1.2.2. The Dynamics of an Expanding Universe
Now that we have specified the metric that holds in a homogeneous and isotropic Universe, the next step is to consider the dynamics of an expanding Universe. Since the Universe has mass, then at all times the expansion must compete against the combined (attractive) gravitational acceleration of that matter. In Newtonian mechanics this acceleration is given by
where is the matter density and is the gravitational potential. This equation is historically known as Poisson's equation.
This simple equation for gravitational acceleration does not apply in the very early Universe due to the presence of very high energy photons. The majority of the mass-energy in the early Universe is in the form of radiation moving at c. The high radiation pressure drags the matter along with it and effectively counters the tendency for the matter to collapse. In this sense, the Universe acts as a relativistic fluid with a pressure term whose behavior is not adequately described by Newtonian mechanics. The details about the Stress-Energy tensor in Einstein's field equations are beyond the scope of this book (for reference see Weinberg 1972; Peebles 1993) but they lead to a generalization of equation 14:
where p / c2 is the pressure (which we subsequently set to p; c=1) and the combined term +3p effectively becomes the gravitational mass density g which produces the net gravitational acceleration of material that decreases the expansion rate.
If we now consider a sphere of radius rs and volume V which has some mean gravitational mass density within it, the total mass of that sphere is given by
The acceleration at the surface of the sphere is given by Newton's law of gravitation as -GM / Rs2. Multiplying equation 13 by the term -G / Rs2 then yields
where s2 refers to the second time derivative of the spatial coordinate, rs (the first time derivative s is a velocity). Equation 17 is a standard equation in General Relativity and it describes the evolution (e.g., expansion or contraction) of a homogeneous and isotropic mass distribution. Within this sphere there is some net energy, En. This energy is V. If the material enclosed in rs moves so that it changes rs then En changes in accordance with how much work is done by the pressure of the fluid on the surface of the sphere. By conservation of energy we then have
Equation 18 states that the change in net energy is exactly equal to the change in volume multiplied by the pressure. Rearranging the terms involving dV and defining the volume as V = (4 / 3) rs3 ( = 3s rs2)
Solving for p in equation 19 and plugging that solution in equation 17 yields the following differential equation
This is a messy differential equation. If we multiply both sides by the term s and choose units such that the quantity (4 / 3)G = 1 and let rs be x then we arrive at the following functional form
Now the right hand side is just
and the left hand side is just
Integrating both sides over time then yields
Switching back to normal units yields the first integral of equation 20:
where K is a constant of integration which we can identify with the curvature term in the Robertson-Walker metric. Equations 17 and 23 are the main equations of this cosmological model.
If we consider the case of a static Universe where rs is, by definition, constant and hence all derivatives are zero then equations 17 and 23 become
Since the mass density must be positive then to satisfy the constraint of a static Universe p must be negative. Since normal matter cannot have negative pressure, Einstein introduced the cosmological constant into the field equations to serve as the source of negative pressure. In the static Universe balances the net gravitational acceleration. But the Universe is not static, it is expanding according to the expansion scale factor R(t) given in equation 13. Our hypothetical sphere radius rs will then be different at some later time, t, such that
Equation 17 now becomes
where the first of the two terms on the right hand side is for the pressure and density of ordinary matter (e.g., stars and galaxies) and the second term includes the contribution of the Cosmological Constant. Equation 26 then describes the relativistic acceleration of the expansion, which in principle, can be dominated by at late times as and p decrease with time while stays nearly constant. This in fact is the physical manifestation of non-zero . In this case, the Universe evolves from being radiation dominated, to being matter dominated, to being vacuum energy dominated.
The second of the two cosmological equations, equation 23, can now be expressed as
Equation 27 demonstrates that the rate of change of the scale factor R(t) is affected by three things: the net gravitational force acting to decelerate the Universe which is determined by the matter density (); a curvature term related to the geometry of the Universe (K = 0 is flat; K = +1 is positive curvature, K = -1 is negative curvature); and a term related to the vacuum energy which acts as a long range repulsive force.
The quantity (t) / R(t) is the rate of change of the scale factor and is parameterized as H, the Hubble constant. H is a measurable quantity. For the case of = 0 if we can measure H and (the present day mass density of the Universe) then we will have solved for K / R(t)2 and hence, under the Robertson-Walker metric, completely specify the geometry of spacetime. In this way, we have formulated a mechanism where observations can fully determine the cosmological model - there are no hidden variables. In the special case where the curvature of the Universe is zero (K = 0) and = 0, we have
The rate of change of with time is given by equation 19 which can be rewritten in terms of the scale factor R(t) as
With zero pressure (p = 0) we have
The only solution to this is that R(t)-3 ( = -3R(t)-4 (t) ). Note that the expression R(t)-3 is also the one that satisfies the condition that the mass within a sphere (4 / 3 (t) R(t)3) as a function of time remains constant. From equation 28 we then have
which is only satisfied if the scale factor R(t) goes as t2/3. The quantity (t) / R(t) = H is now
Thus, in the zero curvature case, the expansion age of the Universe is
If space is devoid of mass (and hence is negatively curved) then we can set = 0 to yield
which has a solution of the form R(t) goes as t and the expansion age tends to H-1. In either case, observations which determine H also then reveal the approximate age of the Universe. The fact that the ages of the oldest stars in the Universe are in the range .67 H-1 - 1.0 H-1 has led some (e.g., Liddle 1996) to claim that this forms another observational pillar for the Hot Big Bang theory. However, its important to emphasize that the expansion age of the Universe is only H-1 in the special case where = 0. For 0, the relationship with H-1 is considerably more complicated as acts to make the expansion rate decrease more slowly as its a repulsive force. In fact, it can be shown that
(see Kolb and Turner 1991; Krauss and Turner 1994) in the case of inflation (see chapter 4) in which + = 1 (where = 0 is the usual case). H0t0 > 1 is satisfied for > 0.74. Hence, to reconcile the possible age problem by invoking non-zero requires a value of which is in significant excess of meaning that at the present epoch the Universal is dominated by Vacuum Energy.
The final useful relation to derive is the expression for the critical density of the Universe. This is defined as the density which is required to eventually halt, by mutual gravitational contraction, the expansion of the Universe. Like the previous derivations, this one again can be done in terms of energy conservation assuming the Universe is a sphere of uniform density. In this case the total mass is given by equation 16 with p = 0. Consider now a galaxy trying to escape the surface of this sphere. Its potential energy is given by
and its kinetic energy is given by
The velocity, v, of this galaxy is determined by the expansion rate expressed by the Hubble law. Thus v = Hrs. To escape this sphere of radius rs, KE must exceed PE so we have the critical condition
we can readily eliminate m and rs2 to arrive at the expression for critical density:
The critical density is not dependent upon the size and mass of the Universe but only on its expansion rate. If the real density of Universe exceeds c then the Universe is destined to collapse. The Universe can not collapse at early times due to entropy production and the associated high radiation pressure. Equation 35 strictly only applies in the matter dominated era.