Dark Matter & Dark Energy

A comprehensive treatment of the dark universe — from dark matter observational evidence, theoretical candidates, and detection strategies to dark energy, the cosmological constant, dynamical models, and the fate of the cosmos.

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Computational Programs

8 interactive Python & Fortran programs — rotation curves, freeze-out, direct detection, N-body simulations, and more

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Video Lectures

78 lectures from TRIUMF, ICTP, KITP & more — dark matter theory, detection, cosmology, and astroparticle physics

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Prerequisites

Particle Physics

Standard Model, WIMPs, beyond-SM candidates, and annihilation cross sections

Cosmology

Friedmann equations, CMB anisotropies, and large-scale structure formation

Statistical Mechanics

Thermal equilibrium, freeze-out calculations, and phase-space distributions

General Relativity

Gravitational lensing, spacetime metrics, and cosmological dynamics

Contents

1. Introduction & Historical Context

2. Galaxy Rotation Curves

3. Dark Matter Halo Profiles

4. Gravitational Lensing

5. CMB & Large-Scale Structure

6. Big Bang Nucleosynthesis

7. Thermal Relic Freeze-Out

8. Dark Matter Candidates

9. Direct Detection

10. Indirect Detection

11. Collider Searches

12. Self-Interacting Dark Matter

13. Dark Sectors & Hidden Valleys

14. N-Body Simulations

15. Small-Scale Problems

16. Modified Gravity Alternatives

17. Discovery & Evidence for Dark Energy

18. The Cosmological Constant

19. The Cosmological Constant Problem

20. Dynamical Dark Energy Models

21. Observational Constraints on Dark Energy

22. The Future of the Universe

23. Current Status & Future Prospects

1. Introduction & Historical Context

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.

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

Historical Timeline

The concept of "missing mass" dates back to the early 20th century:

  • • 1933 — Fritz Zwicky: Studied the Coma Cluster using the virial theorem. He found the velocity dispersion of galaxies implied a total mass ~400× greater than the luminous mass. He called it dunkle Materie (dark matter).
  • • 1939 — Horace Babcock: Measured the rotation curve of Andromeda (M31), finding unexpectedly high velocities at large radii, though he attributed this to mass-to-light ratio variations.
  • • 1959 — Kahn & Woltjer: Argued that the Local Group required unseen mass to be gravitationally bound, given the approach velocity of M31 toward the Milky Way.
  • • 1970s — Vera Rubin & Kent Ford: Systematically measured flat rotation curves in spiral galaxies, providing definitive evidence that galaxies are embedded in massive dark matter halos.
  • • 1970s — Ostriker, Peebles, Yahil: Showed that galaxies, groups, and clusters all require dark matter halos to be dynamically stable.
  • • 1990s–2000s — WMAP/Planck: CMB measurements precisely determined the cosmic matter budget, confirming $\Omega_{\text{DM}}h^2 = 0.120 \pm 0.001$.

The Virial Theorem Argument

Zwicky applied the virial theorem to the Coma Cluster. For a gravitationally bound system in equilibrium:

$$2\langle K \rangle = -\langle U \rangle \quad \Rightarrow \quad M_{\text{virial}} = \frac{5\langle v^2 \rangle R_h}{G}$$

where $\langle v^2 \rangle$ is the mean square velocity of cluster galaxies and $R_h$ is the half-mass radius. The virial mass of the Coma Cluster is $M_{\text{virial}} \sim 10^{15} M_\odot$, far exceeding the luminous mass $M_{\text{lum}} \sim 10^{13} M_\odot$.

Derivation: Virial Mass

For $N$ galaxies of mass $m_i$ in a cluster, the total kinetic energy is:

$$K = \frac{1}{2}\sum_i m_i v_i^2 = \frac{1}{2}M\langle v^2\rangle$$

The gravitational potential energy for a uniform sphere of radius $R$:

$$U = -\frac{3GM^2}{5R} \quad \Rightarrow \quad U = -\frac{GM^2}{R_g}$$

where $R_g = 5R/3$ is the gravitational radius. Applying $2K = -U$:

$$M\langle v^2\rangle = \frac{GM^2}{R_g} \quad \Rightarrow \quad M = \frac{R_g\langle v^2\rangle}{G} = \frac{5R_h\langle v^2\rangle}{G}$$

Using the projected velocity dispersion $\sigma_v^2 = \langle v^2\rangle/3$ (for isotropic orbits):$M_{\text{virial}} = \frac{5 \times 3\sigma_v^2 R_h}{G}$.

What We Know About Dark Matter

  • • Gravitationally interacting: it clusters, forms halos, and lenses light
  • • Electrically neutral: does not emit or absorb photons
  • • Stable: lifetime $\tau_{\text{DM}} \gg t_{\text{universe}}$
  • • Cold or warm: non-relativistic at matter-radiation equality (CDM is favored)
  • • Collisionless: self-interaction cross section $\sigma/m < 1 \text{ cm}^2/\text{g}$ (from Bullet Cluster)
  • • Non-baryonic: BBN constrains $\Omega_b \ll \Omega_m$

2. 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:

$$v(r) \approx v_0 = \text{const} \quad \Rightarrow \quad M(<r) \propto r$$

Derivation: Flat Rotation Curve → ρ ∝ r⁻²

A flat rotation curve means $v(r) = v_0 = \text{const}$. From $v^2 = GM(<r)/r$:

$$M(<r) = \frac{v_0^2 r}{G}$$

The enclosed mass grows linearly with $r$. Differentiating to find the density:

$$\frac{dM}{dr} = 4\pi r^2 \rho(r) = \frac{v_0^2}{G} \quad \Rightarrow \quad \rho(r) = \frac{v_0^2}{4\pi G r^2} \propto r^{-2}$$

This is the singular isothermal sphere profile. The total halo mass diverges logarithmically, so the halo must be truncated at the virial radius $r_{200}$.

Decomposing the Rotation Curve

The total rotation curve is composed of contributions from disk, bulge, gas, and dark matter halo:

$$v_{\text{tot}}^2(r) = v_{\text{disk}}^2(r) + v_{\text{bulge}}^2(r) + v_{\text{gas}}^2(r) + v_{\text{halo}}^2(r)$$

For an exponential disk with scale length $R_d$ and surface density $\Sigma_0$:

$$v_{\text{disk}}^2(r) = 4\pi G \Sigma_0 R_d\, y^2 \left[I_0(y)K_0(y) - I_1(y)K_1(y)\right]$$

where $y = r/(2R_d)$ and $I_n, K_n$ are modified Bessel functions. The halo contribution dominates beyond $\sim 2R_d$.

The Tully-Fisher Relation

The empirical relation between luminosity and asymptotic rotation velocity:

$$L \propto v_{\text{flat}}^\alpha, \quad \alpha \approx 4$$

This tight correlation (baryonic Tully-Fisher relation) constrains the relationship between baryonic and dark matter in galaxies and is a key prediction that any dark matter model must explain.

3. Dark Matter Halo Profiles

Navarro-Frenk-White (NFW) Profile

N-body simulations consistently produce a universal density profile for CDM halos:

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

where $r_s$ is the scale radius and $\rho_0$ is a characteristic density. The concentration parameter $c = r_{200}/r_s$ relates the virial radius to the scale radius. The enclosed mass:

$$M_{\text{NFW}}(<r) = 4\pi\rho_0 r_s^3 \left[\ln\left(1+\frac{r}{r_s}\right) - \frac{r/r_s}{1+r/r_s}\right]$$

The NFW profile has a cuspy inner slope $\rho \propto r^{-1}$ and an outer slope $\rho \propto r^{-3}$.

Einasto Profile

A more flexible parametrization that better fits high-resolution simulations:

$$\ln\frac{\rho(r)}{\rho_{-2}} = -\frac{2}{\alpha}\left[\left(\frac{r}{r_{-2}}\right)^\alpha - 1\right]$$

where $\alpha \approx 0.16-0.20$ for galaxy-mass halos and $r_{-2}$ is the radius where the logarithmic slope equals $-2$. Unlike NFW, the Einasto profile has a finite (non-divergent) central density.

