Dark Matter

Introduction

Dark matter is a hypothetical form of matter that does not emit, absorb, or reflect electromagnetic radiation, making it invisible to telescopes. Despite being undetectable through direct observation, overwhelming evidence from gravitational effects indicates that it comprises approximately 85% of the matter in the universe and about 27% of the total mass-energy content.

$$\Omega_{\text{DM}} \approx 0.27, \quad \Omega_{\text{baryon}} \approx 0.05, \quad \Omega_\Lambda \approx 0.68$$

1. Galaxy Rotation Curves

The most iconic evidence for dark matter comes from galactic rotation curves. For a galaxy with circular velocity $v(r)$ at radius $r$, Newtonian dynamics predicts:

$$v(r) = \sqrt{\frac{GM(<r)}{r}}$$

where $M(<r)$ is the enclosed mass within radius $r$. For visible matter concentrated near the galactic center, we expect:

$$v(r) \propto r^{1/2} \quad \text{(for } r \ll R) \quad \text{and} \quad v(r) \propto r^{-1/2} \quad \text{(for } r \gg R)$$

Observed Flat Rotation Curves

Instead, observations show that rotation curves remain approximately flat at large radii:

$$v(r) \approx v_0 = \text{const}$$

This implies a dark matter density profile:

$$\rho_{\text{DM}}(r) \propto r^{-2}$$

The NFW (Navarro-Frenk-White) profile from simulations:

$$\rho(r) = \frac{\rho_0}{\frac{r}{r_s}\left(1+\frac{r}{r_s}\right)^2}$$

2. Gravitational Lensing

Dark matter's gravitational field bends light from distant sources. The deflection angle for a point mass is:

$$\alpha = \frac{4GM}{c^2b}$$

where $b$ is the impact parameter.

Strong Lensing

Galaxy clusters create multiple images and Einstein rings. The critical surface density is:

$$\Sigma_{\text{crit}} = \frac{c^2}{4\pi G}\frac{D_s}{D_dD_{ds}}$$

where $D_d, D_s, D_{ds}$ are angular diameter distances to lens, source, and between them.

Weak Lensing

Statistical distortions of background galaxies measure the dark matter distribution. The shear is:

$$\gamma = \frac{1}{2\pi}\int \frac{D_{ds}}{D_dD_s}\Sigma(\vec{\theta'})K(\vec{\theta}-\vec{\theta'})d^2\theta'$$

Bullet Cluster

The Bullet Cluster (1E 0657-56) provides smoking gun evidence: weak lensing maps show dark matter (pink) passed through the collision while gas (blue, X-ray) was slowed by interactions. This directly demonstrates dark matter's collisionless nature.

3. CMB and Large-Scale Structure

CMB Acoustic Peaks

The CMB power spectrum shows acoustic oscillations. The ratio of odd to even peak heights constrains the baryon-to-photon ratio:

$$\eta = \frac{n_b}{n_\gamma} = (6.09 \pm 0.06) \times 10^{-10}$$

This gives $\Omega_b h^2 = 0.02237 \pm 0.00015$, far less than the total matter density$\Omega_m h^2 = 0.1430 \pm 0.0011$ from Planck 2018.

Structure Formation

Dark matter provides gravitational wells for structure to form. The matter power spectrum is:

$$P(k) = \langle|\delta_k|^2\rangle = A k^{n_s} T^2(k)$$

where $T(k)$ is the transfer function and $n_s \approx 0.965$ is the spectral index. Without dark matter, structure formation would be too slow to match observations.

Growth of Perturbations

In the linear regime, density perturbations grow as:

$$\delta(t) = D(t)\delta_0$$

where $D(t)$ is the growth factor. For matter domination, $D(t) \propto a(t) \propto t^{2/3}$.

4. Dark Matter Candidates

WIMPs (Weakly Interacting Massive Particles)

Mass range: 10 GeV - 10 TeV. The relic abundance from thermal freeze-out is:

$$\Omega_\chi h^2 \approx \frac{3 \times 10^{-27} \text{ cm}^3\text{s}^{-1}}{\langle\sigma v\rangle}$$

For $\langle\sigma v\rangle \sim 10^{-26}$ cm³s⁻¹ (weak scale cross section), this gives the observed dark matter density - the "WIMP miracle."

Axions

Proposed to solve the strong CP problem. Mass constrained by:

$$m_a \approx 6 \times 10^{-6} \text{ eV} \left(\frac{10^{12} \text{ GeV}}{f_a}\right)$$

Current constraints: $10^{-6}$ eV $\lesssim m_a \lesssim 10^{-2}$ eV.

