Fundamental Equations of the Big Bang

The hot, dense origin of everything — its evidence, its timeline from the Planck epoch to first light, and its open mysteries  ·  Summer 2026

0

What Is the Big Bang — and What It Is Not

Not an explosion in space, but an expansion of space

The Big Bang theory is the well-tested account of how the Universe evolved from an extremely hot, dense early state and has been expanding and cooling ever since. It is worth clearing up what it does not claim. It is not an explosion that went off at a point in pre-existing space — there was no "outside" and no center. Instead, space itself stretches, carrying galaxies apart everywhere at once. And the theory does not describe the instant t = 0; the "singularity" is where our equations break down, not a established event. The Big Bang describes everything after the first sliver of a second, with growing confidence the later we look.

Its credibility rests on several independent pillars of evidence that all point to the same hot origin, and on a remarkably detailed timeline we can compute and, increasingly, observe:

Planck epoch — the first 10⁻⁴³ s, where quantum gravity rules and physics is unknown. Inflation — a burst of expansion that smoothed and seeded the cosmos. Baryogenesis — the tiny matter–antimatter imbalance that left everything we see. Quark–gluon plasma — a fireball of fundamental particles, cooling and freezing out. Nucleosynthesis — the first nuclei, in the first few minutes. Recombination — atoms form, light escapes as the CMB. Dark ages → first light — gravity builds the first stars and galaxies.

As on the companion stellar, solar, cosmology, and black-hole sheets, every equation is paired with a plain-language reading of what it physically asserts, a Use in Research column with key references, and each section ends with the open unknowns — which, for the first instant, remain some of the deepest in science. Toggle the Dark theme at top-right for a dark background.

I

The Pillars of Evidence

5 equations

The Big Bang is accepted not because of one observation but because several completely independent lines of evidence converge on the same hot, expanding origin — and rule out the alternatives.

NameEquationVariablesUse in Research
Hubble–Lemaître Expansion \[ v = H_0\,d \]
The first pillar: distant galaxies recede faster the farther they are, because space is stretching. Run the expansion backward and everything was once together — the original argument for a beginning.
 v = recession velocity; d = distance; H₀ ≈ 70 km/s/Mpc
The discovery that launched the theory; you fit galaxy redshifts vs. distances to confirm uniform expansion and measure its rate.
Key referencesLemaître (1927); Hubble (1929); Riess et al. (2022).
CMB Blackbody \[ B_\nu(T),\qquad T_0 = 2.7255\,\text{K} \]
The second pillar: a near-perfect blackbody glow filling the whole sky — the cooled afterglow of the hot early Universe. No other theory naturally produces it.
 T₀ = present CMB temperature; B_ν = Planck spectrum
The single most decisive piece of evidence; its flawless thermal spectrum is the relic radiation a hot Big Bang must leave behind.
Key referencesGamow (1948); Penzias & Wilson (1965); Mather et al. (1994, FIRAS).
Primordial Light Elements \[ Y_p \approx 0.247,\quad \text{D/H} \approx 2.5\times10^{-5} \]
The third pillar: the Universe should have forged ~25% helium plus traces of deuterium and lithium in its first minutes — before any star existed. Measured abundances match the prediction.
 Y_p = helium mass fraction; D/H = deuterium ratio
You measure these abundances in the most pristine gas available to test the first-minutes physics — a prediction with essentially no free parameters once the baryon density is fixed.
Key referencesAlpher, Bethe & Gamow (1948); Cyburt et al. (2016, review).
Tolman Surface-Brightness Test \[ \Sigma \propto (1+z)^{-4} \]
A sharp discriminator: in a truly expanding Universe, the surface brightness of identical galaxies must dim very steeply with redshift. A static "tired-light" cosmos predicts a much weaker dimming.
 Σ = surface brightness; z = redshift
The test you apply to standard galaxies across redshift to confirm the expansion is real, not an illusion of light losing energy en route.
Key referencesTolman (1930); Sandage & Lubin (2001).
Concordance of the Baryon Density \[ \Omega_b h^2\big|_{\rm BBN} = \Omega_b h^2\big|_{\rm CMB} \approx 0.022 \]
The clincher: two utterly different epochs — nucleosynthesis (minutes) and the CMB (380,000 years) — independently weigh ordinary matter and get the same answer. A stringent self-consistency test.
 Ω_b h² = baryon density; from D/H and from CMB peaks
You cross-check the baryon density from deuterium against the CMB acoustic peaks — agreement validates the whole hot Big Bang framework across orders of magnitude in time.
Key referencesCooke et al. (2018); Planck Collaboration (2020).
Open unknowns · Pillars of Evidence
The Hubble Tension
Why do early- and late-Universe measurements of the expansion rate disagree?
The CMB predicts H₀ ≈ 67 while the local distance ladder gives ~73 — a 5σ clash. It could be a measurement systematic or a crack in the standard model that the Big Bang framework will have to accommodate.
Lithium Problem
Why is the predicted primordial lithium three times what old stars show?
Helium and deuterium are triumphs, but ⁷Li stubbornly disagrees. Stellar depletion or new physics in the first minutes are both candidates — the one persistent flaw in BBN.
II

The Expanding Universe & the Singularity

5 equations

Running the observed expansion backward leads to ever hotter, denser conditions — and formally to a singularity where the known laws fail. These equations describe that extrapolation and where it stops being trustworthy.