Burkert Profile

An empirical profile motivated by observations of dwarf galaxies, which exhibit constant-density cores:

$$\rho_{\text{Burkert}}(r) = \frac{\rho_0 r_0^3}{(r+r_0)(r^2+r_0^2)}$$

This has a constant-density core ($\rho \to \rho_0$ as $r \to 0$), in contrast to the NFW cusp. The discrepancy between simulated cusps and observed cores is the "core-cusp problem."

Isothermal Sphere

The simplest model consistent with flat rotation curves:

$$\rho_{\text{iso}}(r) = \frac{\sigma_v^2}{2\pi G r^2}, \quad v_{\text{circ}} = \sqrt{2}\,\sigma_v = \text{const}$$

where $\sigma_v$ is the 1D velocity dispersion. While unphysical at $r=0$ (infinite density) and $r \to \infty$ (infinite mass), it provides a useful analytic approximation for the intermediate regime.

Mass-Concentration Relation

Simulations predict that halo concentration decreases with mass:

$$c(M, z) \approx A \left(\frac{M}{M_{\text{pivot}}}\right)^B (1+z)^C$$

with typical values $c \sim 5-15$ for galaxy-mass halos at $z=0$. More massive (cluster-scale) halos formed later and are less concentrated ($c \sim 3-5$).

Simulation: NFW Halo Profile & Rotation Curve

The following simulation computes the NFW density profile and the corresponding circular velocity (rotation curve) for a halo with scale radius $r_s = 20$ kpc and scale density $\rho_s = 0.01$ M$_\odot$/pc$^3$:

Python
script.py45 lines

Click Run to execute the Python code

Code will be executed with Python 3 on the server

4. Gravitational Lensing

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

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

where $b$ is the impact parameter. For a general mass distribution, the lens equation relates source position $\beta$ to image position $\theta$:

$$\vec{\beta} = \vec{\theta} - \frac{D_{ds}}{D_s}\vec{\hat{\alpha}}(\vec{\theta}) = \vec{\theta} - \vec{\alpha}(\vec{\theta})$$

Einstein Radius

For a point mass, the Einstein radius defines the characteristic angular scale of lensing:

$$\theta_E = \sqrt{\frac{4GM}{c^2}\frac{D_{ds}}{D_d D_s}}$$

A source aligned behind the lens at $\beta = 0$ is imaged as a ring of radius $\theta_E$ — the Einstein ring.

Derivation: Einstein Radius

Start from the lens equation for a point mass: $\beta = \theta - D_{ls}/(D_s)\cdot\hat{\alpha}$. The physical deflection angle is$\hat{\alpha} = 4GM/(c^2 b)$ where the impact parameter $b = D_l\theta$:

$$\beta = \theta - \frac{D_{ls}}{D_s}\frac{4GM}{c^2 D_l \theta}$$

For a perfectly aligned source ($\beta = 0$), the image forms a ring at angle $\theta_E$:

$$0 = \theta_E - \frac{4GM}{c^2}\frac{D_{ls}}{D_l D_s}\frac{1}{\theta_E} \quad \Rightarrow \quad \theta_E^2 = \frac{4GM}{c^2}\frac{D_{ls}}{D_l D_s}$$

For $\beta \neq 0$, the lens equation becomes quadratic: $\theta^2 - \beta\theta - \theta_E^2 = 0$, giving two images at $\theta_\pm = (\beta \pm \sqrt{\beta^2 + 4\theta_E^2})/2$.

Convergence and Critical Surface Density

The lensing convergence $\kappa$ measures the projected surface mass density relative to the critical value:

$$\kappa(\vec{\theta}) = \frac{\Sigma(\vec{\theta})}{\Sigma_{\text{crit}}}, \quad \Sigma_{\text{crit}} = \frac{c^2}{4\pi G}\frac{D_s}{D_d D_{ds}}$$

Strong lensing occurs where $\kappa \geq 1$, producing multiple images, arcs, and Einstein rings.

Weak Lensing

Statistical distortions of background galaxy shapes map the dark matter distribution. The reduced shear is:

$$g = \frac{\gamma}{1 - \kappa}, \quad \gamma = \gamma_1 + i\gamma_2$$

The shear power spectrum probes the matter power spectrum along the line of sight:

$$C_\ell^{\gamma\gamma} = \int_0^{\chi_H} d\chi \frac{W^2(\chi)}{\chi^2} P_\delta\!\left(\frac{\ell}{\chi}, z(\chi)\right)$$

where $W(\chi)$ is the lensing kernel and $P_\delta$ is the 3D matter power spectrum. This is the basis for cosmic shear surveys (DES, KiDS, HSC, Euclid, Rubin).

The Bullet Cluster

The Bullet Cluster (1E 0657-56) provides direct evidence for dark matter as a particle. In this merging cluster system:

  • • The hot intracluster gas (visible in X-rays) was slowed by ram pressure during the collision
  • • The galaxies, being collisionless, passed through each other
  • • Weak lensing mass reconstruction shows the mass peaks are co-located with the galaxies, not the gas
  • • This spatial offset between baryonic mass (gas) and total mass (lensing) cannot be explained by modified gravity

The Bullet Cluster constrains the dark matter self-interaction cross section to $\sigma/m < 1.25 \text{ cm}^2\text{g}^{-1}$.

Microlensing

Compact dark matter objects (MACHOs, PBHs) can be detected via microlensing of background stars. The optical depth is:

$$\tau = \frac{4\pi G}{c^2}\int_0^{D_s} \rho(D_d) \frac{D_d(D_s - D_d)}{D_s}\,dD_d$$

EROS and MACHO surveys found far fewer microlensing events than expected if the halo were composed of compact objects in the range $10^{-7} - 10 \, M_\odot$, ruling out MACHOs as the dominant dark matter component.

5. CMB & Large-Scale Structure

CMB Acoustic Peaks

The CMB power spectrum encodes the baryon-photon-dark matter interactions before recombination. The angular power spectrum of temperature anisotropies:

$$C_\ell = \frac{2}{\pi}\int_0^\infty k^2 dk\, |\Theta_\ell(k)|^2 P_\mathcal{R}(k)$$

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$\Omega_m h^2 = 0.1430 \pm 0.0011$ (Planck 2018). The difference requires non-baryonic dark matter:

$$\Omega_{\text{DM}} h^2 = \Omega_m h^2 - \Omega_b h^2 = 0.1200 \pm 0.0012$$

How Dark Matter Affects the CMB

  • • Third peak enhancement: Dark matter deepens gravitational potential wells, boosting the third acoustic peak relative to the second — a signature of CDM
  • • Silk damping: Increasing $\Omega_m$ shifts the damping tail to smaller scales
  • • ISW effect: Dark matter controls when dark energy begins to dominate, affecting the ISW plateau at low $\ell$
  • • Lensing of the CMB: Dark matter structures along the line of sight smooth the peaks at high $\ell$

Matter Power Spectrum

Dark matter provides the gravitational scaffolding for structure formation. The matter power spectrum:

$$P(k) = A_s k^{n_s} T^2(k) D^2(z)$$

where $T(k)$ is the transfer function encoding the processing of perturbations through the radiation era,$D(z)$ is the growth factor, and $n_s \approx 0.965$ is the spectral index. The turnover scale corresponds to the horizon size at matter-radiation equality:

$$k_{\text{eq}} = a_{\text{eq}} H_{\text{eq}} \approx 0.01 \, h \, \text{Mpc}^{-1}$$

Baryon Acoustic Oscillations (BAO)

Before recombination, baryons and photons formed a tightly coupled fluid. Acoustic waves propagated outward from initial overdensities at the sound speed:

$$c_s = \frac{c}{\sqrt{3(1+R)}}, \quad R = \frac{3\rho_b}{4\rho_\gamma}$$

At recombination, the sound horizon $r_s \approx 147$ Mpc provides a standard ruler. Dark matter, being decoupled from photons, remained at the center of the overdensity while baryons were carried outward, creating the BAO peak in the correlation function at $\sim 150$ Mpc.

