Cosmological solutions of the Einstein-Friedmann equations ...

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Cosmology known way. However, this seems not to matter as long as we have p 0. That a non-vanishing Λ spoils the geometry ⇔ matter duality, because the Einstein equation remains true no matter on which side of the equation we write the cosmological term. Many believe it has some thing to do with quantum vacuum fluctuations, but no answer can be given why it is so small. In any case if we take the energy momentum tensor as given on the quantum level by the Standard Model (SM) of strong and electroweak interactions we would expect the Higgs field vacuum expectation value (Bose condensate) as well as the quark condensates to contribute to the cosmological constant. Again, the value obtained is about 50 orders of magnitude to big! What tames the cosmological constant to a small non-vanishing value? Exercise: estimate the cosmological constant as induced by the Higgs mechanism and by spontaneous symmetry breaking of chiral symmetry in QCD. A cosmological constant of course affects local gravitation in particular it might affect black holes by changing the Schwarzschild radius etc. In order to study the c○ 2009, F. Jegerlehner ≪❘ Lect. 7 ❘≫ 480
Cosmology physical effect of a non-vanishing Λ we consider the spherically symmetric mass distribution in outer space: ➊ Λ = 0 : Schwarzschild ds 2 = Gµν ≡ 0 ✄ Gµν = Λ gµν 1 − r0 r (cdt) 2 − 1 1 − r0 r dr 2 − r 2 dΩ 2 which is unique, static with boundary condition: flat as r → ∞. It has the Newto- nian approximation with potential: ϕ − m r with m r0 2 and where g00 1 + 2ϕ c 2 c○ 2009, F. Jegerlehner ≪❘ Lect. 7 ❘≫ 481
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Cosmology Cosmological solutions of
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Cosmology Cosmological solutions of
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Cosmology claimed to represent the
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Cosmology A) Empty Worlds, vacuum e
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Cosmology A) Empty Worlds, vacuum e
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Cosmology The general solution is S
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Cosmology (d) Λ > 0 , k = 0: S = e
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Cosmology Terminology: Λ > 0 , k =
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Cosmology B) Matter dominated: ρ >
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Cosmology B) Matter dominated: ρ >
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Cosmology ct = C arcsin √ X −
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Cosmology In each case we have S (t
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Cosmology Einstein-de Sitter univer
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Cosmology Einstein-de Sitter univer
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Cosmology t = 2 3 H−1 (t) (EdS) H
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Cosmology t = 2 3 H−1 (t) (EdS) H
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Cosmology Back to the non-flat geom
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Cosmology Back to the non-flat geom
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Cosmology in terms of the observabl
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Cosmology The conclusion of the abo
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Cosmology C) Radiation dominated: p
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Cosmology ❖ k = 0 : S (t) = 4 ¯
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Cosmology κ ρ(t) = 3 H2 (t) K(t)
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Cosmology Actually, in the radiatio
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Cosmology Actually, in the radiatio
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Cosmology 8π hν ργ(ν) dν = 3
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Cosmology 8π hν ργ(ν) dν = 3
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Cosmology Since ρ(t)S (t) 4 = ρ0S
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Cosmology c○ 2009, F. Jegerlehner
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Cosmology and Dicke and Peebles at
Page 105 and 106: Cosmology T0 = (2.725 ± 0.002) ◦
Page 107 and 108: Cosmology T0 = (2.725 ± 0.002) ◦
Page 109 and 110: Cosmology However, the evolution of
Page 111 and 112: Cosmology However, the evolution of
Page 113 and 114: Cosmology ρ(t) Big Bang radiation
Page 115 and 116: Cosmology The critical energy densi
Page 121 and 122: Cosmology Ω0 = 1 c○ 2009, F. Je
Page 123 and 124: Cosmology Ω0 = 1 is the critical
Page 129 and 130: Cosmology ˙S 2 c 2 + k = κ 3 ρ S
Page 131 and 132: Cosmology ˙S 2 c 2 + k = κ 3 ρ S
Page 133 and 134: Cosmology q0 = κ (ρ0+3 p0) 6 H 2
Page 135 and 136: Cosmology Form of energy equation o
Page 141 and 142: Cosmology in which case, independen
Page 147 and 148: Cosmology (in accord with the unive
Page 149 and 150: Cosmology (in accord with the unive
Page 151 and 152: Cosmology missing part making Ω =
Page 153 and 154: Cosmology missing part making Ω =
Page 155: Cosmology Appendix: Λ 0 The issue
Page 159 and 160: Cosmology and the metric ds 2 = 1
Page 161 and 162: Cosmology and the metric ds 2 = 1
Page 163 and 164: Cosmology With a 2 1 K = 3 Λ ; a
Page 165 and 166: Cosmology With a 2 1 K = 3 Λ ; a
Page 167 and 168: Cosmology Λ < 0: Anti-de Sitter sp
Page 169 and 170: Cosmology Λ < 0: Anti-de Sitter sp
Page 171 and 172: Cosmology Note: z = constant is the
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