Cosmic Microwave Background Spectral Distortions in a Thermodynamic R_H = ct Cosmology

1. Introduction

Planck introduced a system of natural units constructed from universal constants in 1899 [1] and developed the thermodynamics of blackbody radiation in his 1906 lectures [2]. In modern notation the Planck length, time, mass, and temperature are

$$\begin{array}{cccc}\hfill {\ell }_{\mathrm{P}}& =\sqrt{{\displaystyle \frac{\hslash G}{c^3}}},\hfill & \hfill t_{\mathrm{P}}& =\sqrt{{\displaystyle \frac{\hslash G}{c^5}}},\hfill \end{array}$$

(1)

$$\begin{array}{cccc}\hfill m_{\mathrm{P}}& =\sqrt{{\displaystyle \frac{\hslash c}{G}}},\hfill & \hfill T_{\mathrm{P}}& =\sqrt{{\displaystyle \frac{\hslash c^5}{G{k}_{\mathrm{B}}^2}}}.\hfill \end{array}$$

(2)

These quantities provide the microscopic scale in the CMB relation considered below.

Tatum et a [3]l proposed a heuristic connection between the CMB temperature . Haug and Wojnow [4] later derived the relation from the Stefan–Boltzmann law within a black-hole-inspired thermodynamic cosmology. Independently complementary interpretations were developed by Haug and Tatum [5], who related the CMB temperature to the geometric mean of a minimum Hawking–Hubble temperature and a maximum Planck-scale Hawking temperature. Haug [6] subsequently presented the particularly direct statement

$$T_{\mathrm{CMB}}=\sqrt{T_{min}{T}_{max}}$$

(3)

in a subsequent publication .

The resulting temperature formula is

$$T_{\mathrm{CMB}}={\displaystyle \frac{T_{\mathrm{P}}}{8\pi }}\sqrt{{\displaystyle \frac{2{\ell }_{\mathrm{P}}}{R_H}}},$$

(4)

where $R_H=c/H$. Haug has further argued that the corresponding photon radiation-density parameter is exactly

$${\Omega }_{\gamma }={\displaystyle \frac{1}{5760\pi }}\simeq 5.5262\times {10}^{-5}.$$

(5)

The goal of this paper is not to assume that Equations (4) and (5) the complete thermal history. Rather, we ask what additional physics they require and whether CMB spectral distortions can test that physics. This is a natural question because COBE/FIRAS measured a spectrum extremely close to a blackbody and constrained chemical-potential and Compton distortions to $\left|\mu \right|<9\times {10}^{-5}$ and $\left|y\right|<1.5\times {10}^{-5}$ at 95% confidence [7]. Modern distortion theory identifies the blackbody spectrum as an exceptionally sensitive record of energy and photon injection throughout cosmic history [8,9].

2. Which $R_H=ct$ Cosmology?

The notation $R_H=ct$ is used for more than one cosmological construction. Melia and Shevchuk’s well-known $R_h=ct$ universe is a spatially flat Friedmann–Robertson–Walker (FRW) cosmology whose total equation of state satisfies $\rho +3p=0$, or $w=-1/3$, giving $a\left(t\right)\propto t$ and $R_h=c/H=ct$ [10]. Its motivation, matter content, and perturbation framework are specific to that FRW model.

The present paper instead studies the thermodynamic, black-hole-inspired $R_H=ct$ construction used in the cited Haug–Tatum–Wojnow literature. It combines $R_H=ct$ with a critical Friedmann mass or Hubble-sphere black-hole analogy and a Planck-to-Hubble-scale temperature relation. It is therefore a different type, or subclass, of $R_H=ct$ model. The shared relation $R_H=ct$ does not make the two theories dynamically identical. In particular, none of the spectral-distortion conclusions below should automatically be attributed to Melia’s model unless Melia’s framework independently adopts Equation (4) and the same photon-production mechanism.

The redshift prescription is another essential difference. Haug and Tatum define [11]

$$1+z_{\mathrm{HT}}=\sqrt{{\displaystyle \frac{R_{H,0}}{R_H\left(t\right)}}}=\sqrt{{\displaystyle \frac{t_0}{t}}}=\sqrt{{\displaystyle \frac{H\left(t\right)}{H_0}}}.$$

(6)

This is not the ordinary FRW scale-factor redshift $1+z_a=a_0/a$. If $a\propto t$, the two labels satisfy

$$1+z_a={\displaystyle \frac{R_{H,0}}{R_H\left(t\right)}}={(1+z_{\mathrm{HT}})}^2.$$

(7)

3. Two Independent Routes to the Same Temperature Relation

3.1. Stefan–Boltzmann Route

For a blackbody photon gas the radiation energy density is

$$u_{\gamma }=a_{\mathrm{R}}{T}_{\gamma }^4,a_{\mathrm{R}}={\displaystyle \frac{{\pi }^2{k}_{\mathrm{B}}^4}{15{\hslash }^3{c}^3}}.$$

(8)

Haug and Wojnow’s construction applies this law to the energy assigned to the Hubble sphere and obtains Equation (4) [4]. The derivation is conditional on the cosmological energy assignment and on treating the radiation as blackbody radiation; the Stefan–Boltzmann law alone does not fix a cosmological temperature without those assumptions.

