Blog #36, Worksheet #12.1, Problem #1 and 2(d), Dark Matter Halos

1. Linear perturbation theory. In this and the next exercise we study how small fluctuations in the initial condition of the universe evolve with time, using some basic fluid dynamics. In the early universe, the matter/radiation distribution of the universe is very homogeneous and isotropic. At any given time, let us denote the average density of the universe as p(t). Nonetheless, there are some tiny fluctuations and not everywhere exactly the same. So let us define the density at co-moving position r and time t as ρ(x,t) and the relative density contrast. 

In this exercise we focus on the linear theory, namely, the density contrast in the problem remains small enough so we only need consider terms linear in δ. We assume that cold dark matter, which behaves like dust (that is, it is pressureless) dominates the content of the universe at the early epoch. The absence of pressure simplifies the fluid dynamics equations used to characterize the problem.

a) In the linear theory, it turns out that the fluid equations simplify such that the density contrast δ satisfies the following second-order differential equation.

where a(t) is the scale factor of the universe. Notice that remarkably in the linear theory this equation does not contain spatial derivatives. Show that this means that the spatial shape of the density fluctuations is frozen in comoving coordinates, only their amplitude changes. Namely this means that we can factorize

Derive this differential equation.

We begin with the equation \[\ddot{\delta} + \frac{2 \dot{a}}{a} \dot{\delta} = 4 \pi G \bar{\rho} \delta\]We then use the relationship that \[\delta = D \tilde{\delta}\]We then plug in and simplify our equation to get a quadratic with variable D. \[\ddot{D} + \frac{2 \dot{a}}{a} \dot{D} - 4 \pi G \bar{\rho} D = 0 \]

b) Now let us consider a matter dominated flat universe, so that ρ(t) = a^-3ρ(c,0) where ρ(c,0) is the critical density today, 3H^2/8πG as in Worksheet 11.1 (aside: such a universe sometimes is called the Einstein-de Sitter model). Recall that the behaviour of the scale factor of this universe can be written a(t) = (3Ht/2)^2/3 , which you learned in previous worksheets, and solve the differential equation for D(t). Hint: you can use the ansatz D(t) is prop to t^q and plug it into the equation that you derived above; and you will end up with a quadratic equation for q. There are two solutions for q, and the general solution for D is a linear combination of two components: One gives you a growing function in t, denoting it as D(t); another decreasing function in t, denoting it as D(t).

We are then given that \[\bar{\rho} = a^{-3} {\rho}_{c,o}\]Where \[a = (\frac{3 H_0 t}{2})^{\frac{2}{3}} \qquad and \qquad {\rho}_{c,o} = \frac{3 H_0}{8 \pi G}\]We combine these to show \[\bar{\rho} = \frac{1}{6 \pi G t^2}\]We are then asked to approximate \(D\) as \(t^q
\). Plugging this into our quadratic, along with \(\bar{\rho}\), \(a\) and \(\dot{a}\) we get \[0 = (q)(q-1)t^{q-2} + \frac{4q}{3}t^{q-2} - \frac{2}{3}t^{q-2}\]We divide by \(t^{q-2}\) and simplify to get \[q^2 + \frac{1}{3}q - \frac{2}{3} = 0\]Then we solve for q to get \[q = \frac{2}{3}, -1\] Which is the desired positive side and negative side solution that we expected. To solve for a final D, we put our two answers into a linear combination to give us \[D = t^{-1} + t^{\frac{2}{3}}\]where \[D_+ = t^{\frac{2}{3}} \qquad and \qquad D_- = t^{-1}\]

c) Explain why the D component is generically the dominant one in structure formation, and show that in the Einstein-de Sitter model, D(t) prop to a(t).

We can see of the bat that our \(D_+ = t^{\frac{2}{3}}\) is proportion to \(a(t) = (\frac{3 H_0 t}{2})^{\frac{2}{3}}\) and the two will increase proportionally. We can also see that \(D_+ = t^{\frac{2}{3}}\) will increase much faster than \(D_- = \frac{1}{t}\) and thus will dominate the overall D formula.

2d) Plot r as a function of t for all three cases (i.e. use y-axis for r and x-axis for t), and show that in the closed case, the particle turns around and collapse; in the open case, the particle keeps expanding with some asymptotically positive velocity; and in the flat case, the particle reaches an infinite radius but with a velocity that approaches zero.

In the first plot we can see the closed case, where the particle turns and collapses back to an incredibly small radius. 

Eventually this graph will hit an asymptotically high velocity in which the particle is increasing radius at nearly an infinite rate. This is the case when the universe is modeled in an open system.  

Finally for the flat case we can see that the radius is ever expanding but at a decreasing rate that will eventually flatten to a zero velocity.