Cosmology

A comprehensive exploration of the universe's origin, evolution, and ultimate fate—from the Big Bang through cosmic inflation, structure formation, dark matter, dark energy, to the cosmic microwave background and beyond.

Course Overview

Cosmology is the study of the universe as a whole—its origin, evolution, large-scale structure, and ultimate fate. Modern cosmology combines General Relativity with particle physics, quantum field theory, and observational astronomy to understand the cosmos from the Planck scale (10⁻³⁵ m) to the observable universe (10²⁶ m). This course covers the standard ΛCDM model, cosmic inflation, the cosmic microwave background (CMB), large-scale structure formation, dark matter, dark energy, and current observational frontiers.

What You'll Learn

  • • FLRW metric and Friedmann equations
  • • Big Bang nucleosynthesis and recombination
  • • Cosmic inflation and flatness/horizon problems
  • • Cosmic Microwave Background (CMB) physics
  • • Large-scale structure and galaxy formation
  • • Dark matter evidence and candidates
  • • Dark energy and the accelerating universe
  • • Cosmological perturbation theory
  • • Observational cosmology (Planck, WMAP, surveys)
  • • Quantum origin of primordial fluctuations

Prerequisites

Course Structure

Comprehensive cosmology curriculum • From Big Bang to present epoch • Includes Friedmann equations derivation, inflation theory, CMB physics, structure formation • Features world-class video lectures from leading cosmologists • Suitable for graduate students and advanced undergraduates in physics and astronomy

Core Topics

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FLRW Universe & Friedmann Equations

The Friedmann-Lemaître-Robertson-Walker metric describes a homogeneous, isotropic expanding universe. Friedmann equations govern the scale factor evolution, relating expansion rate to energy density and curvature.

FLRW MetricFriedmann EquationsScale Factor
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Big Bang & Thermal History

The early universe underwent a hot, dense phase starting from the Planck epoch. Big Bang nucleosynthesis (BBN) formed light elements, followed by recombination when atoms formed, releasing the CMB at z ≈ 1100.

BBNRecombinationCMB Formation
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Cosmic Inflation

Inflation solves the horizon, flatness, and monopole problems via exponential expansion driven by a scalar field. Quantum fluctuations during inflation seeded all structure in the universe, imprinted in CMB anisotropies.

Inflation TheoryQuantum FluctuationsHorizon Problem
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Cosmic Microwave Background

The CMB is thermal radiation from recombination at T ≈ 3000 K, redshifted to 2.725 K today. Its temperature anisotropies (ΔT/T ≈ 10⁻⁵) probe initial conditions, geometry, and cosmological parameters with exquisite precision.

CMB AnisotropiesPlanck MissionPower Spectrum
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Large-Scale Structure Formation

Gravitational instability amplifies density perturbations, forming galaxies, clusters, and the cosmic web. Dark matter dominates gravitational collapse, while baryonic matter follows to form stars and galaxies.

Structure FormationCosmic WebGalaxy Clusters
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Dark Matter

Dark matter comprises ≈27% of the universe. Evidence from galaxy rotation curves, gravitational lensing, CMB, and structure formation points to non-baryonic, weakly interacting massive particles (WIMPs) or axions.

WIMPsGravitational LensingRotation Curves
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Dark Energy & Accelerating Universe

Dark energy (≈68% of universe) drives cosmic acceleration discovered via Type Ia supernovae (1998). The cosmological constant Λ (vacuum energy) is the simplest explanation, but its value remains a deep mystery.

Dark EnergyCosmological ConstantSNe Ia

Key Equations — Step-by-Step Derivations

Below we derive the twelve foundational equations of modern cosmology, starting from the cosmological principle and building to observational quantities.

1. The FLRW Metric

Starting point: The cosmological principle asserts that the universe is spatially homogeneous and isotropic on large scales. The most general metric compatible with these symmetries is the Friedmann-Lemaître-Robertson-Walker (FLRW) line element.

Step 1 — Maximally symmetric 3-spaces: A homogeneous, isotropic 3-dimensional space has constant curvature characterised by a single parameter $k$. In spherical coordinates the spatial part of the metric is $d\sigma^2 = \frac{dr^2}{1-kr^2} + r^2\,d\Omega^2$, where $d\Omega^2 = d\theta^2 + \sin^2\!\theta\;d\phi^2$ and$k = +1,\,0,\,-1$ for closed, flat, and open geometries respectively.

