The CMB Hubble Sphere Flux

1. Introduction

The black-body radiation problem played a decisive role in the birth of quantum theory. Planck’s radiation law gives the spectral radiance of a black body at temperature T. In frequency form it is

$$B_{\nu }\left(T\right)={\displaystyle \frac{2h{\nu }^3}{c^2}}{\displaystyle \frac{1}{exp\left(h\nu /k_{B}T\right)-1}},$$

(3)

where h is Planck’s constant, $\nu$ is frequency, c is the speed of light, and $k_B$ is Boltzmann’s constant. In wavelength form it can be written as

$$B_{\lambda }\left(T\right)={\displaystyle \frac{2h{c}^2}{{\lambda }^5}}{\displaystyle \frac{1}{exp\left(hc/\left(\lambda k_{B}T\right)\right)-1}}.$$

(4)

Planck’s law gives the full spectral distribution. The Stefan–Boltzmann law is obtained by integrating Planck’s spectrum over all frequencies and over the outward hemisphere.

The quantity commonly written as $\sigma T^4$ is, in radiometry, the radiant exitance. We denote radiant exitance by $M_{\mathrm{e}}$, where the subscript “e” indicates an emitted radiometric quantity. Thus $M_{\mathrm{e}}$ is not a mass; it is the total power emitted per unit surface area into the outward hemisphere. In astrophysics and radiative transfer, the same quantity is often denoted by F and called the emitted radiative flux. Thus, in this paper we use

$$F\equiv M_{\mathrm{e}}=\sigma T^4,$$

(5)

where F and $M_{\mathrm{e}}$ both refer to the black-body radiant exitance, or one-sided emitted radiative flux.

This distinction is useful for the CMB. Locally, the observed CMB is an almost isotropic radiation field. Its net vector flux is therefore approximately zero, because radiation arrives almost equally from all directions. Nevertheless, the CMB temperature defines a black-body energy density and a corresponding one-sided black-body-equivalent flux. That flux, rather than the net vector flux of the observed sky, is the quantity studied here.

The flux law is important because it translates temperature into energy flow. In ordinary thermal physics, it tells us how rapidly a black body radiates energy per unit area. In stellar astrophysics, it connects a star’s effective temperature to its luminosity. In cosmology, when the CMB is treated as a black-body field, it connects the CMB temperature to a characteristic radiation energy density and to a black-body-equivalent flux scale. This provides a compact way to compare Planck-scale and Hubble-scale quantities.

In this note we examine the CMB black-body-equivalent flux inside an assumed $R_H=ct$ cosmological setting, where the Hubble radius is

$$R_H=ct.$$

(6)

Tatum et al [4] heuristically suggested the CMB temperature was given by

$$T_{\mathrm{cmb}}={\displaystyle \frac{\hslash c}{k_{B}4\pi \sqrt{2R_H{l}_p}}}$$

(7)

Haug and Wojnow derived this CMB-temperature relation from the Stefan–Boltzmann law using the Hubble parameter and the Planck scale [5]. This is naturally connected to the observed near-black-body character of the CMB, since the Stefan–Boltzmann law is a black-body radiation law.

The main contribution of the present paper is not the temperature relation itself, but the explicit Hubble-sphere flux obtained from it:

$$F_{\mathrm{cmb}}=\sigma T_{\mathrm{cmb}}^4={\displaystyle \frac{\hslash c^2}{61440{\pi }^2{R}_H^2{l}_p^2}}.$$

(8)

This result can also be written in the geometric-mean form

$$F_{\mathrm{cmb}}=\sigma T_{max}^2{T}_{min}^2,$$

(9)

where

$$T_{max}={\displaystyle \frac{\hslash c}{k_{B}8\pi l_p}},T_{min}={\displaystyle \frac{\hslash c}{k_{B}4\pi R_H}}.$$

(10)

The flux scales as $R_H^{-2}$, or as $t^{-2}$ when $R_H=ct$.

The associated Hubble-sphere black-body-equivalent luminosity is obtained only after multiplying the flux by the Hubble-sphere area. This gives $4\pi R_H^2{F}_{\mathrm{cmb}}=c^5/\left(15360\pi G\right)$, a fixed fraction of the Planck power. This same constant luminosity agrees algebraically with the CMB/internal black-hole luminosity discussed by Haug and Wojnow when the Schwarzschild radius is identified with the Hubble radius, $R_s=R_H$ [6]. The present paper differs in emphasis: it derives the flux first and treats the luminosity as an area-integrated consequence.

