phenomenon 48.15 A standard ruler in the galaxy distribution

open in the book · parts/05-general-relativity-cosmology/07-cosmology.tex:1236 · p. 1333

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

Rests on

depends_on corollary 48.8 Angular-diameter distance and the duality relation ¶
- 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 equation 13.83 eq:mfd-ecsgauss ¶
      - depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶
      - 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 proposition 13.140 Killing's equation ¶
      - depends_on proposition 13.154 Symmetries of the curvature ¶
      - proves proof ch:11-manifolds-tensors-curvature@proof-48 ¶
    - proves proof ch:07-cosmology@proof-2 ¶
  - proves proof ch:07-cosmology@proof-5 ¶
- proves proof ch:07-cosmology@proof-6 ¶
depends_on equation 48.16 eq:cosmo-etherington ¶
depends_on equation 48.34 eq:cosmo-sound-horizon ¶
proves proof ch:07-cosmology@proof-12 ¶

Supports

depends_on phenomenon 103.68 The sum of the neutrino masses is bounded ¶

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:1245
`cites`	→	Detection of the Baryon Acoustic Peak in the Large-Scale Correlation Function of SDSS Luminous Red Galaxies	derived	parts/05-general-relativity-cosmology/07-cosmology.tex:1242
`depends_on`	→	Angular-diameter distance and the duality relation	declared	parts/05-general-relativity-cosmology/07-cosmology.tex:1250
`depends_on`	→	eq:cosmo-etherington	declared	parts/05-general-relativity-cosmology/07-cosmology.tex:1250
`depends_on`	→	eq:cosmo-sound-horizon	declared	parts/05-general-relativity-cosmology/07-cosmology.tex:1250
`depends_on`	←	The sum of the neutrino masses is bounded	declared	parts/11-qft-standard-model/05-flavour-neutrinos.tex:3351
`proves`	←	ch:07-cosmology@proof-12	declared	parts/05-general-relativity-cosmology/07-cosmology.tex:1254