Particle Horizon

The edge of what we can know

Every observer in an expanding universe is surrounded by a sphere — beyond which no light has ever had time to reach them. The radius of that sphere is the particle horizon. Inside it, every galaxy has at least had the opportunity to send us a signal. Outside it, every galaxy is forever causally disconnected from our entire history: no photon, no neutrino, no gravitational wave has ever crossed the boundary.

The horizon is observer-relative. Our particle horizon sits about 46.5 billion light-years out in any direction (proper distance today, ~14.4 Gpc comoving). An astronomer in the Andromeda galaxy has their own 46.5 Gly sphere, slightly offset. The universe itself may be infinite; what is finite is the slice each observer can causally probe.

Definition and derivation

In an FLRW universe, the proper distance light has travelled from emission at t_em to reception at t₀ is

  d_light = a(t₀) · ∫_{t_em}^{t₀} c · dt' / a(t')

The particle horizon at cosmic time t₀ is the limit as t_em → 0 (the Big Bang):

  d_p(t₀) = a(t₀) · ∫₀^{t₀} c · dt' / a(t')
         = c · ∫₀^{t₀} dt' / a(t')              (comoving form, a(t₀) = 1)
         = c · ∫₀^∞ dz / H(z)                   (redshift form)

The integral spans every moment of cosmic history, weighted by 1/a(t'). Early times — when a(t) was small — contribute the most. The horizon is finite because the radiation era kept a(t) tiny only briefly: a ∝ t^(1/2) during radiation, so the integrand 1/a ~ 1/t^(1/2), and ∫₀^t dt'/t'^(1/2) = 2√t is finite. No divergence at t = 0.

In flat ΛCDM with Planck 2018 best-fit parameters (H₀ = 67.4 km/s/Mpc, Ω_m = 0.315, ΩΛ = 0.685, Ω_r ≈ 9.2 × 10⁻⁵):

  d_p(t₀) ≈ 14 140 Mpc = 14.14 Gpc  (comoving)
         ≈ 46.1 Gly  (comoving · 3.262 Gly/Gpc)
         ≈ 46.1 Gly  (proper today; a(t₀) = 1)

Numerical values across cosmic time

The horizon has been growing throughout cosmic history. The value at any epoch is set by integrating from the Big Bang up to that time:

Cosmic time t (Gyr)	Scale factor a	Redshift z	Particle horizon d_p (Gpc comoving)	Proper d_p today (Gly)
3.8 × 10⁻⁴ (CMB)	9.1 × 10⁻⁴	1100	0.28	0.91 (~ 42 Mpc proper at emission)
0.001	1.4 × 10⁻³	721	0.39	1.27
0.5	0.18	4.4	9.8	32.0
1.0	0.26	2.85	11.4	37.2
5.92 (z = 1)	0.5	1.0	13.0	42.4
10.0	0.79	0.26	13.8	45.0
13.787 (today)	1.0	0	14.14	46.1
50	~8.7	−0.88	18.0	~59 (proper at that epoch)
∞ (asymptote)	∞	−1	~19	∞

The horizon was 0.28 Gpc comoving at recombination (z = 1100). Galaxies on opposite sides of our CMB sky lie within this horizon today — but they did not at recombination. That is the horizon problem.

Galaxies inside the horizon receding faster than light

Hubble's law v = H₀ · d says that at proper distance d_H = c/H₀ ≈ 4.45 Gpc ≈ 14 Gly, the recession velocity reaches c. Beyond this Hubble distance, galaxies recede from us faster than light. This is not a violation of special relativity: SR forbids superluminal local motion, not the stretching of intergalactic space. The Hubble distance is well inside the particle horizon, which sits about 3× further out.

How can photons cross the gap from a galaxy receding faster than c? Because as the photon moves toward us, it traverses regions of progressively lower Hubble velocity. The local recession at the photon's current position is what matters, not the recession at the galaxy. Galaxies up to about d_p = 46 Gly today have managed to send light that reached us; the very furthest CMB photons crossed regions that had recession velocities up to 60c at the time of emission, but their light cone propagated through the expanding metric and is finally arriving.

