theorem 48.5 The Friedmann equations
open in the book ·
parts/05-general-relativity-cosmology/07-cosmology.tex:360
· p. 1323
Rests on
- 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
¶
-
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
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 ¶
-
depends_on
definition 13.117
Metric tensor of signature $(p,q)$
¶
-
depends_on
definition 13.153
Contractions
¶
-
depends_on
definition 44.2
Einstein tensor
¶
- proves proof ch:07-cosmology@proof-4 ¶
Supports
- depends_on phenomenon 48.17 The expansion is accelerating ¶
- depends_on phenomenon 48.10 A primordial floor of light elements ¶
-
depends_on
phenomenon 51.6
Acoustic peaks, and a spatially flat universe
¶
- depends_on phenomenon 51.7 E-mode polarization, and no primordial B mode ¶
-
depends_on
theorem 48.14
Linear growth of density perturbations
¶
- depends_on phenomenon 48.16 Flat rotation curves and the missing mass ¶
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 |
→ | Über die Krümmung des Raumes | derived | parts/05-general-relativity-cosmology/07-cosmology.tex:380 |
cites |
→ | Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des Raumes | derived | parts/05-general-relativity-cosmology/07-cosmology.tex:380 |
cites |
→ | Un Univers homogène de masse constante et de rayon croissant rendant compte de la vitesse radiale des nébuleuses extra-galactiques | derived | parts/05-general-relativity-cosmology/07-cosmology.tex:381 |
depends_on |
→ | eq:cosmo-flrw-metric | declared | parts/05-general-relativity-cosmology/07-cosmology.tex:382 |
depends_on |
→ | eq:cosmo-perfect-fluid | declared | parts/05-general-relativity-cosmology/07-cosmology.tex:382 |
depends_on |
→ | Einstein field equations | declared | parts/05-general-relativity-cosmology/07-cosmology.tex:382 |
depends_on |
← | The expansion is accelerating | declared | parts/05-general-relativity-cosmology/07-cosmology.tex:1454 |
depends_on |
← | A primordial floor of light elements | declared | parts/05-general-relativity-cosmology/07-cosmology.tex:749 |
depends_on |
← | Acoustic peaks, and a spatially flat universe | declared | parts/05-general-relativity-cosmology/10-exp-cmb.tex:734 |
depends_on |
← | Linear growth of density perturbations | declared | parts/05-general-relativity-cosmology/07-cosmology.tex:1135 |
proves |
← | ch:07-cosmology@proof-4 | declared | parts/05-general-relativity-cosmology/07-cosmology.tex:386 |