The particle horizon and the available past

A horizon calculation needs an integration domain

At a chosen observation time t, ask how far comoving light could have travelled during the available past history. In a noncompact FLRW background with the stated radial coordinate convention, the comoving particle-horizon radius is r_ph(t) = c∫ from t_i to t of dt′/a(t′), when the relevant lower-limit integral converges.

The corresponding proper radius on the observation-time slice is D_ph(t) = a(t)r_ph(t). The integral follows null propagation through the changing scale factor, while the final multiplication assigns a same-time spatial radius.

The lower limit t_i and the assumed history are part of the model. If an ideal expression is extended toward a formal t_i = 0 boundary, the convergence and physical validity of that extension must be examined separately.

A particle-horizon radius on a time slice is different from the complete past-light-cone boundary of events. It also differs from a direct statement that every source inside the radius emits detectable electromagnetic radiation.

Integrate before multiplying by the final scale factor

Consider separate flat kinematic histories a(u) = uᵖ, with u = t/T > 0 and T a positive time unit. The choices p = 1/2 and p = 2/3 can describe the ideal flat radiation-only and pressureless-matter-only general-relativistic solutions respectively.

For p < 1, the extrapolated past integral gives r_ph/(cT) = ∫ from 0 to u of v⁻ᵖdv = u¹⁻ᵖ/(1 − p). Multiplication by a = uᵖ then gives D_ph/(cT) = u/(1 − p).

Radiation with p = 1/2 therefore gives D_ph = 2ct. Matter with p = 2/3 gives D_ph = 3ct. These are different ideal histories, not two simultaneous radius values assigned to one actual Universe.

At u = 1 both models have a = 1. Their proper and comoving particle-horizon radii are then 2cT and 3cT respectively. Merely multiplying the coordinate age T by c would give cT, which is neither result.

Proper and comoving horizons evolve differently

At u = 1/4 in the radiation model, r_ph/(cT) = 2√u = 1 and a = 1/2, so D_ph/(cT) = 1/2. At u = 4, the same model gives comoving radius 4cT and proper radius 8cT.

For the matter model at u = 1/4, the comoving radius is 3u¹ᐟ³cT ≈ 1.88988cT while the proper radius is 0.75cT. At u = 4, the comoving radius is about 4.76220cT and the proper radius is 12cT.

The table compares proper radii on their stated time slices in the separate histories. It is a model calculation, not a measurement of the observable Universe or a claim that either ideal single-component account describes every actual epoch.

A proper horizon radius greater than ct does not require a photon to exceed local light speed. The global quantity combines light propagation over an evolving metric with a final same-time spatial-distance definition.

Finite coordinate age does not ensure finite reach

Now consider a separate supplied spatially flat coasting geometry, a(u) = u, so p = 1. It is a kinematic comparison; it is not the empty negatively curved Milne model introduced later.

At observation u = 1, start with a positive lower cutoff ε. The comoving integral is r/(cT) = ∫ from ε to 1 of dv/v = ln(1/ε). For ε = 0.01 it is about 4.60517; for ε = 0.0001 it is about 9.21034.

As ε approaches zero, the logarithm grows without bound. Under this ideal noncompact history’s formal past extension, there is no finite particle-horizon radius, even though the coordinate-time interval from zero to T is finite.

The result is about convergence of a declared model integral. It does not establish a verified physical origin, an actual infinite survey or the causal structure of every possible cosmological extension.

Visibility needs more than an available causal path

A causal particle horizon does not guarantee photon transparency, source emission or detector sensitivity. Those physical and practical requirements can restrict what is directly observed inside the causally accessible domain.

Last scattering describes a physical visibility window for the radiation under consideration, rather than automatically equalling the causal particle-horizon boundary. Reprocessed photons and later observables can carry information about earlier processes without being direct images of every earlier event.

A horizon estimate must name its background history, lower limit, spatial assumptions, time slice and proper or comoving convention. A change in the early history can change the integral even if later quantities are held fixed.

The examples show why the definition should come before a quoted size. The radiation and matter integrals converge to different finite radii; the coasting counterexample diverges. Finite age by itself does not choose among those outcomes.

Worked reasoning

Try the reasoning

1. Does a finite coordinate age alone prove the past particle-horizon integral is finite?

   Show the answer and reasoning

   No.

   For the supplied flat coasting history a = u, the integral diverges logarithmically as the lower cutoff approaches zero.

2. How is proper particle-horizon radius obtained from its comoving radius at observation time t?

   Show the answer and reasoning

   Multiply by a(t).

   D_ph(t) = a(t)r_ph(t) in the stated radial convention.

3. Does a location within the causal particle-horizon radius necessarily contain an electromagnetically detectable source?

   Show the answer and reasoning

   No.

   Emission, photon transparency and practical detection add distinct conditions.

What to carry forward

• A particle horizon depends on the available past history and convergence of its null-propagation integral.
• Proper and comoving radii require different final conventions; neither is generally c times age.
• Causal access and photon visibility are distinct, model-dependent questions.