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Open CosmologyPhysics & cosmology: four years, plus a graduate year

Symbols and notation

Every symbol on this page is used somewhere in the lessons. Each one has its meaning, how to say it, and links to the lessons that use it. Symbols are grouped by the year of the programme in which they first appear. The list is built from the lessons themselves, so it grows as the site grows.

The same letter often means different things in different courses: σ is a velocity spread in one lesson and a cross section in another. Where that happens, each meaning says where it applies. In a lesson, tap or click a symbol in a formula to see its meaning there; every lesson also lists its symbols near the end, under “Symbols on this page”. Under every formula, “In words” spells the whole formula out.

The Greek alphabet

Physics borrows Greek letters when it runs out of Latin ones. Here are all 24, with a guide to saying them (where two are given, both are common). Letters used on the site link to their entry below.

CapitalSmallNameSay it
ΑαalphaAL-fa
ΒβbetaBAY-ta or BEE-ta
ΓγgammaGAM-a
ΔδdeltaDEL-ta
Εε, ϵepsilonEP-si-lon
ΖζzetaZAY-ta or ZEE-ta
ΗηetaAY-ta or EE-ta
ΘθthetaTHAY-ta or THEE-ta
Ιιiotaeye-OH-ta
ΚκkappaKAP-a
ΛλlambdaLAM-da
ΜμmuMYOO
ΝνnuNEW
ΞξxiKSEE or ZYE
ΟοomicronOM-i-kron
ΠπpiPIE
ΡρrhoROW
ΣσsigmaSIG-ma
ΤτtauTOW (rhymes with cow)
ΥυupsilonUP-si-lon
Φφ, ϕphiFYE or FEE
ΧχchiKYE
ΨψpsiSIGH or PSY
Ωωomegaoh-MAY-ga or oh-MEE-ga

Symbols that look alike

ν and v
ν is the Greek letter nu (a frequency, or a neutrino); v is the Latin letter v (a speed). Nu comes to a point at the bottom.
ρ and p
ρ (rho) is a density; p is a pressure. Both appear in ρ + 3p/c². Rho has a tail below the line.
κ and k
κ (kappa) and k (curvature) even appear together: for an open universe κ = −k.
χ and x
χ (chi) has a long stroke below the line; x does not.
ω and w
ω (omega) is rounded at the bottom; w is pointed.
ξ and ζ
ξ (xi) has three bumps stacked up; ζ (zeta) is a single curl.
ϕ and φ; ε and ϵ
Two ways of writing the same letter (phi; epsilon). They mean the same thing.
∝ and α
∝ means “is proportional to” and sits on the line like =; α is the letter alpha.
ℓ, l and 1
ℓ is a script l, used for lengths so that it is not mistaken for the number 1.
≈, ∼, ≃ and ∝
≈ and ≃: close in value. ∼: the same order of magnitude (within a factor of a few). ∝: the same shape, up to a constant factor.
Δ and ∇
Δ (capital delta) means “a change in”; ∇ (nabla, upside down) is the vector derivative “del”.
d, ∂ and δ
d: an ordinary derivative or small step (dx). ∂: a partial derivative, other variables held fixed. δ: a small variation, or the density contrast in 8.902.
ħ and h
ħ (h-bar) is Planck's constant divided by 2π.

First used in Year I · Mechanics, electromagnetism and calculus

Letters

H H

Say it: H

Used in 29 lessons

z z

Say it: z

Used in 20 lessons

G G

Say it: G

  • Newton's gravitational constant, G ≈ 6.674 × 10⁻¹¹ m³ kg⁻¹ s⁻².
Used in 26 lessons

k k

Say it: k

Used in 31 lessons

Greek letters

capital delta \Delta

Say it: capital delta

Used in 13 lessons

theta \theta

Say it: theta (sounds like “thay-ta or thee-ta”)

Used in 13 lessons

rho \rho

Say it: rho

  • Density: mass (or energy divided by c²) per unit volume. ρ_c is the critical density, ρ̄ the mean density.
Used in 30 lessons

Named quantities (a letter with a subscript)

Hubble constant H_0

Say it: H nought

  • The Hubble constant: today's value of H, about 67–73 km/s/Mpc.
Used in 13 lessons

comoving distance \ell_c

Say it: ell c

  • A comoving distance: fixed as the universe expands, measured in coordinate units (Guth's notches).
Used in 4 lessons

