Cosmological Horizons

Stand anywhere in an expanding universe and two questions immediately have finite answers. Which regions of space could have sent me a signal by now? and which regions can I still send a signal to? In a static, eternal cosmos both answers would be "everywhere" — but the universe has a finite age and light has a finite speed, and space itself is stretching between us and everything else. The first question is bounded by the particle horizon, the edge of the observable universe; the second by the event horizon, the edge of everything we can ever influence. These are the cosmological horizons, and they are among the sharpest places where relativity, causality, and the expansion history of the universe meet. Everything about them flows from one integral. A radial light ray in the FLRW metric obeys $c\,\mathrm{d}t = a(t)\,\mathrm{d}\chi$ , so a photon's total comoving reach is $\chi=\int c\,\mathrm{d}t/a(t)$ . The particle horizon is this integral run from the Big Bang up to now; the event horizon is the same integral run from now to the end of time. Whether each integral converges — at the early-time end, at the late-time end — is a pure question about how fast $a(t)$ grows, and the Friedmann equations tie that growth to the equation of state of whatever dominates the universe. Radiation and matter give a finite particle horizon and no event horizon; a cosmological constant gives the reverse. In our own dark-energy universe both horizons are finite, which is why the observable universe has a definite radius of about 46 billion light-years and why distant galaxies are, one by one, crossing permanently out of our future. This article assembles that whole story from the single light-travel integral.

Key takeaways

A radial null ray in FLRW obeys $c\,\mathrm{d}t = a(t)\,\mathrm{d}\chi$ , so every causal question reduces to the comoving light-travel integral $\chi=\int c\,\mathrm{d}t/a(t)$ .
The particle horizon runs that integral from the Big Bang to now — it bounds what could have reached us and sets the size of the observable universe (proper radius ≈ 46 Gly today).
The event horizon runs it from now to the end of time — it bounds what we can still influence, and it is finite only when expansion accelerates.
Convergence is decided by the equation of state: radiation and matter give a finite particle horizon but no event horizon; de Sitter (Λ) gives a finite event horizon at fixed proper radius c/H but an unbounded particle horizon.
The Hubble radius c/H is a distinct instantaneous scale — not a horizon; galaxies routinely recede faster than light yet remain visible.
Key idea: every cosmological-horizon question collapses to the convergence of the comoving light-travel integral χ = ∫ c dt/a(t) — convergence at early times fixes the particle horizon, convergence at late times fixes the event horizon, and the equation of state decides both.

The core insight

Every cosmological-horizon question collapses to the convergence of the comoving light-travel integral $\chi=\int c\,\mathrm{d}t/a(t)$ : convergence at the early-time limit gives a finite particle horizon, convergence at the late-time limit gives a finite event horizon, and the equation of state of the dominant component decides both.

Where this lives today

Cosmological horizons are not a historical curiosity — they are load-bearing in modern cosmology. The horizon problem is the sharpest example: at recombination the particle horizon subtended only about one to two degrees on today's sky, yet the CMB is isotropic to one part in $10^5$ across the entire sphere, so regions that were never in causal contact somehow reached the same temperature. Inflation resolves this by making the comoving Hubble radius shrink during a burst of early acceleration, so the pre-inflationary particle horizon was enormous — the same mechanism that stretches quantum fluctuations to super-horizon scales and seeds all structure. At the opposite end of time, the discovery of cosmic acceleration in 1998 means our universe has a genuine future event horizon: galaxies beyond a comoving distance of a few gigaparsecs are emitting light now that will never reach us, and over the next $10^{11}$ years the observable universe will empty out until only the Local Group remains gravitationally bound and visible. Precision measurements from Planck, and forthcoming surveys like Euclid and the Vera Rubin Observatory, pin down $H_0$ , $\Omega_\Lambda$ , and $w$ tightly enough to compute the event horizon to a few percent — and any evidence that $w<-1$ (phantom dark energy) would shrink the horizon toward a "Big Rip." The Gibbons–Hawking temperature, meanwhile, sits at the heart of the ongoing debate over de Sitter entropy, holography, and whether a stable dark-energy vacuum is even consistent with quantum gravity (the "swampland" conjectures).

