The theory of cosmological perturbations in an expanding universe is the main frame to describe the structure formation and to calculate the flactuations measured in the Cosmic Microwave Background radiation (CMB). Even though at large scale the universe appears essentially homogeneous and isotropic, nevertheless at some extent small perturbations had to be present in the primordial universe in order to allow the formation of galaxies, and clusters that we observe at present. The idea is that small primordial perturbations have grown up due to the expansion till to reach the scales of cosmological/astrophysical structures. The theory of cosmological pertubations studies the evolution of such primordial disomogeneities. The gravitational instability is a natural property of gravitation, and the presence of small regions of overdensities yields an attractive force on the matter placed around, which collapsing on the overdensities enhances the process. Nevertheless, such positive feedback, that in a static universe would cause an exponential instability, is partially counterbalanced by the universe expansion. This results in a power law growth of perturbations.

From a mathematical point of view, one must solve Einstein’s equations in a framework of an expanding universe, taking into account at the same time the freedom to choose a generic reference frame (gauge choice). Such freedom is responsible for the appearence of fictitious perturbations, which do not describe any real disomogeneity, but rather the properties of the reference frame chosen.

Let us consider for example a homogeneous and isotropic universe, whose energy density be described by

In a different reference frame, the new time coordinate is related to the previous one by the transformation

In the new frame (see figure), assuming , results

Note that the perturbation has no physical meaning but is only due to the choice of reference frame. In an analogous way one can eliminate real perturbations through the choice of particular frames. Since physical quantities have to be gauge independent, it has been developed a gauge-invariant formalism which will be briefly reviewed in the following.

Let us assume that the space-time is in first approximation homogeneous and isotropic and the deviation from this behaviour is small enough to treat it as a perturbation. In this case it is convenient to split the metric in two contributions: an homogeneous and isotropic one describing the background, and an additional term representing the perturbation.

where η is the conformal time dη=a^{-1} dt. In case of FRW metric we have

where K=0,1,-1 as previously discussed. The Einstein’s equation in natural unit reads

where R^{μ}_{ν} is the Ricci tensor, R the scalar curvature, whereas T^{μ}_{ν} is the energy-momentum tensor. Using the FRW metric the equations become

where the symbol ‘ stands for d/dη .

For an isotropic universe, as described by FRW, the spatial part of Ricci tensor, R_{i}^{j}, is proportional to a δ_{i}^{j}, hence to satisfy the Einstein’s equation the spatial components of energy-momentum tensor have to be diagonal . From the Einstein’s equation one derives the continuity equation for matter

,

namely

By adding the perturbation to the background metric one obtains the complete line element

where the perturbations are encoded in the term .

In the linear approximation different perturbations evolve independently and hence they can be studied separately. The perturbation of a metric can be of three kinds: scalar, vectorial and tensorial. The classification concerns the way in which the different fields involved in transform under the Lorentz group. Vectorial and tensorial perturbations do not show instability. The first ones decay in an expanding universe, whereas the tensorial lead to gravitational waves that do not couple to disomogeneity in the energy density and pressure. Scalar perturbations are more interesting since they can have a role in the growing of disomogeneity and thus affect the dynamics of matter.

There are two ways to form a tensor like δg_{ij} starting from a scalar field:

1) multiplying the tensor γ_{ij} for a scalar field;

2) deriving two times a scalar field by means of the covariant derivative.

Two additional scalar fields are then necessary to fully characterize δg_{μν} . One is required for δg_{00}, whereas δg_{0i} can be derived in terms of the spatial derivative of a second scalar field, respectively.

