Big Bang Study Guide

📖 Core Concepts Big Bang Theory – Universe began from an extremely hot, dense state and has been expanding ever since. Cosmological Principle – On large scales the universe is homogeneous (same everywhere) and isotropic (same in every direction). Perfect‑Fluid Approximation – Cosmic matter/energy behaves like a fluid with no viscosity; pressure \(p\) is proportional to energy density \(\rho\). Friedmann–Lemaître–Robertson–Walker (FLRW) Metric – Geometry of a homogeneous, isotropic expanding universe; leads to the Friedmann equations. Friedmann Equations (for a flat universe, \(k=0\)): \[ \left(\frac{\dot a}{a}\right)^2 = \frac{8\pi G}{3}\rho,\qquad \frac{\ddot a}{a}= -\frac{4\pi G}{3}(\rho+3p) \] where \(a(t)\) is the scale factor, \(\dot a\) its time derivative, \(G\) the gravitational constant. Scale Factor \(a(t)\) – Quantifies how distances between comoving points grow with time; today \(a0\equiv1\). Components of Cosmic Energy Density – 5 % ordinary (baryonic) matter, 27 % dark matter, 68 % dark energy (≈ 73 % of total energy density). Dark Energy – Component with large negative pressure causing the accelerating expansion; often modeled as a cosmological constant \(\Lambda\). Inflation – Ultra‑rapid exponential expansion at \(t\sim10^{-37}\,\text{s}\) that solves the horizon and flatness problems and seeds primordial perturbations. Particle Horizon vs Event Horizon – Particle horizon = farthest light we can see today; event horizon = farthest region we will ever be able to affect because of ongoing expansion. Cosmic Microwave Background (CMB) – Relic black‑body radiation at \(T\simeq2.725\;\text{K}\) released at recombination (\(\sim380\,000\) yr). --- 📌 Must Remember Age of the Universe: \(13.787\pm0.02\) billion yr. CMB Temperature: \(2.725\;\text{K}\) (perfect black‑body). Light‑element Mass Fractions (BBN): He‑4 ≈ 25 %, D ≈ 0.01 %, He‑3 ≈ 0.0001 %, Li‑7 ≈ \(10^{-9}\). Hubble’s Law: \(v = H0 d\) – linear redshift–distance relation confirming expansion. Energy‑density Fractions (ΛCDM): Dark energy ≈ 68 % (≈ 73 % in some measurements), dark matter ≈ 27 %, baryonic matter ≈ 5 %. Critical Density: \(\rhoc = \frac{3H0^2}{8\pi G}\); flatness ⇒ total \(\rho = \rhoc\). Inflation Timing: Begins ≈ \(10^{-37}\,\text{s}\) after the Big Bang, expands space by > \(10^{26}\) in linear size. Recombination Epoch: \(t\approx3.8\times10^5\) yr, \(T\approx3000\;\text{K}\). --- 🔄 Key Processes Inflation (≈ \(10^{-37}\) s) – Scalar field drives exponential growth → smooths curvature & density. Cooling & Nucleosynthesis (seconds‑minutes) – Temperature falls to \(\sim10^9\) K → protons + neutrons form D, He‑4, He‑3, Li‑7. Recombination (≈ 380 kyr) – Electrons bind nuclei → universe becomes transparent → CMB photons free‑stream. Structure Formation (post‑recombination) – Tiny overdensities (∼\(10^{-5}\)) grow via gravity → gas clouds → stars → galaxies → clusters. Accelerated Expansion (≈ 5 Gyr ago to present) – Dark energy dominates → \(\ddot a>0\). --- 🔍 Key Comparisons Dark Matter vs Dark Energy Dark Matter: gravitates, clusters, 27 % of density, pressure ≈ 0. Dark Energy: causes accelerated expansion, 68 % of density, pressure \(p\approx -\rho\) (negative). Open vs Closed Geometry Open: \(\rho < \rhoc\), negative curvature, expands forever. Closed: \(\rho > \rhoc\), positive curvature, could recollapse (absent dark energy). Inflation vs Steady‑State Inflation: early rapid expansion, predicts CMB, solves horizon/flatness. Steady‑State: continuous creation of matter, no CMB, disproven by observations. Recession Speed vs Peculiar Velocity Recession: space itself expanding; can exceed \(c\) without violating relativity. Peculiar: motion through space; always \(<c\). --- ⚠️ Common Misunderstandings “Big Bang was an explosion in space.” – It was an expansion of space everywhere. “Galaxies receding faster than light break relativity.” – The speed refers to metric expansion, not local motion. “Dark energy and dark matter are the same.” – Dark matter pulls; dark energy pushes (negative pressure). “CMB is still hot because it was hot.” – It is now cooled to 2.7 K due to cosmic expansion. --- 🧠 Mental Models / Intuition Balloon Analogy: Dots on a balloon surface move apart as the balloon inflates; the surface has no center → reflects homogeneity and isotropy. Fluid Flow: Treat the universe as a frictionless fluid; pressure and density dictate expansion via Friedmann equations. Rubber‑Sheet Stretching: Inflation stretches tiny quantum ripples into macroscopic density perturbations (seed of galaxies). --- 🚩 Exceptions & Edge Cases Particle Horizon ≠ Event Horizon: We can see beyond the region we will ever be able to influence. Flatness is Approximate: Observations give \(|\Omegak| < 0.005\); a perfectly flat universe is an idealization. Dark Energy Equation of State: While \(\Lambda\) implies \(w = p/\rho = -1\), alternative models allow \(w\neq -1\) (still under investigation). --- 📍 When to Use Which Friedmann Equation – To compute the evolution of \(a(t)\) given a specified \(\rho(t)\) and \(p(t)\). CMB Power Spectrum – To extract \(\Omega{\rm baryon}\), \(\Omega{\rm DM}\), \(\Omega\Lambda\), and curvature. Type Ia Supernova Distance Modulus – Best for probing recent (low‑\(z\)) acceleration and dark‑energy density. Baryon‑Acoustic Oscillations (BAO) – Complementary ruler for measuring expansion history at intermediate redshifts. --- 👀 Patterns to Recognize Linear Redshift–Distance → Hubble’s law holds for nearby galaxies. Black‑body CMB Spectrum → Any deviation suggests non‑standard physics. \(10^{-5}\) Temperature Fluctuations → Signature of primordial density perturbations. Element‑Abundance Ratios (He‑4 ≈ 25 %) → Consistent with a single baryon‑to‑photon ratio. --- 🗂️ Exam Traps Confusing Age Determinations: The Hubble‑constant‑derived age and CMB‑derived age must agree; a mismatch signals misuse of \(H0\) value. Mixing Percentages: Remember dark energy ≈ 68‑73 %, dark matter ≈ 27 %, ordinary matter ≈ 5 % – not the reverse. Redshift ≠ Velocity: For high‑\(z\) objects, redshift reflects space expansion, not simple Doppler shift. Flatness vs Geometry: “Flat” does not mean “no curvature”; it means curvature is negligible within observational limits. Inflation vs Ongoing Expansion: Inflation is a brief, ultra‑rapid phase; the current accelerated expansion is driven by dark energy, not inflation.