NameEquationVariablesUse in Research
Scale Factor & Redshift \[ 1 + z = \frac{a_0}{a} \]
Cosmic size is tracked by the scale factor a; light stretched by a factor (1+z) left when the Universe was that many times smaller. The redshift is a direct readout of the past.
 a = scale factor; z = redshift
The conversion you use constantly to turn an observed redshift into "how small the Universe was then."
Key referencesLemaître (1927); Hubble (1929).
Friedmann Equation \[ H^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3} \]
The master equation governing how fast the Universe expands given its contents. Integrated backward, it implies density and temperature climbing without bound toward early times.
 ρ = density; k = curvature; Λ = cosmological constant
The equation you integrate (forward or back) to reconstruct the entire expansion history from the present contents.
Key referencesFriedmann (1922); Lemaître (1927).
Temperature–Time Relation \[ t \approx \left(\frac{1\,\text{MeV}}{kT}\right)^2 \,\text{s} \]
A cosmic clock-thermometer: in the radiation era, the age in seconds is fixed by the temperature. It lets us say exactly when each event happened from the temperature alone.
 t = cosmic time; kT = thermal energy; (g*-dependent prefactor)
The workhorse you use to date every early-Universe milestone — fusion, neutrino decoupling, particle freeze-outs — straight from its temperature.
Key referencesWeinberg (1977, The First Three Minutes); Kolb & Turner (1990).
Radiation-Era Expansion \[ H = \frac{1}{2t},\qquad a \propto t^{1/2} \]
During the radiation-dominated era the Universe expands at a precise, simple pace set only by its age. This makes the first ~50,000 years exactly solvable.
 H = expansion rate; t = time; a = scale factor
The relation you plug into freeze-out and nucleosynthesis calculations, which all hinge on the competition between reaction rates and \(H\).
Key referencesFriedmann (1922); Peebles (1993).
The Initial Singularity \[ \rho \to \infty \;\;\text{as}\;\; a \to 0 \]
Taken literally, the equations say density and curvature diverge at t = 0. This is not a confirmed event but a signal that general relativity has broken down and quantum gravity is needed.
 ρ = density; a = scale factor → 0
The boundary marker telling you where classical cosmology stops being valid — the motivation for everything in the Planck-epoch section.
Key referencesPenrose (1965); Hawking & Penrose (1970).
Open unknowns · Expansion & Singularity
Was There a Singularity?
Did the Universe truly begin in a singularity, or did something precede it?
Bounces, eternal inflation, and cyclic models all replace the singularity with something else. Without a theory of quantum gravity we cannot say what — if anything — t = 0 means.
Arrow of Time
Why did the Universe begin in such a low-entropy state?
The smooth, ordered early Universe set the thermodynamic arrow of time. Why initial conditions were so special is a profound, unresolved question.
III

The Planck Epoch & the First Instant

5 equations

The earliest moment we can even name is the Planck epoch, where gravity becomes as strong as the quantum effects it acts on. Here our equations give only scales, not physics — this is the frontier of the unknown.

NameEquationVariablesUse in Research
Planck Time \[ t_P = \sqrt{\frac{\hbar G}{c^5}} \approx 5.4\times10^{-44}\,\text{s} \]
The shortest meaningful interval — before it, quantum gravity rules and "time" may not behave as we know it. It marks the absolute earliest moment any current theory can address.
 ħ, G, c = the three fundamental constants
The lower limit on the timeline — you quote the Planck time as where the known-physics story must begin.
Key referencesPlanck (1899); Wheeler (1955).
Planck Temperature \[ T_P = \sqrt{\frac{\hbar c^5}{G k_B^2}} \approx 1.4\times10^{32}\,\text{K} \]
The highest temperature with any physical meaning — the heat of the Universe at the Planck time. Above it, known physics simply has nothing to say.
 k_B = Boltzmann constant; ħ, c, G
The top of the temperature axis on any Big Bang timeline — the reference scale for "grand unified" and quantum-gravity energies.
Key referencesPlanck (1899); Kolb & Turner (1990).
Planck Energy / Mass \[ E_P = \sqrt{\frac{\hbar c^5}{G}} \approx 1.22\times10^{19}\,\text{GeV} \]
The energy at which a particle's quantum wavelength shrinks to its own gravitational radius — where gravity and quantum mechanics must merge into one theory.
 E_P = Planck energy; m_P = E_P/c²
The benchmark energy that defines the quantum-gravity regime; you compare any proposed early-Universe energy scale against it.
Key referencesPlanck (1899); Misner, Thorne & Wheeler (1973).
Grand Unification Scale \[ E_{\rm GUT} \sim 10^{16}\,\text{GeV},\quad t \sim 10^{-36}\,\text{s} \]
The energy where the strong, weak, and electromagnetic forces may merge into one. As the Universe cooled below it, that unity broke — possibly triggering inflation.
 E_GUT = grand-unification energy; t = corresponding time
The scale where particle physics meets cosmology — proposed GUT phase transitions are candidates for both inflation and baryogenesis.
Key referencesGeorgi & Glashow (1974); Langacker (1981, review).
Quantum-Gravity Condition \[ \lambda_{\rm Compton} \sim r_s \;\Rightarrow\; m \sim m_P \]
When a particle's quantum "size" equals its own black-hole radius, gravity can no longer be ignored in quantum physics. That coincidence defines the Planck scale and the limit of present theory.
 λ_Compton = ħ/mc; r_s = 2Gm/c²
The reasoning behind why the Planck scale is special — it tells you where you can no longer treat gravity classically.
Key referencesPlanck (1899); Wheeler (1955, spacetime foam).
Open unknowns · The Planck Epoch
Quantum Gravity
What physics governs the Planck epoch?
String theory, loop quantum gravity, and other approaches all aim here, but none is confirmed. Without a theory uniting gravity and quantum mechanics, the first instant is genuinely beyond us.
Are the Forces Unified?
Do the fundamental forces truly merge at high energy?
Grand unification is elegant and motivates inflation and baryogenesis, but proton decay — its key prediction — has never been seen, leaving the idea unproven.
IV

Inflation

5 equations

A fraction of a second in, the Universe may have undergone a stupendous burst of accelerated expansion. Inflation elegantly explains why the cosmos is so uniform, so flat, and seeded with exactly the right ripples.