Step 2 — Include cosmic time: Introduce the scale factor $a(t)$ that tracks how distances between comoving observers stretch with time. The full spacetime interval becomes:

$$\boxed{\;ds^2 \;=\; -c^2\,dt^2 \;+\; a(t)^2\!\left[\frac{dr^2}{1-kr^2} \;+\; r^2\,d\Omega^2\right]\;}$$

Here $a(t)$ is normalised so that $a(t_0)=1$ today, $r$ is a comoving radial coordinate (constant for objects carried by the Hubble flow), and $t$ is cosmic time measured by comoving clocks. All dynamics of the expanding universe are encoded in $a(t)$.

2. First Friedmann Equation

Goal: Determine how the expansion rate $H \equiv \dot{a}/a$ depends on the energy content.

Step 1 — Einstein Field Equations (EFE): Start from $G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}\,T_{\mu\nu}$. Insert the FLRW metric and a perfect-fluid stress-energy tensor $T^{\mu\nu} = (\rho + p/c^2)\,u^\mu u^\nu + p\,g^{\mu\nu}$.

Step 2 — (0,0) component: The time-time component of the EFE yields (after computing the Christoffel symbols and Ricci tensor for the FLRW metric):

$$\boxed{\;H^2 = \left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3}\,\rho \;-\; \frac{kc^2}{a^2} \;+\; \frac{\Lambda c^2}{3}\;}$$

Newtonian analogy: Consider a thin shell of mass $m$ at the surface of a uniform sphere of radius $R = a\,r$. Energy conservation gives$\tfrac{1}{2}m\dot{R}^2 - \tfrac{G M m}{R} = E$, where $M = \tfrac{4}{3}\pi\rho R^3$. Dividing by $\tfrac{1}{2}mR^2$ and identifying the curvature term with the total energy reproduces the Friedmann equation exactly.

The three terms on the right represent: energy density driving expansion, spatial curvature resisting or aiding it, and the cosmological constant (dark energy) accelerating it.

3. Second Friedmann (Acceleration) Equation

Step 1 — Spatial components of EFE: The (i, i) components of Einstein's equations applied to the FLRW metric give:

$$\frac{\ddot{a}}{a} = -\frac{4\pi G}{3}\!\left(\rho + \frac{3p}{c^2}\right) + \frac{\Lambda c^2}{3}$$

Step 2 — Alternative derivation: Take the time derivative of the first Friedmann equation and substitute the fluid equation (below) to eliminate $\dot{\rho}$. After simplification one obtains the same result:

$$\boxed{\;\frac{\ddot{a}}{a} = -\frac{4\pi G}{3}\!\left(\rho + \frac{3p}{c^2}\right) + \frac{\Lambda c^2}{3}\;}$$

Key insight: Pressure contributes to gravity! For ordinary matter ($\rho + 3p/c^2 > 0$) the expansion decelerates. Only if $\Lambda$ is large enough (dark energy dominates) does the expansion accelerate ($\ddot{a}>0$).

4. Fluid (Continuity) Equation

Starting point: Local conservation of energy-momentum $\nabla_\mu T^{\mu\nu} = 0$.

Step 1: For the $\nu = 0$ component in the FLRW background, expand the covariant derivative. The Christoffel symbols give $\Gamma^0_{ij} = \frac{\dot{a}}{a}\,g_{ij}/c$ and $\Gamma^i_{0j} = \frac{\dot{a}}{a}\,\delta^i_j$.

Step 2: Substituting and simplifying:

$$\boxed{\;\dot{\rho} + 3\,\frac{\dot{a}}{a}\!\left(\rho + \frac{p}{c^2}\right) = 0\;}$$

Physical meaning: Consider a comoving volume $V \propto a^3$. The first law of thermodynamics $dE + p\,dV = 0$ (no heat flow by isotropy) with $E = \rho c^2 V$ gives:

$$\frac{d(\rho\, a^3)}{dt} + \frac{p}{c^2}\,\frac{d(a^3)}{dt} = 0$$

which is identical to the fluid equation. Note: only two of the three equations (Friedmann I, Friedmann II, Fluid) are independent.