Haug and Tatum and related works have discussed geometric-mean CMB-temperature relations connecting minimum and maximum temperatures in the Hubble sphere [7,8]. In addition, Haug has recently derived an exact CMB photon radiation density and a CMB photon number-density relation from $R_H=ct$ cosmology [10]. We therefore also clarify the distinction between the flux $F_{\mathrm{cmb}}$, the radiation energy density $u_{\mathrm{cmb}}$, the photon number density $n_{\gamma }$, and the dimensionless photon radiation-density parameter ${\Omega }_{\gamma }$. These quantities are related, but they are not interchangeable.

2. From Planck’s Law to the Stefan–Boltzmann Flux Law

Planck’s law gives the spectral radiance. The total black-body emitted flux is obtained by integrating radiance over wavelength and over the outward hemisphere:

$$F={\int }_0^{\infty }{\int }_{\mathrm{hemisphere}}{B}_{\lambda }\left(T\right)cos\theta d\Omega d\lambda .$$

(11)

For an isotropic black-body surface, the angular integral is

$${\int }_{\mathrm{hemisphere}}cos\theta d\Omega =\pi .$$

(12)

Therefore

$$F=\pi {\int }_0^{\infty }{B}_{\lambda }\left(T\right)d\lambda .$$

(13)

Substituting Planck’s law in wavelength form gives

$$F=\pi {\int }_0^{\infty }{\displaystyle \frac{2h{c}^2}{{\lambda }^5}}{\displaystyle \frac{1}{exp\left(hc/\left(\lambda k_{B}T\right)\right)-1}}d\lambda .$$

(14)

With the substitution

$$x={\displaystyle \frac{hc}{\lambda k_{B}T}},$$

(15)

the integral becomes

$$F={\displaystyle \frac{2\pi k_B^4{T}^4}{h^3{c}^2}}{\int }_0^{\infty }{\displaystyle \frac{x^3}{e^x-1}}dx.$$

(16)

The standard integral is

$${\int }_0^{\infty }{\displaystyle \frac{x^3}{e^x-1}}dx={\displaystyle \frac{{\pi }^4}{15}}.$$

(17)

Thus

$$F={\displaystyle \frac{2{\pi }^5{k}_B^4}{15h^3{c}^2}}{T}^4.$$

(18)

The Stefan–Boltzmann constant is therefore

$$\sigma ={\displaystyle \frac{2{\pi }^5{k}_B^4}{15h^3{c}^2}}.$$

(19)

Since $h=2\pi \hslash$, this can also be written as

$$\sigma ={\displaystyle \frac{{\pi }^2{k}_B^4}{60{\hslash }^3{c}^2}}.$$

(20)

Consequently,

$$F=\sigma T^4.$$

(21)

3. Assumed CMB Temperature Inside $R_H=ct$ Cosmology

In an $R_H=ct$ cosmology, the Hubble radius is written as

$$R_H=ct.$$

(22)

The CMB temperature in $R_H=ct$ cosmology is given by

$$T_{\mathrm{cmb}}={\displaystyle \frac{\hslash c}{k_{B}4\pi \sqrt{2R_H{l}_p}}},$$

(23)

where $l_p$ is the Planck length and $R_H$ is the Hubble radius.

This expression links the CMB temperature to the geometric mean of the Hubble radius and the Planck length. The dependence is

$$T_{\mathrm{cmb}}\propto {\displaystyle \frac{1}{\sqrt{R_H{l}_p}}}.$$

(24)

Since $l_p$ is constant, the time-dependence in $R_H=ct$ is

$$T_{\mathrm{cmb}}\propto R_H^{-1/2}\propto t^{-1/2}.$$

(25)

The same temperature can also be written in a geometric-mean form. Define

$$T_{max}={\displaystyle \frac{\hslash c}{k_{B}8\pi l_p}},$$

(26)

and

$$T_{min}={\displaystyle \frac{\hslash c}{k_{B}4\pi R_H}}.$$

(27)

Then

$$T_{max}{T}_{min}={\displaystyle \frac{{\hslash }^2{c}^2}{32{\pi }^2{k}_B^2{R}_H{l}_p}}.$$

