Difference between revisions of "Problems of the Hot Universe (the Big Bang Model)"

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=== Problem 1 ===
 
=== Problem 1 ===
 
Determine the size of the Universe at Planck temperature (the problem of the size of the Universe).
 
Determine the size of the Universe at Planck temperature (the problem of the size of the Universe).
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=== Problem 4 ===
 
=== Problem 4 ===
 
Show that in the radiation-dominated Universe there are causally disconnected regions at any moment in the past.
 
Show that in the radiation-dominated Universe there are causally disconnected regions at any moment in the past.
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At the radiation-dominated era $a(t) \sim t^{1/2} $ and $L_p (t) = 2t$. For $t \to 0$ the particle horizon $L_p (t)$ shrinks faster than the scale factor $a(t)$. Therefore at any moment of time in the past one can find regions of size $L_p,$ which are causally disconnected. It contradicts to well-established (with accuracy $10^{ - 5} $) isotropy of CMB. This contradiction is the essence of the horizon problem.</p>
 
At the radiation-dominated era $a(t) \sim t^{1/2} $ and $L_p (t) = 2t$. For $t \to 0$ the particle horizon $L_p (t)$ shrinks faster than the scale factor $a(t)$. Therefore at any moment of time in the past one can find regions of size $L_p,$ which are causally disconnected. It contradicts to well-established (with accuracy $10^{ - 5} $) isotropy of CMB. This contradiction is the essence of the horizon problem.</p>
 
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=== Problem 5 ===
 
=== Problem 5 ===
 
If at present time the deviation of density from the critical one is $\Delta$, then what was the deviation at $t\sim t_{Pl}$ (the problem of the flatness of the Universe)?
 
If at present time the deviation of density from the critical one is $\Delta$, then what was the deviation at $t\sim t_{Pl}$ (the problem of the flatness of the Universe)?
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It should be stressed that the flatness problem does not imply a contradiction between theory and observations, it rather concerns about the reason why Nature chose such a ''strange'' initial condition.</p>
 
It should be stressed that the flatness problem does not imply a contradiction between theory and observations, it rather concerns about the reason why Nature chose such a ''strange'' initial condition.</p>
 
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=== Problem 8 ===
 
=== Problem 8 ===
 
Formulate the horizon problem in terms of the entropy of the Universe.
 
Formulate the horizon problem in terms of the entropy of the Universe.
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It means that early Universe consisted of ${{10}^{88}}$ independent, causally disconnected regions! The Big Bang model lacks a mechanism which could transform such a Universe into the presently observed one (with isotropy on level of $10^{-5}$).</p>
 
It means that early Universe consisted of ${{10}^{88}}$ independent, causally disconnected regions! The Big Bang model lacks a mechanism which could transform such a Universe into the presently observed one (with isotropy on level of $10^{-5}$).</p>
 
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=== Problem 9 ===
 
=== Problem 9 ===
 
Show that the standard model of Big Bang must include the huge dimensionless parameter--the initial entropy of the Universe--as an initial condition.
 
Show that the standard model of Big Bang must include the huge dimensionless parameter--the initial entropy of the Universe--as an initial condition.
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     <p style="text-align: left;">It was shown in Chapter 6 that current value of entropy of the Universe equals to $S \sim 10^{88} $. If the total entropy is conserved during the expansion of the Universe (recall that entropy conservation is embedded into the Friedman equations: $\dot \rho  + 3H\left( {\rho  + p} \right)= 0$ $ \to \;dE + pdV = 0$), then the Universe had to have such huge value of entropy already at the very moment of birth.</p>
 
     <p style="text-align: left;">It was shown in Chapter 6 that current value of entropy of the Universe equals to $S \sim 10^{88} $. If the total entropy is conserved during the expansion of the Universe (recall that entropy conservation is embedded into the Friedman equations: $\dot \rho  + 3H\left( {\rho  + p} \right)= 0$ $ \to \;dE + pdV = 0$), then the Universe had to have such huge value of entropy already at the very moment of birth.</p>
 
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=== Problem 10 ===
 
=== Problem 10 ===
 
Show that any mechanism of generation of the primary inhomogeneities generation in the Big Bang model violates the causality principle.
 
Show that any mechanism of generation of the primary inhomogeneities generation in the Big Bang model violates the causality principle.
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and the above cited condition is deliberately violated for small $a$.</p>
 
and the above cited condition is deliberately violated for small $a$.</p>
 
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=== Problem 11 ===
 
=== Problem 11 ===
 
How should the early Universe evolve in order to make the characteristic size $\lambda_p$ of primary perturbations decrease faster than the Hubble radius $l_H$, if one moves backward in time?
 
How should the early Universe evolve in order to make the characteristic size $\lambda_p$ of primary perturbations decrease faster than the Hubble radius $l_H$, if one moves backward in time?
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$$-\ddot{a}< 0,$$ i.e. expansion of early Universe had to be accelerated. In other words the early Universe had to pass the accelerated (inflationary) expansion phase if generation of primary perturbations (fluctuations) corresponds to any physical mechanism.</p>
 
$$-\ddot{a}< 0,$$ i.e. expansion of early Universe had to be accelerated. In other words the early Universe had to pass the accelerated (inflationary) expansion phase if generation of primary perturbations (fluctuations) corresponds to any physical mechanism.</p>
 
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=== Problem 12 ===
 
=== Problem 12 ===
 
Suppose that in some initial moment the homogeneity scale in our Universe was greater than the causality scale. Show that in the gravitation--dominated Universe this scale relation will be preserved in all future times.
 
