The Expansion of the Universe and the Scale Factor

Take a photograph of the universe on scales larger than a few hundred megaparsecs and it looks the same in every direction and from every vantage point — homogeneous and isotropic. The Cosmological Principle forbids any preferred place or direction, and that single symmetry is astonishingly restrictive: the only freedom left in the geometry is one overall multiplicative factor that scales every separation in lockstep. That factor is the scale factor $a(t)$ , a dimensionless function of cosmic time alone. It is the mathematical form of the sentence "the universe expands uniformly," and it is the one dynamical variable of the entire Friedmann–Lemaître–Robertson–Walker (FLRW) universe. What makes $a(t)$ so powerful is that it cleanly separates two ideas that are easy to confuse. Galaxies at rest in the cosmic rest frame keep fixed comoving coordinates forever; the proper distance between them grows as $d_p=a(t)\,\chi$ . Nothing moves through space — the metric, the very definition of distance, is what changes. From this one relation everything follows: the fractional growth rate $H=\dot a/a$ is the Hubble parameter, the recession law $v=H d_p$ is its time derivative, and the stretching of light in flight gives the cosmological redshift $1+z=1/a$ . Because that stretch is directly readable from a galaxy's spectrum, $a$ (or equivalently $z$ ) is the natural clock and ruler of cosmology. The dynamics of $a(t)$ come from Einstein's equations applied to a perfect fluid — the Friedmann and acceleration equations. Radiation, matter, and dark energy dilute at different rates, so the universe hands dominance from one to the next, and the growth law of $a$ changes with each handover. This article is where that machinery is assembled.

Key takeaways

The scale factor $a(t)$ is a dimensionless function of cosmic time that describes how all cosmic distances grow together under homogeneity and isotropy.
Proper distance is fixed comoving distance times the scale factor, $d_p(t)=a(t)\,\chi$ , so any two comoving observers separate in proportion to $a(t)$ .
Its logarithmic derivative is the Hubble parameter, $H(t)=\dot a/a$ , which equals the Hubble constant $H_0$ today (with the convention $a(t_0)=1$ ).
Expansion stretches light in flight, so redshift reads the scale factor at emission directly: $1+z=1/a$ .
The Friedmann equations govern $a(t)$ ; single-component flat universes give power laws $a\propto t^{1/2}$ (radiation) and $a\propto t^{2/3}$ (matter), and exponential growth $a\propto e^{Ht}$ under a cosmological constant.
Today's universe is dark-energy dominated and accelerating, $\ddot a>0$ — the discovery that won the 2011 Nobel Prize.
Key idea: All cosmic distances scale in lockstep with one dimensionless function $a(t)$ ; proper distance is $d_p=a(t)\chi$ , expansion rate is $H=\dot a/a$ , and redshift is $1+z=1/a$ .

Where this lives today

The scale factor is the working coordinate of modern cosmology, and pinning down its history is a multi-billion-dollar enterprise. The Dark Energy Spectroscopic Instrument (DESI) and its predecessors measure baryon acoustic oscillations — a fixed comoving ruler imprinted at recombination — across tens of millions of galaxies to chart $H(z)$ and hence $a(t)$ from $z\approx0$ out past $z=2$ ; the 2024–2025 DESI results even hint that dark energy may be evolving rather than a pure constant. The James Webb Space Telescope now routinely spectroscopically confirms galaxies at $z>13$ , sampling $a<0.07$ — light from when the universe was under 5% its present size. And the tension that dominates the field is a disagreement about $H_0$ itself: the local distance ladder (SH0ES, Cepheids and supernovae) gives $H_0\approx73$ km/s/Mpc, while the Planck CMB fit assuming standard $a(t)$ history gives $\approx67$ , a $5\sigma$ gap that may signal new physics in how $a$ evolved. Euclid and the Roman Space Telescope are being flown expressly to measure the expansion history and the growth of structure precisely enough to decide whether $w=-1$ exactly — that is, whether the late-time $a(t)$ is truly de Sitter or something subtler.

