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Unformatted text preview: LECTURE VII: THE EARLY UNIVERSE, FLUCTUATIONS, AND THE DEVELOPMENT OF STRUCTURE AST 204 25 Feb 2008 I. The Planck Era and the Notion of Time and Space First thing today, we’ll discuss the fact that there’s an epoch, associated with a density, a temperature, a mass, and a timescale ’earlier’ than which it makes no sense to even discuss the notions of time and space. This is caleld the Planck time . The energy of a photon or other highly relativistic ( γ >> 1) particle is just E = hν = hc/λ . The Schwarzschild radius associated with a black hole of mass m is r s = 2 Gm/c 2 in ordinary units, or, since the energy E associated with a mass m is mc 2 , r s = 2 GE/c 4 . Thus an energetic particle so energetic that its own selfgravity makes a black hole in which its wavelength just fits has a wavelength which satisfies λ = r s = 2 GE c 4 = 2 Gh λc 3 , so this critical wavelength, the Planck length , is l P = λ = radicalbigg 2 Gh c 3 = 5 × 10 − 33 cm The energy, the Planck energy , is E P = hc λ = radicalbigg hc 5 2 G = 4 × 10 16 erg = 2 × 10 19 GeV And, of course, an associated temperature, the Planck temperature T P = E P /k = 3 × 10 32 K. The associated mass, the Planck mass , is m P = E/c 2 = 4 × 10 − 5 g There is a density associated with this notion, namely one Planck mass per Planck length cubed, or ρ P = m P /l 3 P = 3 × 10 92 g/cm 3 and a time which might be either of two expressions–the inverse of the frequency associated with the wavelength l P or the expansion time of the universe associated with ρ P . These two quantities are the same within a factor of two; the Planck time is usually called l p /c = 2 × 10 − 43 sec . The fact that the two notions agree is profound. Can you figure out why they agree? 1 We do not know whether the universe was ever this dense, hot, or rapidly expanding, but we do know that it does not make sense to discuss conditions even more extreme than this. In order to study a more extreme condition (smaller length scales, say), one would have to resolve finer detail, which would involve particles of even shorter wavelength. But Planck energy particles already cannot propagate because they are within their own event hori zon. So the concepts of space and time as a classical arena for events breaks down here, or probably actually well before— it makes no sense to talk about the spacetime continuum on scales shorter than the Planck length and time; the universe is a fundamentally quantum gravitational entity at this epoch. The notion of even what quantum gravity is is rapidly evolving at the present time, and the whole field of fundamental physics is currently doing a grand experiment of an unprecedented kind today. There is much hope that superstring theory will somehow produce the Holy Grail of the Theory of Everything, but the motiva tion now is mathematical beauty rather than any real connection with physics. We will see (hopefully within your lifetimes, but I suspect not within mine) whether this crusade will...
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 Spring '08
 Knapp
 Astronomy, Space, General Relativity, Inflation, Big Bang

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