as N- body simulations, are generally required. In dimensionless notation this hap- pens when∆2 _{≥ 1. There is a semi-analytic derivation that works well but it is}

made under the ansatz that the two-point correlation functions in linear and non-linear regimes are related by a general scaling relation. However these ap- proaches can provide only expectation values of non-linear power spectrum. The most used fitting formula adopted for this non-linear transfer function is the one derived by Eisenstein & Hu (1998).