Trace light rays in conformal coordinates—see particle horizons and the observable universe expand
In general relativity, the causal structure of spacetime is determined by light cones—the paths light rays trace through space and time. In an expanding universe, these light cones define what regions can communicate, what we can observe, and what events can influence each other. The conformal diagram representation makes this causal structure especially clear.
The visualization shows a conformal spacetime diagram where the horizontal axis is comoving distance χ (distance in coordinates that expand with the universe) and the vertical axis is conformal time τ = ∫ dt/a(t). In these coordinates, light rays always travel at 45° angles (±1 slope), making causal structure manifest. The observer's past light cone is the region bounded by null rays extending backward from their worldline.
The past light cone of an observer at (τ_obs, χ=0) consists of all events that could have sent light signals to reach the observer. In comoving coordinates, this is defined by χ(z) = ∫₀z dz'/H(z'), where H(z) = H₀√[Ωₘ(1+z)³ + ΩΛ] is the Hubble parameter. The light cone expands as you look further back in time—more distant regions become causally connected to the observer's past.
The particle horizon is the comoving distance to the furthest particles we can currently observe—the edge of the observable universe. It's calculated as χ_p = ∫₀∞ dz/H(z), the integral extending to the Big Bang. For the standard ΛCDM cosmology (Ωₘ=0.3, ΩΛ=0.7), this gives about 46 billion light-years today, despite the universe being only 13.8 billion years old—space itself has expanded.
The event horizon is the comoving distance to the furthest events that we will ever be able to observe, even waiting forever into the future. In a universe with dark energy (ΩΛ > 0), accelerating expansion creates an event horizon: galaxies beyond this distance are receding faster than light can catch up. For our universe, the event horizon is about 16 billion light-years—we will never see events happening beyond this boundary.
Conformal time τ is defined by dτ = dt/a(t), integrating cosmic time weighted by the inverse scale factor. This coordinate choice "untwists" the light cones so they become straight 45° lines. Physically, τ measures the distance light can travel in comoving coordinates. The relationship between τ and redshift z follows from τ(z) = ∫_z∞ dz'/[(1+z')H(z')].
Comoving distance χ removes the expansion of space, giving distances in coordinates that expand with the Hubble flow. A galaxy at comoving distance χ stays at the same χ coordinate as the universe expands (assuming no peculiar velocity). The physical distance is d = a(t)χ. Light rays in comoving coordinates follow χ = ±(τ - τ_emit), corresponding to the 45° null geodesics in conformal diagrams.
Current Universe (Ωₘ=0.3, ΩΛ=0.7): Notice the past light cone widens as you go back in time—we see a larger comoving volume the further back we look. The particle horizon extends to about 46 Gpc (14,000 Mpc) in comoving distance. The CMB last scattering surface at z≈1100 appears as a nearly horizontal line—we observe it on a thin shell.
Matter-Only Universe (Ωₘ=1.0, ΩΛ=0.0): Set dark energy to zero. The past light cone has a different shape—deceleration changes the causal structure. There is no event horizon in this universe: given enough time, light from any event will eventually reach any observer. The particle horizon is smaller because the early universe expanded more slowly.
Pure Dark Energy (Ωₘ=0.0, ΩΛ=1.0): de Sitter space with exponential expansion. The event horizon is much closer—accelerating expansion quickly makes distant regions unreachable. The past light cone narrows dramatically. If the universe were always dominated by Λ, we could only ever observe a finite comoving volume, even waiting forever.
Observer at High Redshift: Move the observer redshift slider to z=2. The past light cone now shows what a galaxy at z=2 (about 10 billion years ago) could observe. Their particle horizon is smaller—they can see less of the universe because there's been less time for light to travel. Their "now" is a different hypersurface of simultaneity.
CMB Last Scattering: The CMB surface at z≈1100 marks recombination, when the universe became transparent to photons. This surface defines the edge of the visible universe in electromagnetic radiation—everything before is opaque. The angular size of fluctuations on this surface depends on the sound horizon at recombination, about 150 Mpc comoving.
Future Light Cone: Although not shown explicitly, imagine extending light cones forward. For Ωₘ=0.3, ΩΛ=0.7, events happening beyond χ≈5 Gpc today will never be observable, even in the infinite future—they lie outside our event horizon. Galaxies there see us redshift to infinity and fade away, and we see them do the same. Causal contact is permanently lost.
Light cones and horizons define the fundamental limits of observation and causality in cosmology:
Conformal diagrams are powerful tools for understanding spacetime structure. They originated with Roger Penrose's work on black holes and gravitational collapse in the 1960s. By compressing infinity to a finite region while preserving causal structure (light cones remain at 45°), these diagrams make the global geometry of spacetime manifest. In cosmology, they clarify how horizons form, how the observable universe evolves, and how causal structure constrains what observers at different epochs can see. The past light cone is not just a mathematical abstraction—it defines the totality of events that shaped your existence, the boundary of causal influence etched into the fabric of spacetime itself.