NameEquationVariablesUse in Research
Exponential Expansion \[ a(t) \propto e^{Ht} \]
Driven by a near-constant energy, space doubles in size again and again at a steady rate. A tiny, smooth patch balloons to encompass everything we now see — solving the horizon problem.
 H = (nearly constant) inflationary rate
The defining behaviour you assume to explain why opposite sides of the CMB sky share a temperature despite never having been in contact.
Key referencesGuth (1981); Linde (1982); Albrecht & Steinhardt (1982).
Number of e-Folds \[ N = \ln\frac{a_{\rm end}}{a_{\rm start}} \gtrsim 60 \]
Counts how many times space doubled (in natural units) during inflation. At least ~60 are needed to flatten the cosmos and explain its uniformity.
 N = e-folds; a = scale factor
The minimum amount of inflation a model must deliver — you compute N for a candidate potential and check it clears ~60.
Key referencesGuth (1981); Liddle & Lyth (2000).
Slow-Roll Condition \[ \epsilon = \frac{m_P^2}{2}\left(\frac{V'}{V}\right)^2 \ll 1 \]
For inflation to last, the field driving it must roll very slowly down a nearly flat energy hill, keeping the expansion rate almost constant. This parameter measures the slope.
 V = inflaton potential; m_P = Planck mass
The quantity linking an abstract inflaton model to measurable CMB observables — small ε means prolonged inflation and a near-flat spectrum.
Key referencesSteinhardt & Turner (1984); Liddle & Lyth (2000).
Quantum Seeds of Structure \[ \frac{\delta\rho}{\rho} \sim \frac{H^2}{2\pi\,\dot\phi} \]
The astonishing claim that all cosmic structure grew from quantum jitters in the inflaton field, stretched to astronomical size. Galaxies are inflated quantum fluctuations.
 H = inflation rate; φ̇ = field roll speed
The bridge from inflation to the CMB ripples and the galaxy distribution — you compute this spectrum and confront it with observations.
Key referencesMukhanov & Chibisov (1981); Guth & Pi (1982); Bardeen, Steinhardt & Turner (1983).
Reheating Temperature \[ T_{\rm reh} \sim \left(\frac{90}{\pi^2 g_*}\right)^{1/4}\sqrt{\Gamma_\phi m_P} \]
When inflation ends, the inflaton's energy dumps into a hot bath of particles, "reheating" the cold, empty post-inflation Universe into the familiar hot Big Bang.
 Γ_φ = inflaton decay rate; g* = particle species
The temperature at which the hot Big Bang proper begins; it bounds when baryogenesis and nucleosynthesis can occur.
Key referencesAlbrecht et al. (1982); Kofman, Linde & Starobinsky (1997, reheating).
Open unknowns · Inflation
Did Inflation Happen?
Is inflation real, and what field drove it?
It explains many facts elegantly but has never been confirmed, and there is no known particle for the inflaton. Detecting primordial gravitational waves would clinch it.
Primordial Gravitational Waves
Can we detect the spacetime ripples inflation should have made?
A nonzero tensor-to-scalar ratio (CMB B-modes) would reveal the energy scale of inflation. Current limits (r < 0.036) keep tightening, but it may be vanishingly small.
The Multiverse
Does eternal inflation spawn endless other universes?
Many inflation models never fully stop, budding off bubble universes forever. Whether this is physics or untestable speculation is hotly debated.
V

Baryogenesis & the Matter–Antimatter Asymmetry

4 equations

The hot Big Bang should have made matter and antimatter in equal amounts, which would have annihilated to leave nothing but light. That we exist means a tiny imbalance survived — one of the deepest puzzles in physics.

NameEquationVariablesUse in Research
Baryon Asymmetry \[ \eta_B = \frac{n_b - n_{\bar b}}{n_\gamma} \approx 6\times10^{-10} \]
A measure of the matter excess: for every billion matter–antimatter pairs that annihilated, about one extra matter particle was left over. Everything you see is built from that residue.
 n_b, n_b̄ = baryon, antibaryon densities; n_γ = photons
The number any baryogenesis theory must reproduce; it's pinned independently by BBN and the CMB.
Key referencesSakharov (1967); Planck Collaboration (2020).
Sakharov Conditions \[ \Delta B \neq 0,\;\; C\text{/}CP\text{ violation},\;\; \text{out of equilibrium} \]
The three ingredients any mechanism needs to generate a matter excess from a symmetric start: baryon-number violation, a preference for matter over antimatter, and a departure from thermal balance.
 B = baryon number; C, CP = discrete symmetries
The checklist every proposed baryogenesis model is tested against — a model failing any one cannot produce the asymmetry.
Key referencesSakharov (1967); Riotto & Trodden (1999, review).
Survivor Fraction \[ \frac{n_b}{n_\gamma}\bigg|_{\rm today} \approx \eta_B \approx 10^{-9} \]
After matter and antimatter annihilated, the leftover matter became vastly outnumbered by the resulting photons. Today there are ~a billion CMB photons for every atom.
 n_b = baryons; n_γ = photons (~411 cm⁻³)
The fossil photon-to-baryon ratio you read from the CMB and BBN — a direct measure of how complete the early annihilation was.
Key referencesSteigman (1976); Cyburt et al. (2016).
Electroweak Sphalerons \[ \Delta(B+L) \neq 0,\quad \Delta(B-L) = 0 \]
A subtle Standard-Model process that can convert lepton excess into baryon excess at high temperature — central to leading "leptogenesis" theories of the asymmetry.
 B = baryon number; L = lepton number
The mechanism leptogenesis exploits — generate a lepton asymmetry (e.g. via heavy neutrino decays), then let sphalerons reprocess it into baryons.
Key references't Hooft (1976); Kuzmin, Rubakov & Shaposhnikov (1985); Fukugita & Yanagida (1986).
Open unknowns · Baryogenesis
Why Is There Matter at All?
What mechanism generated the matter–antimatter asymmetry?
The Standard Model can't do it — its CP violation is far too small. Leptogenesis, GUT baryogenesis, and electroweak baryogenesis are candidates, all requiring new physics, none confirmed.
Neutrino Nature
Are neutrinos their own antiparticles, enabling leptogenesis?
If neutrinos are "Majorana" particles, heavy partners could have generated the lepton asymmetry. Neutrinoless double-beta-decay experiments are hunting for this clue.
VI

The Hot Plasma & Thermal History

5 equations

For its first minutes the Universe was a seething, near-perfect thermal soup of fundamental particles. As it expanded and cooled, it passed through transitions where particles condensed, annihilated, or froze out, each leaving a trace.

