Curvature is a fundamental property of the universe and figures prominently in Einstein's theory of general relativity as well as in many cosmological theories.
We begin by noting that the metric on any space -like slice å (t) is given by the scale factor a(t) times the constant curvature, e.g. k = (+1, 0, -1). Further we note that the “sectional curvature” is defined:
k /a(t) 2
So when:
‖ k ‖ = 1
The scale factor a(t) is simply the curvature radius. And when k = 0 the scale factor remains arbitrary. Basically, the function a(t) describes the evolution of the universe. It is completely determined by Einstein’s field equation, see e.g.
For a homogeneous and isotropic space -time it reduces to an
ordinary differential equation:
(a’/ a)
2 + k / a
2 = 8 p G r / 3
Where, a’ = da/dt, G is the Newtonian gravitational constant, r is the mass-energy density and units are chosen to make the speed of light c = 1. Note also the first term is the square of the Hubble parameter :
H = a’/ a.
Currently, we estimate the corresponding constant H o » 70 km/ sec/Mpc). If we substitute H = (a’/a) into the simplified Einstein field equation we see that when k =0 the mass –energy density is:
r o = 3 H 2 / 8 p G
Which mass density works out – using the current value of H o to about :
9.3 x 10 -27 kg/ m3
In effect, if we can measure the current mass- energy density (r o ) and the Hubble constant we can obtain the sign of the curvature.
Thus, when k = 1 the mass energy density must be greater than 3 H 2 / 8 p G. Further, if k ≠ 0 we can solve for the curvature radius, e.g.
a = 1/ H Ö k /8 p G r
/ (3 H 2 - 1)
= 1/H Ö (k / W - 1 ) k ≠ 0
Where the density parameter is the dimensionless ratio of the actual density to the critical density:
r c
= 3 H 2 / 8 p G
As a further refinement in formalism, it’s important to point out the
universe contains different forms of mass-energy which contribute to the total,
and which relates to the critical threshold density, e.g..
W = r / r c = r y + r m + r L =
W y + W m + W L
Where r y is the energy density in radiation, r m is the energy density in matter, and r L is a possible vacuum energy. Vacuum energy with density r L = 3 L / 8 p G mimics the cosmological constant, L.
In a universe containing only ordinary matter we would have: W y = W L = 0, and the mass- density scales as:
r = r o (a o / a) 3
Substituting into the original equation:
(a’/ a)
2 + k / a
2 = 8 p G r / 3
Exact solutions can be obtained for a(t). The solutions predict that if W < 1 the universe will expand forever. And if W > 1 the universe will re-contract. If however W = 1 the universe will expand forever but at a rate a’ approaching zero.
It is well to point out here one element of confusion for some cosmologists. That is, assuming that a negatively curved universe must be the infinite hyperbolic 3-space H 3.
This careless conflation has led to the use of the term "open universe" to mean three different things:
1- A negatively curved universe
2- A spatially infinite universe
3- Expanding forever universe
On the positive side, as finite manifolds have become more widely understood - basically via more research, published papers- the terminology of curved space-time has moved toward greater coherence. Specifically, one now sees a general consensus that universes of positive ( k = +1), negative ( k = - 1), and zero ( k = 0) curvature are called "spherical", "hyperbolic" and "flat" respectively.
To fix ideas, the top graphic illustrates the first two curvatures. The one on the left side represents a spherical ( k = +1) space-time and on the right a ( k = -1) or hyperbolic one. In like manner, universes that recollapse are called "closed", those that expand forever are with zero limiting velocity are called "critical" and those that expand forever with positive limiting velocity are called "open".
The primary downside to this cosmological terminology? It conflicts with topologists definitions of closed and open. Well, you can't have everything!
See Also:
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