Cosmic Microwave Background Calculator
CMB Calculator computes temperature, peak frequency, energy density at redshift, plots blackbody spectrum.
Cosmic Microwave Background Overview
The CMB is a snapshot of the early universe, described as a blackbody spectrum with properties scaling with redshift:
Temperature:
\\[ T(z) = T_0 (1 + z) \\]Peak Frequency (Wien’s Law):
\\[ \nu_{\text{peak}} \approx 2.821 \cdot \frac{k T(z)}{h} \\]Energy Density:
\\[ \epsilon = a T(z)^4, \quad a = \frac{4 \sigma}{c} \\]Where:
- \\(T(z)\\): Temperature at redshift \\(z\\) (K)
- \\(T_0\\): Present-day CMB temperature (\\(\approx 2.725 \, \text{K}\\))
- \\(\nu_{\text{peak}}\\): Peak frequency of the blackbody spectrum (GHz)
- \\(\epsilon\\): Energy density (J/m³)
- \\(k\\): Boltzmann constant (\\(1.380649 \times 10^{-23} \, \text{J/K}\\))
- \\(h\\): Planck constant (\\(6.62607015 \times 10^{-34} \, \text{J·s}\\))
- \\(\sigma\\): Stefan-Boltzmann constant (\\(5.670367 \times 10^{-8} \, \text{W/m}^2\text{K}^4\\))
- \\(c\\): Speed of light (\\(2.99792458 \times 10^8 \, \text{m/s}\\))
Example Calculation
Example: \\(z = 0\\)
\\[
T(0) = 2.725 \cdot (1 + 0) \approx 2.725 \, \text{K}
\\]
\\[
\nu_{\text{peak}} \approx 2.821 \cdot \frac{1.380649 \times 10^{-23} \cdot 2.725}{6.62607015 \times 10^{-34}} \approx 160.2 \, \text{GHz}
\\]
\\[
\epsilon = \frac{4 \cdot 5.670367 \times 10^{-8}}{2.99792458 \times 10^8} \cdot (2.725)^4 \approx 4.17 \times 10^{-14} \, \text{J/m}^3
\\]