Cosmology · Horizons and the observable Universe
The Hubble radius is not generally a causal horizon
Trace one exact inward photon in an ideal matter-only background and show how it reaches the observer after beginning outside the Hubble radius.
Preparation: ratios, powers of ten and ordinary arithmetic. Physical quantities and notation are introduced as they are needed.
Back to the chapterAfter reading, you can
- Define the Hubble radius c/H for a positive expansion rate.
- Separate a photon’s local velocity from the changing proper distance to a distant observer.
- Calculate a ray crossing the Hubble radius without treating it as a reflecting or opaque boundary.
Start with the definition of the Hubble radius
For an expanding FLRW background with H > 0, define the Hubble radius R_H = c/H. On a cosmic-time slice, a comoving separation D = R_H has recession derivative HD = c. This identifies a useful scale associated with the current expansion rate.
The definition uses the instantaneous value of H. A causal horizon requires a light-path calculation over an available past or future history. These constructions can coincide in special models, but the definition c/H does not make them generally identical.
For a radial inward photon, write its proper slice distance from the origin as D(t) = a(t)r(t), where dr/dt = −c/a. Differentiating gives dD/dt = HD − c. The two terms come from background expansion and the photon’s inward motion relative to the local comoving frame.
The photon’s local inward speed remains c. The distant same-slice derivative dD/dt can initially be positive if HD > c. That does not mean the photon locally travels away from its inward direction or exceeds the local light speed.
Follow a ray through one ideal history
Take a spatially flat matter-only general-relativistic background with scale factor a = u^(2/3), where u = t/T > 0 and T is a positive time unit. Use a length-valued radial comoving coordinate and write x = r/(cT). No radiation, vacuum component or absorption is included.
Choose the inward null ray x(u) = 3(2 − u^(1/3)) over 1 ≤ u ≤ 8. At u = 1 it starts at x = 3; at u = 8 it reaches x = 0. Direct differentiation gives dx/du = −u^(−2/3) = −1/a, as required for this inward photon.
Its proper slice distance is D/(cT) = a x = 6u^(2/3) − 3u. In this background H = 2/(3t), so R_H/(cT) = 3u/2. Both distances vary while the photon propagates.
At u = 1 the photon is at D = 3 cT, outside R_H = 1.5 cT. Nevertheless the declared exact ray reaches the origin at u = 8. This example directly disproves treating the Hubble radius as a universal permanent barrier to inward light.
A distance maximum is not a local bounce
The proper-distance slope is dD/dt divided by c = 4u^(−1/3) − 3. It is +1 at u = 1, zero at u = 64/27, negative thereafter and −1 at arrival. The positive early slope describes an increasing separation on successive cosmic-time slices.
At u = 64/27 the scale factor is 16/9, x = 2 and D = R_H = (32/9)cT. The ray’s proper separation is momentarily stationary even though the photon still moves locally inward at c.
At u = 27/8, the proper separation has fallen to 3.375 cT while the Hubble radius has grown to 5.0625 cT. At u = 8 the separation vanishes and the Hubble radius is 12 cT. Comparing only the emission-time radius would miss this changing geometry.
No reflection, collision or halt occurs at the distance maximum. It is the instant when the two terms in dD/dt = HD − c cancel. The local null path remains continuous through it.
| u | a | x | D/(cT) | R_H/(cT) | HD/c | (dD/dt)/c | Local photon velocity/c |
|---|---|---|---|---|---|---|---|
| 1 | 1 | 3 | 3 | 1.5 | 2 | +1 | −1 |
| 64/27 | 16/9 | 2 | 32/9 | 32/9 | 1 | 0 | −1 |
| 27/8 | 9/4 | 1.5 | 3.375 | 5.0625 | 2/3 | −1/3 | −1 |
| 8 | 4 | 0 | 0 | 12 | 0 | −1 | −1 |
The same model gives different characteristic radii
If this ideal matter-only history is formally continued back to t → 0, its proper particle-horizon radius is 3ct. Its Hubble radius is instead 3ct/2. The different values follow from a past integral and an instantaneous rate respectively.
That past extension is an ideal mathematical assumption, not verification of a classical description arbitrarily close to an origin. The ray over 1 ≤ u ≤ 8 does not require claiming that the early limiting state has been observed.
For other scale-factor histories, determine causal reach from the relevant null integral and its convergence. The Hubble radius may grow, shrink or behave differently relative to a given photon path. The dust example is a counterexample to a universal barrier claim, rather than a rule for every future.
Some horizon notions can agree in a specially specified spacetime. Such agreement should be demonstrated from that model and the definitions. It should not be assumed merely because several quantities have units of length.
State which separation and clock define a speed
A local inertial observer measures a passing light ray at c. A cosmological recession derivative compares distant events using a chosen cosmic-time slicing. A derivative of proper separation for a propagating photon includes both the expansion term and local propagation.
These different comparisons can give different numerical values without contradiction. Substituting a distant recession derivative into a special-relativistic formula for two local velocities would change the meaning of the calculation.
The figure and table are exact teaching results for a declared matter-only background, not measured photon trajectories or estimates of today’s cosmological radii. T fixes a convenient unit and does not supply a measured age.
For an actual source, ask when its light was emitted and integrate the null path using a justified expansion history. A single current number c/H cannot replace that calculation.
Worked reasoning
Find where the ray’s proper separation stops increasing
At what u does the supplied photon have zero proper-distance slope?
- Set 4u^(−1/3) − 3 = 0, giving u^(1/3) = 4/3.
- Cube to obtain u = 64/27. Then a = 16/9 and x = 2.
- Thus D/(cT) = 32/9 and R_H/(cT) = (3/2)(64/27) = 32/9. The photon is locally inward at c throughout.
At u = 64/27, where D = R_H = (32/9)cT. This is a maximum of slice distance, not a local bounce.
Try the reasoning
The photon begins at D = 3 cT while R_H = 1.5 cT. Does it fail to reach the observer?
Show the answer and reasoning
No; the exact supplied ray arrives at u = 8.
The background and Hubble radius evolve while the photon propagates.
When dD/dt = 0 at the Hubble-radius crossing, has the photon stopped according to a local comoving observer?
Show the answer and reasoning
No.
Its local inward velocity remains −c; the two terms HD and −c cancel only in the distant slice-distance derivative.
In the formal matter-only past extension, are the particle horizon and Hubble radius equal?
Show the answer and reasoning
No. They are 3ct and 3ct/2 respectively.
The particle horizon uses the past light-path integral, while c/H uses the instantaneous rate.
What to carry forward
- The Hubble radius c/H is generally different from a causal horizon.
- An inward photon can initially increase its distant proper separation and still arrive later.
- A zero proper-distance slope does not stop local null propagation.
References
Sources supporting the explanations, physical distinctions and examples in this lesson.