2 8 G a const H 2 1 /a 4 = , = = . 3 a a 4 - - PowerPoint PPT Presentation

Alan Guth, Black-Body Radiation and the Early History of the Universe, Part 3, 8.286 Lecture 17, November 12, 2013, p. 1.

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8 H2 πG = ρ , ρ 3 ∝ 1/a4 = ⇒ ˙ a a 2 const = . a4 Then a da = √ 1 const dt = √ ⇒ a2 = const t + const′ . 2 So, setting our clocks so that const′ = 0, a(t) t (flat radiation-dominated) . ∝ √

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˙ a 1 H(t) = = (flat radiation-dominated) . a 2t

t

ℓp,horizon(t) = a(t)

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dt′

0 a(t′)

= 2ct (flat radiation-dominated) . 8 H2 πG = ρ = 3 ⇒ 3 ρ = . 32πGt2

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Black-body radiation is a gas of massless particles at temperature T. Energy Density: π u = ρc2 kT = g , 30 (¯ hc)3

2 (

)4 where g = 2 for photons. (There are two spin states, or polarizations, for photons: either left-circularly polarized and right-circularly polarized, or x-polarized and y-polarized.) Pressure: p = u 1 3 .

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Alan Guth, Black-Body Radiation and the Early History of the Universe, Part 3, 8.286 Lecture 17, November 12, 2013, p. 2.

Number density: ζ(3) (kT)3 n = g∗ , π2 (¯ hc)3 where ζ(3) is the Riemann zeta function with argument 3, 1 1 1 ζ(3) = + + + 13 23 33 · · · ≈ 1.202 , and g∗ = 2 for photons.

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Entropy density: Entropy is a measure of “disorder,” in the sense that it measures the number of microscopic quantum states that contribute to a given macroscopic state. The second law

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If the system stays close to thermal equilibrium, then entropy is essentially conserved. In the early universe, entropy is essentially conserved for all processes except for inflation. 2π2 k4T 3 s = g . 45 (¯ hc)3

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8.286 The Early Universe

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