Growth of Perturbations

Dark matter perturbations begin growing at matter-radiation equality. In the linear regime:

$$\ddot{\delta} + 2H\dot{\delta} = 4\pi G \bar{\rho}\delta$$

Derivation: Linear Perturbation Growth

Start from the continuity, Euler, and Poisson equations for a pressureless fluid:

$$\frac{\partial\rho}{\partial t} + \nabla\cdot(\rho\vec{v}) = 0, \quad \frac{\partial\vec{v}}{\partial t} + (\vec{v}\cdot\nabla)\vec{v} = -\nabla\Phi, \quad \nabla^2\Phi = 4\pi G\rho$$

Write $\rho = \bar{\rho}(1+\delta)$, linearize ($\delta \ll 1$), and transform to comoving coordinates:

$$\ddot{\delta}_k + 2H\dot{\delta}_k - 4\pi G\bar{\rho}\delta_k = 0$$

During matter domination ($a \propto t^{2/3}$, $H = 2/(3t)$, $4\pi G\bar{\rho} = 2/(3t^2)$), try $\delta \propto t^n$:

$$n(n-1)t^{n-2} + \frac{4}{3t}nt^{n-1} - \frac{2}{3t^2}t^n = 0 \quad \Rightarrow \quad n^2 + \frac{1}{3}n - \frac{2}{3} = 0$$

Solutions: $n = 2/3$ (growing mode, $\delta_+ \propto t^{2/3} \propto a$) and$n = -1$ (decaying mode, $\delta_- \propto t^{-1}$). During radiation domination, subhorizon DM perturbations experience the Mészáros effect: growth is suppressed to logarithmic.

During matter domination, the growing mode is $\delta \propto a(t) \propto t^{2/3}$. Without dark matter to start growing perturbations before recombination, baryons alone would produce$\delta \sim 10^{-3}$ at $z=0$ — far too small for the observed structures. Dark matter perturbations grow from $z_{\text{eq}} \sim 3400$, giving a factor of $\sim 3400$ in growth before today.

6. Big Bang Nucleosynthesis

BBN occurred between $t \sim 1$ s and $t \sim 3$ min after the Big Bang ($T \sim 1$ MeV to $T \sim 0.1$ MeV). The primordial abundances of light elements depend sensitively on the baryon-to-photon ratio:

$$\eta_{10} \equiv 10^{10}\frac{n_b}{n_\gamma} = 273.9 \, \Omega_b h^2 = 6.09 \pm 0.06$$

The Neutron-to-Proton Ratio

Before freeze-out, weak interactions maintain equilibrium:

$$\frac{n}{p} = \exp\left(-\frac{Q}{k_BT}\right), \quad Q = m_n - m_p = 1.293 \text{ MeV}$$

Weak interactions freeze out at $T_f \approx 0.8$ MeV, giving $n/p \approx 1/6$. After neutron decay before nucleosynthesis begins, this drops to $n/p \approx 1/7$.

Primordial Helium Abundance

Nearly all neutrons end up in $^4$He:

$$Y_p = \frac{2(n/p)}{1 + (n/p)} \approx \frac{2 \times 1/7}{1 + 1/7} = 0.25$$

Observed primordial helium abundance: $Y_p = 0.2449 \pm 0.0040$, in excellent agreement.

BBN and Dark Matter

The concordance between BBN-predicted and observed abundances of D, $^3$He, $^4$He, and $^7$Li constrains:

$$\Omega_b h^2 = 0.0224 \pm 0.0001$$

Since the total matter density from dynamics and CMB gives $\Omega_m h^2 \approx 0.143$, the dark matter component must be non-baryonic:

$$\Omega_{\text{DM}} = \Omega_m - \Omega_b \approx 0.27 - 0.05 = 0.22$$

This is one of the strongest arguments that dark matter cannot consist of ordinary baryonic matter (gas, dust, brown dwarfs, planets, etc.).

7. Thermal Relic Freeze-Out

In the early universe, dark matter particles were in thermal equilibrium with the plasma through annihilation and creation processes $\chi\bar{\chi} \leftrightarrow f\bar{f}$. As the universe cooled, the annihilation rate dropped below the expansion rate, and the dark matter "froze out."

The Boltzmann Equation

The evolution of the dark matter number density is governed by:

$$\frac{dn_\chi}{dt} + 3Hn_\chi = -\langle\sigma v\rangle\left(n_\chi^2 - n_{\chi,\text{eq}}^2\right)$$

Derivation: From Number Density to Yield

Define the yield $Y \equiv n_\chi/s$ where $s$ is the entropy density. Since $sa^3 = \text{const}$ (entropy conservation):

$$\dot{n} + 3Hn = s\dot{Y} \quad \text{(using } \dot{s} + 3Hs = 0\text{)}$$

The Boltzmann equation becomes $s\dot{Y} = -\langle\sigma v\rangle s^2(Y^2 - Y_{\text{eq}}^2)$. Change variables from $t$ to $x = m_\chi/T$. Since $T \propto 1/a$ during radiation domination, $dx/dt = Hx$:

$$\frac{dY}{dx} = -\frac{s\langle\sigma v\rangle}{Hx}\left(Y^2 - Y_{\text{eq}}^2\right) \equiv -\frac{\lambda}{x^2}\left(Y^2 - Y_{\text{eq}}^2\right)$$

where $\lambda = m_\chi^3 \langle\sigma v\rangle \sqrt{g_*/90}\, M_{\text{Pl}} / (2\pi^{5/2} g_{*s})$ is dimensionless. This ODE is stiff near freeze-out and requires implicit integrators.

Freeze-Out Temperature

Freeze-out occurs when $\Gamma = n_\chi\langle\sigma v\rangle \sim H$. The freeze-out temperature is approximately:

$$x_f = \frac{m_\chi}{T_f} \approx 20-25$$

This is remarkably insensitive to the dark matter mass. For a 100 GeV WIMP, freeze-out occurs at$T_f \approx 4-5$ GeV.

The WIMP Miracle

The relic abundance for s-wave annihilation:

$$\Omega_\chi h^2 \approx \frac{3 \times 10^{-27} \text{ cm}^3\text{s}^{-1}}{\langle\sigma v\rangle} \approx \frac{0.1 \text{ pb} \cdot c}{\langle\sigma v\rangle}$$

Derivation: Relic Abundance Formula

After freeze-out ($x > x_f$), $Y \gg Y_{\text{eq}}$, so $dY/dx \approx -\lambda Y^2/x^2$. Integrating:

$$\int_{Y_f}^{Y_\infty} \frac{dY}{Y^2} = -\lambda\int_{x_f}^{\infty}\frac{dx}{x^2} \quad \Rightarrow \quad \frac{1}{Y_\infty} - \frac{1}{Y_f} = \frac{\lambda}{x_f}$$

Since $Y_\infty \ll Y_f$, we get $Y_\infty \approx x_f/\lambda$. The present energy density:

$$\rho_\chi = m_\chi n_\chi = m_\chi s_0 Y_\infty = m_\chi s_0 \frac{x_f}{\lambda}$$

Substituting $\lambda$ and using $s_0 = 2891$ cm⁻³, $\rho_c/h^2 = 1.054 \times 10^4$ eV/cm³, and $x_f \approx 20$: $\Omega_\chi h^2 \approx 3 \times 10^{-27} \text{cm}^3\text{s}^{-1}/\langle\sigma v\rangle$.

For a weak-scale cross section $\langle\sigma v\rangle \sim \alpha_W^2/(100 \text{ GeV})^2 \sim 3 \times 10^{-26} \text{ cm}^3\text{s}^{-1}$, this naturally gives $\Omega_\chi h^2 \approx 0.1$ — the observed dark matter density. This numerical coincidence is the "WIMP miracle" and motivated decades of WIMP searches.

