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 \( \bar{\rho} (t) \). 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 ρ(x, t) and the relative density contrast as \[δ(r, t) = \frac{ \rho(r, t) - \bar{\rho} (t)}{\bar{\rho} (t)} \] In this exercise we focus on the linear theory, namely, the density contrast in the problem remains
small enough so we only need 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 \[ \frac{d^2δ}{dt^2} + \frac{2 \dot{a}dδ}{adt} = 4 \pi G \bar{\rho}δ \] 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 \[δ(x, t) = D(t) \tilde{δ}(x) \] where ˜δ(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.
Plugging in the second equation to the first equation gives: \[ \frac{d^2}{dt^2} D(t) \tilde{δ}(x) + \frac{2 \dot{a}d}{adt} D(t) \tilde{δ}(x) - 4 \pi G \bar{\rho} D(t) \tilde{δ}(x) = 0 \] Which is \[ \tilde{δ}(x)( \ddot{D}(t) + \frac{2 \dot{a}}{a} \dot{D}(t) - 4 \pi G \bar{\rho} D(t)) = 0 \] When \( \tilde{δ}(x) = 0\) this is \[ \ddot{D}(t) + \frac{2 \dot{a}}{a} \dot{D}(t) - 4 \pi G \bar{\rho} D(t) = 0 \]
(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, \( \frac{3H_0^2}{8 \pi G} \) 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 \( a(t) = (3H_0t/2)^{2/3} \) , 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; 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).
Combining the given equations allows us to find a new expression for density: \[\bar{\rho}(t) = \left( \frac{3H_0^2}{2} \right)^{-2} \frac{3H_0^2}{8 \pi G} \] We know from previous weeks that \[H_0 = \frac{ \dot{a}}{a} \] and \[ a \propto t^{\frac{2}{3}} \] which means \[ \dot{a} \propto \frac{2}{3} t^{- \frac{1}{3}}\] so \[ H_0 = \frac{ \dot{a}}{a} \propto \frac{2}{3t} \] Combining that with our answer from (a) gives: \[ \ddot{D}(t) + \frac{4 }{3t} \dot{D}(t) - \frac{2}{3t^2} D(t) = 0 \] Using the anasatz \( D(t) \propto t^q \) simplifies this to \[q(q-1)t^{q-2} + \frac{4 }{3t}qt^{q-1} - \frac{2}{3t^2}t^q \] \[t^{q-2}(q^2 - q + \frac{4q}{3} - \frac{2}{3}) = 0 \] Using the quadratic equation, q can be -1 or 2/3 so the solutions to this problem are: \[D_-(t) = C_1 t^{-1} \] \[D_+(t) = C_2 t^{\frac{2}{3}}\]
(c) Explain why the D+ component is generically the dominant one in structure formation, and
show that in the Einstein-de Sitter model, \(D_+(t) \propto a(t) \).
The \(D_+ \) component is generally the dominant one because it is proportional to \(t^{\frac{2}{3} }\), which, when t is large, grows much faster than the shriking \(D_- \propto t^{-1} \). Because both \(D_+(t) \propto t^{\frac{2}{3}} \) and \(a(t) \propto t^{\frac{2}{3}}\), \(D_+(t) \propto a(t) \).
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. We will look at some amazingly
numerical results later in this worksheet. However, in some very special situations, analytical
treatment is possible and provide some insights to some important natures of gravitational collapse.
In this exercise we study such an example.
We consider a spherical patch of uniform over-density. Let us study the motion of a particle
in terms of its distance r from the center of the sphere as a function of time t. Recall that
in Worksheet 9, from Newtonian dynamics, we have derived that this function satisfies the
following equation \[ \frac{1}{ 2} \left( \frac{dr}{dt} \right)^2 - \frac{GM }{r} = C \] where C is a constant. We can study this in a closed case where r = A(1 - cos η) and t =B(η - sin η), the open case where r = A(cosh η - 1) and t =B(sin η - η ), and the flat case where r = \(Aη^2 /2\) and t = \(Bη^3 /6 \).
We found that in the closed case, \[ C = \frac{-A^2}{2B^2} \] in the open case \[ C = \frac{A^2}{B^2} \] and in the flat case \[ C = 0\] Plotting these three together with r as a function of t (with 0 < η < \( 2 \pi \) ) where 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. We can see this in a closed universe:
An open universe:
And a flat universe:
Which is where we are theorized to live.