(28)

Taking the square root gives

$$\sqrt{T_{max}{T}_{min}}={\displaystyle \frac{\hslash c}{\sqrt{32}\pi k_B\sqrt{R_H{l}_p}}}.$$

(29)

Since $\sqrt{32}=4\sqrt{2}$, this is

$$\sqrt{T_{max}{T}_{min}}={\displaystyle \frac{\hslash c}{4\pi k_B\sqrt{2R_H{l}_p}}}.$$

(30)

Thus

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

(31)

4. The CMB Hubble Sphere Flux

The black-body-equivalent CMB flux is

$$F_{\mathrm{cmb}}=\sigma T_{\mathrm{cmb}}^4.$$

(32)

Using

$$\sigma ={\displaystyle \frac{{\pi }^2{k}_B^4}{60{\hslash }^3{c}^2}}$$

(33)

and

$$T_{\mathrm{cmb}}={\displaystyle \frac{\hslash c}{4\pi k_B\sqrt{2R_H{l}_p}}},$$

(34)

we obtain

$$F_{\mathrm{cmb}}={\displaystyle \frac{{\pi }^2{k}_B^4}{60{\hslash }^3{c}^2}}{\left({\displaystyle \frac{\hslash c}{4\pi k_B\sqrt{2R_H{l}_p}}}\right)}^4.$$

(35)

The fourth power of the temperature is

$$T_{\mathrm{cmb}}^4={\displaystyle \frac{{\hslash }^4{c}^4}{{\left(4\pi \right)}^4{k}_B^4{\left(2R_H{l}_p\right)}^2}}.$$

(36)

Since

$${\left(4\pi \right)}^4=256{\pi }^4$$

(37)

and

$${\left(2R_H{l}_p\right)}^2=4R_H^2{l}_p^2,$$

(38)

we have

$$T_{\mathrm{cmb}}^4={\displaystyle \frac{{\hslash }^4{c}^4}{1024{\pi }^4{k}_B^4{R}_H^2{l}_p^2}}.$$

(39)

Therefore

$$F_{\mathrm{cmb}}={\displaystyle \frac{{\pi }^2{k}_B^4}{60{\hslash }^3{c}^2}}{\displaystyle \frac{{\hslash }^4{c}^4}{1024{\pi }^4{k}_B^4{R}_H^2{l}_p^2}}.$$

(40)

Canceling $k_B^4$, simplifying powers of ℏ, c, and $\pi$, gives

$$F_{\mathrm{cmb}}={\displaystyle \frac{\hslash c^2}{61440{\pi }^2{R}_H^2{l}_p^2}}.$$

(41)

This is the CMB Hubble-sphere radiant exitance, or black-body-equivalent one-sided radiative flux, implied by the assumed temperature relation.

5. Geometric-Mean Flux Form

Since

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

(42)

it follows that

$$T_{\mathrm{cmb}}^4=T_{max}^2{T}_{min}^2.$$

(43)

Thus the flux can be written as

$$F_{\mathrm{cmb}}=\sigma T_{\mathrm{cmb}}^4=\sigma T_{max}^2{T}_{min}^2.$$

(44)

Substituting the explicit expressions for $T_{max}$ and $T_{min}$ gives

$$F_{\mathrm{cmb}}=\sigma {\left({\displaystyle \frac{\hslash c}{k_{B}8\pi l_p}}\right)}^2{\left({\displaystyle \frac{\hslash c}{k_{B}4\pi R_H}}\right)}^2.$$

(45)

Using

$$\sigma ={\displaystyle \frac{{\pi }^2{k}_B^4}{60{\hslash }^3{c}^2}},$$

(46)

we again obtain

$$F_{\mathrm{cmb}}={\displaystyle \frac{\hslash c^2}{61440{\pi }^2{R}_H^2{l}_p^2}}.$$

(47)

The geometric-mean form emphasizes that the CMB flux is the Stefan–Boltzmann flux associated with a temperature lying between a maximum Planck-scale temperature and a minimum Hubble-scale temperature.

