174 5 Cosmology and General Relativity
so that:
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This means that according to the Standard Cosmological Model our Universe has evolved from a completely homogeneous region that was 103×28 bigger than a causally connected region. How could it be homogeneous if there was no possibility of communication among the various causally disconnected cells and of establishing thermal equilibrium?
This paradox is named the horizon problem. Another conceptual problem is named the flatness problem. It appears from all our present cosmological data that our Universe is spatially flat, namely that the cosmological parameter Ω is nearly equal to 1. Why is that so? Who prepared once again the initial conditions in such a precise way as to make Ω exactly equal to one?
The answer to both problems can be provided by the scenario of a primeval cosmic inflation.
As we observed in Sect. 5.6.1, during a phase of exponential expansion of the scale factor the horizon scale remains constant so that, assuming the existence of such a phase in our remote past, explains how a single causally connected region could split into many apparently disconnected ones. Similarly as we will see in the sequel we can argue that the Universe always exits flat from an exponential expansion phase irrespectively of the spatial curvature it had when it entered such a phase.
Hence an exponential inflation provides a generic mechanism able to solve the conceptual problems of cosmology. The question is: which kind of matter can provide the means of realizing such an inflationary phase. The answer is simple enough. It suffices to have a microscopic dynamical theory that besides other fields includes also scalar ones, self-interacting through the presence of some potential. In the next section we will see that the requirements on the structure of the potential in order to realize a reasonable inflationary phase are rather mild and generic. This implies that the inflationary universe scenario is robust.
5.8 Cosmic Evolution with a Scalar Field: The Basis for Inflation
The dynamical basis of inflation is fairly simple. The paradigm is provided by the simple model of a scalar field ϕ(x) interacting with gravity and with itself by means of some potential V (ϕ).
Let us write the following action:
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5.8 Cosmic Evolution with a Scalar Field: The Basis for Inflation |
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Varying it respectively in the metric g and in the scalar field ϕ we obtain the following coupled equations:
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If we introduce the homogeneous, isotropic ansatz (5.4.4) for the metric g and if we assume that the scalar field ϕ = ϕ(t) depends only on the cosmic time t , then the two equations (5.8.2) and (5.8.3) are easily worked out and reduce to a very simple form. It suffices to observe that, under the above conditions, the stress energy tensor of the scalar field has the canonical form (5.3.49) of a fluid, with the following identification of the energy density and of the pressure:
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Inserting this result into the Friedman equations (5.4.12) and choosing a vanishing cosmological constant (Λ = 0)4 we obtain:
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is the Hubble function. The differential system is completed by the explicit form of the propagation equation (5.8.3) of the scalar field which, in the chosen metric, is the following one:
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4This choice has the following motivation. In presence of a generic potential for the scalar field, the cosmological constant is redundant. Indeed any constant contribution to V (ϕ) just plays the role of a cosmological constant.
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5 Cosmology and General Relativity |
Equations (5.8.7), (5.8.8), (5.8.10) encode the inflationary Universe paradigm. To explain this let us explore some possible solutions of this differential system.
5.8.1 de Sitter Solution
Consider a value ϕ0 of the scalar field corresponding to an extremum of the scalar potential where this latter attains a finite positive value:
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In this case we can solve (5.8.7), (5.8.8), (5.8.10) by setting:
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(5.8.11)
(5.8.12)
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(5.8.14)
Recalling the results of Sect. 5.5.3 it follows that, for a constant scalar field sitting at an extremal point of the potential, Einstein equations are solved by de Sitter space, which can be represented in the three versions of a closed, open or spatially flat Universe, in any case exponentially expanding. Indeed, differently from the case of all other fluids, characterized by a pressure p > − 13 ρ, the equation of state p = −ρ, satisfied when ϕ˙ = 0, implies, irrespectively of the sign κ of the spatial curvature, a constant positive acceleration of the Universe expansion, which proceeds indefinitely with an exponential asymptotic behavior. For large t the three scale factors a±(t) and a0(t) have the same form and tend to merge. This is quite clear from Friedman equations. The curvature term κ/a2(t) becomes negligible for large values of a(t), which are always attained in an expanding Universe.
Considering the three metrics dsdS2 + , dsdS2 − and dsdS2 0 , we see that, not only they describe the same intrinsic geometry, but they also approximately coincide in an open region Olate of de Sitter space, corresponding to late times and relatively small distances:
MdS Olate = {t & 1, r ' 1, θ, φ } |
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all range