Cosmology

The universe has a history, and we can read it. Over the last thirty years, cosmology stopped being the business of order-of-magnitude arguments and became a precision science, with six numbers — the Lambda-CDM parameters — that fit essentially every observation of the sky. This page is about where those numbers come from, what they mean physically, and where the current cracks in the model are.

1. Why cosmology became a precision science

In 1964, two Bell Labs engineers — Arno Penzias and Robert Wilson — were trying to eliminate a persistent hiss in a horn antenna they wanted to use for satellite communications. They cleaned the instrument, evicted pigeons, scrubbed the "white dielectric material" the pigeons left behind, and the hiss stayed. It turned out to be blackbody radiation at about 3 Kelvin, coming from every direction in the sky with startling uniformity. Theoretical physicists at Princeton had just predicted such radiation as the leftover glow of a hot early universe. Penzias and Wilson had accidentally detected the afterglow of the big bang, and cosmology changed overnight from philosophy to data.

In the half century since, three generations of satellites (COBE, WMAP, Planck), ground-based CMB telescopes (BICEP, ACT, SPT), galaxy surveys (SDSS, BOSS, DES, DESI), type Ia supernova campaigns, and gravitational-wave detectors have turned cosmology into a percent-level enterprise. We now know the age of the universe (13.8 billion years) to about 0.2%, the curvature (essentially flat) to better than 1%, and the fractions of matter and dark energy to a couple of percent each. Most cosmologists today spend their careers arguing about the third decimal place of numbers that were order-of-magnitude guesses thirty years ago.

Why should anyone outside the field care? Three reasons. First, cosmology is the world's largest and oldest particle-physics experiment. The big bang probes energy scales the LHC cannot reach, and its leftovers — baryon asymmetry, light-element abundances, dark matter density — constrain physics far beyond the Standard Model. Second, the same general-relativity equations that govern black holes and gravitational waves govern the universe as a whole; cosmology is where you see GR at work on the biggest possible scale. Third, the "six numbers" model (Lambda-CDM) is one of the cleanest examples of a simple theory winning over complicated alternatives. It is a useful model for thinking about what a mature physical theory looks like.

2. Vocabulary cheat sheet

A small number of symbols show up on every cosmology page. Skim them now and return when the notation looks opaque.

A unit warning: cosmologists measure distances in megaparsecs (1 Mpc $\approx 3.26$ million light-years), densities in "critical density" units, and time in gigayears. Little-$h$ denotes the Hubble parameter in units of 100 km/s/Mpc: $h \approx 0.7$. Many published quantities carry implicit factors of $h$.

3. The Friedmann equations — general relativity meets a uniform universe

The cosmological principle says the universe, on the largest scales, is homogeneous (the same everywhere) and isotropic (the same in every direction). If you put either of those into Einstein's field equations of general relativity, the result is the Friedmann-Lemaître-Robertson-Walker (FLRW) metric and its two dynamical equations — the Friedmann equations.

The second Friedmann equation comes from taking the time derivative and using energy conservation. It tells you the rate of change of $H$:

The third equation you need is the continuity equation — it is the first law of thermodynamics applied to an expanding volume:

4. The hot big bang and big-bang nucleosynthesis

Run the Friedmann equations backward and the universe gets hotter, denser, and younger. Many things happen on the way. The main checkpoints:

• Planck time ($t \sim 10^{-43}$ s, $T \sim 10^{32}$ K). We do not know what happened here; quantum gravity is required, and we do not have a working theory of it. See frontier physics.
• Inflation (somewhere around $t \sim 10^{-36}$ to $10^{-34}$ s). The universe expands by a factor of $\sim e^{60}$ in much less than a microsecond. Makes the universe flat, large, and smooth.
• Electroweak phase transition ($t \sim 10^{-12}$ s, $T \sim 100$ GeV). The Higgs picks a vacuum value and the weak bosons acquire mass.
• Quark-gluon plasma condensing into hadrons ($t \sim 10^{-5}$ s, $T \sim 200$ MeV). Quarks get trapped inside protons and neutrons.
• Big-bang nucleosynthesis (BBN) ($t \sim 1$–$200$ s, $T \sim 1$ MeV to 0.1 MeV). Protons and neutrons fuse into deuterium, helium-3, helium-4, and a trace of lithium-7. The predicted abundances (about 75% hydrogen, 25% helium-4 by mass, with tiny D and Li fractions) match observation to within a few percent. BBN is one of the most precise tests of the hot big bang picture.
• Recombination ($t \sim 380{,}000$ years, $T \sim 3000$ K, $z \approx 1100$). Electrons combine with nuclei into neutral atoms. Photons stop scattering. The CMB is the snapshot at this moment.
• Reionization ($z \sim 6$–$10$). First stars and galaxies light up and re-ionize the intergalactic medium.
• Matter-$\Lambda$ equality ($z \approx 0.3$). $\Lambda$ takes over from matter as the dominant term. Cosmic acceleration kicks in.

5. The cosmic microwave background

At $z \approx 1100$, the universe became cool enough (about 3000 K) for electrons to bind to protons and form neutral hydrogen. Before that, free electrons scattered photons efficiently; after, photons were free to travel. The photons we see today as the CMB are the ones that last scattered at that moment. They have been redshifting ever since, and today they appear as a nearly uniform 2.7255 K blackbody.

Two things about the CMB are spectacular. First, its blackbody spectrum — as measured by COBE/FIRAS in 1990 — is the most perfect blackbody ever found in nature, agreeing with the Planck curve to better than 50 parts per million over the microwave band. That alone rules out any model in which photons are substantially re-processed since recombination; the universe is transparent, and the primordial spectrum survives. Second, the tiny temperature anisotropies — at the $10^{-5}$ level — carry astonishingly rich information, because their power spectrum has a series of acoustic peaks whose positions and heights depend sensitively on every one of the cosmological parameters.

The acoustic peaks are the frozen imprint of sound waves in the pre-recombination plasma. Before recombination, photons and baryons were tightly coupled into a single fluid. Overdensities oscillated: the photon pressure pushed them apart, gravity pulled them back together, and the oscillation period depended on the sound speed in the plasma (about $c/\sqrt 3$). At recombination, photons decoupled and the oscillations froze, leaving a preferred scale — the sound horizon — imprinted on the CMB. That scale is about 150 Mpc today, and you can measure it both in the CMB and (independently) in the clustering of galaxies, where it shows up as baryon acoustic oscillations (BAO).

Planck (2013, 2015, 2018) mapped the CMB with 5 arcminute resolution across the full sky, measured its power spectrum down to the noise floor at several thousand multipoles, and reduced the six Lambda-CDM parameters to uncertainties of a percent or better. WMAP did it first at lower resolution starting in 2003; COBE discovered the anisotropies at the $10^{-5}$ level in 1992, winning George Smoot and John Mather the 2006 Nobel Prize.

6. Interactive: expansion-history explorer

Drag the sliders to change the matter, radiation, and dark-energy density fractions. The chart on the right plots the scale factor $a(t)$ you get by integrating the Friedmann equation with those inputs. You will see how different mixes give radically different cosmic histories.

Things to try:

• Set $\Omega_m = 1$, $\Omega_\Lambda = 0$ (the "Einstein-de Sitter" matter-dominated universe). The curve is a cube-root power law in $t$. That is what most cosmologists believed until 1998.
• Set $\Omega_m = 0.3$, $\Omega_\Lambda = 0.7$. Matter dominates early and $\Lambda$ takes over at late times. The curve bends upward — that is cosmic acceleration, and the shape was the surprise of the century.
• Set $\Omega_\Lambda > 1$ with matter small. The universe eventually goes into pure de Sitter expansion: exponential growth, $a(t) \propto e^{Ht}$. That is also roughly what inflation looks like.

7. Inflation — solving problems the hot big bang cannot

The hot big bang model, as described so far, has two embarrassments. First, the horizon problem: different patches of the CMB sky that you see today were never in causal contact with each other in the standard expansion history. Yet they have the same temperature to one part in $10^5$. Why?