3.2. Geometric Mean of Limiting Hawking Temperatures

For a Schwarzschild black hole of radius $R_s=2GM/c^2$, the Hawking temperature is [12]

$$T_{\mathrm{Haw}}\left(R_s\right)={\displaystyle \frac{\hslash c}{4\pi k_{\mathrm{B}}{R}_s}}.$$

(9)

Haug and Tatum identify a minimum temperature with the Hubble-radius scale,

$$T_{min}=T_{\mathrm{Haw},\mathrm{H}}={\displaystyle \frac{\hslash c}{4\pi k_{\mathrm{B}}{R}_H}},$$

(10)

and a maximum temperature with a Planck-mass Schwarzschild radius $R_{s,\mathrm{P}}=2{\ell }_{\mathrm{P}}$,

$$T_{max}=T_{\mathrm{Haw},\mathrm{P}}={\displaystyle \frac{\hslash c}{8\pi k_{\mathrm{B}}{\ell }_{\mathrm{P}}}}={\displaystyle \frac{T_{\mathrm{P}}}{8\pi }}.$$

(11)

Their geometric mean gives

$$\begin{array}{cc}\hfill \sqrt{T_{min}{T}_{max}}& ={\displaystyle \frac{\hslash c}{4\sqrt{2}\pi k_{\mathrm{B}}\sqrt{{\ell }_{\mathrm{P}}{R}_H}}}\hfill \\ \hfill & ={\displaystyle \frac{T_{\mathrm{P}}}{8\pi }}\sqrt{{\displaystyle \frac{2{\ell }_{\mathrm{P}}}{R_H}}},\hfill \end{array}$$

(12)

which is Equation (4), see [6]. The numerical factors depend on the adopted circumference, Schwarzschild-radius, and Hawking-temperature conventions; they are part of the physical ansatz and should be stated explicitly.

4. The Radiation-Density Parameter

With $R_H=c/H$, Equation (4) gives

$$T_{\gamma }^4={\displaystyle \frac{T_{\mathrm{P}}^4{\ell }_{\mathrm{P}}^2}{1024{\pi }^4}}{\displaystyle \frac{H^2}{c^2}}.$$

(13)

Using the Planck definitions and Equation (8), one finds

$$u_{\gamma }={\displaystyle \frac{c^2{H}^2}{15360{\pi }^{2}G}}.$$

(14)

The critical mass density and critical energy density are

$${\rho }_{\mathrm{cr}}={\displaystyle \frac{3H^2}{8\pi G}},u_{\mathrm{cr}}={\rho }_{\mathrm{cr}}{c}^2.$$

(15)

Therefore

$${\Omega }_{\gamma }\equiv {\displaystyle \frac{u_{\gamma }}{u_{\mathrm{cr}}}}={\displaystyle \frac{1}{5760\pi }}.$$

(16)

This reproduces Haug’s claimed exact result [13]. It also clarifies its logical status: once Equation (4), the blackbody law, and the conventional critical density are adopted, Equation (16) follows algebraically. Thus ${\Omega }_{\gamma }$ is a useful consistency relation and normalization, but is not an independent prediction of photon kinetics.

5. Background Evolution and the Required Photon Source

For $R_H=ct$,

$$H={\displaystyle \frac{1}{t}},a\left(t\right)\propto t.$$

(17)

Equation (4) then implies

$$T_{\gamma }\propto t^{-1/2}\propto a^{-1/2},T_{\gamma }=T_0{(1+z_a)}^{1/2}=T_0(1+z_{\mathrm{HT}}),$$

(18)

where Equation (7) was used. Thus the temperature is nonlinear in the ordinary scale-factor redshift but linear in the Haug–Tatum redshift. This differs dynamically from the standard adiabatic law $T_{\gamma }\propto a^{-1}$ even though both theories can be written as $T=T_0(1+z)$ when their respective redshift variables are used.

For a Planck spectrum,

$$u_{\gamma }\propto T_{\gamma }^4\propto a^{-2},n_{\gamma }={\displaystyle \frac{2\zeta \left(3\right)}{{\pi }^2}}{\left({\displaystyle \frac{k_{\mathrm{B}}{T}_{\gamma }}{\hslash c}}\right)}^3\propto a^{-3/2}.$$

(19)

Consequently the photon number in a comoving volume grows as

$$N_{\gamma }^{\mathrm{com}}=a^3{n}_{\gamma }\propto a^{3/2}.$$

(20)

To express this departure from freely redshifting radiation, we use the zeroth and first energy moments of the photon Boltzmann equation. In a homogeneous and isotropic background there is no preferred spatial direction, so only scalar energy and number transfer rates are required; a momentum-transfer vector would vanish in the cosmic rest frame. For a collisionless photon gas, expansion alone gives ${\dot{u}}_{\gamma }+4H{u}_{\gamma }=0$ and ${\dot{n}}_{\gamma }+3H{n}_{\gamma }=0$. The factors $4H$ and $3H$ have different origins: photon energy density is diluted by the increase of volume and by the redshift of each photon, whereas number density is diluted only by volume expansion.

We therefore introduce homogeneous source terms $Q_{\gamma }$ and ${\Psi }_{\gamma }$ through

$$\begin{array}{cc}\hfill {\dot{u}}_{\gamma }+4H{u}_{\gamma }& =Q_{\gamma },\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill {\dot{n}}_{\gamma }+3H{n}_{\gamma }& ={\Psi }_{\gamma }.\hfill \end{array}$$

(22)

$Q_{\gamma }$ has units of energy per physical volume per unit cosmic time and represents the net energy transferred into the photon sector after emission, absorption, and scattering have been combined. Likewise, ${\Psi }_{\gamma }$ has units of photon number per physical volume per unit time and represents net photon production. Positive values denote injection or creation and negative values denote removal. These quantities are macroscopic moments of a frequency-dependent collision operator; specifying them does not yet specify the spectrum of the injected photons. Compton scattering, for example, can contribute to $Q_{\gamma }$ while giving no contribution to ${\Psi }_{\gamma }$, whereas double-Compton scattering and bremsstrahlung can change both moments.

