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-ti
it is possible to rewrite the FRW metric using χ as a coordinate, namely
where the function r2(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 Pμ = m uμ with uμ standing for the 4-velocity. The components of Pμ satisfy the mass shell condition P2 = – m2 , 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 ≡ P0 .We can also introduce the physical momentum, which measures the rate of change in physical distances as pi =a(t) Pi
By using the connection we can compute how the momentum components Pi 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 tem with energy Eem and is then received at tobs. The energy which is measured is then Eobs = Eem a(tem)/a(tobs). 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 a0 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 tobs
Since the time difference tobs - t = d/c with d the physical distance of the source, we get the famous Hubble’s law
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 Rμν 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, Tμν 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
from which we get
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 ρR ≈ a-4 for ω=1/3 (Radiation), ρM ≈ a-3 for ω=0 (Matter), ρΛ ≈ 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)≈ t2/3. 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
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 horizon χp(t), defined as the comoving distance travelled by light between a very early initial time ti 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
tem 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) = a0/(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 zLS ≈ 103. For a universe MD the comoving distance of this surface is therefore
Such value would increase if we had included the contribution of ΩΛ (for ΩΛ = 1 one would get χLS = zLS/H0. 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 dL we have to take into account that:
1) the signal received have propagated over a physical distance a(tobs) 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(tobs)2 r2(χ)
2) the energy of photons received is smaller than at the emission, by a factor a(tem)/a(tobs) due to redshift
3) the arrival rate is also lower by the same factor. In fact we know that
Moreover a radiation emitted at tem+δt will be received at the time tobs+δt’ such that
Thus the flux Φ results
and the luminosity distance results
For a spatially flat metric (k=0) and for ΩR << ΩM this expression simplifies for small redshift, namely
Expanding at small redshifts up to the first order in z, assuming k=0 and ΩM +ΩΛ=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 H0 and q which intervein in enlarged analysis of cosmological parameters.
6. Dark Matter
11. Cosmic Rays - I
12. Cosmic Rays - II