The hot Big-Bang scenario and its problems

Observation suggests that we live in a homogeneous and isotropic patch of the Universe. The hypothesis of homogeneity and isotropy of the Universe translate into the fact that the line element can be written under the form 

The expansion of the Universe in encoded into the time dependence of the scale factor ad ). The quantity t corresponds to the cosmic time. One can easily show that observers situated at xl = Const follow geodesics, therefore cosmic time corresponds to the proper time of such observers. It is assumed that any matter element is at rest or almost at rest with respect to the spatial part of the coordinate system. For this reason, the xl are called comoving coordinates, and any distance measured using line element ijdx dx3 is called comoving distance. 

It is very convenient to introduce the conformal time, 7, defined as 

The interpretation of this quantity is the following consider a photon, or any massless particle, which travels along the direction. By definition the line element for such a particle is null, therefore one has drj = d -. Since there is a correspondence between p and t through Eq. (7.6), Ar/ can therefore be seen as the maximal distance that can be traveled by any particle between time /,i(//) and taiv + Ar/). When r/ varies from — oo to some finite value today, any observer has been in causal contact with any observer in the past. On the contrary, if p has varied between two finite values, then only a fine region of the Universe has been in contact with the observer. Equivalently if asymptotically p goes to infinity, an observer can send signal to any observer, whereas if p reaches a finite value in the future, only a finite set of observer can receive 

The matter content of the Universe is described as an ensemble of perfect fluids of density pj and pressure Pf. An important quantity is the equation of state parameter Wf defined as 

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