Constants

pi \pi

Say it: pi

  • The ratio of a circle's circumference to its diameter, π ≈ 3.14159.
Used in 34 lessons

Other symbols

script l \ell

Say it: ell

  • A length or distance; ℓ_p is a physical distance and ℓ_c a comoving one.
Used in 13 lessons

degree \circ

Say it: degrees

  • Degrees of angle (as in 45°).
Used in 4 lessons

Relations

approximately equal \approx

Say it: is approximately

  • Is approximately equal to.
Used in 43 lessons

proportional to \propto

Say it: is proportional to

  • Is proportional to: doubling one doubles the other.
Used in 28 lessons

identical to \equiv

Say it: is defined as

  • Is defined as (or is identically equal to).
Used in 16 lessons

not equal \neq, \ne

Say it: is not equal to

  • Is not equal to.
Used in 4 lessons

arrow \to, \rightarrow

Say it: goes to

  • Goes to (tends to), or becomes.
Used in 19 lessons

implies \Rightarrow, \Longrightarrow

Say it: which gives

  • Implies: what follows is a consequence of what came before.
Used in 21 lessons

if and only if \Longleftrightarrow, \Leftrightarrow

Say it: is equivalent to

  • If and only if: each statement implies the other.
Used in 3 lessons

Operators and functions

times \times

Say it: times

  • Times (multiplication). Between two vectors: the cross product.
Used in 37 lessons

dot \cdot

Say it: times

  • Times (multiplication). Between two vectors: the dot product.
Used in 17 lessons

sine \sin

Say it: sine

  • Sine of an angle.
Used in 15 lessons

cosine \cos

Say it: cos

  • Cosine of an angle.
Used in 10 lessons

tangent \tan

Say it: tan

  • Tangent of an angle, sin/cos.
Used in 4 lessons

natural logarithm \ln

Say it: log

  • Natural logarithm (base e ≈ 2.718).
Used in 5 lessons

Calculus

integral \int

Say it: the integral of

  • Integral: the sum of infinitely many small pieces, such as the area under a curve.
Used in 16 lessons

limit \lim

Say it: the limit

  • Limit: the value approached as the variable written below approaches its target.
Used in 4 lessons

Accents (marks on a letter)

vector arrow \vec

Say it: vector

  • A vector: a quantity with a size and a direction.
Used in 17 lessons

hat \hat

Say it: hat

  • In 8.902: A Fourier-transformed quantity, such as δ̂(k) or Ŵ(k).
  • A unit vector: a vector of length 1 that only gives a direction (î, r̂).
Used in 10 lessons

dot \dot

Say it: dot

  • Rate of change with time: ȧ = da/dt.
Used in 18 lessons

double dot \ddot

Say it: double dot

  • Second rate of change with time: ä = d²a/dt².
Used in 10 lessons

First used in Year IV · Cosmology

Letters

a a

Say it: a

  • On Why a static universe cannot work: The scale factor a(t); in the integral limits a is just a number, and in d a⃗ (with an arrow) a small piece of surface area.
  • On Non-Euclidean geometry: A constant that sets the length scale of the disk.
  • On The spacetime interval: A label for an event (events a and b).
  • In 8.286: The scale factor a(t): how much the universe has expanded. Physical distance = a × comoving distance (Guth's units: metres per notch).
  • In 8.902: The scale factor a(t): how much the universe has expanded (a = 1 today, 1/(1 + z) in the past).
Used in 40 lessons

c c

Say it: c

Used in 38 lessons

T T

Say it: T

Used in 12 lessons

Greek letters

alpha \alpha

Say it: alpha

Used in 5 lessons

beta \beta

Say it: beta (sounds like “bay-ta or bee-ta”)

Used in 3 lessons

gamma \gamma

Say it: gamma

Used in 9 lessons

delta \delta

Say it: delta

Used in 6 lessons

epsilon \varepsilon, \epsilon

Say it: epsilon

  • A small starting time, later taken to zero.
Used in 1 lesson

zeta \zeta

Say it: zeta (sounds like “zay-ta or zee-ta”)

