The Expanding Universe and Hubble's Law

Cosmology is the attempt to treat the universe as a single physical system whose history and structure follow from the same laws that govern a laboratory. The attempt only becomes tractable because the universe, viewed on scales larger than a few hundred megaparsecs, reduces to something simple: it looks the same everywhere and in every direction. That simplicity, elevated to a working assumption, reduces the entire spatial geometry and its time evolution to a single function, the scale factor $a(t)$. This lesson sets up the cosmological principle, defines comoving and proper distance in terms of the scale factor, derives Hubble's law and the cosmological redshift as direct consequences, recovers the expansion dynamics from a Newtonian energy argument, and closes with Olbers' paradox, whose resolution was the first hint that the universe is neither static nor infinitely old.

The cosmological principle

The foundational assumption of modern cosmology is the cosmological principle: on sufficiently large scales the universe is homogeneous (the same at every place) and isotropic (the same in every direction). Homogeneity is invariance under translation; isotropy about every point is the stronger statement, and isotropy holding everywhere implies homogeneity. The two are logically distinct — a universe with a uniform magnetic field is homogeneous but not isotropic — but the observed universe appears to satisfy both.¹

The evidence is empirical, not a priori. Redshift surveys such as the Sloan Digital Sky Survey show that galaxies cluster into filaments and voids up to scales of order $100\text{Mpc}$, but that averaging over larger volumes washes the structure out: the mean density in one $300\text{Mpc}$ box matches another to a fraction of a percent. Isotropy is even sharper. The cosmic microwave background, the relic radiation from the hot early universe, has the same temperature in every direction to one part in $10^5$ once the dipole from our own motion is removed. A homogeneous and isotropic distribution of matter with an isotropic radiation bath is what the cosmological principle asserts.

The scale factor and comoving coordinates

A homogeneous, isotropic universe can expand or contract, but it can do so only in one way: uniformly, preserving the shape of every configuration while rescaling all distances by a common time-dependent factor. Any position-dependent rescaling would introduce a preferred location and violate homogeneity. This single degree of freedom is the scale factor $a(t)$, a dimensionless function of cosmic time normalized so that its present value is $a(t_0)=a_0=1$.

To separate the expansion from the fixed pattern of galaxies, introduce comoving coordinates. Assign each galaxy a coordinate $\vec{x}$ that does not change as the universe expands — the galaxies are at rest in the comoving frame, apart from small peculiar velocities. The physical, or proper, distance between two comoving points separated by comoving distance $x$ is

The comoving separation $x$ is a fixed label; all of the time dependence lives in $a(t)$. A useful picture is a rubber sheet or a rising loaf of raisin bread: the raisins keep their coordinates in the dough, but the dough stretches and the physical distance between any two raisins grows in proportion to the expansion.

Hubble's law

Differentiate the proper distance with respect to cosmic time. Because the comoving separation $x$ is constant,

The rate of change of proper distance — the recession velocity of one comoving point relative to another — is proportional to the proper distance between them. The proportionality constant is the Hubble parameter

so that $v=H(t)d$. Evaluated at the present epoch this is Hubble's law,

with $H_0$ the Hubble constant. Note that $H_0$ is a constant in space, not in time: the subscript denotes its value now, and $H(t)$ generally changes over cosmic history. The linear relation is exact and unavoidable for uniform expansion; any other distance dependence would single out an origin. Every observer riding along with the expansion sees exactly the same law with the same $H_0$, so the recession is not evidence that we sit at a center. It is not motion through space but the growth of space between comoving observers.²

Hubble measured the relation in 1929 from the redshifts and estimated distances of nearby galaxies, finding the recession velocity proportional to distance. His slope was some seven times too steep because his distance calibration was in error, but the linearity was correct. The modern value, from the Cepheid–supernova distance ladder and from the CMB, is

the range reflecting a persistent tension between early-universe and late-universe determinations that later lessons return to. The units make the meaning transparent: a galaxy $1\text{Mpc}$ away recedes at about $70\text{km}{\text{s}}^{-1}$, one at $100\text{Mpc}$ at about $7000\text{km}{\text{s}}^{-1}$.

Cosmological redshift

A photon travelling from a distant galaxy to us moves through space that is stretching, and its wavelength stretches with it. Consider a light wave emitted at time $t_e$ with wavelength ${\lambda }_e$ and observed now at $t_0$ with wavelength ${\lambda }_0$. Successive wave crests are emitted a comoving distance apart; as they travel, that comoving separation is fixed, but the proper wavelength scales with $a(t)$. Carrying the argument through the null geodesic of the photon (done rigorously from the metric in the next lesson) gives the exact result

Defining the redshift $z$ through the fractional wavelength shift,

using $a_0=1$. This is the single most important relation in observational cosmology: a measured redshift is a direct readout of the scale factor at the time the light was emitted. A galaxy observed at $z=1$ emitted its light when the universe was half its present size; the CMB at $z\approx 1100$ shows us the universe when it was about a thousandth of its current scale.