NameEquationVariablesUse in Research
Temperature–Redshift \[ T = T_0\,(1+z) \]
The Universe was hotter in the past in exact proportion to how much smaller it was. Expansion is literally what cooled the primordial fireball to today's 2.7 K.
 T₀ = 2.725 K; z = redshift
The relation that lets you assign a temperature — and hence the active physics — to any past epoch.
Key referencesGamow (1948); Alpher & Herman (1948).
Radiation Energy Density \[ \rho_r = \frac{\pi^2}{30}\,g_*(T)\,\frac{(k_BT)^4}{(\hbar c)^3} \]
The energy packed into all the relativistic particles, counting how many species are present. It dominated the early Universe and set the expansion rate during the first 50,000 years.
 g* = relativistic species; T = temperature
The density you feed into the Friedmann equation for the radiation era; the species count g* drops in steps as the Universe cools.
Key referencesKolb & Turner (1990, The Early Universe).
Quark–Hadron Transition \[ kT_{\rm QCD} \approx 150\,\text{MeV},\quad t \sim 10^{-5}\,\text{s} \]
When the Universe cooled below this temperature, free quarks and gluons condensed into protons, neutrons, and other hadrons — the moment ordinary matter's building blocks first formed.
 kT_QCD = QCD transition energy; t = time
A transition recreated in miniature at RHIC and the LHC, where heavy-ion collisions make quark–gluon plasma — letting us study early-Universe matter in the lab.
Key referencesHagedorn (1965); Borsányi et al. (2016, lattice QCD).
Equilibrium Number Density \[ n \propto (mT)^{3/2}\,e^{-mc^2/kT} \quad (kT \ll mc^2) \]
Once it gets too cold to create a particle's mass, that species becomes exponentially rare. This Boltzmann suppression is what removes antimatter and most heavy particles as the Universe cools.
 m = particle mass; T = temperature
The starting point for any relic-abundance calculation — you track equilibrium densities until a species can no longer keep up and freezes out.
Key referencesKolb & Turner (1990); Dodelson (2003).
Freeze-Out Condition \[ \Gamma(T) = n\langle\sigma v\rangle = H(T) \]
A species stays in equilibrium only while its reactions outpace the expansion. When the Universe expands faster than particles can interact, their abundance "freezes" at whatever it was.
 Γ = reaction rate; H = expansion rate; ⟨σv⟩ = cross section
The single criterion behind every relic: neutrinos, the light elements, and (if it exists) thermal dark matter all freeze out when Γ falls below H.
Key referencesZeldovich (1965); Lee & Weinberg (1977).
Open unknowns · The Hot Plasma
What Is Dark Matter?
Did dark matter freeze out of the primordial plasma, and what is it?
The freeze-out mechanism elegantly gives the right abundance for a weakly-interacting particle (the "WIMP miracle"), but decades of searches have found nothing. Axions, sterile neutrinos, and others remain in play.
Phase Transitions
Were the early transitions smooth or violent, and did they leave relics?
A sharply first-order transition could have produced gravitational waves or magnetic fields. The nature of the electroweak and QCD transitions affects what fossils survive.
VII

Neutrino Decoupling & the Cosmic Neutrino Background

4 equations

About one second in, neutrinos stopped interacting and have streamed freely ever since — an even older relic than the CMB. This cosmic neutrino background is a direct fossil of the first second.

NameEquationVariablesUse in Research
Neutrino Decoupling \[ kT_{\rm dec} \approx 1\,\text{MeV},\quad t \approx 1\,\text{s} \]
When weak interactions became too slow to keep neutrinos coupled, they decoupled and began free-streaming — fixing their distribution one second after the start.
 kT_dec = decoupling energy; t = time
The moment that sets the neutrino background's properties; its closeness to nucleosynthesis is why neutrinos influence the helium yield.
Key referencesWeinberg (1972); Dolgov (2002, review).
Neutrino–Photon Temperature \[ \frac{T_\nu}{T_\gamma} = \left(\frac{4}{11}\right)^{1/3} \approx 0.714 \]
Because electrons and positrons annihilated after neutrinos decoupled, they reheated the photons but not the neutrinos. The neutrino background ended up slightly colder than the CMB.
 T_ν = neutrino temperature; T_γ = photon temperature
A precise, parameter-free prediction; you use it to compute the neutrino background's temperature and its contribution to the radiation density.
Key referencesWeinberg (1972); Lesgourgues & Pastor (2006, review).
Relic Neutrino Density \[ n_\nu = \frac{3}{11}\,n_\gamma \approx 112\,\text{cm}^{-3}\;\text{per species} \]
Space is awash in relic neutrinos — hundreds in every cubic centimetre, passing through everything. They are the second-most-abundant particle in the Universe after photons.
 n_ν = neutrino density; n_γ = photon density
The target abundance for proposed direct-detection experiments (like PTOLEMY) and the input for neutrino-mass effects on structure.
Key referencesWeinberg (1972); PTOLEMY Collaboration (2019).
Effective Neutrino Number \[ N_{\rm eff} \approx 3.046 \]
Counts the relativistic "neutrino-like" energy in the early Universe — slightly above 3 because annihilation wasn't quite instantaneous. Extra value would betray hidden particles.
 N_eff = effective species; 3 standard neutrinos
A sensitive probe of new physics measured by both BBN and the CMB; an excess could even relieve the Hubble tension.
Key referencesMangano et al. (2005); Planck Collaboration (2020).
Open unknowns · Neutrino Background
Detecting the Cν B
Can we ever directly detect the cosmic neutrino background?
It is the oldest relic in principle observable, but the relic neutrinos are so low-energy that capturing them (e.g. on tritium, as PTOLEMY aims) is at the edge of feasibility.
Neutrino Masses
What are the absolute neutrino masses, and how do they shape structure?
Massive relic neutrinos suppress small-scale structure, so the sky can weigh them where labs cannot. Upcoming surveys should detect the total mass.
VIII

Big Bang Nucleosynthesis

5 equations

In the window from about one second to a few minutes, the Universe was a cosmic fusion reactor, forging the lightest nuclei. Their leftover amounts are a precise, computable fossil of those first minutes.

NameEquationVariablesUse in Research
Neutron-to-Proton Ratio \[ \frac{n_n}{n_p} = e^{-\Delta m c^2/kT} \]
Neutrons are slightly heavier than protons, so as the Universe cooled it became harder to make them. The frozen ratio almost single-handedly fixes how much helium forms.
 Δm c² = 1.293 MeV; T = temperature
The starting point of every BBN calculation; you track it through freeze-out and neutron decay to predict the helium abundance.
Key referencesHayashi (1950); Wagoner, Fowler & Hoyle (1967).
Deuterium Bottleneck \[ kT \lesssim \frac{B_D}{\ln(1/\eta_B)},\quad B_D = 2.22\,\text{MeV} \]
Helium can't form until deuterium does, but the huge photon-to-baryon ratio destroys deuterium until the Universe is far cooler than its binding energy. This delay sets the clock for nucleosynthesis.
 B_D = deuterium binding energy; η_B = baryon-to-photon ratio
Explains why nucleosynthesis waits until ~3 minutes despite the binding energy allowing it much earlier — a key timing input to the reaction network.
Key referencesPeebles (1966); Smith, Kawano & Malaney (1993).
Primordial Helium \[ Y_p = \frac{2(n_n/n_p)}{1 + n_n/n_p} \approx 0.247 \]
Almost every surviving neutron ends up locked in a helium nucleus, so ~a quarter of all ordinary matter by mass is helium — produced before any star existed.
 Y_p = helium mass fraction; n_n/n_p ≈ 1/7
A near-parameter-free prediction you compare to pristine-gas measurements — one of the Big Bang's most precise successes.
Key referencesAlpher, Bethe & Gamow (1948); Aver et al. (2015).
Deuterium Baryometer \[ \text{D/H} \propto \eta_B^{-1.6} \]
Deuterium is fragile and easily burned to helium, so the more ordinary matter there was, the less survives. The leftover deuterium therefore precisely weighs all the ordinary matter.
 D/H = deuterium ratio; η_B = baryon density
The most sensitive measure of the cosmic baryon density — you measure D/H in pristine quasar absorbers and invert.
Key referencesEpstein, Lattimer & Schramm (1976); Cooke et al. (2018).
Effective Species (BBN) \[ Y_p \;\text{increases with}\; N_{\rm eff} \]
More relativistic species speed up the early expansion, freezing the neutron ratio higher and making more helium. So the helium yield doubles as a count of particle species in the first second.
 N_eff = relativistic species; Y_p = helium
Turns the helium abundance into a probe of new physics — you compare the measured Y_p to predictions for different N_eff.
Key referencesSteigman, Schramm & Gunn (1977); Cyburt et al. (2016).
Open unknowns · Nucleosynthesis
The Lithium Problem
Why is predicted ⁷Li three times the amount seen in old stars?
Helium and deuterium match beautifully, but lithium does not. Stellar depletion, nuclear-rate errors, or new physics in the first minutes are all candidates — BBN's lone persistent failure.
Hidden Radiation
Is N_eff exactly 3, or is extra radiation present?
A small excess would reveal sterile neutrinos or other light particles in the first second — and could help reconcile the Hubble tension.
IX

