Wolfram Alpha:

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Geometries of the Universe
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Unit 1 Sphere in Spherical Coordinates
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The 1-sphere is just a circle swept out by angle φ

dS2 = dr2 + r2dφ2

It is conventional to refer to dφ as dΩ1 - the metric for a circle.
Thus:

dS2 = dr2 + r2dΩ12

In cartesian coordinates this is equivalent to writing:

x2 + y2 = 1

Where,

x = cosφ
y = sinφ

Unit 2 Sphere in Spherical Coordinates
---------------------------------------

The 2-sphere can be formed by stacking 1-spheres on top of each
other where the circle diameters continuously vary from 0 to 1 and
then back to 0.  θ ranges 0 -> π

dS2 = dθ2 + sin2θdφ2

= dθ2 + sin2θdΩ12

= dΩ22

In cartesian coordinates this is equivalent to writing:

x2 + y2 + z2 = 1

Where,

x = sinθcosφ
y = sinθsinφ
z = cosθ

Unit 3 Sphere in Spherical Coordinates
---------------------------------------

The 3-sphere can be visualized as a 2-sphere that can be
inflated/deflated just like a ballon with r ranging from 0 -> π

dS2 = dr2 + sin2r[dθ2 + sin2θdφ2]

= dr2 + sin2r[dθ2 + sin2θdΩ12]

= dr2 + sin2rdΩ22

= dΩ32

In cartesian coordinates this is equivalent to writing:

x2 + y2 + z2 + w2 = 1

Where,

x = sinrsinθcosφ
y = sinrsinθsinφ
z = sinrcosθ
w = cosr

FRW Universes
-------------

Positive spacial curvature:

For a 3 sphere, we can incorporate the expansion of space by
modifying the metric as follows:

dS2 = a(t)2[dr2 + sin2rdΩ22]

dτ2 = dt2 - dS2

dτ2 = dt2 - a(t)2[dr2 + sin2rdΩ22]

Negative spacial curvature (Hyperbolic plane):

For the negative curvature case we have to use the ideas of
hyperbolic geometry.  It turns out that in this regime the
metric is given by:

dS2 = a(t)2[dr2 + sinh2rdΩ22]

= dH32

Note:  sinhr = (er - e-r)/2  r: 0 -> ∞

dτ2 = dt2 - a(t)2H32

Flat spacial curvature:

dS2 = a(t)2[dr2 + r2dΩ22]

dτ2 = dt2 - a(t)2[dr2 + r2dΩ22]

We can summarize all of the above cases as follows:

dτ2 = dt2 - a(t)2[dr2 + ξ(r)2dΩ22]   ... 1.

Where,

ξ(r) = r       k = 0
ξ(r) = sinr    k = +1
ξ(r) = sinhr   k = -1

The above FRW metrics fullfill the COSMOLOGICAL PRINCIPLE that the
distribution of matter in the universe is homogeneous and isotropic
when viewed on a large enough scale.

All three geometries are classes of Riemannian geometry that are
based on three possible states for parallel lines

- Never meeting (flat or Euclidean)
- Must cross (spherical)
- Always divergent (hyperbolic)

Alternatively, one can think of triangles where for a flat universe
the angles of a triangle sum to 180 degrees, in a closed universe
the sum must be greater than 180, in an open universe the sum
must be less than 180.

Note:  These geometries should not in any way be construed as
objects embedded in a higher dimension like a sphere in space.
Rather, they are they geometries that are intrinsic to space itself.

Note,  this curvature is similar to spacetime curvature due to stellar
masses except that in this case it is the entire mass of the universe
that determines the curvature.

Measuring Curvature
-------------------

One way to visualize positive and negative curvature is via the use
of STEREGRAPHIC PROJECTIONS.

In flat space angles subtended by objects at a distance, r, with a
diameter, d, vary as θ = d/r.  In positively curved space, angles
subtended by objects vary as d/sinr.  In negatively curved space angles
subtended by objects vary as d/sinhr.  This is equivalent to saying that,
compared to flat space, objects in positively curved space appear to
be larger while objects in negatively curved space appear to be smaller.

Similarly, if the universe is flat it would be expected that that the
number of galaxies out to a distance, r, increases proportionally to
r2.  If the universe has positive curvature it would be expected that
the number of galaxies increases more slowly than r2 (N ∝ sin2r).  If
the universe has negative curvature it would be expected that the
number of galaxies increases faster than N ∝ r2 (N ∝ sinh2r).  The
curvature of the universe is revealed by whether the number of galaxies
per volume increases more slowly or more quickly than the flat space
case.

The number of galaxies at different distances (or redshifts) has been
as has the size of prominent ripples in the CMB.  In both cases the
indications are that space seems to be flat.

The Critical Density
--------------------

The general case of the FRW equation can be written as:
.
(a/a)2 + k/a2 = 8πGρR/3 + 8πGρM/3 + Λ/3
.
(a/a)2 + k/a2 = 8πGρ/3

Define the critical density, ρc, such that

Ω = ρ/ρc

The critical density corresponds to the case where k = 0 and so:

ρc = 3H2/8πG

This is calculated today to be 9.47 x 10-27kg/m3

A value of Ω < 1 corresponds to a universe that expands forever (open
with k < 0).  A value of Ω > 1 corresponds to a universe that will
eventually stop expanding and collapse (closed with k > 0).  Setting
Ω = 1 (i.e. k = 0) we get:

1 = ΩR + ΩM + ΩΛ

Where R stands for radiation and M stands for mass.

Determination of the Ω's is an ongoing process in cosmology. Direct
observations indicate that the radiation component can be ignored.
To get values for the remaining terms, physicists construct complex
mathematical models to describe experimental results.  One such
model is the Lambda-CDM* model.  Basically, the model describes the
the existence and structure of the CMB, the accelerating expansion of
the universe observed in the redshifted light from distant galaxies and
supernovae, the abundances of the light elements hydrogen, helium
and lithium and the large scale distribution of galaxies.  It is based on
the FRW metric and the FRW equations and their solutions.  The inputs
consists of a number of different parameters that include the baryonic
density, the dark matter density and the dark energy density.  An initial
'guess' is made for the values of each of these parameters, and the
model output is compared against physical observations.  The process is
iterative as parameters are adjusted to improve the fit to the observed
data.  Values of the Ω's at the present time are indicated to be:

* Cold Dark Matter

ΩR = 0
ΩM ~ 0.3
ΩΛ ~ 0.7
k ~ 0

General Relativity and the FRW Equation
---------------------------------------

The FRW equations can be derived from Einstein's field equations using
the metric shown in equation 1.  From these metrics it is possible to
calculate the Einstein tensor and set it equal to the pressure and energy
of the matter in the universe.  The Einstein field equations are:

Rμν - (1/2)gμνR + gμνΛ= 4πGTμν

From the 00 component we get:

R00 - (1/2)g00R + gμνΛ = 4πGρ   ρ = T00

This leads to (assuming c = 1):
.
(a/a)2 + k/a2 - Λ/3 = 8πGρ/3

and, using this equation and the trace of Einstein's field equations,
we get:
..
(a/a) - Λ/3 = -4πG(ρ + 3p)/3

The second equation states that both the energy density and the
.
pressure, p, cause the expansion rate of the universe, a, to decrease.
This is a consequence of gravitation, with pressure playing a similar
role to that of mass density.   The cosmological constant, on the other
hand, causes an acceleration in the expansion of the universe.
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