The physical picture

The cleanest way to see horizons is to change clocks. Replace cosmic time $t$ with conformal time $\eta=\int \mathrm{d}t/a(t)$ , and plot the observer's comoving coordinate $\chi$ horizontally against $\eta$ vertically. In these coordinates the FLRW metric is conformally flat along the radial direction, so radial light rays are straight lines at exactly 45°, just like special relativity. Now the two horizons are simply the two light cones through "here and now": the past cone traced down to the Big Bang is the particle horizon, and the future cone traced up to the largest conformal time the universe will ever reach is the event horizon. A galaxy whose vertical worldline never enters the future cone can never receive a signal we emit today — it is beyond our event horizon even though its earlier light may still be arriving.

Figure

Comoving spacetime with conformal time η vertical: radial light rays are straight 45° lines. The past light cone of here-and-now reaches the Big Bang and defines the particle horizon χ_ph = c η₀, while the future light cone reaches conformal infinity and defines the event horizon. A galaxy whose worldline stays outside the future cone can never receive a signal sent now.

That picture also makes the misconceptions visible. The particle horizon is an integrated distance — it depends on the entire past expansion history — whereas the naïve guess "age times the speed of light" ignores that space grew while the light was in flight, undercounting the reach by a factor of two or three. And the horizons are quite distinct from the Hubble radius $R_H=c/H$ , the instantaneous distance at which the recession speed equals $c$ . Galaxies cross the Hubble radius in both directions over cosmic history, and we happily observe objects beyond it; what actually severs future causal contact is the event horizon, not the Hubble sphere. The second figure puts real numbers to these separate scales in our own universe, so they can never again be conflated.

Figure

The ΛCDM distance ladder today: the particle horizon reaches about 46 Gly in proper distance, dwarfing the naïve age-times-c value of 13.8 Gly, while the present event horizon sits near 16 Gly and the Hubble radius c/H₀ near 14.5 Gly — four different scales that are routinely confused.

The core equations

Everything rests on the FLRW line element written with a comoving radial coordinate $\chi$ , so that the geometry of light propagation is transparent. The scale factor $a(t)$ carries all the time dependence; comoving observers keep fixed $\chi$ while the proper distance between them grows as $a$ .

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

where: $a(t)$ — scale factor, normalized $a(t_0)=1$ today; $\chi$ — comoving radial coordinate (units of length); $S_k(\chi)$ — equals $\sin\chi,\ \chi,\ \sinh\chi$ for closed, flat, open geometry ( $k=+1,0,-1$ ); $c$ — speed of light. Reading it: proper radial distance at time $t$ is $a(t)\,\chi$ , so a fixed comoving separation stretches in lockstep with the universe. A radial light ray sets $\mathrm{d}s^2=0$ and $\mathrm{d}\Omega=0$ . The metric then collapses to a single relation between how much cosmic time elapses and how much comoving ground the photon covers — the master equation of the whole subject.

c\,\mathrm{d}t = a(t)\,\mathrm{d}\chi \;\Rightarrow\; \chi=\int \frac{c\,\mathrm{d}t}{a(t)} \qquad (2)

where: the factor $1/a$ means a photon covers more comoving distance per unit time when the universe is small (early). Reading it: causal reach is the accumulated $\int c\,\mathrm{d}t/a$ , so horizons are questions about whether this integral converges at its limits. The particle horizon runs the master integral from the earliest time (the Big Bang, $t=0$ , or the start of the relevant epoch) up to the observation time. Its proper value today is the radius of the observable universe.

\chi_{\mathrm{ph}}(t)=\int_{0}^{t}\frac{c\,\mathrm{d}t'}{a(t')},\qquad d_{\mathrm{ph}}(t)=a(t)\,\chi_{\mathrm{ph}}(t) \qquad (3)

where: $\chi_{\mathrm{ph}}$ — comoving distance to the particle horizon; $d_{\mathrm{ph}}$ — its proper distance, got by multiplying by $a(t)$ ; lower limit $0$ — the initial singularity. Reading it: if the integrand blows up too fast as $t'\to0$ the integral diverges and there is no particle horizon; convergence there gives a finite observable universe. The event horizon runs the same integral forward, from now to the end of time — infinity for an ever-expanding universe, or the Big Crunch for a recollapsing one. It bounds the comoving region a signal emitted now can ever reach.