It is also useful to define fractional source rates

$${\Gamma }_E\equiv {\displaystyle \frac{Q_{\gamma }}{u_{\gamma }}},{\Gamma }_N\equiv {\displaystyle \frac{{\Psi }_{\gamma }}{n_{\gamma }}}.$$

(23)

An arbitrary pair $({\Gamma }_E,{\Gamma }_N)$ does not preserve a blackbody. Since $u_{\gamma }\propto T^4$ and $n_{\gamma }\propto T^3$, an infinitesimal displacement from one Planck spectrum to another requires

$${\Gamma }_E={\displaystyle \frac{4}{3}}{\Gamma }_N.$$

(24)

If this relation fails after photon-production processes become inefficient, the excess energy relative to photon number appears as a chemical potential or a more general spectral distortion.

Using Equations (17) and (19) yields

$$Q_{\gamma }=2H{u}_{\gamma },{\Psi }_{\gamma }={\displaystyle \frac{3}{2}}H{n}_{\gamma }.$$

(25)

Thus ${\Gamma }_E=2H$ and ${\Gamma }_N=3H/2$, which satisfy Equation (24) exactly. Continuous photon creation or energy transfer is therefore not optional if the universe is to remain blackbody while obeying Equation (4), but the required background rates are thermodynamically compatible with a succession of Planck spectra. In covariant language $Q_{\gamma }$ is the time-like projection of an interaction four-vector $Q_{\gamma }^{\mu }$ in the radiation rest frame, while ${\Psi }_{\gamma }$ is the divergence of the photon-number current. Total stress–energy conservation requires the component supplying the photons to carry the opposite interaction four-vector. The remaining theoretical task is to identify that donor component and a microphysical collision process that produces Equation (25) with the required frequency dependence.

6. The $\mu$-Distortion Era

Spectral distortions arise when a process changes photon energy or number after complete thermalization becomes inefficient. Compton scattering redistributes energy but conserves photon number, whereas double-Compton scattering and bremsstrahlung change photon number. At very high redshift these interactions restore a Planck spectrum. At approximately $5\times {10}^4\lesssim z\lesssim 2\times {10}^6$ in the standard thermal history, Comptonization is efficient while photon creation is increasingly inefficient, producing a Bose–Einstein-like spectrum with chemical potential $\mu$. At lower redshift Comptonization itself becomes inefficient and energy release produces intermediate and y-type distortions [8,9]. The classic physical foundations include the work of Sunyaev and Zeldovich [14].

For small perturbations in the $\mu$ era, the chemical potential generated by changes in energy and photon number can be written approximately as

$$\mu \simeq 1.401\left({\displaystyle \frac{\Delta u_{\gamma }}{u_{\gamma }}}-{\displaystyle \frac{4}{3}}{\displaystyle \frac{\Delta n_{\gamma }}{n_{\gamma }}}\right){\mathcal{J}}_{\mu },$$

(26)

where ${\mathcal{J}}_{\mu }$ is a thermalization visibility factor. For pure energy release with negligible photon injection, this reduces to the familiar estimate $\mu \simeq 1.4\Delta u_{\gamma }/u_{\gamma }$, modulated by thermalization.

The background source demanded by Equation (25) obeys

$${\displaystyle \frac{Q_{\gamma }}{u_{\gamma }}}-{\displaystyle \frac{4}{3}}{\displaystyle \frac{{\Psi }_{\gamma }}{n_{\gamma }}}=2H-{\displaystyle \frac{4}{3}}\left({\displaystyle \frac{3}{2}}H\right)=0.$$

(27)

This is the thermodynamic adiabaticity condition for preserving a Planck spectrum: the mean photon energy and photon number change in precisely the ratio required to move from one blackbody to another. It provides a possible deeper interpretation of the temperature and density relations. Their consistency with the observed blackbody is conditional on the exact cancellation in Equation (27) being realized spectrally, not merely after frequency integration.

To parameterize departures, write

$${\displaystyle \frac{Q_{\gamma }}{u_{\gamma }}}=2H(1+{ϵ}_E),{\displaystyle \frac{{\Psi }_{\gamma }}{n_{\gamma }}}={\displaystyle \frac{3}{2}}H(1+{ϵ}_N).$$

(28)

Then the instantaneous nonadiabatic combination is

$${\displaystyle \frac{Q_{\gamma }}{u_{\gamma }}}-{\displaystyle \frac{4}{3}}{\displaystyle \frac{{\Psi }_{\gamma }}{n_{\gamma }}}=2H({ϵ}_E-{ϵ}_N).$$

(29)

A first-order estimate of the accumulated chemical potential is therefore

$$\mu \simeq 2.802\int ({ϵ}_E-{ϵ}_N){\mathcal{J}}_{\mu }\left(t\right)H\left(t\right)dt,$$

(30)

before including the full frequency-dependent Green function. The very small FIRAS limits imply that an order-unity source operating over a Hubble time cannot have a generic energy–number mismatch. Future sensitivity to $\mu$ near ${10}^{-8}$ would test the cancellation far more stringently.

7. Frequency-Dependent Kinetic Derivation

We now derive the previously unspecified source term at the homogeneous blackbody level. Let $n(\nu ,t)$ be the photon occupation number. In a spatially homogeneous and isotropic expanding universe its kinetic equation is

$${\displaystyle \frac{\partial n}{\partial t}}-H\nu {\displaystyle \frac{\partial n}{\partial \nu }}=C_{\mathrm{C}}\left[n\right]+C_{\mathrm{DC}}\left[n\right]+C_{\mathrm{BR}}\left[n\right]+C_{\mathrm{new}}\left[n\right],$$

(31)

where $C_{\mathrm{C}}$, $C_{\mathrm{DC}}$, and $C_{\mathrm{BR}}$ are the Compton, double-Compton, and bremsstrahlung collision terms. Define

$$x\equiv {\displaystyle \frac{h\nu }{k_{\mathrm{B}}{T}_{\gamma }\left(t\right)}},n_{\mathrm{Pl}}\left(x\right)={\displaystyle \frac{1}{e^x-1}}.$$

(32)