Beyond Thermal WIMPs

  • • Co-annihilation: If another particle has mass close to $m_\chi$, it participates in freeze-out, modifying the effective cross section
  • • Sommerfeld enhancement: Long-range forces boost the annihilation cross section at low velocities: $\langle\sigma v\rangle \propto 1/v$
  • • Forbidden channels: Endothermic reactions that open only at high $T$
  • • Secluded annihilation: $\chi\chi \to \phi\phi$ where $\phi$ later decays to SM particles
  • • Freeze-in (FIMPs): Particles never reach equilibrium; produced slowly from SM bath, with relic density $\Omega \propto \langle\sigma v\rangle$ (opposite to freeze-out)

Simulation: Thermal Relic Freeze-Out (Fortran)

This Fortran program computes the freeze-out temperature and relic abundance for a thermal WIMP, scanning over different masses with the canonical annihilation cross section:

Fortran
program.f9051 lines

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

8. Dark Matter Candidates

WIMPs (Weakly Interacting Massive Particles)

Mass range: 1 GeV – 100 TeV. The canonical candidate from thermal freeze-out. Well-motivated by supersymmetry (neutralino $\tilde{\chi}^0_1$), extra dimensions (Kaluza-Klein dark matter), and other BSM scenarios.

$$\sigma_{\text{SI}} \sim \frac{g^4}{16\pi^2 m_{\text{mediator}}^4}m_\chi^2 m_N^2 \sim 10^{-46} - 10^{-48} \text{ cm}^2$$

The spin-independent (SI) cross section is the primary target for direct detection experiments. Current limits from LZ: $\sigma_{\text{SI}} < 9.2 \times 10^{-48} \text{ cm}^2$ at $m_\chi = 36$ GeV.

Axions

Originally proposed to solve the strong CP problem in QCD. The Peccei-Quinn symmetry breaking scale $f_a$determines the axion mass:

$$m_a = \frac{f_\pi m_\pi}{f_a}\frac{\sqrt{m_u m_d}}{m_u + m_d} \approx 6 \times 10^{-6} \text{ eV} \left(\frac{10^{12} \text{ GeV}}{f_a}\right)$$

Axions are produced non-thermally via:

  • • Vacuum realignment: the axion field oscillates coherently when $m_a(T) \sim H$, giving $\Omega_a h^2 \sim (f_a/10^{12} \text{ GeV})^{7/6}$
  • • String decay: if PQ symmetry breaks after inflation, axion strings form and radiate axions
  • • Domain wall decay: contributes if $N_{\text{DW}} = 1$

Detection principle: axion conversion in a strong magnetic field via the Primakoff effect:

$$\mathcal{L}_{a\gamma\gamma} = -\frac{g_{a\gamma\gamma}}{4}F_{\mu\nu}\tilde{F}^{\mu\nu}a = g_{a\gamma\gamma}\vec{E}\cdot\vec{B}\,a$$

Experiments: ADMX, HAYSTAC, ABRACADABRA, CASPEr, MADMAX, IAXO.

Sterile Neutrinos

Right-handed neutrinos with mass in the keV range are warm dark matter candidates. They mix with active neutrinos:

$$\nu_\alpha = \cos\theta\,\nu_{\text{active}} + \sin\theta\,\nu_s$$

Production mechanisms:

  • • Dodelson-Widrow: Non-resonant production from active neutrino oscillations, $\Omega_s \propto \sin^2(2\theta)\,m_s^5$
  • • Shi-Fuller: Resonant production in the presence of a lepton asymmetry, allowing smaller mixing angles

Decay signature: $\nu_s \to \nu_\alpha + \gamma$ produces an X-ray line at $E = m_s/2$. The claimed 3.5 keV line from galaxy clusters (2014) remains unconfirmed.

Primordial Black Holes (PBHs)

Formed from large density fluctuations in the early universe. The mass at formation is approximately:

$$M_{\text{PBH}} \sim \frac{c^3 t}{G} \sim M_\odot \left(\frac{t}{10^{-5} \text{ s}}\right)$$

Constraints on PBH abundance $f_{\text{PBH}} = \Omega_{\text{PBH}}/\Omega_{\text{DM}}$:

  • • $M < 10^{15}$ g: Hawking evaporation (extragalactic gamma-ray background)
  • • $10^{-11} - 10^{-7}\,M_\odot$: Femtolensing of GRBs and microlensing surveys
  • • $10^{-7} - 10\,M_\odot$: EROS/OGLE microlensing of Magellanic Clouds
  • • $10 - 100\,M_\odot$: CMB spectral distortions, dynamical constraints, LIGO merger rates
  • • $> 10^3\,M_\odot$: Dynamical friction in dwarf galaxies, wide binary disruption

Gravitinos

The supersymmetric partner of the graviton. Mass set by the SUSY-breaking scale:

$$m_{3/2} = \frac{F}{\sqrt{3}M_{\text{Pl}}}$$

If the gravitino is the LSP (lightest supersymmetric particle), it is stable and its abundance depends on the reheating temperature: $\Omega_{3/2}h^2 \propto T_R$. This constrains $T_R \lesssim 10^{9-10}$ GeV to avoid overclosure.

Asymmetric Dark Matter

Motivated by the coincidence $\Omega_{\text{DM}} \sim 5\Omega_b$. If dark matter carries a conserved quantum number analogous to baryon number, the dark matter density is set by an asymmetry rather than an annihilation cross section:

$$\frac{\Omega_{\text{DM}}}{\Omega_b} = \frac{m_\chi}{m_p}\frac{\eta_\chi}{\eta_b} \approx 5 \quad \Rightarrow \quad m_\chi \approx 5\,\text{GeV}$$

This predicts a dark matter mass near 5 GeV and that dark matter annihilation in the present universe is suppressed (no symmetric component left).

9. Direct Detection Experiments

Direct detection searches for the nuclear recoil produced by WIMP-nucleus elastic scattering. The local dark matter density is approximately:

$$\rho_0 \approx 0.3 \text{ GeV/cm}^3 \approx 0.008 \, M_\odot/\text{pc}^3$$

Kinematics

The recoil energy for elastic scattering:

$$E_R = \frac{\mu^2 v^2}{m_N}(1 - \cos\theta_{\text{CM}}), \quad \mu = \frac{m_\chi m_N}{m_\chi + m_N}$$

Maximum recoil: $E_R^{\max} = 2\mu^2 v^2/m_N$. For a 100 GeV WIMP on xenon ($A=131$) with $v = 220$ km/s: $E_R^{\max} \approx 50$ keV — requiring very low-threshold detectors.

Differential Event Rate

$$\frac{dR}{dE_R} = \frac{\rho_0}{m_\chi m_N}\frac{\sigma_0}{2\mu^2} F^2(E_R) \int_{v_{\min}}^{v_{\text{esc}}} \frac{f(\vec{v})}{v}\,d^3v$$

Derivation: Differential Event Rate

The number of scattering events per unit time, per unit detector mass, per unit recoil energy is:

$$\frac{dR}{dE_R} = N_T \int_{v > v_{\min}} \frac{d\sigma}{dE_R}(v, E_R)\, v\, \frac{\rho_0}{m_\chi}f(\vec{v})\,d^3v$$

For elastic scattering, the differential cross section is constant in $E_R$ (isotropic in CM):

$$\frac{d\sigma}{dE_R} = \frac{m_N \sigma_0}{2\mu^2 v^2}F^2(E_R)$$

where $\sigma_0$ is the zero-momentum cross section. The minimum velocity to produce recoil $E_R$:

$$E_R^{\max} = \frac{2\mu^2 v^2}{m_N} \quad \Rightarrow \quad v_{\min}(E_R) = \sqrt{\frac{m_N E_R}{2\mu^2}}$$

Substituting and integrating: $dR/dE_R = (N_T \rho_0 \sigma_0)/(2m_\chi \mu^2)\,F^2(E_R)\,\eta(v_{\min})$where $\eta(v_{\min}) = \int_{v > v_{\min}} f(\vec{v})/v\,d^3v$ is the mean inverse speed.

Spin-Independent vs Spin-Dependent

The WIMP-nucleus cross section has two contributions:

$$\sigma_{\text{SI}} \propto A^2 \mu^2 \quad \text{(coherent, scales as } A^2\text{)}$$
$$\sigma_{\text{SD}} \propto J(J+1)\mu^2 \quad \text{(spin-dependent, scales with nuclear spin)}$$

Heavy targets (Xe, Ge, W) are optimal for SI searches due to the $A^2$ enhancement. Odd-nucleon targets ($^{19}$F, $^{127}$I, $^{73}$Ge) are needed for SD searches.

