The Scale Factor and Hubble Parameter

Almost every quantity in cosmology is really a statement about one function. The cosmological principle forces the geometry of the universe into the FLRW form, in which all of the time dependence of space lives in a single dimensionless number, the scale factor $a(t)$ . Two galaxies pinned to fixed comoving coordinates a comoving distance $\chi$ apart have a proper separation $D(t)=a(t)\chi$ , so the entire kinematics of cosmic expansion — every recession velocity, every redshift, every horizon — reduces to how one number grows with cosmic time. This is why the ten-function problem of a general spacetime collapses into ordinary differential equations for $a(t)$ . The Hubble parameter is what you get when you ask how fast that number grows, measured as a fraction of itself: $H(t)\equiv\dot a/a=\mathrm{d}\ln a/\mathrm{d}t$ . It has units of inverse time and is a property of a whole time slice, not of any one galaxy — every comoving observer measures the same $H$ , as homogeneity demands. Differentiate $D=a\chi$ and Hubble's law $\dot D=HD$ falls out in one line. Three things organize what follows. First, $H$ is not a constant: its present value $H_0$ is one number, but the full history $H(t)$ is fixed by what fills the universe through the Friedmann equations, giving $a\propto t^{1/2}$ , $t^{2/3}$ , or $e^{Ht}$ for radiation, matter, and a cosmological constant. Second, the observable that carries $a(t)$ to us is redshift, through $1+z=1/a$ . Third, the mismatch between local and early-universe measurements of $H_0$ — the Hubble tension — is a sharp open problem in physics.

Key takeaways

The scale factor $a(t)$ is a dimensionless function of cosmic time that sets how all proper distances between comoving observers grow, normalized to $a(t_0)=1$ , with $D(t)=a(t)\chi$ .
The Hubble parameter $H(t)\equiv\dot a/a$ is the instantaneous fractional expansion rate; its value now is the Hubble constant $H_0\approx67\text{–}74\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ .
Hubble's law $\dot D=HD$ follows in one line from $D=a\chi$ : recession velocity proportional to distance, no special centre, as homogeneity requires.
Cosmological redshift measures the expansion factor directly, $1+z=1/a(t_e)$ , making $z$ a more physical time coordinate than $t$ .
The dynamics of $a(t)$ are fixed by the Friedmann equation and the energy content; each component dilutes as $\rho\propto a^{-3(1+w)}$ , giving power-law or exponential $a(t)$ .
Anchor values: Hubble time $1/H_0\approx14\ \mathrm{Gyr}$ , Hubble distance $c/H_0\approx4.3\ \mathrm{Gpc}$ , $\Lambda$ CDM age $13.8\ \mathrm{Gyr}$ , present deceleration $q_0\approx-0.53$ .
Key idea: All of cosmic expansion is encoded in one function $a(t)$ ; the Hubble parameter $H=\dot a/a$ is its fractional growth rate, and proper distances obey $D(t)=a(t)\chi$ so that $\dot D=HD$ .

Where this lives today

The scale factor and its rate are the front line of a live controversy: measuring $H_0$ splits into two camps. The local distance ladder (Cepheid-calibrated Type Ia supernovae, the SH0ES programme) reads $H_0\approx73\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ from the nearby Hubble flow, while the early-universe inference from the Planck CMB, propagated forward through $\Lambda$ CDM, gives $H_0\approx67.4\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ . The roughly five-sigma gap is the Hubble tension, and it has survived a decade of cross-checks, including JWST's re-verification of the Cepheid calibration. New handles keep arriving — gravitational-wave "standard sirens" like GW170817 measure $H_0$ geometrically with no distance ladder (early value near $70$ ), and BAO surveys such as DESI map $H(z)$ across redshift, hinting that dark energy may not be an exactly constant $\Lambda$ . Whether the tension dissolves into a systematic or forces new physics, the whole question is a debate about the single function $a(t)$ and the number $\dot a/a$ this article defines.