The physical picture

Draw a coordinate grid on a rubber sheet and paint dots on it for galaxies. Now stretch the sheet uniformly. Every dot keeps the same grid label — its comoving coordinates never change — yet the physical distance between any two dots grows by the same factor everywhere. That universal stretch factor is $a(t)$ . Nothing is "moving through" space in the ordinary sense: it is the metric, the rule that converts grid labels into measured distances, that evolves. This is why very distant galaxies can recede faster than light without violating relativity — the growth of $a$ is not local motion, and no signal overtakes another. Photons ride the same stretch. A wave crest laid down on the rubber sheet is pulled longer in proportion to $a$ , so light emitted when the universe was half its present size arrives with twice the wavelength — a redshift $z=1$ . The scale factor is therefore a cosmic odometer you can read off any spectrum: measure $z$ , and you know $a=1/(1+z)$ at the moment that light departed. The first figure makes the two halves of this picture one image — distances doubling and a photon's wavelength doubling by the same factor.

Figure

A comoving grid stretching from a = 0.5 to a = 1: the galaxies keep fixed grid labels A, B, C while every proper distance doubles, d_p = a·χ growing 500 Mpc → 1000 Mpc, and a photon in flight is stretched by the same factor, λ_obs = 2 λ_em, so its light arrives redshifted by z = 1

Which growth law $a(t)$ actually obeys is dictated by whatever dominates the energy budget, and that changes over cosmic history. Radiation, whose energy also redshifts away, dilutes fastest ( $\rho\propto a^{-4}$ ) and decelerates the expansion most strongly; pressureless matter dilutes as $a^{-3}$ ; a cosmological constant has constant energy density and drives runaway exponential growth. The story of the universe is the story of $a$ handing dominance from radiation to matter to dark energy, and the second figure shows the resulting curve — decelerating for its first several billion years, then bending upward into accelerated expansion.

Figure

The ΛCDM scale factor a(t) for a flat universe with Ω_m = 0.3, Ω_Λ = 0.7 and H₀ = 70 km/s/Mpc: growth decelerates while matter dominates, turns to acceleration near a ≈ 0.6 (z ≈ 0.67, t ≈ 7.6 Gyr), passes a = 1 today at t₀ ≈ 13.8 Gyr, and runs away exponentially into the dark-energy future

The core equations

The starting point is the split between the two kinds of distance. A comoving distance $\chi$ is a fixed label carried by every galaxy at rest in the cosmic rest frame; the physical, ruler-measured proper distance is that label multiplied by the scale factor at the moment of measurement.

d_p(t)=a(t)\,\chi \qquad (1)

where: $d_p(t)$ — proper (physical) distance at cosmic time $t$ ; $a(t)$ — dimensionless scale factor; $\chi$ — comoving distance, constant for a comoving pair. Reading: fix $\chi$ , let $a$ grow, and every proper distance grows by the same factor — the entire content of "uniform expansion." Equation (1) is one component of the full spacetime geometry. The FLRW metric writes the interval in terms of a fixed comoving spatial part scaled by $a(t)^2$ , with a single curvature constant $k$ fixing whether space is open, flat, or closed.

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

where: $\mathrm{d}s$ — spacetime interval; $c$ — speed of light; $k$ — curvature constant ( $-1,0,+1$ , or units of inverse length squared); $r$ — comoving radial coordinate; $\mathrm{d}\Omega^2=\mathrm{d}\theta^2+\sin^2\theta\,\mathrm{d}\phi^2$ . Reading: all the time dependence of the geometry is funnelled into the single scalar $a(t)$ multiplying an unchanging comoving grid. The instantaneous fractional growth rate of the scale factor is the Hubble parameter — not a constant in general, but a function of time whose present value is the Hubble constant $H_0$ .

