Why structure formation needs cold dark matter

Perturbations grow only above the Jeans length, or above the Jeans mass $M_J = \frac{4}{3}\pi\bar\rho\lambda_J^3$. For baryons the sound speed changes dramatically at recombination:

• Before recombination the electrons are tightly coupled to the photons, so the baryons and photons form a single fluid whose pressure is that of the radiation, $p = \rho c^2/3$:

• After recombination the baryons are an ordinary ideal gas, with $p = \rho kT/m$ and $T = 2.7\ \text{K}\,(1+z)$ [the notes print $p = \rho/(mkT)$, a typo for $\rho kT/m$]:

The Jeans mass therefore drops enormously when the photon pressure support is removed:

Before recombination, anything smaller than a supercluster just oscillates. Baryonic structure can only start forming after $z \sim 1000$.

From $z \sim 1000$ to today the scale factor grows by about $10^3$, and in the matter era $\delta \propto a$. The CMB shows $\delta \sim 10^{-5}$ at recombination, so baryons alone would reach only $\delta_\text{today} \sim 10^{-5} \times 10^3 = 10^{-2}$ — far less than the clumping we see in the low-redshift universe, where galaxies and clusters have $\delta \gg 1$. Growth of ordinary matter alone cannot produce today's structure. The way out: dark matter must have clumped earlier, and after recombination the baryons fell into the dark matter's gravitational wells.

Cold dark matter (CDM) has two properties that make this work:

• It is very cold, with a tiny velocity dispersion $\sigma$, so its Jeans mass is tiny: collapse is possible on essentially all scales.
• It does not interact with radiation, so photon pressure doesn't hold it up: it can grow before recombination.

In the notes' words, cold dark matter is "needed to make structure formation work".

Perturbations come on all length scales, and their growth also depends on whether they are larger or smaller than the horizon. A full general-relativistic treatment shows that modes outside the horizon always grow — there is no causal contact to resist them:

Once a mode enters the horizon, its growth changes. The physical horizon grows, for Ω = 1, as

(the same horizon distances as in 8.286). For baryons before recombination, the Jeans length $\lambda_J = \frac{c}{\sqrt{G\bar\rho}}\sqrt{\pi/3}$ (eq. 398, with $c_s = c/\sqrt3$) is larger than the horizon. So as soon as a baryonic mode enters the horizon, it starts to oscillate. The Jeans mass reaches about $10^{16}\,M_\odot$ by $z \sim 1000$, so every baryonic mode smaller than that entered the horizon before recombination, and stopped growing.

Silk damping

Worse, those modes are erased. Before decoupling photons can't stream freely — they keep Thomson-scattering off free electrons — but their mean free path grows towards recombination. Below about $10^{16}\,M_\odot$ perturbations oscillate under photon pressure, and photons can diffuse out of the potential wells, dragging the baryons with them: electrons through Thomson scattering, protons through their Coulomb attraction to the electrons. The net effect: baryonic perturbations below about $10^{12}\,M_\odot$ — the Silk mass — are damped away.

Dark matter: frozen, not erased

Cold dark matter has essentially zero Jeans mass, so all its modes can grow. But a CDM mode that enters the horizon during the radiation era stalls: $\delta$ stays roughly constant. The reason is that the expansion is then faster than the collapse:

So modes entering the horizon during radiation domination are frozen (a fuller treatment gives a slow, logarithmic growth), but — unlike the baryons — not damped. A mode entering after matter–radiation equality grows all the time; one entering earlier stagnates until matter domination and then grows. After recombination the baryons fall into the CDM wells and catch up. So the notes make cold dark matter the main engine of structure formation: only it has time to build perturbations large enough, and without it there would be no structure to speak of.

• Before recombination baryons and photons form one fluid with $c_s = c/\sqrt3$ and Jeans mass ≳ $10^{16}\,M_\odot$; afterwards $c_s \approx 5$ km/s and $M_J$ drops to ~$10^5$–$10^6\,M_\odot$.
• Baryons alone could grow only from $z \sim 1000$, by ~$10^3$: from $10^{-5}$ to ~$10^{-2}$ — far too little.
• Before decoupling, baryonic modes inside the horizon oscillate, and Silk damping erases those below ~$10^{12}\,M_\odot$.
• Cold dark matter has almost no Jeans mass and ignores radiation; its sub-horizon modes are frozen (not erased) in the radiation era and grow from matter–radiation equality. Baryons then fall into its wells: cold dark matter is needed for structure formation.