Fundamental Equations of Cosmology

What Is Cosmology — and How Can Equations Describe Everything?

The science of the Universe as a single object

Cosmology is the audacious attempt to treat the entire Universe — all of space, time, matter, and energy — as one physical system we can write equations for. It works only because of a sweeping assumption backed by observation: on the largest scales the Universe looks the same everywhere and in every direction (the "cosmological principle"). Average over enough galaxies and the lumpiness washes out, leaving a smooth cosmos that a handful of equations can capture.

The story those equations tell is staggering. About 13.8 billion years ago the Universe was unimaginably hot and dense; it has been expanding and cooling ever since, and that expansion is now speeding up. The equations below trace this history outward in time:

Inflation — a fraction of a second of explosive expansion that smoothed and flattened everything. The hot Big Bang — a fireball of particles cooling through nuclear reactions. Nucleosynthesis — the first light elements forged in the first few minutes. Recombination — atoms form and the Universe turns transparent, releasing the cosmic microwave background. Structure formation — gravity pulls tiny ripples into galaxies and the cosmic web. Dark energy — a mysterious push now driving accelerating expansion.

As on the companion stellar and solar sheets, every equation is paired with a plain-language reading of what it physically asserts, and each section ends with the open unknowns — and in cosmology those are profound: we have precise equations for a Universe that is 95% made of dark stuff we cannot identify. Toggle the Dark theme at top-right for a dark background.

The Expanding Universe

6 equations

The founding discovery of modern cosmology: distant galaxies are rushing away from us, and the farther they are, the faster they recede. Space itself is stretching, carrying galaxies along like raisins in rising dough.

Name	Equation	Variables	Use in Research
Hubble–Lemaître Law	\[ v = H_0\,d \] The headline fact of cosmology: a galaxy twice as far away flees twice as fast. It's not that galaxies are flying through space — space itself is stretching between them, like dots drawn on an inflating balloon all drifting apart. This simple line is the first hard evidence that the Universe had a beginning.	`v` = recession speed; `d` = distance; `H₀` ≈ 70 km/s/Mpc	The relation you fit to galaxy redshifts and distances to measure H₀; running it backward is the original argument for the Big Bang. Key referencesLemaître (1927); Hubble (1929); Riess et al. (2022).
Cosmological Redshift	\[ 1 + z = \frac{\lambda_{\rm obs}}{\lambda_{\rm emit}} = \frac{a_0}{a} \] As light crosses the expanding Universe, its waves get stretched along with space, sliding toward the red end of the spectrum — the more stretch, the older and more distant the light. Measuring this "redshift" is how we clock how much the Universe has grown since the light set out.	`z` = redshift; `λ` = wavelength; `a` = scale factor (size of space)	The master observable — almost everything in cosmology is plotted against redshift, which you read directly off spectral lines. Key referencesSlipher (1917); Hubble (1929).
Scale Factor & Hubble Parameter	\[ H(t) = \frac{\dot a}{a} \] Cosmologists track the Universe's size with a single number, the "scale factor" a — set to 1 today. The Hubble parameter is just how fast a is growing relative to its current size: the Universe's expansion rate, which changes over cosmic time.	`a` = scale factor; `ȧ` = its growth rate; `H` = expansion rate	The expansion rate is the cosmos's clock; measuring \(H(z)\) versus redshift is how surveys reconstruct the contents of the Universe. Key referencesFriedmann (1922); Hubble (1929).
Hubble Distance & Time	\[ D_H = \frac{c}{H_0} \approx 4300\,\text{Mpc},\quad t_H = \frac{1}{H_0} \approx 14\,\text{Gyr} \] A rough yardstick for the size and age of the observable Universe, built from just the expansion rate and the speed of light. It's no accident that the Hubble time lands close to the true 13.8-billion-year age — they're deeply related.	`c` = speed of light; `H₀` = present expansion rate	The natural unit of cosmic size and age; you express comoving distances in terms of the Hubble distance. Key referencesHogg (1999, distance measures).
Peculiar vs Hubble Velocity	\[ v_{\rm tot} = H_0\,d + v_{\rm pec} \] A galaxy's measured speed has two parts: the smooth flow from cosmic expansion, plus its own local "peculiar" motion as gravity tugs it toward neighbours. Near galaxies (like Andromeda, which is actually approaching us) the local tug can win — expansion only dominates on large scales.	`v_pec` = local motion; `H₀d` = expansion flow	The reason you need distant objects for cosmology — and the basis of peculiar-velocity surveys that map local mass. Key referencesTully & Fisher (1977); Davis & Peebles (1983).
How Densities Dilute	\[ \rho_m \propto (1+z)^3,\qquad \rho_r \propto (1+z)^4 \] As space expands, matter thins out as its volume grows. Radiation thins out even faster — it loses an extra factor because its waves also get stretched and lose energy. This is why the early, small Universe was radiation-dominated, while matter and then dark energy took over later.	`ρ_m` = matter density; `ρ_r` = radiation density; `z` = redshift	The scalings you plug into the Friedmann equation to track which component drives expansion at each epoch. Key referencesPeebles (1993, Principles of Physical Cosmology).

Open unknowns · Expanding Universe

The Hubble Tension

Why do two trusted ways of measuring the expansion rate H₀ disagree?

The local distance ladder gives ~73 km/s/Mpc while the early-Universe CMB predicts ~67 — a 5-sigma clash that won't go away. Either there's an unknown error in one method, or new physics is hiding in the cosmic recipe. It's the hottest controversy in cosmology.

Is Expansion Perfectly Smooth?

Does the Universe really expand at the same rate in every direction?

The cosmological principle assumes uniformity, but some surveys hint at large-scale flows and mild anisotropies. Whether these are real or statistical flukes tests the very foundation cosmology is built on.

Defining the Hubble Flow

How far out must we look before local motions stop contaminating the pure cosmic expansion?

Galaxies' peculiar velocities, driven by nearby mass, blur the clean Hubble law. Mapping these flows precisely is essential to nailing H₀ and to weighing the local Universe.