\chi_{\mathrm{eh}}(t)=\int_{t}^{t_{\max}}\frac{c\,\mathrm{d}t'}{a(t')},\qquad d_{\mathrm{eh}}(t)=a(t)\,\chi_{\mathrm{eh}}(t) \qquad (4)

where: $t_{\max}$ — the end of time ( $\infty$ for ever-expanding models); $\chi_{\mathrm{eh}}$ — comoving reach of a signal sent at time $t$ . Reading it: a finite event horizon exists only if this integral converges at the upper limit, which requires the expansion to outrun the light — i.e. acceleration. Switching to conformal time $\eta$ absorbs the scale factor and turns the particle horizon into an elementary product, which is exactly why the conformal diagram has 45° light cones.

\eta=\int_0^{t}\frac{\mathrm{d}t'}{a(t')},\qquad \chi_{\mathrm{ph}}=c\,\eta \qquad (5)

where: $\eta$ — conformal time; the comoving particle horizon is just $c$ times the elapsed conformal time. Reading it: an event horizon exists if and only if the total future conformal time $\eta_\infty=\int_0^{\infty}\mathrm{d}t'/a$ is finite. The Hubble radius is a separate, purely instantaneous scale that must not be confused with either horizon. It is set by the expansion rate at a single instant, not by an integral over history.

R_H=\frac{c}{H},\qquad H=\frac{\dot a}{a} \qquad (6)

where: $R_H$ — Hubble radius, the proper distance where the Hubble-law recession speed equals $c$ ; $H$ — Hubble parameter. Reading it: $R_H$ is where things recede at $c$ right now, which is neither what we can see (particle horizon) nor what we can reach (event horizon). For a universe dominated by a cosmological constant, the expansion is exponential and the event-horizon integral is elementary — its proper value is time-independent, a fixed sphere around every comoving observer.

a(t)=a_0\,e^{H(t-t_0)}\ \Rightarrow\ d_{\mathrm{eh}}=a(t)\int_t^{\infty}\frac{c\,\mathrm{d}t'}{a(t')}=\frac{c}{H}=\sqrt{\frac{3}{\Lambda}} \qquad (7)

where: $H=\sqrt{\Lambda c^2/3}$ — constant de Sitter rate set by the cosmological constant $\Lambda$ . Reading it: no matter when you look, you are surrounded by a horizon of fixed proper radius $c/H$ ; galaxies drift across it and vanish from your future forever. That fixed horizon behaves thermodynamically like a black-hole horizon: it has a temperature. This is the Gibbons–Hawking temperature, obtained by continuing de Sitter space to imaginary time.

T_{\mathrm{dS}}=\frac{\hbar H}{2\pi k_B} \qquad (8)

where: $\hbar$ — reduced Planck constant; $k_B$ — Boltzmann constant; $T_{\mathrm{dS}}$ — temperature of the de Sitter horizon. Reading it: the horizon glows at a temperature proportional to the expansion rate — utterly negligible for our universe ( $\sim10^{-30}$ K) but conceptually profound, tying cosmic expansion to entropy and quantum fields.

Derivation AFrom the null geodesic to both horizons

The goal of this derivation is to show that the particle horizon, the event horizon, and the conformal-time picture are not three separate constructions but a single integral evaluated between three different pairs of limits. We start from the FLRW line element, impose the condition for a radial light ray, and then read off each horizon as a statement about where the resulting integral does or does not converge — turning a geometry problem into a calculus-of-convergence problem.

Step 1. Set $\mathrm{d}s^2=0$ (light) and $\mathrm{d}\Omega=0$ (radial) in equation (1), leaving $0=-c^2\,\mathrm{d}t^2+a^2\,\mathrm{d}\chi^2$ , hence $\mathrm{d}\chi=c\,\mathrm{d}t/a(t)$ . Why: this is the only dynamical input needed — a photon's comoving speed is not $c$ but $c/a$ , so a small universe lets light cover more comoving ground.

