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1.3. Cosmological Parameters

Based on (7) we can define some very important parameters like the Critical Density, which is the density necessary to obtain a flat Universe (Lambda = 0):

Equation 15 (15)

and the Cosmological density parameter Omega, which is a unit-less measure of the density of the Universe:

Equation 16 (16)

Furthermore, the constant of proportionality in Hubble's law, the Hubble constant, is:

Equation 17 (17)

Note that the necessity of parametrizing with h was due to earlier discordant determinations of H0. Today most studies converge to a value of ~ 0.7 (see section 2.4).

A convenient representation of these interrelations can be produced by re-writing Friedmann's equation, in the matter dominated era (using 10), as following:

Equation 18 (18)

were the contribution to the total density parameter from the curvature and Lambda terms is:

Equation 19 (19)

Note that H(z) is called Hubble function. It is evident that at the present epoch we obtain from (18) that E(0) = 1 and thus:

Equation 20 (20)

which also holds for any epoch (evaluated directly from 7). Note that we can have a flat Universe (Omegak = 0) while having Omegam < 1 (as suggested by many different observations).

The Age of the Universe: Using (18), evaluated at the present epoch, we have R dot / R0 = H0 E(z) / (1 + z) and from dR/R0 = -dz / (1 + z)2 we obtain the age of the Universe:

Equation 21 (21)

For example, in an Einstein-de Sitter universe (OmegaLambda = Omegak = 0) we have:

Equation 22 (22)

while for a OmegaLambda > 0 model we obtain:

Equation 23 (23)

We therefore see that if OmegaLambda > 0 we have that the age of the Universe is larger than what is predicted in an Einstein-de Sitter Universe.

The Lambda neq 0 Universe: Due to the recent exciting observational indications for a positive cosmological constant and the important consequences that this has to our understanding of the Cosmos, we will briefly present this model.

Originally, the cosmological Lambda-parameter was introduced ad hoc by Einstein in his field equations in order to get a static solution (R dot = R ddot = 0). From (8) he derived R = (k / Lambdac)1/2 and inserting this into (7) he obtained, in a matter dominated (p = 0) Universe:

Equation 24 (24)

where Lambdac is the critical value of Lambda for which R dot = R ddot = 0. However, it was found that his solution was unstable and that small perturbations of the value of Lambdac would change drastically the behaviour of R. From (7) we see that if k leq 0, then R dot2 is always nonnegative for Lambda > 0, and thus the universe expands for ever, while if Lambda < 0 then the universe can expand and then recontract again (as in the k = 1, Lambda = 0 case).

The recent SNIa observations (see section 3.2) and the CMB power-spectrum results (see section 3.1) have shown that the Standard Cosmological paradigm should be considered that of a flat, OmegaLambda appeq 0.7, Omegam appeq 0.3 model. Thus we will consider such a model in the following discussion. Evaluating (7) at the present epoch, changing successively variables: x = R3/2, y = x(Omegam / OmegaLambda)1/2 R0-3/2 and theta = sinh -1 y and then integrating, we obtain:

Equation 25 (25)


Equation 26 (26)

Figure 1

Figure 1. The expansion of the Universe in an Einstein de-Sitter (EdS) and in the preferred Lambda model. We indicate the inflection point beyond which the expansion accelerates. It is evident that in this model we live in the accelerated regime and thus the age of the Universe is larger than the Hubble time (H0-1).

It is interesting to note that in this model there is an epoch which corresponds to a value of R = RI, where the expansion slows down and remains in a quasi-stationary phase for some time, expanding with R ddot > 0 thereafter (see Fig.1). At the quasi-stationary epoch, called the inflection point, we have R ddot = 0 and thus from (7) by differentiation we have:

Equation 27 (27)

Now from (25) and (27) we have that the age of the universe at the inflection point is:

Equation 28 (28)

The Hubble function at tI is:


so if t0 > tI we must have H0 < HI.

This is an important result because it indicates that introducing an OmegaLambda-term, and if we live at a time that fulfils the condition t0 > tI, we can increase the age of the universe to comfortably fit the globular cluster ages while keeping the value of Omegam < 1 and also a flat (Omegak = 0) space geometry. From (28) and (23) and for the preferred values OmegaLambda = 0.7 and Omegam = 0.3 we indeed obtain t0 / tI appeq 1.84 (see also Fig.1), which implies that we live in the accelerated phase of the Universe. Note that in order for the present time (t0) to be in the accelerated phase of the expansion we must have: OmegaLambda > 1/3.

Importance of k and Lambda terms in global dynamics: Due to the recent interest in the Lambda > 0, k = 0 Universes, it is important to investigate the dynamical effects that this term may have in the evolution of the Universe and thus also in the structure formation processes (see Fig.2). We realize these effects by inspecting the magnitudes of the two terms in the right hand side of (7). We have the density term:


and from (20) we have

Equation 29 (29)

By equating the above two terms we can find the redshift at which they have equal contributions to the dynamics of the Universe. Evidently this happens only in the very recent past:

Equation 30 (30)

Observations suggest that Omegam appeq 0.3 and OmegaLambda appeq 0.7, and therefore we have zc appeq 0.3, which implies that the present dynamics of the universe are dominated by the Lambda-term, although for the largest part of the history of the Universe the determining factor in shaping its dynamical evolution is the matter content.

Figure 2

Figure 2. The strength of the three factors shaping the recent dynamics of the Universe. Compare the strength of the rho and Lambda term (k = 0) and of the rho and k term. We have assumed H0 = 72 km s-1 Mpc-1, Omegam = 0.3 and OmegaLambda = 0.7.

Similar results are found for the k-term in Lambda = 0 models. In this case we have from (29) that


and thus the redshift at which the density and curvature terms have equal impact in the global dynamics, is:


We see that as z increases the density term grows faster than the curvature term which is important only very recently. A similar line of argument shows that also in the radiation dominated era the Lambda and k terms do not affect the dynamics of the Universe.

Density parameter as a function of z: From (18) and eliminating the curvature term, using Omegak = 1 - Omegam - OmegaLambda, we obtain the time evolution of the density parameter Omegam:

Equation 31 (31)

and using (15) and (10) we have:

Equation 32 (32)

It is easy to see that whatever the value of Omegam, at large z we always have Omega(z) = 1.

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