Friedmann Equations & the Cosmic Budget

6 equations

Apply Einstein's general relativity to a smooth, expanding Universe and out come the Friedmann equations — the master rules that say how fast the cosmos expands and whether that expansion speeds up or slows down, given everything inside it.

Name	Equation	Variables	Use in Research
Friedmann Equation	\[ H^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3} \] The single most important equation in cosmology: it says the expansion rate is driven by everything the Universe contains — matter and radiation, the curvature of space, and the mysterious cosmological constant. Tell it what's in the cosmos and it tells you how the cosmos grows.	`ρ` = total density; `k` = curvature; `Λ` = cosmological constant	The equation you integrate (as `astropy.cosmology` or CAMB/CLASS do) to get every distance, age, and growth prediction. Key referencesFriedmann (1922); Lemaître (1927).
Acceleration Equation	\[ \frac{\ddot a}{a} = -\frac{4\pi G}{3}\!\left(\rho + \frac{3p}{c^2}\right) + \frac{\Lambda c^2}{3} \] This decides whether expansion is slowing or speeding up. Strangely, pressure adds to gravity's pull here — so ordinary matter brakes the expansion. But something with strong negative pressure (dark energy) flips the sign and pushes space apart, which is exactly what we observe today.	`ä` = acceleration of expansion; `p` = pressure; `ρ` = density	The equation whose sign you measure with supernovae to test whether expansion accelerates — the dark-energy discovery. Key referencesFriedmann (1922); Riess et al. (1998); Perlmutter et al. (1999).
Fluid (Continuity) Equation	\[ \dot\rho + 3H\!\left(\rho + \frac{p}{c^2}\right) = 0 \] Energy bookkeeping for an expanding Universe: as space grows, the density of its contents drops, and how fast depends on the stuff's pressure. It's what makes matter thin out as 1/volume but radiation fade even faster — the rule behind cosmic history's changing of the guard.	`ρ` = density; `p` = pressure; `H` = expansion rate	The energy-conservation rule that, with an equation of state, gives the density scalings feeding the Friedmann equation. Key referencesPeebles (1993); Dodelson (2003, Modern Cosmology).
Critical Density	\[ \rho_c = \frac{3H_0^2}{8\pi G} \approx 9\times10^{-27}\,\text{kg m}^{-3} \] The exact density that makes space perfectly flat — the dividing line between a Universe that eventually recollapses and one that expands forever. Astonishingly small (about five hydrogen atoms per cubic metre), yet our Universe sits right at it.	`H₀` = expansion rate; `G` = gravitational constant	The reference density every cosmic component is quoted relative to (as an Ω); the flat-universe dividing line. Key referencesPeebles (1993); standard texts.
Density Parameter	\[ \Omega = \frac{\rho}{\rho_c},\qquad \Omega_m + \Omega_\Lambda + \Omega_r + \Omega_k = 1 \] Each ingredient's density measured as a fraction of the critical value, so the pieces must add up to one. Today the cosmic recipe is roughly 5% ordinary matter, 27% dark matter, 68% dark energy — and almost zero curvature, meaning space is flat.	`Ω_m` = matter; `Ω_Λ` = dark energy; `Ω_r` = radiation; `Ω_k` = curvature	The compact inventory you fit to data — the headline "what the Universe is made of" numbers. Key referencesPlanck Collaboration (2020).
Equation of State	\[ w = \frac{p}{\rho c^2}:\quad w_{\rm matter}=0,\; w_{\rm rad}=\tfrac13,\; w_\Lambda=-1 \] A single number describing how each ingredient pushes: ordinary matter has no pressure (w=0), radiation pushes outward (w=⅓), and dark energy bizarrely has strong negative pressure (w=−1), behaving like a tension that drives space apart. Measuring whether dark energy's w is exactly −1 is a major goal.	`w` = pressure-to-density ratio; sets how density dilutes	The parameter surveys measure to test whether dark energy is a true constant or something dynamical. Key referencesChevallier & Polarski (2001); DESI Collaboration (2024).

Open unknowns · Friedmann & Budget

The Dark Sector

What actually is the ρ that dominates the Friedmann equation?

95% of the cosmic energy budget is dark matter and dark energy — quantities we measure precisely but can't identify. The Friedmann equation works beautifully while leaving its two biggest terms physically unexplained.

Is Space Exactly Flat?

Is the curvature Ω_k precisely zero, or just very small?

Measurements peg curvature near zero, as inflation predicts — but a tiny nonzero value would have deep implications. Pinning it down to the next decimal place tests our origin story.

Is Dark Energy Constant?

Is the dark-energy equation of state exactly w = −1?

Recent DESI data hint that w might drift with time rather than stay fixed. If confirmed, dark energy is not a simple cosmological constant but something dynamical — rewriting the Universe's future.

III

Cosmic Distances & Horizons

6 equations

In an expanding Universe, "distance" splinters into several different meanings depending on how you measure it. Untangling them is essential to interpreting every observation of the distant cosmos.

