The Physics of μ-Type Spectral Distortions
The cosmic microwave background (CMB) is renowned for its near-perfect blackbody spectrum. Yet, subtle deviations from this equilibrium shape can arise when energy is injected into the photon-baryon plasma after the initial release of CMB photons but before recombination. These deviations are encoded as spectral distortions, typically categorized as μ-type or y-type depending on the epoch and thermalization efficiency. Our calculator focuses on μ-type distortions, which are generated when energy is dumped into the plasma at redshifts roughly between 5×104 and 2×106. In this window, Compton scattering is efficient enough to redistribute photon energies and establish a Bose–Einstein distribution characterized by a nonzero chemical potential μ, yet photon-number–changing processes such as double Compton scattering and bremsstrahlung become slow, preventing full relaxation to a Planck spectrum.
Quantifying the size of the μ distortion provides a sensitive probe of processes in the early universe, including decaying or annihilating particles, dissipation of small-scale acoustic modes, primordial black hole evaporation, or cosmic string interactions. Many of these phenomena inject energy at levels much smaller than the CMB energy density, making distortions tiny yet potentially measurable with future satellite missions. The canonical approximation for the distortion amplitude is , valid when the injection occurs within the aforementioned redshift range and the fractional energy change is small. Outside this window, the universe either fully thermalizes the injection () or the distortion transitions toward a y-type signature (), with Compton scattering insufficient to establish a Bose–Einstein spectrum.
To capture the suppression of μ distortion near the boundaries of this epoch, it is common to multiply the simple estimate by window functions that approximate the efficiency of thermalization. A widely used fit is . Our calculator employs this approximate expression, which smoothly suppresses μ outside the optimal redshift range.
The presence of a nonzero μ modifies the photon occupation number from the Planck distribution to a Bose–Einstein distribution . In the Rayleigh–Jeans limit (), the relative brightness temperature shift is . Our tool reports this fractional temperature change alongside μ, offering intuition about observational signatures. For example, a μ of 10−8 corresponds to a brightness temperature shift of about 4.6×10−9, well below the sensitivity of current instruments but within reach of proposed missions like PIXIE or PRISM.
The table below illustrates how μ depends on energy injection fraction and redshift. We fix ΔE/E at 10−5 and vary the injection redshift. The exponential window dramatically attenuates μ outside the optimal range.
| z | μ | ΔT/T |
|---|---|---|
| 1×104 | ≈0 | ≈0 |
| 1×105 | 1.4×10−5 | 6.4×10−6 |
| 5×106 | ≈0 | ≈0 |
These values demonstrate the sharp redshift dependence: injections deep in the μ-era yield distortions proportional to ΔE/E, while earlier or later injections effectively vanish. Other parameters, such as the injection’s duration or spectrum, can modify the result. Short, localized bursts integrate in the same way as gradual energy releases provided the total ΔE/E is identical; however, injection outside the μ-era may produce a mixture of μ and y distortions or even more complex spectral shapes.
Spectral distortions offer a promising window into otherwise inaccessible physics. Dissipation of primordial acoustic waves on small scales—so-called Silk damping—should inject about ΔE/E ≈ 10−8, generating μ ≈ 10−8. Exotic energy injection mechanisms, such as decaying dark matter or primordial black hole evaporation, could produce larger signals. Because the μ distortion integrates energy injection over redshift rather than depending on precise timing, it serves as a calorimeter for the early universe. Upcoming missions aim to measure μ down to 10−8, potentially constraining models of inflation, dark matter, and high-energy particle physics.
Our calculator provides an accessible way to explore these ideas. By adjusting the fractional energy injection and redshift, users can assess whether a given process might produce a detectable μ distortion. The script computes μ using the smooth exponential window described above and outputs both the μ value and the corresponding brightness temperature shift. Results are categorized as "undetectable" when μ < 10−9, "marginal" for 10−9 ≤ μ < 10−7, and "potentially observable" when μ ≥ 10−7. Because all calculations occur client-side, the tool is well suited for classroom demonstrations, quick feasibility checks, or integration into other educational materials.
It is worth noting that spectral distortions do not simply vanish once generated; they persist to the present day unless erased by subsequent thermalization, which is inefficient at late times. Thus, any nonzero μ today carries a fossil record of energy release long before recombination. This persistence makes spectral distortions complementary to anisotropy measurements: whereas the CMB power spectrum probes fluctuations at recombination and later, μ encodes information from much earlier epochs, extending the cosmic timeline accessible to observation.
In summary, μ-type spectral distortions provide a unique lens on early-universe processes that fall between primordial nucleosynthesis and recombination. Their dependence on energy injection and redshift is compactly captured by the relation implemented here. By experimenting with different scenarios in this calculator, users can develop intuition about how various cosmological events translate into spectral imprints on the microwave background, offering guidance for future observational searches.