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lecture2 - AST204 February 6 2008 LECTURE II THE REDSHIFT...

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AST204 February 6, 2008 LECTURE II: THE REDSHIFT AND THE WHISPERS OF CREATION: THE COSMIC MICROWAVE BACKGROUND I. The Past and Future of the Universe Last time we derived the Friedman equation, which describes the expansion of the universe: ˙ R 2 - 8 πG ¯ ρR 2 / 3 = 2 ε (2) and is exact for the whole universe, because it is the same for all shells and is exact for small, nearby shells. This result is consistent with (and indeed demands) the cosmological principle. Shells of radius smaller than some radius of interest expand more slowly, and shells larger expand more rapidly, so shells never cross. We may take as origin anywhere in the universe as we have seen, and the expansion is spherically symmetric about that place, so the density is just inversely proportional to the volume of a sphere centered on the (arbitrary) origin: m = 4 πr 3 ¯ ρ/ 3 = 4 πR 3 u 3 ρ/ 3 = const., so, since u is constant (and, of course, is 4 π/ 3) we can write R 3 ¯ ρ = const, and the density is proportional to 1 /R 3 , a function of time alone, and is spatially constant for all time if it is at any time. Thus the density, Hubble constant, and anything else one can measure about the expansion are homogeneous for all time. II. Looking Back: How old is the universe? If we look at the Friedman equation, it is clear that the second term is proportional to 1 /R , since we just showed that the density is proportional to 1 /R 3 . So as we look back in time to times when R was smaller, the gravitional term becomes larger and larger. Since the difference between the gravitational and kinetic terms is constant, the kinetic term also gets larger and larger, and so the time derivative ˙ R gets larger and larger. Thus the recession velocity of any particle, which is just proportional to ˙ R , decreases with time, which is exactly what one expects with gravitation; the universe is, in this simple model, decelerating . The other side of that coin is that the velocities get higher and higher as one goes into the past, and so the scale factor MUST go to zero at some finite time in the past, and indeed at a time less long ago than 1 /H , since the motion is accelerated. Thus the conclusion is inescapable that there was indeed a Big Bang, a time when the density was infinite and the scale factor zero. It is also clear that since the kinetic and potential terms both become very large in the distant past, their difference, the total energy, becomes negligible in comparison to either, 1
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so the total energy plays essentially no role in the early evolution of the universe. This is certainly not the case in the late evolution, as we shall see now. III. Looking Forward: The Cosmic Energy–Will the Universe Expand Forever? It should be clear from its construction that the value and even the units of R can be anything; all we demand is that R ( t ) u = r ( t ), the physical distance, measured in any convenient units. We have done nothing which would demand that ε has a particular sign, and indeed it can be positive, negative, or zero. First consider the case in which it is zero. This says that our shell has zero energy—its kinetic energy is just balanced by its
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