1. Linear perturbation theory. In the early universe, the matter/radiation distribution of the universe is very homogenous and isotropic. At any given time, let us denote the average density of the universe as \(\bar{\rho}(t)\). Nonetheless, there are some tiny fluctuations and not everywhere exactly the same. So let us define the density at comoving position r and time t as \(\rho(x,t)\) and the relative density contrast as \[\delta(r,t)\equiv\frac{\rho(r,t)-\bar{\rho}(t)}{\bar{\rho}(t)}.\]In this exercise we focus on the linear theory, namely, the density contrast in the problem remains small enough so we only need to consider terms linear in \(\delta\). 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 \(\delta\) satisfies the following second-order differential equation: \[\frac{d^2\delta}{dt^2}+\frac{2\dot{a}}{a}\frac{d\delta}{dt}=4\pi G\bar{\rho}\delta,\]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 and only their amplitude changes. Namely this means that we can factorize \[\delta(x,t)=D(t)\tilde{\delta}(x),\]where \(\tilde{\delta}(x)\) is arbitrary and independent of time, and D(t) is a function of time and valid for all x. D(t) is not arbitrary and must satisfy a differential equation. Derive this differential equation.
We can check that this is a solution by plugging the third equation into the second.
\(\frac{d\delta}{dt}=\dot{D}(t)\tilde{\delta}(x)\)
\(\frac{d^2\delta}{dt^2}=\ddot{D}(t)\tilde{\delta}(x)\)
\(\ddot{D}(t)\tilde{\delta}(x)+\frac{2\dot{a}}{a}\dot{D}(t)\tilde{\delta}(x)=4\pi G\bar{\rho}D(t)\tilde{\delta}(x)\)
\(\ddot{D}(t)+\frac{2\dot{a}}{a}\dot{D}(t)=4\pi G\bar{\rho}D(t)\)
This is a solution to the differential equation.
b) Now let us consider a matter-dominated flat universe, so that \(\bar{\rho}(t)=a^{-3}\rho_{c,0}\) where \(\rho_{c,0}\) is the critical density today, \(3H^2_0/8\pi G\) as in WS11.1. Recall that the behavior of the scale factor of this universe can be written \(a(t)=(3H_0t/2)^{2/3}\), which you learned in previous worksheets, and solve the differential equation for D(t). Hint: you can use the ansatz \(D(t)\propto t^q\) and plug it into the equation that you derived above; you will end up with a quadratic equation for q, There are two solutions, 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 \(D_-(t)\).
By this ansatz, we can say:
\(D(t)=t^q\)
\(\dot{D}(t)=qt^{q-1}\)
\(\ddot{D}(t)=q(q-1)t^{q-2}\)
We can then plug into the differential equation.
\((q)(q-1)t^{q-2}+\frac{2\dot{a}}{a}qt^{q-1}=4\pi G\bar{\rho}t^q\)
We can multiply by \(t^{2-q}\) and substitute in today's critical density.
\((q)(q-1)+\frac{2\dot{a}}{a}qt-4\pi Ga^{-3}\rho_{c,0}t^2=0\)
We plug in the critical density and \(\frac{2}{3t}\) for \(\frac{\dot{a}}{a}\).
\(q^2+\frac{1}{3}q-\frac{2}{3}=0\)
\(q=\frac{2}{3}, -1\)
c) Explain why the \(D_+\) component is generically the dominant one in structure formation, and show that in the Einstein-de Sitter model, \(D_+(t)\propto a(t)\).
Since this is a linear combination, it means that \(D(t)=D_+(t)+D_-(t)=At^{2/3}+Bt^{-1}\).
\(D_+(t)\propto t^{2/3}\propto a(t)\). This term will grow as time increases, since
t's exponent is positive.
\(D_-(t)\propto t^{-1}\). This term will decrease as time increases, since
t's exponent is negative.
As a result, \(D_+(t)\) will come to dominate as time moves forward.
2. Spherical collapse. Gravitational instability makes initial small density contrasts grow in time. When the density perturbation grows large enough, the linear theory, such as the one presented in the above exercise, breaks down. Generically speaking, non-linear and non-perturbative evolution of the density contrast have to be dealt with in numerical calculations. However, in some very special situations, analytical treatment is possible and provides some insights to some important natures of gravitational collapse. In this exercise we study such an example.
d) Plot r as a function of t for all three cases (open, closed, and flat universe), and show that in the closed case, the particle turns around and collapses; 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.
Closed case: blue
\(r=A(1-cos\eta)\)
\(t=B(\eta-sin\eta)\)
The particle's radius increases, but then decreases until the particle collapses to zero again.
Open case: red
\(r=A(cosh\eta-1)\)
\(t=B(sinh\eta-\eta)\)
The particle's radius appears to continue to increase.
Flat case: green
\(r=A\eta^2/2\)
\(t=B\eta^3/6\)
The particle's radius appears to increase with a decreasing velocity.
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The x-axis is time and the y-axis is radius.
A and
B were arbitrarily assigned values of 1, but in practice these values depend on the mass of the patch of over-density.