The physical picture

Picture galaxies as raisins fixed in a rising loaf, or as dots painted on a coordinate grid. The dots never move through the dough — their comoving coordinates are frozen — yet every pair recedes from every other because the grid itself stretches. The scale factor $a(t)$ is that stretch, and the proper distance between two galaxies is their fixed comoving separation multiplied by $a$ . Because the stretch is uniform, doubling the separation doubles the recession speed: that proportionality is Hubble's law, and it needs no centre. The first figure freezes this into two snapshots and attaches a number.

Figure

A comoving grid of galaxies at two epochs: galaxies stay pinned to fixed comoving nodes while the scale factor grows from a = 0.5 to a = 1, so the proper distance D = a·χ between galaxies A and B doubles from 200 to 400 Mpc, and light emitted then arrives redshifted by 1 + z = 1/a = 2

The grid never picks out a centre — run the same movie from galaxy B and it looks identical. What the second figure adds is dynamics: the shape of $a(t)$ is not free. Feed the Friedmann equation the actual contents of our universe and it returns one specific curve — a scale factor that decelerates while matter dominates and then, once the cosmological constant takes over, inflects into accelerating growth, with hard-numbered milestones.

Figure

The computed expansion history of our universe: the ΛCDM scale factor a(t) for H₀ = 67.7 km s⁻¹ Mpc⁻¹, Ω_m = 0.315, Ω_Λ = 0.685 decelerates through the matter era, inflects to acceleration at a ≈ 0.61 near 7.6 Gyr, and reaches a = 1 today at 13.8 Gyr, whereas a matter-only universe of the same H₀ would be only 9.3 Gyr old

The core equations

Everything begins with the metric: homogeneity and isotropy force the line element into the FLRW form, in which a single function $a(t)$ multiplies the entire spatial part and the constant $k$ selects one of three geometries.

\mathrm{d}s^2=-c^2\,\mathrm{d}t^2+a^2(t)\left[\frac{\mathrm{d}r^2}{1-kr^2}+r^2\,\mathrm{d}\Omega^2\right] \qquad (1)

where: $a(t)$ — dimensionless scale factor, normalized $a(t_0)=1$ ; $t$ — cosmic (proper) time read by comoving observers; $k$ — spatial curvature constant $+1,0,-1$ (or carrying dimensions $1/L^2$ ); $r,\Omega$ — comoving radial and angular coordinates. Reading it: fix the coordinates and only $a(t)$ can change, so all cosmic evolution is squeezed into one function. The Hubble parameter is the logarithmic time derivative of the scale factor — the growth rate measured as a fraction of the current size. Writing it as $\mathrm{d}\ln a/\mathrm{d}t$ makes plain that it is a rate, not a velocity, with units of inverse time.

H(t)\equiv\frac{\dot a}{a}=\frac{\mathrm{d}\ln a}{\mathrm{d}t} \qquad (2)

where: $\dot a$ — the cosmic-time derivative $\mathrm{d}a/\mathrm{d}t$ ; $H$ — Hubble parameter, units of inverse time; $H_0\equiv H(t_0)$ — its present value, the Hubble constant. Reading it: because $a$ is common to the whole slice, so is $H$ — it labels an epoch, not a place. Combine the metric's proper distance with the definition of $H$ and Hubble's law appears immediately: a comoving object sits at fixed $\chi$ , so differentiating $D=a\chi$ gives a recession velocity proportional to distance.

D(t)=a(t)\,\chi,\qquad \dot D=\frac{\dot a}{a}\,D=H D \qquad (3)

where: $D$ — proper distance to a comoving object; $\chi$ — comoving distance, time-independent; $\dot D$ — proper recession velocity. Reading it: double the distance, double the recession speed, same $H$ for every observer — the signature of uniform expansion. The observable that delivers $a(t)$ to a telescope is redshift. As light travels through the stretching geometry its wavelength is dragged along with the grid, so received over emitted wavelength is exactly the ratio of scale factors.