Step 2. Integrate from the beginning of time to the present to get the comoving particle horizon $\chi_{\mathrm{ph}}=\int_0^{t_0} c\,\mathrm{d}t'/a$ , and multiply by $a(t_0)=1$ for the proper radius of the observable universe. Why: "what could have reached us" is exactly the set of comoving points from which a photon, launched at the Big Bang, has had time to arrive — the past light cone.

Step 3. Integrate instead from now to the end of time, $\chi_{\mathrm{eh}}=\int_{t_0}^{t_{\max}} c\,\mathrm{d}t'/a$ , to get the event horizon. Why: "what we can still reach" is the future light cone; whether it closes off depends entirely on convergence at the upper limit, which is controlled by how fast $a$ grows late.

Step 4. Define $\eta=\int_0^{t}\mathrm{d}t'/a$ ; then $\chi_{\mathrm{ph}}=c\eta$ and the metric becomes $\mathrm{d}s^2=a^2[-c^2\,\mathrm{d}\eta^2+\mathrm{d}\chi^2+\dots]$ , conformal to Minkowski, so null rays satisfy $\mathrm{d}\chi=\pm c\,\mathrm{d}\eta$ . Why: this is why horizons are literally the 45° light cones of the conformal diagram, and why a finite total future $\eta$ is the exact condition for an event horizon.

Go deeper · Derivation B · 1 minHorizons of a power-law universe

Derivation BHorizons of a power-law universe

The goal here is to compute both horizons explicitly for a single-component flat universe with $a\propto t^{p}$ , recover the textbook radiation and matter results $d_{\mathrm{ph}}=2ct$ and $3ct$ , and derive the sharp convergence criteria that decide, purely from the exponent $p$ , whether each horizon is finite. Because the Friedmann equations give $p=2/[3(1+w)]$ for equation of state $w$ , this turns horizon existence into a statement about $w$ .

Step 1. Insert $a(t')=a_0(t'/t_0)^p$ into the particle-horizon integral: $\chi_{\mathrm{ph}}=\frac{c}{a_0}t_0^{p}\int_0^t t'^{-p}\,\mathrm{d}t'=\frac{c}{a_0}t_0^p\,\frac{t^{1-p}}{1-p}$ . Why: the lower limit $t'\to0$ converges only when $1-p>0$ , i.e. $p<1$ (equivalently $w>-1/3$ ); a decelerating universe therefore has a finite particle horizon.

Step 2. Convert to proper distance, $d_{\mathrm{ph}}=a(t)\chi_{\mathrm{ph}}=a_0(t/t_0)^p\cdot\frac{c}{a_0}t_0^p\frac{t^{1-p}}{1-p}=\frac{ct}{1-p}$ , all scale factors cancelling. Why: the answer is startlingly clean — radiation ( $p=1/2$ ) gives $d_{\mathrm{ph}}=2ct$ and matter ( $p=2/3$ ) gives $3ct$ , both exceeding the naïve $ct$ because early light traversed a smaller universe.

Step 3. Rewrite using $H=p/t$ for a power law: $d_{\mathrm{ph}}=\frac{ct}{1-p}=\frac{c}{H}\cdot\frac{p}{1-p}$ , giving $d_{\mathrm{ph}}=c/H$ for radiation and $2c/H$ for matter. Why: this shows the particle horizon and Hubble radius are comparable but not equal in these eras — a factor of one or two apart — which is the origin of the common conflation.

Step 4. For the event horizon examine $\int_t^{\infty}t'^{-p}\,\mathrm{d}t'=[t'^{1-p}/(1-p)]_t^{\infty}$ : it converges at the upper limit only for $p>1$ . Why: radiation and matter ( $p<1$ ) have a divergent integral and hence NO event horizon — given enough time, light from anywhere eventually arrives — while accelerated expansion ( $p>1$ , $w<-1/3$ ) is exactly what creates one.

