13.1 Fundamentals of Big Bang cosmology

Observational Evidence and Theoretical Foundations

The Big Bang theory is the leading model for how the universe began and evolved. It rests on three major pieces of observational evidence, each independently pointing to a hot, dense origin roughly 13.8 billion years ago. Understanding these pillars, along with cosmic inflation and the cosmological principle, forms the backbone of modern cosmology.

Evidence for Big Bang Theory

Hubble's Law and the expanding universe. In the late 1920s, Edwin Hubble observed that distant galaxies are redshifted, meaning their light is stretched toward longer wavelengths. The farther away a galaxy is, the faster it appears to recede. This relationship is captured by Hubble's Law:

$v = H_0 \, d$

where $v$ is the recession velocity, $H_0$ is the Hubble constant (currently measured at roughly 67–73 km/s/Mpc, depending on the method), and $d$ is the distance to the galaxy. The direct implication: space itself is expanding. If you run the expansion backward in time, everything converges to an extremely hot, dense state.

Cosmic Microwave Background Radiation (CMB). Discovered accidentally by Arno Penzias and Robert Wilson in 1965, the CMB is a nearly uniform glow of microwave radiation filling all of space at a temperature of about 2.725 K. It's the remnant thermal radiation from the epoch of recombination, when the universe cooled enough for neutral atoms to form and photons to travel freely. The CMB's near-perfect blackbody spectrum and tiny temperature anisotropies (on the order of $\Delta T / T \sim 10^{-5}$) match Big Bang predictions with remarkable precision.

Primordial nucleosynthesis and light element abundances. During the first 3 to 20 minutes after the Big Bang, the universe was hot and dense enough for nuclear fusion to occur. This produced light elements in specific ratios: roughly 75% hydrogen, 25% helium-4 by mass, with trace amounts of deuterium, helium-3, and lithium-7. These predicted abundances closely match what astronomers observe in regions of the universe that have undergone minimal stellar processing. No other model reproduces these ratios as accurately.

Cosmic Inflation in the Early Universe

Standard Big Bang cosmology, on its own, leaves several puzzles unexplained. Cosmic inflation, proposed by Alan Guth in 1981 and refined by others, addresses them through a brief period of extraordinarily rapid expansion.

What happened: Starting at roughly $10^{-36}$ seconds after the Big Bang and lasting until about $10^{-32}$ seconds, space underwent exponential expansion, increasing in size by a factor of at least $\sim 10^{26}$ in that tiny fraction of a second.

Problems inflation solves:

• Horizon problem: Regions of the CMB on opposite sides of the sky have nearly identical temperatures, yet in a standard expansion they would never have been in causal contact. Inflation solves this because those regions were close together before inflation stretched them apart.
• Flatness problem: The universe's spatial geometry is measured to be very close to flat ($\Omega \approx 1$). Without inflation, even a slight deviation from flatness in the early universe would have grown enormously over time. Inflation drives the geometry toward flatness, much like inflating a balloon makes a small patch of its surface appear flat.
• Magnetic monopole problem: Grand unified theories predict the production of magnetic monopoles in the early universe, yet none have been observed. Inflation dilutes their density to undetectable levels by expanding the volume of space so dramatically.

Seeding structure: Quantum fluctuations during inflation were stretched to macroscopic scales, producing slight density variations. These became the seeds for all large-scale structure: galaxy clusters, superclusters, filaments, and voids. The pattern of CMB anisotropies measured by satellites like COBE, WMAP, and Planck matches inflationary predictions.

Universe Evolution and Cosmological Principles

Universe Evolution Since the Big Bang

The history of the universe can be divided into a sequence of epochs, each defined by the dominant physical processes at that time:

1. Planck epoch (0 to $10^{-43}$ s): All four fundamental forces (gravity, strong, weak, electromagnetic) are thought to be unified. Current physics cannot describe this era because general relativity and quantum mechanics have not yet been reconciled.
2. Grand Unification epoch ($10^{-43}$ to $10^{-36}$ s): Gravity separates as a distinct force. The strong, weak, and electromagnetic forces remain unified under a grand unified theory (GUT).
3. Inflationary epoch ($10^{-36}$ to $10^{-32}$ s): Exponential expansion of space occurs, as described above. The strong force separates from the electroweak force at the start of this epoch.
4. Quark epoch ($10^{-12}$ to $10^{-6}$ s): The universe is a quark-gluon plasma. Temperatures are too high for quarks to bind into hadrons.
5. Hadron epoch ($10^{-6}$ s to 1 s): The universe cools enough for quarks to combine into hadrons (protons and neutrons). Most hadron-antihadron pairs annihilate, leaving a slight excess of matter over antimatter.
6. Lepton epoch (1 s to 10 s): Leptons (electrons, neutrinos) and their antiparticles dominate. Neutrinos decouple from matter during this period.
7. Nucleosynthesis (3 to 20 minutes): Protons and neutrons fuse into light nuclei: deuterium, helium-3, helium-4, and lithium-7. After about 20 minutes, the universe expands and cools too much for further fusion.
8. Recombination and photon decoupling (~380,000 years): The temperature drops to roughly 3,000 K, allowing electrons to combine with nuclei to form neutral atoms. Photons are no longer constantly scattered and stream freely. This is the light we detect today as the CMB, now cooled to 2.725 K by the expansion of the universe.
9. Dark Ages (~380,000 years to ~150 million years): The universe is transparent but contains no luminous sources. Matter slowly clumps under gravity.
10. Reionization (~150 million to ~1 billion years): The first stars and galaxies ignite, and their ultraviolet radiation reionizes the surrounding neutral hydrogen.
11. Present day (~13.8 billion years): The universe contains large-scale structure organized into filaments and voids, with galaxies, stars, and planetary systems.

The Cosmological Principle and Its Implications

The cosmological principle states that, on sufficiently large scales (hundreds of megaparsecs and above), the universe is both homogeneous (the same everywhere) and isotropic (the same in every direction). There is no special location or preferred direction in the cosmos.

This is not just a philosophical assumption. It's supported by observations:

• Large-scale galaxy surveys (such as the Sloan Digital Sky Survey) show that matter is distributed roughly uniformly when you zoom out far enough, even though it's clumpy on smaller scales (stars, galaxies, clusters).
• CMB uniformity confirms isotropy: the temperature is the same in all directions to about 1 part in 100,000.

Why does this matter for cosmology? The cosmological principle dramatically simplifies the mathematics. If the universe is homogeneous and isotropic, its large-scale geometry can be described by a single metric, the Friedmann-Lemaître-Robertson-Walker (FLRW) metric. This metric depends on just a few parameters: the scale factor $a(t)$ (which tracks how distances in the universe change over time) and the curvature constant $k$. The Friedmann equations, derived from general relativity applied to the FLRW metric, then govern how the universe expands or contracts based on its energy content. Without the cosmological principle, building tractable cosmological models would be far more difficult.