5. Equation of State & Density Scaling

Closure relation: A barotropic equation of state relates pressure to density:

$$\boxed{\;p = w\,\rho\,c^2\;}$$

where $w$ is constant for each component: $w=0$ (non-relativistic matter / dust), $w=\tfrac{1}{3}$ (radiation / relativistic particles), $w=-1$ (cosmological constant / vacuum energy).

Derivation of scaling laws: Substitute $p = w\rho c^2$ into the fluid equation: $\dot{\rho}/\rho = -3(1+w)\,\dot{a}/a$. Integrate:

$$\rho \;\propto\; a^{-3(1+w)}$$
Component$w$ScalingPhysics
Matter (dust)$0$$\rho_m \propto a^{-3}$Number density dilutes as volume expands
Radiation$1/3$$\rho_r \propto a^{-4}$Volume dilution + wavelength redshift
Cosmological constant$-1$$\rho_\Lambda = \text{const}$Vacuum energy density is fixed

6. Critical Density & Density Parameters

Step 1: Set $k = 0$ and $\Lambda = 0$ in the first Friedmann equation. The density required for a spatially flat universe is:

$$\boxed{\;\rho_c \;=\; \frac{3H_0^2}{8\pi G} \;\approx\; 1.88 \times 10^{-26}\;h^2 \;\text{kg\,m}^{-3} \;\approx\; 9.47 \times 10^{-27}\;\text{kg\,m}^{-3}\;}$$

Step 2 — Dimensionless density parameters: Define $\Omega_i \equiv \rho_i / \rho_c$ for each component:

$$\Omega_m = \frac{\rho_m}{\rho_c}, \qquad \Omega_r = \frac{\rho_r}{\rho_c}, \qquad \Omega_\Lambda = \frac{\Lambda c^2}{3H_0^2}, \qquad \Omega_k = -\frac{kc^2}{a_0^2 H_0^2}$$

Step 3 — Cosmic sum rule: Dividing the first Friedmann equation evaluated today by $H_0^2$:

$$\boxed{\;\Omega_m + \Omega_r + \Omega_\Lambda + \Omega_k = 1\;}$$

Planck 2018 values: $\Omega_m \approx 0.315$, $\Omega_r \approx 9.1 \times 10^{-5}$, $\Omega_\Lambda \approx 0.685$, $|\Omega_k| < 0.002$ — the universe is flat to within observational precision.

7. Hubble's Law (Hubble-Lemaître Law)

Derivation from FLRW: The proper distance to a comoving object at coordinate $\chi$ is $d_p(t) = a(t)\,\chi$. Differentiate:

$$v = \dot{d}_p = \dot{a}\,\chi = \frac{\dot{a}}{a}\,(a\,\chi) = H\,d_p$$
$$\boxed{\;v = H_0\,d\;}$$

This is exact for all distances (not just small ones), though for $v > c$ it describes the recession rate of the metric expansion, not a physical velocity through space.

The Hubble Tension

A $\sim 5\sigma$ discrepancy exists between early-universe and late-universe measurements of $H_0$:

  • • Planck CMB (early universe): $H_0 = 67.4 \pm 0.5$ km/s/Mpc
  • • SH0ES Cepheids + SNe Ia (late universe): $H_0 = 73.0 \pm 1.0$ km/s/Mpc

Whether this reflects new physics beyond $\Lambda$CDM or unresolved systematics is one of the most actively debated questions in modern cosmology.

8. Redshift–Scale Factor Relation

Step 1: A photon emitted at time $t_{\text{emit}}$ with wavelength $\lambda_{\text{emit}}$ travels along a null geodesic ($ds^2 = 0$). Consider two successive wave crests separated by $\delta t_{\text{emit}}$ at emission and $\delta t_{\text{obs}}$ at observation.

