phenomenon 48.13 The anisotropy spectrum of the relic radiation

open in the book · parts/05-general-relativity-cosmology/07-cosmology.tex:999 · p. 1330

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

Rests on

depends_on equation 48.24 eq:cosmo-temperature-scaling ¶
depends_on phenomenon 48.12 The relic microwave background ¶
- 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-9 ¶
proves proof ch:07-cosmology@proof-10 ¶

Supports

depends_on phenomenon 48.18 The Hubble tension ¶

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`	→	Planck 2018 results. VI. Cosmological parameters	derived	parts/05-general-relativity-cosmology/07-cosmology.tex:1007
`cites`	→	Primeval Adiabatic Perturbation in an Expanding Universe	derived	parts/05-general-relativity-cosmology/07-cosmology.tex:1012
`cites`	→	Perturbations of a Cosmological Model and Angular Variations of the Microwave Background	derived	parts/05-general-relativity-cosmology/07-cosmology.tex:1010
`cites`	→	Structure in the COBE differential microwave radiometer first-year maps	derived	parts/05-general-relativity-cosmology/07-cosmology.tex:1004
`cites`	→	Small-scale fluctuations of relic radiation	derived	parts/05-general-relativity-cosmology/07-cosmology.tex:1012
`depends_on`	→	eq:cosmo-temperature-scaling	declared	parts/05-general-relativity-cosmology/07-cosmology.tex:1015
`depends_on`	→	The relic microwave background	declared	parts/05-general-relativity-cosmology/07-cosmology.tex:1015
`depends_on`	←	The Hubble tension	declared	parts/05-general-relativity-cosmology/07-cosmology.tex:1671
`proves`	←	ch:07-cosmology@proof-10	declared	parts/05-general-relativity-cosmology/07-cosmology.tex:1018