Large-scale structure and evolution of the universe: Big Bang, cosmic microwave background, Hubble expansion, dark matter, dark energy, ΛCDM concordance model.
cosmology
Big Bang (hot dense early universe)
The observable universe expanded from a hot, dense, nearly-uniform state ~13.8 Gyr ago. Supported by Hubble expansion, CMB, primordial…
Cosmic microwave background
Near-uniform ~2.725 K blackbody radiation filling the universe; relic photons from recombination (~380 000 yr after Big Bang).
Hubble expansion
Distant galaxies recede with velocity v = H₀·d on average; interpreted as expansion of the metric, not motion through space.
Dark matter
Inferred non-luminous matter, ~27% of the cosmic energy budget, required to explain galaxy rotation curves, structure formation, lensing.…
Dark energy
Component (~68% of cosmic energy budget) driving the accelerating expansion of the universe. Its nature — cosmological constant, dynamical…
ΛCDM concordance model
Six-parameter cosmological model with cold dark matter and cosmological constant Λ; current best fit to CMB, BAO, lensing, SNIa,…
Friedmann equations
Evolution equations of the cosmological scale factor a(t): H² = (8πG/3)ρ − k/a² + Λ/3 and ä/a = −(4πG/3)(ρ + 3p) + Λ/3.
Cosmological inflation
Hypothesised early-universe phase of ~60 e-folds of exponential expansion driven by a slow-rolling inflaton. Solves horizon, flatness, and…
Big Bang nucleosynthesis
Primordial synthesis of light nuclei (D, ³He, ⁴He, ⁷Li) in the first ~20 min. Observed abundances constrain baryon density and number of…
Recombination (z ≈ 1100)
Epoch ~380,000 yr after Big Bang at T ≈ 3000 K when electrons recombined with protons; photons decoupled to form the CMB.
Baryon asymmetry & Sakharov conditions
Observed η_B ~ 6×10⁻¹⁰. Requires B-violation, C+CP violation, out-of-equilibrium. SM CP violation insufficient; leptogenesis + sphaleron…
Slow-roll inflation
Scalar field dominated by potential → quasi-exponential expansion. Solves horizon/flatness problems. Slow-roll parameters ε,η;…
Baryon acoustic oscillations
Sound horizon at decoupling (~150 Mpc) imprints standard ruler in galaxy correlation function. SDSS, BOSS, DESI — probe of H(z), w(z).
σ₈ / H₀ tensions
Cosmological parameter discrepancies between early-universe (CMB) and late-universe (SNe Ia, galaxy clustering, lensing) probes. 5σ H₀…
Primordial scalar power spectrum
P_ζ(k) = A_s (k/k_*)^(n_s-1); Planck: A_s ≈ 2.1×10⁻⁹, n_s ≈ 0.965; red tilt consistent with single-field slow-roll.
Tensor-to-scalar ratio r
Amplitude of primordial gravitational waves vs scalar perturbations; r < 0.036 (BICEP/Keck 2021); constrains inflation energy scale to ≲…
Baryon acoustic oscillations (BAO)
Sound horizon r_s ≈ 150 Mpc imprinted at recombination; standard ruler in galaxy clustering (SDSS, BOSS, DESI); probes dark energy.
CMB anisotropy power spectrum
C_ℓ multipoles: acoustic peaks from baryon-photon fluid; first peak ℓ≈220 → flat universe; Silk damping at high ℓ; integrated Sachs–Wolfe…
Silk damping
Photon diffusion in baryon fluid before recombination erases fluctuations below λ_Silk ≈ 10 Mpc; exponential CMB suppression at high ℓ.
Sachs–Wolfe effect
Gravitational potential Φ imprints ΔT/T = Φ/3 on CMB (large scales); late-time ISW probes dark energy via decaying potentials.
Big Bang nucleosynthesis (BBN)
First 20 min: ⁴He mass fraction Y_p ≈ 0.245, D/H ≈ 2.5×10⁻⁵; agreement over 9 orders of magnitude fixes Ω_b h² ≈ 0.0224.
Recombination at z ≈ 1100
e⁻ + p⁺ → H + γ when kT ≈ 0.26 eV (not 13.6 eV: Saha + photon/baryon ratio); CMB last-scattering surface formed.
Linear structure formation
δ̈ + 2H δ̇ - 4πG ρ̄ δ = 0 in matter era → D₊ ∝ a; transfer function from inflation; nonlinear below ~10 Mpc today.
Press–Schechter halo mass function
dn/dM ∝ (ρ̄/M²)(δ_c/σ(M)) exp(-δ_c²/(2σ²))|d ln σ/d ln M|; Sheth–Tormen correction for ellipsoidal collapse. Cluster abundance cosmology.
Horizon problem
CMB isotropic over 10⁵ causally disconnected patches at last scattering; solved by inflation (exponential causal patch expansion).
Flatness problem
|Ω-1| grows like a² in radiation era; today Ω≈1 ⇒ |Ω-1|≲10⁻⁶⁰ at Planck time unless inflation drives Ω→1.
Monopole problem
GUT phase transition would produce ~1/horizon magnetic monopoles; unobserved; inflation dilutes them below detection.
Cold dark matter (CDM)
Non-relativistic collisionless DM with Ω_c ≈ 0.27; required for structure formation, galaxy rotation curves, cluster lensing.
Dark energy equation of state w
p = w ρ; cosmological constant w = -1; quintessence w varying; DESI 2024 hints at w₀ ≈ -0.9, w_a ≈ -0.4 (evolving).
Hubble tension
Local H₀ ≈ 73 km/s/Mpc (SH0ES) vs CMB-inferred ≈ 67 (Planck) — 5σ discrepancy. Early dark energy, N_eff, modified gravity candidates.
σ₈ / S₈ tension
Weak-lensing (KiDS, DES) finds lower S_8 = σ_8 √(Ω_m/0.3) than CMB-predicted; ~2-3σ hint of suppressed late-time structure.
Cosmic variance
Finite-mode sampling limits how well low-ℓ observables can be measured even with perfect detectors; ΔC_ℓ/C_ℓ = √(2/(2ℓ+1)).
Eternal inflation & multiverse
Quantum fluctuations of inflaton continually produce new inflating regions; self-reproducing multiverse; measure problem unresolved.
FLRW flat matter-dominated solution (cosmology)
Cosmological application of L0 Riemannian metric and curvature tensor. The flat (k=0) matter-dominated FLRW metric ds² = −dt² +…
Raychaudhuri focusing equation (cosmological congruences)
Cosmological application of L0 geodesic and Levi-Civita connection. Raychaudhuri's 1955 equation for the expansion scalar θ = ∇_μ u^μ of a…