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Expansion of the Universe
--------------------------
Mass Dominated Period
-------------------------
. .
. .
. M .
. .
. + .m (x = 1)
. 0 .
. .
. .
. .
<-- D -->
Consider the case of a mass, m sitting on the surface
of a sphere with mass, M, that is being ejected from M
at velocity, v, we can use the conservation of energy
and Newton's law that states:
1. A spherically symmetric body affects external
objects gravitationally as though all of its
mass were concentrated at a point at its
center.
2. If the body is a spherically symmetric shell
(i.e., a hollow ball), no net gravitational
force is exerted by the shell on any object
inside, regardless of the object's location
within the shell.
We can write:
mv^{2}/2 - GMm/D = E
Where mv^{2}/2 is the KE of m and GMm/D is the gravitational
PE due to M. This equation can be written as:
v^{2} - 2GM/D = 2E/m
Now D = ax where a is the scale factor and x is a
distance. For x = 1:
. .
v = D = xa
So,
.
x^{2}a^{2} - 2GM/xa = 2E/m
Now rewrite M in terms of mass density, M = (4/3)πx^{3}a^{3}ρ:
.
∴ x^{2}a^{2} - (2G/xa)(4/3)πx^{3}a^{3}ρ = 2E/m
.
∴ x^{2}a^{2} - (8/3)πGx^{2}a^{2}ρ = 2E/m
.
∴ x^{2}(a/a)^{2} - (8/3)πGx^{2}ρ = 2E/a^{2}m
Dimensionally,
[m^{2}.(m^{2}/s^{2})/m^{2}] - [(m^{3}/kg.s^{2}).m^{2}.kg/m^{3}] = [(kg.m^{2}/s^{2})/m^{2}.kg]
[(m^{2}/s^{2})] - [(m^{2}/s^{2})] = [(1/s^{2})]
In order to make this equation correct, the RHS
needs to multiplied by a factor of x^{2}. Thus,
.
x^{2}(a/a)^{2} - (8/3)πGx^{2}ρ = 2Ex^{2}/a^{2}m
.
(a/a)^{2} - (8/3)πρG = 2E/a^{2}m
= κ/a^{2} (where κ = 2E/m)
Dimension check:
[(m/s)/m]^{2} - [(kg/m^{3}).(m^{3}/kg.s^{2})] = [(kg.m^{2}/s^{2})/(kg.m^{2})] ✓
[s^{-1}]^{2} - [s^{-1}]^{2} = [s^{-1}]^{2}
Note that κ is a dimensionless.
This is the FRIEDMANN-ROBERTSON-WALKER equation for the
MASS DOMINATED PERIOD.
ρ is a function of the scale factor (hence time). As
the volume expands, the density will decrease if the
amount of mass in the universe is assumed to be constant
(conserved). To account for this, ρ can be written
as:
ρ = ν/a^{3} where ν is now fixed.
k = 0 Case
----------
Note when the RHS is equal to 0 we get the familiar
formula for the escape velocity. The energy is just
enough to prevent the sphere from re-collapsing but
not enough to make it expand further. Therefore,
.
(a/a)^{2} = (8/3)πρG
Reinstate ρ = ν/a^{3} and let (8/3)πGν = α. Therefore,
.
(a/a)^{2} = α/a^{3}
.
a = √α/√a
da/dt = √α/√a
∴ dt/da = √a/√α
With solution:
t = (2/3)a^{3/2}/√α
Rearranging:
a = (3√α/2)^{2/3}t^{2/3}
Therefore, during the mass dominated period, the
universe grew at a rate according to:
a ∝ t^{2/3}
Radiation Dominated Period
---------------------------
This occurred before the mass dominated period.
The Universe has 10^{9} photons for every proton. Photons
are massless but have energy by virtue of their motion.
E_{p} = hν = c/λ (de Broglie)
As the universe expands, λ, gets stretched and so the
energy (and temperature) diminish by a factor of 1/a.
There will now be an extra a in the denominator of
the FRW equation. Thus,
.
(a/a)^{2} = (8/3)πMG/a^{4}
Solve this equation in the same way as before to get:
p = 1/2. So,
a = (6πMG)^{1/3}t^{1/2}
.