In summary, the most general form of a scalar perturbation of the metric can be written by using four scalar fields: φ, ψ, B and E

In terms of such expression the complete line element reads

vectorial perturbations can be written in terms of two vector fields, hereafter denoted by S_{i} and F_{i} , which satisfy the following conditions

Since each vector field can be split into a solenoidal part (divergenceless) plus the gradient of a scalar, if such conditions were not satisfied the scalar component would be in general not vanishing, and hence the perturbation would result not purely vectorial. The perturbation to the metric for the vectorial component results

The tensorial perturbations are constructed in terms of a symmetric tensor h_{ij}, which satisfies the conditions

Such conditions ensure that h_{ij }does not contain terms behaving as scalars or tensors. The metric for tensorial perturbations reads

Let us consider a generic coordinate transformation

that is characterized by the space-time function ξ^{μ}. In a particular point of space-time the metric tensor can be written in the reference frame by using the transformation law

where we have considered the linear terms in δg and ξ only. In the new frame we can separate the metric in the background and perturbation as

where denotes the FRW metric in the reference frame . Comparing the two previous expressions for the metric and considering that

we get the following law of transformation

It is possible to rewrite the spatial components of the vector ξ as

where is a vector divergenceless and a scalar function.

Let us choose an unperturbed FRW metric, spatially flat (K=0). From the gauge transformation condition we get

From which it is possible to obtain the transformation law for each kind of perturbation.

- The gauge transformation for a scalar perturbation -

By using the form of the perturbed metric in this case we have

Combining φ, ψ, B and E it is possible to define an infinite number of gauge-independent variables. The most simple combinations of these fields are

The variables Φ and Ψ have been introduced by Bardeen for the first time, they have an intuitive physical meaning and have simple EoM.

The gauge freedom can be used to fix additional conditions on the function φ, ψ, B and E, this by choosing in a proper way and .

A particular choice of reference frame consists in fixing B=E=0 (longitudinal gauge). In this case the condition E=0 fixes , which consequently by using the condition B=0 determies univocally . In this gauge the scalar variables of the metric take the form

In the longitudinal gauge φ, ψ correspond to the gauge-invariant variables Φ, Ψ. Finally, in the longitudinal gauge the metric takes the form

Whenever the spatial part of energy-momentum tensor is diagonal we have or , hence it is sufficient a single variable to describe the perturbation of the metric, which in this case can be regarded as a generalization of Newton gravitational potential. This is at the basis of the alternative name given to the longitudinal gauge, i.e. *conformal newtonian gauge.* Observing the form of the metric, the gauge-invariant variables have a clear physical meaning they represent the amplitude of metric perturbations in the conformal newtonian gauge.

For vectorial transformations the metric reads

and the variables S_{i} and F_{i} transform as follow

Obviously, the variable

is gauge-invariant, hence only two of the four independent S_{i} and F_{i }characterize the physical perturbation.

For tensorial transformations the metric reads

and h_{ij} is invariant for coordinate transformation. Such field describes in a gauge-invariant way the gravitational waves.

In order to derive the EoM for the perturbations is necessary to linearize the Einstein’s equations

for small disomogeneities in a FRW. The Einstein tensor becomes

where . As already stated before in order to satisfy the Einstein equation the energy-momentum tensor must obey the following conditions

For small perturbations one has

and once linearized the equations become

However, neither nor are gauge invariant, but combining them with the metric perturbations one can obtain the following gauge-invariant quantities

In an analogous way for we have

However, it is possible to rewrite the EoM for the perturbations in the following form

In particular the term can be rewritten as a function of the gaige-invariant variables Φ, Ψ.

The general form of the gauge-invariant EoM for cosmological perturbations

The Equations have been obtained without fixing any gauge condition and hold in any reference frame. In order to complete the set of equations it is necessary to know the EoM for the matter fields.