For the proposed law ${\dot{T}}_{\gamma }/T_{\gamma }=-H/2$. At fixed physical frequency,

$${\left.{\displaystyle \frac{\partial x}{\partial t}}\right|}_{\nu }=-x{\displaystyle \frac{{\dot{T}}_{\gamma }}{T_{\gamma }}}={\displaystyle \frac{H}{2}}x,\nu {\displaystyle \frac{\partial x}{\partial \nu }}=x.$$

(33)

Substitution of $n_{\mathrm{Pl}}\left(x\right)$ into the left-hand side of Equation (31) gives

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial n_{\mathrm{Pl}}}{\partial t}}-H\nu {\displaystyle \frac{\partial n_{\mathrm{Pl}}}{\partial \nu }}& =\left({\displaystyle \frac{H}{2}}x-Hx\right){\displaystyle \frac{\partial n_{\mathrm{Pl}}}{\partial x}}\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill & =-{\displaystyle \frac{H}{2}}x{\displaystyle \frac{\partial n_{\mathrm{Pl}}}{\partial x}}\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill & ={\displaystyle \frac{H}{2}}x{n}_{\mathrm{Pl}}(1+n_{\mathrm{Pl}}).\hfill \end{array}$$

(36)

Ordinary thermal collision terms vanish on a Planck spectrum when the electrons have the same temperature as the photons. The net new collision term required to make Equation (4) an exact blackbody solution is consequently

$$C_{\mathrm{new}}^{\left(0\right)}(x,t)={\displaystyle \frac{H\left(t\right)}{2}}x{n}_{\mathrm{Pl}}\left(x\right)\left[1+n_{\mathrm{Pl}}\left(x\right)\right].$$

(37)

Equation (37) is not merely a frequency-integrated constraint. It specifies the spectral shape of the net source on the background. Equivalently,

$$C_{\mathrm{new}}^{\left(0\right)}=-{\displaystyle \frac{H}{2}}{\displaystyle \frac{\partial n_{\mathrm{Pl}}}{\partial lnx}},$$

(38)

so the source continuously shifts the occupation function through frequency space at half the rate needed to compensate ordinary cosmological redshifting. Given the assumed $T\left(t\right)$ and the vanishing of the standard collision terms on equilibrium, this net background term is uniquely fixed by the kinetic equation. A microscopic theory may decompose it into emission, absorption, and scattering contributions in more than one way.

7.1. Covariant Effective Interaction

An explicit covariant effective realization is supplied by the “adiabatic” gravitational particle-creation formalism. This terminology means constant entropy per photon, not constant entropy in a comoving volume. Lima, Trevisani, and Santos showed that its relativistic Boltzmann equation preserves a Planck spectrum while modifying the temperature–redshift law [15]. Let $u^{\mu }$ be the cosmological four-velocity, $h^{\mu \nu }=g^{\mu \nu }+u^{\mu }{u}^{\nu }$ the spatial projector, and

$$N_{\gamma }^{\mu }=n_{\gamma }{u}^{\mu },T_{\gamma }^{\mu \nu }=u_{\gamma }{u}^{\mu }{u}^{\nu }+{\displaystyle \frac{u_{\gamma }}{3}}{h}^{\mu \nu }.$$

(39)

Introduce the scalar photon-creation rate ${\Gamma }_{\gamma }$ through

$${\nabla }_{\mu }{N}_{\gamma }^{\mu }=n_{\gamma }{\Gamma }_{\gamma }.$$

(40)

Constant specific entropy implies the covariant energy-transfer law

$${\nabla }_{\mu }{T}_{\gamma }^{\mu \nu }=Q_{\gamma }^{\nu },Q_{\gamma }^{\nu }={\displaystyle \frac{4}{3}}{u}_{\gamma }{\Gamma }_{\gamma }{u}^{\nu },$$

(41)

with no momentum transfer in the radiation rest frame. The scalar projections of Equations (40) and (41) are

$$\begin{array}{cc}\hfill {\dot{n}}_{\gamma }+3H{n}_{\gamma }& =n_{\gamma }{\Gamma }_{\gamma },\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill {\dot{u}}_{\gamma }+4H{u}_{\gamma }& ={\displaystyle \frac{4}{3}}{u}_{\gamma }{\Gamma }_{\gamma }.\hfill \end{array}$$

(43)

Since $n_{\gamma }\propto T_{\gamma }^3$, either equation gives

$${\displaystyle \frac{{\dot{T}}_{\gamma }}{T_{\gamma }}}=-H+{\displaystyle \frac{{\Gamma }_{\gamma }}{3}}.$$

(44)

The required ${\dot{T}}_{\gamma }/T_{\gamma }=-H/2$ is obtained for

$${\Gamma }_{\gamma }={\displaystyle \frac{3}{2}}H.$$

(45)

The corresponding gravitationally modified Liouville operator may be written

$${\displaystyle \frac{\partial n}{\partial t}}-H\nu {\displaystyle \frac{\partial n}{\partial \nu }}=-{\displaystyle \frac{{\Gamma }_{\gamma }}{3}}\nu {\displaystyle \frac{\partial n}{\partial \nu }}+C_{\mathrm{micro}}\left[n\right].$$

(46)

On setting ${\Gamma }_{\gamma }=3H/2$ and $n=n_{\mathrm{Pl}}$, the first term on the right is exactly Equation (37). Equations (40)–(46) therefore give a covariant phenomenological interaction that realizes the required spectrum. They do not constitute a fundamental quantum-field interaction or specify which sector supplies the energy. Total conservation requires another component X satisfying

$${\nabla }_{\mu }{T}_X^{\mu \nu }=-Q_{\gamma }^{\nu }.$$

(47)

Thus the construction is covariantly closed only after a donor sector and its stress–energy are supplied.