H(t)\equiv\frac{\dot a}{a},\qquad H_0=H(t_0) \qquad (3)

where: $H(t)$ — Hubble parameter, the fractional expansion rate; $\dot a$ — time derivative of $a$ ; $H_0$ — Hubble constant, present value ( $\approx 70$ km/s/Mpc); $t_0$ — present cosmic time. Reading: $H$ is a rate with units of inverse time, and $1/H_0\approx 14$ Gyr sets the rough age and size scale of the observable universe. Because a photon's wavelength is stretched in step with $a$ , the redshift of a spectral line reports the scale factor at the epoch of emission. With the standard convention $a(t_0)\equiv 1$ , this is the single most useful equation in observational cosmology.

1+z=\frac{a(t_0)}{a(t_{\rm em})}=\frac{1}{a},\qquad (a(t_0)\equiv 1) \qquad (4)

where: $z$ — cosmological redshift; $t_{\rm em}$ — time of emission; $a$ — scale factor at emission. Reading: a measured $z$ is a measured $a$ ; there is no separate "distance" or "time" needed to convert — redshift is the scale factor. The dynamics come from Einstein's equations for a homogeneous perfect fluid. The Friedmann equation relates the square of the expansion rate to the total energy density, the spatial curvature, and the cosmological constant.

\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: $G$ — Newton's constant; $\rho$ — total density (in mass units); $k$ — curvature constant; $\Lambda$ — cosmological constant. Reading: given the energy content, this first-order equation determines $a(t)$ up to the normalisation and an integration constant. Differentiating (5) with the fluid equation gives the acceleration (second Friedmann) equation, which tells you whether the expansion is speeding up or slowing down.

\frac{\ddot a}{a}=-\frac{4\pi G}{3}\left(\rho+\frac{3p}{c^2}\right)+\frac{\Lambda c^2}{3} \qquad (6)

where: $\ddot a$ — second time derivative of $a$ ; $p$ — pressure of the cosmic fluid. Reading: ordinary matter and radiation ( $\rho+3p/c^2>0$ ) decelerate the expansion; only a component with sufficiently negative pressure ( $p<-\rho c^2/3$ ) can give $\ddot a>0$ . Each ingredient dilutes at its own rate as space expands, which is why the universe passes through distinct eras. Radiation loses an extra factor of $a$ to redshifting; matter simply thins out; the cosmological constant does not dilute at all.

\rho(a)=\rho_{r,0}\,a^{-4}+\rho_{m,0}\,a^{-3}+\rho_{\Lambda} \qquad (7)

where: $\rho_{r,0}$ — present radiation density; $\rho_{m,0}$ — present matter density; $\rho_\Lambda$ — constant dark-energy density. Reading: at small $a$ the $a^{-4}$ term wins (radiation era); at intermediate $a$ , $a^{-3}$ (matter era); today the constant term dominates (dark-energy era). Finally, the recession law is nothing but the time derivative of (1) — the Hubble–Lemaître law falls out with $H$ identified as $\dot a/a$ .

v_{\rm rec}(t)=\dot a\,\chi=\frac{\dot a}{a}\,d_p=H(t)\,d_p \qquad (8)

where: $v_{\rm rec}$ — proper recession velocity; $d_p$ — proper distance; $H(t)$ — Hubble parameter. Reading: recession velocity is strictly proportional to proper distance at any instant, and there is no upper bound on $v_{\rm rec}$ — sufficiently distant galaxies recede faster than light.

Derivation ACosmological redshift as the stretching of a null geodesic

The goal is to show that the redshift of light is not a Doppler shift of local motion but a direct consequence of the metric growing while a photon is in flight, giving the clean result $1+z=1/a$ . We track two successive wave crests along a radial null path through the FLRW metric, use the fact that both crests traverse the same comoving distance, and let the emission and observation intervals shrink to a single period. The payoff is that redshift becomes a ruler for the scale factor itself.

Step 1. For light, the interval vanishes, $\mathrm{d}s^2=0$ , so along a radial path the metric (2) gives $c\,\mathrm{d}t=a(t)\,\mathrm{d}\chi$ . Why: this is the photon's equation of motion in comoving coordinates — coordinate time and comoving distance are tied together by the current value of $a$ .