Name	Equation	Variables	Use in Research
Comoving Distance	\[ D_C = c\int_0^z \frac{dz'}{H(z')} \] The distance between us and a galaxy measured on a grid that expands with the Universe — so it stays fixed even as space stretches. It factors out the expansion, giving the "true" separation that cosmologists compute everything else from.	`H(z)` = expansion rate vs redshift; `c` = speed of light	The base distance every cosmological calculation starts from — computed by integrating the expansion history (one call in `astropy.cosmology`). Key referencesHogg (1999, distance measures); Peebles (1993).
Luminosity Distance	\[ D_L = (1+z)\,D_C \] How far away an object seems based on how faint it looks. Because expansion dims and redshifts the light, distant objects appear fainter — and thus "farther" by this measure — than their true separation. It's the distance you use with standard candles like supernovae.	`z` = redshift; `D_C` = comoving distance	The distance you use with standard candles; plotting it against redshift is the supernova Hubble diagram that revealed dark energy. Key referencesHogg (1999); Riess et al. (1998); Perlmutter et al. (1999).
Angular-Diameter Distance	\[ D_A = \frac{D_C}{1+z} \] How far away something seems based on how big it looks. Bizarrely, beyond a certain redshift very distant objects start looking bigger again, because we see them as they were when the Universe — and they — were much closer to us. Distance in an expanding cosmos defies intuition.	`z` = redshift; `D_C` = comoving distance	The distance you pair with standard rulers (the CMB sound horizon, BAO) to constrain the geometry and expansion history. Key referencesHogg (1999); Etherington (1933, reciprocity).
Distance Modulus (with redshift)	\[ \mu = 5\log_{10}\!\left(\frac{D_L}{10\,\text{pc}}\right) \] The familiar "fainter means farther" rule, but using the luminosity distance so it works across cosmic scales. Plotting this against redshift for supernovae builds the "Hubble diagram" whose subtle curve betrays dark energy.	`D_L` = luminosity distance; `μ` = apparent minus absolute magnitude	The "fainter means farther" relation extended across the Universe; its subtle curvature with redshift encodes dark energy. Key referencesRiess et al. (1998); Perlmutter et al. (1999); Betoule et al. (2014).
Lookback Time	\[ t_L = \int_0^z \frac{dz'}{(1+z')\,H(z')} \] How long ago the light we now see actually left its source — so looking out into space is literally looking back in time. A galaxy at high redshift is a snapshot of the young Universe; telescopes are time machines.	`z` = redshift; `H(z)` = expansion history	Converts a measured redshift into the epoch you're observing — the basis for studying galaxy evolution as a function of cosmic time. Key referencesHogg (1999); standard texts.
Particle Horizon	\[ D_{\rm hor} = c\,a_0\!\int_0^{t} \frac{dt'}{a(t')} \approx 46\,\text{Gly} \] The edge of the observable Universe — the farthest anything's light could possibly have reached us since the beginning. Surprisingly it's ~46 billion light-years away, not 13.8, because space has expanded while the light was in transit. Beyond it lies more Universe we simply can't see yet.	`a(t)` = scale factor history; `c` = speed of light	Defines how much Universe is in principle observable, and frames the horizon problem that inflation was invented to solve. Key referencesRindler (1956); Guth (1981).

Open unknowns · Distances & Horizons

Calibrating the Ladder

Are the rungs of the cosmic distance ladder cross-calibrated well enough to trust H₀?

Each step — parallax, Cepheids, supernovae — passes calibration to the next, and a hidden offset anywhere biases the whole scale. This is a prime suspect in the Hubble tension.

Standard Candle Drift

Do Type Ia supernovae stay truly standard across cosmic time?

If their brightness subtly changes with the age or chemistry of their host galaxies, the inferred expansion history — and dark energy itself — could shift. Keeping them "standard" is an ongoing worry.

What Lies Beyond?

How much Universe exists past our horizon, and is it like ours?

We can never directly observe beyond the particle horizon. Whether the cosmos is finite or infinite, and uniform far past what we see, may be permanently beyond reach.

Thermal History & the Hot Big Bang

5 equations

Run the expansion backward and the Universe gets hotter and denser, until it was a seething fireball of particles and radiation. Its history is a story of cooling through a sequence of transitions, each leaving a relic we can detect today.

Name	Equation	Variables	Use in Research
Temperature–Redshift	\[ T = T_0\,(1+z),\qquad T_0 = 2.725\,\text{K} \] The Universe was hotter in the past in direct proportion to how much smaller it was. Today's cosmos sits at a frosty 2.7 degrees above absolute zero, but at redshift 1000 it was a glowing 3000 K, and near the beginning, unimaginably hot. Expansion is literally what cooled the cosmos.	`T₀` = present CMB temperature; `z` = redshift	The thermometer of cosmic history — it tells you the temperature, and hence which physical processes were active, at any redshift. Key referencesGamow (1948); Alpher & Herman (1948).
Radiation Energy Density	\[ \rho_r c^2 = a_{\rm rad}\,T^4 \propto (1+z)^4 \] The energy packed into radiation soars as the fourth power of temperature, so the early hot Universe was utterly dominated by light, not matter. Only as it expanded and cooled did matter eventually take charge — a handover that shaped how galaxies could form.	`a_rad` = radiation constant; `T` = temperature	Shows why the early Universe was radiation-dominated and sets the photon energy density that the CMB measures today. Key referencesAlpher & Herman (1948); Peebles (1966).
Matter–Radiation Equality	\[ 1 + z_{\rm eq} = \frac{\Omega_m}{\Omega_r} \approx 3400 \] The turning point when matter's density finally overtook radiation's, about 50,000 years after the Big Bang. Before it, radiation pressure smoothed things out; after it, gravity could start clumping matter into the seeds of galaxies. This moment is imprinted on the CMB.	`Ω_m, Ω_r` = matter, radiation densities	The epoch whose horizon size imprints the turnover in the matter power spectrum you fit to galaxy surveys. Key referencesEisenstein & Hu (1998); Planck Collaboration (2020).
Relativistic Degrees of Freedom	\[ \rho_r = \frac{\pi^2}{30}\,g_*(T)\,\frac{(kT)^4}{(\hbar c)^3} \] A tally of how many kinds of particle are buzzing around as radiation at a given temperature. As the Universe cooled and heavy particles "froze out," this count dropped in steps — and tracking it is how physicists reconstruct the particle content of the first second.	`g*` = effective particle species; `T` = temperature	The particle census you track through the early Universe to compute expansion rate, entropy, and relic abundances. Key referencesKolb & Turner (1990, The Early Universe).
Freeze-Out Condition	\[ \Gamma \sim H \] A particle reaction keeps a species in balance only as long as reactions happen faster than the Universe expands. When expansion wins, the species "freezes out" and its abundance is locked in. This simple race sets the leftover amounts of helium, neutrinos, and possibly dark matter.	`Γ` = reaction rate; `H` = expansion rate	The simple race (reaction rate vs. expansion) that fixes every relic abundance — neutrinos, light elements, and possibly dark matter. Key referencesZeldovich (1965); Lee & Weinberg (1977); Kolb & Turner (1990).

Open unknowns · Thermal History

Matter–Antimatter Asymmetry

Why is the Universe made of matter, with almost no antimatter?

The hot Big Bang should have made equal amounts that annihilated to nothing. A tiny imbalance — about one extra matter particle per billion — survived to build everything, and what caused it ("baryogenesis") is unknown.