  • The Riemann zeta function; ζ(3) ≈ 1.202.
Used in 1 lesson

kappa \kappa

Say it: kappa

  • κ = −k, positive for an open universe.
Used in 1 lesson

lambda \lambda

Say it: lambda

Used in 8 lessons

capital lambda \Lambda

Say it: lambda

Used in 5 lessons

mu \mu

Say it: mu (sounds like “myoo”)

Used in 8 lessons

nu \nu

Say it: nu (sounds like “new”)

Used in 8 lessons

xi \xi

Say it: xi (sounds like “ksee or zye”)

Used in 3 lessons

sigma \sigma

Say it: sigma

Used in 6 lessons

tau \tau

Say it: tau (sounds like “rhymes with cow”)

Used in 10 lessons

phi \phi, \varphi

Say it: phi (sounds like “fye”)

Used in 12 lessons

chi \chi

Say it: chi (sounds like “kye”)

Used in 3 lessons

psi \psi

Say it: psi (sounds like “sigh”)

Used in 7 lessons

omega \omega

Say it: omega

Used in 4 lessons

capital omega \Omega

Say it: omega

  • The density parameter Ω = ρ/ρ_c. Subscripts pick one ingredient: Ω_m matter, Ω_vac (or Ω_Λ) vacuum energy, Ω_r radiation, Ω_k curvature.
Used in 17 lessons

capital gamma \Gamma

Say it: gamma

  • A Christoffel symbol, Γ^i_jk, built from derivatives of the metric.
Used in 1 lesson

Named quantities (a letter with a subscript)

little h h_0

Say it: h nought

  • H₀ in units of 100 km/s/Mpc: h₀ ≈ 0.67–0.7.
Used in 4 lessons

today t_0

Say it: t nought

  • Today: the present time, about 13.8 billion years after the Big Bang in the real universe (simple models on some pages give other ages).
Used in 11 lessons

temperature today T_0

Say it: T nought

  • Today's temperature of the cosmic background radiation, 2.7255 K.
Used in 3 lessons

redshift of equality z_eq

Say it: z equality

  • The redshift of matter–radiation equality, about 3400.
Used in 2 lessons

time of equality t_eq

Say it: t equality

  • The time of matter–radiation equality, some 50 000–75 000 years after the Big Bang.
Used in 1 lesson

critical density \rho_c

Say it: rho c

  • The critical density, 3H²/(8πG): the density of a flat universe (about 9 × 10⁻²⁷ kg/m³ today).
Used in 13 lessons

matter density \rho_m

Say it: rho m

Used in 6 lessons

radiation density \rho_r

Say it: rho r

  • The mass density of radiation (its energy density divided by c²).
Used in 3 lessons

vacuum density \rho_vac

Say it: rho vac

  • The mass density of the vacuum energy (its energy density divided by c²); constant in time.
Used in 2 lessons

false-vacuum density \rho_f

Say it: rho f

  • The mass density of the false vacuum during inflation; constant in time.
Used in 3 lessons

initial density \rho_i

Say it: rho i

  • The density when a = 1, so that ρ = ρᵢ/a³ for matter.
Used in 3 lessons

normal density \rho_n

Say it: rho n

  • The density of everything except the vacuum energy: ρₙ = ρ_m + ρ_r.
Used in 2 lessons

matter density parameter \Omega_m, \Omega_m,0

Say it: omega m

  • The matter density as a fraction of the critical density, ρ_m/ρ_c; ',0' means today (about 0.3).
Used in 8 lessons

vacuum density parameter \Omega_vac, \Omega_vac,0, \Omega_\Lambda

Say it: omega vac

  • The vacuum-energy density as a fraction of the critical density; ',0' means today (about 0.7).
Used in 6 lessons

radiation density parameter \Omega_r, \Omega_rad,0

Say it: omega r

  • The radiation density as a fraction of the critical density (about 9 × 10⁻⁵ today).
Used in 4 lessons

curvature density parameter \Omega_k,0, \Omega_k

Say it: omega k

  • The curvature term written like a density parameter, −kc²/(a²H²); ',0' means today. Zero for a flat universe.
Used in 5 lessons

