Geometry of Friedmann-Lemaître-Robertson-Walker Spacetimes  

Metric > s.a. FLRW spacetimes; cosmologies and relativistic cosmological models [geodesics]; spherical symmetry.
* Idea: A homogeneous and isotropic metric, characterized by one of three types of 3D constant curvature spatial geometries (spatially open k = –1, spatially flat k = 0, or spatially closed k = 1), and an arbitrary function a(t) representing the fiducial size of the universe at time t.
* Proper time gauge: The line element is of the form

ds2 = –dτ2 + a(τ)2 [dχ2 + f 2(χ) (dθ2 + sin2θ dφ2)] = –dt2 + a(t)2 [dr2/(1–kr2) + r22] ,

where f 2(χ) = sin2χ if k = 1, χ2 if k = 0, and sinh2χ if k = –1.
* Conformal gauge: The line element is of the form (using the same definitions for f(χ))

ds2 = a2(t) [–dt2 + dχ2 + f 2(χ) dΩ2] .

* Singularities: For k > 0, a point; For k ≤ 0, an infinite manifold; > s.a. singularities [metric extension].
* Useful quantities: The Hubble expansion factor, defined as H:= a·/a; It satisfies \(1\over6\)R = \(\ddot a\)/a + H2 = k/a2.
@ Metric and coordinates: Rindler GRG(81); Lachièze-Rey A&A(00)ap [embedding in 5M]; Ibison JMP(07) [conformal forms], a0704 [static form]; Grøn & Johannesen a0911-wd [conformally-flat spacetime coordinates].
@ Singularities: Gruszczak a1011 [differentially singular boundary]; het Lam & Prokopec a1606 [non-singular past-geodesically incomplete spacetimes].
@ Extending the spacetime: Schröter a0906; Stoica IJTP(15)-a1112; Belbruno CMDA(13)-a1205; Stoica IJTP(16), comment Fernández-Jambrina IJTP(16)-a1603; Gielen & Turok PRD(17)-a1612 [Feynman propagator].

Connection > s.a. geodesics; holonomy; Raychaudhuri Equation; relativistic cosmological models [geodesics].
* In conformal time gauge: The non-equivalent, non-vanishing components of the metric connection / Christoffel symbols are

Γ000 = Γ011 = a·/a
Γ122 = –f f '
Γ101 = Γ202 = Γ303 = a·/a
Γ022 = f 2 a·/a
Γ133 = –f f ' sin2θ
Γ212 = Γ313 = f '/f
Γ033 = f 2 sin2θ a·/a
Γ233 = –sin θ cos θ
Γ323 = cot θ .

Other Geometric Quantities and Topics > s.a. horizons; types of singularities [including sudden singularities]; world function.
* Scalar curvature: Using conformal time,

R = 6 \(\ddot a\)/a – 2 (2ff '' + f '2–1)/(af)2 .

@ References: Ellis & van Elst gq/97-fs [geodesic deviation]; Chen & Van der Veken JMP(07) [non-degenerate surfaces]; Goulart & Novello G&C(08) [Weyl tensor, stability]; Cunningham et al a1705 [exact geodesic distances].
> Related topics: see bianchi models.

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