Physics of the First Second

What happened at energies far beyond any collider, in the Universe's first instant?

The earliest moments involved physics we can only theorize about — possible phase transitions, exotic particles, and the freeze-out of dark matter. We have no direct probe of this era.

Extra Radiation?

Is there more relativistic energy in the early Universe than the known particles provide?

The "effective number of neutrinos" (N_eff) tests for hidden radiation — extra neutrino species or other light particles. Small anomalies could signal new physics and even ease the Hubble tension.

Big Bang Nucleosynthesis

5 equations

In the first few minutes, the Universe was a cosmic nuclear reactor that forged the lightest elements. The leftover amounts of hydrogen, helium, and a trace of lithium are a stunningly precise fossil of those minutes — and one of the Big Bang's strongest pillars.

Name	Equation	Variables	Use in Research
Baryon-to-Photon Ratio	\[ \eta = \frac{n_b}{n_\gamma} \approx 6\times10^{-10} \] There are roughly a billion photons for every atom of ordinary matter in the Universe — a measure of how outnumbered matter is by light. This one tiny number sets exactly how much of each light element the Big Bang could cook, making it the master dial of early-Universe chemistry.	`n_b` = baryon density; `n_γ` = photon density	The master dial of Big Bang nucleosynthesis; matching the BBN-required value to the CMB's is a stringent consistency test of the hot Big Bang. Key referencesWagoner, Fowler & Hoyle (1967); Cyburt et al. (2016, review).
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, leaving about one neutron for every seven protons. That frozen ratio almost single-handedly decides how much helium the Universe ends up with.	`Δm` = neutron–proton mass difference; `T` = temperature	The frozen ratio that almost single-handedly sets how much helium the Big Bang produces. Key referencesHayashi (1950); Cyburt et al. (2016).
Primordial Helium	\[ Y_p = \frac{2(n_n/n_p)}{1 + n_n/n_p} \approx 0.247 \] Almost every neutron from the early Universe got locked into helium, predicting that about a quarter of all ordinary matter by mass should be helium — before any star ever formed. Measuring exactly this fraction in pristine gas is a triumphant confirmation of the Big Bang.	`Y_p` = helium mass fraction; `n_n/n_p` = neutron-proton ratio	A parameter-free prediction you compare against helium measured in pristine extragalactic gas — a pillar of the Big Bang. Key referencesPeebles (1966); Aver, Olive & Skillman (2015).
Deuterium as a Baryometer	\[ \frac{\text{D}}{\text{H}} \propto \eta^{-1.6} \] Deuterium (heavy hydrogen) is fragile and easily destroyed, and the more ordinary matter there was, the more got burned away. So measuring how much deuterium survives in ancient gas precisely weighs all the ordinary matter in the Universe — and it agrees with the CMB.	`D/H` = deuterium-to-hydrogen ratio; `η` = baryon-to-photon ratio	The most sensitive "baryometer" — you measure deuterium in pristine gas to weigh all ordinary matter, then check it against the CMB. Key referencesCooke et al. (2018); Pitrou et al. (2018).
Effective Neutrino Number	\[ N_{\rm eff} \approx 3.046 \] Counts the species of lightweight particles streaming through the early Universe — essentially, how many flavours of neutrino plus anything extra. The exact value affects how fast the Universe expanded during nucleosynthesis, so it doubles as a search for hidden particles.	`N_eff` = effective relativistic species; 3 standard neutrinos	A search for hidden light particles — both BBN and the CMB constrain it, and an excess could ease the Hubble tension. Key referencesMangano et al. (2005); Planck Collaboration (2020).

Open unknowns · Nucleosynthesis

The Lithium Problem

Why do old stars contain only a third of the lithium-7 the Big Bang should have made?

Helium and deuterium predictions are triumphs, but primordial lithium stubbornly comes out three times too high. Stellar destruction or new physics? It's the one persistent crack in an otherwise flawless theory.

Hidden Light Particles

Is N_eff exactly the standard value, or is extra radiation lurking?

A small excess would reveal sterile neutrinos or other unknown particles present in the first minutes — and might help reconcile the Hubble tension.

Pristine Gas Hunt

Can we find truly primordial gas, untouched by any star?

Every abundance measurement must correct for later stellar pollution. Finding cleaner samples would sharpen the test of nucleosynthesis and the baryon density.

Recombination & the Cosmic Microwave Background

5 equations

About 380,000 years after the Big Bang, the Universe cooled enough for electrons and protons to join into atoms. Suddenly light could travel freely, and that first liberated glow — stretched to microwaves today — is the oldest light we can ever see.

Name	Equation	Variables	Use in Research
Saha Recombination	\[ \frac{n_e n_p}{n_H} = \left(\frac{2\pi m_e kT}{h^2}\right)^{3/2}\!e^{-\chi/kT} \] This decides when the cooling Universe's free electrons and protons could finally stick together as hydrogen atoms. Once they did, light stopped bouncing off free electrons and streamed out — the moment the fog cleared and the cosmos became transparent.	`χ` = hydrogen ionization energy (13.6 eV); `T` = temperature	The calculation that fixes when the Universe turned transparent and released the CMB — its precision now limits cosmic-parameter accuracy. Key referencesSaha (1920); Peebles (1968); Zeldovich, Kurt & Sunyaev (1968).
Surface of Last Scattering	\[ z_* \approx 1090,\qquad t_* \approx 380{,}000\,\text{yr} \] The redshift and age at which light last bounced off matter before flying free forever. Looking at the CMB, we're seeing a wall of light from this exact moment — a baby photo of the Universe at just 380,000 years old, the farthest back our telescopes can ever look.	`z` = redshift of last scattering; `t` = cosmic age then	The redshift you assign the CMB — the farthest in time light can take us, and the surface all CMB analysis is referenced to. Key referencesSunyaev & Zeldovich (1970); Planck Collaboration (2020).
CMB Blackbody Spectrum	\[ B_\nu(T),\qquad T_0 = 2.7255\,\text{K} \] The cosmic microwave background is the most perfect blackbody glow ever measured — light from the hot early Universe, cooled by expansion to just 2.7 degrees above absolute zero. Its flawless thermal shape is overwhelming proof the cosmos really began hot and dense.	`T₀` = present CMB temperature; `B_ν` = Planck spectrum	The measured spectrum whose perfection is decisive proof of a hot, dense origin — it killed steady-state cosmology. Key referencesPenzias & Wilson (1965); Mather et al. (1994, FIRAS).
CMB Photon Density	\[ n_\gamma \approx 411\,\text{cm}^{-3} \] Space is awash in relic light — about 400 microwave photons in every cubic centimetre, even in the emptiest void. They outnumber atoms a billion to one, a faint, all-pervading echo of the Big Bang passing through you right now.	`n_γ` = photon number density (per cm³)	The photon count that, with the baryon density, fixes η — and a vivid reminder the early Universe is all around us. Key referencesstandard texts; Fixsen (2009).
Sound Horizon	\[ r_s = \int_0^{t_*}\! c_s\,(1+z)\,dt \approx 147\,\text{Mpc} \] Before atoms formed, pressure waves (literally sound) rippled through the hot plasma, and this is the farthest such a wave could travel before being frozen in at recombination. That fixed length acts as a cosmic "standard ruler" stamped on both the CMB and the distribution of galaxies.	`c_s` = sound speed in the plasma; `t*` = recombination time	The standard ruler underpinning precision cosmology — the same length appears in the CMB peaks and in the galaxy distribution (BAO). Key referencesHu & Sugiyama (1996); Eisenstein et al. (2005, BAO detection).

