Problems of the Hot Universe (the Big Bang Model)

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

Problem 2

Determine the number of causally disconnected regions at 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 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 at 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 is preserved at future.

Problem 13

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

Problem 14

Suppose there be some initial homogeneous matter distribution of the Universe. 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 the present?

Problem 15

Estimate the present density of relic 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, by 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 conservation of the momentum. Explain this seeming contradiction.