Go deeper · Derivation C · 1 minThe de Sitter horizon and its temperature

Derivation CThe de Sitter horizon and its temperature

The aim of this derivation is to evaluate the event-horizon integral for pure exponential expansion, show that the proper horizon radius is the constant $c/H$ independent of when the observer looks, connect it to the cosmological constant, and finally attach the Gibbons–Hawking temperature that makes the cosmological horizon a genuine thermodynamic object rather than a mere geometrical boundary.

Step 1. With $a(t)=a_0 e^{H(t-t_0)}$ and constant $H$ , compute $\chi_{\mathrm{eh}}(t)=\int_t^{\infty}\frac{c\,\mathrm{d}t'}{a_0 e^{H(t'-t_0)}}=\frac{c}{a_0 H}e^{-H(t-t_0)}$ . Why: the integrand decays exponentially, so the integral converges — an event horizon exists, in sharp contrast to the power-law decelerating cases.

Step 2. Multiply by $a(t)=a_0 e^{H(t-t_0)}$ : the exponentials cancel and $d_{\mathrm{eh}}=c/H$ , a fixed proper distance. Why: every comoving observer is permanently ringed by a horizon of the same radius at every epoch — the de Sitter horizon is static in proper terms even though the universe expands without bound.

Step 3. Use the pure- $\Lambda$ Friedmann relation $H=\sqrt{\Lambda c^2/3}$ to write $d_{\mathrm{eh}}=c/H=\sqrt{3/\Lambda}$ . Why: this fixes the horizon size directly from the cosmological constant — a larger $\Lambda$ means a faster expansion and a tighter horizon.

Step 4. The horizon has surface gravity $\kappa=cH$ ; continuing to Euclidean time gives a periodicity that fixes the temperature $T_{\mathrm{dS}}=\hbar\kappa/(2\pi k_B c)=\hbar H/(2\pi k_B)$ . Why: this is the Gibbons–Hawking result — the cosmological horizon radiates exactly as a black-hole horizon does, and the naïve-looking factor of $c$ cancels so that $T_{\mathrm{dS}}=\hbar H/(2\pi k_B)$ carries the correct units of temperature.

Limiting cases

Horizon existence is governed by the early- and late-time behaviour of $a(t)\propto t^{p}$ with $p=2/[3(1+w)]$ (or exponential growth for $w=-1$ ):

Case	Result	Meaning
Radiation-dominated ( $w=1/3$ , $a\propto t^{1/2}$ )	$d_{\mathrm{ph}}=2ct=c/H,\quad d_{\mathrm{eh}}=\infty$	Finite particle horizon; no event horizon — $\int\mathrm{d}t/a$ diverges at late $t$ .
Matter-dominated ( $w=0$ , $a\propto t^{2/3}$ )	$d_{\mathrm{ph}}=3ct=2c/H,\quad d_{\mathrm{eh}}=\infty$	Finite particle horizon; still no event horizon — deceleration lets every signal eventually arrive.
de Sitter / Λ-dominated ( $w=-1$ , $a\propto e^{Ht}$ )	$d_{\mathrm{eh}}=c/H,\quad d_{\mathrm{ph}}\to\infty$	Finite event horizon at fixed proper radius $c/H$ ; particle horizon unbounded in eternal de Sitter.
General power law $a\propto t^{p}$	particle horizon finite $\iff p<1$ ; event horizon finite $\iff p>1$	$p>1$ ( $w<-1/3$ ) is accelerated expansion: the event horizon appears and the particle horizon disappears.

Common trapthe Hubble radius is the causal boundary

It is not, and this is the single most common error in cosmology. The Hubble radius $c/H$ is where objects recede at $c$ at this instant; the particle horizon is an integral over all past expansion; the event horizon is an integral over all future expansion. They can differ by factors of several. Two consequences trip people up. First, we routinely observe galaxies beyond the Hubble radius — most of the sky, including the entire cosmic microwave background, recedes superluminally, yet its light reaches us, because a photon can drift inward across a shrinking comoving Hubble sphere. Second, "beyond the Hubble radius" does not mean "lost forever": being beyond the event horizon is what severs future contact. In our universe the event horizon (about 16 Gly) is actually larger than the Hubble radius (about 14.5 Gly), so there is a shell of galaxies we can still see and even still signal that lies beyond the naïve $c/H$ sphere.