Open unknowns · Recombination & CMB

Reionization

When and how did the first stars re-ionize the Universe's hydrogen?

After recombination the Universe went dark, then the first stars and galaxies lit up and stripped electrons back off the gas. Exactly when this happened, and what sources drove it, is a frontier JWST is now probing.

Spectral Distortions

Does the CMB hide tiny departures from a perfect blackbody?

Faint "distortions" would record energy releases in the early Universe — from decaying particles to the first structures. They're predicted but still below detection; a future mission could reveal a whole new window.

Recombination Precision

Is the detailed physics of recombination modelled accurately enough for next-gen data?

Tiny errors in how atoms formed shift the inferred cosmic parameters. As CMB measurements sharpen, the recombination calculation itself must keep pace.

VII

CMB Anisotropies

5 equations

The microwave background is not perfectly smooth: it carries faint temperature ripples of one part in 100,000. These tiny patterns are the seeds of all cosmic structure and the single richest dataset in cosmology, encoding the Universe's full recipe.

Name	Equation	Variables	Use in Research
Anisotropy Amplitude	\[ \frac{\Delta T}{T} \sim 10^{-5} \] The CMB's temperature varies by only about one part in a hundred thousand across the sky — fantastically smooth, yet not perfectly so. Those whisper-faint warm and cool spots are the primordial seeds that gravity later grew into galaxies, clusters, and us.	`ΔT` = temperature fluctuation; `T` = mean temperature	The faint fluctuation level you measure across the CMB sky — the primordial density seeds of all structure. Key referencesSmoot et al. (1992, COBE); Bennett et al. (2013, WMAP).
Acoustic Peak Scale	\[ \theta_A = \frac{r_s}{D_A},\qquad \ell_1 \approx 220 \] The hot early plasma rang like a bell, and the loudest note left warm/cool spots of a characteristic size on the sky. Comparing that known physical size to how big it appears reveals the geometry of space — and it told us the Universe is flat.	`r_s` = sound horizon; `D_A` = angular-diameter distance; `ℓ` = angular scale	The standard-ruler-on-the-sky measurement: comparing the sound horizon's known size to its apparent angle gives the geometry of space. Key referencesde Bernardis et al. (2000, BOOMERANG); Planck Collaboration (2020).
Sachs–Wolfe Effect	\[ \frac{\Delta T}{T} = \frac{1}{3}\frac{\Phi}{c^2} \] Light climbing out of a denser region loses a little energy and arrives slightly cooler, so the CMB's big cold spots mark where matter was piled up. It directly links the temperature map to the gravitational landscape of the infant Universe.	`Φ` = gravitational potential; `ΔT/T` = temperature shift	Connects the CMB's large-scale temperature map to the gravitational potential landscape of the infant Universe. Key referencesSachs & Wolfe (1967).
Angular Power Spectrum	\[ C_\ell = \langle \|a_{\ell m}\|^2 \rangle \] A way to measure how strong the CMB ripples are at every angular size — from broad swathes of sky down to tiny patches. The resulting curve of peaks and troughs is cosmology's Rosetta Stone: its exact shape pins down nearly every parameter of the Universe at once.	`ℓ` = angular scale (multipole); `a_ℓm` = ripple amplitudes	The single curve you fit (with CAMB/CLASS) to extract nearly all six ΛCDM parameters at once — cosmology's richest dataset. Key referencesBond & Efstathiou (1987); Hu & Dodelson (2002, review); Planck (2020).
Baryon Loading	\[ \frac{\text{peak}_1}{\text{peak}_2} \propto \Omega_b \] Ordinary matter (baryons) weighs down the oscillating plasma, making the compressions stronger than the rarefactions — so it boosts the odd-numbered peaks of the CMB spectrum over the even ones. The relative peak heights literally weigh how much normal matter the Universe holds.	`Ω_b` = baryon density; peak height ratios	The peak-height pattern you read to weigh ordinary matter from the CMB alone — a check on BBN's baryon density. Key referencesHu & Sugiyama (1995); Page et al. (2003).

Open unknowns · CMB Anisotropies

Large-Scale Anomalies

Are the odd features on the CMB's largest scales real, or flukes?

The "cold spot," a hemispherical power asymmetry, and aligned low multipoles all look slightly off from predictions. With only one sky to observe, telling genuine anomalies from chance is genuinely hard.

Lensing Amplitude

Why does the CMB appear slightly more gravitationally lensed than expected?

The mild "A_L anomaly" hints either at a statistical fluctuation or at a real tension in the model's matter content — a clue cosmologists watch closely.

Primordial Non-Gaussianity

Were the initial ripples purely random, or do they carry subtle correlations?