Step 2. Integrate from emission at $t_{\rm em}$ to observation at $t_0$ : the comoving distance the photon covers is $\chi=\int_{t_{\rm em}}^{t_0} c\,\mathrm{d}t/a(t)$ . A crest emitted one period later, at $t_{\rm em}+\delta t_{\rm em}$ and observed at $t_0+\delta t_0$ , must cover the same comoving distance $\chi$ , because the source and observer keep fixed comoving coordinates. Why: comoving distance between two comoving observers does not change — that is the definition of comoving — so the two integrals are equal.

Step 3. Setting the two integrals equal and cancelling the common interior, the leftover edge terms give $\delta t_0/a(t_0)=\delta t_{\rm em}/a(t_{\rm em})$ , since $a$ is essentially constant over one wave period. Why: subtracting two integrals that share an overlapping range leaves only the thin slivers at each end, which is where the period information lives.

Step 4. A period is $\delta t=1/\nu=\lambda/c$ , so $\lambda_0/\lambda_{\rm em}=\delta t_0/\delta t_{\rm em}=a(t_0)/a(t_{\rm em})=1+z$ . With $a(t_0)\equiv 1$ this is $1+z=1/a$ . Why: the observed wavelength is longer than the emitted one by exactly the factor the universe has grown — redshift measures the scale factor at emission, no cosmological model required.

Go deeper · Derivation B · 1 minPower-law growth for a single-component flat universe

Derivation BPower-law growth for a single-component flat universe

The goal is to solve the Friedmann equation for a spatially flat universe filled with one fluid obeying $p=w\rho c^2$ , and to show that the scale factor grows as a pure power of time, $a\propto t^{2/[3(1+w)]}$ , which specialises to the radiation ( $t^{1/2}$ ) and matter ( $t^{2/3}$ ) laws. We first get the dilution of density from the fluid equation, feed it into the flat Friedmann equation, separate variables, and integrate from the initial singularity. This single formula organises the whole early expansion history.

Step 1. The fluid (continuity) equation $\dot\rho+3(\dot a/a)(\rho+p/c^2)=0$ with $p=w\rho c^2$ integrates to $\rho\propto a^{-3(1+w)}$ . Why: this is where the different dilution rates come from — radiation ( $w=1/3$ ) gives $a^{-4}$ , matter ( $w=0$ ) gives $a^{-3}$ , and a cosmological constant ( $w=-1$ ) gives a constant.

Step 2. Drop curvature and $\Lambda$ ( $k=0$ , single component), so the Friedmann equation (5) becomes $(\dot a/a)^2=\tfrac{8\pi G}{3}\rho\propto a^{-3(1+w)}$ . Why: with only one ingredient and flat space, the expansion rate is fixed entirely by how that ingredient dilutes.

Step 3. Take the square root and collect powers: $\dot a\propto a^{-\frac{1}{2}(1+3w)}$ , which separates as $a^{\frac{1}{2}(1+3w)}\,\mathrm{d}a\propto \mathrm{d}t$ . Why: the equation is now a simple separable ODE in $a$ and $t$ , integrable by inspection for any $w\neq -1$ .

Step 4. Integrate with $a=0$ at $t=0$ , fixing the Big Bang as the origin: $a^{\frac{3}{2}(1+w)}\propto t$ , hence $a(t)\propto t^{\frac{2}{3(1+w)}}$ . Why: the integration constant is set by the singularity, and the exponent reproduces $t^{1/2}$ for radiation and $t^{2/3}$ for matter — the Einstein–de Sitter law — exactly as observed to bracket the early universe.