1+z=\frac{a(t_0)}{a(t_{\mathrm{emit}})}=\frac{1}{a(t_{\mathrm{emit}})} \qquad (4)

where: $z$ — cosmological redshift of the observed light; $t_0$ — time of observation, today; $t_{\mathrm{emit}}$ — time of emission. Reading it: $z$ gives the expansion factor since emission — $z=1$ means the universe was half its present size at emission. The shape of $a(t)$ is dictated by dynamics through the first Friedmann equation — the Einstein field equation on the FLRW metric — which ties $H^2$ to the energy density, the spatial curvature, and the cosmological constant.

H^2=\left(\frac{\dot a}{a}\right)^2=\frac{8\pi G}{3}\rho-\frac{kc^2}{a^2}+\frac{\Lambda c^2}{3} \qquad (5)

where: $\rho$ — total energy density (as a mass density) of the cosmic fluid; $G$ — Newton's constant; $\Lambda$ — cosmological constant. Reading it: energy density and $\Lambda$ drive expansion while positive curvature ( $k>0$ ) brakes it; given the contents, this equation fixes $H$ at every epoch. How fast the expansion decelerates or accelerates is captured by the second time derivative, packaged dimensionlessly as the deceleration parameter, whose sign is the whole story.

q(t)\equiv-\frac{\ddot a\,a}{\dot a^2}=-\frac{\ddot a}{aH^2} \qquad (6)

where: $q$ — dimensionless deceleration parameter; $q>0$ deceleration, $q<0$ acceleration; $\ddot a$ — second cosmic-time derivative of $a$ . Reading it: the sign convention builds in a minus, so a decelerating universe has positive $q$ — the 1998 discovery that $q_0<0$ is why the name now looks perverse. Rolling each component's density scaling into the Friedmann equation gives the workhorse of observational cosmology: $H$ as an explicit function of redshift, the integrand behind every distance and lookback-time.

H(z)=H_0\sqrt{\Omega_r(1+z)^4+\Omega_m(1+z)^3+\Omega_k(1+z)^2+\Omega_\Lambda} \qquad (7)

where: $H_0$ — present Hubble constant; $\Omega_r,\Omega_m,\Omega_k,\Omega_\Lambda$ — present density parameters for radiation, matter, curvature, and dark energy, summing to one. Reading it: each power of $(1+z)$ records how a component dilutes, so at high $z$ radiation, then matter, then $\Lambda$ successively dominate $H(z)$ . Two combinations of $H_0$ set the natural scales of the observable universe: a timescale and a length scale, obtained by dimensional analysis alone.

t_H=\frac{1}{H_0}\approx14\ \mathrm{Gyr},\qquad D_H=\frac{c}{H_0}\approx4.3\ \mathrm{Gpc} \qquad (8)

where: $t_H$ — Hubble time, a rough estimate of the age of the universe; $D_H$ — Hubble distance, the scale beyond which the recession speed exceeds $c$ . Reading it: order-of-magnitude anchors; the true age and horizon follow from integrating $H(z)$ .

Derivation AThe expansion law and the meaning of the fractional rate

The goal is to show, with no dynamics assumed, that the single geometric statement $D=a\chi$ already forces recession velocity to be strictly proportional to distance, and to make precise the sense in which the Hubble velocity is a stretching rate rather than a motion through space. We differentiate at fixed comoving coordinate, then separate the Hubble flow from peculiar motion.

Step 1. Place a galaxy at fixed comoving distance $\chi$ ; the FLRW metric says its proper distance at cosmic time $t$ is $D(t)=a(t)\chi$ , because the spatial line element is just $a(t)$ times the comoving one. Why: comoving coordinates are defined precisely so that free-falling galaxies keep constant coordinate labels, so all of the time dependence sits in $a$ .

