CMB Blackbody Spectrum and T ∝ 1/a

Derivation

CMB Blackbody Spectrum and T ∝ 1/a

D-344 Home PU-308 Threads light · chance · symmetry Depends on planck-radiation-law, Cosmological Redshift from the Scale Factor

Statement

In a homogeneous, isotropic universe expanding adiabatically with scale factor \(a(t)\), a radiation field that is Planckian at one epoch remains exactly Planckian at every later epoch, with its temperature falling as the inverse scale factor: \(T(t)\propto 1/a(t)\), equivalently \(T = T_0\,(1+z)\). No spectral distortion is introduced by free expansion alone.

Why it matters

The cosmic microwave background is the most perfect blackbody ever measured, with fractional deviations from a Planck curve below \(10^{-4}\). Yet the radiation was last in thermal contact with matter at recombination, \(z\approx 1100\), and has simply free-streamed since. This result explains the paradox: expansion preserves the blackbody form, so a spectrum thermalised once stays thermal forever, merely cooling. The relation \(T\propto 1/a\) is the thermometer that converts a redshift into a temperature and underpins big-bang nucleosynthesis, recombination, and the whole thermal history of the cosmos.

It also fixes the scaling of the radiation energy density, \(\rho_\gamma\propto a^{-4}\), which determines when the universe was radiation-dominated and thereby sets the epoch of matter–radiation equality.

Assumptions

Homogeneity and isotropy (FLRW geometry).Without a single scale factor common to all directions and positions, different photons would redshift by different factors and a single global temperature could not be defined.

Adiabatic expansion: no photon creation or destruction.If Compton up-scattering, decays, or energy injection added or removed photons, the occupation number of each mode would change and the spectrum could develop a chemical-potential (\(\mu\)) or Compton-\(y\) distortion instead of staying pure Planck.

Photons free-stream on null geodesics with momentum redshifting as \(p\propto 1/a\).This is the kinematic input; drop it and there is no relation between the photon frequency at two epochs, so no temperature scaling follows.

The phase-space distribution is collisionless-invariant (Liouville's theorem) once decoupled.If the coarse-grained distribution function were not conserved along trajectories, the occupation number \(N(p)\) would not simply track the redshifted momentum and the Planck form need not be preserved.

Zero chemical potential for photons (Bose–Einstein with \(\mu=0\)).Photon number is not separately conserved microscopically; only frequent number-changing interactions (double Compton, bremsstrahlung) at early times drive \(\mu\to 0\). Drop this and the equilibrium form is \((e^{(h\nu-\mu)/k_BT}-1)^{-1}\), a distorted spectrum.

Derivation

We track the specific energy density \(u_\nu\,d\nu\) of a Planck field from an epoch with scale factor \(a_i\) and temperature \(T_i\) to a later epoch with scale factor \(a\). Symbols are rearranged fully before any numbers appear.

\[ u_\nu(\nu_i,T_i)\,d\nu_i = \frac{8\pi h}{c^3}\,\frac{\nu_i^{\,3}}{e^{h\nu_i/k_B T_i}-1}\,d\nu_i \]

Planck's law for the spectral energy density at the initial epoch (assumed prior result). A

\[ \nu = \nu_i\,\frac{a_i}{a},\qquad p=\frac{h\nu}{c}\propto \frac{1}{a} \]

Cosmological redshift: every photon's frequency scales inversely with \(a\) (prior result). Because the geometry is isotropic, the same factor applies to all modes. A

\[ d\nu = d\nu_i\,\frac{a_i}{a}\quad\Longrightarrow\quad d\nu_i = d\nu\,\frac{a}{a_i} \]

Differentiate the linear map of step 2; a frequency band is compressed by the same factor as its centre. A

\[ u_\nu(\nu)\,d\nu = \left(\frac{a_i}{a}\right)^{4} u_\nu(\nu_i)\,d\nu_i \]