Go deeper · Derivation C · 1 minDe Sitter exponential expansion from a cosmological constant

Derivation CDe Sitter exponential expansion from a cosmological constant

The goal is to show that when a constant vacuum energy dominates, the Friedmann equation forces a constant Hubble rate and therefore exponential growth of the scale factor, $a(t)=a_i\,e^{H_\Lambda(t-t_i)}$ . This is the late-time attractor of our universe and the engine of inflation in the very early universe, so the same one-line solution describes both the far future and the first instants. We set matter and radiation to zero, read off the constant right-hand side, and integrate the logarithmic derivative.

Step 1. With $k=0$ and $\rho_m,\rho_r\to 0$ , only the constant term survives in (5): $(\dot a/a)^2=\Lambda c^2/3\equiv H_\Lambda^2$ . Why: because a cosmological constant does not dilute as space expands, the right-hand side is genuinely time-independent, unlike every other component.

Step 2. Take the positive root for an expanding universe: $\dot a/a=H_\Lambda=\text{const}$ , i.e. $\mathrm{d}(\ln a)/\mathrm{d}t=H_\Lambda$ . Why: a constant $H$ is the defining signature of de Sitter space — the expansion rate never changes.

Step 3. Integrate to $\ln a=H_\Lambda t+\text{const}$ , so $a(t)=a_i\,e^{H_\Lambda(t-t_i)}$ with $H_\Lambda=c\sqrt{\Lambda/3}$ . Why: exponential growth means every proper distance doubles in a fixed time $\ln 2/H_\Lambda$ , no matter how large it already is — the runaway that drives cosmic acceleration.

Limiting cases

For a spatially flat universe dominated by a single component with equation of state $p=w\rho c^2$ , the Friedmann equation integrates to simple, memorable behaviours.

Case	Result	Meaning
Radiation ( $w=1/3$ )	$a(t)\propto t^{1/2}$	Density dilutes as $\rho\propto a^{-4}$ ; dominates the early universe, $z\gtrsim 3400$ ; $H=1/(2t)$ .
Matter ( $w=0$ )	$a(t)\propto t^{2/3}$	Pressureless dust, $\rho\propto a^{-3}$ ; the Einstein–de Sitter growth law; $H=2/(3t)$ .
Cosmological constant ( $w=-1$ )	$a(t)\propto e^{Ht}$	Constant density $\rho_\Lambda$ ; de Sitter exponential expansion, dominant today and into the far future; $H$ constant.
Empty / curvature ( $\rho\to 0$ , $k=-1$ )	$a(t)\propto t$	Coasting Milne universe; a useful zero-gravity reference, $\ddot a=0$ .
General $w\neq -1$ (flat)	$a(t)\propto t^{2/[3(1+w)]}$	Recovers radiation and matter; a power-law expansion from a singularity requires $w>-1$ .

Common trapThe expansion stretches everything, including atoms, rulers, and the solar system

It does not. The scale factor governs distances between objects that are freely floating in the cosmic rest frame — galaxies and clusters held apart only by the expansion itself. Any system bound by a force stronger than the cosmic pull simply does not expand: atoms are held by electromagnetism, planets and stars by their own gravity, galaxies and even galaxy clusters by their collective gravitational binding. A hydrogen atom is the same size today as it was at $z=1000$ ; the metre stick in your lab is not quietly growing. Bound systems reach an equilibrium size set by their own physics and stay there while the empty space between distant, unbound systems balloons. So "space expands" is a statement about the large-scale mean metric between comoving galaxies, not a universal solvent that inflates rulers — which is precisely why redshift is measured against fixed atomic wavelengths in the first place.

Worked examples

1 · Reading the scale factor and wavelength stretch from redshift (fully worked)

A galaxy is observed at redshift $z=1$ and a quasar at $z=6$ . For each, find the scale factor at emission, the factor by which the universe has grown since, and the factor by which the light's wavelength has been stretched.

Step 1. Use $a=1/(1+z)$ directly (equation 4). For the galaxy, $a=1/(1+1)=\mathbf{0.5}$ ; for the quasar, $a=1/(1+6)=1/7\approx\mathbf{0.143}$ . These are the sizes of the universe, relative to today, at the moments the two beams departed.

