Problems of the Hot Universe (the Big Bang Model)
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).
At adiabatic expansion $aT \approx const$ and thus $a_{Pl} T_{Pl} \approx a_0 T_0 $. Then it follows that at Planck temperature $T_{Pl} \sim 10^{32} K$ the region which the observable Universe ($ a_0 \sim 10^{28} \mbox{cm}$, $T_0 \sim 1K $) originated from had the size of order of $10^{ - 4} \mbox{cm}$, which is $29$ orders of magnitude greater than the Planck length. What is the reason of such discrepancy?
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 maximum size of the region where the causality relations can be established is the particle horizon: $$ L_p (t) = a(t)\int_0^t {\frac{dt'}{a(t')}}. $$ At the radiation-dominated era $a(t) \sim t^{1/2} $ and $L_p (t) = 2t$. When $t \to 0$ the particle horizon $L_p (t)$ shrinks faster than the scale factor $a(t)$. Therefore for any two comoving points there is a time in the past when they are out of casual contact yet. It contradicts the well-established (with accuracy $10^{ - 5} $) isotropy of CMB. This discrepancy is the essence of the horizon problem.
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)?
It is well known that the total energy density in the Universe is very close to the critical one: $\Omega \approx 1$. This condition is satisfied with at least $2\%$ accuracy. The problem of flatness of the Universe lies in the fact that the density of early Universe had to be incredibly close to the critical one in order to provide such correspondence at present. To convince oneself in this fact use the first Friedman equation in the form $$ \Omega - 1 = \frac{k}{\dot a^2 }. $$ For matter-dominated Universe $a \sim t^{2/3}$, and $\dot a \sim t^{ - 1/3} $, then $a\dot a^2 = const$, therefore \begin{equation} \label{prob_of_mod_2_sol_1} \Omega - 1 \sim ka \sim kt^{2/3}. \end{equation} Under assumption that the matter-domination era lasted from the time of equality between densities of matter and radiation $t_{eq} \approx 50\,000$ years to present time $t_0 \approx 1.4 \times 10^{10}$ years, it follows from (\ref{prob_of_mod_2_sol_1}) that \begin{equation} \label{prob_of_mod_2_sol_2} \frac{\Omega \left( {t_{eq} } \right) - 1}{\Omega \left( {t_0 } \right) - 1} = \frac{a\left( {t_{eq} } \right)}{a\left( {t_0 } \right)} = \left( {\frac{t_{eq} }{t_0 }} \right)^{2/3} = \left( {\frac{50\,000}{1.4 \times 10^{10}}} \right)^{2/3} \approx 2 \times 10^{ - 4}. \end{equation} If at present $\Omega \left( {t_0 } \right) - 1$ is less than $\Delta$, then it follows from (\ref{prob_of_mod_2_sol_2}) that at $t = t_{eq} $ after the Big Bang $\Omega - 1$ was less than $2\Delta \times 10^{ - 4}$. Let us proceed even further back in time. Assume that at time $t_{Pl} < t < t_{eq} $ the Universe was dominated by radiation. Then $a \sim t^{1/2} ,\,\,\dot a \sim t^{ - 1/2} $ and therefore $$ \Omega - 1 \sim ka^2 \sim kt. $$ Then $$ \begin{array}{l} \displaystyle\frac{\Omega \left( {t_{Pl} } \right) - 1}{\Omega \left( {t_0 } \right) - 1} = \left( {\frac{a\left( {t_{Pl} } \right)}{a\left( {t_{eq} } \right)}} \right)^2 \frac{a\left( {t_{eq} } \right)}{a\left( {t_0 } \right)}=\frac{t_{Pl}}{t_{eq}} \left( {\frac{t_{mr}}{t_0 }} \right)^{2/3} \approx 10^{ -60}, \\ \\ \displaystyle\Omega \left( {t_{Pl} } \right) - 1 \approx \Delta \times 10^{ - 60}. \\ \end{array} $$ It means that if the present value of relative density $\Omega $ is close to unity with accuracy of a few percent, than at Planck time it could not differ from unity more than by $10^{ - 60} $. It should be stressed that the flatness problem does not imply a contradiction between theory and observations, it rather concerns the reason why Nature chose such a strange initial condition.
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.