Sterile Neutrinos

Right-handed neutrinos with mass:

$$m_s \sim \text{keV} - \text{MeV}$$

Mix with active neutrinos through:

$$\sin^2(2\theta) \sim 10^{-10} - 10^{-5}$$

Primordial Black Holes

Formed in the early universe from density fluctuations. Mass constraints from:

  • • Microlensing: $10^{-7}M_\odot < M_{\text{PBH}} < 10M_\odot$ largely excluded
  • • CMB distortions: $M_{\text{PBH}} < 10^{15}$ g excluded
  • • LIGO detections: $10M_\odot - 100M_\odot$ window remains open

5. Direct Detection Experiments

Direct detection searches for WIMP-nucleus elastic scattering. The recoil energy is:

$$E_R = \frac{\mu^2v^2}{m_N}(1-\cos\theta)$$

where $\mu = m_\chi m_N/(m_\chi + m_N)$ is the reduced mass.

Event Rate

$$\frac{dR}{dE_R} = \frac{\rho_\chi}{m_\chi m_N}\sigma_0 F^2(E_R)\int_{v_{\min}}^{v_{\max}} v f(v) dv$$

where $\rho_\chi \approx 0.3$ GeV/cm³ is the local dark matter density, $F(E_R)$ is the nuclear form factor, and $f(v)$ is the velocity distribution.

Leading Experiments

  • • XENON1T/XENONnT: Liquid xenon TPC, $\sigma_{\text{SI}} > 10^{-47}$ cm²
  • • LUX-ZEPLIN (LZ): 7-tonne LXe, $\sigma_{\text{SI}} > 10^{-48}$ cm²
  • • SuperCDMS: Cryogenic Si and Ge detectors, low-mass WIMP search
  • • DAMA/LIBRA: Claims annual modulation signal (controversial)

6. Indirect Detection

Dark matter annihilation produces Standard Model particles. The annihilation rate per volume is:

$$\Gamma = \frac{1}{2}\langle\sigma v\rangle n_\chi^2 = \frac{1}{2}\langle\sigma v\rangle \left(\frac{\rho_\chi}{m_\chi}\right)^2$$

Gamma Rays

The differential photon flux from a region is:

$$\frac{d\Phi}{dE_\gamma} = \frac{\langle\sigma v\rangle}{8\pi m_\chi^2}\frac{dN_\gamma}{dE_\gamma}\times J(\Delta\Omega)$$

where $J(\Delta\Omega) = \int_{\Delta\Omega}d\Omega\int_{\text{l.o.s}}\rho^2(r)dl$ is the J-factor.

Searches

  • • Fermi-LAT: Galactic center excess (disputed), dwarf spheroidals
  • • H.E.S.S., VERITAS: TeV gamma rays from Galactic halo
  • • IceCube: Neutrinos from Sun and Earth (WIMP capture)
  • • AMS-02, PAMELA: Cosmic ray positron excess

7. Modified Gravity Alternatives

MOND (Modified Newtonian Dynamics)

Proposes modification of Newton's law at low accelerations:

$$a = \begin{cases} \frac{GM}{r^2} & \text{if } a \gg a_0 \\ \sqrt{\frac{GMa_0}{r^2}} & \text{if } a \ll a_0 \end{cases}$$

where $a_0 \approx 1.2 \times 10^{-10}$ m/s². This produces flat rotation curves without dark matter.

TeVeS (Tensor-Vector-Scalar)

Relativistic extension of MOND by Jacob Bekenstein. Adds vector and scalar fields to GR:

$$S = S_{\text{Einstein}} + S_{\text{scalar}} + S_{\text{vector}} + S_{\text{matter}}$$

Challenges for Modified Gravity

  • • Bullet Cluster: clear spatial separation of lensing mass from baryons
  • • CMB acoustic peaks: require precise baryon-photon ratio inconsistent with no DM
  • • Structure formation: timing and scales don't match without cold dark matter
  • • Galaxy cluster mass profiles: lensing, X-ray, and dynamical masses all require DM

8. Current Status and Future Prospects

Evidence Summary

Dark matter is supported by independent observations across vastly different scales:

  • • Galactic scales: rotation curves, velocity dispersions
  • • Cluster scales: lensing, dynamics, X-ray gas
  • • Cosmological scales: CMB, BAO, large-scale structure

Open Questions

  • • What is dark matter made of?
  • • Why is $\Omega_{\text{DM}} \sim \Omega_b$ (same order of magnitude)?
  • • Core-cusp problem: simulations predict cusps, observations show cores
  • • Missing satellites problem: simulations predict more dwarf galaxies than observed
  • • Too-big-to-fail problem: massive subhalos should host bright dwarfs

Future Experiments

  • • XENONnT, LZ, DARWIN: Next-generation direct detection (ton-scale)
  • • CTA (Cherenkov Telescope Array): Improved gamma-ray sensitivity
  • • Euclid, LSST (Rubin): Weak lensing surveys
  • • LISA: Primordial black hole constraints from gravitational waves
  • • CMB-S4: Improved small-scale power spectrum measurements