Step 2. The universe has grown by the reciprocal since emission: $1/a=1+z$ . Since the galaxy's light left, all cosmic distances have doubled ( $1+z=2$ ); since the quasar's light left, they have grown sevenfold ( $1+z=7$ ). A structure that was 100 Mpc across in comoving-to-proper terms then spans twice or seven times as much now.

Step 3. Wavelength stretches by exactly the same factor, $\lambda_{\rm obs}=(1+z)\lambda_{\rm em}$ . A hydrogen Lyman-α line emitted at $121.6$ nm reaches us at $2\times121.6=\mathbf{243\ nm}$ from the galaxy and at $7\times121.6=\mathbf{851\ nm}$ from the quasar — carried from the ultraviolet all the way into the near-infrared. The quasar's light left when the universe was one-seventh its present size, and every wavelength in its spectrum arrives seven times longer.

2 · Superluminal recession and the Hubble distance

With $H_0=70$ km/s/Mpc, find the recession velocity of a galaxy at proper distance $d_p=5000$ Mpc, and the "Hubble distance" at which the recession velocity formally equals $c$ .

Step 1. Apply the Hubble law (equation 8) at the present epoch: $v_{\rm rec}=H_0\,d_p=(70\ \text{km/s/Mpc})(5000\ \text{Mpc})=\mathbf{3.5\times10^{5}\ km/s}$ . That is $1.17\,c$ — the galaxy recedes faster than light.

Step 2. This does not violate relativity: the recession is the growth of $a(t)$ , not motion through space, and no local signal exceeds $c$ . Setting $v_{\rm rec}=c$ defines the Hubble distance, $d_H=c/H_0=(3.0\times10^5\ \text{km/s})/(70\ \text{km/s/Mpc})=\mathbf{4290\ Mpc}\approx4.3$ Gpc.

Step 3. Galaxies beyond $d_H$ recede superluminally, yet we still see many of them — light emitted in the past crossed comoving distances while $H$ was different, and the Hubble distance is not the same as the observable horizon. A galaxy at 5000 Mpc recedes at 1.17 c, and the crossover to superluminal recession sits at the Hubble distance ≈ 4.3 Gpc.

3 · Age of a matter-dominated universe and a dark-energy doubling time

In the matter era, $a\propto t^{2/3}$ . Use it to estimate the age of an Einstein–de Sitter universe with $H_0=70$ km/s/Mpc, and compare with the doubling time of a dark-energy-dominated future.

Step 1. For $a\propto t^{2/3}$ , $H=\dot a/a=2/(3t)$ , so $t_0=2/(3H_0)$ . With $1/H_0=13.97$ Gyr, $t_0=\tfrac23(13.97)=\mathbf{9.3\ Gyr}$ — notably younger than the real 13.8 Gyr, because pure matter decelerates too hard; the missing time is supplied by the late dark-energy phase.

Step 2. In the dark-energy future $a\propto e^{H_\Lambda t}$ with $H_\Lambda\approx H_0\sqrt{\Omega_\Lambda}=70\sqrt{0.7}\approx58.6$ km/s/Mpc, so $1/H_\Lambda\approx16.7$ Gyr. Distances double every $t_{\rm double}=\ln 2/H_\Lambda\approx0.69\times16.7=\mathbf{11.6\ Gyr}$ .

Step 3. The two numbers frame the cosmic story: a decelerating matter era that would have aged the universe to only ~9 Gyr, handing over to an accelerating era that now doubles every separation roughly every 12 Gyr. Matter alone gives t₀ ≈ 9.3 Gyr; the dark-energy future doubles all distances every ≈ 11.6 Gyr.

Physics of the portrait

The scale factor is where a symmetry principle, a metric, and a century of measurement converge: one dimensionless function that a mathematician can constrain from homogeneity alone, and an astronomer can read straight off a spectrum.