Second, the flatness problem: in the Friedmann equation, the curvature term scales as $1/a^2$, while the matter and radiation terms scale as $1/a^3$ and $1/a^4$. So the curvature grows in relative importance as the universe expands. For the universe to be approximately flat today, it had to be almost unimaginably flat early on — fine-tuned to something like $10^{-60}$ at the Planck time. Why?

Alan Guth's 1980 insight, building on earlier ideas by Starobinsky, was that both problems disappear if the very early universe went through a brief period of exponential expansion. During that phase, a single small causally-connected patch gets stretched to encompass the whole observable universe today (solving the horizon problem), and any initial curvature is diluted away (solving the flatness problem). The mechanism is a scalar field — the "inflaton" — whose potential energy is roughly constant during slow-roll, giving an effective $\Lambda$ many orders of magnitude larger than the present one.

Inflation also explains, as a bonus, the primordial density fluctuations that later seeded galaxies and the CMB anisotropies. Quantum fluctuations of the inflaton get stretched outside the Hubble horizon during inflation and become classical, scale-invariant density perturbations. The CMB power spectrum has a nearly but not exactly scale-invariant tilt — the measured scalar spectral index is $n_s \approx 0.965$, close to but not exactly 1, as slow-roll predicts. That percent-level deviation from scale invariance is one of the cleanest experimental fingerprints of inflation.

What inflation has not yet done is produce a detected signal of primordial gravitational waves. Inflation predicts a tensor power spectrum from quantum fluctuations of the metric itself, characterized by a tensor-to-scalar ratio $r$. Upper limits on $r$ are around $0.03$ as of 2025 (BICEP/Keck + Planck). A detection would directly measure the energy scale of inflation and would be strong confirmation. A non-detection keeps ruling out models.

8. Dark matter and dark energy — the 95%

Dark matter

The evidence for dark matter is independent across half a dozen unrelated observations, and that independence is what makes the case so airtight.

• Galaxy rotation curves. Vera Rubin's 1970s measurements of spiral galaxies showed that stars in the outskirts orbit at almost the same speed as stars deep inside. Newtonian gravity with only the visible mass would predict a Keplerian falloff. Something extra — five or ten times the visible mass, distributed in a roughly spherical halo — has to be there.
• Galaxy clusters. Fritz Zwicky noticed as early as 1933 that galaxies in the Coma cluster move faster than the cluster's visible mass can gravitationally bind. The required "missing mass" is about the same ratio Rubin found in individual galaxies.
• Gravitational lensing. Background galaxies are distorted into arcs and multiple images by the gravity of foreground clusters. The mass required to reproduce the images is several times the visible mass. The Bullet Cluster (2006) spectacularly showed the lensing mass and the X-ray hot-gas mass in different places — a smoking gun that the gravitating stuff is not just hidden ordinary matter.
• CMB acoustic peaks. The ratio of heights of the first three CMB peaks depends on how much non-interacting "dust" (cold dark matter) versus photon-pressure-sensitive baryons there is. The peak pattern fixes $\Omega_b \approx 0.049$ and $\Omega_{\text{cdm}} \approx 0.265$ cleanly.
• Large-scale structure. The growth of cosmic structure from the tiny CMB fluctuations to today's galaxies and filaments requires non-relativistic matter to clump, and there is not nearly enough baryonic matter to do it on time.

What dark matter is made of, nobody knows. The leading candidates are weakly interacting massive particles (WIMPs), axions, sterile neutrinos, and primordial black holes. Direct-detection experiments like LUX-ZEPLIN, XENONnT, and PandaX have pushed WIMP cross-sections below $10^{-47}\,\text{cm}^2$ at the 100 GeV scale. Axion haloscopes (ADMX) have covered a small slice of parameter space and continue to expand. So far nobody has found a dark-matter particle, but the observational case for something being there is overwhelming.

Dark energy and the cosmological constant

In 1998, two independent teams (Saul Perlmutter's Supernova Cosmology Project and Brian Schmidt / Adam Riess's High-Z Supernova Team) measured distances to several dozen type Ia supernovae and found that the distant ones were fainter than expected in a matter-dominated universe. The simplest interpretation: the expansion of the universe is accelerating. Both teams shared the 2011 Nobel Prize. The fitted fraction of dark energy came out to $\Omega_\Lambda \approx 0.7$, and subsequent CMB, BAO, and weak-lensing measurements have independently converged on the same number.