Recombination & the Cosmic Microwave Background

5 equations

For 380,000 years the Universe was an opaque plasma. Then it cooled enough for atoms to form and light to fly free — releasing the oldest light we can see, now stretched into microwaves.

NameEquationVariablesUse in Research
Saha Equilibrium — and Its Failure \[ \frac{n_e n_p}{n_H} = \left(\frac{2\pi m_e kT}{h^2}\right)^{3/2}\!e^{-\chi/kT} \]
Equilibrium ionization balance: it correctly delays recombination to T ≈ 3700 K (0.3 eV, not 13.6 eV — the billion-to-one photon excess keeps reionizing hydrogen), but it fails as recombination proceeds, because every direct recombination to the ground state emits a photon that immediately ionizes another atom. The real process is bottlenecked.
 χ = 13.6 eV; n_e, n_p, n_H = densities
The first approximation — and a teaching case in why equilibrium thermodynamics is not enough: Saha predicts ionization crashing to zero, while the true freeze-out leaves the residual electrons that enable all later gas chemistry.
Key referencesSaha (1920); Peebles (1968); Zeldovich, Kurt & Sunyaev (1968).
Peebles Equation \[ \frac{dx_e}{dt} = -C\left[\alpha_B\,n_H x_e^2 - \beta\,(1-x_e)\,e^{-E_{21}/kT}\right] \]
The non-equilibrium kinetics that actually governs recombination. Hydrogen can only reach the ground state through two slow doors — escape of redshifting Lyman-α photons, or the forbidden 2s→1s two-photon decay — and the Peebles C-factor is the probability of getting through them.
 x_e = ionization fraction; α_B = case-B recombination; β = photoionization; C = Lyα-escape + 2γ factor
The engine inside RECFAST and its successors. Planck-era precision demanded better: HyRec and CosmoRec track hundreds of atomic levels, because a 0.1% error in x_e(z) biases the damping tail enough to shift H₀ and n_s.
Key referencesPeebles (1968); Seager, Sasselov & Scott (2000, RECFAST); Ali-Haïmoud & Hirata (2011, HyRec); Chluba & Thomas (2011, CosmoRec).
The Visibility Function \[ g(\eta) = -\dot\tau\,e^{-\tau};\qquad z_* = 1089.9,\;\; \Delta z \approx 80 \]
The probability distribution of where each CMB photon last scattered. Its peak defines the last-scattering surface; its width says the "surface" is really a shell ~115,000 years deep. The CMB is a photograph with a finite exposure time.
 τ = Thomson optical depth; g(η) = last-scattering probability density
The kernel through which every line-of-sight CMB code (CAMB, CLASS) projects the photon perturbations onto the sky — anisotropy formation is an integral against g(η).
Key referencesSunyaev & Zeldovich (1970); Hu & Sugiyama (1995); Planck Collaboration (2020).
Sound Horizon & Acoustic Scale \[ r_s = \!\int_0^{\eta_*}\! \frac{c\,d\eta}{\sqrt{3(1+R)}},\;\; R \equiv \frac{3\rho_b}{4\rho_\gamma};\quad \theta_* = \frac{r_s}{D_A} \]
The distance a pressure wave travels in the photon–baryon fluid before recombination. Baryons load the fluid (R grows), slowing the sound speed below c/√3. Frozen at recombination, r_s becomes the Universe's standard ruler — visible as the acoustic peaks and, later, as BAO in the galaxy distribution.
 R = baryon loading; η* = conformal time at recombination; θ* = angular acoustic scale
θ* = (1.04109±0.0003)×10⁻² is the single best-measured quantity in cosmology (0.03%). Every claim about flatness, H₀ from the CMB, and BAO distances is a use of this ruler — which is also why "early-time" H₀-tension solutions all try to shrink r_s.
Key referencesPeebles & Yu (1970); Sunyaev & Zeldovich (1970); Hu & Dodelson (2002, review); Planck Collaboration (2020).
Sachs–Wolfe, Loading & Damping \[ \frac{\Delta T}{T}\Big|_{\rm SW} = \frac{\Phi}{3c^2};\qquad k_D^{-2} \sim \int \frac{d\eta}{n_e\sigma_T a}\;\Rightarrow\;\ell_D \approx 1350 \]
The anatomy of the power spectrum in two equations: on large scales, anisotropy is gravitational redshift out of potential wells (Sachs–Wolfe — overdense spots are cold, the famous sign surprise); on small scales, photons random-walk out of perturbations during recombination's finite thickness, erasing power exponentially beyond the diffusion (Silk) scale.
 Φ = potential at last scattering; k_D = diffusion wavenumber; ℓ_D = damping multipole
Between the SW plateau (ℓ ≲ 100), the acoustic peaks, and the damping tail (ℓ ≳ 1000) the spectrum's shape overconstrains the parameter set — the damping tail alone (ACT, SPT) independently checks N_eff, Y_p, and n_s against the peak-derived values.
Key referencesSachs & Wolfe (1967); Silk (1968); Hu & Sugiyama (1995); Planck Collaboration (2020).
Open unknowns · Recombination & CMB
Large-Scale Anomalies
Are the odd features on the CMB's largest scales real or flukes?
The cold spot, hemispherical asymmetry, and aligned low multipoles look slightly off from predictions. With only one sky to observe, distinguishing genuine anomalies from chance is genuinely hard.
Spectral Distortions
Does the CMB hide tiny departures from a perfect blackbody?
Faint distortions would record energy releases in the first months to centuries — from decaying particles to the first structures. They are predicted but still below detection.
X

Anisotropy & Spectral-Distortion Physics

5 equations

The CMB is not one observable but several: temperature, polarization, lensing, frequency spectrum — each encoding different fireball physics. This is where the Big Bang becomes a sub-percent precision laboratory, and where the next decade's instruments will dig.

