Chapter 11

Although on very large scales the universe is homogeneous, the universe is actually quite clumpy! A way to make this more quantitative is to consider the average energy density inside a spherical volume

For sufficiently large volumes, $\bar{\varepsilon}$ will equal the average density of the universe, e.g., when $r$ is large enough to contain a volume in which the universe is homogeneous, $\bar{\varepsilon}_m = \Omega_m \varepsilon_{\rm crit}$. On smaller scales, however, the energy density will be slightly larger or smaller depending on where the volume is in the universe. Quantitatively,

This quantity, $\delta(\vec{r},t)$, is called the overdensity parameter. If $\delta > 0$, then that region of the universe will be gravitationally unstable to collapse (since the additional matter, compared to the surroundings closer to $\bar{\varepsilon}$, will attract itself). The density inside a sphere of radius $R(t)$ is $\rho(t) = \bar{\rho}(t)(1+\delta(t)) = 3(M+\Delta M)/(4\pi R^3)$. Only the extra mass will cause gravitational collapse, so Newton's law of gravitation tells us

Since mass is conserved inside the sphere (no stuff flows in/out of the imaginary spherical surface), $M = \rho V = 4\pi \bar{\rho}[1+\delta(t)] R(t)^3 / 3 \rightarrow 3M/(4\pi \bar{\rho}) = R_0^3 = R(t)^3[1+\delta(t)]$. At early times, density fluctuations will be small compared to the average density, and so $\delta \ll 1$, allowing us to make a linear approximation of the expression for $R(t)$:

Because density variations are so small, we can also say that $R_0 \approx R$ and thus $\ddot{R}/R \approx -\ddot{\delta}/3$. Combining this expression for $\ddot{R}/{R}$ with the one from above, we have $\ddot{\delta} = 4\pi G \bar{\rho}\delta$, which you may recognize as a harmonic oscillatory type differential equation with the constants out front determine the characteristic frequency $\omega^2$. This frequency can be inverted to provide a timescale, which we call the dynamical timescale $t_{\rm dyn} = (4\pi G \bar{\rho}\delta)^{-1/2}$. Happily, the radius of the sphere is irrelevant — only the density determines how quickly a given overdensity grows. The mathematical solution to the equation is $\delta(t) = A_1 e^{t/t_{\rm dyn}} + A_2 e^{-t/t_{\rm dyn}}$, which tells us that the overdensity grows exponentially over time (once $t>t_{\rm dyn}$, only the first term will matter).

Jeans Length

This exponential collapse will be stopped by the build up of pressure, if the pressure is able to act in time. In other words, if the gas collapses faster than sound waves (which communicate pressure to outer layers of the gas) can travel out, then the collapse will continue. If the collapse is slower than sound waves can travel, then pressure (a force per area) can counteract the gravitational force of collapse. These times vary depending on the size of the region (since sound takes longer to travel farther) — on sizes larger than a certain length scale, collapse can proceed, but inside that scale pressure has time to build up and counteract collapse. This length scale is called the Jeans Length:

where $c_s = c\sqrt{w}$ is the sound speed of the gas (and $w$ is our good friend the equation of state parameter).

Think back to the era of recombination and decoupling, when baryonic matter and radiation stopped interacting (decoupled) because electrons got tied up in neutral atoms. Before decoupling, radiation allowed pressure waves to propagate quickly ($w=1/3$), and we can define the Jeans mass as the mass of an overdensity that can collapse (enclosed by the Jeans length):

After decoupling, however, the equation of state parameter drops dramatically to the non-relativistic value (we generally consider $w \sim 0$, but it's really a small number given by $w \approx kT / (\mu c^2)$). Matter can't communicate with radiation anymore, so if matter collapses, radiation can't communicate that info quickly to matter farther away; communication is now done through matter-matter interactions (i.e., collisions) at a much slower speed. This lowers the Jeans length and the Jeans mass for a collapsing region:

Before decoupling, the amount of mass inside a "collapsing" region is huge — much larger than any collapsed structures today (massive galaxy clusters, the most massive collapsed objects, are $\sim 10^{15}~{\rm M}_\odot$). After decoupling, relatively tiny structures (the mass of stellar globular clusters today) can collapse, and this is the time when structure formation can truly begin.

Growth of Structure

What we didn't account for in the first section is the expansion of the universe, which slows down gravitational collapse. Considering our sphere from before,

since $\varepsilon_m \propto a^{-3}$. Again approximating $R(t)$ in the linear regime where $\delta(t)\ll1$ and taking two derivatives, looking only at the contribution of the overdensity (ignoring the homogeneous component), and substituting in $H= \dot{a}/a$, we find a similar differential equation to before, but this time with an extra term:

The first equation is a Newtonian derivation, similar to what was done at the top of the page, while the second equation is the result from GR including some substitutions. After decoupling, the universe is matter-dominated, and we know now the universe is flat, so $\Omega_m = 1$ and $H = 2/3t$ and

The solution again has two terms, but only one gets bigger with time, and $\delta \propto t^{2/3} \propto a(t) \propto (1+z)^{-1}$. Decoupling occurs at $z = 1090$, so if the universe only contained baryonic matter, linear growth would cause overdensities to increase by a factor of $10^3$. However, dark matter is never coupled to radiation and can start collapsing earlier, once the universe is matter dominated at $z_{\rm rm} = 3440$.

Power Spectrum of Density Fluctuations

The spatial distribution of density perturbations can also be represented by a Fourier transform with wavelength $\lambda = 2\pi / k$ (where wavelength is the size of the region within which we have an overdensity of size $\delta_k$). At any given location then, the total overdensity is the sum of the sizes of all these components, and the power spectrum is given by the mean square amplitude of the components: $P(k) = \langle | \delta_{\vec{k}} |^2 \rangle$. If we assume the density fluctuations are random and independent as a function of $k$, we have what's called a Gaussian Field where the probability of any value of $\delta$ is given by

The parameter $\sigma_\delta$ can be computed from the power spectrum. Specifically,

and $P(k) \propto k^n$ is an assumption commonly made, because with this form and $n=1$ you get nice mathematical behavior for density fluctuations. Also, inflation predicts this form for the power spectrum, although the flavor of inflation can give you different values of $n$.

This $\sigma_\delta$ is not readily observable; instead, we can choose a sphere of some size and ask how much mass is contained within it compared to the universal average $\langle M \rangle = \frac{4\pi}{3} r^3 \rho_{\rm m,0} = 1.67 \times 10^{11} {\rm M}_\odot \left( \frac{r}{1~{\rm Mpc}} \right)^3$. Considering fluctuations from this average mass:

For $n \sim 1$, the root mean square deviation from the average less as the size (or enclosed mass) increases. By convention, the value of $r$ used to parameterize $\delta M/M = \sigma_r$ is chosen to be that on scales of $r=8~{\rm Mpc}$ and is called $\sigma_8$, the amplitude of density fluctuations. The larger $\sigma_8$ is, the larger the probability of getting a larger density fluctuation is, so big values of $\sigma_8$ can generate more large structures within a given volume, like clusters of galaxies.

Both $\sigma_8$ and $n$, called the scalar spectral index, can be measured by detailed analysis of the CMB temperature fluctuation spectrum; current measurements give values for these dimensionless parameters of $\sigma_8 \sim 0.8$ and $n \sim 0.965$.