Spotlight #38 – Cosmology & Dark Matter: CMB, Inflation, N-Body Simulations and Large-Scale Structure

Modern cosmology rests on three observational pillars: the cosmic microwave background (CMB), the large-scale distribution of galaxies, and the abundance of light elements from Big Bang nucleosynthesis. Together they support the ΛCDM (Lambda Cold Dark Matter) model — a universe 13.8 Gyr old, flat to within 0.2%, composed of 5% baryons, 27% cold dark matter, and 68% dark energy. This post examines the physics behind each constraint.

1. Inflationary Cosmology

The standard Hot Big Bang model faces three fine-tuning problems: the horizon problem (why is the CMB temperature uniform to 1 part in 10⁵ across regions causally disconnected at recombination?), the flatness problem (why is the spatial curvature parameter |Ω_k| < 0.002?), and the magnetic monopole problem (Grand Unified Theories predict relic monopole densities far exceeding observations). Alan Guth (1981) proposed cosmic inflation — an epoch of near-exponential expansion — that solves all three simultaneously.

Friedmann equation:
  H² = (ȧ/a)² = (8πG/3)ρ − k/a² + Λ/3

Inflationary expansion (slow-roll approximation):
  a(t) ≈ a_i exp(H_inf t)   where H_inf ≈ constant
  Doubling time: t_2 = ln2 / H_inf ≈ 10−³&sup7; s  (GUT scale inflation)

Slow-roll parameters:
  ε = −Ḣ/H² = (M_Pl²/2)(V'/V)²   (flatness of potential)
  η = M_Pl² V''/V   (curvature of potential)
  Inflation: ε < 1,  ends when ε = 1

Number of e-folds:
  N = ∫ H dt ≈ 50–60  (required to solve horizon + flatness problems)
  Causal patch grows from sub-Planck scale to > Hubble radius

Quantum fluctuations → primordial power spectrum:
  P_s(k) = A_s (k/k_0)^(n_s − 1)
  n_s = 1 − 6ε + 2η ≈ 0.965  (Planck 2018; n_s=1 = Harrison-Zeldovich)
  A_s ≈ 2.1 × 10−&sup9;  at pivot scale k_0 = 0.05 Mpc−¹

Tensor modes (primordial gravitational waves):
  Tensor-to-scalar ratio r = 16ε
  Planck+BICEP/Keck 2021: r < 0.036  (at 95% CL)
          

2. The Cosmic Microwave Background

The CMB is thermal radiation emitted when the universe cooled below ~3000 K at redshift z ≈ 1100 and hydrogen recombined, making the universe transparent. Its nearly perfect 2.725 K blackbody spectrum and small temperature anisotropies ΔT/T ∼ 10⁻⁵ encode the initial conditions of the universe and the physics of the photon-baryon plasma.

Recombination:
  z_rec ≈ 1100,  T_rec ≈ 3000 K,  t_rec ≈ 380 000 yr
  CMB temperature today: T_0 = 2.7255 K

Angular power spectrum C_ℓ:
  ΔT/T(θ) = Σ_(ℓm) a_(ℓm) Y_(ℓm)(θ,φ)
  C_ℓ = ⟨|a_(ℓm)|²⟩   (rotationally invariant)

Acoustic peaks (Sachs-Wolfe effect + BAO):
  1st peak: ℓ ≈ 220  (sound horizon at recombination, θ ≈ 1°)
    → universe is flat: Ω_k ≈ 0
  2nd peak: ℓ ≈ 540  (suppressed by baryon loading)
    → baryon density Ω_b h² ≈ 0.0224
  3rd peak: ℓ ≈ 810  (dark matter supports potential wells)
    → Ω_DM h² evidence

Sound horizon at last scattering:
  r_s = ∫_0^(t_rec) c_s(t)/ a(t) dt ≈ 147 Mpc  (comoving)
  c_s = c / √(3(1 + R))   where R = 3ρ_b/(4ρ_γ)

Integrated Sachs-Wolfe effect (ISW):
  Late-time ISW: photons gain energy in growing potential wells
  (ΛCDM: causes correlation between CMB and galaxy density at ℓ < 20)

Planck 2018 parameters (ΛCDM best-fit):
  H_0 = 67.4 ± 0.5 km/s/Mpc  (Hubble tension: local = 73.2 ± 1.3)
  Ω_b h² = 0.02237,  Ω_DM h² = 0.1200,  Ω_Λ = 0.685
  n_s = 0.9649,  A_s = 2.1 × 10−&sup9;,  τ = 0.054
          

3. Dark Matter: Candidates and Detection

Dark matter constitutes ~85% of all matter in the universe. Its existence is inferred from galaxy rotation curves, gravitational lensing (most dramatically the Bullet Cluster where baryonic gas and dark matter separated during a cluster merger), CMB anisotropies, and large-scale structure. Despite 40 years of experimental effort, no non-gravitational signal has been confirmed.