Step 2: Since the comoving distance is the same for both crests: $\int_{t_{\text{emit}}}^{t_{\text{obs}}} \frac{c\,dt}{a(t)} = \int_{t_{\text{emit}}+\delta t_{\text{emit}}}^{t_{\text{obs}}+\delta t_{\text{obs}}} \frac{c\,dt}{a(t)}$. For small $\delta t$ this gives $\delta t_{\text{obs}} / \delta t_{\text{emit}} = a(t_{\text{obs}}) / a(t_{\text{emit}})$.

Step 3: Since $\lambda \propto \delta t$ and defining redshift $z \equiv (\lambda_{\text{obs}} - \lambda_{\text{emit}})/\lambda_{\text{emit}}$:

$$\boxed{\;1 + z = \frac{a(t_{\text{obs}})}{a(t_{\text{emit}})} = \frac{a_0}{a(t_{\text{emit}})} = \frac{1}{a(t_{\text{emit}})}\;}$$

(using the convention $a_0 = a(t_{\text{today}}) = 1$). This is a purely geometric result — it holds for any FLRW spacetime regardless of the matter content.

9. CMB Temperature Scaling

Step 1 — Photon gas thermodynamics: The CMB has a near-perfect blackbody spectrum. For a thermal photon gas, the energy density is $u \propto T^4$ (Stefan-Boltzmann law) and $\rho_r \propto a^{-4}$ from the equation of state.

Step 2 — Adiabatic expansion preserves blackbody form: Each photon frequency redshifts as $\nu \propto 1/a$, and the number density dilutes as $a^{-3}$. The Planck distribution $B_\nu \propto \nu^3/(e^{h\nu/k_BT}-1)$ retains its form if and only if $T \propto 1/a$.

Step 3: Combining with the redshift-scale factor relation:

$$\boxed{\;T(z) = T_0\,(1+z), \qquad T_0 = 2.7255 \pm 0.0006\;\text{K}\;}$$

At recombination ($z \approx 1100$): $T_{\text{rec}} \approx 3000$ K — just cool enough for electrons and protons to form neutral hydrogen, rendering the universe transparent. The CMB photons we observe today have been freely streaming since that epoch.

10. Luminosity Distance

Step 1 — Definition: Define $d_L$ so that the observed flux from a source of luminosity $L$ satisfies the inverse-square law: $F = L / (4\pi d_L^2)$.

Step 2 — Two redshift factors: In an expanding universe the received flux is reduced by two powers of $(1+z)$: one from the photon energy redshifting ($E \propto 1/(1+z)$), and one from the reduced photon arrival rate ($dt_{\text{obs}}/dt_{\text{emit}} = 1+z$).

Step 3: The photons are spread over a sphere of comoving radius $d_M$ (the transverse comoving distance), so at the observer the proper area is $(a_0\,d_M)^2 = d_M^2$. Including both $(1+z)$ factors:

$$\boxed{\;d_L = (1+z)\,d_M\;}$$

where the comoving distance is $d_M = c\!\int_0^z \frac{dz'}{H(z')}$ (for a flat universe, $k=0$).

Application: Type Ia supernovae are standardisable candles with known intrinsic luminosity $L$. Measuring their apparent brightness gives $d_L(z)$, which in 1998 revealed the accelerating expansion of the universe (Riess et al.; Perlmutter et al.).

11. Angular Diameter Distance

Step 1 — Definition: For an object of known proper size $\ell$ subtending angle $\delta\theta$, define $d_A \equiv \ell / \delta\theta$.

Step 2: At the time of emission the object's proper size relates to its comoving size via the scale factor: $\ell = a(t_{\text{emit}})\,d_M\,\delta\theta$. Therefore:

$$\boxed{\;d_A = \frac{d_M}{1+z}\;}$$

Etherington reciprocity relation: Combining the two distance measures:

$$d_L = (1+z)^2\,d_A$$

This is a model-independent identity valid in any metric theory of gravity. Notably, $d_A$ has a maximum — very distant objects can appear larger than closer ones (the angular size turnover), a distinctive signature of expanding space.

12. Age of the Universe

Step 1: From the definition $H = \dot{a}/a$ and the substitution $a = 1/(1+z)$, we have $dt = -dz/[(1+z)H(z)]$.

