172 5 Cosmology and General Relativity
Let us now evaluate χ (z) for a matter dominated universe (w = 0) with negative spatial curvature (Ω0 < 1). In this case, the defining integral yields:
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z |
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χ−(z) = H0 |
|1 − Ω0| |
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H (z ) |
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dz |
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− |
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1 − Ω0 |
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coth−1 |
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tanh−1 |
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zΩ0 + 1 |
(5.6.45) |
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Ω0) |
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If we perform the same calculation for a matter dominated (w = 0) closed universe (Ω0 > 1) we obtain instead the result:
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χ+(z) = H0 |
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H (z ) |
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dz |
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tan−1 |
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z + 1 |
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cot−1( |
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(5.6.46) |
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Ω0 − 1 |
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Let us now calculate R±(χ±(z)). With some algebraic effort we can verify that in both case the result is the same namely:
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= (1 + z)Ω02 |
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R |
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(z) |
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2√|Ω0 − 1| |
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zΩ0 |
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2)( 1 |
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zΩ0 |
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(5.6.47) |
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Using once again (5.6.42) and inserting the above results into (5.6.43) we conclude that:
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Ω02(1 + z) |
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(z) |
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D(z, Ω0) |
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zΩ0 |
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(5.6.48) |
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The interesting point is that in the limit Ω0 → 1 we exactly retrieve D0(z): |
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lim |
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(z) |
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(5.6.49) |
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Ω0 1 D(z, Ω0) = D0 |
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This allows us to consider D(z, Ω0) as the spatial distance of any source at red-shift z for any possible value of the cosmological parameter Ω0.
5.7 Conceptual Problems of the Standard Cosmological Model
As we anticipated in Sect. 4.6, the great success of the Standard Cosmological Model based on the principles of homogeneity and isotropy does not remove a fundamental conceptual problem which can be summarized in the following question: why is our Universe so much homogeneous and isotropic?
The main source of the problem is the quality of Einstein equations that, in the course of time evolution enlarge perturbations and anisotropies rather than damping
5.7 Conceptual Problems of the Standard Cosmological Model |
173 |
them. Therefore if our Universe is so much homogeneous at the present time it must have been even more so in the past and this is quite unnatural. Who has prepared such fine tuned homogeneous initial conditions? The paradox becomes evident if we compare the extension of the Universe with the causal horizon at various times. The argument is masterly presented in chapter five of [9] and we just follow the reasoning of that author.
Let us consider as initial time that fixed by the Planck scale which corresponds
to:
tPlanck 10−43 s |
(5.7.1) |
The present time is instead fixed by the Hubble scale H0 and we have: |
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t0 14 billion years 1017 s |
(5.7.2) |
The present size of the homogeneous region covers all the visible Universe and therefore is of the order of the present horizon scale namely:
hom(t0) = hor(t0) ct0 1028 cm |
(5.7.3) |
Since, as we said, anisotropies and inhomogeneities cannot be washed away by the expansion of the Universe when it proceeds according to power laws, then assuming that this was the case, it follows that the size of the homogeneous region at the Planckian time must have been the following:
hom(tPlanck) = hom(t0) |
a(tPlanck) |
(5.7.4) |
a(t0) |
and we can compare it to the size of a causally connected region at the same time, which is the horizon scale at Planckian time:
hor(tPlanck) ctPlanck 10−32 cm ≡ Planck |
(5.7.5) |
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In this way we obtain: |
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hom(tPlanck) |
= 1060 × |
a(tPlanck) |
(5.7.6) |
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a(t0) |
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How can we estimate the ratio a(tPlanck) ? The answer is simple: from the temperatures
a(t0)
of the black-body radiation. Because of the cosmological red-shift the ratio between scale factors is proportional to the inverse ratio of radiation temperatures:
a(tPlanck) T0
(5.7.7)
a(t0) TPlanck
At the present time we have T0 1 K while at the Planckian time the radiation temperature must have been the temperature equivalent of the Planck length, namely
−Planck1 . This yields:
TPlanck 1032 K |
(5.7.8) |