Omega today \Omega_0

Say it: omega nought

  • Today's value of Ω, the total density as a fraction of the critical density.
Used in 1 lesson

physical distance \ell_p

Say it: ell p

  • A physical (proper) distance: what a ruler would measure at one moment, ℓ_p = a(t) ℓ_c.
Used in 4 lessons

horizon distance \ell_p,horizon

Say it: ell horizon

  • The horizon distance: how far light could have travelled since the Big Bang, as a physical distance.
Used in 2 lessons

Planck length \ell_P

Say it: ell P

  • The Planck length, √(ħG/c³) ≈ 1.6 × 10⁻³⁵ m: where quantum gravity should matter.
Used in 1 lesson

spectral index n_s

Say it: n s

  • The spectral index (tilt) of the primordial power spectrum, P ∝ kⁿ; measured n_s ≈ 0.965.
Used in 3 lessons

photon temperature T_\gamma

Say it: T gamma

  • The temperature of the photons (the CMB).
Used in 1 lesson

neutrino temperature T_\nu

Say it: T nu

  • The temperature of the neutrinos, (4/11)^(1/3) of the photon temperature after electron–positron annihilation.
Used in 1 lesson

photon energy density u_\gamma

Say it: u gamma

  • The energy density of the photons, proportional to T⁴.
Used in 3 lessons

photon number density n_\gamma

Say it: n gamma

  • The number of photons per unit volume, proportional to T³.
Used in 2 lessons

electron mass m_e

Say it: m e

  • The electron mass; m_e c² ≈ 0.511 MeV.
Used in 3 lessons

luminosity distance d_L

Say it: d L

  • The luminosity distance: the distance that would make the observed brightness obey the inverse-square law.
Used in 2 lessons

Schwarzschild radius r_s

Say it: r s

  • The Schwarzschild radius, 2GM/c²: the horizon of a black hole.
Used in 1 lesson

vacuum pressure p_vac

Say it: p vac

  • The pressure of the vacuum energy, p_vac = −ρ_vac c²: negative.
Used in 2 lessons

vacuum expansion rate H_vac

Say it: H vac

  • The expansion rate a universe dominated by vacuum energy settles to, √(8πGρ_vac/3).
Used in 2 lessons

maximum wavenumber k_max

Say it: k max

  • The largest wavenumber included: a cut-off at the smallest length scale.
Used in 2 lessons

Constants

h-bar \hbar

Say it: h bar

  • The reduced Planck constant, ħ = h/2π ≈ 1.055 × 10⁻³⁴ J·s.
Used in 9 lessons

Other symbols

infinity \infty

Say it: infinity

  • Infinity: larger than any number.
Used in 9 lessons

Relations

tilde (similar) \sim

Say it: is of order

  • Is of the order of: roughly equal, to within a factor of a few.
Used in 7 lessons

much less than \ll

Say it: is much less than

  • Is much less than.
Used in 8 lessons

much greater than \gg

Say it: is much greater than

  • Is much greater than.
Used in 6 lessons

less than about \lesssim

Say it: is less than about

  • Is less than, or roughly equal to.
Used in 5 lessons

greater than about \gtrsim

Say it: is greater than about

  • Is greater than, or roughly equal to.
Used in 3 lessons

less than or equal \le, \leq

Say it: is less than or equal to

  • Is less than or equal to.
Used in 6 lessons

element of \in

Say it: is in

  • Is an element of (belongs to) a set.
Used in 2 lessons

Operators and functions

plus or minus \pm

Say it: plus or minus

  • Plus or minus: both signs, or an uncertainty (67.3 ± 1.2).
Used in 12 lessons

minus or plus \mp

Say it: minus or plus

  • Minus or plus: the opposite sign to a ± in the same formula.
Used in 2 lessons

dagger \dagger

Say it: dagger

  • Conjugate transpose of a matrix: transpose it and take complex conjugates (U†).
Used in 2 lessons

maximum \max

Say it: max

  • Maximum: the largest value.
Used in 4 lessons

determinant \det

Say it: the determinant of

  • Determinant of a matrix.
Used in 1 lesson

cotangent \cot

Say it: cot

  • Cotangent of an angle, cos/sin.
Used in 1 lesson

arcsine \arcsin

Say it: arc sine

  • Inverse sine: the angle whose sine is the given number.
Used in 2 lessons

arccosine \arccos

Say it: arc cos

  • Inverse cosine: the angle whose cosine is the given number.
Used in 1 lesson

hyperbolic sine \sinh

Say it: sinh (sounds like “shine or sinch”)