Different theories of the Universe's birth predict tiny deviations from perfect randomness. Detecting them would sharply narrow down what actually happened in the first instant.

VIII

Inflation & the Very Beginning

5 equations

The standard Big Bang leaves puzzles: why is the Universe so uniform and so flat? Inflation answers them with a wild idea — a fleeting burst of exponential expansion in the first split-second that stretched a tiny patch into our entire visible cosmos.

Name	Equation	Variables	Use in Research
The Horizon Problem	\[ a(t) \propto e^{Ht}\;\;(\text{inflation}) \] Opposite sides of the sky have nearly identical temperatures, yet in a normal Big Bang they could never have touched to even out. Inflation solves this by proposing the Universe briefly expanded exponentially fast, blowing up one tiny, already-uniform patch to encompass everything we see.	`H` = (nearly constant) inflationary expansion rate	The mechanism invoked to explain why causally-disconnected CMB patches share the same temperature. Key referencesGuth (1981); Linde (1982); Albrecht & Steinhardt (1982).
Number of e-folds	\[ N = \ln\!\frac{a_{\rm end}}{a_{\rm start}} \gtrsim 60 \] A measure of how much inflation stretched space, counted in doublings (well, e-foldings). At least 60 are needed — meaning space ballooned by a factor of more than 10²⁶ in a sliver of a second, flattening any curvature the way blowing up a balloon flattens its surface.	`N` = e-folds of expansion; `a` = scale factor	The amount of inflation required, in doublings — the number model-builders must achieve to flatten and smooth the Universe. Key referencesGuth (1981); Liddle & Lyth (2000, Cosmological Inflation).
Slow-Roll Parameters	\[ \epsilon = \frac{m_P^2}{2}\!\left(\frac{V'}{V}\right)^2 \ll 1 \] For inflation to last, the field driving it must roll very slowly down its energy hill — like a ball creeping down a nearly flat slope. These parameters measure how gentle that slope is, and they connect the abstract theory to things we can actually measure in the CMB.	`V` = inflaton potential; `m_P` = Planck mass	The small numbers you compute from a candidate inflaton potential to predict observables (n_s, r) and confront with the CMB. Key referencesSteinhardt & Turner (1984); Liddle & Lyth (2000).
Scalar Spectral Index	\[ n_s \approx 0.965 \] Inflation predicts the primordial ripples should be almost the same strength on all scales, but not exactly — slightly stronger on larger scales. Measuring this gentle tilt (n_s just below 1) is a stunning confirmation of inflation's prediction and rules out the simplest "perfectly equal" alternative.	`n_s` = tilt of the primordial spectrum; 1 = scale-invariant	A clean, falsifiable inflationary prediction — the slight "tilt" of the primordial spectrum, measured precisely by the CMB. Key referencesMukhanov & Chibisov (1981); Planck Collaboration (2020).
Tensor-to-Scalar Ratio	\[ r = \frac{P_t}{P_s} \] Inflation should also have shaken spacetime itself, launching primordial gravitational waves. This ratio measures how strong they are — and detecting them (via a special swirl pattern in CMB polarization) would be the smoking-gun proof that inflation really happened. So far we have only upper limits.	`P_t` = gravitational-wave power; `P_s` = density-ripple power	The signature of primordial gravitational waves — the "smoking gun" CMB B-mode experiments are chasing to confirm inflation. Key referencesKamionkowski, Kosowsky & Stebbins (1997); BICEP/Keck Collaboration (2021).

Open unknowns · Inflation

Did Inflation Happen?

Is inflation real, and what field drove it?

Inflation elegantly explains many facts but has never been directly confirmed, and there is no known particle for the "inflaton." Detecting primordial gravitational waves would clinch it; until then it remains a brilliant hypothesis.

Primordial Gravitational Waves

Can we detect the spacetime ripples inflation should have made?

A nonzero tensor-to-scalar ratio r would be revolutionary, opening a window onto energies a trillion times beyond any collider. Experiments are closing in, but it may be vanishingly small.

Before Inflation & the Multiverse

What, if anything, came before inflation — and is ours one of many universes?

Many inflation models imply "eternal inflation" spawning endless bubble universes. Whether this multiverse is physics or untestable speculation is fiercely debated.

Structure Formation

5 equations

Gravity is an amplifier. Over billions of years it took the CMB's faint one-in-100,000 ripples and grew them into galaxies, clusters, and the vast filamentary cosmic web — the largest structures in existence.

Name	Equation	Variables	Use in Research
Growth of Perturbations	\[ \ddot\delta + 2H\dot\delta = 4\pi G\rho_m\,\delta \] The tug-of-war that built everything: gravity (right side) pulls slightly denser regions together, while cosmic expansion (the middle term) fights to pull them apart. Where gravity wins, ripples grow into galaxies. This equation governs how the cosmic web slowly assembled.	`δ` = density contrast; `H` = expansion rate; `ρ_m` = matter density	The equation behind every structure-formation calculation and the linear-theory backbone of N-body codes. Key referencesLifshitz (1946); Peebles (1980, LSS of the Universe).
Linear Growth (matter era)	\[ \delta \propto a \] While matter ruled the Universe, density ripples grew in lockstep with the expanding scale factor — steadily, but only as fast as space itself grew. This gentle pace is actually too slow to build galaxies from the tiny CMB seeds using normal matter alone, which is a major hint that dark matter exists.	`δ` = density contrast; `a` = scale factor	The growth rate you use to evolve the CMB seeds forward — and the argument that baryons alone can't make galaxies in time. Key referencesPeebles (1982); Davis, Efstathiou, Frenk & White (1985).
Matter Power Spectrum	\[ P(k) = \langle \|\delta_k\|^2 \rangle \] A way to measure how clumpy the Universe is on every scale at once — from giant superclusters to individual galaxies. Its shape is a treasure map of cosmic ingredients, bending at the scale set by matter-radiation equality and rippled by the same sound waves seen in the CMB.	`k` = spatial scale (wavenumber); `δ_k` = ripple amplitude	The statistic you measure from galaxy maps to test cosmology — its shape encodes the matter content and primordial spectrum. Key referencesPeacock & Dodds (1994); Tegmark et al. (2004, SDSS).
Clustering Amplitude σ₈	\[ \sigma_8 \approx 0.81 \] A single number capturing how lumpy the Universe is today, measured in spheres about 26 million light-years across. It's the standard yardstick for cosmic clumpiness — and intriguingly, different methods of measuring it don't quite agree.	`σ₈` = density fluctuation on 8 h⁻¹ Mpc scales	The standard "clumpiness" normalization — and the focus of a possible tension between early- and late-Universe measurements. Key referencesHeymans et al. (2021, KiDS); DES Collaboration (2022).
Baryon Acoustic Oscillations	\[ r_{\rm BAO} \approx 150\,\text{Mpc (comoving)} \] The same sound waves frozen into the early plasma left a preferred separation between galaxies — a faint but real tendency for pairs to sit about 500 million light-years apart. This built-in "standard ruler" lets astronomers measure cosmic distances and trace the expansion history with great precision.	`r_BAO` = acoustic scale imprinted on galaxies	The galaxy-survey standard ruler — a geometry-based probe of \(H(z)\) and dark energy independent of the supernova ladder. Key referencesEisenstein et al. (2005); DESI Collaboration (2024).

