Thermal radiation and a cooling cosmic background

Temperature describes a distribution

Thermal radiation contains photons over a range of frequencies. The temperature of a blackbody labels the whole equilibrium-shaped distribution; it does not assign one frequency, energy or individual temperature to every photon.

For zero photon chemical potential, the mean occupation per photon mode is f = 1/[exp(x) − 1], where x = hν/(k_B T). Here ν is frequency in hertz, T is temperature in kelvin, h is Planck’s constant and k_B converts temperature to an energy scale.

Cosmic expansion can redshift an already established blackbody while preserving its form. An observed thermal shape need not mean that thermalizing collisions continue at the time of observation. Establishing the original thermal distribution and propagating it later are separate questions.

The background radiation spectrum supplies evidence to test thermal-history models. The calculation below uses entirely supplied values and no instrument samples, current measured background temperature or claimed physical photon-origin event.

A mode and the temperature scale redshift together

Choose q = a/a_ref from 1 to 4 and an initially homogeneous blackbody with reference temperature T_ref = 1200 K. Assume collisionless photon propagation in an FLRW background, with no absorption, spectral injection or later entropy transfer to this photon population.

Then T = T_ref/q and the frequency of a freely redshifting tracked mode is ν = ν_ref/q. Choose ν_ref = 2k_B T_ref/h, approximately 5.000789 × 10¹³ Hz. This selects one mode whose reference x is 2; other modes have their own x values.

Because both ν and T divide by q, the tracked ratio x stays exactly 2 within this model. Its occupation remains 1/(e² − 1) ≈ 0.156518. Frequency decreases without changing that mode’s dimensionless location in the rescaled Planck distribution.

This statement does not hold the occupation fixed at every fixed laboratory frequency while the spectrum cools. It follows corresponding redshifting modes. At a fixed numerical ν, the dimensionless x changes when T changes.

Cooling and dilution make different powers

The number density of the freely propagating photons falls as q⁻³ because the physical volume of a comoving region grows as q³. The energy of each tracked photon falls as q⁻¹, adding another factor to energy-density dilution.

Consequently n_γ/n_γ,ref = q⁻³ and ε_γ/ε_γ,ref = q⁻⁴, where ε is energy density. These ratios also agree with the blackbody relations n_γ proportional to T³ and ε_γ proportional to T⁴ under the stated conditions.

At q = 2, temperature and each tracked photon frequency are halved, number density is one eighth of its reference value and energy density is one sixteenth. A statement about one of these changes cannot substitute for all the others.

The reference temperature is a teaching input. Exact definitions of h and k_B do not make a measured cosmic temperature or a proposed epoch exact. The table isolates scaling, rather than identifying an observed cosmic event.

Entropy transfer changes the bath’s temperature relation

A thermal bath can contain several particle species that exchange energy. When species populations change and heat the remaining bath, a simple T proportional to 1/a relation need not hold across the whole episode.

For an isolated equilibrium bath with conserved comoving entropy, one instead tracks a T g_*s^(1/3) = constant. The symbol g_*s counts effective entropy degrees of freedom. The energy-density count g_* is a different quantity in general and should not be substituted without justification.

As a chosen endpoint account, if a doubles while g_*s falls from 8 to 1 under those conditions, the temperature ratio is (8/1)^(1/3)/2 = 1. The bath does not cool between those two supplied endpoints despite expansion.

That endpoint example is not an actual instantaneous particle transition or a measured history. A real calculation must specify which species share a bath, their distributions and energy or entropy exchanges. Independently decoupled populations can have different temperatures.

A preserved shape is a conditional prediction

The spectrum’s shape, amplitude and distortions provide distinct information. A mechanism that injects energy without enough thermalizing interactions can change the spectrum rather than merely shifting one fitted temperature.

The supplied example begins with a zero-chemical-potential Planck distribution and excludes such processes. It demonstrates how that specific initial form propagates. It does not derive the initial equilibrium from the scaling law alone.

To compare with observations, use calibrated spectral measurements and a model that includes relevant foregrounds, detector response and possible distortions. A toy table should not acquire fabricated measurement errors or the precision of a real instrument.

When reconstructing a thermal history, name the population whose temperature is quoted and the interval over which its scaling assumptions apply. This keeps radiation cooling, entropy redistribution and changing particle contents from being combined into an unsupported all-era rule.

Worked reasoning

Try the reasoning

1. Does the chosen tracked frequency describe every photon in the blackbody?

   Show the answer and reasoning

   No.

   It selects one mode from a distribution spanning many frequencies.

2. For the stated conserved-entropy endpoint example, a doubles and g_*s changes from 8 to 1. What is T_after/T_before?

   Show the answer and reasoning

   1.

   The factor from the degree-of-freedom change is 2 and cancels the factor of 2 from expansion.

3. Does a preserved thermal-shaped spectrum prove ongoing thermalizing collisions?

   Show the answer and reasoning

   No.

   An initially Planck spectrum can retain that form under collisionless homogeneous redshift with the stated exclusions.

What to carry forward

• Temperature labels a spectrum, while photon number density and energy density have different dilution powers.
• Corresponding redshifting modes preserve hν/(k_B T) in the supplied collisionless example.
• Entropy transfer and the identity of the bath must be stated before extending a temperature law.