The assumption that our Universe is spatially homogeneous and isotropic implies that it is possible to choose a suitable set of coordinates in which the space- time metric is

where *t* is the “physical time”, while *r*, θ and Φ are the spatial comoving coordinates which labels the points of the 3-dimensional constant time slice. They are not sensitive to the expansion dynamics and define comoving observers, namely those which have constant values of *r*, θ and Φ as time flows.

The physical distances are weighted by the scale factor *a(t)*, and are thus increasing with time in an expanding Universe. The parameter k specifies the spatial curvature of the model. By a suitable redefinition of the radial variable r it can be reduced to the canonical values *k = +1, 0, −1* corresponding to a constant positive curvature 3-space (a 3- sphere), a flat 3-dimensional plane and a negative curvature 3-dimensional space, respectively. For the flat case using cartesian coordinates the FRW metric becomes

By using the conformal time η defined as

the metric then becomes

In terms of the comoving distance χ(t) travelled by the photon in the time interval t-t_{i}

it is possible to rewrite the FRW metric using χ as a coordinate, namely

where the function r^{2}(χ) is

In general the freely falling motion of a particle in a given spacetime metric is given by the geodesic equation

where λ is some monotonically increasing variable along the particle path and the Christoffel symbols are related to first derivatives of the metric as.

In the FRW metric and for k=0 the only non vanishing components are the following

with the dot denoting the derivative with respect to physical time. For a particle of mass *m* one can introduce the energy-momentum four vector as P^{μ} = m u^{μ} with u^{μ }standing for the 4-velocity. The components of P^{μ} satisfy the mass shell condition P^{2} = – m^{2} , and also obey the geodesic equation

The geodesic equation allows to compute the evolution in time of the energy and momentum of particles, and gives one of the main observational consequences of an expanding Universe, the redshift of energy/momentum of photons, and more generally the fact that physical linear momentum of all particles decreases as 1/a as the Universe expands.

Using the mass shell condition for k=0 (the general case can be treated in close analogy using χ as radial variable)

where E ≡ P^{0} .We can also introduce the *physical* momentum, which measures the rate of change in physical distances as p^{i} =a(t) P^{i }

so that

By using the connection we can compute how the momentum components P^{i} changes with the expansion, namely

from which we see that

; ;

In the case of photons, this result gives a simple description of the observed redshift of wavelength. If t a photon is emitted by some astrophysical source (galaxy, star, etc.) at t_{em }with energy E_{em }and is then received at t_{obs}. The energy which is measured is then E_{obs }= E_{em }a(t_{em})/a(t_{obs}). In terms of the radiation wavelength

which defines the redshift *z *. The redshift of a given object is related to the scale factor at emission time through the relation

where a_{0} is the scale factor today, usually set to unity in the case of a flat Universe.

For small redshifts we can expand up to the first order the previous expression around present time t_{obs}

Since the time difference *t _{obs }- t = d/c* with

The Hubble constant has dimension t^{-1} and its order of magnitude is of a few tenth of km s^{-1 }Mpc^{-1}. It is usually expressed in terms of the adimensional parameter *h*

The dynamics of the expansion is completely encoded in the function a(t) which appears in the metric, and it is the dynamical variable in the Einstein equations

where R_{μν} and R are the Ricci tensor and scalar, respectively. Moreover Λ is the Einstein cosmological constant and G the Newton constant. In natural units G defines a mass scale, the Planck mass scale

Finally, T_{μν} stands for the energy-stress tensor of all matter species filling the Universe. This tensor is symmetric and is covariantly conserved

In many cases one can approximate a gas of particles as a perfect fluid with no viscosity, thus the energy-stress tensor reads

If the fluid has vanishing velocity in the comoving frame we get

The characteristic of the fluid filling the universe is encoded into its equation of state P=P(ρ). For relativistic particles (usually denoted as “radiation”) P=ρ/3, while for nonrelativistic particles (dust or simply “matter”), P = 0. In these relevant cases the equation of state is a simple linear relationship P = ω ρ. Even the cosmological term can be seen as a further contribution to the energy-stress tensor

that implies …

this means a fluid with ω = -1. The energy-stress tensor gets contribution from all particles filling the universe can be seen,at some extent, as a gas with Bose-Einstein or Fermi-Dirac statistics (this is not true in the inflationary and accelerating stages, which will not be considered for the moment). In this case the energy-stress tensor can be given in terms of the distribution of particles in phase space. In particular we obtain

and

from which we get

and

For massless particles *E=p* hence we get *P=ρ/3*, whereas for non-relativistic particles *E ≈ m* and thus *ρ ≈ m n* and