Open unknowns · Structure Formation

The S₈ Tension

Why does the Universe look slightly less clumpy than the CMB predicts?

Weak-lensing surveys tend to find a lower σ₈ than extrapolated from the early Universe. Like the Hubble tension, it could be a measurement issue — or a hint that dark matter or dark energy behaves unexpectedly.

Small-Scale Structure

Does the standard cold-dark-matter model predict the right number of small galaxies?

Long-standing puzzles ("missing satellites," "too big to fail," "core-cusp") concern whether simulations over-predict tiny structures. They may signal warm or interacting dark matter — or just messy galaxy-formation physics.

Early Massive Galaxies

Why is JWST finding such big galaxies so soon after the Big Bang?

Some early galaxies look more massive and mature than models comfortably allow. Whether this strains the cosmic model or just our understanding of fast early star formation is hotly debated.

Dark Matter & Dark Energy

5 equations

The deepest embarrassment and greatest opportunity in physics: 95% of the Universe is made of two things we cannot identify. Dark matter holds galaxies together; dark energy is tearing the cosmos apart. We see their effects everywhere and their nature nowhere.

Name	Equation	Variables	Use in Research
Flat Rotation Curves	\[ v(r) = \sqrt{\frac{G M(r)}{r}} \to \text{constant} \] Stars at the edge of a galaxy orbit just as fast as those near the center — which is impossible if the galaxy's mass is only the visible stars, since outer stars should slow down like distant planets. The fix: galaxies are embedded in vast halos of unseen "dark matter." This was the first hard evidence for it.	`v` = orbital speed; `M(r)` = mass within radius r	The galaxy-scale measurement that first forced dark matter on astronomers — you fit it to weigh a galaxy's invisible halo. Key referencesRubin & Ford (1970); Bosma (1981); van Albada et al. (1985).
Dark Energy Density	\[ \rho_\Lambda = \frac{\Lambda c^2}{8\pi G} \] If dark energy is Einstein's cosmological constant, it's an energy woven into empty space itself — the same density everywhere, never diluting as the Universe grows. That unchanging push is why it stayed negligible early on but now dominates and accelerates the expansion.	`Λ` = cosmological constant; `ρ_Λ` = vacuum energy density	The simplest dark-energy model — a constant vacuum energy — and the \(\Lambda\) you fit as one number in ΛCDM. Key referencesEinstein (1917); Carroll (2001, review).
Evolving Dark Energy	\[ w(a) = w_0 + w_a(1 - a) \] A way to test whether dark energy is truly constant or slowly changing over cosmic time. If the measured numbers stray from w₀=−1, wₐ=0, dark energy is something more exotic than a constant — a possibility recent surveys have begun to hint at.	`w₀` = present value; `w_a` = rate of change; `a` = scale factor	The two-parameter form surveys fit to test whether dark energy deviates from a constant. Key referencesChevallier & Polarski (2001); Linder (2003); DESI Collaboration (2024).
The Vacuum Energy Problem	\[ \frac{\rho_\Lambda^{\rm theory}}{\rho_\Lambda^{\rm obs}} \sim 10^{120} \] When physicists try to compute the energy of empty space from quantum theory, they get a number about 10¹²⁰ times too large — the worst prediction in the history of physics. Why the real value is so absurdly tiny, but not quite zero, is one of the deepest mysteries in all of science.	ratio of predicted to observed vacuum energy	The most infamous gap between theory and observation in physics — a standing challenge to any fundamental theory. Key referencesWeinberg (1989); Carroll (2001).
The Coincidence Problem	\[ \Omega_\Lambda \approx 2\,\Omega_m \;\;(\text{today}) \] Matter thins out as the Universe expands while dark energy stays constant, so for almost all of cosmic history one or the other dominated overwhelmingly. Yet right now, when we happen to be looking, they're roughly comparable — a curious coincidence that may be a clue, or may be chance.	`Ω_Λ` = dark energy; `Ω_m` = matter (today)	The "why now?" puzzle that motivates dynamical dark-energy (quintessence) models over a bare constant. Key referencesSteinhardt (1997); Zlatev, Wang & Steinhardt (1999).

Open unknowns · Dark Matter & Energy

What Is Dark Matter?

What particle (or modification of gravity) makes up the invisible 27%?

Decades of searches — underground detectors, colliders, the sky — have found no dark-matter particle. Candidates span WIMPs, axions, primordial black holes, and sterile neutrinos. Its identity is the biggest open question in physics.

What Is Dark Energy?

Is dark energy a constant vacuum energy, a new field, or a failure of general relativity?

We can't explain why it has the value it does, nor whether it's truly constant. It could even mean Einstein's gravity needs modifying on cosmic scales — a possibility taken increasingly seriously.

Is Gravity the Problem?

Could modified gravity replace dark matter or dark energy entirely?

Theories like MOND explain some galaxy data without dark matter but struggle with clusters and the CMB. Whether any modification can replace the dark sector remains an active, contested line of research.