Worked example: today's particle horizon

For flat ΛCDM with Planck parameters, split the integral by epoch:

  d_p = c · ∫₀^∞ dz / H(z)
      = (c/H₀) · ∫₀^∞ dz / √(Ω_r(1+z)⁴ + Ω_m(1+z)³ + ΩΛ)

  (c/H₀) = 4448 Mpc

Decompose by z-interval (numerical):
   z ∈ [0,    1]:     ∫ ≈ 0.7459 → d_p contribution = 3318 Mpc
   z ∈ [1,    3]:     ∫ ≈ 0.7027 → d_p contribution = 3125 Mpc
   z ∈ [3,    10]:    ∫ ≈ 0.7218 → d_p contribution = 3211 Mpc
   z ∈ [10,   100]:   ∫ ≈ 0.7290 → d_p contribution = 3243 Mpc
   z ∈ [100,  1000]:  ∫ ≈ 0.2347 → d_p contribution = 1044 Mpc
   z ∈ [1000, ∞]:     ∫ ≈ 0.0454 → d_p contribution =  202 Mpc

  d_p(total) ≈ 14 144 Mpc = 14.14 Gpc comoving
           = 46.1 Gly  (× 3.262 Gly/Gpc)

Notice how much of the horizon comes from low and intermediate z. Half the radius (~7 Gpc) accumulates between z = 0 and z ≈ 1.5; only ~1.4 Gpc comes from the radiation era (z > 1000). The mathematical reason the integral converges is that ∫ dz/(1+z)² is finite as z → ∞ — radiation-era H(z) grows as (1+z)², and 1/H falls fast enough.

Particle horizon vs cosmic event horizon

The two "horizons" of cosmology look symmetric in name but are very different. The particle horizon is the past-light-cone limit: the largest comoving distance from which a signal has had time to reach us. The cosmic event horizon is the future-light-cone limit: the largest comoving distance from which a signal emitted now will ever reach us in the infinite future.

Horizon	Defined as	Today's value (flat ΛCDM)	Limit as t → ∞
Hubble distance d_H	c/H₀	4.45 Gpc · 14.5 Gly	asymptotes to c/√(Λ/3) ≈ 5.3 Gpc
Particle horizon d_p	c · ∫₀^t₀ dt'/a(t')	14.4 Gpc · 46.1 Gly	asymptotes to ~19 Gpc
Event horizon d_e	c · ∫_{t₀}^∞ dt'/a(t')	4.9 Gpc · 16 Gly proper today	shrinks (in comoving) to 0
CMB last-scattering distance	D_C(z=1100)	14.0 Gpc · 45.7 Gly	—

In a Λ-free matter-dominated universe both d_p and d_e diverge; in a Λ-dominated universe both are bounded. We live in a transitional epoch where d_p is still growing (~20 Mpc per Gyr) and d_e is finite — galaxies currently inside d_e but outside d_p will become observable over the next 50 Gyr or so; galaxies outside d_e will never become observable from any signal emitted from now on.

Inflation and the horizon problem

One of the foundational puzzles for cosmology before 1980 was the homogeneity of the CMB. The temperature is uniform across the sky to one part in 10⁵, despite the fact that, in a matter+radiation universe extrapolated back to the Big Bang, opposite points of the CMB sky never lay within each other's particle horizon at recombination. They could not possibly have exchanged photons or thermalized.

Quantitatively: at recombination (t = 380 000 yr), the particle horizon was ~0.28 Gpc comoving. Two CMB pixels on opposite sides of our sky lie 28 Gpc apart in comoving coordinates today — 100 times the recombination-era horizon. They had no causal contact at any time before recombination.

