The cosmic microwave background, usually called the CMB, is the relic radiation left over from the hot early
Universe. Today it fills space almost uniformly and behaves to excellent approximation like a blackbody with temperature
\(T_0 \approx 2.725\ \mathrm{K}\). Because the Universe expands, radiation is stretched as it travels. That stretching
changes the typical photon wavelength and therefore changes the temperature scale associated with the CMB at different
cosmic epochs.
Temperature and redshift
In an expanding Universe, photon wavelengths scale with the cosmic scale factor, and the CMB temperature scales inversely.
This leads to a very simple and important relation:
CMB temperature law.
\[
\begin{aligned}
T(z) &= T_0(1+z)
\end{aligned}
\]
Here \(z\) is the redshift. When \(z=0\), we recover the present-day CMB temperature \(T_0\). When \(z\) is larger,
the CMB was hotter in the past. For example, if \(z=10\), then the CMB temperature was about eleven times larger than today.
This is why the CMB becomes a much more energetic radiation field when we look back toward the early Universe.
Wien’s law and the peak wavelength
A blackbody spectrum has a characteristic peak. Wien’s displacement law connects that peak wavelength to temperature:
Wien’s law.
\[
\begin{aligned}
\lambda_{\text{peak}} &= \frac{b}{T}
\end{aligned}
\]
The constant \(b\) is Wien’s displacement constant. This law shows that higher temperature means a shorter peak wavelength.
For the present CMB temperature, the peak lies in the millimeter range, which is why the radiation is called
“microwave background.” As redshift increases and the temperature rises, the peak moves to shorter wavelengths.
Peak shift with redshift.
\[
\begin{aligned}
\lambda_{\text{peak}}(z) &= \frac{b}{T_0(1+z)} \\
&= \frac{\lambda_{\text{peak},0}}{1+z}
\end{aligned}
\]
This second form is especially useful because it shows the peak shift directly. If the temperature grows by a factor
\(1+z\), then the peak wavelength shrinks by the same factor.
Blackbody spectrum
The detailed shape of the spectrum is given by Planck’s law:
Planck spectrum in wavelength form.
\[
\begin{aligned}
B_{\lambda}(\lambda,T) &= \frac{2hc^2}{\lambda^5}\cdot \frac{1}{e^{hc/(\lambda kT)} - 1}
\end{aligned}
\]
In the calculator, this formula is used to preview the spectral shape for the present CMB and for the selected redshifted
temperature. The graph is normalized to each curve’s own peak so that both shapes remain visible even when one temperature
is much larger than the other. That makes the visualization easier to read, especially when comparing today’s microwave
spectrum with a much hotter early-universe spectrum.
Worked examples
For the present-day Universe, \(z=0\), so the temperature remains
Today.
\[
\begin{aligned}
T(0) &= T_0(1+0) \\
&= 2.725\ \mathrm{K}
\end{aligned}
\]
Then Wien’s law gives a peak wavelength near
Present peak wavelength.
\[
\begin{aligned}
\lambda_{\text{peak},0} &= \frac{b}{2.725} \\
&\approx 1.06\ \mathrm{mm}
\end{aligned}
\]
This matches the standard statement that the present CMB peaks in the microwave band.
Now consider a redshift near recombination, for example \(z \approx 1089\). Then
Recombination-era temperature.
\[
\begin{aligned}
T(1089) &= 2.725(1+1089) \\
&\approx 2.97\times 10^{3}\ \mathrm{K}
\end{aligned}
\]
The corresponding peak wavelength is much shorter:
Recombination-era peak.
\[
\begin{aligned}
\lambda_{\text{peak}}(1089) &= \frac{b}{2.97\times 10^{3}} \\
&\approx 9.76\times 10^{-7}\ \mathrm{m} \\
&\approx 0.976\ \mathrm{\mu m}
\end{aligned}
\]
So by recombination the peak has shifted from the present millimeter range down to about a micrometer, which lies in the
near-infrared. This is an excellent reminder that the same relic radiation looked very different in the early Universe.
Physical interpretation
The CMB is often called the “Big Bang remnant,” but more precisely it is the radiation released when the Universe cooled
enough for neutral atoms to form and photons began to travel freely. As the Universe expanded afterward, those photons were
stretched to longer wavelengths, and the radiation cooled from thousands of kelvin down to today’s \(2.725\ \mathrm{K}\).
The simple relation \(T(z)=T_0(1+z)\) captures this cooling in a compact way.
Important limitation
This calculator is a clean educational preview. It focuses on the temperature scaling law, Wien’s law, and the blackbody
shape. It does not attempt to include all observational complications of real CMB analysis such as anisotropies,
polarization, detector bandpasses, foreground contamination, or the distinction between different spectral conventions.
The plotted spectrum is a normalized preview meant to highlight the shift in shape and peak location.
| Concept |
Main relation |
Meaning |
| Present CMB temperature |
\(T_0 \approx 2.725\ \mathrm{K}\) |
Observed temperature of today’s cosmic microwave background |
| Temperature scaling |
\(T(z) = T_0(1+z)\) |
The CMB was hotter at higher redshift |
| Wien peak |
\(\lambda_{\text{peak}} = b/T\) |
Higher temperature shifts the peak to shorter wavelength |
| Peak ratio |
\(\lambda_{\text{peak}}(z)/\lambda_{\text{peak},0} = 1/(1+z)\) |
Direct comparison with the present-day peak |
| Planck spectrum |
\(B_{\lambda}(\lambda,T) = \frac{2hc^2}{\lambda^5}\frac{1}{e^{hc/(\lambda kT)}-1}\) |
Blackbody spectral shape used for the graph preview |
| Recombination benchmark |
\(z \approx 1089 \Rightarrow T \approx 2970\ \mathrm{K}\) |
The CMB peak shifts from millimeter wavelengths to the near infrared |