  • Hyperbolic sine, (eˣ − e⁻ˣ)/2.
Used in 4 lessons

hyperbolic cosine \cosh

Say it: cosh (sounds like “kosh”)

  • Hyperbolic cosine, (eˣ + e⁻ˣ)/2.
Used in 2 lessons

hyperbolic tangent \tanh

Say it: tanh (sounds like “tanch”)

  • Hyperbolic tangent, sinh/cosh.
Used in 1 lesson

logarithm \log

Say it: log

  • Logarithm; log₁₀ is the base-10 logarithm.
Used in 6 lessons

Calculus

partial \partial

Say it: partial

  • Partial derivative: the rate of change with one variable, holding the others fixed.
Used in 8 lessons

nabla (del) \nabla

Say it: del

  • Del: ∇f is the gradient, ∇·v the divergence, ∇×v the curl, ∇² the Laplacian.
Used in 4 lessons

loop integral \oint

Say it: the loop integral of

  • An integral around a closed loop.
Used in 1 lesson

sum \sum

Say it: the sum of

  • Sum: add up the terms over the range written below and above.
Used in 2 lessons

Accents (marks on a letter)

bar \bar, \overline

Say it: bar

Used in 5 lessons

tilde \tilde

Say it: tilde

  • A modified version of a quantity (ã, k̃); its meaning is given where it is introduced.
Used in 7 lessons

First used in Year V · Graduate year: astrophysics and general relativity

Named quantities (a letter with a subscript)

solar mass M_\odot

Say it: solar masses

  • One solar mass, about 1.99 × 10³⁰ kg.
Used in 3 lessons

Jeans length \lambda_J

Say it: lambda J

  • The Jeans length: perturbations larger than this grow under gravity, smaller ones oscillate.
Used in 2 lessons

Jeans mass M_J

Say it: M J

  • The Jeans mass: the mass inside a sphere whose radius is the Jeans length, (4/3)πρ̄λ_J³.
Used in 2 lessons

speed of sound c_s

Say it: c s

  • The speed of sound in the gas.
Used in 2 lessons

sound or dispersion speed v_s

Say it: v s

  • The speed that resists collapse: the sound speed for baryons, the velocity dispersion for dark matter, c/√3 for radiation.
Used in 1 lesson

collapse threshold \delta_c

Say it: delta c

  • The critical linear density contrast: a region has collapsed when linear theory gives it δ ≈ 1.69.
Used in 2 lessons

sigma eight \sigma_8

Say it: sigma 8

  • The rms density fluctuation in spheres of radius 8 h⁻¹ Mpc: the standard measure of how clumpy matter is (about 0.8).
Used in 2 lessons

sigma R \sigma_R

Say it: sigma R

  • The rms density fluctuation after smoothing over spheres of radius R(M).
Used in 1 lesson

Hubble time \tau_H

Say it: tau H

  • The expansion time scale, 1/H.
Used in 1 lesson

collision time \tau_coll

Say it: tau collision

  • The mean time between annihilations, 1/(n_eq⟨σv⟩).
Used in 1 lesson

equilibrium density n_eq, n_c,eq

Say it: n equilibrium

  • The number density in thermal equilibrium (n_c,eq: the same, comoving).
Used in 1 lesson

Constants

Sun symbol \odot

Say it: solar

  • The Sun: M_⊙ is one solar mass, about 1.99 × 10³⁰ kg.
Used in 3 lessons

Relations

congruent \cong

Say it: is approximately

  • Is approximately equal to (written ≅ in the source).
Used in 1 lesson

greater than or equal \ge, \geq

Say it: is greater than or equal to

  • Is greater than or equal to.
Used in 1 lesson

equilibrium arrows \rightleftharpoons

Say it: goes both ways to

  • A reaction that runs in both directions (equilibrium).
Used in 1 lesson

Operators and functions

angle brackets \langle, \rangle

Say it: the average of

  • The average (mean value) of what is inside, ⟨…⟩.
Used in 2 lessons