Alan Guth's 1981 inflation hypothesis solves this. During inflation (10⁻³⁶ to 10⁻³² s), the universe expanded by a factor of 10²⁶ or more. A pre-inflation patch the size of an atomic nucleus — which was in causal contact — got stretched to cover the entire observable universe and far beyond. The CMB pixels we see today all originated from a single thermalised microscopic patch. Inflation rewrites the horizon: the effective particle horizon for late-time observers extends back to before inflation, far enough to include everything we now see.

What lies inside vs just outside the horizon

Catalog of contents, ordered outward from us:

• z = 0–0.1 (d ≤ 1.3 Gly): the Local Group, Virgo cluster, modern surveys.
• z = 1–3 (d ≈ 11–21 Gly): cosmic noon, peak star formation, the bulk of luminous galaxies.
• z = 6–14 (d ≈ 28–45 Gly): cosmic dawn, JWST frontier, end of reionization.
• z = 1100 (d ≈ 45.7 Gly): the CMB last-scattering surface — the farthest light we can see directly. Sits at 99% of the particle horizon radius.
• z ≈ 6 × 10⁹ (d ≈ 46.0 Gly): the cosmic neutrino background — never directly detected, but in principle observable with future ν-cosmology probes.
• z ≈ 10²⁶ (d ≈ 46.1 Gly): inflationary gravitational waves — potentially detectable through B-mode CMB polarisation; never directly observed yet.
• Just outside d_p: galaxies whose light has not yet had time to reach us, but whose proper position today differs from ours only by their peculiar motion since recombination.

Numerical pitfalls

• "Observable universe = 46 Gly" is comoving, not light-travel. Light has been in transit for 13.8 Gyr but the source has been receding the whole time. The 46 Gly is where the source sits now (proper distance today = comoving distance for a(t₀) = 1), not how far light has actually travelled.
• Hubble distance ≠ particle horizon. d_H = c/H₀ ≈ 4.45 Gpc is where v_recession = c. Particle horizon is ~3× larger. Confusing the two is the most common cosmology-101 error.
• Recession ≠ proper motion. Galaxies inside the particle horizon but beyond the Hubble distance recede from us at v > c. This is consistent with GR — only local v < c is required by special relativity. The recession is metric expansion, not motion.
• Particle horizon depends on Ω_r at high z. The radiation contribution is small (~200 Mpc out of 14 100), but ignoring Ω_r gives noticeably different early-universe behaviour. For precision work always include radiation.
• Particle horizon ≠ age × c. The light has been in flight for 13.8 Gyr, but the radius is 46 Gly, not 13.8 Gly. The factor of ~3 comes from cosmic expansion during the photon's flight.
• Curvature. If Ω_k ≠ 0, the transverse comoving distance differs from the radial one — the particle horizon becomes ellipsoidal. Flatness (Ω_k = 0 to 0.5% precision) makes this academic in our universe.

A brief history of the horizon concept

The notion of a cosmological horizon was first carefully defined by Wolfgang Rindler in 1956, in "Visual Horizons in World Models." Rindler distinguished the particle horizon (past light cone) from the event horizon (future light cone) and showed that whether either is finite depends on the equation of state and the lifetime of the universe. In a matter-only Einstein-de Sitter universe, the particle horizon at age t₀ is d_p = 3 c t₀, three times the naïve light-travel distance — an early demonstration that expansion lets us see far further than a static-universe estimate.

With the 1965 discovery of the CMB by Penzias and Wilson, the horizon problem became sharp: the CMB uniformity could not be explained by causal physics in standard hot Big Bang cosmology. Robert Dicke and others highlighted this in the 1970s. Alan Guth's 1981 inflation paper showed how exponential expansion could solve it. The 1992 COBE detection of CMB anisotropies — the 1-in-10⁵ fluctuations seeded by inflation — and the precise WMAP and Planck measurements of the early 2000s converted "inflation" from speculation into the standard early-universe paradigm. Today the particle horizon is a Planck-mission-measured quantity: 14.140 ± 0.040 Gpc, accurate to 0.3%.