Step 2. Differentiate at fixed $\chi$ : since $\chi$ does not change for a comoving object, $\dot D=\dot a\,\chi$ . Substitute $\chi=D/a$ to eliminate the coordinate and obtain $\dot D=(\dot a/a)D=HD$ , which is Hubble's law with $H=\dot a/a$ . Why: replacing $\chi$ by $D/a$ is what converts a statement about a fixed coordinate into the observable proportionality between velocity and distance.

Step 3. $HD$ is the rate at which a proper distance grows as the metric stretches, not a locally measured speed. A galaxy at $D=c/H$ has $\dot D=c$ , and beyond it $\dot D>c$ , with no contradiction because nothing overtakes a passing light ray locally. Why: the superluminal recession is the growth of a global coordinate difference, a regime special relativity does not govern.

Step 4. Add a peculiar velocity. A real galaxy has a small motion $v_{\mathrm{pec}}$ on top of the Hubble flow, so its redshift combines $HD$ with $v_{\mathrm{pec}}$ ; over many galaxies the peculiar terms cancel and $\dot D=HD$ remains. Why: this is why $H_0$ is measured from large samples reaching beyond the local $\sim500\ \mathrm{km/s}$ peculiar velocities.

Go deeper · Derivation B · 1 minScale-factor evolution for a single-component flat universe

Derivation BScale-factor evolution for a single-component flat universe

The aim is to solve the Friedmann equation for the cleanest case — a spatially flat universe filled with one fluid of constant equation of state $w$ , with $p=w\rho c^2$ — and to show that the answer is always a power law in cosmic time, degenerating to an exponential exactly when $w=-1$ . We dilute the density, insert into the flat Friedmann equation, and integrate — reproducing radiation, matter, and de Sitter at once.

Step 1. Local energy conservation in FLRW is the continuity equation $\dot\rho+3H(\rho+p/c^2)=0$ ; with $p=w\rho c^2$ it becomes $\dot\rho/\rho=-3(1+w)\dot a/a$ . Why: the $3H$ factor is the fractional rate at which a comoving volume grows, so this equation just says energy density is diluted by expansion plus the work done by pressure.

Step 2. Integrate to get the dilution law $\rho\propto a^{-3(1+w)}$ — radiation $\rho\propto a^{-4}$ , matter $\rho\propto a^{-3}$ , cosmological constant $\rho=$ const. Why: the extra power of $a$ for radiation over matter is the redshifting of each photon's energy on top of the volume dilution both share.

Step 3. Insert into the flat ( $k=0$ , $\Lambda$ absorbed into the fluid) Friedmann equation $\dot a^2/a^2=(8\pi G/3)\rho\propto a^{-3(1+w)}$ , so $\dot a\propto a^{-(1+3w)/2}$ . Separate variables as $a^{(1+3w)/2}\,\mathrm{d}a\propto\mathrm{d}t$ and integrate. Why: reducing the dynamics to a single separable first-order equation is exactly what the one-function FLRW simplification buys us.

Step 4. For $w>-1$ the integral gives $a^{3(1+w)/2}\propto t$ , hence $a(t)\propto t^{2/[3(1+w)]}$ ( $t^{1/2}$ radiation, $t^{2/3}$ matter). For $w=-1$ the density is constant, so $H=\sqrt{8\pi G\rho/3}$ is constant and $\dot a/a=H$ integrates to $a(t)=a_0\,e^{H(t-t_0)}$ , the de Sitter exponential. Why: the algebra breaks at $w=-1$ precisely because the exponent $2/[3(1+w)]$ diverges — the power law smoothly hands off to exponential growth as $w\to-1$ .

Go deeper · Derivation C · 1 minCosmological redshift from the scale factor

Derivation CCosmological redshift from the scale factor

The goal is to derive $1+z=1/a(t_e)$ directly from the geometry, tracking a light ray and two successive wave crests through the expanding FLRW space, so that redshift is exposed as a stretching of wavelength by the same factor $a$ that stretches every other proper length. We use the null condition and the equality of comoving distance for both crests.

