phenomenon 48.17 The expansion is accelerating

Rests on

• depends_on equation 48.7 eq:cosmo-friedmann-2 ¶
• depends_on equation 48.15 eq:cosmo-luminosity-distance ¶
• depends_on proposition 48.7 Luminosity distance ¶
  • depends_on equation 48.13 eq:cosmo-chi-of-z ¶
  • depends_on equation 48.2 eq:cosmo-redshift-scale-factor ¶
  • depends_on theorem 48.3 The Robertson–Walker metric ¶
    • depends_on proposition 13.162 Curvature induced on the quadric ¶
      • depends_on definition 13.119 Induced metric ¶
        • depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶
      • depends_on equation 13.83 eq:mfd-ecsgauss ¶
      • depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶
        • depends_on definition 13.145 Affine connection ¶
        • depends_on definition 13.147 Torsion ¶
        • depends_on proposition 7.105 Clairaut–Schwarz ¶
        • proves proof ch:11-manifolds-tensors-curvature@proof-42 ¶
      • proves proof ch:11-manifolds-tensors-curvature@proof-49 ¶
    • depends_on remark 13.161 rem:mfd-schur ¶
    • depends_on theorem 13.160 Maximal symmetry forces constant curvature ¶
      • depends_on definition 13.153 Contractions ¶
        • depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶ ↺
        • depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶ ↺
      • depends_on proposition 13.140 Killing's equation ¶
        • depends_on definition 13.139 Killing vector ¶
        • depends_on definition 13.149 Metric compatibility ¶
        • depends_on proposition 13.127 Component formulas ¶
        • depends_on theorem 13.150 Levi-Civita connection and contorsion ¶
        • depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶ ↺
        • proves proof ch:11-manifolds-tensors-curvature@proof-37 ¶
      • depends_on proposition 13.154 Symmetries of the curvature ¶
        • depends_on definition 13.149 Metric compatibility ¶ ↺
        • depends_on proposition 13.157 Bianchi identities ¶
        • depends_on theorem 13.150 Levi-Civita connection and contorsion ¶ ↺
        • depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶ ↺
        • proves proof ch:11-manifolds-tensors-curvature@proof-43 ¶
      • proves proof ch:11-manifolds-tensors-curvature@proof-48 ¶
    • proves proof ch:07-cosmology@proof-2 ¶
  • proves proof ch:07-cosmology@proof-5 ¶
• depends_on theorem 48.5 The Friedmann equations ¶
  • depends_on equation 48.3 eq:cosmo-flrw-metric ¶
  • depends_on equation 48.5 eq:cosmo-perfect-fluid ¶
  • depends_on postulate 44.3 Einstein field equations ¶
    • depends_on definition 44.2 Einstein tensor ¶
      • depends_on definition 13.153 Contractions ¶ ↺
  • proves proof ch:07-cosmology@proof-4 ¶
• proves proof ch:07-cosmology@proof-14 ¶

Supports

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