Origins · 1 minscale factor and how this was discovered

scale factor

the one dynamical variable a(t) of the FLRW universe · relativistic cosmology and the expanding universe

The scale factor is the mathematical residue of a physical idea that took a decade of argument to accept: that the universe is not static. In 1917 Einstein applied general relativity to the cosmos and, expecting an eternal, unchanging universe, inserted the cosmological constant $\Lambda$ by hand to hold it still; the same year Willem de Sitter found a curious empty solution that expanded. In 1922 the Russian mathematician Alexander Friedmann took the FLRW symmetry seriously and derived the equations governing a time-dependent $a(t)$ , showing that expanding and contracting universes are the generic solutions — Einstein at first dismissed the work as an error, then retracted the objection. Independently, in 1927, the Belgian priest and physicist Georges Lemaître rederived the expanding solution and, crucially, connected it to data: he predicted a linear velocity–distance relation, $v=H d$ , and estimated $H$ from the galaxy redshifts Vesto Slipher had been collecting since 1912. Two years later Edwin Hubble published the measured relation, and the expansion became observational fact. Howard Robertson and Arthur Walker put the geometry on rigorous footing in 1935–36, proving that $\mathrm{d}s^2=-c^2\mathrm{d}t^2+a(t)^2\,\mathrm{d}\Sigma^2$ is the unique homogeneous, isotropic metric — so $a(t)$ is not a modelling choice but a theorem. The story's last turn came in 1998, when two teams tracking Type Ia supernovae found $\ddot a>0$ : the expansion is accelerating, $\Lambda$ (or something like it) is real after all, and Einstein's "greatest blunder" returned as dark energy. The 2011 Nobel Prize in Physics honoured that discovery, closing the loop Friedmann opened.

History

1917

— Einstein adds Λ for a static universe; de Sitter finds an empty expanding solution

1922

— Friedmann derives the equations governing a time-varying scale factor

1927

— Lemaître links expansion to galaxy redshifts and predicts v = H d

1929

— Hubble publishes the linear redshift–distance relation

1998

— Type Ia supernovae reveal ä > 0: the expansion is accelerating

Going further

Texts. Ryden, Introduction to Cosmology, 2nd ed., ch. 3–5 (scale factor, redshift, single-component solutions) · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., ch. 29 · Dodelson, Modern Cosmology, 2nd ed., ch. 2 (FLRW and the expansion history) · Mo, van den Bosch & White, Galaxy Formation and Evolution, ch. 3 · Binney & Tremaine, Galactic Dynamics, 2nd ed., cosmology appendix.

Provenance. The metric, redshift relation, and single-fluid power laws follow directly from the FLRW geometry and the Friedmann/fluid equations as presented in Ryden (ch. 3–5), Dodelson (ch. 2), and Carroll & Ostlie (ch. 29). A verifier should confirm: (1) the single-fluid law $a\propto t^{2/[3(1+w)]}$ reduces to $t^{1/2}$ and $t^{2/3}$ ; (2) the redshift derivation $1+z=1/a$ from the null geodesic; (3) matter–radiation equality $z_{\rm eq}\approx3400$ (Planck 2018); (4) the ΛCDM curve values ( $\Omega_m=0.3$ , $\Omega_\Lambda=0.7$ , $H_0=70$ : acceleration onset $a\approx0.60$ / $z\approx0.67$ / $t\approx7.6$ Gyr, $t_0\approx13.8$ Gyr); (5) the worked numbers $v_{\rm rec}=1.17c$ at 5000 Mpc, $d_H\approx4290$ Mpc, EdS age $9.3$ Gyr, doubling time $11.6$ Gyr. No non-standard numbers introduced.

Prerequisites. The Cosmological Principle · Comoving and Proper Coordinates · The FLRW Metric · Homogeneity and Isotropy in Cosmology. Leads to. The Friedmann Equations · Cosmological Redshift · The Hubble Constant · Critical Density and Density Parameters.

Ask the tutor: Ask me to walk through how the three growth laws stitch together into the single ΛCDM curve, or to work any redshift into its scale factor, look-back time, and recession velocity.

