Blog #34, Worksheet #11.1, Problem #4, Flatness Problem of The Big Bang Model

4. The flatness problem of the big bang model. Despite the success of the Big Bang model mentioned in the lecture, there are also problems with it. These problems mostly have to do with the initial conditions for the Big Bang model, and provide the motivations for scientists to seek deeper answers for the origin of the Big Bang. In this exercise we study one of those problems. 

a) Recall the Friedman equation we learned in a previous lecture \[H^2 = \frac{8 \pi G \rho}{3} - \frac{k c^2}{a^2}\]. Rewrite this in terms of the density parameter \(\Omega\). 

We began with the given equation \[H^2 = \frac{8 \pi G \rho}{3} - \frac{k c^2}{a^2}\]Solving this for \(\rho\) we get \[\rho = \frac{3 H^2}{8 \pi G} + \frac{3 k c^2}{8 a^2 \pi G}\]However we were told that when we assume that k = 0, we get our critical \(\rho\) which is \[\rho_c  = \frac{3H^2}{8 \pi G}\]We substitute in our critical rho to get \[\rho = \rho_c + \frac{k c^2}{a^2 H^2} \rho_c \] We then solve for the desired \(\Omega\) ratio of \(\frac{\rho}{\rho_c}\) and get \[\Omega = 1 + \frac{k c^2}{a^2 H^2}\]as expected.

b) Consider the epoch of our universe that is dominated by matter. We have already computed the time-dependence of a and H in such a universe (refer to Worksheet 9.1). Use this result and a) to show the time-dependence of Ω.

To show the time dependence we have to relate our \(\Omega\) ratio to the time dependence of a. Getting rid of the constants in the equation above, we know \[\Omega \propto (a \cdot H)^{-2} \qquad and \qquad H = \frac{\dot{a}}{a}\] meaning \[\Omega \propto {\dot{a}}^{-2}\] From a previous worksheet we found that \[a \propto t^{\frac{2}{3}} \qquad and \qquad \dot{a} \propto t^{-  \frac{1}{3}}\]Thus we have \[\Omega \propto t^{\frac{2}{3}} \]

c) Today, experiments have measured the density parameter of our universe is within 0.005 : Ω - 1 : 0.005, namely Ω is close to 1 today. Use your above result to determine the range of Ω - 1 at the time of CMB formation, using the fact that today the age of universe is about 13.7 billion years and the age of universe when the CMB formed was about 380, 000 years. From this exercise we see that in the early universe, the density parameter was very close to one. This is one of the issues of the Big Bang model – why is this special initial condition chosen? We provide one possible solution to this question in the next exercise.

Beginning with the equation we just solved for \[\Omega - 1 = \frac{kc^2}{a^2 H^2}\]and plugging in the time dependence we just found we are left with \[\Omega - 1 = C_0 t^{\frac{2}{3}}\]We then use the given information for \(\Omega -1\) and t to solve for \(C_0\). We solve that the time constant for light in this model is \[C_0 \thickapprox 8.73 x10^{-10} \]We then use this constant and the given time when the CMB was taken t = 380,000 to solve for the approximate density constant. \[\Omega - 1 = (8.73 x 10^{-10})(380,000)^{- \frac{2}{3}} = \pm 5 x 10^{-6} \]Because this answer is approximately zero, we can approximate that the density constant, \(\Omega\) at that time fits the estimation that \[\Omega \thickapprox 1 \]