Cosmology · Cosmic contents and background radiation

Relic thermal radiation and the microwave background

Read a thermal spectrum by its axes, compare two temperatures and connect the microwave background’s measured shape with an earlier radiation field.

Preparation: ratios, powers of ten and ordinary arithmetic. Physical quantities and notation are introduced as they are needed.

Back to the chapter

After reading, you can

Describe a blackbody spectrum as a distribution of radiation energy across frequencies.
Predict how undisturbed expansion changes a thermal spectrum and its temperature.
Distinguish a measured sky spectrum from an analytic teaching curve.

The temperature belongs to the whole spectral shape

Imagine sorting radiation by frequency into equally wide frequency bins. Each bin contains some energy per volume. A spectrum shows how that energy is distributed across frequencies. A broad peak does not mean that every photon has the same energy or that there are no photons outside the peak.

A blackbody spectrum is the particular thermal distribution described by Planck’s law. Its temperature sets its shape and overall energy density. The cosmic microwave background, or CMB, has a spectrum close to this form, with a temperature about 2.7 K. Kelvin measures absolute temperature; 2.7 K is a low temperature, not a negative one.

This statement describes a mean background spectrum. Small variations with direction and other signals superposed on the sky are distinct measurements. They require their own maps, foreground treatment and uncertainty analysis, introduced in the next chapter.

An ideal blackbody curve is a model with one temperature. Establishing that the sky closely follows that curve requires calibrated observations over many spectral bins. A chosen curve that looks smooth is not itself evidence that the real radiation is thermal.

Compare cooler radiation on the same axes

Choose a reference temperature T_ref and a dimensionless frequency x = hν/(k_B T_ref). Here ν is frequency, h is Planck’s constant and k_B is Boltzmann’s constant, which converts temperature into an energy scale. For the comparison, x uses the same reference temperature for both curves.

Plot energy density per unit frequency interval, with one shared vertical normalization. An ideal thermal spectrum at T_ref has its frequency peak at x ≈ 2.82. At half that temperature, the peak is at x ≈ 1.41 and its height on these same axes is one eighth as large. Both the horizontal shift and the lower height matter.

The energy summed across all frequencies falls even faster: at half the temperature, the total radiation energy density is one sixteenth. A peak height is energy density per frequency interval; total energy density is the area under that spectrum. Their different changes are consistent because the cooler distribution is also narrower along the frequency axis.

Do not normalize each curve to its own peak when making this energy comparison. That would hide the height change. Nor should the vertical axis be relabelled photon count: energy weighting changes a spectrum’s shape. A spectrum per wavelength interval has another peak location because equal wavelength bins are not equal frequency bins.

Ideal blackbody comparisons with one shared energy-per-frequency normalization; these are analytic examples, not CMB measurements
Temperature	Peak x = hν/(k_B T_ref)	Peak height relative to reference peak	Total energy density relative to reference
T_ref	2.82 approximately	1	1
T_ref/2	1.41 approximately	1/8	1/16

Two ideal Planck energy-per-frequency spectra share the same axes and the same fixed vertical normalization, with the reference-temperature peak set to one. The linear horizontal coordinate is dimensionless frequency x = hν/(k_B T_ref), shown from 0 to 10; the linear vertical coordinate is spectral energy density per unit frequency, divided by the reference peak density per frequency. The solid T_ref curve peaks at x = 2.82144 with height one. The dashed T_ref/2 curve peaks at x = 1.41072 with height 1/8. Thin dotted guides identify model peak positions, not observations. Integrated across all frequencies, the cooler spectrum has 1/16 the energy density; the displayed curves have tails outside the plotted interval. The plotted function is Fθ(x) = [x³/(exp(x/θ) − 1)]/[x_peak³/(exp(x_peak) − 1)], with θ = T/T_ref. These are analytic examples, not FIRAS measurements, photon-number spectra, wavelength spectra or detector fluxes. Original Ark teaching model.

Expansion can preserve a spectrum while cooling it

Suppose the radiation initially has a blackbody spectrum and then propagates through a smooth expanding background without a later disturbance. If the scale factor doubles, each wavelength doubles and each photon’s energy halves. The volume following that background grows by eight.