∴ H = (a/a) = 1/(2t)
Therefore, during the radiation dominated period, the
universe grew according to a power law of 1/2.
Vacuum Dominated Period
-----------------------
First, we need to talk about pressure and energy density
for radiation dominated period.
Consider many photons sloshing backwards and forwards
inside a 1D box:
|--------------------<>|<- wall
<------ L ------>
F = dp/dt
The change in photon momentum is 2p as it bounces off
the wall. The bouncing creates a pressure on the wall.
Time between collisions = 2L/c
F = 2p/(2L/c) = pc/L = E/L = ρ in 1D.
Now consider a cube with side, a:
-->x
------
/ /|
-----/ |-> P (pressure)
| |A|
| | /
| |/
-----
<- L ->
F = E/L Therefore P = E/LA => P = ρ But this is only
for one direction. To get the correct answer for 3D we
need to divide by 3, Thus,
P = ρ_{R}/3
Now consider the above box again. Let's move the face,
A, of the cube by an amount dL. The work done is PAdL
or PdV.
The decrease in the internal energy of the box is -PdV.
Now E inside the box is:
E = ρV
Therefore,
dE = ρdV + Vdρ = -PdV
or,
Vdρ = -(P + ρ)dV
Now introduce the parameter, w, as P = ρw
Vdρ = -(w + 1)ρdV
dρ = -(w + 1)ρdV/V
dρ/ρ = -(w + 1)dV/V
lnρ = -(w + 1)lnV after integration
= ln{1/V^{w+1}} + c'
Where, c' is the constant of integration.
ρ = c'/V^{w+1}
= c'/a^{3(w+1)} since V = a^{3}
Rewrite the FRW equation (a/a)^{2} + k/a^{2} = (8/3)πρG, as
follows:
.
(a/a)^{2} + k/a^{2} = (8/3)πGc'/a^{3(1 + w)}
We can now use w to describe any combination of periods.
Radiation Dominated:
w = 1/3 => (8/3)πGc'/a^{4} => (8/3)πGρ_{R}
Mass Dominated:
w = 0 => (8/3)πGc'/a^{3} => (8/3)πGρ_{M}
Vacuum Dominated (Dark Energy):
w = -1 => c' => (8/3)πGρ_{0}
The quantity 8πGρ_{o} is the COSMOLOGICAL CONSTANT, Λ.
de SITTER SPACE
---------------
A universe where the only source of energy comes
from the vacuum (ρ_{0} ≠ 0, ρ_{M} = ρ_{R} = 0) is referred
to as de SITTER SPACE. Thus,
.
(a/a)^{2} = H_{0}^{2} = Λ/3 - k/a^{2}
Where Λ/3 = (8/3)πGρ_{o}. For the k = 0 case we get:
.
(a/a) = H_{0}
∴ da/dt = aH_{0}
This has the solution:
a = Aexp(H_{0}t)
Therefore, in the vacuum dominated phase the Universe
is growing exponentially. The metric becomes:
dτ^{2} = dt - Aexp(2H_{0}t)(dx^{2} + dy^{2} + dz^{2})
Recall that:
v = H_{0}D = D√(8πGρ_{0}/3)
For v = c this gives:
D_{H} = c/√(8πGρ_{0}/3)
Since ρ_{0} is a constant, D_{H} is fixed. This distance
corresponds to the distance at which we cannot see
beyond because the universe is expanding at the speed
of light. D_{H} is referred to as the COSMIC HORIZON.
The implication of this is that as the universe
continues to expand, the objects that we see in
the night sky today will eventually slip through
the cosmic horizon and become invisible. Another
way of saying this is that the light emitted at
the horizon is effectively redshifted to infinite
wavelengths. Observers in the distant future will
look up at the heavens and see a different universe
than we see today. Ultimately, the night sky will
become completely black!
The radius of the cosmic horizon is approximately
46.6 billion light years. The radius of the surface
of last scattering is about 45.7 billion light years.
Eventually the SOLS will pass through the cosmic
horizon and the CMB will disappear.
Note: The age of the universe is estimated to be
13.8 billion years. However, the radius of the
observable universe is not this number. In the real
universe, spacetime is expanding and the distances
obtained by multiplying the speed of light multiplied
by this amount of time has no direct physical meaning.
