Friedmann
Equation and Solutions| Friedmann
Equation and Solutions |
In General > s.a. cosmological
expansion and acceleration; gravitational
thermodynamics.
* Idea: A 1-parameter
family of solutions of Einstein's equation, describing dust-filled homogeneous
and isotropic spacetimes.
* History: Proposed in 1922 by A Friedmann.
* Metric: The line element
(in proper-time gauge) is of the FRW form
ds2 = –dt2 +
a2(t) [dr2/(1–kr2) +
r2d
2]
,
where a is interpreted as a characteric size of the universe.
* Friedmann equation:
The equation of motion for a that one gets from the scalar constraint
of general relativity,
(a·/a)2 =
8
G
/3 –k/a2 + 
/3
,
with a·/a = H0,
the Hubble constant and the cosmological constant plays
the role of an effective energy density, together with the appropriate matter
equations.
* Labelling parameter: The conserved quantity
c:= ((4/3)G)–2/3 – ((8/3)G – a·2)
,
(interpreted as binding energy), such that
- c < 0: Open universe, negative curvature; a 
t2/3
for t → 0,
a t for
t → 
;
- c = 0: Open universe, zero curvature; a 
t2/3;
- c < 0: Closed universe, positive curvature; a t2/3 for
t → 0,
a = amax at t =
c–3/2.
* Spatial curvature: The spatial Ricci tensor is given by
Rac =
(4G – a··– 3 a·2) hac =
2 a–2 chac , R =
16G – 6 a·2 .
References
@ General: Friedmann ZP(22); Lemaître ASSB(33)
[translation GRG(97)];
Cai & Kim JHEP(05)ht [from
thermodynamics]; 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 = (
–1)];
Jordan AJP(05)jul-ap/04 [Newtonian
approach].
@ Related topics: Gluck et al a0910 [as an active dynamical system]
> Related topics:
see early-universe cosmology; FRW
spacetime;
cosmology and perturbations in
general relativity; singularities [b-boundary].
Modified Versions
* Idea: Modifications
can be motivated by theoretical (e.g., quantum gravity corrections, leading
to
changes
in
the lhs), or phenomenological
considerations (e.g., dark energy, leading to changes in the rhs).
@ 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]; Akbar &
Cai PLB(06)ht
[scalar-tensor and f(R), from thermodynamics]; Andrew et al
ht/06/CQG
[with added Gauss-Bonnet term, higher D]; Ling JCAP(07)gq/06 [rainbow
gravity]; > s.a. FRW spacetime, higher-order
gravity.
@ In non-commutative geometry:
Guzmán et al a0812; Malekolkalami
& Farhoudi PLB-a0911 [non-minimal
scalar field].
@ From quantum gravity: Singh PRD(06)gq [lqg
and Randall-Sundrum braneworlds]; Bojowald et al PRD(07)-a0706 [lqg
+ scalar, effective
equations]; Battisti PRD-a0807
[bounce from deformed Heisenberg algebra]; Janssen et al a0807,
Janssen & Prokopec a0807 [one-loop
quantum corrections]; > s.a. FRW
quantum cosmology.
> Other variations:
see branes; cosmological
models [various types of matter, bounces, Cardassian]; dark
energy.
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send feedback and suggestions to bombelli at olemiss.edu – modified 16
nov 2009