Energy-density dilution of the band: photon number in it is conserved so the number density dilutes as \(a^{-3}\), and each photon's energy redshifts as \(a^{-1}\); together \(u\propto a^{-4}\). B

\[ u_\nu(\nu)\,d\nu = \left(\frac{a_i}{a}\right)^{4}\frac{8\pi h}{c^3}\,\frac{\nu_i^{\,3}}{e^{h\nu_i/k_B T_i}-1}\,\underbrace{d\nu\,\frac{a}{a_i}}_{d\nu_i} \]

Insert the Planck form (step 1) and the change of variables (step 3) into the transport law (step 4). A

\[ \nu_i^{\,3} = \nu^3\left(\frac{a}{a_i}\right)^{3},\qquad \frac{h\nu_i}{k_B T_i}=\frac{h\nu}{k_B}\cdot\frac{a/a_i}{T_i}=\frac{h\nu}{k_B\,T_i(a_i/a)} \]

Express the initial-epoch frequency \(\nu_i=\nu(a/a_i)\) in the numerator and, crucially, inside the exponent, where it can be absorbed into a rescaled temperature. B

\[ u_\nu(\nu)\,d\nu=\left(\frac{a_i}{a}\right)^{4}\!\left(\frac{a}{a_i}\right)^{3}\!\left(\frac{a}{a_i}\right)\frac{8\pi h}{c^3}\,\frac{\nu^{3}}{e^{h\nu/k_B T}-1}\,d\nu,\quad T\equiv T_i\frac{a_i}{a} \]

Collect the scale-factor powers from the prefactor (\(-4\)), the numerator (\(+3\)) and the Jacobian (\(+1\)); they sum to zero. B

\[ \left(\frac{a_i}{a}\right)^{4-3-1}=\left(\frac{a_i}{a}\right)^{0}=1 \]

The three scalings cancel identically — the exact cancellation is why the blackbody form is preserved rather than merely approximately. A

\[ u_\nu(\nu)\,d\nu=\frac{8\pi h}{c^3}\,\frac{\nu^{3}}{e^{h\nu/k_B T}-1}\,d\nu,\qquad T=T_i\,\frac{a_i}{a} \]

The result is Planck's law again, at a new temperature. Since \(a_i,T_i\) are fixed, \(T\propto 1/a\). A

\[ N(p)=\frac{1}{e^{p_i c/k_B T_i}-1}=\frac{1}{e^{pc/k_B T}-1},\qquad p_i=p\,\frac{a}{a_i} \]

Equivalent phase-space argument: the collisionless occupation number of each mode is conserved (Liouville), and substituting the redshifted momentum reproduces a Planck occupation at \(T=T_i(a_i/a)\) directly, independent of the energy-density bookkeeping above. C

Result

\[ u_\nu(\nu)=\frac{8\pi h}{c^3}\frac{\nu^3}{e^{h\nu/k_BT}-1},\qquad \boxed{\,T(t)=T_i\,\frac{a_i}{a(t)}\propto \frac{1}{a}\,},\qquad T=T_0(1+z) \]

Reading. Free adiabatic expansion is a similarity transformation on a Planck spectrum: it slides the whole curve to lower frequencies and lower amplitude while keeping the dimensionless shape variable \(x=h\nu/k_BT\) fixed. The spectrum stays a blackbody; only the single label \(T\) changes, and it changes as \(1/a\). Because \(1+z=a_0/a\), a measured redshift is a direct measurement of past CMB temperature.

Units check. \(T_i(a_i/a)\): \(a\) is dimensionless, so \(T\) keeps units of kelvin, \([\mathrm{K}]\). The prefactor \(8\pi h/c^3\) has units \(\mathrm{J\,s}\cdot(\mathrm{s^3/m^3})=\mathrm{J\,s^4/m^3}\); times \(\nu^3\;[\mathrm{s^{-3}}]\) gives \(\mathrm{J\,s/m^3}\), i.e. energy density per unit frequency (\(\mathrm{J\,m^{-3}\,Hz^{-1}}\)), as required. The exponent \(h\nu/k_BT\) is \((\mathrm{J\,s})(\mathrm{s^{-1}})/[(\mathrm{J\,K^{-1}})(\mathrm K)]=1\), dimensionless.

