The standard cosmological model

The study of the global structure of the Universe represents one of the most exciting research fields in modern cosmology. In the last twenty years or so the collection of a huge amount of observational data has greatly contributed to improve our knowledge of ``cosmography'', so as to adequately test theoretical models about the origin and evolution of the Universe. The currently accepted view is that the cosmic structures (like galaxies, galaxy clusters and superclusters) observed today represent the result of gravitational evolution, starting from almost homogeneous initial conditions, with fluctuations of the energy density of the order $\delta \sim 10^{-5}$, which have subsequently grown by gravitational instability. This picture represents nowaday the standard cosmological model and is supported by a large variety of observations.

The idea that the Universe should be uniform led to the formulation of the Cosmological Principle, on which most of the current cosmogonic pictures are based. In one of its versions, the Cosmological Principle states that the Universe is homogeneous and isotropic in its spatial part. Under this assumption about the symmetry of the space-like hypersurfaces, it is possible to show (see, e.g., ref.[35]) that a system of coordinates can always be found in which the line element is written as

$$ds^2 = c^2 dt^2-a^2(t) \left[{dr^2\over 1-kr^2}+r^2 (d\vartheta^2+ \sin^2\vartheta d\varphi^2)\right] .$$ (1)

With a suitable definition of the units of $r$, in the above expression the curvature constant $k$ can be considered to have only three possible values; $k= 0$ for a spatially flat Universe, $k=+1$ for a closed (positive curvature) Universe and $k=-1$ for an open (negative curvature) Universe. The quantity $a(t)$ represents the cosmic expansion factor. It gives the rate at which two points of fixed comoving coordinates $(r_1,\vartheta_1,\varphi_1)$ and $(r_2,\vartheta_2,\varphi_2)$ increase their mutual physical distance as $a(t)$ increases. Its time dependence can be worked out by solving Einstein's equations for the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric of eq.(1).

If the matter content of the Universe can be described by a perfect fluid, such equations reduce to the system of two equations

$$\left({\dot a\over a}\right)^2 \equiv  H^2   =   {8\pi G\over 3} \rho+ {\Lambda \over 3}-{k\over a^2}$$ (2)

$$-{\ddot a\over a}   =   {4\pi G\over 3} (\rho +3p) ,$$ (3)

which are usually called Friedmann's equations. In eq.(2) we have also included the cosmological constant term $\Lambda$, which a number of recent observations indicate representing a substantial fraction of the total energy content of the Universe [26]. From a heuristic point of view, such equations can be seen as the equivalent of the energy conservation principle and of the second law of dynamics in classical (non relativistic) mechanics. Following the expression of the FLRW metric, two points at distance $d=a(t)r$ ($r$ is the fixed comoving distance) will move apart with a velocity $v=\dot a r=Hd$, so as to reproduce the Hubble law for the recession velocities of galaxies. Determinations of the Hubble constant $H$ by using redshift-independent methods to measure galaxy distances give us

$$H = 100 h {\rm km s^{-1} Mpc^{-1}} ,$$ (4)

where the Hubble parameter $h$ is introduced to parametrize uncertainties in the determination of the distance-scale of the Universe. While about ten years ago different determinations of the Hubble constant were discrepant by about a factor two, more recent calibrations indicate that it should be comprised in the range $0.6\raise -2.truept\hbox{\rlap{\hbox{$\sim$}}\raise5.truept \hbox{$<$} }h\raise -2.truept\hbox{\rlap{\hbox{$\sim$}}\raise5.truept \hbox{$<$} }0.7$.

Based on eq.(2), we define the critical density $\rho_c=3H^2/ 8\pi G=1.9\times 10^{-29} h^2 {\rm g cm^{-3}}$, such that density values $\rho$ above, below or equal to $\rho_c$ refer to closed, open or flat universes (in the absence of a cosmological constant term), respectively. Measurements of the cosmic mean density are usually expressed through the density parameter $\Omega_m\equiv \rho / \rho_c$. Current limits on its present value, $\Omega_m$, are

$$0.2 <   \Omega_m  \raise -2.truept\hbox{\rlap{\hbox{$\sim$}}\raise5.truept \hbox{$<$} } 1 ,$$ (5)

although some recent observations point toward low values, $\Omega_m\simeq 0.3$ [2].

