Where dark matter comes from — freeze-out, hot and cold

At early enough times — the notes give $kT \approx 100$ GeV, which is about $10^{15}$ K — $kT \ge m_xc^2$ for the leading dark matter candidates. [The notes print "$T > 10^{12}$ K $\cong$ 100 GeV"; $10^{12}$ K is only about 86 MeV, and 100 GeV corresponds to $1.2\times10^{15}$ K.] Once the particles are non-relativistic but still in thermal equilibrium, their number density is set by the temperature alone:

where $g$ counts the internal states (spin). Equilibrium means creation and annihilation balance, for example through

The annihilation rate per unit volume is $n^2\langle\sigma v\rangle$, with $\langle\sigma v\rangle$ the annihilation cross section times relative speed, averaged over the thermal velocities. In equilibrium the creation rate $\psi$ matches it:

[The notes print only the right-hand side, "$= n_\text{eq}^2\langle\sigma v\rangle$".]

If equilibrium lasted for ever, the Boltzmann factor $e^{-mc^2/kT}$ would drive the number to zero once $kT \ll mc^2$: no particles left, no relic abundance. What saves them is that the annihilation rate $n^2\langle\sigma v\rangle$ also falls as the universe expands and dilutes them.

Without any reactions, the particles are simply diluted, $n \propto a^{-3}$, so the comoving density $n_c$ stays constant:

[The notes print the exponent as $-3$; with $n \propto a^{-3}$, the comoving density is $n(a/a_0)^{+3}$.] With reactions, annihilations remove particles and creation replaces them, pushing $n_c$ towards its equilibrium value:

[As written, this suppresses a factor $(a_0/a)^3$ on the right, which turns comoving densities back into physical ones in the collision rate. It is this factor that makes the $n_{c,\text{eq}}$ in the prefactor of eq. (471) the physical $n_\text{eq}$ used in $\tau_\text{coll}$ below, so read it as included.] Changing the variable from $t$ to $a$ with $\dot a = aH$:

The prefactor is a ratio of two time scales: the expansion time $\tau_H = 1/H$ and the mean time between annihilations, $\tau_\text{coll} = 1/(n_\text{eq}\langle\sigma v\rangle)$:

This gives two regimes:

• Early times, $\tau_\text{coll} \ll \tau_H$: the huge prefactor forces the bracket to zero, so $n_c \approx n_{c,\text{eq}}$. Reactions are fast; the particles track equilibrium.
• Late times, $\tau_\text{coll} \gg \tau_H$: the right-hand side vanishes, so $n_c \approx$ constant $\approx n_{c,\text{eq}}(z_\text{freeze})$. Particles no longer find each other.

The switch happens at the redshift $z_\text{freeze}$ where $\tau_\text{coll} \sim \tau_H$. From then on the comoving number is frozen: that is the relic abundance.

Running the argument backwards: the observed amount of dark matter fixes $\langle\sigma v\rangle$ (the relic abundance depends on the mass only weakly, through $x_f$). The notes point out that the mass and cross section this suggests land at the electroweak scale — a particle of order 100 GeV with a weak-interaction-sized cross section, freezing out at a temperature some 20–30 times below its mass — exactly what particle physicists had proposed for other reasons as WIMPs (weakly interacting massive particles). This coincidence is called the WIMP miracle. The notes add a sober caveat: so far, nothing has been detected.

Were the particles still moving at nearly the speed of light when they froze out? A particle becomes non-relativistic roughly when

which splits relics into two kinds:

• $t_\text{nr} > t_\text{freeze}$ — still relativistic at freeze-out: a hot relic, hot dark matter (light neutrinos are the classic example).
• $t_\text{nr} < t_\text{freeze}$ — already slow at freeze-out: a cold relic, cold dark matter (WIMPs, which freeze out at $x_f \approx 20$–$30$, are cold).

Hot dark matter: free-streaming

Fast particles don't stay inside small overdensities — they stream straight out of them. By analogy with the Jeans length, with the particle speed $v$ in place of the sound speed, the scale below which this smooths out perturbations is the free-streaming length

While $v \approx c$, every scale smaller than the horizon is wiped out. The particles do slow down, because their peculiar momentum redshifts with the expansion:

($d$ the distance), and free-streaming effectively ends once they become non-relativistic, $mc^2 \sim 3kT$. The scale erased is roughly the (comoving) horizon at that time:

[The notes print the normalising masses as 2 keV and 3 keV and write $l_h \sim ct_h$. Recomputing in the radiation era — $t \propto T^{-2} \propto m^{-2}$ and comoving horizon $\propto t(1 + z) \propto m^{-1}$ — the quoted numbers fit masses in eV and the comoving length $ct_h(1 + z_h)$: a 2 eV particle turns non-relativistic at $t \approx 3\times10^{12}$ s, and for 3 eV, $ct(1 + z) \approx 55$ Mpc. For keV masses the times are a million times shorter and the lengths a thousand times smaller. The worked example below uses the full comoving horizon $2ct(1 + z)$, twice the notes' $ct(1+z)$; at this level of estimate the factor 2 doesn't matter.]