NameEquationVariablesUse in Research
Matter–Radiation Equality \[ 1+z_{\rm eq} = \frac{\Omega_m}{\Omega_r} \approx 3400;\qquad k_{\rm eq} \approx 0.010\,\text{Mpc}^{-1} \]
The moment matter overtakes radiation as the dominant energy — 50,000 years in, well before recombination. Perturbations entering the horizon earlier are suppressed (radiation pressure resists collapse); later ones grow freely. The transition is permanently imprinted as the turnover of the matter power spectrum.
 z_eq = equality redshift; k_eq = horizon wavenumber at equality
The shape parameter of all large-scale structure: P(k) rises as k up to k_eq and falls as k⁻³ beyond it (the Mészáros suppression, ~k⁻⁴ relative). Galaxy surveys measure the turnover to weigh Ω_m h independently of the CMB.
Key referencesMészáros (1974); Bardeen et al. (1986, transfer functions); Eisenstein & Hu (1998).
Polarization: E-Modes & TE \[ (Q\pm iU)(\hat n) \to E_{\ell m},\,B_{\ell m};\qquad C_\ell^{TE} {\lt} 0 \;\text{at}\;\ell\sim150 \]
Thomson scattering of a local temperature quadrupole polarizes the CMB at the few-µK level. Decomposed into parity-even E and parity-odd B patterns: scalars make only E; primordial gravitational waves make both. The TE anticorrelation at ℓ ≈ 150 is the smoking gun that fluctuations existed outside the horizon — adiabatic, superhorizon, as inflation requires.
 Q, U = Stokes parameters; E, B = gradient/curl modes
Polarization doubles the information content of the CMB and is the entire future of the field: E-modes sharpen parameters, the reionization bump fixes τ, and a primordial B-mode detection would be the discovery of inflation's gravitational waves.
Key referencesKamionkowski, Kosowsky & Stebbins (1997); Zaldarriaga & Seljak (1997); Peiris et al. (2003, WMAP TE); BICEP/Keck (2021).
The Neutrino Phase Shift \[ \delta\ell \simeq -57\,\frac{R_\nu}{1+R_\nu},\qquad R_\nu = \frac{\rho_\nu}{\rho_\gamma+\rho_\nu} \]
Free-streaming neutrinos travel at c — faster than the sound waves in the plasma — and their gravity tugs the acoustic oscillations slightly ahead in phase, shifting every peak by the same small amount. A particle-physics measurement performed by sound waves.
 R_ν = neutrino fraction of radiation; δℓ = peak shift
The cleanest signature that the cosmic neutrino background really exists as free-streaming radiation at z ~ 1100: no fluid mimics it. The same phase shift has now been detected independently in the BAO of galaxy surveys.
Key referencesBashinsky & Seljak (2004); Follin et al. (2015); Baumann et al. (2019, BOSS phase shift).
Spectral Distortions (μ, y) \[ \mu \approx 1.4\!\int\! \frac{d(Q/\rho_\gamma)}{dz}dz \;\;(z\sim10^5\text{–}10^6);\qquad |\mu| {\lt} 9\times10^{-5} \]
Energy injected into the photon bath cannot fully rethermalize once double-Compton and bremsstrahlung become inefficient — it freezes in as a chemical-potential (μ) or Compton (y) distortion of the blackbody. The CMB spectrum is a calorimeter of everything that ever heated the fireball.
 Q = injected energy; μ, y = distortion amplitudes (FIRAS bounds)
The unopened information channel: decaying dark matter, primordial black hole evaporation, and dissipating small-scale acoustic waves all leave distortions. ΛCDM itself predicts a guaranteed μ ≈ 2×10⁻⁸ from Silk damping — a target, not a hope.
Key referencesSunyaev & Zeldovich (1970); Fixsen et al. (1996, FIRAS); Chluba & Sunyaev (2012); Kogut et al. (2011, PIXIE).
Recombination Radiation \[ \sim\!6.1\;\text{photons per H atom} \;\Rightarrow\; \Delta I_\nu/I_\nu \sim 10^{-9} \;\text{ripples} \]
Every hydrogen and helium atom that formed emitted line photons — Lyman, Balmer, two-photon continua — which still ride the CMB spectrum today as nK-level wiggles across GHz frequencies. The fireball's own emission-line spectrum, redshifted by 1100.
 ΔI_ν = spectral ripple amplitude; lines from H and He recombination
A guaranteed signal (no model freedom: just atomic physics and Ω_b) that would let us watch recombination spectroscopically — measuring T and x_e at z = 1100 directly, with no reliance on anisotropies whatsoever.
Key referencesZeldovich, Kurt & Sunyaev (1968); Rubiño-Martín et al. (2006); Chluba & Ali-Haïmoud (2016, CosmoSpec).
Open unknowns · Anisotropy & Distortion Physics
The Damping-Tail Tension
Why do ACT and SPT pull parameters in different directions?
The two ground-based experiments' damping-tail spectra show mild but persistent disagreements (in n_s, lensing amplitude) with each other and with Planck. Either sub-percent systematics or the first crack in the six-parameter model — current data cannot say which.
Lensing Amplitude
Is the CMB more gravitationally lensed than ΛCDM predicts?
Planck's A_L ≈ 1.18±0.07 prefers extra smoothing of the peaks at ~2.5σ — formally favoring closed geometry in CMB-only fits. BAO restores flatness, but the internal preference remains unexplained: fluke, foreground, or physics.
Will the Distortion Era Open?
Can a FIRAS-successor finally fly?
PIXIE-class missions have been proposed for 15 years without selection, despite guaranteed ΛCDM targets. Whether spectral distortions — arguably the richest unmined CMB observable — get measured this generation is a programmatic question as much as a scientific one.
XI

The Dark Ages, First Light & Reionization

4 equations

After the CMB the Universe went dark — neutral, starless gas in expanding gloom. Gravity slowly amplified the primordial seeds until the first stars ignited, ending the cosmic dark ages and re-ionizing the Universe.