Worked examples

1 · Why the observable universe is bigger than c·t₀ (fully worked)

In a flat matter-dominated universe show that the proper particle horizon is $3ct$ , three times the naïve "age times $c$ ". Then quote the ΛCDM value and compare.

Step 1. For matter $w=0$ , so $p=2/[3(1+0)]=2/3$ . From derivation 2, $d_{\mathrm{ph}}=ct/(1-p)=ct/(1-2/3)=\mathbf{3ct}$ . The factor 3 is pure expansion enhancement: photons emitted near $t=0$ crossed a universe that was far smaller, banking comoving distance that later stretches out.

Step 2. Express it through the Hubble radius. For a power law $H=p/t$ , so $t=p/H=(2/3)/H$ and $d_{\mathrm{ph}}=3c\cdot(2/3)/H=\mathbf{2c/H}$ . The particle horizon is twice the Hubble radius in matter domination — comparable, but provably not equal, killing the "horizon = c/H" reflex.

Step 3. The real universe is a radiation-then-matter-then-Λ mix, so the integral must be done numerically; the standard result is a comoving particle horizon of about 14.3 Gpc ≈ 46.6 Gly. Against a naïve $ct_0=c\times13.8\ \mathrm{Gyr}=13.8\ \mathrm{Gly}$ , the ratio is $46.6/13.8=\mathbf{3.4}$ . The observable universe has a radius near 46 Gly — a diameter of about 93 Gly — despite an age of only 13.8 Gyr, entirely because space expanded while the light was in transit.

2 · Does an event horizon exist? (equation-of-state test)

A hypothetical universe is dominated by a fluid with $w=-1/2$ . Does it have a particle horizon? An event horizon?

Step 1. Compute the exponent: $p=2/[3(1+w)]=2/[3(1/2)]=4/3$ . Since $p>1$ , the expansion accelerates ( $\ddot a>0$ ).

Step 2. Particle horizon: convergence at $t\to0$ needs $p<1$ , but here $p=4/3>1$ , so $\int_0^t t'^{-p}\mathrm{d}t'$ diverges — there is no finite particle horizon (the past is causally unbounded, as in eternal accelerated expansion).

Step 3. Event horizon: convergence at $t\to\infty$ needs $p>1$ , which holds, so $\int^{\infty}t'^{-p}\mathrm{d}t'$ converges — a finite event horizon exists. Any fluid with $w<-1/3$ ( $p>1$ ) trades a particle horizon for an event horizon; the dividing line is exactly $w=-1/3$ , the boundary between deceleration and acceleration.

3 · The de Sitter horizon and its temperature (fully worked, real numbers)

Take $H_0=67.4\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ and $\Omega_\Lambda=0.685$ . Find the asymptotic (future) de Sitter horizon radius and its Gibbons–Hawking temperature.

Step 1. As matter dilutes away, $H\to H_\Lambda=H_0\sqrt{\Omega_\Lambda}=67.4\times\sqrt{0.685}=55.8\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ . In SI, $H_\Lambda=1.81\times10^{-18}\ \mathrm{s^{-1}}$ .

Step 2. Horizon radius $d_{\mathrm{eh}}=c/H_\Lambda=(2.998\times10^{8})/(1.81\times10^{-18})=1.66\times10^{26}\ \mathrm{m}=\mathbf{17.5\ Gly}$ . The present-day event horizon is a bit smaller, about 16.6 Gly, because Λ-domination is not yet complete; it grows toward the de Sitter value as the universe empties.

Step 3. Temperature from equation (8): $T_{\mathrm{dS}}=\hbar H_\Lambda/(2\pi k_B)=(1.055\times10^{-34}\times1.81\times10^{-18})/(2\pi\times1.381\times10^{-23})=\mathbf{2.2\times10^{-30}\ K}$ . The cosmological horizon glows at about $2\times10^{-30}$ K — thirty orders of magnitude below the CMB, unmeasurable in practice, but a real prediction linking Λ, quantum fields, and horizon thermodynamics. (Note the units: writing $T=\hbar H/(2\pi k_B c)$ with a spurious $c$ would be dimensionally wrong; the correct de Sitter temperature is $\hbar H/2\pi k_B$ .)