Annual Modulation

Earth's orbital motion around the Sun modulates the WIMP flux with a period of one year:

$$v_{\text{lab}}(t) = v_\odot + v_\oplus \cos\gamma \cos\left[\omega(t - t_0)\right]$$

where $v_\odot \approx 232$ km/s, $v_\oplus \approx 30$ km/s, and $t_0 \approx$ June 2nd. DAMA/LIBRA has observed a modulation signal at $12.9\sigma$ for over 20 years, but this result is in strong tension with null results from other experiments.

Leading Experiments

  • • LUX-ZEPLIN (LZ): 7-tonne active mass liquid Xe TPC in the Sanford Underground Research Facility. World-leading SI limit: $\sigma < 9.2 \times 10^{-48}$ cm² at 36 GeV
  • • XENONnT: 5.9-tonne LXe at LNGS. Competitive SI limits and electron-recoil searches for light mediators
  • • PandaX-4T: 3.7-tonne LXe at CJPL, China. Independent confirmation of LZ/XENON results
  • • SuperCDMS: Cryogenic Ge and Si detectors at SNOLAB. Optimized for sub-GeV dark matter via phonon detection
  • • CRESST-III: CaWO₄ scintillating bolometers. Very low threshold ($\sim 30$ eV), sensitive to sub-GeV WIMPs
  • • DARWIN/XLZD: Next-generation 50-tonne LXe detector. Will reach the neutrino fog ($\sigma \sim 10^{-49}$ cm²)

The Neutrino Fog

At $\sigma_{\text{SI}} \sim 10^{-49}$ cm², coherent neutrino-nucleus scattering (CE$\nu$NS) from solar, atmospheric, and diffuse supernova neutrinos becomes an irreducible background:

$$\frac{d\sigma_{\nu}}{dE_R} = \frac{G_F^2 Q_W^2 m_N}{4\pi}\left(1 - \frac{m_N E_R}{2E_\nu^2}\right)$$

where $Q_W = N - (1-4\sin^2\theta_W)Z$ is the weak nuclear charge. Beyond the neutrino fog, directional detection (tracking the recoil direction) or timing information is needed to distinguish WIMPs from neutrinos.

10. Indirect Detection

Dark matter annihilation or decay in astrophysical environments produces Standard Model particles. The annihilation rate per volume:

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

The factor of 1/2 applies for self-conjugate (Majorana) dark matter; for Dirac particles it is 1/4.

Gamma Rays

The differential photon flux from dark matter annihilation:

$$\frac{d\Phi_\gamma}{dE_\gamma} = \frac{\langle\sigma v\rangle}{8\pi m_\chi^2}\sum_f \text{Br}_f\frac{dN_\gamma^f}{dE_\gamma} \times \underbrace{\int_{\Delta\Omega}d\Omega\int_{\text{l.o.s}}\rho^2(r)\,dl}_{J\text{-factor}}$$

The J-factor quantifies the astrophysical contribution. Prime targets:

  • • Galactic Center: Highest J-factor ($\sim 10^{23-25}$ GeV²/cm⁵) but complex astrophysical backgrounds
  • • Dwarf spheroidals: Low backgrounds, well-measured J-factors from stellar kinematics ($\sim 10^{18-19}$ GeV²/cm⁵)
  • • Galaxy clusters: Large dark matter reservoirs; also searched via radio (synchrotron from DM-produced $e^\pm$)

The Galactic Center Excess

Fermi-LAT data reveals an excess of gamma rays from the inner Galaxy peaking at $\sim 2-3$ GeV, consistent with a 30-50 GeV WIMP annihilating into $b\bar{b}$ with$\langle\sigma v\rangle \sim 2 \times 10^{-26}$ cm³/s. However, the excess may also be explained by an unresolved population of millisecond pulsars. This remains one of the most debated signals.

Cosmic Ray Positrons

AMS-02 and earlier PAMELA data show a rising positron fraction above 10 GeV:

$$\frac{\Phi(e^+)}{\Phi(e^+) + \Phi(e^-)} \text{ rises from 5\% at 10 GeV to } \sim 15\% \text{ at 300 GeV}$$

While potentially from dark matter annihilation, this is more likely explained by nearby pulsars (Geminga, Monogem). Dark matter interpretations require $\langle\sigma v\rangle \gg 3 \times 10^{-26}$ cm³/s, which is in tension with other constraints.

Neutrinos

WIMPs captured in the Sun or Earth annihilate, producing neutrinos detectable by IceCube and Super-Kamiokande. The capture rate in the Sun:

$$C_\odot \approx 1.3 \times 10^{25} \text{ s}^{-1} \left(\frac{\rho_0}{0.3 \text{ GeV/cm}^3}\right)\left(\frac{\sigma_{\text{SD}}}{10^{-40} \text{ cm}^2}\right)$$

In equilibrium, the annihilation rate equals half the capture rate, making the neutrino signal proportional to the scattering cross section rather than the annihilation cross section.

11. Collider Searches

If dark matter couples to Standard Model particles, it can be produced in high-energy collisions. Dark matter particles escape the detector, appearing as missing transverse energy ($\slashed{E}_T$).

Mono-X Searches

The primary strategy at the LHC is to look for dark matter production recoiling against a visible object X:

  • • Mono-jet: $pp \to \chi\bar{\chi} + j$ (highest sensitivity)
  • • Mono-photon: $pp \to \chi\bar{\chi} + \gamma$
  • • Mono-Z/W: $pp \to \chi\bar{\chi} + Z/W$
  • • Mono-Higgs: $pp \to \chi\bar{\chi} + h$

Simplified Models

Modern collider searches use simplified models with a mediator connecting the SM to dark matter:

$$\mathcal{L} \supset g_q \bar{q}\Gamma q\, Z' + g_\chi \bar{\chi}\Gamma\chi\, Z'$$

where $Z'$ is a vector or axial-vector mediator with couplings $g_q$ to quarks and $g_\chi$ to dark matter. The mediator can be produced on-shell or off-shell depending on$m_{Z'}$ versus $\sqrt{s}$.

Complementarity

Collider, direct, and indirect detection probe different aspects of the dark matter interaction:

  • • Colliders: Production cross section, constrain mediator mass and coupling. Best for light DM ($m_\chi < m_{Z'}/2$)
  • • Direct detection: Elastic scattering cross section. Best for heavy DM with coherent nuclear coupling
  • • Indirect detection: Annihilation cross section. Best for high-density environments

For a given simplified model, constraints from all three approaches can be compared in the$(m_\chi, m_{\text{med}})$ plane, with direct detection typically dominating for $m_\chi > 10$ GeV and colliders dominating for $m_\chi < 10$ GeV.

12. Self-Interacting Dark Matter (SIDM)

SIDM models introduce significant dark matter self-scattering to address small-scale structure problems. The key parameter is the self-interaction cross section per unit mass:

$$\frac{\sigma_{\text{self}}}{m_\chi} \sim 0.1 - 10 \text{ cm}^2/\text{g} \approx 0.2 - 20 \text{ barn/GeV}$$

Physics of Self-Interaction

Self-scattering thermalizes the inner halo, converting cusps into isothermal cores. The scattering rate in the halo center:

$$\Gamma_{\text{scatter}} = \frac{\rho_0}{m_\chi}\sigma_{\text{self}}\, v_{\text{rms}} \sim \frac{1}{t_{\text{age}}}$$

For one scattering per Hubble time in dwarf galaxies ($\rho \sim 0.1$ GeV/cm³,$v \sim 30$ km/s), this requires $\sigma/m \sim 1$ cm²/g.

