Past exam of the astrophysics course of the University of Oxford / 2016 / 4

The density contrast is $\delta (\mathbf{x},t)=[\rho (\mathbf{x},t)-{\rho }_0(t)]/{\rho }_0(t)$, and ${\delta }_k(t)$ is one spatial Fourier mode with comoving wavevector $\mathbf{k}$. The scale factor is $a(t)$, $H=\dot{a}/a$ is the Hubble parameter, $c_s$ is the physical sound speed, ${\rho }_0$ the mean matter density, and $G$ the gravitational constant. For each fixed $k$ the displayed equation is an ordinary differential equation in cosmic time. It compares pressure restoration, self-gravity and expansion-induced damping:The pressure term resists compression; self-gravity enhances an overdensity; the $2H\dot{\delta }$ term slows the motion relative to a static background. The Newtonian treatment is appropriate to small perturbations on subhorizon scales within the specified fluid model, rather than an arbitrary gauge-independent treatment of superhorizon perturbations.

The physical Jeans length follows by equating pressure and gravity:The comoving Jeans wavenumber is $a{k}_{J,\mathrm{phys}}$. Perturbations well below the physical Jeans length are pressure-supported acoustic oscillations whose amplitude is modified by expansion; those above it admit gravitational growth when the background evolution permits. Pressureless matter in a matter-dominated universe is the example below. Merely having a large wavelength is not sufficient to apply this Newtonian equation outside the horizon.

For a gas of particle mass $m$ and temperature $T$, write $c_s^2=\Gamma k_{B}T/m$, with $\Gamma =1$ for the isothermal sound-speed convention or the appropriate adiabatic heat-capacity ratio otherwise. ThenAt fixed density it is larger for hotter or lighter particles, and it decreases with increasing density. Numerical coefficients differ between sound-crossing and self-gravitating-sphere conventions; the expression here uses the Fourier instability boundary of the stated equation.

In an Einstein-de Sitter universe, the spatially flat Friedmann-Lemaître-Robertson-Walker metric is $d{s}^2=-c^{2}d{t}^2+a^2(t)d{\mathbf{x}}^2$. Conservation of pressureless matter gives ${\rho }_0{a}^3=\mathrm{constant}$; the Friedmann equation then gives $a\propto t^{2/3}$, $H=2/(3t)$ and ${\rho }_0=1/(6\pi G{t}^2)$. With $c_s=0$,Trying $\delta =t^p$ gives $p^2+p/3-2/3=(p-2/3)(p+1)=0$. Hence the Einstein-de Sitter density-growth modes areThe growing mode increases only as a power law because the expansion dilutes and slows the collapsing matter.

Conformal time is defined by $d\eta =dt/a(t)$, so the metric becomes $d{s}^2=a^2(-c^{2}d{\eta }^2+d{\mathbf{x}}^2)$. Choosing its origin at the big bang gives $\eta \propto t^{1/3}$ and $a\propto {\eta }^2$. ThereforeEquivalently the equation is ${\delta }^{\prime \prime }+2{\delta }^{\prime }/\eta -6\delta /{\eta }^2=0$, where primes denote conformal-time derivatives.

Before and through cosmological recombination, photons diffuse between hotter and cooler parts of the tightly coupled photon–baryon fluid. Radiative viscosity and heat transport erase short-wavelength acoustic perturbations. As free electrons disappear, the photon mean free path grows, enhancing this diffusion before decoupling. Silk damping can be represented by a factor approximately $e^{-k^2/k_D^2}$ in the linear acoustic amplitude. The comoving Silk wavelength may be defined as ${\lambda }_D=2\pi /k_D$; the associated diffusion-length convention is $r_D=1/k_D$.

A random-walk estimate is $r_D^2\sim \int c{\lambda }_{\gamma }dt/a^2$ up to transport and baryon-loading factors, with physical ${\lambda }_{\gamma }=(n_e{\sigma }_T{)}^{-1}$. Scales much shorter than the damping wavelength are strongly suppressed. This is photon diffusion, not the Jeans boundary and not erasure of every cold-dark-matter perturbation. After decoupling, photon free streaming is a distinct part of the evolution.