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

d*s*^{2} = −d*t*^{2}
+ *a*^{2}(*t*)
[d*r*^{2}/(1−*kr*^{2})
+ *r*^{2}dΩ^{2}] ,

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

*R*_{ac}
= (4π*Gρ* − \(\ddot a\) −
3 *a*^{·}^{2})
*h*_{ac} =
2 *a*^{−2}* ch*_{ac} ,
*R* = 16π*G* *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} =
8π*Gρ*/3 − *k*/*a*^{2} + Λ/3 ,

with *a*^{·}/*a* = *H*_{0},
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)π*G* *ρ*)^{−2/3}
− ((8/3)π*G* *ρ*
− *a*^{·}^{2}) ,

(which can be interpreted as binding energy), such that:

- __c____ < 0__:
Open universe, with negative spatial curvature and eternal expansion,
*a* ~ *t*^{2/3} for *t*
→ 0, *a* ~ *t* for *t* → ∞;

- __c____ = 0__:
Open universe, with zero spatial curvature and eternal expansion, *a*
∝ *t*^{2/3} for all *t*;

- __c____ < 0__:
Closed universe, with positive spatial curvature and eventual recollapse,
*a* ~ *t*^{2/3} for *t*
→ 0, *a* = *a*_{max} 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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