phenomenon 51.6 Acoustic peaks, and a spatially flat universe

open in the book · parts/05-general-relativity-cosmology/10-exp-cmb.tex:714 · p. 1381

phenomenon-tested: no experiment declares a \tests edge to it and it is not listed in untested.txt

Rests on

depends_on phenomenon 65.3 Thomson scattering ¶
- proves derived proof ch:08-radiation-scattering@proof-2 ¶
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 definition 13.48 Differentiable manifold ¶
      - depends_on definition 13.84 Tensor ¶
  - 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 13.90 Mixed tensor ¶
      - depends_on definition 13.82 Vector field ¶
    - depends_on definition 13.147 Torsion ¶
      - depends_on definition 13.145 Affine connection ¶ ↺
    - depends_on proposition 7.105 Clairaut–Schwarz ¶
      - depends_on definition 7.20 Continuity at a point ¶
      - depends_on theorem 7.35 Mean value theorem ¶
      - proves proof ch:05-real-analysis@proof-64 ¶
    - 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 13.120 Isometry ¶
      - depends_on equation 13.267 eq:mfd-lie-def ¶
    - depends_on definition 13.149 Metric compatibility ¶
      - depends_on definition 13.145 Affine connection ¶ ↺
      - depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶ ↺
      - depends_on definition 13.146 Parallel transport and autoparallels ¶
    - depends_on proposition 13.127 Component formulas ¶
      - depends_on definition 13.89 Contravariant and covariant tensors ¶
      - depends_on definition 13.90 Mixed tensor ¶ ↺
      - depends_on equation 13.267 eq:mfd-lie-def ¶ ↺
      - proves proof ch:11-manifolds-tensors-curvature@proof-28 ¶
    - depends_on theorem 13.150 Levi-Civita connection and contorsion ¶
      - depends_on definition 13.145 Affine connection ¶ ↺
      - depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶ ↺
      - depends_on definition 13.149 Metric compatibility ¶ ↺
      - depends_on definition 13.147 Torsion ¶ ↺
      - proves proof ch:11-manifolds-tensors-curvature@proof-41 ¶
    - 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 equation 13.246 eq:mfd-leibniz-forms ¶
      - depends_on equation 13.245 eq:mfd-nilpotency ¶
      - depends_on theorem 13.156 Cartan structure equations ¶
      - proves proof ch:11-manifolds-tensors-curvature@proof-46 ¶
    - 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 ¶
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:10-exp-cmb@proof-5 ¶

Supports

depends_on phenomenon 51.7 E-mode polarization, and no primordial B mode ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`cites` (evidence)	→	Planck 2018 results. VI. Cosmological parameters	derived	parts/05-general-relativity-cosmology/10-exp-cmb.tex:720
`cites` (evidence)	→	Planck 2018 results. VI. Cosmological parameters	derived	parts/05-general-relativity-cosmology/10-exp-cmb.tex:729
`cites` (history)	→	Primeval Adiabatic Perturbation in an Expanding Universe	derived	parts/05-general-relativity-cosmology/10-exp-cmb.tex:724
`cites` (history)	→	Cosmic Black-Body Radiation and Galaxy Formation	derived	parts/05-general-relativity-cosmology/10-exp-cmb.tex:720
`cites` (history)	→	Small-scale fluctuations of relic radiation	derived	parts/05-general-relativity-cosmology/10-exp-cmb.tex:724
`cites` (evidence)	→	A flat Universe from high-resolution maps of the cosmic microwave background radiation	derived	parts/05-general-relativity-cosmology/10-exp-cmb.tex:720
`depends_on`	→	Thomson scattering	declared	parts/05-general-relativity-cosmology/10-exp-cmb.tex:734
`depends_on`	→	The Robertson–Walker metric	declared	parts/05-general-relativity-cosmology/10-exp-cmb.tex:734
`depends_on`	→	The Friedmann equations	declared	parts/05-general-relativity-cosmology/10-exp-cmb.tex:734
`depends_on`	←	E-mode polarization, and no primordial B mode	declared	parts/05-general-relativity-cosmology/10-exp-cmb.tex:842
`proves`	←	ch:10-exp-cmb@proof-5	declared	parts/05-general-relativity-cosmology/10-exp-cmb.tex:738