The covariant conservation of stress-energy tensor, for an ideal fluid , using *ν=0* gives

*which can be solved by ρ = a ^{-3(1+ω)}. In particular one gets ρ_{R} ≈ a^{-4} for ω=1/3 (Radiation), ρ_{M} ≈ a^{-3} for ω=0 (Matter), ρ_{Λ} ≈ const. for ω=-1 (Cosmological constant).* The 0-0 component of the Einstein equations gives (Friedmann Equation)

that together with the previous equation allows for determining *a=a(t)*. Note that the off-diagonal components of the equation do not give contribution due to the isotropy of space-time, and that the i – i component does not add further conditions. Differentiating the Friedmann Equation and using the relation for *ρ we have*

For an Universe radiation dominated, i.e. the main contribution to ρ comes from relativistic particles, one gets a(t)≈ √t. If the Universe is dominated by non–relativistic particles, i.e. matter dominated, one gets a(t)≈ t^{2/3}. A cosmological constant yields an exponentially growing scale factor.

Defining the critical density as

the Friedmann equation can be rewritten in the form

where . From the previous equation one can see that the spatial curvature is fixed by the value ρ. A critical Universe, namely with Ω=1, is spatially flat. Defining the spatial contribution to the energy density as

we have the sum rule . The critical density today (a=1) is

by using this quantity we can measure the contributions of Radiation, Matter, Curvature and Cosmological Constant in terms of

Hence the Friedmann equation gives

For small redshifts the expansion rate can be Taylor expanded. Up to the first order the deceleration parameter

reads

To define a distance in an expanding Universe is more tricky than in a static background. To determine how far is a source from the observation point it is necessary to take into account the cosmological expansion between the emission and observation times. According to the particular operational definition of distance one gets different results (luminosity distance, angular distance, etc.).

Let usconsider the so-called comoving *particle horizo*n χ_{p}(t), defined as the comoving distance travelled by light between a very early initial time t_{i} and some time t. Since photons propagate along null geodesics we get in FRW

where we have assumed a vanishing initial scale factor. This quantity represents the largest distance from which an observer can receive information at a given t. Analogously, the light received today from a source with redshift z has travelled the comoving distance

t_{em} is the emission time, and can be obtained in terms of z if we know the expansion history of the Universe, namely inverting the relation a(t) = a_{0}/(1+z(t)).

CMB Photons were emitted from a thin surface corresponding to their last interaction with charged particles. Such surface of last scattering surface is placed at redshift z_{LS }≈ 10^{3}. For a universe MD the comoving distance of this surface is therefore

Such value would increase if we had included the contribution of Ω_{Λ }(for Ω_{Λ }= 1 one would get χ_{LS }= z_{LS}/H_{0}. In case of a particle with mass m, such as neutrinos, we have

As stated before, the distance we get depends on the operational definition we use. If we look at a point–like source, like galaxies, stars etc, of which we know the absolute luminosity L (denoted as standard candle), the physical distance d of the source can be defined by measuring the energy flux Φ observed at the detector

Such relationship allows to define the luminosity distance as

To compute the expression of d_{L} we have to take into account that:

1) the signal received have propagated over a physical distance a(t_{obs}) χ , with χ the comoving distance travelled by photons to reach us. Since we assume an isotropic emission, the sources are homogeneously distributed over a thin spherical shell with surface 4 π a(t_{obs})^{2} r^{2}(χ)

2) the energy of photons received is smaller than at the emission, by a factor a(t_{em})/a(t_{obs}) due to redshift

3) the arrival rate is also lower by the same factor. In fact we know that

Moreover a radiation emitted at t_{em}+δt will be received at the time t_{obs}+δt’ such that

which implies

Thus the flux Φ results

and the luminosity distance results

For a spatially flat metric (k=0) and for Ω_{R} << Ω_{M} this expression simplifies for small redshift, namely

Expanding at small redshifts up to the first order in z, assuming k=0 and Ω_{M} +Ω_{Λ}=1, we get again the Hubble law. Up to the second order we have instead

By using SN Ia as standard candels, one can obtain precise measurements of H_{0} and q which intervein in enlarged analysis of cosmological parameters.

*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

Progetto "Campus Virtuale" dell'Università degli Studi di Napoli Federico II, realizzato con il cofinanziamento dell'Unione europea. Asse V - Società dell'informazione - Obiettivo Operativo 5.1 e-Government ed e-Inclusion