7.2. Carnot-Engine Interpretation of the Photon Source

Haug has proposed that an extremal Reissner–Nordström black-hole Hubble sphere can be interpreted as a reversible Carnot engine operating between a maximum Planck-scale Hawking temperature and a minimum Hubble-scale Hawking temperature [16]. The proposal is presently a working-paper hypothesis rather than a peer-reviewed microscopic theory. Nevertheless, it supplies a useful macroscopic interpretation of the source terms derived above. The limiting temperatures are

$$T_{max}=T_{\mathrm{Haw},\mathrm{P}},T_{min}=T_{\mathrm{Haw},\mathrm{H}}\propto R_H^{-1},$$

(48)

and the proposed reversible intermediate temperature is

$$T_{\gamma }=\sqrt{T_{max}{T}_{min}}\propto R_H^{-1/2}.$$

(49)

Thus the Carnot construction reproduces the temperature scaling required by Equation (4), but the stronger question is whether it also reproduces the required heat, entropy, and photon-number flows.

Consider a comoving volume $V\propto a^3$. Since $u_{\gamma }\propto a^{-2}$,

$$U_{\gamma }=u_{\gamma }V\propto a,{\dot{U}}_{\gamma }=H{U}_{\gamma }.$$

(50)

For radiation $p_{\gamma }=u_{\gamma }/3$ and $\dot{V}=3HV$, so

$$p_{\gamma }\dot{V}=H{U}_{\gamma }.$$

(51)

The first law for the CMB working fluid therefore requires

$${\dot{Q}}_{\mathrm{CMB}}={\dot{U}}_{\gamma }+p_{\gamma }\dot{V}=2H{U}_{\gamma }.$$

(52)

Dividing by V gives ${\dot{Q}}_{\mathrm{CMB}}/V=2H{u}_{\gamma }$, exactly the energy source $Q_{\gamma }$ in Equation (25). The Carnot picture can consequently identify the otherwise abstract donor term with heat transferred from the horizon-scale engine to the CMB working fluid.

The blackbody entropy in the same volume is

$$S_{\gamma }={\displaystyle \frac{4U_{\gamma }}{3T_{\gamma }}}\propto a^{3/2},{\dot{S}}_{\gamma }={\displaystyle \frac{3}{2}}H{S}_{\gamma }.$$

(53)

Using Equation (52),

$${\displaystyle \frac{{\dot{Q}}_{\mathrm{CMB}}}{T_{\gamma }}}={\displaystyle \frac{2H{U}_{\gamma }}{T_{\gamma }}}={\displaystyle \frac{3}{2}}H{S}_{\gamma }={\dot{S}}_{\gamma }.$$

(54)

Moreover, Equation (20) gives

$${\dot{N}}_{\gamma }={\displaystyle \frac{3}{2}}H{N}_{\gamma },{\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{S_{\gamma }}{N_{\gamma }}}\right)=0.$$

(55)

The Carnot heat flow is therefore compatible with adiabatic photon creation in the precise sense of constant entropy per photon.

The required mean heat supplied per created photon follows without further assumptions:

$${\displaystyle \frac{{\dot{Q}}_{\mathrm{CMB}}}{{\dot{N}}_{\gamma }}}={\displaystyle \frac{4}{3}}{\displaystyle \frac{U_{\gamma }}{N_{\gamma }}}.$$

(56)

For a Planck spectrum,

$${\displaystyle \frac{U_{\gamma }}{N_{\gamma }}}={\displaystyle \frac{{\pi }^4}{30\zeta \left(3\right)}}{k}_{\mathrm{B}}{T}_{\gamma }\simeq 2.701k_{\mathrm{B}}{T}_{\gamma },$$

(57)

and hence

$${\displaystyle \frac{{\dot{Q}}_{\mathrm{CMB}}}{{\dot{N}}_{\gamma }}}={\displaystyle \frac{2{\pi }^4}{45\zeta \left(3\right)}}{k}_{\mathrm{B}}{T}_{\gamma }\simeq 3.601k_{\mathrm{B}}{T}_{\gamma }.$$

(58)

This is a quantitative constraint on any microscopic implementation of the Carnot analogy. Matching only the total heat flow is insufficient: created photons must collectively realize the full spectral operator Equation (37), whose moments yield both Equations (52) and (55).

There is also an important entropy condition. The CMB entropy increases according to Equation (53), whereas a reversible Carnot engine has zero total entropy production. Compatibility therefore requires a compensating reservoir or horizon entropy current,

$${\dot{S}}_{\mathrm{res}}=-{\dot{S}}_{\gamma },{\dot{S}}_{\mathrm{total}}={\dot{S}}_{\gamma }+{\dot{S}}_{\mathrm{res}}=0.$$

(59)

Haug’s proposed zero net entropy change for the extremal Hubble-sphere engine could play this role, but the working paper does not derive the necessary local entropy current, reservoir stress tensor, transition amplitudes, or frequency-dependent photon emissivity. The combined interpretation is therefore: the Carnot model supplies a candidate macroscopic heat source; adiabatic creation supplies the covariant number and entropy balance; and Equation (37) specifies the spectrum that a fundamental interaction must generate.

7.3. Number and Energy Moments

The photon number and energy densities are

$$\begin{array}{cc}\hfill n_{\gamma }& ={\displaystyle \frac{8\pi }{c^3}}{\int }_0^{\infty }{\nu }^{2}n(\nu ,t)d\nu ,\hfill \end{array}$$

(60)

$$\begin{array}{cc}\hfill u_{\gamma }& ={\displaystyle \frac{8\pi h}{c^3}}{\int }_0^{\infty }{\nu }^{3}n(\nu ,t)d\nu .\hfill \end{array}$$

(61)

The number source generated by Equation (37) is

$$\begin{array}{cc}\hfill {\Psi }_{\gamma }& ={\displaystyle \frac{8\pi }{c^3}}{\int }_0^{\infty }{\nu }^2{C}_{\mathrm{new}}^{\left(0\right)}d\nu \hfill \end{array}$$