Suppose that in some initial moment the homogeneity scale in our Universe was greater than the causality scale. Show that in the gravitation--dominated Universe this scale relation will be preserved in all future times.
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For a Universe dominated by gravity $\dot a_i  > \dot a_0 $. Therefore the scale of inhomogeneity will always remain greater than that of causality.</p>
 
For a Universe dominated by gravity $\dot a_i  > \dot a_0 $. Therefore the scale of inhomogeneity will always remain greater than that of causality.</p>
 
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=== Problem 13 ===
 
=== Problem 13 ===
 
If the presently observed CMB was strictly homogeneous, then in what number of causally independent regions would constant temperature be maintained at Planck time?
 
If the presently observed CMB was strictly homogeneous, then in what number of causally independent regions would constant temperature be maintained at Planck time?
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Therefore in order to reproduce the presently observed homogeneity of CMB the condition $\delta \rho /\rho  \sim 10^{ - 5} $ must be satisfied in $10^{84}$ causally disconnected regions at Planck time.</p>
 
Therefore in order to reproduce the presently observed homogeneity of CMB the condition $\delta \rho /\rho  \sim 10^{ - 5} $ must be satisfied in $10^{84}$ causally disconnected regions at Planck time.</p>
 
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=== Problem 14 ===
 
=== Problem 14 ===
 
Suppose an initial homogeneous matter distribution in the Universe is given. The initial velocities must obey Hubble law (otherwise the initially homogeneous matter distribution will be quickly destroyed). What should the accuracy of the initial velocity field homogeneity be in order to preserve the homogeneous matter distribution until present time?
 
Suppose an initial homogeneous matter distribution in the Universe is given. The initial velocities must obey Hubble law (otherwise the initially homogeneous matter distribution will be quickly destroyed). What should the accuracy of the initial velocity field homogeneity be in order to preserve the homogeneous matter distribution until present time?
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The obtained inequality means that velocities of initial Hubble flow should be tuned so that the huge negative potential energy of matter could compensate the huge positive kinetic energy. A tiny deviation of order $10^{ - 54} \% $  in initial velocity distribution would dramatically change history of the Universe.</p>
 
The obtained inequality means that velocities of initial Hubble flow should be tuned so that the huge negative potential energy of matter could compensate the huge positive kinetic energy. A tiny deviation of order $10^{ - 54} \% $  in initial velocity distribution would dramatically change history of the Universe.</p>
 
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Revision as of 12:48, 10 June 2013


Although inflation is remarkably successful as a phenomenological model for
the dynamics of the very early universe, a detailed understanding of the physical
origin of the inflationary expansion has remained elusive.
D. Baumann, L. McAllister.


The Horizon Problem

Problem 1

Determine the size of the Universe at Planck temperature (the problem of the size of the Universe).



Problem 2

Determine the number of causally disconnected regions at the redshift $z$, represented in our causal volume today.



Problem 3

What is the expected angular scale of CMB isotropy (the horizon problem)?



Problem 4

Show that in the radiation-dominated Universe there are causally disconnected regions at any moment in the past.


The Flatness Problem

Problem 5

If at present time the deviation of density from the critical one is $\Delta$, then what was the deviation at $t\sim t_{Pl}$ (the problem of the flatness of the Universe)?



Problem 6

Show that both in the radiation--dominated and matter--dominated epochs the combination $a^2 H^2$ is a decreasing function of time. Relate this result to the problem of flatness of the Universe.




Problem 7

Show that both in the radiation--dominated and matter--dominated cases $x=0$ is an unstable fixed point for the quantity \[x\equiv\frac{\Omega-1}{\Omega}.\]


The Entropy Problem

Problem 8

Formulate the horizon problem in terms of the entropy of the Universe.



Problem 9

Show that the standard model of Big Bang must include the huge dimensionless parameter--the initial entropy of the Universe--as an initial condition.


The Primary Inhomogeneities Problem

Problem 10

Show that any mechanism of generation of the primary inhomogeneities generation in the Big Bang model violates the causality principle.



Problem 11

How should the early Universe evolve in order to make the characteristic size $\lambda_p$ of primary perturbations decrease faster than the Hubble radius $l_H$, if one moves backward in time?



Problem 12

Suppose that in some initial moment the homogeneity scale in our Universe was greater than the causality scale. Show that in the gravitation--dominated Universe this scale relation will be preserved in all future times.



Problem 13

If the presently observed CMB was strictly homogeneous, then in what number of causally independent regions would constant temperature be maintained at Planck time?



Problem 14

Suppose an initial homogeneous matter distribution in the Universe is given. The initial velocities must obey Hubble law (otherwise the initially homogeneous matter distribution will be quickly destroyed). What should the accuracy of the initial velocity field homogeneity be in order to preserve the homogeneous matter distribution until present time?



Problem 15

Estimate the present density of relict monopoles in the framework of the model of the Hot Universe.



Problem 16

The cyclic model of the Universe is interesting because it avoids the intrinsic problem of the Big Bang model--the initial singularity problem. However, as it often happens, avoiding old problems, the model produces new ones. Try to determine the main problems of the cyclic model of the Universe.



Problem 17

In the Big Bang model the Universe is homogeneous and isotropic. In this model the momentum of a particle decreases as $p(t) \propto a(t)^{ - 1} $ as the Universe expands. At first sight it seems that due to homogeneity of the Universe the translational invariance must ensure the conservation of the momentum. Explain this seeming contradiction.