Step 1. For a radial light ray $\mathrm{d}s^2=0$ and $\mathrm{d}\Omega=0$ , so $0=-c^2\mathrm{d}t^2+a^2\mathrm{d}r^2$ , giving $c\,\mathrm{d}t/a(t)=\mathrm{d}r$ . Why: dividing by $a$ is the key step — it turns cosmic-time intervals into fixed comoving-distance increments, which is what lets two crests be compared.

Step 2. A crest emitted at $t_e$ and the next at $t_e+\delta t_e$ both travel the same comoving distance $\chi=\int c\,\mathrm{d}t/a$ , arriving at $t_0$ and $t_0+\delta t_0$ . Subtracting the two integrals, the overlap cancels and for small intervals $\delta t_0/a(t_0)=\delta t_e/a(t_e)$ . Why: the comoving distance to a fixed source is frozen, so any change in travel time between crests is bookkeeping about the endpoints alone.

Step 3. The crest spacings $\delta t$ are the wave periods, and wavelength is $\lambda=c\,\delta t$ , so $\lambda_{\mathrm{obs}}/\lambda_{\mathrm{emit}}=a(t_0)/a(t_e)$ . Defining $1+z\equiv\lambda_{\mathrm{obs}}/\lambda_{\mathrm{emit}}$ and using $a(t_0)=1$ gives $1+z=1/a(t_e)$ . Why: this shows cosmological redshift is not a Doppler shift from a velocity but a direct readout of how much the universe has expanded since emission — the same $a$ , seen in the light.

Limiting cases

The behaviour of $a(t)$ is fixed by what dominates the energy budget through the equation of state $p=w\rho c^2$ . For a single-component flat universe, the solutions and deceleration parameters $q=(1+3w)/2$ are:

Case	Result	Meaning
Radiation ( $w=1/3$ )	$a\propto t^{1/2}$ , $H=1/(2t)$ , $q=1$	Early universe; density dilutes as $\rho\propto a^{-4}$ ; strongly decelerating.
Matter ( $w=0$ )	$a\propto t^{2/3}$ , $H=2/(3t)$ , $q=1/2$	Einstein–de Sitter behaviour; $\rho\propto a^{-3}$ ; the classic decelerating dust cosmology.
Cosmological constant ( $w=-1$ )	$a\propto e^{Ht}$ , $H=$ const, $q=-1$	de Sitter expansion; constant $H$ , steady acceleration; drives inflation and the late universe.
General $w$ , flat, $w>-1$	$a\propto t^{2/[3(1+w)]}$	Power-law expansion; accelerates ( $q<0$ ) whenever $w<-1/3$ .
Empty, curved ( $k=-1$ )	$a\propto t$ , $q=0$	Milne model; coasting expansion, a massless reference.

Common trapthe Hubble constant is not constant in time

Two traps hide in the vocabulary. First, the "Hubble constant" $H_0$ is constant only across space on today's time slice — the same for every comoving observer, by homogeneity — but $H(t)=\dot a/a$ generally changes with time, and was vastly larger early on (formally $H\to\infty$ as $t\to0$ ). The time-varying object is properly the Hubble parameter; $H_0$ is just its present snapshot. Second, the recession "velocity" $v=HD$ is the growth rate of a proper distance, not a local motion: for $D>c/H$ it exceeds $c$ without violating relativity, because no local signal outruns light — the superluminal number is a coordinate statement about a stretching metric. Galaxies beyond the Hubble distance recede faster than light yet remain observable, because photons they emitted long ago crossed regions where recession was slower.

Worked examples

1 · Hubble time, Hubble distance, and a recession velocity (confidence builder)

Take $H_0=70\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ . Find the Hubble time $1/H_0$ in gigayears, the Hubble distance $c/H_0$ in gigaparsecs, and the recession velocity and approximate redshift of a galaxy at proper distance $D=100\ \mathrm{Mpc}$ .