(62)

$$\begin{array}{cc}\hfill & ={\displaystyle \frac{H}{2}}{\displaystyle \frac{8\pi }{c^3}}{\left({\displaystyle \frac{k_{\mathrm{B}}{T}_{\gamma }}{h}}\right)}^3{\int }_0^{\infty }{x}^3{n}_{\mathrm{Pl}}(1+n_{\mathrm{Pl}})dx.\hfill \end{array}$$

(63)

Since $n_{\mathrm{Pl}}(1+n_{\mathrm{Pl}})=-{\partial }_x{n}_{\mathrm{Pl}}$, integration by parts gives

$${\int }_0^{\infty }{x}^3{n}_{\mathrm{Pl}}(1+n_{\mathrm{Pl}})dx=3{\int }_0^{\infty }{x}^2{n}_{\mathrm{Pl}}dx.$$

(64)

Hence

$${\Psi }_{\gamma }={\displaystyle \frac{3}{2}}H{n}_{\gamma }.$$

(65)

The energy moment similarly obeys

$${\int }_0^{\infty }{x}^4{n}_{\mathrm{Pl}}(1+n_{\mathrm{Pl}})dx=4{\int }_0^{\infty }{x}^3{n}_{\mathrm{Pl}}dx,$$

(66)

and therefore

$$Q_{\gamma }=2H{u}_{\gamma }.$$

(67)

Thus the explicit frequency-dependent source reproduces both background continuity equations rather than satisfying only one chosen moment.

7.4. Linearized Distortion Equation

Write the occupation number and new collision term as

$$n=n_{\mathrm{Pl}}+\Delta n,C_{\mathrm{new}}=C_{\mathrm{new}}^{\left(0\right)}+\Delta C_{\mathrm{new}}.$$

(68)

Subtracting the exact background equation yields

$${\displaystyle \frac{\partial \Delta n}{\partial t}}-H\nu {\displaystyle \frac{\partial \Delta n}{\partial \nu }}={\mathcal{L}}_{\mathrm{C}}[\Delta n]+{\mathcal{L}}_{\mathrm{DC}}[\Delta n]+{\mathcal{L}}_{\mathrm{BR}}[\Delta n]+\Delta C_{\mathrm{new}},$$

(69)

where the $\mathcal{L}$ operators are the standard collision terms linearized about the Planck distribution. A small frequency-independent chemical potential has

$$n_{\mathrm{BE}}\left(x\right)={\displaystyle \frac{1}{e^{x+\mu }-1}},\Delta n_{\mu }=-\mu n_{\mathrm{Pl}}(1+n_{\mathrm{Pl}})+\mathcal{O}\left({\mu }^2\right).$$

(70)

Taking the energy and number moments of Equation (69) gives the model-independent source combination

$$\dot{\mu }+{\Gamma }_{\mu }\mu \simeq 1.401\left({\displaystyle \frac{\Delta Q_{\gamma }}{u_{\gamma }}}-{\displaystyle \frac{4}{3}}{\displaystyle \frac{\Delta {\Psi }_{\gamma }}{n_{\gamma }}}\right),$$

(71)

where ${\Gamma }_{\mu }$ summarizes the chemical-potential relaxation produced mainly by double-Compton and bremsstrahlung processes. The solution observed at a later time $t_0$ is

$$\mu \left(t_0\right)\simeq 1.401{\int }^{t_0}dt\left({\displaystyle \frac{\Delta Q_{\gamma }}{u_{\gamma }}}-{\displaystyle \frac{4}{3}}{\displaystyle \frac{\Delta {\Psi }_{\gamma }}{n_{\gamma }}}\right)exp\left[-{\int }_t^{t_0}{\Gamma }_{\mu }\left(t^{\prime }\right)d{t}^{\prime }\right].$$

(72)

The exponential is the $\mu$-visibility function in time-domain form. If the fractional energy and number departures are parameterized by ${ϵ}_E$ and ${ϵ}_N$ as above, Equation (72) reduces to Equation (30). The exact background source Equation (37) has $\Delta C_{\mathrm{new}}=0$ and produces no distortion. Any microscopic source whose spectrum differs from Equation (37) generally has a nonzero projection onto the $\mu$ shape and residual distortion eigenmodes.

7.5. Implicit Boundaries of the Distortion Eras

The often quoted standard $\mu$-era redshift interval is not fundamental; it is obtained by comparing microscopic rates with the expansion rate. A self-consistent calculation in the present model must evaluate

$${\mathcal{R}}_{\mathrm{C}}\left(T\right)={\displaystyle \frac{{\Gamma }_{\mathrm{C}}(T,n_e)}{H\left(T\right)}},{\mathcal{R}}_{\mathrm{DC}}\left(T\right)={\displaystyle \frac{{\Gamma }_{\mathrm{DC}}(T,n_e)}{H\left(T\right)}},{\mathcal{R}}_{\mathrm{BR}}\left(T\right)={\displaystyle \frac{{\Gamma }_{\mathrm{BR}}(T,n_e)}{H\left(T\right)}}.$$

(73)

The temperature law itself gives the exact expansion–temperature relation

$$H\left(T\right)={\displaystyle \frac{32{\pi }^{2}c}{T_{\mathrm{P}}^2{\ell }_{\mathrm{P}}}}{T}^2=H_0{\left({\displaystyle \frac{T}{T_0}}\right)}^2.$$

(74)

The end of complete thermalization is determined implicitly by the failure of photon-number relaxation, approximately ${\Gamma }_{\mu }\left(T_{\mathrm{th}}\right)\simeq H\left(T_{\mathrm{th}}\right)$, while the transition from $\mu$-type to intermediate or y-type evolution is determined by inefficient Comptonization, approximately ${\Gamma }_{\mathrm{C}}\left(T_{\mu y}\right)\simeq H\left(T_{\mu y}\right)$. Their corresponding Haug–Tatum redshifts are

$$1+z_{\mathrm{HT},\mathrm{th}}={\displaystyle \frac{T_{\mathrm{th}}}{T_0}},1+z_{\mathrm{HT},\mu y}={\displaystyle \frac{T_{\mu y}}{T_0}}.$$

(75)

The associated scale-factor labels are their squares, $1+z_a={(1+z_{\mathrm{HT}})}^2$. Numerical values cannot be obtained from Equation (4) alone because the rates require $n_e\left(T\right)$, the baryon abundance, the ionization history, and the electron temperature. Since Equation (20) makes the comoving photon number time dependent, even conserved baryon number implies a time-dependent baryon-to-photon ratio. Big-bang nucleosynthesis, recombination, and the cosmological recombination radiation are therefore linked tests of the same source mechanism.

