Dark matter Study Guide

📖 Core Concepts Dark Matter (DM) – Matter that does not emit, absorb, or reflect light; only detectable via its gravitational influence. Energy‑density scaling – Matter (including DM) dilutes as \(\rho \propto a^{-3}\); radiation as \(\rho \propto a^{-4}\); cosmological constant as \(\rho \propto a^{0}\). Cosmic inventory – \(\sim5\%\) ordinary (baryonic) matter, \(\sim26.8\%\) dark matter, \(\sim68.2\%\) dark energy. DM ≈ 85 % of all matter. Cold, Warm, Hot DM – Classification by particle speeds in the early Universe; determines free‑streaming length and smallest structures that can form. Virial theorem & mass‑to‑light discrepancy – Velocity dispersion in galaxies/clusters exceeds that predicted from visible mass → unseen mass (DM). Key particle candidates – WIMPs (≈ GeV–TeV, weak‑scale annihilation cross‑section \(\sim3\times10^{-26}\,\text{cm}^{3}\,\text{s}^{-1}\)), axions (µeV–meV, produced by misalignment), primordial black holes (PBHs). --- 📌 Must Remember \(\Omega{\text{dm}} \approx 0.258\); \(\Omega{\text{b}} \approx 0.048\). Flat rotation curves ⇒ mass interior to radius \(r\) grows \(\propto r\). Free‑streaming length sets the minimum halo mass; cold ≪ dwarf‑galaxy scale, warm ≈ dwarf scale, hot ≫ dwarf scale. WIMP “miracle”: thermal relic with \(\langle\sigma v\rangle \approx 3\times10^{-26}\,\text{cm}^{3}\,\text{s}^{-1}\) yields the observed DM density. Axion‑photon conversion (Primakoff effect) is the basis of haloscope searches. Bullet Cluster – Lensing mass peaks offset from X‑ray gas → DM is collisionless. Neutrino fog – Solar/atmospheric neutrinos produce an irreducible background for WIMP direct detection. --- 🔄 Key Processes Virial Mass Estimate Measure velocity dispersion \(\sigma\). Apply \(M \approx \frac{5\sigma^{2}R}{G}\) (spherical approximation). Freeze‑out of a WIMP Early Universe: \(n\langle\sigma v\rangle > H\) → particles stay in equilibrium. As \(T\) drops, expansion rate \(H\) exceeds interaction rate → “freeze‑out”. Relic density \(\Omega{\chi}h^{2}\propto 1/\langle\sigma v\rangle\). Axion Misalignment Production Axion field \(a\) initially displaced from minimum. When \(H \sim m{a}\), field begins coherent oscillations → behaves as cold DM. Gravitational Lensing Mass Mapping Strong lensing: identify multiple images/arcs → solve lens equation for mass distribution. Weak lensing: measure average shear \(\gamma\) of background galaxies → reconstruct projected mass (convergence \(\kappa\)). Direct‑Detection Signal Generation DM particle scatters off nucleus → recoil energy \(E{R}= \frac{\mu^{2}v^{2}}{m{N}}(1-\cos\theta)\). Detect scintillation, ionization, or phonons from recoil. --- 🔍 Key Comparisons Cold vs Warm vs Hot DM Velocity: \(v{\text{cold}} \ll c\) ; \(v{\text{warm}} \sim 10^{-3}c\) ; \(v{\text{hot}} \approx c\). Free‑streaming: short vs intermediate vs long → impacts smallest halo mass. WIMPs vs Axions Mass: GeV–TeV vs µeV–meV. Interaction: Weak‑scale nuclear scattering vs axion‑photon conversion in magnetic fields. Searches: Underground nuclear recoil detectors vs resonant microwave cavities. Direct vs Indirect Detection Goal: Observe scattering on Earth vs detect annihilation/decay products (γ, e⁺, ν). Signal type: Low‑energy recoils vs high‑energy astrophysical photons/particles. Particle DM vs Modified Gravity (MOND) Explanation: Extra mass component vs change in Newton’s law at \(a<a{0}\). Key test: Bullet Cluster (supports particle DM, challenges MOND). --- ⚠️ Common Misunderstandings “Dark matter is just unseen ordinary matter.” – BBN and CMB fix baryon density at 5 %; the remaining 26 % must be non‑baryonic. “WIMPs have been ruled out.” – Experiments have set stronger limits, but viable parameter space (especially below the neutrino floor) remains. “Axions are dark energy.” – Axions are a cold‑dark‑matter candidate; dark energy is a separate component with \(\rho=\text{const}\). “Hot DM can explain galaxies.” – Hot DM free‑streams over galaxy scales, suppressing small‑scale structure; observations require cold (or warm) DM. “All MACHOs are excluded.” – Microlensing rules out MACHOs as the dominant DM, but they may still exist in small fractions. --- 🧠 Mental Models / Intuition Scaffolding analogy – Think of DM as the invisible steel framework of a building; stars and gas are the décor that hangs on it. Free‑streaming as “cosmic smoothing” – Fast particles blur out density bumps; slower particles let them survive, allowing small halos to form. Freeze‑out as “traffic jam” – Early Universe: particles constantly collide (traffic dense). As the Universe expands, collisions become rare and the “traffic” freezes in place, leaving a relic density. --- 🚩 Exceptions & Edge Cases Self‑interacting DM – Can modify halo cores (cored vs cuspy) without changing large‑scale clustering. Primordial Black Holes – Non‑particle DM; constraints depend heavily on assumed mass spectrum (monochromatic vs extended). Neutrino floor – Below \(\sim10^{-48}\,\text{cm}^{2}\) nuclear cross‑section, solar/atmospheric neutrinos mimic WIMP recoils, setting a practical detection limit. --- 📍 When to Use Which Estimate total cluster mass → Use velocity dispersion (virial theorem) or X‑ray temperature or gravitational lensing; compare all three for consistency. Choose DM candidate for a given scale → Cold DM for hierarchical formation; Warm DM if you need to suppress dwarf‑galaxy numbers; Hot DM only for large‑scale “pancake” structures (disfavored). Select detection technique → Nuclear recoil: best for WIMPs ≳ 10 GeV. Axion haloscope: optimal for \(\mu\)eV–meV masses. Gamma‑ray/CR: target high‑density regions (Galactic Center, dwarf spheroidals). Interpret rotation curves → If acceleration \(<a{0}\) and no DM halo model fits, consider MOND as an alternative hypothesis (but test against lensing evidence). --- 👀 Patterns to Recognize Flat rotation curves + high mass‑to‑light ratios → Classic DM signature. Offset between X‑ray gas and lensing mass → Collisionless DM (Bullet Cluster). Acoustic peak pattern in CMB – First peak height set by baryons, third peak boosted by DM; mismatched peaks imply missing DM. Suppressed small‑scale power in Lyman‑α forest → Warm DM or large free‑streaming length. --- 🗂️ Exam Traps Confusing baryonic “missing mass” with DM – Remember BBN limits baryons to 5 % of critical density. Assuming all WIMPs have been excluded – Limits are strong but not absolute; the neutrino floor remains a frontier. Mixing up free‑streaming length with interaction cross‑section – Free‑streaming is set by particle speed, not scattering strength. Attributing the CMB acoustic peaks solely to dark energy – Peaks diagnose both baryon and DM densities; dark energy mainly affects late‑time expansion. Choosing MOND for galaxy clusters – MOND fits rotation curves of spirals but fails to explain cluster lensing and the Bullet Cluster. ---