Limiting cases

No expansion (\(a\to a_i\)): \(T\to T_i\), the spectrum is unchanged — the trivial identity limit.
Rayleigh–Jeans tail (\(h\nu\ll k_BT\)): \(u_\nu\to 8\pi\nu^2 k_BT/c^3\); expansion still preserves the form, with \(T\propto1/a\), so the tail amplitude scales as \(a^{-3}\) at fixed \(\nu\).
Wien tail (\(h\nu\gg k_BT\)): \(u_\nu\to (8\pi h\nu^3/c^3)e^{-h\nu/k_BT}\); the exponential edge shifts to lower \(\nu\) as \(1/a\).
Total energy density: integrating gives \(\rho_\gamma=aT^4\)-type \(\propto T^4\propto a^{-4}\) (Stefan–Boltzmann), the same power that governs radiation domination.
Number density: \(n_\gamma\propto T^3\propto a^{-3}\), just the dilution of a fixed photon count in an expanding volume — consistent with number conservation.

Breaks when

Energy injection distorts the spectrum. Decaying particles, dissipating acoustic waves, or a hot electron gas add energy or photons. Before \(z\sim 2\times10^6\), double-Compton and bremsstrahlung re-thermalise it; between \(z\sim 10^5\) and recombination they cannot restore photon number, so a Bose–Einstein \(\mu\)-distortion survives. At lower \(z\), inverse-Compton scattering off hot electrons (galaxy clusters) produces a \(y\)-distortion — the Sunyaev–Zel'dovich effect — and \(T\propto1/a\) no longer describes that line of sight.
Anisotropic or inhomogeneous expansion. If the geometry is not FLRW (strong tidal fields, an anisotropic Bianchi model, or propagation through deep potential wells) the redshift becomes direction-dependent, photons from different modes cool by different factors, and a single blackbody temperature ceases to exist.
Photon number not conserved along the path. Photon–axion or photon–dark-photon conversion, or absorption and re-emission by dust, removes or reshuffles photons and breaks the exact cancellation of scale-factor powers.
Relativistic species become non-relativistic. The clean \(T\propto1/a\) holds for a decoupled gas of massless photons; when a component's rest mass matters (e.g. neutrinos near \(T\sim m_\nu\)) its "temperature" no longer scales simply as \(1/a\).

Failure modes

Confusing \(T\propto1/a\) with \(\rho\propto1/a\). The temperature scales as \(a^{-1}\), but the energy density scales as \(a^{-4}\) and the number density as \(a^{-3}\); students often quote one power for all three.
Thinking each photon's energy drops but "temperature" is unrelated. The temperature is not an independent bookkeeping variable — it is fixed by the requirement that the redshifted spectrum still be Planck, which forces \(T\propto\nu\propto1/a\).
Using \(T=T_0/(1+z)\) instead of \(T=T_0(1+z)\). Higher redshift means the earlier, hotter universe; the temperature rises with \(z\).
Assuming a redshifted Planck curve is only approximately Planck. The preservation is exact for free adiabatic expansion; the cancellation \(4-3-1=0\) is algebraic, not a small-distortion expansion.
Forgetting the Jacobian \(d\nu_i=d\nu\,(a/a_i)\). Transforming \(u_\nu\) as if \(d\nu\) were invariant gives the wrong power of \(a\) and a spurious distortion.
Applying \(T\propto1/a\) to a \(y\)-distorted or line spectrum. The scaling law is a statement about the blackbody form; it does not describe individual spectral lines, which redshift but do not "cool".

Discussion

The deep reason the blackbody survives is a scale symmetry. A Planck spectrum depends on frequency only through the dimensionless combination \(x=h\nu/k_BT\); its entire shape is the universal function \(x^3/(e^x-1)\). Expansion multiplies every \(\nu\) by \(a_i/a\), and preserving the shape requires only that \(T\) be multiplied by the same factor so that \(x\) is invariant mode-by-mode. The three thermodynamic quantities then follow from dimensional counting: \(x\) fixed forces \(T\propto\nu\propto a^{-1}\), so \(n_\gamma\propto T^3\propto a^{-3}\) and \(\rho_\gamma\propto T^4\propto a^{-4}\). This connects the symmetry thread (self-similarity of the Planck curve under simultaneous rescaling of \(\nu\) and \(T\)) to the light thread (null-geodesic redshift) and the chance thread (the Bose–Einstein statistics that produce the Planck form in the first place).