It is worth noting that while the universe as a whole
is expanding, it can be contracting locally due to
gravity.
Summary
-------
Period | Density | Expansion | P | w
----------+----------+-------------+------+--------
Radiation | ∝ 1/a^{4} | ∝ t^{1/2} | ρ_{R}/3 | w = 1/3
Mass | ∝ 1/a^{3} | ∝ t^{2/3} | 0 _{ }| w = 0
Vacuum |∝constant^{ }| ∝ exp(H_{0}t)_{ }| -ρ_{0} | w = -1
Where H_{0} = √(Λ/3) = √(8πGρ_{0}/3)
The Universe grew fairly rapidly during the radiation
period, slowed down during the mass period and grows
exponentially during the vacuum period. This is
illustrated below.
Geometry of the Universe
------------------------
To begin this discussion it is necessary to consider
the geometry and topology of n-spheres. n-spheres
can be viewed in terms of embedding spaces. However,
n-spheres can also be described without an embedding
if we only consider their boundaries or surfaces.
For example, the description of a circle on a plane
requires 2D space, (x,y) or (r,φ) but the boundary
does not need to be embedded in the same 2D space to
have a meaning. A person living on a circle can travel
around the entire circle by going backwards or forwards.
To them, the world is a just 1-dimensional space. With
this in mind it is possible to describe n-spheres in
terms of spatial metrics as follows:
Metric for the 0-Sphere
-----------------------
The 0-sphere is the pair of points at the ends
of a line segment, r. The spatial metric is:
ds^{2} = dr^{2}
Metric for the 1-Sphere
-----------------------
The 1-sphere is a circle defined in terms of
polar coordinates (r,φ) where φ ranges from
0 to 2π. The spatial metric is:
ds^{2} = dr^{2} + r^{2}dφ^{2}
= dr^{2} + r^{2}dΩ_{1}^{2}
For the unit 1-sphere, r = 1. Therefore:
ds^{2} = dφ^{2}
= dΩ_{1}^{2}
Metric for the 2-Sphere
-----------------------
The 2-sphere represents the 2 dimensional surface
of conventional sphere defined in 3D space. The
2-sphere can be looked at as the 'stacking' of
1-spheres on top of each other where the circle
diameters vary continuously from 0 to 1 and then
back to 0 in accordance with sinθ. All you need
to reach every spot on the surface of a sphere is
to be able to go left or right, up or down. Thus,
a person living on the surface of a 2-sphere would
see the world in 2 dimensions.
The spatial metric is:
ds^{2} = dθ^{2} + sin^{2}θdΩ_{1}^{2} = dΩ_{2}^{2}
Metric for the 3-Sphere
-----------------------
A 3-sphere is the 4D analog of a 2-sphere. Although
a 2-sphere exists in 3D space, its surface is 2
dimensional. Similarly, a 3-sphere has a 3 dimensional
surface which is embedded into 4D space. Since there
is no such space in our physical world it is impossible
to imagine it in the same way as we imagine 3d space.
Our minds just don't have any experience of being in
4d space. In this sense, the 4th dimension of the
3-sphere is unobservable, therefore unknown. However,
it is possible to do math in 4D, and, in fact, in
any-dimensional space. Note: 4D in this context
refers to space only and should not be confused with
4D spacetime.
The 3-sphere is also referred to as a hypersphere. It
can be viewed as the nesting of concentric 2-spheres
where the sphere diameters vary continuously from 0
to 1 and then back to 0 in accordance with sinχ.