It is technically possible dark energy is not a cosmological constant but a slowly rolling scalar field (quintessence), in which case $w$ would drift slightly from $-1$. DESI's 2024–2025 BAO data hinted at a possible time-varying $w$, though the significance is still modest. This remains one of the most active observational questions.

9. Large-scale structure and the cosmic web

Zoom out from the Milky Way and galaxies are not randomly distributed. They string together along filaments, concentrate at nodes where filaments meet, and leave vast empty voids in between. The pattern — the cosmic web — was predicted by cold-dark-matter N-body simulations in the 1980s (the Millennium Simulation is the famous visualization) and has since been mapped by galaxy redshift surveys (2dF, SDSS, BOSS, DESI).

The growth of this structure is a linear-theory calculation for small overdensities and a simulation problem for large ones. In linear theory, a density contrast $\delta \equiv (\rho - \bar\rho)/\bar\rho$ evolves under gravity and Hubble friction:

The cosmological principle — homogeneous and isotropic on the largest scales — is an averaging statement. Locally, the universe is extremely lumpy; galaxies are concentrated matter, voids are nearly empty. But when you average over scales of a few hundred megaparsecs and up, the differences wash out. The observed two-point correlation function of galaxies, and the BAO bump at 150 Mpc, confirm the same scale-invariant small-fluctuation picture that inflation predicts.

10. A toy Friedmann integrator

Here is a short script that integrates the Friedmann equation numerically for a Lambda-CDM universe. You can change the density parameters and see how the scale factor evolves.

import numpy as np

# Lambda-CDM parameters (Planck 2018, approximate)
H0   = 67.4                 # km/s/Mpc
Om   = 0.315
Or   = 9.2e-5               # radiation (photons + neutrinos)
Ol   = 1.0 - Om - Or         # flat universe

def H_over_H0(a):
    # E(a) = sqrt(Om/a^3 + Or/a^4 + Ol) for flat LambdaCDM
    return np.sqrt(Om / a**3 + Or / a**4 + Ol)

# Integrate dt = da / (a H) from a_start -> a_end to get age(a)
def age_of_universe(a_target=1.0, N=10000):
    a = np.linspace(1e-8, a_target, N)
    integrand = 1.0 / (a * H_over_H0(a))
    # Trapezoidal rule, in units of 1/H0
    t = np.trapz(integrand, a)
    # Convert from 1/H0 to Gyr: 1/H0 = (1/67.4) * (Mpc / km s^-1) ~ 14.5 Gyr
    Gyr_per_inv_H0 = 977.8 / H0
    return t * Gyr_per_inv_H0

print(f"Age today:               {age_of_universe(1.0):.2f} Gyr")
print(f"Age at recombination:    {age_of_universe(1/1100):.2e} Gyr")
print(f"Matter-Lambda equality:  a = {(Om/Ol)**(1/3):.3f}")
print(f"Matter-radiation equal:  a = {Or/Om:.2e}")

import math

H0, Om, Or = 67.4, 0.315, 9.2e-5
Ol = 1.0 - Om - Or

def E(a):
    return math.sqrt(Om / a**3 + Or / a**4 + Ol)

# Simple midpoint integration
def age(a_target, N=20000):
    da = (a_target - 1e-8) / N
    t = 0.0
    for i in range(N):
        a = 1e-8 + (i + 0.5) * da
        t += da / (a * E(a))
    return t * 977.8 / H0

print(f"age today = {age(1.0):.2f} Gyr")

Things the output makes concrete:

• Matter-radiation equality sits at $a \approx 3 \times 10^{-4}$ ($z \approx 3400$). Before that, radiation dominated; after, matter did.
• Matter-$\Lambda$ equality sits at $a \approx 0.77$ ($z \approx 0.3$). That is when the expansion flipped from decelerating to accelerating.
• The age of the universe comes out at $13.8$ Gyr — the number you see on every Wikipedia sidebar.

11. Cheat sheet

See also