Chapter 3: THE FRIEDMANN MODELS

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gravitational effects of uniformly distributed distributions of mass allow us to consider only the mass enclosed within the sphere of radius x = R(τ)ω OA centered on O whan calculating the gravitational attraction between O and A.. O x A Thus if x is the distance OA, then we have (3.2) &&x =− 4 π G ρ x 3 Now, since the comoving distance to A from O, ω OA , is by definition constant during the expansion, we can write: x = Rω ⇒ x& = R& ω ⇒ x&& = R&& ω OA OA OA We may thus rewrite our dynamical equation in terms of R(τ) alone, thus eliminating reference to any particular galaxy or distance x. (3.3) R && 4π =− Gρ R 3 So, the dynamics of our shell of material are the same as the dynamics of the whole Universe. In a sense, this is then the justification for the Newtonian approach since we can make our radius r arbitrarily small so that curvature and light travel time effects are completely irrelevant. To integrate this equation we need to know how ρ behaves as functions of R(τ), i.e. we need to know the equation of state of the matter-energy in the Universe. For the time being lets assume it is simple pressure-less matter (known as “dust”) that evolves as ρ ∝ R -3 . We adopt the convention of putting in square brackets quantities that are invariant under the expansion. If we multiply both sides by & R and set [ρR 3 ] = constant, then we have
(3.4) & &&& 4πG R RR R 3 =− ρ 2 3 R R& 2 8πG = R 3 1 ρ + 2ξ 3 R R& 2 8πG = ρR 2 + 2 ξ 3 Here, ξ is a constant of integration that, in this Newtonian analysis, corresponds to the binding energy of the system: If ξ > 0, then we have solutions with &R 2 → 2ξ as R→ ∞. R(τ) is proportional to τ and the Universe is unbound and expands forever. If ξ < 0, we get R & = 0 at finite R so the Universe is bound and at some point the expansion is halted and the Universe recollapses. If ξ = 0, then we have the asymptotic limit R & → 0 as R→ ∞ Note that this has exactly the same form as the Relativistic Friedmann equation (3.1) with Λ= 0 if we associate the ξ binding energy with the curvature term. Let us now return to the equation of state and consider possibilities other than the familiar one considered above, where we assumed that the density is dominated by dust with ρ ∝ R -3 . An obvious possibility is that the density of the Universe is dominated by electromagnetic radiation, e.g. the cosmic background radiation field. As the Universe expands, the density of this radiation field will drop as R -4 . We get a factor of R -3 from the decreasing number density of photons, but an additional factor of R -1 from the fact that all the photons lose energy due to the redshift. There are other more exotic possibilities. As we’ll see below, we could imagine a scalar density field that does not change its density as the Universe expands, i.e. ρ = constant (more than that – in fact we have evidence that this is the case!). We will refer to this as a false vacuum density. Other possibilities exist. A few years ago there was some interest in hypothetical Universes that were dominated by the mass from “cosmic strings” – strange entities that have a constant density per unit length regardless of how much they are stretched. These give ρ ∝ R -2 . It is easy to see that putting these other ρ(R) dependencies into our Newtonian equation (3.3) will not yield the right form of (3.4) upon integration unless we introduce a small fudge. To see how to proceed, it is useful to consider the ρ(R) dependence in thermodynamic terms. We can assume that the expansion is adiabatic (i.e. that there is no inflow or outflow of heat). The 1st Law of Thermodynamics gives us that the change in the internal energy (ρV) equals the pdV work done by the system in expanding: 2 (3.5) c d( ρV) =−pdV ⇒ dρ =− ( ρ + pc) 2 3 dR R If the pressure can be neglected, then
Page 1: Chapter 3: THE FRIEDMANN MODELS In
Page 5 and 6: 3.2 A poor-man’s General Relativi
Page 7 and 8: R& 2 3 2 0 8πG = 3 RR& = R R R 3 2
Page 9 and 10: (a) Matter vs. Radiation in the den
Page 11 and 12: (3.29) Conditions for a flat Univer
Page 13 and 14: Now, since from the Robertson-Walke
Page 15 and 16: Note that if the density was exactl
Page 17 and 18: Now, using the definition of z, R0
Page 19 and 20: k ( RA) 2 ( Ω −1) = 2 ⎛ c ⎞

Chapter 3: THE FRIEDMANN MODELS

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