The spatial metric is:
ds^{2} = dχ^{2} + sin^{2}χdΩ_{2}^{2} = dΩ_{3}^{2}
For the 2-Sphere embedded in 3D flat space:
ds^{2} = dx^{2} + dy^{2} + dz^{2}
This can be written in terms of spherical coordinates
x = χcosθ, y = χsinθcosφ and z = χsinθsinφ as:
- - - -
^{ } | 1 0 ^{ } 0 ^{ } || dr^{2} |
ds^{2} = | 0 χ^{2} 0 ^{ } || dθ^{2} |
^{ } | 0 0 χ^{2}sin^{2}θ || dφ^{2} |
- - - -
= dχ^{2} + χ^{2}(dχ^{2} + sin^{2}θdφ^{2})
= dχ^{2} + χ^{2}(dχ^{2} + sin^{2}θdΩ_{1}^{2})
= dχ^{2} + χ^{2}dΩ_{2}^{2}
= dΩ_{3}^{2}
Positive Spacial Curvature, k = +1
----------------------------------
ds^{2} = dχ^{2} + sin^{2}χdΩ_{2}^{2}
^{ } = dΩ_{3}^{2}
Flat Spacial Curvature, k = 0
-----------------------------
ds^{2} = dx^{2} + dy^{2} + dz^{2}
≡ dχ^{2} + χ^{2}dΩ_{2}^{2}
Negative Spacial Curvature, k = -1
----------------------------------
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:
ds^{2} = dχ^{2} + sinh^{2}χdΩ_{2}^{2}
^{ } = dℋ_{3}^{2}
dℋ_{3}^{2} is called the hyperbolic plane.
Note: sinhχ = (e^{χ} - e^{-χ})/2 where χ is not an angle
but a distance. Unlike sinχ, which is periodic with
π, sinhχ is not and proceeds exponentially to ∞ as χ
increases.
The FRW Metric
--------------
For an n-sphere, we can incorporate the expansion
of space by modifying the metric as follows:
ds^{2} = a(t)^{2}dΩ_{n-1}^{2}
The spacetime metric is:
dτ^{2} = dt^{2} - a(t)^{2}dΩ_{n-1}^{2}
We can now summarize all of the above cases as
follows:
dτ^{2} = dt^{2} - a(t)^{2}[dχ^{2} + ξ(χ)^{2}dΩ_{2}^{2}]
Where,
ξ^{2}(χ) = χ^{2} k = 0
ξ^{2}(χ) = sin^{2}χ k = +1
ξ^{2}(χ) = sinh^{2}χ k = -1
Now, let r ≡ ξ(χ).
r = χ case: dr^{2} = dχ^{2}
∴ dχ^{2} = dr^{2}/(1 - 0.r^{2})
r = sinχ case: dr^{2} = cos^{2}χdχ^{2}
∴ dχ^{2} = dr^{2}/(1 - sin^{2}χ)
= dr^{2}/(1 - 1.r^{2})
r = sinhχ case: dr^{2} = cosh^{2}χdχ^{2}
∴ dχ^{2} = dr^{2}/(1 + sinh^{2}χ)
= dr^{2}/(1 + 1.r^{2})
In each case the denominator can be written as:
(1 + kr^{2})
Where,
k = 0 (flat), +1 (spherical) or -1 (hyperbolic).
Using the above.
ds^{2} = dt^{2} - a^{2}(t)(dχ^{2} + ξ^{2}(χ)[dθ^{2} + sin^{2}dθdφ^{2}]
= dt^{2} - a^{2}(t)(dr^{2}/(1 - kr^{2}) + r^{2}[dθ^{2} + sin^{2}dθdφ^{2}]
In matrix form the FRW metric is:
- - - -
_{ } | 1 0 ^{ } 0 0 ^{ } || dt^{2} |
g_{μν} = | 0 -a^{2}(t)/(1 - kr^{2}) 0 0 ^{ } || dr^{2} |
_{ } | 0 0 -a^{2}(t)r^{2} 0 ^{ } || dθ^{2} |
_{ } | 0 0 ^{ } 0 -a^{2}(t)r^{2}sin^{2}θ || dφ^{2} |
- - - -
The FRW metric fullfills the COSMOLOGICAL PRINCIPLE
that the distribution of matter in the universe is
homogeneous and isotropic when viewed on a large
enough scale.
General Relativity and the FRW Equation
---------------------------------------
The FRW equations can be derived from Einstein's
field equations using the above metric.