Velocity-Dependent Cross Sections

A light mediator $\phi$ with mass $m_\phi$ produces velocity-dependent scattering analogous to Yukawa scattering:

$$\sigma_T \propto \begin{cases} \text{const} & v \ll \alpha_\chi m_\phi/m_\chi \\ v^{-4} & v \gg \alpha_\chi m_\phi/m_\chi \end{cases}$$

This naturally gives large cross sections in dwarfs (low $v$) and small cross sections in clusters (high $v$), reconciling constraints across scales. Resonant scattering through bound states can further enhance the cross section at specific velocities.

13. Dark Sectors & Hidden Valleys

Dark matter may inhabit a rich "dark sector" with its own forces and particles, connected to the Standard Model through portal interactions.

Portal Interactions

The SM admits a limited number of renormalizable portals to a dark sector:

$$\text{Vector: } \frac{\epsilon}{2}F_{\mu\nu}F'^{\mu\nu}, \quad \text{Higgs: } \lambda |H|^2|\Phi|^2, \quad \text{Neutrino: } y_N L H N$$
  • • Vector portal (dark photon): A new U(1) gauge boson $A'$ kinetically mixed with the photon. Mass $m_{A'} \sim$ MeV–GeV, mixing $\epsilon \sim 10^{-3}-10^{-6}$
  • • Higgs portal: A scalar $\Phi$ coupling to the Higgs. Invisible Higgs decay: $h \to \chi\chi$, currently constrained to Br$(h \to \text{inv}) < 0.11$
  • • Neutrino portal: Heavy neutral leptons mixing with active neutrinos

Dark Photons

The dark photon Lagrangian:

$$\mathcal{L} = -\frac{1}{4}F'^{\mu\nu}F'_{\mu\nu} + \frac{m_{A'}^2}{2}A'_\mu A'^\mu + \frac{\epsilon}{2}F^{\mu\nu}F'_{\mu\nu} + e' A'_\mu J^\mu_{\text{dark}}$$

After diagonalization, SM particles acquire a coupling $\epsilon e$ to the dark photon. Searches include fixed-target experiments (HPS, APEX, DarkQuest), $e^+e^-$ colliders (BaBar, Belle II), beam dumps (NA62, SHiP), and direct detection experiments.

Atomic Dark Matter

If the dark sector contains both a dark "proton" and dark "electron" with a dark electromagnetic force, bound states form:

$$E_{\text{binding}} = \frac{\alpha_D^2 \mu_D}{2}, \quad a_D = \frac{1}{\alpha_D \mu_D}$$

where $\alpha_D$ is the dark fine structure constant and $\mu_D$ is the reduced mass of the dark atom. Dark atoms can cool via dark photon emission, potentially forming a dark disk in galaxies.

14. N-Body Simulations & Halo Formation

Cosmological N-body simulations follow the gravitational evolution of dark matter from linear perturbations to highly nonlinear structures. The fundamental equation solved:

$$\frac{d^2\vec{x}_i}{dt^2} = -\nabla\Phi(\vec{x}_i), \quad \nabla^2\Phi = 4\pi G a^2 (\rho - \bar{\rho})$$

Major Simulations

  • • Millennium (2005): $10^{10}$ particles in a 500 Mpc/h box. First large-scale simulation to resolve the halo mass function and clustering
  • • Bolshoi (2011): $8 \times 10^9$ particles in 250 Mpc/h, resolving substructure in Milky Way-mass halos
  • • IllustrisTNG (2018): Magneto-hydrodynamic simulation including baryons, star formation, AGN feedback. Multiple box sizes
  • • FIRE (2014–): Zoom-in simulations with resolved ISM physics. Key for studying baryonic effects on DM profiles
  • • Uchuu (2021): $2.1 \times 10^{12}$ particles in a 2 Gpc/h box — largest N-body simulation

The Halo Mass Function

The Press-Schechter formalism predicts the number density of halos of mass $M$:

$$\frac{dn}{dM} = \frac{\bar{\rho}}{M}\frac{d\ln\sigma^{-1}}{dM}f(\sigma)$$

where $\sigma(M)$ is the rms density fluctuation in spheres of mass $M$ and $f(\sigma)$ is the multiplicity function. For the original Press-Schechter:

$$f_{\text{PS}}(\sigma) = \sqrt{\frac{2}{\pi}}\frac{\delta_c}{\sigma}\exp\left(-\frac{\delta_c^2}{2\sigma^2}\right)$$

with critical overdensity $\delta_c = 1.686$. Improved fitting functions (Sheth-Tormen, Tinker) better match simulation results.

Substructure & Merger Trees

Dark matter halos form hierarchically: smaller halos merge to form larger ones. Each Milky Way-mass halo contains $\sim 10^4 - 10^5$ subhalos. The subhalo mass function:

$$\frac{dN}{dM_{\text{sub}}} \propto M_{\text{sub}}^{-\alpha}, \quad \alpha \approx 1.9$$

Tidal stripping gradually removes mass from subhalos as they orbit within the host, with disruption timescales depending on the orbital parameters and subhalo concentration.

15. Small-Scale Problems of CDM

While $\Lambda$CDM is spectacularly successful on large scales, several tensions exist on galactic and sub-galactic scales:

Core-Cusp Problem

CDM simulations predict cuspy inner profiles ($\rho \propto r^{-1}$ for NFW), while observations of dwarf galaxies and LSB galaxies favor constant-density cores ($\rho \approx \text{const}$). The inner slope $\gamma$ where $\rho \propto r^{-\gamma}$:

  • • CDM simulations: $\gamma \approx 0.8 - 1.4$
  • • Observed dwarf galaxies: $\gamma \approx 0 - 0.5$

Missing Satellites Problem

CDM predicts ~500 subhalos with $V_{\max} > 10$ km/s within a Milky Way halo, but only ~60 satellite galaxies are observed. Solutions:

  • • Reionization suppresses gas accretion in small halos ($M < 10^9 M_\odot$)
  • • Many ultra-faint dwarfs remain undiscovered (DES, LSST discovering more)
  • • Galaxy formation is stochastic in low-mass halos

Too-Big-to-Fail Problem

The most massive CDM subhalos are too dense to host the observed Milky Way satellites. These halos are "too big to fail" at forming bright galaxies, yet their predicted kinematics don't match any observed dwarf.

Diversity Problem

Galaxies of similar halo mass show a much wider range of rotation curve shapes than CDM-only simulations predict. Some galaxies at $V_{\max} \sim 80$ km/s have rising rotation curves (cores), while others have steeply rising curves (cusps).

Baryonic Solutions

Baryonic feedback processes can address many of these issues within standard CDM:

  • • Supernova feedback: Repeated gas outflows create fluctuating gravitational potential, transforming cusps into cores
  • • Dynamical friction: Gas clumps transfer energy to dark matter via dynamical friction
  • • UV background: Reionization prevents gas cooling in halos with $v_{\text{circ}} < 20$ km/s
  • • Tidal stripping: Ram pressure and tidal effects remove gas and reduce stellar masses of satellites

16. Modified Gravity Alternatives

MOND (Modified Newtonian Dynamics)

Milgrom (1983) proposed that Newton's second law is modified at low accelerations:

$$\mu\!\left(\frac{|\vec{a}|}{a_0}\right)\vec{a} = -\vec{\nabla}\Phi_N$$

where $a_0 \approx 1.2 \times 10^{-10}$ m/s² and $\mu(x) \to 1$ for $x \gg 1$,$\mu(x) \to x$ for $x \ll 1$. In the deep-MOND regime:

$$v_{\text{flat}}^4 = GMa_0 \quad \Rightarrow \quad L \propto v^4 \quad \text{(baryonic Tully-Fisher)}$$

Derivation: MOND → Baryonic Tully-Fisher

In the deep-MOND regime ($a \ll a_0$), the effective gravitational acceleration is $a = \sqrt{a_N a_0}$where $a_N = GM/r^2$ is the Newtonian acceleration:

$$\frac{v^2}{r} = a = \sqrt{\frac{GM}{r^2}\cdot a_0} = \frac{\sqrt{GMa_0}}{r}$$

Therefore $v^2 = \sqrt{GMa_0}$, which gives:

$$v^4 = GMa_0 \quad \Rightarrow \quad M = \frac{v^4}{Ga_0} \propto v^4$$

This is exactly the baryonic Tully-Fisher relation $M_b \propto v_{\text{flat}}^4$, a prediction with zero free parameters. Note that $v$ is independent of $r$ — MOND predicts flat rotation curves automatically in the deep-MOND regime.

