5.5 Friedman Equations for the Scale Factor and the Equation of State |
159 |
5.5.3 Embedding Cosmologies into de Sitter Space
The Lorentzian manifold dS, named de Sitter space is identified with the following coset manifold:
dS = |
SO(1, 4) |
(5.5.30) |
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SO(1, 3) |
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and therefore admits the 10-parameter group of isometries: |
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SO(1, 4) |
(5.5.31) |
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The entire coset manifold can be identified as an algebraic locus in R5, namely as the set of points satisfying the following quadratic equation:
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Y02 − Yi2 = −H0−2 |
(5.5.32) |
i=1 |
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where H0 is a real number, whose physical interpretation will be that of Hubble constant.
Using rescaled variables Y I = Y I /H0, we see that de Sitter space corresponds to the manifold H(+4,1) in the language of Sect. 5.2.2. Here we introduce other coordinates for the coset manifold by solving explicitly the constraint (5.5.32), in various ways.
One parametric solution of the above algebraic equation is as follows:
Y0 = H0−1 sinh H0t |
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Y1 |
= H0−1 cosh H0t cos R |
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Y2 |
= H0−1 cosh H0t sin R cos θ |
(5.5.33) |
Y3 |
= H0−1 cosh H0t sin R sin θ cos φ |
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Y4 |
= H0−1 cosh H0t sin R sin θ sin φ |
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and the pull-back of the Lorentzian metric in R5:
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ds52 = dY02 − dYi2 |
(5.5.34) |
i=1 |
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on the locus (5.5.32) by means of the parameterization (5.5.33) leads to the following metric:
dS+ |
= − |
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cosh2 H0t |
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H02 |
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ds2 |
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dt |
2 |
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dR2 |
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sin2 R dθ |
2 |
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sin2 |
θ dφ2 |
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= −dt2 |
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H0 0 |
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1 − r2 + r2 dθ 2 |
+ sin2 θ dφ2 |
(5.5.35) |
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cosh H |
t |
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dr2 |
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160 |
5 Cosmology and General Relativity |
where in the last step we have further posed:
sin R = r |
(5.5.36) |
In this way we have shown that de-Sitter space can be identified with a cosmological model characterized by:
κ = 1; |
a(t) = |
cosh H0t |
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(5.5.37) |
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H0 |
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The corresponding Hubble function and acceleration parameters are: |
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H (t) = H0 tanh(H0t) |
(5.5.38) |
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a(t¨ ) |
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H 2 |
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(5.5.39) |
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a(t) = |
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0 |
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An alternatively equally good parametric solution of the quadric (5.5.32) is given by:
Y0 = H0−1 sinh H0t cosh R |
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Y1 |
= H0−1 sinh H0t sinh R cos θ |
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Y2 |
= H0−1 sinh H0t sinh R sin θ cos φ |
(5.5.40) |
Y3 |
= H0−1 sinh H0t sinh R sin θ sin φ |
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Y4 |
= H0−1 cosh t |
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The pull back of the flat metric (5.5.34) on the locus (5.5.32) by means of this second parameterization (5.5.40) is:
dsdS2 |
− = −dt |
2 |
+ H0−2 sinh2 H0t dR2 + sinh2 R dθ 2 + sin2 θ dφ2 |
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= −dt |
2 |
+ |
H0 0 |
t |
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1 + r2 + r2 dθ 2 |
+ sin2 θ dφ2 |
(5.5.41) |
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sinh H |
2 |
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dr2 |
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where in the last step we have further posed: |
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sinh R = r |
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(5.5.42) |
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In this way we have shown that de-Sitter space can be also identified with a cosmological model characterized by:
κ = −1; |
a(t) = |
sinh H0t |
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(5.5.43) |
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H0 |
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The corresponding Hubble function and acceleration parameter are: |
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H (t) = H0 |
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(5.5.44) |
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tanh(H0t) |
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5.5 Friedman Equations for the Scale Factor and the Equation of State |
161 |
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a(t¨ ) |
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H |
2 |
(5.5.45) |
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a(t) |
0 |
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A third possibility to obtain a parametric solution of the quadric (5.5.32) is the following one. First redefine
U = Y0 − Y4; V = Y0 + Y4
and rewrite the quadric and the associated Lorentzian metric as follows:
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U V − Yi2 = −H0−2
i=1
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ds(25) = −dU dV + dYi2
i=1
Then solve parametrically (5.5.47) as shown below:
U |
H |
−1 |
ρ |
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= |
0 |
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Yi = H0−1 |
ρxi |
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V = H0− |
− ρ + ρ−→ |
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(5.5.46)
(5.5.47)
(5.5.48)
(5.5.49)
By means of this parameterization the pull-back of the Lorentz metric (5.5.48) on the locus (5.5.47) becomes:
dS0 |
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0− |
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dρ2 |
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−→ |
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ρ2 + ρ |
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ds2 |
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H |
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2 |
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2 |
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x |
2 |
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= −dt |
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H0 0 |
t |
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2 |
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dr2 + r2 dθ 2 + sin2 θ dφ2 |
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exp H |
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where in the last step we have set:
ρ = exp H0t
x1 = r cos θ
x2 = r sin θ cos φ
x3 = r sin θ sin φ
(5.5.50)
(5.5.51)
In this way we have shown that de Sitter space can also be seen as a cosmological metric characterized by:
κ = 0; |
a(t) = |
exp H0t |
(5.5.52) |
H0 |