Cosmic microwave background Study Guide

📖 Core Concepts Cosmic Microwave Background (CMB) – relic black‑body radiation filling the observable universe; today ≈ 2.725 K. Decoupling / Recombination – when protons + electrons formed neutral H (z ≈ 1100), making the universe transparent and releasing photons that become the CMB. Surface of Last Scattering – spherical shell from which the observed CMB photons were emitted at decoupling. Anisotropy – tiny temperature variations (ΔT/T ≈ 10⁻⁵) that encode early‑universe physics. Power Spectrum & Multipole ℓ – spherical‑harmonic decomposition; \(C{\ell}\) measures variance at angular scale ≈ 180°/ℓ. Acoustic Peaks – standing‑wave modes in the photon‑baryon plasma; the first peak fixes curvature, later peaks constrain matter content. Polarization Modes – E‑mode (gradient) from Thomson scattering of a quadrupole; B‑mode (curl) from primordial gravitational waves or lensing of E‑modes. --- 📌 Must Remember CMB temperature: 2.725 K (black‑body). Photon density today: ≈ 411 cm⁻³; energy density ≈ 0.260 eV cm⁻³ (4.17 × 10⁻¹⁴ J m⁻³). Dipole amplitude: 3.362 mK → solar system motion. First acoustic peak at ℓ ≈ 220 → flat geometry. Baryon density from second peak, dark‑matter density from third peak. Planck 2018 Hubble constant: \(H{0}=67.4\pm0.5\) km s⁻¹ Mpc⁻¹. Matter density parameter: \(\Omega{m}=0.315\pm0.007\). Dark‑energy fraction ≈ 68 %; ordinary matter ≈ 4.9 %; dark matter ≈ 26.8 %. Tensor‑to‑scalar ratio upper limit \(r \lesssim 0.06\) (Planck + ground‑based). --- 🔄 Key Processes Early‑universe plasma → recombination Expansion cools plasma → electrons + protons → neutral H → photons free‑stream. Acoustic oscillation cycle Gravity pulls matter inward → photon pressure pushes outward → standing wave; mode’s phase at decoupling sets peak height. Silk (diffusion) damping Growing photon mean free path → exponential suppression of small‑scale fluctuations. Polarization generation Thomson scattering of photons off free electrons in a quadrupole temperature field → linear polarization (E‑mode). Secondary anisotropies (post‑decoupling) Reionization: rescattering → large‑scale polarization. Sunyaev–Zel’dovich: hot electrons up‑scatter CMB photons → spectral distortion. Integrated Sachs–Wolfe: evolving potentials shift photon energies → large‑scale temperature change. --- 🔍 Key Comparisons E‑mode vs. B‑mode – E‑mode: gradient‑type, produced by scalar density perturbations; B‑mode: curl‑type, requires tensor perturbations (inflationary GWs) or lensing of E‑modes. Primary vs. Secondary anisotropy – Primary: imprinted at decoupling (acoustic peaks); Secondary: generated later (reionization, SZ, ISW). Adiabatic vs. Isocurvature perturbations – Adiabatic: peaks at ℓ ratios 1 : 2 : 3 …; Isocurvature: peaks at 1 : 3 : 5 …; observations favor adiabatic. --- ⚠️ Common Misunderstandings “CMB temperature is 3 K” – the precise measured value is 2.725 K; 3 K is a rounded historical estimate. Dipole anisotropy as cosmological – it is kinematic, caused by the Solar System’s motion, not primordial structure. B‑mode detection = proof of inflation – lensing‑induced B‑modes can mimic the signal; foreground dust (e.g., BICEP2 case) can also contaminate. Higher ℓ = higher frequency – ℓ indexes angular scale, not photon frequency; ℓ ≈ 220 corresponds to 1° on the sky. --- 🧠 Mental Models / Intuition “Sound waves frozen in the sky” – Think of the photon‑baryon fluid as a drumhead; at recombination the drum stops vibrating, leaving a snapshot of the standing‑wave pattern (the acoustic peaks). “CMB as a photograph” – The surface of last scattering is like a cosmic photo‑negative; each point records temperature/polarization at a single epoch, later stretched by cosmic expansion. “Multipole ladder” – ℓ = 1 (dipole) = whole‑sky tilt; ℓ ≈ 200 = degree‑scale spots; ℓ ≈ 2000 = arc‑minute features. --- 🚩 Exceptions & Edge Cases Silk damping tail – At ℓ > 1500 the power spectrum drops sharply; not a failure of ΛCDM but a physical diffusion limit. Low‑ℓ anomalies – Quadrupole (ℓ = 2) power lower than expected; alignment with ecliptic (“axis of evil”) – still debated, may be foreground or statistical fluke. Reionization bump – Large‑scale (ℓ < 10) E‑mode polarization rise due to scattering off free electrons after the first stars formed. --- 📍 When to Use Which Estimating curvature → use first acoustic peak position (ℓ ≈ 220). Deriving baryon density → analyze second peak height relative to the first. Constraining dark‑matter density → look at third peak amplitude. Testing inflationary tensors → focus on large‑scale B‑mode power (ℓ ≈ 80) after foreground cleaning. Assessing late‑time effects → compute ISW contribution at low ℓ; cross‑correlate with large‑scale structure surveys. --- 👀 Patterns to Recognize Peak spacing ≈ constant – regular spacing of acoustic peaks signals a sound horizon set by early‑universe physics. Even vs. odd peak heights – odd peaks (compression) enhanced by baryons; even peaks (rarefaction) relatively suppressed. Damping tail exponential – rapid fall‑off beyond ℓ ≈ 1500 indicates Silk damping. E‑mode ↔ temperature correlation – TE cross‑spectrum peaks line up with temperature peaks, confirming Thomson scattering origin. --- 🗂️ Exam Traps Choosing “dipole” as a cosmological anisotropy – the dipole is kinetic, not primordial. Assuming B‑mode detection automatically confirms inflation – must rule out lensing and Galactic dust first. Confusing ℓ with frequency – ℓ indexes angular scale; higher ℓ = smaller angular features, not higher photon energy. Interpreting low‑ℓ anomalies as proof of new physics – they could be statistical variance or foreground residuals; most textbooks treat them as open questions. Mixing up photon density (411 cm⁻³) with matter density – photons outnumber matter particles by 10⁹, but their energy density is still tiny compared to matter/​dark energy today.