Friedmann Equation and Solutions  

Friedmann Equation > s.a. cosmological expansion and acceleration.
* Idea: The form taken by Einstein's equation in the case of homogeneous and isotropic spacetimes, proposed in 1922 by A Friedmann.
* Geometry: The spatial metric reduces to a single degree of freedom a(t), interpreted as a characteric size of the universe, in terms of which the line element

ds2 = –dt2 + a2(t) [dr2/(1–kr2) + r22] ,

(in proper-time gauge), where k = –1, 0, 1; The spatial Ricci tensor and scalar curvature are given by

Rac = (4πG ρ – \(\ddot a\) – 3 a·2) hac = 2 a–2 chac ,   R = 16πG ρ – 6 a·2 ;

For more on the geometry, > see geometry of FLRW spacetimes.
* The equation: The Friedmann equation is the equation of motion for a that one gets from the scalar constraint of general relativity,

(a·/a)2 = 8πG ρ/3 – k/a2 + Λ/3 ,

with a·/a = H0, the Hubble constant, and the cosmological constant Λ plays the role of an effective energy density; This needs to be solved, together with the appropriate dynamical equations for the coupled matter, which enters the Friedmann equation through the matter energy density ρ.
@ General references: Friedmann ZP(22); Lemaître Nat(31)may, ASSB(33) [translation GRG(97)]; Nemiroff & Patla AJP(08)mar-ap/07 [with various types of matter]; Aldrovandi et al IJMPD(08)-a0705-in [effect of matter interactions].
@ Simple treatment: Faraoni AJP(99)aug [simple, p = (γ–1) ρ]; Jordan AJP(05)jul-ap/04 [Newtonian approach].
@ Related topics: Gluck et al IJBC(11)-a0910 [as an active dynamical system]; Rubio & Salamanca a1402 [recast as harmonic oscillator equation].
> Related topics: see early-universe cosmology; gravitational thermodynamics; thermodynamic systems.

Friedmann Solutions > s.a. cosmology and perturbations in general relativity; singularities [b-boundary].
* With perfect-fluid matter: The solutions (for vanishing cosmological constant) are parametrized by the single conserved quantity

c:= ((4/3)πG ρ)–2/3 – ((8/3)πG ρa·2) ,

(which can be interpreted as binding energy), such that:
- c < 0: Open universe, with negative spatial curvature and eternal expansion, a ~ t2/3 for t → 0, a ~ t for t → ∞;
- c = 0: Open universe, with zero spatial curvature and eternal expansion, at2/3 for all t;
- c < 0: Closed universe, with positive spatial curvature and eventual recollapse, a ~ t2/3 for t → 0, a = amax at t = π c–3/2.
@ References: Baumgärtel a1110 [classification of types of solutions, non-vanishing Λ]; Faraoni PLB(12) [k = 0, barotropic fluids, map between exact solutions].
> With other types of matter: see FLRW spacetime; cosmological models in general relativity and in modified gravity [including Cardassian]; early-universe models [bounces].

Modified Versions > s.a. brane-world cosmology; dark energy.
* Idea: Modifications can be motivated by theoretical (e.g., quantum gravity corrections, leading to changes in the lhs), or phenomenological considerations (different types of matter, including dark energy, leading to changes in the right-hand side).
@ From averaging of inhomogeneities: Paranjape & Singh GRG(08)ap/06 [and observation]; > s.a. expansion; relativistic cosmology.
@ From modified gravity: Meng & Wang CQG(03), CQG(04) [from R–1 gravity]; Andrew et al GRG(07)ht/06 [with added Gauss-Bonnet term, higher D]; Sheykhi PRD(10)-a1004, Sheykhi & Teimoori GRG(12) [entropic corrections]; Cai JHEP(12)-a1207 [higher dimensional, emergent from holography].
@ In non-commutative geometry: Guzmán et al PLB(11)-a0812; Malekolkalami & Farhoudi PLB(09)-a0911 [non-minimal scalar field].
@ Quantum corrections: Janssen et al a0807, Janssen & Prokopec AP(10)-a0807 [one-loop correction]; Kuzmichev & Kuzmichev APPB(13)-a1307, a1411; Awad & Ali CEJP(14)-a1403 [Verlinde's entropic force proposal and gup]; Myrzakulov et al PRD(15)-a1412 [inflation from higher-derivative quantum gravity effective action]; Viaggiu MPLA(16)-a1511.
@ Other quantum gravity: Singh PRD(06)gq [lqg and Randall-Sundrum braneworlds]; Bojowald et al PRD(07)-a0706 [lqg + scalar, effective equations]; Battisti PRD(09)-a0807 [bounce from deformed Heisenberg algebra]; Majumder ASS(11)-a1105, Jalalzadeh et al PRD(14)-a1403 [generalized uncertainty principle]; > s.a. FLRW quantum cosmology.
@ Other variations: Choudhury et al JCAP(12)-a1106 [with delay]; Sheykin & Paston IJMPcs(16)-a1511 [in Regge-Teitelboim theory of embeddings].
> In other modified-gravity theories: see cosmology in gravitational theories [including Hořava-Lifshitz gravity]; FLRW spacetime; gravitational thermodynamics; higher-order gravity; Rainbow Gravity.


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