Thermodynamically, the same result is the statement that comoving photon entropy is conserved. Blackbody entropy density is \(s\propto T^3\), and \(S=sV\propto T^3 a^3\); adiabatic (isentropic) expansion holds \(S\) fixed, giving \(T^3a^3=\text{const}\), i.e. \(T\propto1/a\). The blackbody is the maximum-entropy photon distribution at fixed energy, and free expansion is reversible, so the field cannot leave that maximum-entropy manifold — it can only slide along it to lower \(T\). This is why a spectrum that thermalised once, at very early times, needs no ongoing thermal contact to stay Planckian: the CMB was last scattered at recombination yet remains a near-perfect blackbody today precisely because expansion is entropy-conserving.

The cleanest formulation is kinetic. The photon occupation number \(N(\mathbf p)\) obeys the collisionless Boltzmann (Liouville) equation once scattering ceases, so \(N\) is constant along phase-space trajectories. In FLRW the trajectories are \(p\propto1/a\), and a distribution that is a function of \(p/T\) at one time remains a function of \(p/T\) at all times provided \(T\propto1/a\). Remarkably, this holds even during decoupling: Thomson scattering is elastic and isotropic in the electron rest frame, conserving photon number and energy in that frame, so it drives no departure from Planck. The COBE-FIRAS bound \(|\mu|<9\times10^{-5}\), \(|y|<1.5\times10^{-5}\) is thus an experimental confirmation that the universe expanded adiabatically, with negligible energy injection, across a factor of \(\sim10^3\) in scale factor since recombination.

Common misconceptions. The photons are not "losing energy to friction" or "tiring"; the redshift is a purely geometric stretching of wavelength with the metric, and it is exactly compensated in the spectral bookkeeping by the dilution of number and the compression of frequency bands. The CMB is also not a relic of a hot surface we are looking back at like a fire — it is a bath filling all space, and its temperature is a property of the radiation field everywhere, not a distance to a source.

Worked examples

Example 1 — CMB temperature at recombination.

\[ T(z)=T_0\,(1+z) \]

Write \(1+z=a_0/a\) and apply \(T\propto1/a\) with today's value \(T_0\). A

\[ T_0=2.725\ \mathrm{K},\qquad z_\text{rec}\approx 1100 \]

Insert the measured present temperature and the recombination redshift. A

\[ T=2.725\ \mathrm{K}\times(1+1100)=2.725\times1101\ \mathrm{K} \]

Arithmetic. A

\[ T_\text{rec}\approx 3.00\times10^{3}\ \mathrm{K} \]

Reading. The recombination bath was about 3000 K — hydrogen recombines at a temperature well below its 13.6 eV ionisation energy because photons vastly outnumber baryons (\(n_\gamma/n_b\sim10^9\)), so the exponential Wien tail keeps enough ionising photons until \(k_BT\sim0.3\ \mathrm{eV}\).

Units check. Kelvin \(\times\) dimensionless \(=\) kelvin.

Example 2 — where the spectrum peaks, then and now.

\[ \lambda_\text{peak}\,T=b,\qquad b=2.898\times10^{-3}\ \mathrm{m\,K} \]

Wien's displacement law for a blackbody; the peak tracks temperature. A

\[ \lambda_\text{peak,0}=\frac{b}{T_0}=\frac{2.898\times10^{-3}\ \mathrm{m\,K}}{2.725\ \mathrm{K}} \]

Evaluate at today's temperature. A

\[ \lambda_\text{peak,0}\approx 1.06\times10^{-3}\ \mathrm{m}=1.06\ \mathrm{mm} \]

Arithmetic — a microwave, hence the name. A

\[ \lambda_\text{peak,rec}=\frac{\lambda_\text{peak,0}}{1+z}=\frac{1.06\ \mathrm{mm}}{1101} \]

Since \(\lambda\propto a\propto1/(1+z)\) and \(T\propto1/a\), the peak wavelength shrinks by \((1+z)\) into the past. B

\[ \lambda_\text{peak,rec}\approx 9.6\times10^{-7}\ \mathrm{m}\approx 0.96\ \mathrm{\mu m} \]

Reading. At recombination the CMB peaked in the near-infrared, just beyond visible red — the early universe glowed. Expansion has stretched that same photon population by a factor \(\sim1100\) into the microwave band we detect today, consistent with \(T_\text{rec}/T_0=1101\) from Example 1.