Once the equation of state (i.e., the relation between energy density and pressure) is specified, the two Friedmann equations can be solved for the expansion factor $a(t)$. In the following we recall some particular solutions:

$$\Omega=1, p=0 {\rm (matter dominated)}       \Rightarrow      a(t) \propto t^{2/3}$$

$$\Omega=1, p=\rho /3 {\rm (radiation dominated)}       \Rightarrow      a(t) \propto t^{1/2}$$

$$\Omega=0  {\rm (free expansion)}       \Rightarrow      a(t) \propto t$$ (6)

$$p=-\rho  {\rm (vacuum dominated)}      \Rightarrow      a(t) \propto \exp(Ht) .$$

Note that the $\Omega_m=1$ case approximates the expansion in non-flat geometries at sufficiently early times, when the curvature term in eq.(2) becomes negligible. Viceversa, the $\Omega_m=0$ case represents the asymptotic expansion of an open Universe, when a very large value of $a(t)$ makes the density term so small that it gives negligible deceleration ( $\ddot a\simeq 0$).

One of the fundamental consequences of the Cosmological Principle is the prediction that the Universe has undergone in the past a hot phase, during which the cosmic temperature took a much higher value than that, $T_o\simeq 2.7 K$, which is today observed for the cosmic microwave background. The resulting cosmological framework of the hot Big Bang in a spatially homogeneous and isotropic Universe is so widely accepted that it received the denomination of Standard Model (not to be confused with the Standard Model for electroweak interactions !). Indications point in favour of this model and the most striking and direct supports can be summarized as follows.

i)

The observed proportionality between the recession velocity of galaxies and their distance (Hubble law), which is a natural consequence of assuming the FLRW metric of eq.(1).

ii)

The detection and the high degree of isotropy of the cosmic microwave background radiation, which is the evidence of a primordial hot stage of the Universe, characterized by a high degree of homogeneity [20].

iii)

The observed light element abundances, which match remarkably well the predictions of primordial nucleosynthesis, that represents an unavoidable step in the evolution of the hot Universe [34].

Although the assumption of a homogeneous and isotropic Universe is correct at an early stage of the Universe or today at sufficiently large scales, nevertheless it is manifestly violated at scales below the typical correlation length of density fluctuations, where the structure of the Universe is much more complex. However, this does not represent a problem for the Cosmological Principle, which, instead, would be in trouble if we were observing non negligible anisotropies at scales comparable to the horizon size.

Observations of the Universe on scales similar to the typical galaxy dimension, $\sim 10$ kpc, reveal large inhomogeneities and the current view is that below such scales non-gravitational forces, associated to the baryonic component of the Universe, are dynamically dominant. On the other hand, scales $R\gg 10$ kpc are considered relevant to the Large Scale Structure. The main difference between small (galaxy) and the large scale lies essentially in the dynamics giving rise to structure formation. Indeed, the galaxy mass is determined by the capacity of the baryonic content to cool down during gravitational collapse, as the density increases. A quantitative analysis shows that, for masses $M \raise -2.truept\hbox{\rlap{\hbox{$\sim$}}\raise5.truept \hbox{$>$} }10^{12}M_{\odot}$, the heat produced during the initial collapse prevents a further compression [27]. Apart from the details of the heat production and dissipation, it is clear that, while the efficiency of the dissipation in a proto-object of dimension $R$ is proportional to $R^2$ (i.e., to the extension of its surface), the heat production is proportional to $R^3$ (i.e., to the mass of the object). Then, it is easy to understand that a characteristic scale $R^*$ must exist, such that above $R^*$ the rate of heat production is greater than the dissipation rate, which prevents the gravitational collapse from proceeding. The precise value of $R^*$ depends on the geometry of the collapse process, on environmental effects, and on the dissipation characteristics of the collapsing material. Detailed analysis give $R^*$ values that are very similar to the typical scale of normal galaxies. The study of dissipative processes, which determine the internal structure and dynamics of galaxies, are then essential in understanding the origin of galaxies. However, such analysis can be very difficult and the details of the genesis and evolution of structures below the galaxy scales are widely debated.