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

Both descriptions can be made to agree on local observables, but the "expanding space" picture is the one that generalises. In the FLRW metric the galaxies sit still at fixed comoving coordinates and the distance function $d_p=a(t)\chi$ grows because $a$ grows — nothing travels through space. This is what lets recession velocities exceed $c$ without paradox and what makes redshift a stretching of wavelength rather than a classical Doppler shift. You can reinterpret the nearby, low-redshift regime as ordinary recession velocities, and it matches, but that picture quietly fails at large $z$ where $v>c$ . The honest statement is that $a(t)$ describes the evolving geometry, and "space expands" is shorthand for that.

If everything expands, why don't atoms, planets, and rulers grow too?

Because the expansion only governs systems held apart by nothing but the cosmic flow. Atoms are bound by electromagnetism, stars and planets by gravity, galaxies and clusters by their own collective gravity — each finds an equilibrium size set by its internal forces and stays there. The scale factor multiplies the comoving separation of freely floating, unbound objects; a bound system is not comoving, so $d_p=a\chi$ simply does not apply to it. That is why a hydrogen line has the same rest wavelength now as in the early universe, and why we can measure redshift at all: the ruler (atomic physics) is fixed while the space between distant galaxies stretches.

What does the absolute value of a(t) actually mean?

On its own, nothing — only ratios of the scale factor are physical. The metric contains $a(t)$ multiplied by the comoving coordinate, so rescaling $a\to\lambda a$ and $\chi\to\chi/\lambda$ leaves every proper distance, and every observable, unchanged. What you can measure is $a(t_2)/a(t_1)$ , equivalently a redshift. To remove the ambiguity everyone adopts the convention $a(t_0)=1$ today, which makes $a$ the fraction of its present size the universe had at any epoch and turns equation (4) into the tidy $1+z=1/a$ . For curved universes one sometimes instead normalises so the comoving curvature radius is unity — always state which convention you are using.

Does the universe expand into something, and is there a centre of the expansion?

No on both counts, under the Cosmological Principle. The FLRW metric describes a universe that is homogeneous — the same everywhere — so there is no edge to expand into and no special central point from which everything flees. Every comoving observer sees the identical Hubble law: all other galaxies recede from them, with velocity proportional to distance, exactly as they would from us. The rubber-sheet or rising-loaf analogies capture this: on the surface, every raisin sees every other raisin moving away, and no raisin is the centre. The expansion is an intrinsic growth of internal distances, not an explosion into a pre-existing void.

How is a(t) actually measured, given we only see one instant?

By using redshift as a look-back coordinate and standard rulers and candles to tie redshifts to times. Every galaxy's spectrum gives $a=1/(1+z)$ at the epoch its light left, so a survey spanning $0<z<7$ is literally a survey across $1>a>0.125$ . To convert those into times and rates you need the expansion history $H(z)$ , which comes from standard candles (Type Ia supernovae fix luminosity distance versus $z$ ), standard rulers (baryon acoustic oscillations fix a fixed comoving length versus $z$ ), and the CMB (which anchors the geometry at $z\approx1100$ ). Fitting all of these to the Friedmann equation reconstructs $a(t)$ and its density parameters — the curve drawn in the second figure.

Why do radiation, matter, and dark energy dilute at different rates?

Because expansion does different things to each. Matter is just conserved particles spread through a growing volume $\propto a^3$ , so $\rho_m\propto a^{-3}$ . Radiation thins the same way and each photon's energy redshifts by an extra factor of $a$ (its wavelength stretches), giving $\rho_r\propto a^{-4}$ — which is why radiation loses the race and dominates only at very early times. A cosmological constant is a property of the vacuum itself: doubling the volume simply gives twice as much vacuum with the same energy density, so $\rho_\Lambda$ is constant and eventually wins. These three scalings, plugged into the Friedmann equation, are what carve cosmic history into radiation, matter, and dark-energy eras.