NameEquationVariablesUse in Research
Growth of Structure \[ \ddot\delta + 2H\dot\delta = 4\pi G\bar\rho_m\,\delta \;\;\Rightarrow\;\; \delta \propto a \;(\text{matter era}) \]
The linear growth equation: gravity drives collapse against the friction of expansion (the 2Hδ̇ Hubble drag). In the matter era the winning mode grows exactly as the scale factor — a thousandfold from recombination to today, no more.
 δ = density contrast; H = expansion rate; ρ̄_m = mean matter density
The equation that proves dark matter exists from growth alone: baryonic seeds at recombination (δ_b ~ 10⁻⁵, suppressed by photon coupling) could grow only to ~10⁻² by now. Structure exists because dark matter — decoupled from photons — started growing at equality, 50,000 years earlier, and the baryons fell into its ready-made wells.
Key referencesLifshitz (1946); Peebles (1980); Barkana & Loeb (2001, review).
Jeans Mass at Recombination \[ M_J \propto T^{3/2}\rho^{-1/2} \sim 10^{5}\,M_\odot \]
Once atoms formed and gas pressure dropped, the smallest clouds that gravity could collapse fell dramatically — setting the mass scale of the very first bound structures.
 T = gas temperature; ρ = density
The threshold mass you compute to predict the scale of the first collapsed gas clouds, where Population III stars formed.
Key referencesPeebles & Dicke (1968); Bromm & Larson (2004, review).
21-cm Brightness Temperature \[ \delta T_b \approx 27\,x_{\rm HI}(1+\delta)\left(1-\frac{T_\gamma}{T_S}\right)\!\sqrt{\frac{1+z}{10}}\;\,\text{mK} \]
The dark ages' only observable, in full: neutral hydrogen appears in absorption or emission against the CMB depending on whether its spin temperature T_S is colder or hotter than the radiation. Every term is physics — x_HI tracks reionization, δ traces structure, and T_S records the first stars' Lyman-α light (Wouthuysen–Field coupling) and the first X-ray heating.
 x_HI = neutral fraction; T_S = spin temperature; T_γ = CMB temperature; δ = overdensity
The roadmap of cosmic dawn is the predicted sign history of δT_b: absorption as gas cools below the CMB after thermal decoupling (z ~ 200–30), deeper absorption when the first stars switch on Lyα coupling (z ~ 25–15), swing to emission with X-ray heating, fade to zero as reionization destroys x_HI. EDGES, SARAS, HERA, and ultimately SKA and lunar-farside arrays all chase pieces of this one curve.
Key referencesField (1958); Wouthuysen (1952); Madau, Meiksin & Rees (1997); Furlanetto, Oh & Briggs (2006); Bowman et al. (2018); Singh et al. (2022, SARAS 3).
Reionization Optical Depth \[ \tau = \int n_e\,\sigma_T\,c\,dt \approx 0.054 \]
The first stars and galaxies re-ionized the hydrogen, and the freed electrons slightly scattered the CMB. The amount of scattering dates when the lights came on.
 n_e = electron density; σ_T = Thomson cross section
A CMB-measured number that pins the epoch of reionization — you fit it jointly with the other parameters in the power spectrum.
Key referencesPlanck Collaboration (2020); Robertson et al. (2015).
Open unknowns · Dark Ages & First Light
The First Stars
When and how did the first (Population III) stars form, and were they top-heavy?
Metal-free gas cools differently, likely making very massive stars — but none has ever been observed. JWST is pushing toward the era, but the first generation remains unseen.
Reionization Sources
What re-ionized the Universe — small galaxies, quasars, or something else?
The photon budget is tight. Whether faint early galaxies leak enough ionizing light, or rarer luminous sources dominate, is actively debated.
The 21-cm Signal
Is the claimed cosmic-dawn 21-cm absorption real?
EDGES reported an unexpectedly deep signal in 2018; if confirmed it may hint at exotic gas cooling or new physics, but it awaits independent verification.
XII

Distances, Lookback & Horizons

5 equations

Everything on this sheet is dated and located with the machinery below — the integrals that convert a redshift into a time, a distance, and an angle. They are also where the Big Bang's deepest early puzzle (the horizon problem) is stated precisely. The companion observable-universe sheet treats the full horizon system; these are the working essentials.