The Christoffel symbols are found from:
Γ_{μν}^{ρ} = (1/2)g^{ρσ}(∂_{μ}g_{νσ} + ∂_{ν}g_{μσ} - ∂_{σ}g_{μν})
Example:
Γ_{11}^{0} = (1/2)g^{00}(∂_{1}g_{10} + ∂_{1}g_{10} - ∂_{0}g_{11})
= -(1/2)g_{00}∂_{0}g_{11}
= -(1/2)[-∂_{t}a^{2}/(1 - kr^{2})
_{.}
= aa/(1 - kr^{2})
The Ricci tensor is found from:
R_{μν} = ∂_{ρ}Γ_{μν}^{ρ} - ∂_{ν}Γ_{μρ}^{ρ} + Γ_{μν}^{ρ}Γ_{ρσ}^{σ} - Γ_{μσ}^{ρ}Γ_{νρ}^{σ}
Therefore,
_{..}
R_{00} = -3(a/a)
_{..} _{.}
R_{11} = (a.a + 2aa + 2k)/(1 - kr^{2})
_{..} _{.}
R_{22} = (a.a + 2aa + 2k)r^{2}
_{..} _{.}
R_{33} = (a.a + 2aa + 2k)r^{2}sin^{2}θ
The Scalar Curvature is found from:
R = g^{μν}R_{μν}
Therefore,
_{..} _{.}
R = -6(a/a) + (a^{2}/a^{2}) + (k/a^{2})
The Einsten Field Equations are given by:
G_{μν} = R_{μν} - (1/2)g_{μν}R
G_{00} = R_{00} - (1/2)g_{00}R
_{..} _{..} _{.}
= -3(a/a) + (1/2)6((a/a) + a^{2}/a^{2} + k/a^{2})
_{.}
= 3(a^{2} + k)/a^{2}
The Stress-Energy Tesnsor is:
- -
_{ } | ρ 0 0 0 |
T_{μν} = | 0 -p 0 0 |
_{ } | 0 0 -p 0 |
_{ } | 0 0 0 -p |
- -
Therefore,
_{.}
3(a^{2} + k)/a^{2} = 8πGρ + Λ
Or,
_{.}
(a/a)^{2} + k/a^{2} = 8πGρ/3 + Λ/3
This is the FRIEDMANN EQUATION.
Next, find the trace inverted form of the EFE.
This form is more appropriate for some types of
calculations.
R_{μν} - (1/2)g_{μν}R + g_{μν}Λ = 8πGT_{μν} ... 1.
g^{μν}R_{μν} - (1/2)g^{μν}g_{μν}R + g^{μν}g_{μν}Λ = g^{μν}8πGT_{μν}
Now g_{μν}g^{μν} = D where D is the spacetime dimension.
Therefore,
R - (1/2)4R + 4Λ = 8πGT
-R + 4Λ = 8πGT
Multiply this by (1/2)g_{μν} to get:
(-1/2)g_{μν}R + 2g_{μν}Λ = (1/2)g_{μν}8πGT
∴ (-1/2)g_{μν}R = -2g_{μν}Λ + (1/2)g_{μν}8πGT
Substitute this into 1 to get:
R_{μν} - g_{μν}Λ = 8πGT_{μν} - (1/2)g_{μν}8πGT
∴ R_{μν} = 8πG(T_{μν} - (1/2)g_{μν}T) + g_{μν}Λ
R_{00} = 8πG(T_{00} - (1/2)g_{00}T) + g_{00}Λ
_{..}
-3(a/a) = 8πG(ρ - (1/2)(ρ - 3p) - Λ
or,
_{..}
(a/a) = -4πG(ρ + 3p)/3 - Λ/3
This is often called the FRIEDMANN ACCELERATION EQUATION.
This equation states that both the volumetric mass density,
ρ, and the pressure, p, cause the rate of expansion of the
universe to decrease. On the other hand, Λ, acts in
the opposite way to increase the rate of expansion. The
connection between the EFE and Newton is shown below:
. .
. .
. M .
. .
. + .m (x = 1)
. 0 .
. .
. .
. .
<-- D -->
<-- R -->
D = aR
_{.}
v = aR
_{ ..}
A = a.R (acceleration)
_{ ..}
F = -GmM/R^{2} = a.R
_{..}
a/a = -MG/a^{3}R^{3}
.