MOND's successes: it predicts the baryonic Tully-Fisher relation, explains rotation curves of disk galaxies from baryonic mass alone with no free parameters per galaxy, and predicts the observed relation$a_{\text{obs}} = f(a_{\text{bar}})$ (the radial acceleration relation).

Relativistic Extensions

  • • TeVeS (Bekenstein 2004): Tensor-Vector-Scalar gravity. Adds a unit timelike vector field and a scalar field to GR. Largely ruled out by the gravitational wave event GW170817 (speed of gravity = speed of light)
  • • AQUAL: Modified Poisson equation: $\nabla \cdot [\mu(|\nabla\Phi|/a_0)\nabla\Phi] = 4\pi G\rho$
  • • Emergent gravity (Verlinde 2016): Dark matter effects arise from the entanglement entropy of de Sitter space
  • • Superfluid dark matter: DM behaves as a superfluid in galaxy cores, reproducing MOND, but as CDM particles in clusters

Challenges for Modified Gravity

  • • Bullet Cluster: Clear spatial separation of lensing mass from baryons — requires actual dark matter
  • • CMB acoustic peaks: Precise peak ratios require non-baryonic matter, not modified gravity
  • • Structure formation: MOND cannot reproduce the matter power spectrum without additional dark components
  • • Galaxy clusters: MOND still requires ~2× more mass than visible in clusters — "residual missing mass"
  • • Gravitational wave speed: GW170817 rules out many tensor-vector-scalar theories

17. Discovery & Evidence for Dark Energy

Dark energy is a mysterious form of energy that permeates all of space and causes the expansion of the universe to accelerate. Discovered in 1998 through observations of distant Type Ia supernovae, dark energy comprises approximately 68% of the total energy density of the universe.

$$\Omega_\Lambda \approx 0.68, \quad \Omega_m \approx 0.32, \quad \Omega_r \ll 1$$

Supernova Cosmology

Type Ia supernovae serve as standardizable candles. The distance modulus is:

$$\mu = m - M = 5\log_{10}(d_L/10\text{ pc})$$

where $d_L(z)$ is the luminosity distance. For flat $\Lambda$CDM:

$$d_L(z) = (1+z)\frac{c}{H_0}\int_0^z \frac{dz'}{\sqrt{\Omega_m(1+z')^3 + \Omega_\Lambda}}$$

Key Result (1998)

High-redshift SNe Ia appeared ~25% fainter than expected in a matter-dominated universe, indicating accelerated expansion. The deceleration parameter:

$$q_0 = -\frac{\ddot{a}a}{\dot{a}^2}\bigg|_{t=t_0} = \frac{\Omega_m}{2} - \Omega_\Lambda \approx -0.52$$

Negative $q_0$ means acceleration!

Additional Evidence

  • • CMB: Flat universe ($\Omega_{\text{tot}} = 1$) + matter density requires dark energy
  • • BAO: Standard ruler measurements confirm accelerated expansion
  • • Large-scale structure: Growth rate slower than matter-only universe
  • • Age of universe: Old globular clusters require dark energy for consistency

18. The Cosmological Constant

Einstein Field Equations with $\Lambda$

$$R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}$$

The cosmological constant $\Lambda$ has dimensions of (length)$^{-2}$. In terms of energy density:

$$\rho_\Lambda = \frac{\Lambda c^2}{8\pi G} \approx 6 \times 10^{-10} \text{ J/m}^3$$

Equation of State

The cosmological constant corresponds to a perfect fluid with equation of state:

$$w = \frac{p}{\rho} = -1$$

This negative pressure drives acceleration. The Friedmann equation becomes:

$$H^2 = \frac{8\pi G}{3}\rho + \frac{\Lambda}{3} - \frac{k}{a^2}$$

Vacuum Energy Interpretation

In quantum field theory, the vacuum has energy density from zero-point fluctuations:

$$\rho_{\text{vac}} = \sum_k \frac{1}{2}\hbar\omega_k = \int_0^{\Lambda_{\text{cutoff}}} \frac{k^3dk}{2\pi^2}\hbar\omega_k$$

With Planck-scale cutoff $\Lambda_{\text{cutoff}} \sim \ell_P^{-1}$:

$$\rho_{\text{vac}} \sim \frac{\hbar c}{\ell_P^4} \sim 10^{113} \text{ J/m}^3$$

This is 122 orders of magnitude larger than the observed dark energy density!

19. The Cosmological Constant Problem

Why is the observed vacuum energy density so incredibly small compared to theoretical predictions? This is considered one of the worst fine-tuning problems in physics.

The Fine-Tuning

$$\frac{\rho_{\Lambda,\text{obs}}}{\rho_{\text{Planck}}} \sim 10^{-123}$$

Every contribution to the vacuum energy (QED, QCD, electroweak, etc.) must cancel to ~120 decimal places.

The Coincidence Problem

Why is $\rho_\Lambda \sim \rho_m$ today? Matter density evolves as $\rho_m \propto a^{-3}$while $\rho_\Lambda = \text{const}$:

$$\frac{\rho_\Lambda}{\rho_m} = \frac{\Omega_\Lambda}{\Omega_m}(1+z)^3 \sim 2(1+z)^3$$

We happen to live at the special epoch when these are comparable. At earlier times, matter dominated; at later times, dark energy will dominate completely.

Proposed Solutions

  • • Anthropic principle: Only universes with small $\Lambda$ form structures and observers
  • • Supersymmetry: Boson/fermion contributions cancel (but SUSY must be broken)
  • • Multiverse: Landscape of vacua with different $\Lambda$ values
  • • Modified gravity: Dark energy is geometric, not material

20. Dynamical Dark Energy Models

Equation of State Parameter w(z)

Generalize beyond $w = -1$:

$$w(z) = \frac{p_{DE}(z)}{\rho_{DE}(z)}$$

Common parameterizations (CPL):

$$w(a) = w_0 + w_a(1-a) = w_0 + w_a\frac{z}{1+z}$$

Current constraints (Planck + SNe + BAO): $w_0 = -1.03 \pm 0.03$, $w_a = -0.3^{+0.5}_{-0.7}$

Quintessence

A dynamical scalar field $\phi$ with potential $V(\phi)$. The action is:

$$S = \int d^4x\sqrt{-g}\left[\frac{1}{2}\partial_\mu\phi\partial^\mu\phi - V(\phi)\right]$$

Energy density and pressure:

$$\rho_\phi = \frac{1}{2}\dot{\phi}^2 + V(\phi), \quad p_\phi = \frac{1}{2}\dot{\phi}^2 - V(\phi)$$

Equation of state:

$$w_\phi = \frac{\frac{1}{2}\dot{\phi}^2 - V(\phi)}{\frac{1}{2}\dot{\phi}^2 + V(\phi)}$$

For $w \approx -1$, need $\dot{\phi}^2 \ll V(\phi)$ (slow-roll).

Example Potentials

$$V(\phi) = M^4\left(\frac{\phi}{\phi_0}\right)^{-\alpha} \quad \text{(Inverse power-law)}$$
$$V(\phi) = M^4 e^{-\lambda\phi/M_P} \quad \text{(Exponential)}$$
$$V(\phi) = M^4\left[1 + \cos\left(\frac{\phi}{f}\right)\right] \quad \text{(Pseudo-Nambu-Goldstone)}$$

K-essence

Non-canonical kinetic term:

$$\mathcal{L} = K(X, \phi), \quad X = -\frac{1}{2}\partial_\mu\phi\partial^\mu\phi$$

Phantom Energy

Models with $w < -1$. Lead to "Big Rip" singularity at finite time:

$$a(t) \to \infty \text{ as } t \to t_{\text{rip}} < \infty$$

All bound structures (galaxies, stars, atoms) eventually torn apart.