So hot dark matter erases all structure below $l_h$. For eV-mass particles that is around a hundred megaparsecs — the scale of superclusters.

Cold dark matter: bottom-up

Cold dark matter is already slow at freeze-out, so its free-streaming scale is very small and perturbations survive on almost all scales. In the power spectrum, hot dark matter cuts off $P(k)$ sharply beyond the free-streaming wavenumber, while cold dark matter keeps the gentle $k^{-3}$ fall-off of the transfer function.

The two give opposite orders of events:

• With hot dark matter only very large structures can collapse first; they would then have to fragment into galaxies — top-down formation.
• With cold dark matter small halos form first and merge into larger ones, which keep the smaller ones inside as subhalos — bottom-up (hierarchical) formation, as on the Press–Schechter page.

Observations not discussed in these notes — the clustering of galaxies, the Lyman-α forest (8.902 Lecture 21), the ages of galaxies — favour bottom-up, which is why the standard model uses cold dark matter.

The notes close the section with a brief history in temperature and time; here it is in summary, from the earliest moments to today:

• $10^{19}$ GeV, $10^{-43}$ s — the Planck era, where quantum gravity would be needed. [$10^{-43}$ s is the Planck time itself; the radiation-era relation, counting $g_* \approx 100$ particle species at such temperatures, would put $10^{19}$ GeV at about $10^{-45}$ s. At this point both are only indicative.]
• $10^{16}$ GeV, $10^{-38}$ s — a possible grand-unified (GUT) phase transition; before it, strong and electroweak forces would be indistinguishable (8.286's monopole and inflation lectures).
• $10^{12}$ GeV, $10^{-30}$ s — a Peccei–Quinn transition, if that mechanism solves the strong CP problem (it would also produce axions, another dark matter candidate).
• 10s–100s GeV, $10^{-8}$ s — WIMPs freeze out. [With $g_* \approx 100$ species at these temperatures (the worked example's 3.36 applies only after electron–positron annihilation, below about 0.1 MeV), 10–100 GeV correspond to about $10^{-9}$–$10^{-11}$ s; $10^{-8}$ s fits $kT \approx 4$ GeV, where a 100 GeV WIMP actually freezes out.]
• 100–300 MeV, $10^{-5}$ s — quark–hadron transition: quarks and gluons are bound into protons and neutrons.
• 10 MeV–0.1 MeV, seconds to minutes — Big Bang nucleosynthesis: deuterium, helium-3, helium-4 and lithium-7 form (the next lecture).
• ~keV, ~1 day — photon-number-changing processes stop; from then on the number of photons is conserved, preserving the blackbody spectrum. [By the radiation-era relation of the worked example, 1 keV is reached after about two weeks; one day corresponds to $kT \approx 4$ keV.]
• ~3 eV, $10^4$–$10^5$ yr — matter–radiation equality; before it radiation dominates the energy density. [With today's parameters equality is at $1 + z \approx 3400$, i.e. $T \approx 9000$ K, $kT \approx 0.8$ eV, some $5$–$7\times10^4$ years — 8.286's matter–radiation equality page estimates 68 000–75 000 years and quotes 47 000 years from a fuller calculation.]
• ~eV, ~400 000 yr — electrons and protons combine into hydrogen; the CMB is released. [More precisely $kT \approx 0.26$ eV ($T \approx 3000$ K) at about 380 000 years; recombination happens well below 1 eV because photons vastly outnumber baryons.]
• $10^{-3}$ eV, $10^9$ yr — first stars and galaxies.
• $10^{-4}$ eV, $10^{10}$ yr — today ($T = 2.7$ K, $kT = 2.3\times10^{-4}$ eV).

• In the hot early universe particles with $mc^2 \lesssim kT$ are created and annihilated in equilibrium, with $n_\text{eq} = g(mkT/2\pi\hbar^2)^{3/2}e^{-mc^2/kT}$ once non-relativistic.
• Annihilation competes with expansion: the ratio $\tau_H/\tau_\text{coll}$ decides. When $\tau_\text{coll} \sim \tau_H$ the particles freeze out and their comoving density is fixed — the relic abundance, lower for larger $\langle\sigma v\rangle$.
• The observed dark matter density points to weak-scale masses and cross sections: the WIMP miracle. No WIMP has been detected.
• Hot relics (still relativistic at freeze-out) free-stream and erase structure below $l_h \propto m^{-1}$ (~100 Mpc for eV masses), giving top-down formation; cold relics allow bottom-up, hierarchical growth, which observations support.