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Given the results from observational cosmology discussed in the previous two sections we are now able to present a concrete picture of the universe, as opposed to the traditional array of models with very different properties. Taking a more comprehensive view, in Table 2 we present a set of cosmological parameters (without errors) that might be taken as the "best estimates" based on various observational and theoretical studies. [37]

This set of parameters describes a flat universe the dynamics of which is dominated by two mysterious forms of energy, most prominently, the cosmological constant. So then, the long-standing debate over whether or not the cosmological term should be included in Einstein's theory is over; not only should it be included, it dominates the universe. Although the debate over the existence of the cosmological constant has ended, the debate over its physical implications has just begun. Further comments about this debate will be discussed in the conclusion.

Table 2. Our best estimates of the cosmological parameters for the present-day universe and the primary sources we used to obtain them. If "theory" is listed as the source, then we derived the value from other estimates using the stated equation.

Parameter Value Primary Sources

Hubble constant H0 = 72 km . s-1 . Mpc-1 [13]
Cosmological Constant OmegaLambda = 0.70 [16, 33]
Matter Omegam = 0.30 [16, 33]
Baryonic matter Omegab = 0.04 [33]
Dark matter OmegaCDM = 0.26 theory: Eq. (61)
Curvature Omegak = 0.00 [16, 20-22, 33]
Deceleration parameter q0 = - 0.55 theory: Eq. (39)

The other mysterious form of energy listed in Table 2, OmegaCDM, is dark matter where "CDM" stands for "cold dark matter." Recall that ordinary matter made up of atomic nuclei only contributes to the baryon content of the universe with Omegab approx 0.04. However, since the total matter content is Omegam approx 0.30, the rest of the matter in the universe must be in some exotic, unseen form which is why we call it dark matter

Equation 61 (61)

We have known about dark matter for several decades now, having been first discovered through anomalous rotation curves of galaxies. [3] The results from the CMB anisotropies only help to confirm that not only does dark matter exist, but that it comprises roughly 90% of the matter in the universe.

Given values of the cosmological parameters, we can now solve for the dynamics of the universe. The Friedmann equations (16) and (17) for our present universe (pm = 0, k = 0) can be combined to give

Equation 62 (62)

This equation can be solved exactly giving the result [38]

Equation 63 (63)

where A = Omegam, 0 / OmegaLambda, 0 approx 0.43 and tLambda = (4/3 Lambda c2)1/2 approx 3.4 × 1017 s. The cosmic scale factor is plotted in Fig. 5 and compared to the purely de Sitter universe described by Eq. (20). From this comparison, we see that, today, the qualitative behavior of our universe is that of a de Sitter universe except that the presence of matter has caused the universe to expand less than in the de Sitter case.

Figure 5

Figure 5. The cosmic scale factor for our universe compare to the de Sitter model.

With a(t) in hand, we can now write a precise metric for the universe

Equation 64 (64)

This tells us that we can visualize the universe as an expanding Euclidean sphere with the expansion governed by a(t) as given in Eq. (63). Note, however, that in this visualization the universe is represented as the entire volume of the sphere and not just the surface.

Another interesting feature that emerges from this picture is that if Lambda is truly constant, the universe would have once been matter-dominated. To see why this is, recall that because the size of the universe changes, the density parameters are functions of time. As we go back in time, the universe gets smaller so that the energy density of matter rhom gets larger while the energy density associated with Lambda, see Eq. (15), remains constant. Using Eq. (22) we can see that the ratio of matter-to-cosmological constant is

Equation 65 (65)

Therefore, at some finite time in the past the universe was such that Omegam / OmegaLambda > 1. Since the expansion of a matter-dominated universe would be decelerating, this implies that the universe underwent a transition from decelerated expansion to accelerated expansion. This behavior is reflected in the deceleration parameter as a function of time, which, given the current cosmological parameters becomes

Equation 66 (66)

Figure 6 is a plot of q(t) and shows that the deceleration parameter was once positive and that a transition to q(t) < 0 occurred around the time at which Omegam / OmegaLambda = 1.

Figure 6

Figure 6. The deceleration parameter of our universe. The sign of q(t) switches from positive to negative at around the same time that the universe goes from matter-dominated to Lambda-dominated.

Having a specific model of the universe allows us to determine specific answers to questions that cosmologists have been asking for decades. While we cannot address all such questions in this paper we will tackle a few of the most common: (a) What is the age of the universe? (b) Will the universe expand forever or will the expansion eventually stop followed by a re-collapse? (c) Where is the edge of the observable universe?

The age of the universe can be calculated by integrating dt from now, z = 0, back to the beginning z = infty . For our universe, the steps leading to Eq. (36) produces

Equation 67 (67)

Making the definition x ident 1 + z, the present age of the universe is given by

Equation 68 (68)

The solution to Eq. (68) is complex. Taking only the real part gives

Equation 69 (69)

The question of whether or not the universe will expand forever is determined by the asymptotic behavior of a(t). Since sinh(x) diverges as x rightarrow infty, it is clear that the universe will continue to expand indefinitely unless some presently unknown physical process drastically alters its dynamics.

Finally, concerning the question of the size of the observable universe, there are two types of horizons that might fit this description, the particle horizon and the event horizon. The particle horizon is the position of the most distant event that can presently be seen, that is, from which light has had enough time to reach us since the beginning of the universe. Unfortunately, since current evidence suggests that the universe was not always dominated by the cosmological constant, we cannot extend the current model back to the beginning. We can, however, extend it into the future. The event horizon is the position of the most distant event that we will ever see. If we consider a photon moving radially toward us from this event, then Eq. (31) describes its flight. Since we are interested in those events that will occur from now t0, onward, Eq. (31) leads to

Equation 70 (70)

where rEH is the radial coordinate of our event horizon. Performing a numerical solution to the integral yields

Equation 71 (71)

This result suggests that 16 billion light years is the furthest that we will ever be able to see. As far as we are aware, the most distant object ever observed (besides the CMB) is currently the galaxy RD1 at a redshift of z = 5.34, which places it approximately 12.2 billion light years away. [39]

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