Step 1. Put $H_0$ in SI. With $1\ \mathrm{Mpc}=3.086\times10^{19}\ \mathrm{km}$ , $H_0=2.27\times10^{-18}\ \mathrm{s^{-1}}$ , so $1/H_0=4.41\times10^{17}\ \mathrm{s}$ . Dividing by $3.156\times10^{16}\ \mathrm{s\,Gyr^{-1}}$ gives $1/H_0=\mathbf{13.97\ Gyr}$ , the familiar $14\ \mathrm{Gyr}$ .

Step 2. The Hubble distance is $c/H_0=(2.998\times10^5)/70\ \mathrm{Mpc}=4283\ \mathrm{Mpc}=\mathbf{4.28\ Gpc}$ . Equivalently $c/H_0=c\,t_H=13.97\ \mathrm{Gly}$ — one fact in two units.

Step 3. Hubble's law gives the recession velocity: $v=H_0 D=(70)(100)=\mathbf{7000\ km\,s^{-1}}$ . Since $v\ll c$ , the low-redshift approximation $z\approx v/c$ holds: $z\approx7000/299\,792=\mathbf{0.023}$ — the near-linear redshift-to-distance map behind the low- $z$ Hubble law.

Step 4. Sanity check: at $D=c/H_0$ the same law gives $v=c$ , so the Hubble distance is where recession reaches light speed; our galaxy sits at only $2.3\%$ of that, safely linear. So $1/H_0=14\ \mathrm{Gyr}$ , $c/H_0=4.3\ \mathrm{Gpc}$ , and the galaxy recedes at $7000\ \mathrm{km/s}$ with $z\approx0.023$ .

2 · From a measured redshift to scale factor, age, and acceleration

A quasar is observed at redshift $z=1$ . Find the scale factor and universe size at emission and the wavelength stretch. Then, treating the expansion as matter-dominated (Einstein–de Sitter) with $H_0=70\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ , estimate the present age, and finally compute the present deceleration parameter for the real $\Lambda$ CDM mix $\Omega_m=0.315$ , $\Omega_\Lambda=0.685$ .

Step 1. The redshift fixes the scale factor: $a(t_e)=1/(1+z)=1/2$ . The universe was half its present linear size when this light left, and every wavelength has since been stretched by $1+z=\mathbf{2}$ — ultraviolet emitted then arrives in the visible.

Step 2. In a matter-dominated flat universe $a\propto t^{2/3}$ , so $H=2/(3t)$ and the present age is $t_0=2/(3H_0)=\tfrac23\times13.97\ \mathrm{Gyr}=\mathbf{9.31\ Gyr}$ . This is younger than the oldest stars — a real crisis for matter-only cosmology before dark energy.

Step 3. The fix is $\Lambda$ : dark energy lengthens the age because the universe expanded more slowly early on before accelerating. Integrating $\mathrm{d}t=\mathrm{d}a/(aH)$ with $H(a)=H_0\sqrt{\Omega_m a^{-3}+\Omega_\Lambda}$ gives $t_0=13.8\ \mathrm{Gyr}$ , older than any star — and $1/H_0=14\ \mathrm{Gyr}$ lands close by a near-cancellation of the early slow-down and late speed-up.

Step 4. The deceleration parameter for a mix is $q_0=\tfrac12\sum_i\Omega_i(1+3w_i)=\tfrac12(\Omega_m-2\Omega_\Lambda)$ . With the given values $q_0=\tfrac12(0.315-1.370)=\mathbf{-0.53}$ . The negative sign is the acceleration discovered in 1998: $\Lambda$ outweighs matter today. So $a(t_e)=1/2$ and $\lambda$ stretches by $2$ ; matter-only gives $9.3\ \mathrm{Gyr}$ , real $\Lambda$ CDM gives $13.8\ \mathrm{Gyr}$ , and $q_0=-0.53$ .