Units check. \(\mathrm{m\,K}/\mathrm{K}=\mathrm m\); dividing by dimensionless \((1+z)\) keeps metres.

Problems

Compute the CMB temperature at the epoch of reionisation, \(z=6\), given \(T_0=2.725\ \mathrm{K}\).
Solution
\(T=T_0(1+z)=2.725\times7=19.08\ \mathrm{K}\). The bath was about 19 K at \(z=6\) — still cold, which is why reionisation requires stellar/quasar UV photons rather than the CMB itself.
Show explicitly that the radiation energy density scales as \(a^{-4}\), and evaluate the ratio of \(\rho_\gamma\) at recombination to today.
Solution
Stefan–Boltzmann for a blackbody gives \(\rho_\gamma=\frac{4\sigma}{c}T^4\propto T^4\). With \(T\propto1/a\), \(\rho_\gamma\propto a^{-4}\). Ratio \(=\left(\frac{T_\text{rec}}{T_0}\right)^4=(1101)^4\approx1.47\times10^{12}\). The early radiation field was over a trillion times denser in energy.
Using \(n_\gamma=\dfrac{2\zeta(3)}{\pi^2}\left(\dfrac{k_BT}{\hbar c}\right)^3\), state how \(n_\gamma\) scales with \(a\) and find \(n_\gamma\) at \(z=1100\) given \(n_{\gamma,0}\approx411\ \mathrm{cm^{-3}}\).
Solution
\(n_\gamma\propto T^3\propto a^{-3}=(1+z)^3\). So \(n_\gamma=411\times(1101)^3\ \mathrm{cm^{-3}}=411\times1.335\times10^{9}\approx5.5\times10^{11}\ \mathrm{cm^{-3}}\). The photon number in a comoving volume is unchanged; only the physical density grows into the past, exactly as number conservation demands.
The universe expands so that all distances double, \(a\to2a\). What happens to (a) the CMB temperature, (b) the peak wavelength, (c) the energy density?
Solution
(a) \(T\propto1/a\Rightarrow T\to T/2\): halved. (b) \(\lambda_\text{peak}\propto1/T\propto a\Rightarrow\) doubled. (c) \(\rho_\gamma\propto a^{-4}\Rightarrow\rho_\gamma\to\rho_\gamma/16\): reduced sixteenfold. The dimensionless shape variable \(x=h\nu/k_BT\) for each mode is unchanged, so it is still a blackbody.
Prove from the invariance of \(x=h\nu/k_BT\) along a photon geodesic that a spectrum which is Planck at one epoch is Planck at all epochs, and identify precisely which physical assumption guarantees \(x=\text{const}\).
Solution
The Planck occupation number of a mode is \(N=\left(e^{x}-1\right)^{-1}\) with \(x=h\nu/k_BT\). Under free-streaming the collisionless Boltzmann equation gives \(dN/dt=0\) along the trajectory (Liouville's theorem), so \(N\), and hence \(x\), is conserved for that mode. The geodesic gives \(\nu\propto1/a\); constancy of \(x=h\nu/k_BT\) then forces \(T\propto\nu\propto1/a\). Because every mode carries the same \(N(x)\) at all times, the assembled spectrum is again \(\left(e^{h\nu/k_BT}-1\right)^{-1}\), a Planck curve at the new \(T\). The guaranteeing assumption is that expansion is adiabatic and collisionless (no photon creation/destruction and \(\mu=0\)); if energy or photons were injected, \(N\) would not be conserved along the trajectory and \(x\) would drift, producing a \(\mu\)- or \(y\)-distortion instead.