Galaxy rotation curves (Rubin & Ford 1970):
  Expected: v_c(r) = √(GM(r)/r) → falls as r−¹² outside visible disk
  Observed: v_c(r) ≈ const ≈ 200 km/s out to >50 kpc
  Implies M(r) ∝ r → dark matter halo with ρ ∝ 1/r (isothermal) or NFW

NFW (Navarro-Frenk-White) halo profile:
  ρ(r) = ρ_s / [(r/r_s)(1 + r/r_s)²]
  r_s = scale radius,  ρ_s = characteristic density
  Concentration parameter c = r_200/r_s  (c ≈ 10 for MW-mass halos)

WIMP (Weakly Interacting Massive Particle):
  m ≈ 10 GeV–10 TeV,  <σv⟩ ≈ 3 × 10−²&sup6; cm³/s (thermal relic)
  LHC: no SUSY signals yet
  Direct detection: LZ (LUX-ZEPLIN, 7 T LXe): no signal as of 2024
  Indirect: Fermi-LAT gamma-ray searches → upper limits on <σv⟩

Axion:
  m_a ≈ 10−&sup6;–10−³ eV (QCD axion window)
  Peccei-Quinn mechanism solves strong-CP problem
  ADMX experiment: microwave cavity resonant conversion B·a → photon
  ABRACADABRA: lumped-element axion detection

Primordial Black Holes (PBHs):
  Formed in radiation era from density fluctuations
  Mass window: 10−¹³–45 M_sun (gravitational microlensing+GW constraints)
  Asteroid-mass PBHs (10−¹&sup7;–10−¹¹ M_sun): still viable dark matter candidate
          

4. N-Body Structure Formation

The dark matter density field evolves under gravity from tiny initial fluctuations (Δρ/ρ ∼ 10⁻⁵ at recombination) into the rich web of halos, filaments, sheets, and voids seen today. Modern cosmological N-body simulations follow 10¹⁰–10¹¹ particles using adaptive tree-code gravity and periodic boundary conditions.

Linear perturbation theory (matter-dominated era):
  &ddot;δ + 2Hδ̇ − 4πGρ_m δ = 0
  Growing mode: δ(t) ∝ D(t) (linear growth factor)
  D(z) ≈ H(z) ∫_z^∞ (1+z')/H(z')³ dz'

Jeans length (collapse threshold):
  λ_J = c_s √(π/(Gρ))
  Dark matter: c_s ≈ 0 → all scales collapse (cold DM)
  Free streaming length λ_FS: neutrinos and warm DM suppress small-scale power

Press-Schechter halo mass function (1974):
  dn/dM = √(2/π) (ρ_m/M²) |d ln σ/d ln M| (δ_c/σ) exp(−δ_c²/(2σ²))
  δ_c ≈ 1.686 (spherical collapse threshold)
  σ(M) = rms density fluctuation in sphere of mass M

TreePM gravity:
  Short-range: oct-tree Barnes-Hut O(N log N) force summation
  Long-range: particle-mesh (PM) FFT on 3D grid
  Softening length ε ≈ 1–5% of mean particle separation (avoids 2-body scattering)

Millennium Simulation (Springel 2005):
  2160³ ≈ 10^10 dark matter particles, 500 Mpc/h box
  Illustris-TNG (2018): 300³ Mpc, full baryonic physics (gas, stars, BH feedback)
          

5. Baryon Acoustic Oscillations

Before recombination, baryons and photons were tightly coupled via Thomson scattering, forming a photon-baryon plasma that supported sound waves. At recombination, the photons decoupled and the sound waves stalled, leaving a frozen imprint — the baryon acoustic oscillation scale r_s ≈ 147 Mpc — that acts as a “standard ruler” in the galaxy distribution.

BAO scale at low redshift:
  Comoving sound horizon r_s ≈ 147 Mpc  (same as in CMB)
  Appears as a ring in the 2-point correlation function ξ(r)
  Peak in ξ(r) near r ≈ 150 Mpc comoving

Angular diameter distance D_A(z):
  D_A = (1/(1+z)) ∫_0^z c/H(z') dz'
  BAO measures D_A(z)/r_s and H(z)·r_s along/transverse to LOS

Hubble parameter:
  H(z)² = H_0² [Ω_m(1+z)³ + Ω_r(1+z)&sup4; + Ω_k(1+z)² + Ω_Λ f_DE(z)]
  Dark energy equation of state: w = p/ρc² (ΛCDM: w = −1)

DESI Year-1 BAO (2024) results:
  Consistent with flat ΛCDM to <2%, 11.5 million galaxies
  Mild hint: w_0 = −0.55, w_a = −1.32 (dynamical dark energy, 2.5σ)
  Euclid (launched 2023): will map 1.5 × 10^9 galaxies to z < 2

Baryon fraction cross-check:
  f_b = Ω_b / Ω_m ≈ 0.16  (from CMB + BBN)
  Observed in X-ray clusters: f_gas ≈ 0.12 (+ stars ≈ 0.04) → consistent
          

6. The Cosmic Web

Dark matter collapses preferentially along sheets and filaments before contracting into compact halos at their intersections — the Zeldovich approximation predicts this pancaking behaviour from the eigenvalues of the deformation tensor. Galaxies form inside dark matter halos; the resulting pattern of galaxy clustering, when observed in surveys spanning billions of light years, reveals the cosmic web.

Zel'dovich approximation (1970):
  x(q, t) = q − D(t) ∇ψ(q)
  ψ(q) = gravitational potential; D(t) = linear growth factor
  Collapse along three axes: first along shortest eigenvector (pancakes)

T-web classification (Hahn 2007):
  Tidal tensor T_ij = ∂²φ/∂x_i∂x_j
  # positive eigenvalues: 0 → voids, 1 → sheets, 2 → filaments, 3 → halos

Cosmic web statistics:
  Halos (knots):    ~5% of volume, ~80% of halo mass
  Filaments:        ~5% of volume, ~60% of diffuse gas
  Sheets:           ~15% of volume
  Voids:            ~75% of volume, <1% of matter

SDSS Baryon Oscillation Spectroscopic Survey (BOSS):
  1.3 million galaxies, 0.2 < z < 0.75
  Power spectrum P(k) measured at 1% precision → dark energy constraints

Euclid Wide Survey (2025–2030):
  1/3 of sky, 10^9 sources, spectroscopic z for 35 million
  Target: w ±0.01, σ_8 ±0.005
  First Euclid data release (DR1): June 2026 (15 million galaxies)
          

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