Physics of the portrait

The scale factor and the Hubble parameter are where a theorist's differential equation and an observer's redshift catalogue meet, and one figure sits exactly at that junction: a Belgian priest who both solved for $a(t)$ and read the expansion rate off the sky, two years before the relation acquired someone else's name.

Origins · 1 minGeorges Lemaître and how this was discovered

Georges Lemaître

1894–1966 · the expanding-universe solution and the first empirical expansion rate

Georges Lemaître was a Belgian Catholic priest and mathematical physicist who did more than anyone to turn the abstract $a(t)$ of relativistic cosmology into a claim about the real sky. Trained partly under Eddington and Shapley, he independently rediscovered the expanding solutions of Einstein's equations that Alexander Friedmann had found in 1922, but went a decisive step further. In his 1927 paper he derived that a homogeneous universe must expand, that galaxy spectra should therefore be redshifted in proportion to distance, and — using the meagre data then available — he extracted a numerical expansion rate, the quantity we now call the Hubble constant. This was two years before Edwin Hubble's 1929 paper; the crucial paragraphs were notoriously omitted from the 1931 English translation, which is part of why the credit long went one way. Lemaître pushed the logic backward to its limit: an expanding universe was once arbitrarily small and dense, a "primeval atom" whose disintegration began cosmic history — the conceptual seed of the Big Bang. Einstein, who had introduced $\Lambda$ to keep the universe static, at first resisted, reportedly telling him "your calculations are correct, but your physics is abominable," and only later embraced the expanding picture. Lemaître also anticipated that a cosmological constant could make the expansion accelerate — vindicated seventy years later — and he lived just long enough to learn, in 1966, of the discovery of the cosmic microwave background, the relic glow his primeval atom had predicted.

History

1917

— Einstein introduces the cosmological constant Λ to force a static universe; de Sitter finds an empty expanding solution

1922

— Friedmann derives the expanding and contracting homogeneous solutions and the equation governing a(t)

1927

— Lemaître links an expanding universe to galaxy redshifts and estimates the expansion rate

1929

— Hubble publishes the linear velocity–distance relation from measured galaxy recessions

1998

— supernova surveys find the expansion is accelerating, so q₀ is negative

Going further

Texts. Misner, Thorne & Wheeler, Gravitation, ch. 27 · Wald, General Relativity, ch. 5 · Carroll, Spacetime and Geometry, ch. 8 (eqs. 8.50–8.59) · Hartle, Gravity, ch. 18 · Schutz, A First Course in General Relativity, ch. 12 · Dodelson, Modern Cosmology, ch. 1 · Peebles, Principles of Physical Cosmology, ch. 5. Key numerical benchmarks: the dilution law $\rho\propto a^{-3(1+w)}$ and solution $a\propto t^{2/[3(1+w)]}$ follow from continuity plus the flat Friedmann equation; the redshift relation $1+z=1/a$ follows from the radial null geodesic. At $H_0=70\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ the anchors are $1/H_0=13.97\ \mathrm{Gyr}$ and $c/H_0=4.28\ \mathrm{Gpc}$ ; the Einstein–de Sitter age is $t_0=2/(3H_0)=9.31\ \mathrm{Gyr}$ ; and $q_0=\tfrac12(\Omega_m-2\Omega_\Lambda)=-0.53$ for $\Omega_m=0.315$ , $\Omega_\Lambda=0.685$ . The $\Lambda$ CDM curve in the second figure uses $H_0=67.7$ , $\Omega_m=0.315$ , $\Omega_\Lambda=0.685$ , giving an age of $13.8\ \mathrm{Gyr}$ and acceleration onset at $a\approx0.61$ . The $H_0$ tension values are Planck 2018 ( $67.4$ ) and SH0ES ( $\approx73$ ).

Prerequisites. The Cosmological Principle · The FLRW Metric · Comoving and Proper Coordinates. Leads to. Cosmological Redshift · The Friedmann Equations · Critical Density and the Density Parameters · Cosmological Horizons.