Entropy density in the present Universe by order of magnitude equals to photon number density $\left( n_{\gamma }\sim 400\ \mbox{ cm}^{-3} \right)$ and therefore is \[S\sim n_{\gamma }R_{0}^{3}\sim 10^{88}.\] Let us now calculate the entropy of the early Universe. As specific entropy is $s\sim n_{\gamma }\sim T^3,$ then inside the horizon \[S\sim R_{H}^{3}{{T}^{3}}\sim {{H}^{-3}}{{T}^{3}}.\] As was shown above, in the early Universe $H\sim T^2/M_{Pl}$ and therefore \[S\sim\left( \frac{M_{Pl}}{T} \right)^{3}.\] Thus at Planck epoch inside the horizon one has $T\sim {{M}_{Pl}}$ and \[S\sim 1.\] 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 the level of $10^{-5}$).
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.
It was shown in Problem 8 that current value of the entropy of the observable 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.
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.
The wavelength $\lambda _p $ corresponding to a given perturbation grows as any other length scale according to the law $\lambda _p \propto a$. The Hubble radius is $$R_H = H^{ - 1} = \left( {\frac{\dot a}{a}} \right)^{ - 1}. $$ If $a(t) \propto t^q $, then $R_H \propto a^{1/q} $. Therefore $$ \left( {\frac{\lambda _p }{R_H }} \right) \propto a^{(q-1)/ q}. $$ For radiation $(q = 1/2)$, as well as for matter $(q = 2/3)$, one has $q < 1$, thus we come to the conclusion that the primary fluctuations had to be correlated on scales considerably larger than the Hubble radius. Therefore any mechanism of primary inhomogeneities generation in the Big Bang model comes to contradiction with the causality principle. Indeed, if the inhomogeneities generation would occur according to the causal mechanism then the corresponding length scales must evidently lie within the Hubble sphere, i.e. $\lambda _p < R_H $. However for example for $q = 1/2$ (as for radiation) one obtains $$ {\frac{\lambda _p }{R_H }} \propto a^{ - 1} $$ and thшы condition is obviously violated for small $a$.
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?
The required condition can be presented in the form $$ - \frac{d}{dt}\left( {\frac{\lambda _p }{R_H }} \right) < 0. $$ Taking into account that $\lambda _p \propto a$ and $$R_H = H^{ - 1} = \frac{a}{\dot a},$$ one can see that this condition is equivalent to the requirement $$-\ddot{a}< 0,$$ i.e. expansion of early Universe had to be accelerated. In other words the early Universe had to pass the accelerated (inflationary) expansion phase for the generation of primary perturbations (fluctuations) to correspond to any physical mechanism.
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.
At present our Universe is homogeneous and isotropic at scales of order of $ct_0 $. Initial size of the inhomogeneity is $$ l_i \sim ct_0 \frac{a_i}{a_0 }. $$ Let us compare it with the corresponding causality scale $l_{caus} \sim ct_i $: $$ \frac{l_i }{l_{caus} } \sim \frac{t_0 a_i }{t_i a_0 }. $$ Assuming that the scale factor grows as a power law function of time $\dot a \sim a/t$ one obtains $$ \frac{l_i }{l_{caus} } \sim \frac{\dot a_i }{\dot a_0 }. $$ 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.
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?
Use the relation obtained in the previous problem $$ \frac{l_i }{l_{caus} } \sim \frac{t_0 a_i }{t_i a_0 }. $$ Replace $t_i \to t_{Pl} $ and take into account that $aT \sim const$ and $$ \frac{a_{Pl} }{a_0 } \sim \frac{T_0 }{T_{Pl} } \sim 10^{ - 32} $$ to obtain $$ \frac{l_i }{l_{caus} } \sim \frac{10^{17}} {10^{-43}} 10^{ - 32} \sim 10^{28}. $$ 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 the 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?
Consider spherically symmetric distribution of matter. As total energy is conserved, $$ \begin{array}{l} \displaystyle E^{tot} = E_i^K + E_i^P = E_0^K + E_0^P,\\ \displaystyle E_i^K = E_0^K \left( {\frac{\dot a_i }{\dot a_0 }} \right)^2,\\ \displaystyle \frac{E^{tot} }{E_i^K} = \frac{E_i^K + E_i^P }{E_i^K } = \frac{E_0^K + E_0^P }{E_0^K }\left( {\frac{\dot a_0 }{\dot a_i }} \right)^2.\\ \end{array} $$ As $E_0^K \sim \left| {E_0^P } \right|$ and $$ \left( {\frac{\dot a_0 }{\dot a_i }} \right)^2 \le 10^{ - 28}, $$ (see the previous problem) $$ \frac{E^{tot} }{E_i^K } \le 10^{ - 56}. $$ 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 the order of $10^{ - 54} \% $ in the initial velocity distribution would dramatically change the history of the Universe.
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.
