Frequency–Redshift Relation of the Cosmic Microwave Background - Ralf Hofmann; Janning Meinert (Astronomy, 2023)
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 <semantics>x=hνkBT</semantics>. Since the literature on extractions of the CMB’s T-z relation always assumes (i) <semantics>ν(z)=(1+z)ν(z=0)</semantics>, where <semantics>ν(z=0)</semantics> is the observed frequency in the heliocentric rest frame, the finding (ii) <semantics>T(z)=(1+z)T(z=0)</semantics> just confirms the expected blackbody nature of the interacting CMB at <semantics>z>0</semantics>. In contrast to the emission of isolated, directed radiation, whose frequency–redshift relation (<semantics>ν</semantics>-z relation) is subject to (i), a non-conventional <semantics>ν</semantics>-z relation <semantics>ν(z)=f(z)ν(z=0)</semantics> of pure, isotropic blackbody radiation, subject to adiabatically slow cosmic expansion, necessarily has to follow that of the T-z relation <semantics>T(z)=f(z)T(z=0)</semantics> and vice versa. In general, the function <semantics>f(z)</semantics> 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 <semantics>f(z)=<mfenced separators="" open="(" close=")">1/4</mfenced>1/3(1+z)</semantics> for <semantics>z≫1</semantics>, and <semantics>f(z)</semantics> is non-linear for <semantics>z∼1</semantics>.