Lecture Notes for Astronomy 321, W 2004 1 Stellar Energy ...

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Figure 13: Energy (kT ) per particle versus time. Radiation dominated the energy density until the time of photon decoupling, t dec ≈ 10 13 s. 38
7 Vacuum energy and Inflation In Section 5 we introduced Einstein’s field equations, Eq. 26, including the comsological constant (Λ) term. Since g µν represents flat spacetime, this term is uniformly distributed. For the Friedmann equation, Eq. 27, this term gets included as Λ/3 on the RHS. Since this term has no R dependence, then for sufficiently large R, a non-zero Λ will eventually dominate the RHS of Eq. 27. In this case, the Friedmann equation becomes Ṙ = (Λ/3)R ⇒ dR R = √Λ/3 dt ⇒ R(t) ∝ e αt , (33) √ where α ≡ Λ/3, as in Table 1. Now, we make the connection between this funny Λ term in Einstein’s equations and the idea of vacuum energy from quantum field theory (QFT). In quantum mechanics, the uncertainty principle states that there can be a non-conservation of energy ∆E over a time ∆t such that ∆E∆t ≥ ¯h/2 . In QFT, this must occur with particle pairs such as e + e − appearing and disappearing, consistent with the uncertainty relation above. On average, there is a net contribution which is non-zero. This represents the vacuum energy. So the vacuum is not a state of nothing, it is simply the lowest energy level, or ground state. Hence, Einstein’s general relativity will couple to this vacuum energy. In addition, in QFT we expect the vacuum to have a uniform energy density, just like the Λ field. Hence, we identify vacuum energy with a non-zero Λ. Hence, this allows us to possibly connect episodes of spacetime evolution influenced by Λ with fundamental elementary particles and fields. Such is the case for inflation, discussed below. We complete the identification of Λ with vacuum energy by including it as a contribution to ρ in the Friedmann equation: ρ v = Λ 8πG . We used ρ v in this form in the previous section. We will not get into specific particle physics manifestations for inflation. We will motivate it using the observations of the CMBR and then discuss the properties the inflationary era must possess. 39
Page 1 and 2: Lecture Notes for Astronomy 321, W
Page 3 and 4: 1.3 Four forces of nature The elect
Page 5 and 6: Figure 1: Binding energy per nucleo
Page 7 and 8: 2 Stellar Energy Generation - PP-ch
Page 9 and 10: 2.2 The pp interaction: Coulomb bar
Page 11 and 12: compared to the quantum barrier pen
Page 13 and 14: 0.01 0.009 0.008 0.007 0.006 I (MeV
Page 15 and 16: Figure 8: Yet another high-temperat
Page 17 and 18: part due to the solar neutrinos dis
Page 19 and 20: came up with a viable detector cons
Page 21 and 22: neutrino species have small, but fi
Page 23 and 24: urning in the next sub-shell outwar
Page 25 and 26: solar luminosity, which is comparab
Page 27 and 28: As it turns out, the neutral curren
Page 29 and 30: 4.6 Further collapse and black hole
Page 31 and 32: approaches zero. This distance R s
Page 33 and 34: 5.3 The Friedmann equation and a us
Page 35 and 36: Figure 12: Time evolution of the sc
Page 37: Table 1: Properties of cosmic fluid
Page 41 and 42: 7.2 Inflationary scenario We saw ab
Page 43 and 44: 8 Fluids and 2nd Friedmann equation
Page 45 and 46: Figure 15: Energy (kT ) per particl
Page 47 and 48: decay modes, and hence N ν . The c
Page 49 and 50: Figure 18 gives this critical densi

Lecture Notes for Astronomy 321, W 2004 1 Stellar Energy ...

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