Simulation: Dark Energy Equation of State & Hubble Diagram

This simulation compares the Hubble diagram (distance modulus vs. redshift) for different dark energy models and shows the evolution of the dark energy equation of state parameter $w(z)$:

Python
script.py71 lines

Click Run to execute the Python code

Code will be executed with Python 3 on the server

21. Observational Constraints on Dark Energy

Distance Measurements

  • • SNe Ia: Probe $d_L(z)$ out to $z \sim 2$
  • • BAO: Standard ruler at sound horizon scale $r_s \approx 150$ Mpc
  • • CMB: Angular diameter distance to last scattering $z \sim 1100$

Growth Rate Measurements

The growth rate of structure:

$$f(z) = \frac{d\ln\delta}{d\ln a} \approx \Omega_m(z)^\gamma$$

where $\gamma \approx 0.55$ for $\Lambda$CDM. Modified gravity models predict different $\gamma$.

Current Best Fit (Planck 2018)

$$\Omega_\Lambda = 0.6847 \pm 0.0073$$
$$w = -1.028 \pm 0.031 \quad \text{(constant } w \text{)}$$

Remarkably consistent with $w = -1$ (cosmological constant).

Modified Gravity as Dark Energy

f(R) gravity generalizes the Einstein-Hilbert action:

$$S = \frac{1}{16\pi G}\int d^4x\sqrt{-g}\,f(R)$$

The DGP braneworld model provides self-acceleration without dark energy:

$$H^2 = \frac{8\pi G}{3}\rho + \frac{H}{r_c}$$

where $r_c$ is the crossover scale. Gravitational wave speed measurements (GW170817) strongly constrain many modified gravity alternatives.

22. The Future of the Universe

Scenario 1: $\Lambda$CDM ($w = -1$)

Exponential expansion continues forever:

$$a(t) \propto e^{H_\Lambda t}, \quad H_\Lambda = \sqrt{\frac{\Lambda}{3}}$$

All distant galaxies eventually recede beyond the cosmic horizon. The universe becomes cold, dark, and empty. "Heat death" on a timescale of ~10$^{100}$ years.

Scenario 2: Phantom Energy ($w < -1$)

Big Rip singularity at finite time:

$$t_{\text{rip}} - t_0 = \frac{2}{3|1+w|H_0\sqrt{\Omega_{DE}}}$$

For $w = -1.5$, the rip occurs in ~20 billion years. Galaxies are torn apart, then stars, planets, atoms, and nuclei.

Scenario 3: Quintessence ($-1 < w < -1/3$)

Accelerated expansion but slower than $\Lambda$CDM. Dark energy density may decrease over time.

Event Horizon

In an accelerating universe, there is a cosmological event horizon at comoving distance:

$$r_h = a\int_t^\infty \frac{dt'}{a(t')} = \int_0^{a_{\max}/a_0} \frac{da'}{a'H(a')}$$

For de Sitter space: $r_h = c/H_\Lambda \approx 16$ Gly. We can never observe events beyond this!

23. Current Status & Future Prospects

Evidence Summary

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

  • • Galactic scales: rotation curves, velocity dispersions, satellite dynamics, stellar streams
  • • Cluster scales: gravitational lensing (strong + weak), X-ray gas temperatures, galaxy dynamics, SZ effect
  • • Cosmological scales: CMB power spectrum, BAO, Lyman-$\alpha$ forest, type Ia supernovae, cosmic shear

The $S_8$ Tension

A growing discrepancy between CMB-derived and weak-lensing-derived measurements of the clustering amplitude:

$$S_8 \equiv \sigma_8\sqrt{\Omega_m/0.3}$$
  • • Planck CMB: $S_8 = 0.832 \pm 0.013$
  • • DES Y3 + KiDS-1000: $S_8 \approx 0.76 \pm 0.02$

This 2-3$\sigma$ tension could indicate new dark matter physics (e.g., DM-neutrino interactions, decaying DM, warm DM), baryonic feedback effects, or systematic errors in the surveys.

Open Questions

  • • What is the fundamental nature of dark matter?
  • • Why is $\Omega_{\text{DM}} \sim 5\Omega_b$ (cosmic coincidence)?
  • • Does dark matter have non-gravitational interactions?
  • • Is there a dark sector with multiple particles and forces?
  • • Can small-scale problems be fully resolved by baryonic physics?
  • • What produced the dark matter (thermal freeze-out, freeze-in, asymmetric, non-thermal)?
  • • Is there a connection between dark matter and the matter-antimatter asymmetry?

Next-Generation Experiments

Direct Detection

  • • DARWIN/XLZD (50t LXe, neutrino fog)
  • • SuperCDMS SNOLAB (sub-GeV)
  • • NEWS-G (light DM with SPC)
  • • SENSEI, DAMIC-M (electron recoils)

Indirect Detection

  • • CTA (TeV gamma rays)
  • • IceCube-Gen2 (high-E neutrinos)
  • • SWGO (Southern Wide-field Gamma-ray Observatory)
  • • e-ASTROGAM (MeV gamma rays)

Collider & Axion

  • • HL-LHC (Run 4+, higher luminosity)
  • • FCC-hh (100 TeV, future)
  • • ADMX-G2, MADMAX, IAXO (axions)
  • • ABRACADABRA, CASPEr (ultra-light axions)

Cosmological Surveys

  • • Euclid (weak lensing + BAO)
  • • Vera C. Rubin LSST (deep sky survey)
  • • DESI (BAO + RSD)
  • • CMB-S4 (small-scale CMB + lensing)

Open Questions in Dark Energy

  • • Is dark energy truly a cosmological constant, or does $w$ evolve?
  • • Why is $\rho_\Lambda$ so small yet non-zero (cosmological constant problem)?
  • • Why does dark energy dominate now (coincidence problem)?
  • • Is dark energy related to quantum vacuum energy?
  • • Could dark energy be explained by modified gravity?

Dark Energy Experiments

  • • Euclid (ESA): 3D galaxy survey, weak lensing, BAO out to $z \sim 2$
  • • LSST/Vera Rubin: 10-year survey, billions of galaxies, improved SNe statistics
  • • DESI: Spectroscopic survey, precise BAO and growth rate measurements
  • • CMB-S4: Next-generation CMB polarization, improved lensing
  • • Roman Space Telescope: High-precision SNe Ia, weak lensing, BAO

Goal: Measure $w(z)$ to ~1% precision and $w_a$ to ~10%, discriminating between models.

Key References

  • • Bertone, Hooper & Silk, "Particle Dark Matter: Evidence, Candidates and Constraints", Phys. Rept. 405 (2005)
  • • Jungman, Kamionkowski & Griest, "Supersymmetric Dark Matter", Phys. Rept. 267 (1996)
  • • Bullock & Boylan-Kolchin, "Small-Scale Challenges to the ΛCDM Paradigm", ARAA 55 (2017)
  • • Tulin & Yu, "Dark Matter Self-interactions and Small Scale Structure", Phys. Rept. 730 (2018)
  • • Planck Collaboration, "Planck 2018 Results: VI. Cosmological Parameters", A&A 641 (2020)
  • • Schumann, "Direct Detection of WIMP Dark Matter: Concepts and Status", JPG 46 (2019)
  • • Perlmutter et al., "Measurements of Omega and Lambda from 42 High-Redshift Supernovae", ApJ 517 (1999)
  • • Riess et al., "Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant", AJ 116 (1998)
  • • Weinberg, "The Cosmological Constant Problem", Rev. Mod. Phys. 61 (1989)
  • • Copeland, Sami & Tsujikawa, "Dynamics of Dark Energy", Int. J. Mod. Phys. D 15 (2006)

Related Courses

Cosmology

The cosmic context: dark matter's role in structure formation and the expanding universe

Particle Physics

Dark matter candidates: WIMPs, axions, and physics beyond the Standard Model

General Relativity

Gravitational lensing, galaxy dynamics, and the mass-energy tensor

Observational Cosmology

CMB, galaxy surveys, weak lensing, and direct detection experiments

Nuclear & Particle Astrophysics

Big Bang nucleosynthesis and the non-baryonic nature of dark matter

Quantum Field Theory

Thermal field theory, freeze-out calculations, and dark sector models