Frequency–Redshift Relation of the Cosmic Microwave Background

We point out that a modified temperature–redshift relation (T-z relation) of the cosmic microwave background (CMB) cannot be deduced by any observational method that appeals to an a priori thermalisation to the CMB temperature T of the excited states in a probe environment of independently determined redshift z. For example, this applies to quasar-light absorption by a damped Lyman-alpha system due to atomic as well as ionic fine-splitting transitions or molecular rotational bands. Similarly, the thermal Sunyaev-Zel’dovich (thSZ) effect cannot be used to extract the CMB’s T-z relation. This is because the relative line strengths between ground and excited states in the former and the CMB spectral distortion in the latter case both depend, apart from environment-specific normalisations, solely on the dimensionless spectral variable $x=\frac{h\nu }{k_{B}T}$. Since the literature on extractions of the CMB’s T-z relation always assumes (i) $\nu \left(z\right)=(1+z)\nu (z=0)$, where $\nu (z=0)$ is the observed frequency in the heliocentric rest frame, the finding (ii) $T\left(z\right)=(1+z)T(z=0)$ just confirms the expected blackbody nature of the interacting CMB at $z>0$. In contrast to the emission of isolated, directed radiation, whose frequency–redshift relation ($\nu$-z relation) is subject to (i), a non-conventional $\nu$-z relation $\nu \left(z\right)=f\left(z\right)\nu (z=0)$ of pure, isotropic blackbody radiation, subject to adiabatically slow cosmic expansion, necessarily has to follow that of the T-z relation $T\left(z\right)=f\left(z\right)T(z=0)$ and vice versa. In general, the function $f\left(z\right)$ is determined by the energy conservation of the CMB fluid in a Friedmann–Lemaitre–Robertson–Walker universe. If the pure CMB is subject to an SU(2) rather than a U(1) gauge principle, then $f\left(z\right)={\left(1/4\right)}^{1/3}(1+z)$ for $z\gg 1$, and $f\left(z\right)$ is non-linear for $z\sim 1$.