NameEquationVariablesUse in Research
Lookback Time & Cosmic Age \[ t_L(z) = \int_0^z \frac{dz'}{(1+z')H(z')};\qquad t(z) = t_0 - t_L \]
Converts a redshift into how long ago the light departed — and hence the Universe's age at emission. The integrand weights each epoch by its expansion rate, so the first instants compress into almost no lookback time at enormous redshift.
 t_L = lookback time; t(z) = age at redshift z; t₀ = 13.80 Gyr
The timestamp on every observation in this sheet: it is how "z = 1090" becomes "380,000 years," and how JWST redshifts become ages that strain galaxy-formation models.
Key referencesHogg (1999); Planck Collaboration (2020); Carniani et al. (2024).
Comoving Distance \[ D_C = c\int_0^z \frac{dz'}{H(z')} \]
The expansion-corrected distance — where the source is "now," on the grid that stretches with space. It always exceeds c·t_L because the space behind the photon kept growing while it traveled; no speed limit is violated.
 D_C = comoving distance; H(z) = expansion history
The natural coordinate of cosmology: power spectra, BAO rulers, and horizon scales (r_s = 144 Mpc, k_eq = 0.01 Mpc⁻¹) are all comoving quantities, immune to the expansion that complicates proper distances.
Key referencesHogg (1999); Davis & Lineweaver (2004).
Particle Horizon & the Horizon Problem \[ D_P(t) = a(t)\,c\!\int_0^t\frac{dt'}{a(t')};\qquad \theta_{\rm hor}(z_*) \approx 1.2^\circ \]
The maximum distance causal influence can have traveled since t = 0. Evaluated at recombination it indicts the vanilla Big Bang: regions of the CMB sky separated by more than ~1° had never been in causal contact — yet share a temperature to 10⁻⁵. That is the horizon problem inflation exists to solve.
 D_P = particle horizon; θ_hor = its angular size on the CMB sky
Today's particle horizon (46.5 Gly) defines the observable universe; the recombination-era one defines the problem. Inflation's e^N stretch (section IV) makes the entire CMB sky one pre-inflation causal patch — the quantitative motivation for the ≳60 e-folds.
Key referencesRindler (1956); Guth (1981); Davis & Lineweaver (2004).
Angular-Diameter & Luminosity Distance \[ D_A = \frac{D_C}{1+z};\qquad D_L = (1+z)\,D_C = (1+z)^2 D_A \]
The two operational distances — from apparent size and apparent brightness — and the Etherington relation locking them together in any metric theory. D_A's strangest property: it turns over at z ≈ 1.6, so the most distant objects look bigger, not smaller.
 D_A = angular-diameter distance; D_L = luminosity distance
D_A converts the sound horizon into the acoustic peak angle (sections IX–X); D_L underlies the supernova Hubble diagram that found acceleration. The (1+z)⁴ ratio between them is exactly the Tolman surface-brightness dimming of section I — three pillars, one identity.
Key referencesEtherington (1933); Hogg (1999); Riess et al. (1998); Perlmutter et al. (1999).
Conformal Time \[ \eta = \int_0^t \frac{c\,dt'}{a(t')};\qquad \eta_* \approx 280\,\text{Mpc},\;\; \eta_0 \approx 14.2\,\text{Gpc} \]
The clock in which light moves at 45° regardless of expansion — the natural time of the radiation era. Causal physics before recombination happens within η, which is why the sound horizon (≈ η*/√3, baryon-corrected) and every "horizon entry" statement are conformal-time facts.
 η = conformal time (comoving units); η* = at recombination
The working coordinate of every Boltzmann code and inflation calculation: modes "enter the horizon" when kη ~ 1, acoustic oscillations are cos(kr_s) in conformal time, and the whole pre-recombination Universe fits in 280 comoving Mpc of it.
Key referencesPenrose (1963); Mukhanov, Feldman & Brandenberger (1992); Dodelson (2003).
Open unknowns · Distances & Lookback
The H₀ Tension, Again
Every distance and age on this sheet inherits an unresolved 8% dispute.
All integrals here run over H(z). Local calibration (73 km/s/Mpc) and the CMB-inferred value (67.4) disagree at ~5σ — so t₀, D_C(z), and the horizon sizes carry a systematic the error bars don't show. The resolution (new physics at recombination? calibration?) will re-date the Universe.
Ages vs. Distances
Could object ages independently arbitrate the expansion history?
Stellar-age "cosmic chronometers" measure H(z) without distance ladders, and the oldest globular clusters bound t₀ from below at ~13.5 Gyr — uncomfortably close to the Planck age. Sharpened age dating (Gaia, JWST) could turn chronology into a decisive cosmological probe.
Distance Duality at High z
Does the Etherington relation survive precision tests?
Any violation of D_L = (1+z)²D_A would betray photon non-conservation (axion mixing, dust we don't model) or non-metric gravity. Current tests hold at ~5%; GW standard sirens vs. EM distances will test it without any astrophysical calibration at all.
XIII

Synthesis: The Cosmic Timeline

4 equations

All the physics above assembles into a single, testable chronology — from a fraction of a second to 13.8 billion years. These relations tie the whole story together.

NameEquationVariablesUse in Research
Master Temperature–Time Law \[ kT \approx \frac{1\,\text{MeV}}{\sqrt{t/\text{s}}} \]
A single relation that places every early event on the timeline: give it a time and it returns the temperature, or vice versa. The backbone of the whole chronology.
 kT = thermal energy; t = time in seconds
The quick conversion you carry in your head to locate any process — fusion, decoupling, transitions — on the cosmic clock.
Key referencesWeinberg (1977); Kolb & Turner (1990).
Age of the Universe \[ t_0 = \int_0^\infty \frac{dz}{(1+z)\,H(z)} \approx 13.8\,\text{Gyr} \]
Summing all of cosmic history from the Big Bang to now gives 13.8 billion years — a number that independent stellar clocks confirm.
 H(z) = expansion history; z = redshift
The headline age, computed from the expansion integral and cross-checked against the oldest stars and clusters.
Key referencesPlanck Collaboration (2020); Valcin et al. (2021).
Three-Way Baryon Concordance \[ \Omega_b h^2\big|_{\rm D/H} = \big|_{\rm CMB} = \big|_{\rm ^4He} \approx 0.022 \]
Three independent fossils — deuterium, the CMB, and helium — separated by enormous spans of cosmic time, all agree on how much ordinary matter exists. This concordance is the Big Bang's strongest internal proof.
 Ω_b h² = baryon density from three probes
The consistency check that validates the whole framework — you confirm three epochs return the same baryon density.
Key referencesCyburt et al. (2016); Planck Collaboration (2020).
The Cosmic Budget Today \[ \Omega_\Lambda \approx 0.69,\;\; \Omega_{\rm DM} \approx 0.26,\;\; \Omega_b \approx 0.05 \]
The Big Bang's endpoint, measured today: a Universe dominated by dark energy and dark matter, with ordinary atoms a mere 5%. The hot beginning led to a profoundly dark present.
 Ω_Λ = dark energy; Ω_DM = dark matter; Ω_b = baryons
The final inventory the whole timeline produces — and the starting point for every question about the Universe's future.
Key referencesPlanck Collaboration (2020).
Open unknowns · The Cosmic Timeline
What Came Before?
Is the Big Bang truly the beginning, or part of something larger?
Eternal inflation, bounces, and cyclic models all propose a "before." Whether the question is even answerable — or testable — is itself unresolved.
The Dark 95%
What are the dark matter and dark energy the Big Bang left behind?
The framework predicts their amounts precisely yet identifies neither. They dominate the cosmic budget and the Universe's fate, and remain the biggest open questions in physics.
The Ultimate Fate
How does the story end?
If dark energy is constant, expansion accelerates forever into a cold "heat death." If it evolves, fates range from a gentle fade to a violent "Big Rip" — hinging on the equation of state we cannot yet pin down.
Big Bang reference values: age \(t_0\) = 13.8 Gyr; \(T_{\rm CMB}\) = 2.7255 K, \(n_\gamma\) ≈ 411 cm⁻³; baryon-to-photon \(\eta_B\) ≈ 6.1×10⁻¹⁰; \(\Omega_b h^2\) ≈ 0.0224; recombination \(z_*\) ≈ 1090 (380,000 yr); matter–radiation equality \(z_{\rm eq}\) ≈ 3400; neutrino decoupling ~1 s (1 MeV); nucleosynthesis ~3 min (0.1 MeV); quark–hadron transition ~10⁻⁵ s (150 MeV); \(T_\nu\) = 1.95 K; \(N_{\rm eff}\) ≈ 3.046; \(Y_p\) ≈ 0.247, D/H ≈ 2.5×10⁻⁵; \(n_s\) ≈ 0.965; Planck time 5.4×10⁻⁴⁴ s, Planck energy 1.22×10¹⁹ GeV.