Step 2 — Express H(z): Define $E(z) \equiv H(z)/H_0$. From the first Friedmann equation with each component scaling appropriately:

$$E(z) = \sqrt{\,\Omega_m(1+z)^3 + \Omega_r(1+z)^4 + \Omega_\Lambda + \Omega_k(1+z)^2\,}$$

Step 3 — Integrate from the Big Bang to today:

$$\boxed{\;t_0 = \frac{1}{H_0}\int_0^{\infty} \frac{dz}{(1+z)\,E(z)}\;}$$

Numerical result: For the Planck 2018 best-fit parameters ($H_0 = 67.4$ km/s/Mpc, $\Omega_m = 0.315$, $\Omega_\Lambda = 0.685$):

$$t_0 \approx 13.80 \pm 0.02 \;\text{Gyr}$$

Note that this is slightly larger than the naive Hubble time $1/H_0 \approx 14.5$ Gyr because the universe has been accelerating recently, meaning it was expanding more slowly in the past.

Summary of Key Cosmological Equations

#EquationExpressionPhysical Meaning
1FLRW Metric$ds^2 = -c^2dt^2 + a^2\!\left[\frac{dr^2}{1-kr^2}+r^2d\Omega^2\right]$Geometry of homogeneous expanding space
2Friedmann I$H^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3}$Expansion rate from energy content
3Friedmann II$\frac{\ddot{a}}{a} = -\frac{4\pi G}{3}(\rho+\frac{3p}{c^2})+\frac{\Lambda c^2}{3}$Acceleration / deceleration of expansion
4Fluid Equation$\dot{\rho}+3H(\rho+p/c^2)=0$Energy conservation in expanding space
5Equation of State$p = w\rho c^2;\;\rho\propto a^{-3(1+w)}$Pressure-density relation and scaling
6Critical Density$\rho_c = 3H_0^2/(8\pi G);\;\sum\Omega_i=1$Boundary between open and closed universe
7Hubble's Law$v = H_0\,d$Recession velocity proportional to distance
8Redshift–Scale Factor$1+z = 1/a(t_{\text{emit}})$Wavelength stretches with expansion
9CMB Temperature$T = T_0(1+z) = 2.725(1+z)$ KPhoton temperature scales with redshift
10Luminosity Distance$d_L = (1+z)\,d_M$Distance from standard candle brightness
11Angular Diameter Dist.$d_A = d_M/(1+z)$Distance from standard ruler angular size
12Age of Universe$t_0 = H_0^{-1}\!\int_0^\infty\!\frac{dz}{(1+z)E(z)}$Cosmic age from integrating expansion history

📺 Video Lectures

Explore profound connections between Einstein's general relativity, quantum mechanics, and the origin of the universe's large-scale structure.

Einstein's Cosmos and the Quantum: Origin of Space, Time, and Structure

Comprehensive lecture on how quantum fluctuations in the early universe, amplified by cosmic inflation, seeded the large-scale structure we observe today. Bridges general relativity with quantum field theory in curved spacetime.

Key Topics:

  • Einstein's field equations and spacetime geometry
  • Quantum fluctuations and their amplification during inflation
  • CMB anisotropies as a window into quantum origins
  • Formation of large-scale structure from quantum seeds
  • Interface between GR and quantum field theory
  • The role of cosmological constant and dark energy

Related Courses

General Relativity

Foundation: Einstein field equations, FLRW metric, and spacetime curvature

Black Holes

Schwarzschild, Kerr solutions, Hawking radiation, and astrophysical observations

Quantum Field Theory

Essential for inflation: quantum fluctuations in curved spacetime

Quantum Gravity

Beyond classical cosmology: string theory, loop quantum gravity, and the Planck epoch

"The nitrogen in our DNA, the calcium in our teeth, the iron in our blood, the carbon in our apple pies were made in the interiors of collapsing stars. We are made of starstuff."— Carl Sagan

Learning Path & Prerequisites

Prerequisite
Foundation
Core
Advanced
Application
General Relativity
Statistical Mechanics
Quantum Field Theory
Particle Physics
FLRW Universe
Thermal History
Cosmic Inflation
CMB Physics
Structure Formation
Dark Sector
Cosmological Pert.
Observational
Beyond ΛCDM
Astrophysics
Early Universe
Quantum Gravity

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