On the contrary, on scales much larger than the galaxy ones, it is possible to study the formation and evolution of cosmic structure only on the basis of the gravitational dynamics. This evolution follows initially a linear pattern, while later, when the fluctuation amplitude increases sufficiently, it undergoes non-linear phases. For this reason, the large scale dynamics is not so easy to understand. However, on such scales the problem is much better determined and one's hope is to solve adequately the dynamical picture.

On such scales the essential observation is that galaxies have a spatial distribution with highly non-random characteristics. They show a strong tendency to group together forming clusters, while clusters themselves are clumped into ``superclusters'' on even larger scales. The resulting hierarchical appearance of the galaxy distribution suggests the presence of a sort of scale-invariance, which is also supported by several quantitative statistical analyses. The classical example is represented by the 2-point correlation function, which is observed to decline with a power-law shape, having the same slope for both galaxies and clusters, although at different scales and with different amplitudes. Such scaling properties for the object distribution is one of the most relevant characteristics that must be accounted for by any galaxy formation model. The hierarchical arrangement of the clustering is even more remarkable if we consider that it extends from the small scales, where gravitational dynamics are in the non-linear regime, up to large scales where linearity still holds. Therefore, a detailed statistical representation of the clustering displayed by the distribution of galaxies and galaxy systems is fundamental in order to compare the present Universe with the predictions of theoretical models for structure formation.

Instead of using positions of luminous objects, an investigation of the large scale matter distribution in the Universe can be efficiently realized also by observing the effects of the background gravitational field on galaxy peculiar motions. A direct estimate of the radial peculiar velocity of a galaxy at distance $d$ is obtained by subtracting the Hubble velocity, $H_od$, from the observed recessional velocity, once a redshift-independent estimate of $d$ is available. This kind of distance measurement is usually based on intrinsic relations between intrinsic structural parameters of galaxies, such as the famous Tully-Fisher relation for spirals [33] (which relates the absolute luminosity and the observed rotation velocities), the Faber-Jackson relation for ellipticals [13] (which relates the absolute luminosity and the internal velocity dispersion) and the $D_n$-$\sigma$ relation for ellipticals [22] (which relates a suitably defined apparent diameter $D_n$ to the line-of-sight velocity dispersion $\sigma$). An exciting development in this field has been the recent completion of large galaxy redshift surveys and the availability of a considerable amount of redshift-independent distance estimates (see ref.[31] for a review). As a consequence, a lot of theoretical work has been devoted to find methods for extracting the large-scale three-dimensional velocity and mass density fields from measurements of radial peculiar velocities. At large scales peculiar motions are related to the gravitational potential field by linear dynamical equations. In this regime, it makes sense to address the problem of reconstructing the matter distribution from the observed galaxy motions. Since the linearity of the gravitational clustering at large scales should have preserved the initial shape of the primordial fluctuation spectrum, the reconstruction procedure could furnish precise indications about the initial conditions. Several attempts in this direction have been already pursued (see, e.g., refs.[4]), with quite promising results, despite the rather limited and sparse amount of available data.

A further very efficient way to probe the nature of primordial fluctuations is represented by the investigation of the temperature fluctuations in the CMB. Such fluctuations are expected to be originated at the recombination time (corresponding to a redshift $z_{rec}\sim 1000$), when matter and radiation decouple. After that epoch, the Universe became transparent to the electromagnetic radiation. For this reason, inhomogeneities in the CMB should reflect the matter fluctuations just before decoupling. In past years, many efforts have been devoted to detect such anisotropies, with the result of continuously pushing down the lower limits for their amplitude. Only very recently, the COBE satellite succeeded in detecting a significant signal for CMB temperature fluctuations at the angular scale $\vartheta =7^\circ$. More recently, the BOOMERANG [3] and MAXIMA [15] experiments have measure the CMB anisotropies at much smaller scales. The resulting spectrum of CMB fluctuations indicates that the spatial curvature of the Universe is close to zero. Such results provide a fundamental support to the picture that the presently observed structures have grown from very small initial perturbations in the Friedmann background. In a few years more refined measurements from the MAP [17] and Planck [18] satellites at even smaller angular scales should be able to further restrict the number of allowed initial condition models.