6. Scaling with the Hubble Radius

The expression

$$F_{\mathrm{cmb}}={\displaystyle \frac{\hslash c^2}{61440{\pi }^2{R}_H^2{l}_p^2}}$$

(48)

shows immediately that

$$F_{\mathrm{cmb}}\propto R_H^{-2}.$$

(49)

Since $R_H=ct$, this also gives

$$F_{\mathrm{cmb}}\propto t^{-2}.$$

(50)

Thus the black-body-equivalent CMB flux decreases with the square of cosmic time in the assumed $R_H=ct$ framework.

The CMB radiation energy density corresponding to this flux is obtained from the black-body relation

$$u=a{T}^4,$$

(51)

where

$$a={\displaystyle \frac{4\sigma }{c}}.$$

(52)

Equivalently,

$$u_{\mathrm{cmb}}={\displaystyle \frac{4F_{\mathrm{cmb}}}{c}}.$$

(53)

Therefore

$$u_{\mathrm{cmb}}={\displaystyle \frac{\hslash c}{15360{\pi }^2{R}_H^2{l}_p^2}}.$$

(54)

This gives

$$u_{\mathrm{cmb}}\propto R_H^{-2}\propto t^{-2}.$$

(55)

7. Radiation Energy Density, Photon Number Density, and the Density Parameter

The radiation energy density should not be confused with the CMB photon number density, nor with the dimensionless photon radiation-density parameter. These three quantities answer different questions.

The radiation energy density measures energy per unit volume and has units of $\mathrm{J}/{\mathrm{m}}^3$. For black-body radiation,

$$u_{\mathrm{cmb}}=a{T}_{\mathrm{cmb}}^4={\displaystyle \frac{4\sigma }{c}}{T}_{\mathrm{cmb}}^4.$$

(56)

Using the temperature relation assumed here, this gives

$$u_{\mathrm{cmb}}={\displaystyle \frac{\hslash c}{15360{\pi }^2{R}_H^2{l}_p^2}}.$$

(57)

With $l_p^2=\hslash G/c^3$, this can also be written as

$$u_{\mathrm{cmb}}={\displaystyle \frac{c^4}{15360{\pi }^{2}G{R}_H^2}}.$$

(58)

This is an energy density. It is directly related to the black-body-equivalent flux by

$$F_{\mathrm{cmb}}={\displaystyle \frac{c}{4}}{u}_{\mathrm{cmb}}.$$

(59)

The photon number density instead measures the number of photons per unit volume and has units of ${\mathrm{m}}^{-3}$. For a black body,

$$n_{\gamma }={\displaystyle \frac{2\zeta \left(3\right)}{{\pi }^2}}{\left({\displaystyle \frac{k_B{T}_{\mathrm{cmb}}}{\hslash c}}\right)}^3,$$

(60)

where $\zeta \left(3\right)$ is Apéry’s constant. Thus the two dimensional densities have different temperature dependence:

$$u_{\mathrm{cmb}}\propto T_{\mathrm{cmb}}^4,n_{\gamma }\propto T_{\mathrm{cmb}}^3.$$

(61)

They are related through the average black-body photon energy,

$$\langle E_{\gamma }\rangle ={\displaystyle \frac{u_{\mathrm{cmb}}}{n_{\gamma }}}={\displaystyle \frac{{\pi }^4}{30\zeta \left(3\right)}}{k}_B{T}_{\mathrm{cmb}}\simeq 2.701k_B{T}_{\mathrm{cmb}}.$$

(62)

Using the temperature relation assumed here,

$$T_{\mathrm{cmb}}={\displaystyle \frac{\hslash c}{k_{B}4\pi \sqrt{2R_H{l}_p}}},$$

(63)

the photon number density becomes

$$n_{\gamma }={\displaystyle \frac{2\zeta \left(3\right)}{{\pi }^2}}{\left({\displaystyle \frac{1}{4\pi \sqrt{2R_H{l}_p}}}\right)}^3,$$

(64)

and therefore

$$n_{\gamma }={\displaystyle \frac{\zeta \left(3\right)}{64\sqrt{2}{\pi }^5{\left(R_H{l}_p\right)}^{3/2}}}.$$

(65)

This scales as

$$n_{\gamma }\propto R_H^{-3/2}\propto t^{-3/2},$$

(66)

whereas the radiation energy density and the black-body-equivalent flux scale as $R_H^{-2}$.