7.6. Numerical Benchmark with Constant Baryon-to-Photon Ratio

To make the rate calculation definite, consider a benchmark closure rather than leaving $n_e\left(T\right)$ unspecified. We assume: (i) constant $\eta \equiv n_b/n_{\gamma }=6.10\times {10}^{-10}$; (ii) primordial helium mass fraction $Y_p=0.245$; (iii) complete ionization over the temperatures relevant below; (iv) $T_e=T_{\gamma }$; (v) $H_0=66.8943\pm 0.0287\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$, the value reported by Haug and Tatum from their PantheonPlusSH0ES analysis [17], and $T_0=2.7255\mathrm{K}$. The numerical calculation uses the central $H_0$ value; its quoted statistical uncertainty is negligible compared with the approximations in the rate model. Constant $\eta$ requires baryons to share the creation rate ${\Gamma }_b=3H/2$; this is an additional physical assumption, not a consequence of the CMB formula. It gives

$$n_e\left(T\right)=\left(1-{\displaystyle \frac{Y_p}{2}}\right)\eta {\displaystyle \frac{2\zeta \left(3\right)}{{\pi }^2}}{\left({\displaystyle \frac{k_{\mathrm{B}}T}{\hslash c}}\right)}^3.$$

(76)

Define $\theta =k_{B}T/\left(m_e{c}^2\right)$ and the Thomson rate $\dot{\tau }=n_e{\sigma }_{T}c$. The nonrelativistic Comptonization rate used here is

$${\Gamma }_{\mathrm{C}}=4\theta \dot{\tau }.$$

(77)

For double-Compton and bremsstrahlung we use the low-frequency emission coefficients summarized by Chluba [18],

$$\begin{array}{cc}\hfill {\Lambda }_{\mathrm{DC}}& ={\displaystyle \frac{4\alpha }{3\pi }}{\theta }^2{\displaystyle \frac{4{\pi }^4/15}{1+14.16\theta }},\hfill \end{array}$$

(78)

$$\begin{array}{cc}\hfill {\Lambda }_{\mathrm{BR}}& ={\displaystyle \frac{\alpha {\lambda }_e^3}{2\pi \sqrt{6\pi }}}{\theta }^{-7/2}\sum _i{Z}_i^2{n}_i{g}_{\mathrm{ff}},\hfill \end{array}$$

(79)

where ${\lambda }_e=h/\left(m_{e}c\right)$ and the numerical benchmark takes $g_{\mathrm{ff}}=1$ and ${\sum }_i{Z}_i^2{n}_i\simeq n_e$. These coefficients enter the collision term per unit Thomson optical depth; the corresponding coefficient rates are $\dot{\tau }{\Lambda }_{\mathrm{DC}}$ and $\dot{\tau }{\Lambda }_{\mathrm{BR}}$. They should not individually be mistaken for the full chemical-potential relaxation rate, which also depends on Compton transport through frequency space.

The numerical rates are shown in Table 1. The redshift column uses the Haug–Tatum relation $1+z_{\mathrm{HT}}=T/T_0$. For reference, $\left(\dot{\tau }{\Lambda }_{\mathrm{BR}}\right)/H$ is respectively $1.56\times {10}^{-7}$, $4.94\times {10}^{-7}$, $1.56\times {10}^{-6}$, and $4.94\times {10}^{-6}$ in these four rows. Double-Compton emission therefore dominates the benchmark chemical-potential thermalization at high temperature, while bremsstrahlung remains important primarily at low dimensionless frequency because its collision term scales as $x^{-3}$.

For a more meaningful era boundary, integrate the Comptonization depth using $dt=-2dT/\left(HT\right)$:

$$y_{\gamma }\left(T\right)=2{\int }_{T_0}^T{\displaystyle \frac{\theta \left(T^{\prime }\right)n_e\left(T^{\prime }\right){\sigma }_{T}c}{H\left(T^{\prime }\right)}}{\displaystyle \frac{d{T}^{\prime }}{T^{\prime }}}.$$

(80)

The criterion $y_{\gamma }=1$ gives

$$T_{\mu y}\simeq 2.83\times {10}^6\mathrm{K},z_{\mathrm{HT},\mu y}\simeq 1.04\times {10}^6.$$

(81)

To lowest order, the chemical-potential optical depth is

$${\tau }_{\mu }\left(T\right)=2{\gamma }_N{\int }_{T_0}^T{\displaystyle \frac{\theta x_c{n}_e{\sigma }_{T}c}{H}}{\displaystyle \frac{d{T}^{\prime }}{T^{\prime }}},x_c=\sqrt{{\displaystyle \frac{{\Lambda }_{\mathrm{DC}}+{\Lambda }_{\mathrm{BR}}}{\theta }}},$$

(82)

with ${\gamma }_N\simeq 0.772$. Numerical quadrature of Equation (82) gives ${\tau }_{\mu }=1$ at

$$T_{\mathrm{th}}\simeq 2.64\times {10}^7\mathrm{K},z_{\mathrm{HT},\mathrm{th}}\simeq 9.70\times {10}^6.$$

(83)