It is essential to distinguish this cosmological redshift from a Doppler shift. The galaxy is essentially at rest in comoving coordinates; it is not moving through space away from us in the ordinary kinematic sense. The wavelength grew because the space the photon traversed grew. For small distances the two pictures agree numerically — expanding $1+z=a_0/a(t_e)$ to first order in lookback time recovers $z\approx H_{0}d/c$, the Doppler form $v=cz$ — but at large $z$ the naive Doppler interpretation fails, and redshifts $z>1$ do not imply superluminal motion.

A Newtonian derivation of expansion dynamics

The full dynamics of $a(t)$ follow from general relativity, but a Newtonian argument reproduces the same equation for pressureless matter and makes the physics transparent. This works because of a theorem due to Newton and Birkhoff: in a spherically symmetric mass distribution the gravitational field at radius $r$ depends only on the mass interior to $r$, and the exterior shells exert no net force. In a homogeneous universe we may therefore isolate a sphere, analyze it with Newtonian gravity, and appeal to homogeneity to argue the result holds everywhere.

Consider a sphere of comoving radius $x$ centered on an arbitrary point, with proper radius $R(t)=a(t)x$ and enclosing a fixed mass $M=\frac{4}{3}\pi R^3\rho$. A test galaxy of mass $m$ on the surface feels only the interior mass. Its energy — kinetic plus gravitational potential — is conserved:

Substitute $R=ax$ and $\dot{R}=\dot{a}x$, and divide by $\frac{1}{2}m{x}^2$:

The constant $-2E/(m{x}^2)$ must, by homogeneity, be independent of which sphere we chose; write it as $-k{c}^2$ with $k$ a constant. Then

which is precisely the Friedmann equation, derived rigorously from general relativity in a later lesson. The Newtonian energy constant $E$ reappears there as the spatial curvature: a bound sphere ($E<0$, $k>0$) corresponds to a closed universe that recollapses, an unbound one ($E>0$, $k<0$) to an open universe that expands forever, and the marginal case ($E=0$, $k=0$) to a spatially flat universe poised between them. The Newtonian picture correctly captures the matter-dominated dynamics; it misses the pressure contribution to gravity and the behaviour of radiation and vacuum energy, which require the relativistic treatment.³

Olbers' paradox

Why is the sky dark at night? In a universe that is infinite, static, and eternally old, every line of sight would eventually terminate on the surface of a star, and the whole sky would blaze with the surface brightness of a stellar photosphere. The argument is quantitative. In a homogeneous static universe of number density $n$ of stars each of radius $R_{\star }$, the number of stars in a shell of radius $r$ and thickness $\mathbf{d}r$ is $4\pi r^{2}n\mathbf{d}r$. Each subtends a solid angle $\propto R_{\star }^2/r^2$, so the light received from the shell, or equivalently the sky area it covers, is independent of $r$: the falloff of flux as $1/r^2$ exactly cancels the growth of shell volume as $r^2$. Integrating over all shells to infinity gives a divergent total — an infinitely bright sky. Even accounting for nearer stars blocking farther ones, the whole sky should shine at stellar surface brightness.

The dark night sky therefore rules out the infinite–static–eternal universe. The resolution has two parts, both supplied by an expanding universe of finite age. First, the universe has a finite age $t_0$, so light can only have reached us from within a finite horizon of radius $\sim c{t}_0$; more distant stars have not had time to send us their light, truncating the integral. Second, the expansion redshifts distant starlight, so photons from remote sources arrive with reduced energy and at a reduced rate, dimming the far contributions further. The finite age is the dominant effect: it is not that there is too little matter, but that we see only a finite, and only a finitely old, portion of it. Olbers' paradox is thus the earliest and simplest cosmological observation, and its resolution already contains the two ingredients — finite age and expansion — that the rest of this module develops quantitatively.⁴

Summary

The cosmological principle — large-scale homogeneity and isotropy — reduces the expanding universe to a single function of time, the scale factor $a(t)$, normalized to $a_0=1$ today. Proper distances scale as $d=ax$ with comoving separation $x$ fixed, and differentiating gives Hubble's law $v=Hd$ with $H=\dot{a}/a$, a linear relation that holds for every comoving observer and implies no center. The wavelength of light stretches with the same factor, so the cosmological redshift measures the scale factor at emission, $1+z=1/a(t_e)$, and is not a Doppler shift through space. A Newtonian energy argument on a comoving sphere reproduces the Friedmann equation $(\dot{a}/a{)}^2=8\pi G\rho /3-k{c}^2/a^2$, with the energy constant reappearing as spatial curvature. Olbers' paradox — the dark night sky — already excludes an infinite, static, eternal universe and points to the finite age and expansion this module now makes precise. The next lesson replaces the Newtonian sphere with the Robertson–Walker metric, deriving the geometry, the distances, and the redshift rigorously.