At high energy it is reasonable thinking to describe the matter in terms of fields. In many models of early universe are hypothesized stages dominated by scalar field dynamics. The main characteristics of a scalar dynamics concerns the possibility to have a not vanishing constant field background without breaking Poincaré invariance. If the field is trapped in a potential minimum, at a constant energy density would correspond a de Sitter phase. In this phase the universe scale factor would grow exponentially (better known as inflation). If we consider the matter field described by a scalar field minimally coupled with gravity its action reads

where V(φ) is the potential of scalar field. The corresponding energy-momentum tensor

In a homogeneous and isotropic universe, where the scalar perturbation of the metric have been previously described, even the scalar field can be approximated as almost homogeneous. In this case it is possible to decompose in two parts, namely where is the homogneous part, whereas is a small perturbation. In the same way the energy-momentum tensor can be decomposed in a background term plus a perturbation

where is a linear function of the perturbation of the scalar field and of the metric .

By using the previous expressions one can obtain the background components of the energy-momentum tensor

and the perturbation contribution up to the first order

In order to write the EoM for the cosmological perturbation it is necessary to obtain the gauge-invariant perturbation of the energy-momentum tensor

where Φ is the potential gauge-invariant and is the gauge-invariant part of the scalar field perturbation.

The gauge-invariant EoM for cosmological perturbations in a universe dominated by a scalar field dynamics can be obtained by substituting these last expressions in the general equations for cosmological perturbations (see slide 11). From the equation i-j (i ≠ j) one gets Φ = Ψ, hence we have

where . Combining such relations we get the EoM for , however it results much simpler to obtain the EoM by linearizing the Klein-Gordon equation for the field . Hence we get

It is possible to get the time evolution of the fluctuations from the previous equations and in particular one gets

The previous equation can be rewritten as follows

where the solutions can be easily found in the asymptotic limit by considering plane waves perturbations .

For perturbation of small wavelength

it is possible to neglect the last term of the equation and really obtain . For perturbations of large wavelength

the second term of the equation can be neglected so obtaing

where C_{1}, C_{2} and A are integration constants.

The expressions for Φ and can be obtained by using the definition of u and its EoM. For small wavelength we have

whereas for large wavelength we get

where “the dot” denotes the derivation with respect to physical time .

Among the main applications of the theory of cosmological perturbations plays a main role the study of inflationary models. As already stated in the previous trasparencies the great advantage of inflation is to allow, for perturbations produced inside the Hubble radius, to grow up till to reach scales of the order galaxies and clusters. All this is possible if the inflationary period is larger than 65 H^{-1}. Inflationary models characterized by a single inflationary period during which the Hubble parameter is constant predict a Harrison-Zel’dovich spectrum (all modes are amplified in the same way). By using the results obtained for fluctuations with small wavelenght, the amplitude of metric perturbation Φ results proportional to . During inflation, when

the variation of Φ is negligible. In particular for a quadratic potential the amplitude is exactly constant. After a sufficient expansion, the perturbations with wave number k reach the regime of large wavelength, and hence

where the expansions have been obtained by means of iterative integrations by parts. Note that the series result to be of asymptotic type.

During inflation, when

it is possible to consider the first term only. Moreover, taking into account that in this period we also have

and using the EoM for the background we have

At the end of the inflationary period, the scalar field decays in relativistic dof and the universe scale factor grows up with time according to a power law . From this it follows that

which is almost independent of time. In this case, A is the same constant already appearing in the previous two relations from which one gets

In the previous equation it is necessary evaluate

and the Hubble parameter H at the time in which

namely, when the wavelength of the perturbation crosses the Hubble radius.

By using such formalism is not necessary to make any assumption on the duration of reheating, and on its particular mechanism, whenever perturbations leave the Hubble radius before reheating to reenter only at a later time. Moreover it is necessary to assume that the duration of reheating be negligible with respect to the inflationary epoch as well.

*1*. Rèsumé of standard cosmology in FRWL

*2*. Thermodynamics of the expanding universe

*5*. Baryogenesis

*6*. Dark Matter

*7*. Primordial Nucleosynthesis: theory and experimental data

*8*. Theory of classical cosmological perturbations

*9*. Theory of Quantum Cosmological Perturbations

*10*. A Brief Introduction to Cosmic Microwave Background Anisotropy Formation

*11*. Cosmic Rays - I

*12*. Cosmic Rays - II

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