Ask the tutor: Ask me to start from the Friedmann equation and integrate $\mathrm{d}t=\mathrm{d}a/(aH)$ for the real $\Lambda$ CDM mix, showing exactly how the deceleration-then-acceleration history conspires to make the age 13.8 Gyr land so close to the Hubble time.

Is space itself really expanding, or are the galaxies just flying apart?

Both predict the same low-redshift observations, but the "stretching space" picture generalizes correctly. In FLRW the coefficient $a(t)$ multiplies the spatial line element, so the proper distance between comoving galaxies grows even though their coordinates are fixed — that is what "space expanding" means operationally. The flying-apart picture works locally but breaks down beyond $c/H$ , where recession exceeds $c$ and no single inertial frame contains both galaxies. "Expansion of space" is shorthand for the time dependence of the spatial metric — a coordinate-independent fact.

Does the expansion pull apart atoms, planets, and galaxies too?

No. The Hubble flow only describes objects held apart solely by the average cosmic geometry. Anything bound by a stronger local force — an atom by electromagnetism, a galaxy by its own gravity — sits in a locally overdense, non-expanding patch and does not participate. The raisins do not swell, only the dough between them does. Bound systems decoupled from the Hubble flow long ago and keep a fixed size; only the vast, underdense gaps between galaxy clusters grow with $a(t)$ . Only a future "Big Rip" — dark energy overwhelming local binding forces — would change that.

Why is the Hubble constant sometimes 14 billion years and sometimes the age of the universe?

Because $1/H_0$ is the age the universe would have if it had always expanded at today's rate — a straight-line extrapolation of the present $a(t)$ back to $a=0$ . The real age comes from integrating $\int\mathrm{d}a/(aH)$ , which differs because $H$ was larger in the past. It is a coincidence of our epoch that the two nearly agree: matter-era deceleration made the early expansion faster while recent $\Lambda$ -driven acceleration made it slower, and for the observed parameters these almost cancel, leaving $13.8\ \mathrm{Gyr}$ close to $1/H_0=14\ \mathrm{Gyr}$ .

How is cosmological redshift different from an ordinary Doppler shift?

An ordinary Doppler shift is set by a relative velocity at emission. Cosmological redshift instead accumulates over the whole journey: $1+z=a(t_0)/a(t_e)$ depends on the total expansion between emission and reception, not on any single velocity. For nearby galaxies the two coincide, $z\approx v/c$ , which is why the low- $z$ Hubble law looks Doppler-like. But at high $z$ they diverge — a photon from $z=1100$ (the CMB) is not reporting a velocity of $1100c$ ; it reports that the universe has expanded by a factor $1101$ since emission. Cleanest statement: the wavelength was stretched by exactly the factor by which space stretched.

What actually fixes the convention a(t₀) = 1?

It is a free choice; setting $a=1$ today is simply the most convenient normalization. Only ratios of scale factors are physical — both $1+z=a(t_0)/a(t_e)$ and $D=a\chi$ involve $a$ only through comparisons — so we may rescale $a$ by any constant, absorbing the change into comoving distance and, if $k\neq0$ , into the value of $k$ . Choosing $a(t_0)=1$ makes today's comoving and proper distances coincide and makes $1+z=1/a(t_e)$ read cleanly. Some texts keep $k=\pm1$ and let $a$ carry length; the physics is identical.

If every galaxy sees itself at the centre of the expansion, where is the real centre?

There is none, and that is the whole point of homogeneity. Hubble's law $\dot D=HD$ holds about every comoving point at once: stand on any galaxy and all the others recede with velocity proportional to distance, with the same $H$ . This works because the expansion is a uniform stretching of the entire grid, not an explosion from a point — vector-add the recession galaxy A sees and you recover exactly what galaxy B reports. The Big Bang did not happen at a location in space; it happened everywhere at once, at $a\to0$ . Asking for the centre of the expansion is like asking for the centre of the surface of a sphere.