Chapter 3: THE FRIEDMANN MODELS

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1 (3.53) z) = { q z + ( q −1)( 1+ 2q z −1} Z q ( 2 0 0 0 q0 (1 + z) This reduces in three simple cases to: q 0 = 0 k = −1 Z q z(1 + 0.5z) ( z) = (1 + z) (3.54) q q 0 0 = 0.5 = 1 k = 0 k = 1 ⎛ Z q ( z) = 2⎜1 − ⎝ z Z q ( z) = 1+ z 1 ⎞ ⎟ 1+ z ⎠ These expressions are very useful. The q 0 = 1 case is noteworthy primarily because it produces a luminosity distance D L (see section 2.26) that is proportional to z. Thus the magnitude-redshift relation (the classical Hubble diagram) is a straight line in this cosmology. It should be stressed that all of the foregoing relations apply only to a pressureless matter-dominated Universe, like ours at the present epoch, because they were based on the particular form of R(t) that is produced in such a model. The various relations above are sufficient to derive a number of interesting quantities. For instance, one often encounters the comoving volume element, dV c /dz. This describes the incremental increase in comoving volume (i.e. in which galaxies have constant number density assuming that their numbers are conserved) with redshift and is required for instance when calculating the expected number of faint galaxies seen within a survey of a given surface area on the sky since dN/dz is proportional to dV c /dz. Consider a cone of solid angle dΞ. If this projects to a physical area A on a sphere of constant radius at a redshift z, then (3.55) dV dz dV dz c c 2 dw 2 = A( 1+ z) with A = dΞDθ dz ⎛ = ⎜ ⎝ c H 0 ⎞ ⎟ ⎠ 3 Z (1 + z) 2 q ( z) dΞ 1+ Ω z 0 3.7 The interrelation between the curvature, the density and the expansion rate Remember that all solutions to the Friedmann equation must have the same relationship (3.17) between the curvature and the expansion rate and the density:
k ( RA) 2 ( Ω −1) = 2 ⎛ c ⎞ ⎜ ⎟ ⎝ H ⎠ with Ω= 8 πG ρ 3H 2 Here ρ is the density, be it that of normal matter density, radiation density or false vacuum energy density, or the sum of all three if appropriate. Indeed, in the general case we have: k ( RA) 2 ( Ωtot = ⎛ c ⎜ ⎝ H −1) 2 ⎞ ⎟ ⎠ with Ω tot 8πG = 2 3H ∑ i ρ i This inter-relationship, which can be written in several different ways, including that of the original Friedmann equation (3.1), is the heart of the General Relativistic Friedmann-type models. Having been brought up since the cradle on Newtonian ideas, we tend to think of densities and velocities as the key quantities to focus on, with gravity acting on the density to decellerate the Universe and so on. We think intuitively of Newtonian concepts such as escape speeds, binding energies, etc. For most of us, the curvature is then added on as an uncomfortable afterthought. This Newtonian approach is a useful and pragmatic way to approach questions such as the appearance of distant objects and the physical evolution of the contents of the Universe. Indeed, as we saw, we can derive perfectly correct expressions describing the expansion of R(τ). However, in a fundamental way it is really the other way around. The one quantity that can never change during the expansion of a homogeneous Universe is the comoving curvature k/A 2 . It is the curvature that describes the Universe. Once the curvature is defined, the density at any epoch follows from the expansion rate and vice versa from (3.17). As the Universe expands, the equation of state tells us how the density will change (e.g. as R -3 , R -4 and R 0 in the three cases above) and, as if by magic (but not really of course), the expansion rate is decelerated or accelerated in the ways that we calculated above to compensate. In the evolution of the Universe, the effective equation of state (i.e. that of the dominant density component) has changed and may change again. Each of these changes brings a different form of R(τ) and of ρ(R) with smooth transitions in each so as to preserve the correct relationship between them. However, the topology of the Universe, represented in the homogeneous case by the comoving curvature scalar k/A 2 , remains constant.
Page 1 and 2: Chapter 3: THE FRIEDMANN MODELS In
Page 3 and 4: (3.4) & &&& 4πG R RR R 3 =− ρ 2
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: Now, using the definition of z, R0

Chapter 3: THE FRIEDMANN MODELS

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