= -(4/3)πGρ(a) [V = (4/3)πD^{3} = (4/3)πa^{3}R^{3}]
= -(4/3)πGν/a^{3}
D(t) = a(t)D_{0}
D_{0} = D(t_{0}) => a(t_{0}) = 1
_{.}
[v] = [aR] = [(1/t)L]
Measuring Curvature of the Universe
-----------------------------------
In flat space angles subtended by objects at a
distance, r, with a diameter, d, vary as:
A ^
/ x \
/ x x \
r / x x d = true size
/ x x \
/dθ x v
o------------ B
r
d^{2} = dr^{2} + r^{2}dθ^{2}
dr^{2} = 0 (r same at A and B)
∴ dθ = d/r
In positively curved space, angles subtended by
objects vary as:
d^{2} = sin^{2}rdθ^{2}
∴ dθ = d/sinr
In negatively curved space angles subtended by
objects vary as:
d^{2} = sinh^{2}rdθ^{2}
∴ dθ = d/sinhr
These are just really statements that the sum of
the angles of a triangle is > 180° for k = +1,
< 180° for k = -1 and = 180° for k = 0.
These results can be summarized as follows (d = 1):
r 1/r 1/sinr 1/sinhr
-- ----- ------ -------
0 ∞ ∞ ∞
π/8 2.55 2.61 2.48
2π/8 1.27 1.41 1.15
3π/8 0.85 1.08 0.68
4π/8 0.64 1.00 0.43
5π/8 0.51 1.08 0.29
6π/8 0.42 1.41 0.19
7π/8 0.36 2.62 0.13
π 0.32 ∞ 0.09
If we know the distance, r, and the true size of
a very distant object (i.e. we pick a object that
has similar characteristics to an object, like our
own galaxy, whose size is known) we can determine
if the subtended angle is =, >, or < than the value
for flat space at the same distance.
Another, indication of curvature is the number of
objects observed as a function of distance. If
the universe is flat it would be expected that
the number of objects increases proportionally to
r^{2}. If the universe has positive curvature it
would be expected that the number of galaxies
increases more slowly and then turns over and
starts to decrease again in accordance with sin^{2}r.
If the universe has negative curvature it would
be expected that the number of galaxies increases
faster than in flat space since now N ∝ sinh^{2}r.
This can be seen in the following table:
r r^{2} sin^{2}r sinh^{2}r
-- ---- ----- ------
0 0 0 0
π/8 0.15 0.15 0.16
2π/8 0.62 0.50 0.75
3π/8 1.39 0.85 2.16
4π/8 2.47 1.00 5.30
5π/8 3.86 0.85 12.20
6π/8 5.55 0.50 27.35
7π/8 7.56 0.15 60.58
π 9.87 0 133.48
These scenarios can be illustated schematically
as follows. Hyperbolic space is difficult to
visualize, but the drawing on the right by MC
Escher for dℋ_{2}^{2} is a good proxy.
The Critical Density
--------------------
The general case of the FRW equation can be
written as:
.
(a/a)^{2} = 8πGρ_{R}/3 + 8πGρ_{M}/3 + Λ/3 - k/a^{2}
H_{0}^{2} = 8πGρ_{R}/3 + 8πGρ_{M}/3 + Λ/3 - k/a^{2}
1 = 8πGρ_{R}/3H_{0}^{2} + 8πGρ_{M}/3H_{0}^{2} + Λ/3H_{0}^{2} - k/a^{2}H_{0}^{2}
1 = Ω_{R} + Ω_{M} + Ω_{0} - k/a^{2}H_{0}^{2}
The ratio 3H_{0}^{2}/8πG is called the CRITICAL DENSITY,
ρ_{C}. It corresponds to the case where k = 0 (at
the escape velocity). This is calculated today
to be 9.47 x 10^{-27}kg/m^{3}. Ω = ρ/ρ_{C} is the DENSITY
PARAMETER.
If Ω_{R} + Ω_{M} + Ω_{0} = 1 then k = 0
If Ω_{R} + Ω_{M} + Ω_{0} > 1 then k = -1
If Ω_{R} + Ω_{M} + Ω_{0} < 1 then k = +1
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, complex mathematical models
have been developed that mimic experimental results.
One such model is the Lambda-CDM (Cold Dark Matter)
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 (see below)
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:
Ω_{R} = 0
Ω_{M} ~ 0.3
Ω_{Λ} ~ 0.7
This, along with astronomical measurements described
before, leads to the conclusion that k ~ 0. In other
words, the Universe is remarkably flat.