Together these changes transform the original blackbody into another blackbody at half the temperature. Its total energy per volume is one sixteenth of the original value, matching the table. Free expansion can therefore carry a thermal spectrum forward without requiring the photons to keep colliding in thermal equilibrium at every later instant.

The conditions are essential. Injecting energy, producing or absorbing photons, or mixing emissions can change the spectrum. Scattering that redistributes photon energies is also different from the photon-producing and photon-absorbing processes needed to establish full thermal equilibrium in the first place.

Early thermalization and recombination-era last scattering are separate stages. The near-thermal spectral distribution was established much earlier than the stage when most background photons began their long nearly unobstructed journeys toward us. Last scattering describes an interaction history, not the invention of a thermal curve at that instant.

The real spectrum is an instrument comparison

The COBE Far Infrared Absolute Spectrophotometer, FIRAS, compared radiation from the sky with calibrated reference radiation. Instrument response, calibration and foreground emission from our Galaxy had to be treated before interpreting the cosmic spectrum. The 1996 full-data analysis found close agreement with a blackbody.

The inferred temperature is therefore a measured quantity with uncertainty. A widely used historical combined estimate reported by Fixsen in 2009 is 2.72548 ± 0.00057 K. That combines measurements described in that paper; it is distinct from its own FIRAS/WMAP recalibration result and is not a new 2026 measurement.

The rounded value about 2.7 K is enough for the mechanism in this lesson. Quoting extra digits is useful only when their calibration, source and uncertainty matter to the question. It does not make the analytic comparison above a set of FIRAS data points.

Close agreement also does not mean that all departures are mathematically zero. A measurement can constrain small spectral distortions only to its sensitivity and after accounting for other signals. Learning how a residual is defined and measured comes later; inventing error bars around a teaching curve would erase that distinction.

A relic carries evidence about a previous state

A relic is something whose present properties retain information about an earlier physical state. The CMB spectrum links a present low-temperature radiation field with an earlier hotter one through a tested expansion and interaction history. It is not a picture of the first photon or of the beginning of time.

When reading a spectral plot, name the horizontal variable, the quantity per interval on the vertical axis, and the normalization. Then ask which marks are measurements and which lines are model predictions. These habits let a simple curve teach a physical mechanism without asking it to supply observations it does not contain.

On a first reading, compare the two rows in the table and follow the length doubling. The detailed Planck formula and statistical mechanics explain why these relations hold, but they are a return path rather than a prerequisite for recognizing the cooling of relic thermal radiation.

Worked reasoning

Follow the spectrum through a length doubling

An initially thermal photon field expands undisturbed until the scale factor doubles. What happens to its temperature, frequency-peak location and total energy density?

A wavelength doubling halves each photon’s frequency and energy.
The spectrum retains its thermal form with T_new = T_old/2. Its peak in energy per frequency interval shifts to half the previous frequency.
The photon number per volume falls by eight and the energy per photon by two, giving energy density one sixteenth of its original value.

The temperature and frequency-peak location halve; the total radiation energy density becomes one sixteenth.

Try the reasoning

Both thermal curves are rescaled separately to have peak height 1. Can their displayed peak heights still show the physical energy-per-frequency change?
Show the answer and reasoning
No. Separate rescaling removes that comparison.
A shared vertical scale is required to retain the one-eighth peak-height relation.
May an energy-per-frequency spectrum’s peak simply be relabelled as the energy-per-wavelength peak?
Show the answer and reasoning
No. The binning variable changes the spectral distribution.
Equal frequency intervals correspond to unequal wavelength intervals, so the relevant distribution and peak must be transformed.
Must the radiation remain in frequent collisions forever to preserve an initially blackbody spectrum?
Show the answer and reasoning
No, not in the stated undisturbed expanding-background model.
Redshift and dilution preserve the form with a lower temperature. Later interactions or injection can instead alter it.

What to carry forward

A thermal spectrum distributes radiation energy over frequencies; temperature describes the distribution.
Undisturbed expansion preserves an initially blackbody form with T proportional to 1/a.
Measured CMB spectra require calibration and foreground analysis; the analytic comparison is a teaching model.

References

Sources supporting the explanations, physical distinctions and examples in this lesson.