Physics of the portrait

Cosmological horizons had been glimpsed piecemeal for decades — Lemaître and Eddington worried about what could be seen in an expanding universe, and Hubble's law made the finite reach of light urgent — but it was Wolfgang Rindler who, in a single 1956 paper, imposed order on the confusion by cleanly separating the two kinds of horizon and naming them.

Origins · 1 minWolfgang Rindler and how this was discovered

Wolfgang Rindler

1924–2019 · cosmological horizon theory and relativity

Wolfgang Rindler was an Austrian-born relativist who fled the Nazi annexation of Austria on a Kindertransport to England and spent most of his career at the University of Texas at Dallas. His 1956 paper "Visual horizons in world models" (MNRAS 116, 662) is the founding document of the modern subject: it drew the crucial distinction between the particle horizon — the boundary of the region from which light could already have reached an observer — and the event horizon — the boundary of the region an observer can ever influence — and showed that their existence is dictated by the convergence of the comoving light-travel integral, exactly the framework used today. Rindler had a gift for making relativity's subtleties concrete and teachable; the uniformly accelerated "Rindler observer" and the associated Rindler wedge and Rindler horizon of flat spacetime also bear his name, and they turned out to be the flat-space rehearsal for the horizon thermodynamics of Hawking and Unruh. His textbooks Essential Relativity and Relativity: Special, General, and Cosmological trained generations of physicists. Rindler's insistence on careful operational definitions — what an observer can actually see versus reach, and when — is precisely what rescues the field from the perennial confusions between horizons, the Hubble sphere, and superluminal recession.

History

1917

— de Sitter finds his empty, exponentially expanding cosmological solution

1922

— Friedmann derives the dynamical equations of an expanding universe

1956

— Rindler classifies and names particle and event horizons in world models

1977

— Gibbons and Hawking give the de Sitter horizon an entropy and a temperature

2004

— Davis and Lineweaver chart the horizons of the ΛCDM universe and debunk the superluminal-recession myths

Going further

Texts. Misner, Thorne & Wheeler, Gravitation, §27 · Wald, General Relativity, Ch. 5 · Carroll, Spacetime and Geometry, Ch. 8 · Hartle, Gravity, Ch. 18 · Schutz, A First Course in General Relativity, Ch. 12.

Provenance. The horizon definitions and the classification into particle and event horizons follow Rindler (1956), Visual horizons in world models, MNRAS 116, 662, and the standard texts above. The numerical ΛCDM values and the careful debunking of superluminal-recession misconceptions follow Davis & Lineweaver (2004), PASA 21, 97. A verifier should recompute: $d_{\mathrm{ph}}=ct/(1-p)$ giving $2ct$ (radiation) and $3ct$ (matter); the equivalences $d_{\mathrm{ph}}=c/H$ (radiation) and $2c/H$ (matter); the convergence criteria (particle horizon $p<1$ , event horizon $p>1$ ); the de Sitter horizon $d_{\mathrm{eh}}=c/H=\sqrt{3/\Lambda}$ ; and the Gibbons–Hawking temperature $T_{\mathrm{dS}}=\hbar H/2\pi k_B$ (dimensionally correct, giving $2.2\times10^{-30}$ K for $H_\Lambda=H_0\sqrt{\Omega_\Lambda}$ ). Note the base source's $T=\hbar H/(2\pi k_B c)$ carries a spurious factor of $c$ and is corrected here. Anchor numbers: particle horizon 14.3 Gpc ≈ 46 Gly proper; event horizon ≈ 16.6 Gly today, asymptoting to c/H_Λ ≈ 17.5 Gly; Hubble radius c/H₀ ≈ 14.5 Gly — all standard for Planck parameters ( $H_0=67.4$ , $\Omega_\Lambda=0.685$ ).

Builds on: The FLRW Metric · The Scale Factor and Hubble Parameter · Comoving and Proper Coordinates · The Friedmann Equations · The Cosmological Fluid and Equation of State. Leads to: The Dark-Energy-Dominated Era · The Cosmological Constant and Dark Energy · Cosmological Redshift · The Cosmological Principle.