Cosmic Fine-Tuning

Why are the cosmic parameters so improbably suited to a complex Universe?

Small changes to dark energy or the matter budget would have prevented galaxies, stars, or life. Whether this reflects deeper physics, a multiverse, or pure luck is unresolved — and partly philosophical.

The ΛCDM Model & Cosmic Parameters

6 equations

Everything above converges into one remarkably successful "standard model" of cosmology: ΛCDM (a cosmological constant Λ plus cold dark matter). With just six numbers it fits an enormous range of data — even as its biggest ingredients remain mysteries.

Name	Equation	Variables	Use in Research
Expansion History	\[ H(z)^2 = H_0^2\!\left[\Omega_r(1+z)^4 + \Omega_m(1+z)^3 + \Omega_\Lambda\right] \] The complete recipe for how fast the Universe expanded at every moment, adding up all its ingredients. Each term fades at its own rate, so radiation ruled first, then matter, and now dark energy — this one equation contains the entire past and future of cosmic expansion.	`Ω_r, Ω_m, Ω_Λ` = radiation, matter, dark energy fractions	The full ΛCDM expansion law you actually integrate (in `astropy`, CAMB, CLASS) for every distance, age, and growth prediction. Key referencesHogg (1999); Planck Collaboration (2020).
Age of the Universe	\[ t_0 = \int_0^\infty \frac{dz}{(1+z)\,H(z)} \approx 13.8\,\text{Gyr} \] Add up all of cosmic history from the Big Bang to now and you get 13.8 billion years. Remarkably, this number derived purely from the expansion equation agrees with the ages of the oldest stars and star clusters — independent clocks telling the same time.	`H(z)` = expansion history; `z` = redshift	The Universe's age, derived purely from the expansion integral and cross-checked against independent stellar clocks. Key referencesPlanck Collaboration (2020); Valcin et al. (2021, GC ages).
Flatness	\[ \Omega_{\rm total} = 1.000 \pm 0.002 \] All the cosmic ingredients add up to exactly the critical density, meaning space is geometrically flat — parallel laser beams would never converge or diverge. Inflation predicted this, and the CMB confirmed it to remarkable precision.	`Ω_total` = sum of all density parameters	The measured flatness of space — a precision CMB result and a direct confirmation of an inflationary prediction. Key referencesGuth (1981); Planck Collaboration (2020).
Deceleration Parameter	\[ q_0 = \frac{1}{2}\Omega_m - \Omega_\Lambda \approx -0.55 \] A single number for whether cosmic expansion is speeding up or slowing down today. It came out negative, meaning the Universe is accelerating — the shocking 1998 discovery, encoded in one number, that revealed dark energy and won a Nobel Prize.	`Ω_m` = matter; `Ω_Λ` = dark energy; negative = accelerating	A single number capturing whether expansion accelerates today — the quantity the 1998 supernova teams measured. Key referencesRiess et al. (1998); Perlmutter et al. (1999).
The Cosmic Budget	\[ \Omega_\Lambda \approx 0.69,\;\; \Omega_{\rm DM} \approx 0.26,\;\; \Omega_b \approx 0.05 \] The Universe's full ingredient list: about 69% dark energy, 26% dark matter, and just 5% ordinary matter — the stuff of stars, planets, and people. Everything you have ever seen or touched is a rounding error in the cosmic budget.	`Ω_Λ` = dark energy; `Ω_DM` = dark matter; `Ω_b` = baryons	The famous pie chart — and a result so robust because the CMB, BAO, supernovae, and lensing all return it. Key referencesPlanck Collaboration (2020).
Six-Parameter ΛCDM	\[ \{\Omega_b h^2,\,\Omega_c h^2,\,\theta_*,\,\tau,\,A_s,\,n_s\} \] Astonishingly, just six numbers — the densities of normal and dark matter, the sound-horizon angle, the reionization depth, and the amplitude and tilt of the primordial ripples — are enough to fit virtually all cosmological data. That such a vast Universe needs so few parameters is one of science's great surprises.	the six base parameters fit to CMB + large-scale structure	The complete standard model in six numbers — what you actually fit to the data; all other quantities are derived from these. Key referencesBond, Efstathiou & Tegmark (1997); Planck Collaboration (2020).

Open unknowns · The ΛCDM Model

Is ΛCDM Cracking?

Are the Hubble and S₈ tensions signs that the standard model is incomplete?

ΛCDM fits almost everything, yet two independent tensions persist. They may dissolve with better data — or they may be the first cracks pointing toward new physics beyond the six-parameter model.

Neutrino Masses

What are the absolute masses of neutrinos, and can cosmology weigh them?

Neutrinos subtly suppress cosmic structure, so the sky can weigh them where labs can't. Upcoming surveys should detect the total mass — a rare case of cosmology measuring a fundamental particle property.

The Ultimate Fate

Will the Universe expand forever, and how does it end?

If dark energy is constant, expansion accelerates forever into a cold "heat death." But if it evolves, fates range from a gentle fade to a violent "Big Rip." The answer hinges on pinning down w.

Why These Numbers?

Is there a deeper theory that explains the six parameters rather than just measuring them?

ΛCDM describes the Universe but doesn't explain why its parameters take the values they do. A more fundamental theory — perhaps uniting gravity and quantum mechanics — may be needed.

Cosmic reference values (Planck 2018, flat ΛCDM): H₀ ≈ 67–73 km/s/Mpc (tension); age t₀ = 13.8 Gyr; Ω_Λ ≈ 0.685, Ω_m ≈ 0.315 (of which baryons Ω_b ≈ 0.049, dark matter ≈ 0.265); Ω_k ≈ 0 (flat); T_CMB = 2.7255 K; n_γ ≈ 411 cm⁻³; baryon-to-photon η ≈ 6.1×10⁻¹⁰; z of recombination ≈ 1090; z of matter–radiation equality ≈ 3400; sound horizon r_s ≈ 147 Mpc; σ₈ ≈ 0.81; n_s ≈ 0.965; critical density ρ_c ≈ 8.5×10⁻²⁷ kg m⁻³ (~5 protons m⁻³); observable radius ≈ 46.5 Gly.