The Dark Matter Problem

1. Galactic Rotation Curves

For a test mass orbiting at radius $r$ in a spherically symmetric potential, the circular velocity is

$$v_c(r) = \sqrt{\frac{G\,M(r)}{r}}\,.$$

Beyond the visible disc, where $M(r)$ should be approximately constant, Keplerian orbits predict $v_c \propto r^{-1/2}$. Instead, observations by Rubin and Ford (1970) showed that $v_c$ remains flat out to tens of kpc, implying

$$M(r) \propto r \quad\Longrightarrow\quad \rho_{\rm DM}(r) \propto r^{-2}$$

at large radii. This isothermal profile is reproduced by N-body simulations with cold dark matter, which generically produce NFW halos:

$$\rho_{\rm NFW}(r) = \frac{\rho_s}{(r/r_s)(1+r/r_s)^2}\,.$$

2. Gravitational Lensing

General relativity predicts that a mass $M$ deflects a passing photon by the Einstein angle

$$\theta_E = \sqrt{\frac{4GM}{c^2}\frac{D_{LS}}{D_L D_S}}\,,$$

where $D_L$, $D_S$, $D_{LS}$ are angular-diameter distances to the lens, source, and from lens to source. Weak-lensing surveys reconstruct the projected mass distribution of galaxy clusters, consistently finding total masses 5–10 times larger than the baryonic content. The Bullet Cluster (1E 0657-558) provides a spectacular demonstration: after a cluster merger, the gravitational potential (traced by lensing) is offset from the X-ray gas, proving that most of the mass is collisionless.

3. CMB Acoustic Peaks

Before recombination at $z \approx 1100$, baryons and photons form a tightly coupled fluid. Dark matter, being non-interacting, does not participate in the acoustic oscillations. The baryon-to-dark-matter ratio leaves distinct signatures in the CMB power spectrum:

$$\frac{\ell(\ell+1)C_\ell}{2\pi} \propto \left[\Omega_b,\;\Omega_c,\;H_0,\;n_s,\;\tau\right]\,.$$

Specifically, increasing $\Omega_b h^2$ boosts odd peaks relative to even peaks (baryon loading), while increasing $\Omega_c h^2$ enhances all peaks uniformly by deepening the potential wells before decoupling.

4. Large-Scale Structure

The matter power spectrum encodes the growth of density perturbations:

$$P(k) = A\,k^{n_s}\,T^2(k)\,D_+^2(z)\,,$$

where $T(k)$ is the transfer function and $D_+(z)$ is the linear growth factor. Without dark matter, baryonic perturbations are erased on scales below the Silk damping length $\lambda_S \sim 10$ Mpc. Dark matter perturbations, entering the horizon before matter-radiation equality, grow logarithmically but survive Silk damping, seeding the structure we observe today.

5. Particle Candidates

6. Detection Strategies

Dark matter searches proceed along three complementary axes. Direct detection experiments look for nuclear recoils from DM-nucleus scattering in ultra-pure underground detectors. The event rate scales as

$$R \propto \frac{\rho_0\,\sigma_{\chi N}}{m_\chi\,m_N}\int_{v_{\min}}^{v_{\rm esc}} \frac{f(v)}{v}\,d^3v\,,$$

where $\rho_0 \approx 0.3$ GeV/cm$^3$ is the local DM density. Indirect detection seeks the products of DM annihilation or decay ($\gamma$-rays, neutrinos, positrons, antiprotons) in astrophysical environments:

$$\Phi_\gamma \propto \frac{\langle\sigma v\rangle}{8\pi\,m_\chi^2}\frac{dN_\gamma}{dE}\int_{\rm l.o.s.}\rho^2\,d\ell\,,$$

where the integral along the line of sight defines the $J$-factor. Collider searches at the LHC look for missing transverse energy signatures from pair-produced DM particles: $pp \to \chi\bar\chi + X$, where $X$ is a visible recoil system (monojet, monophoton, mono-$Z$).

7. Phase-Space Constraints

The Tremaine-Gunn bound limits the mass of fermionic dark matter from below using the phase-space density observed in dwarf spheroidal galaxies:

$$m_f \gtrsim \left(\frac{9\pi\hbar^3}{G^{1/2}\,g_f\,r_c^2\,\sigma_v\,\rho_0^{1/2}}\right)^{1/4} \sim \text{few hundred eV}\,,$$

which excludes standard active neutrinos ($m_\nu \lesssim 0.1$ eV) as the dominant DM component. For bosonic DM such as axions, no such bound applies since bosons can occupy the same quantum state, forming a Bose-Einstein condensate.

8. Cosmological Abundance Constraint

Any viable dark matter candidate must satisfy the Planck constraint on the total DM abundance. The general relic density formula from thermal freeze-out is

$$\Omega_{\rm DM}h^2 = \frac{1.07\times10^9\;\text{GeV}^{-1}}{M_{\rm Pl}}\frac{x_f}{\sqrt{g_*(x_f)}}\frac{1}{a+3b/x_f}\,,$$

where $\langle\sigma v\rangle = a + bv^2 + \cdots$ is the thermally averaged cross section expanded in powers of relative velocity, and $x_f = m/T_f \approx 20\text{--}25$. For non-thermal production mechanisms (misalignment, decay), the relic density depends on model-specific parameters as we discuss in the following chapters.

9. N-body Simulations and Halo Profiles

Cosmological N-body simulations (Millennium, Bolshoi, IllustrisTNG) solve the gravitational evolution of $\sim 10^{10}$ particles to map the nonlinear growth of structure. CDM halos are well described by the NFW profile with a concentration parameter

$$c(M,z) = \frac{r_{200}}{r_s} \approx 10\left(\frac{M}{10^{12}M_\odot}\right)^{-0.1}(1+z)^{-0.5}\,,$$

where $r_{200}$ is the virial radius enclosing 200 times the critical density. Small-scale challenges to CDM — the cusp-core, missing satellites, and too-big-to-fail problems — may indicate either baryonic feedback effects or departures from cold, collisionless dark matter (warm DM, self-interacting DM, fuzzy DM).