17.2. Spherical collapse of overdensities in the Universe¶
Now that we understand how density perturbations grow from their initial small values into large overdensities, we can consider when these overdensities become so large that they collapse into a gravitationally-bound halo, how this process occurs, and what its dynamical end state is. We do that in this and the next section.
We start by considering the collapse of a region of uniform density that is spherically symmetric and has a density higher than the mean cosmological background matter density \(\rho_m(t)\); the overdensity therefore has the form of a top-hat. In Newtonian dynamics, the equation of motion for a spherical shell is \begin{equation}\label{eq-spher-collapse-eq} \ddot{r} = -\frac{GM}{r^2}\,, \end{equation} where \(M\) is the mass contained within the shell (see Chapter 2.2). In an expanding Universe, we should pause and consider whether this equation applies. Fortunately, Birkhoff’s theorem in the general theory of relativity states that the gravitational field inside a spherical uniform shell is zero, just like in Newton’s first shell theorem. In relativistic cosmology, the consequence of this is that a spherical, overdense region follows the same Friedmann equations as the Universe as a whole, but with a mean density that is higher than that of the entire Universe. Equation (17.80) then follows from the second Friedmann equation (17.2), with \(a \rightarrow r\) and \(\rho = M/[4\pi r^3/3]\) in the simple case of a matter-only Universe (that is, without radiation, dark energy, or any other non-matter component). Spherical collapse is especially simple in a matter-only Universe, so we will start by discussing this case. Our goal is to compute how an initial overdensity grows in time, estimate the time it takes to complete the gravitational collapse, and compute the density, mass, and other properties of the collapsed object that forms.
To be able to solve Equation (17.80), we make the approximation that shells inside the overdense region that start at different radii never cross each other. That is, a shell labeled ‘1’ that starts out at \(r_1(t=0) > r_2(t=0)\) always remains at larger radius than a second shell labeled ‘2’ inside the first shell: \(r_1(t) > r_2(t)\) for \(t > 0\). This implies that the mass contained within a shell at radius \(r_i\) remains constant, because no mass passes through the shell. The mass in Equation (17.80) is then constant and for any given shell, the dynamics of the shell is equivalent to that of an object in a point-mass gravitational potential. Even though a collapsing overdensity is a highly dynamical, time-dependent system, under the approximation that we are working in, the potential felt by a shell does not explicitly depend on time and the mechanical energy of the shell is therefore conserved. We denote the specific energy by \(E = v^2/2 - GM/r\).
We assume that the collapse is spherical and the motion of the shell is therefore entirely radial. The dynamics of the collapse is then equivalent to a Keplerian orbit with eccentricity \(e=1\). We discussed such orbits in Chapter 4.2.2. Because we are considering shells that collapse, the motion of the shell is bound and \(E < 0\). In this case, it is convenient to express the orbit in the parametric form that uses the eccentric anomaly (Equations 4.38 and 4.39), which we write as \begin{align}\label{eq-spher-collapse-parametrized} r & = \frac{r_{\mathrm{max}}}{2}\,\left(1-\cos \eta\right)\,;\quad t = \frac{t_c}{2\pi}\,\left(\eta-\sin \eta\right)\,, \end{align} where \(\eta\) is the eccentric anomaly for a Keplerian orbit, \(r_\mathrm{max}\) is the maximum radius attained by the shell, and \(t_c\) is the time until the shell reaches \(r=0\). Using the properties of Keplerian orbits discussed in Chapter 4.2.2, we have that \begin{align}\label{eq-spher-collapse-Ermax-rmaxtc} E & = -\frac{GM}{r_{\mathrm{max}}}\,;\quad r^3_{\mathrm{max}} = \frac{2GM}{\pi^2}\,t_c^2\,. \end{align} We can also express the integration constants \((r_{\mathrm{max}},t_c)\) in terms of the initial conditions at an early time. We assume that at time \(t_i\) the uniform overdensity was \(\rho(t_i) = \bar{\rho}_m(t_i)\,(1+\delta_i)\), where \(\bar{\rho}_m(t_i)\) is the mean background matter density at time \(t_i\) and \(\delta_i \ll 1\), and that the shell was at radius \(r_i\) with velocity \(v_i\). These are not all independent quantities; we will derive relations between them. The mass within the shell is constant and can be expressed as \begin{align} 2GM & = \frac{8\pi}{3}\,r_i^3\,\bar{\rho}_m(t_i)\,\left(1+\delta_i\right)= r_i^3\,\Omega_{i,m}\,H_i^2\,\left(1+\delta_i\right)\, \end{align} where \(H_i\) and \(\Omega_{i,m}\) are the Hubble parameter and the matter density parameter, respectively, at time \(t_i\). Using the first of Equations (17.82) and the standard expression for the energy, we can then write \(r_i/r_\mathrm{max}\) as \begin{equation}\label{eq-spher-collapse-ri-over-rmax} \frac{r_i}{r_\mathrm{max}} = 1-\frac{\left(v_i/[H_i\,r_i]\right)^2}{\Omega_{i,m}\,\left(1+\delta_i\right)}\,. \end{equation} The velocity \(v_i\) can be written in terms of \((t_i,r_i,\delta_i)\) by working out the conservation of mass. That is, we have that \(r_i^3\,\bar{\rho}_m\,\left(1+\delta_i\right) = \mathrm{constant}\) and the time derivative of this is therefore zero. Using the chain rule and the fact that \(\mathrm{d}\bar{\rho}_m /\mathrm{d}t_i = -3\,\bar{\rho}_m\,H_i\) (from the cosmological background evolution), we can re-arrange that derivative to give \begin{equation} v_i = \frac{\mathrm{d}r_i}{\mathrm{d}t_i} = H_i\,r_i\left(1-\frac{1}{3H_i\,t_i}\,\frac{\delta_i}{1+\delta_i}\,\frac{\mathrm{d} \ln \delta_i}{\mathrm{d} \ln t_i}\right)\,. \end{equation} For a matter-only Universe, we have that \(H_i\,t_i = 2/3\) and \(\delta_i \propto t_i^{2/3}\) at small \(t_i\). Assuming that \(\delta\) is small and that higher-order \(\delta\) terms can be ignored, we therefore have \begin{equation} v_i = H_i\,r_i\left(1-\frac{\delta_i}{3}\right)\,. \end{equation} Substituting this in Equation (17.84), we find that \begin{equation} \frac{r_i}{r_\mathrm{max}} = 1-\frac{\left(1-\delta_i/3\right)^2}{\Omega_{i,m}\,\left(1+\delta_i\right)}\,. \end{equation} Or to lowest order in \(\delta\) \begin{equation}\label{eq-spher-collapse-rioverrmax} \frac{r_i}{r_{\mathrm{max}}} = 1-\left(1-\frac{5}{3}\,\delta_i\right)\,\Omega_{i,m}^{-1}\,. \end{equation} Using the second of Equations (17.82), the time to collapse can then similarly be written as \begin{equation}\label{eq-spher-collapse-tcti} t_c^2 = \left(\frac{3\pi}{2}\right)^2\,\frac{t_i^2}{\left(1-\left[1-5\delta_i/3\right]\,\Omega_{i,m}^{-1}\right)^3}\,\frac{1}{\Omega_{i,m}\,\left(1+\delta_i\right)}\,. \end{equation}
At the initial time, both \(\delta_i \ll 1\) and \(1-\Omega_{i,m} \ll 1\) and we can therefore simplify Equation (17.89) to \begin{equation}\label{eq-spher-collapse-tcti-2} t_c^2 = \left(\frac{3\pi}{2}\right)^2\,\frac{t_i^2}{\left(1-\left[1-5\delta_i/3\right]\,\Omega_{i,m}^{-1}\right)^3}\,. \end{equation} Note that \(\delta_i\) and \(1-\Omega_{i,m}\) may be similar in size and we, therefore, cannot easily simplify this expression more. One quantity that is of much interest is the final linear overdensity \(\delta_c\) of the collapsed region. By this we mean not the actual physical overdensity of the collapsed object, but the overdensity that it would get to if it simply followed the prediction from linear cosmological perturbation theory. In a flat, matter-only Universe \((\Omega_{i,m} = 1)\), we have that \(\delta_i \propto t_i^{2/3}\) at all times and therefore we can compute this as \begin{align} \delta_c & = \delta_i\,\left(\frac{t_c}{t_i}\right)^{2/3} = \delta_i\,\left(\frac{3\pi}{2}\right)^{2/3}\,\frac{1}{\left(1-\left[1-5\delta_i/3\right]\right)} = \frac{3}{5}\,\left(\frac{3\pi}{2}\right)^{2/3} \approx 1.686\,.\label{eq-spher-collapse-deltac-eds} \end{align} Computing this for \(\Omega_{i,m} \neq 1\) in a matter-only Universe is more complicated, because the detailed linear growth of perturbations needs to be taken into account; we leave this calculation as an exercise. In the end one finds that \(\delta_c\) depends only on the matter density parameter \(\Omega_{m}(t_c)\) at the collapse time, but does so only very weakly: \begin{equation}\label{eq-spher-collapse-deltac-open} \delta_c \approx 1.686\,\left[\Omega_{m}(t_c)\right]^{0.0185}\,. \end{equation}
However, our Universe not only contains matter, it also contains dark energy with present-day density parameter \(\Omega_\Lambda \approx 0.7\). The derivation above does not hold in the case of dark energy, because Equation (17.80) needs to be modified to include the effect of dark energy (which adds a force \(\propto r\) to the right-hand side of this equation). We will not discuss this derivation here, but will simply note that the result for \(\delta_c\) assuming a flat, matter and dark-energy Universe (\(\Omega_{i,m} + \Omega_{\Lambda,i} = 1\)) is \begin{equation}\label{eq-spher-collapse-deltac-LCDM} \delta_c \approx 1.686\,\left[\Omega_{m}(t_c)\right]^{0.0055}\,. \end{equation}
Thus, the linear overdensity of collapsed objects in the Universe at the time that their collapse is complete is \(\delta_c \approx 1.686\), independent of the size or mass of the collapsed object and only weakly dependent on the density content of the Universe. This is a remarkable and powerful result! Without considering any complexities due to the non-linear gravitational interaction of bodies in the Universe, we can thus predict that regions of the Universe have collapsed at a given time \(t_c\) if their overdensity computed using linear cosmological perturbation theory at that time exceeds the critical value of \(\delta_c = 1.686\).
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