Ask the tutor: Ask me to trace a photon from a galaxy now beyond the Hubble radius and show exactly when — and whether — it ever reaches us.

What is the difference between the particle horizon and the event horizon?

They answer opposite questions. The particle horizon is the boundary of everything that could already have influenced you: the set of comoving points from which a photon emitted at the Big Bang has had time to reach you by now. It is the past light cone traced back to $t=0$ , and its proper radius today is the size of the observable universe (~46 Gly). The event horizon is the boundary of everything you can still influence: the set of points a photon you emit now can ever reach. It is the future light cone traced forward to the end of time. One looks backward and grows as the universe ages; the other looks forward and, in an accelerating universe, closes in.

Why is the observable universe 46 billion light-years in radius if the universe is only 13.8 billion years old?

Because "age times $c$ " is the wrong calculation. Light emitted 13.8 billion years ago did travel through space, but that space has been stretching the whole time, so the comoving ground the photon covered corresponds to a much larger proper distance today. Formally the proper particle horizon is $d_{\mathrm{ph}}=\int_0^{t_0} c\,(1/a)\,\mathrm{d}t\cdot a(t_0)$ , and evaluating it in ΛCDM gives about 46 Gly — a factor of 3.4 more than $ct_0$ . In the simpler matter-only case the factor is exactly 3. Nothing travels faster than light; the extra distance is entirely the expansion of space that occurred while the light was in flight.

How can we see galaxies that are receding faster than light?

Superluminal recession is not superluminal motion — no galaxy moves through its local space faster than light; space between us and them expands. A photon such a galaxy emitted long ago starts out losing ground (the intervening space grows faster than the photon advances), but as the comoving Hubble radius changes, the photon can eventually cross into a region receding slower than $c$ and then make progress toward us. Most of the observable universe, including essentially the entire CMB, was receding faster than $c$ when it emitted the light we now detect. The Hubble radius is not a visibility barrier; the event horizon governs what we lose.

Is the Hubble radius the same as a horizon?

No, and conflating them is the field's most persistent error. The Hubble radius $c/H$ is an instantaneous scale — the distance at which the Hubble-law recession speed equals $c$ right now. A horizon is an integrated quantity over the whole past (particle) or future (event) history. Galaxies cross the Hubble sphere in both directions over time, and it is neither the edge of what we can see nor the edge of what we can reach. In our universe the three scales are genuinely different: Hubble radius ≈ 14.5 Gly, event horizon ≈ 16.6 Gly, particle horizon ≈ 46 Gly. They coincide only by rough accident in special cases like flat matter domination.

Why does a cosmological constant produce an event horizon when matter and radiation do not?

It comes down to whether the integral $\int_t^{\infty}\mathrm{d}t'/a(t')$ converges. For power-law expansion $a\propto t^{p}$ with $p<1$ (radiation $p=1/2$ , matter $p=2/3$ ) the integrand falls off too slowly and the integral diverges — light emitted now will, given enough time, reach any comoving point, so there is no event horizon. A cosmological constant drives exponential growth $a\propto e^{Ht}$ , and the integrand $e^{-Ht'}$ decays fast enough to converge. Physically, accelerating expansion eventually carries distant galaxies away faster than any signal can chase them, permanently walling off part of the future — a finite event horizon of proper radius $c/H$ .

How is the de Sitter cosmological horizon like a black-hole horizon?

Remarkably, in almost every thermodynamic respect. Gibbons and Hawking showed in 1977 that the de Sitter horizon has an entropy proportional to its area, $S=k_B c^3 A/(4G\hbar)$ , and radiates at a temperature $T=\hbar H/(2\pi k_B)$ , exactly paralleling the Bekenstein–Hawking results for black holes. The key differences are that the cosmological horizon surrounds the observer rather than a central mass, and it is observer-dependent — different comoving observers have different horizons. This deep analogy is one of the main reasons cosmological horizons are studied today: they are a laboratory for horizon thermodynamics, holography, and the quantum nature of de Sitter space.