Under this explicit benchmark, the approximate distortion eras are therefore

$$\begin{array}{cccc}\hfill T& \gtrsim 2.64\times {10}^7\mathrm{K}:\hfill & \hfill & \mathrm{efficient}\mathrm{blackbody}\mathrm{restoration},\hfill \end{array}$$

(84)

$$\begin{array}{cccc}\hfill 2.83\times {10}^6\mathrm{K}\lesssim T& \lesssim 2.64\times {10}^7\mathrm{K}:\hfill & \hfill & \mu \text{-}\mathrm{dominated}\mathrm{evolution},\hfill \end{array}$$

(85)

$$\begin{array}{cccc}\hfill T& \lesssim 2.83\times {10}^6\mathrm{K}:\hfill & \hfill & \mathrm{intermediate}/y\text{-}\mathrm{type}\mathrm{evolution}.\hfill \end{array}$$

(86)

The values are leading-order estimates, not precision CosmoTherm outputs. Relativistic corrections, accurate Gaunt factors, helium charge states, pairs, and a self-consistent recombination calculation will shift them.

The alternative closure of conserved baryon number gives $n_b\propto a^{-3}\propto T^6$ and hence $\eta \propto T^3$. It cannot be combined with complete ionization down to the instantaneous Compton crossing: a formal fully ionized estimate gives $T\simeq 5.3\times {10}^2$K, where the plasma would actually be neutral. This failure demonstrates why a numerical answer is not unique until the baryon creation and ionization sectors are specified. The constant-$\eta$ benchmark is internally usable over the hot, fully ionized era but commits the model to continuous baryon as well as photon production.

8. Other Observable Consequences

The same framework suggests several tests beyond a single $\mu$ parameter.

• Residual distortions. A source that satisfies the integrated cancellation Equation (27) can still have a non-Planckian injection spectrum. The residual frequency dependence may distinguish it from ordinary heating.
• y and intermediate distortions. If the source persists after efficient Comptonization ends, its low-redshift contribution will not relax to a Bose–Einstein form and should generate intermediate or y-like signals.
• Cosmological recombination radiation. The modified $T\left(z\right)$ law and photon-to-baryon evolution alter atomic transition rates and the mapping of emitted line frequencies to the present spectrum.
• Acoustic damping. The standard prediction $\mu \sim 2\times {10}^{-8}$ from dissipation of small-scale perturbations depends on the expansion, sound horizon, diffusion scale, and primordial power spectrum. Each must be recomputed rather than transplanted from $\Lambda$CDM.
• Temperature–redshift observations. In Haug–Tatum variables the prediction is $T_{\gamma }=T_0(1+z_{\mathrm{HT}})$, while in scale-factor variables it is $T_{\gamma }=T_0{(1+z_a)}^{1/2}$. Tests must specify which operational redshift is assigned to the observations and simultaneously test the nonstandard mapping Equation (7).

9. Discussion

The exact cancellation in Equation (27) is the main conceptual result. It shows that the unusual scaling $T_{\gamma }\propto a^{-1/2}$ need not automatically create a homogeneous $\mu$ distortion if photons are produced with exactly the blackbody-preserving energy-to-number ratio. Conversely, it exposes a strong requirement that is hidden when one considers only the present temperature or ${\Omega }_{\gamma }$.

The result should not be interpreted as a completed microphysical derivation. Equation (37) determines the net source spectrum required on the homogeneous Planck background, but it does not identify the particles, fields, or interactions that generate that collision term. The near-perfect FIRAS blackbody constrains departures $\Delta C_{\mathrm{new}}$ from this required form. A compelling thermodynamic $R_H=ct$ theory would need to identify the source sector, show conservation of total stress–energy, calculate its fluctuations, and fit spectral, anisotropy, nucleosynthesis, and recombination data jointly.

The distinction from Melia’s model is also essential. Melia’s $R_h=ct$ cosmology supplies a particular FRW dynamics and linear scale factor [10]; the Haug–Tatum–Wojnow construction studied here adds black-hole and Planck-scale thermodynamic identifications. Agreement on $R_H=ct$ alone does not entail agreement on $T\left(z\right)$, photon production, or spectral distortions.

10. Conclusions

Within the stated thermodynamic assumptions, the CMB temperature relation and the Stefan–Boltzmann law yield ${\Omega }_{\gamma }=1/\left(5760\pi \right)$. This compact result can deepen the analysis of CMB spectral distortions by revealing the source terms needed to maintain it. In a linearly expanding universe the temperature law requires

$$T_{\gamma }\propto a^{-1/2},Q_{\gamma }=2H{u}_{\gamma },{\Psi }_{\gamma }={\displaystyle \frac{3}{2}}H{n}_{\gamma }.$$

(87)

These rates obey the blackbody-preserving energy–number relation at the homogeneous level. The kinetic equation fixes the required net background source to Equation (37), whose number and energy moments reproduce both rates exactly. The covariant adiabatic-creation model Equations (40) and (46) supplies an effective realization with ${\Gamma }_{\gamma }=3H/2$, although a fundamental donor interaction remains to be identified. Using Haug and Tatum’s value of $H_0$, the explicit constant-$\eta$ benchmark gives $T_{\mu y}\simeq 2.83\times {10}^6$K and $T_{\mathrm{th}}\simeq 2.64\times {10}^7$K, corresponding to Haug–Tatum redshifts $z_{\mathrm{HT},\mu y}\simeq 1.04\times {10}^6$ and $z_{\mathrm{HT},\mathrm{th}}\simeq 9.70\times {10}^6$. Spectral distortions therefore do not immediately rule out the background formula, but they demand a highly specific photon source and a stated baryon-creation closure. Any mismatch $\Delta C_{\mathrm{new}}$ is propagated by Equation (69) into a calculable $\mu$-, residual-, or y-type signal. Precision predictions now require a fundamental donor sector, accurate atomic and plasma rates, and a numerical solution of the linearized kinetic equation.