Haug has recently derived an exact CMB photon radiation-density result for this type of $R_H=ct$ cosmological model, writing the dimensionless photon radiation-density parameter as

$${\Omega }_{\gamma }={\displaystyle \frac{1}{5760\pi }}.$$

(67)

This is not a photon number density and it is not the dimensional radiation energy density $u_{\mathrm{cmb}}$. It is a dimensionless ratio. In the notation used here, it corresponds to comparing the CMB radiation energy density with the critical energy density of the Hubble sphere,

$${\Omega }_{\gamma }={\displaystyle \frac{u_{\mathrm{cmb}}}{{\rho }_c{c}^2}},$$

(68)

where the critical mass-energy density in the Hubble sphere is

$${\rho }_c{c}^2={\displaystyle \frac{3c^4}{8\pi G{R}_H^2}}.$$

(69)

Substituting

$$u_{\mathrm{cmb}}={\displaystyle \frac{c^4}{15360{\pi }^{2}G{R}_H^2}}$$

(70)

gives

$${\Omega }_{\gamma }={\displaystyle \frac{c^4/\left(15360{\pi }^{2}G{R}_H^2\right)}{3c^4/\left(8\pi G{R}_H^2\right)}}={\displaystyle \frac{1}{5760\pi }}.$$

(71)

Thus, in this model, the relative photon radiation density is exact and constant, while the dimensional energy density $u_{\mathrm{cmb}}$ still decreases as $R_H^{-2}$ and the photon number density $n_{\gamma }$ decreases as $R_H^{-3/2}$. This distinction is important: $u_{\mathrm{cmb}}$ measures energy per volume, $n_{\gamma }$ measures photon count per volume, and ${\Omega }_{\gamma }$ measures the CMB radiation energy density relative to the critical energy density.

8. Hubble-Sphere Luminosity

A further quantity of interest is obtained by multiplying the black-body-equivalent flux by the surface area of the Hubble sphere,

$$A_H=4\pi R_H^2.$$

(72)

The corresponding Hubble-sphere luminosity is

$$L_H=4\pi R_H^2{F}_{\mathrm{cmb}}.$$

(73)

Substituting the flux gives

$$L_H=4\pi R_H^2{\displaystyle \frac{\hslash c^2}{61440{\pi }^2{R}_H^2{l}_p^2}}.$$

(74)

The factor $R_H^2$ cancels, and one obtains

$$L_H={\displaystyle \frac{\hslash c^2}{15360\pi l_p^2}}.$$

(75)

Using the Planck-length relation

$$l_p^2={\displaystyle \frac{\hslash G}{c^3}},$$

(76)

this becomes

$$L_H={\displaystyle \frac{c^5}{15360\pi G}}.$$

(77)

The Planck power is

$$P_{\mathrm{Pl}}={\displaystyle \frac{c^5}{G}}.$$

(78)

Therefore

$$L_H={\displaystyle \frac{1}{15360\pi }}{P}_{\mathrm{Pl}}.$$

(79)

Thus, although the flux falls as $R_H^{-2}$, the black-body-equivalent Hubble-sphere luminosity obtained from $4\pi R_H^2{F}_{\mathrm{cmb}}$ is independent of $R_H$. This cancellation is a direct consequence of combining a flux proportional to $R_H^{-2}$ with a spherical area proportional to $R_H^2$.

9. Relation to the Haug–Wojnow Black-Body CMB Luminosity

The luminosity derived here is closely related to the black-body CMB luminosity discussed by Haug and Wojnow [6]. In that work, the starting point is the Stefan–Boltzmann luminosity of a spherical black-body system,

$$L=4\pi R^2\sigma T^4,$$

(80)

where the luminosity radius is set equal to the Schwarzschild radius, $R=R_s$. They then apply the CMB or internal black-body temperature associated with a Schwarzschild black hole and obtain a luminosity that is independent of the black-hole mass.

The present paper arrives at the same constant in a slightly different way. We first derive the black-body-equivalent Hubble-sphere radiant exitance,

$$F_{\mathrm{cmb}}=\sigma T_{\mathrm{cmb}}^4={\displaystyle \frac{\hslash c^2}{61440{\pi }^2{R}_H^2{l}_p^2}},$$

(81)

and only then multiply by the Hubble-sphere area,

$$L_H=4\pi R_H^2{F}_{\mathrm{cmb}}.$$

(82)

This gives

$$L_H={\displaystyle \frac{c^5}{15360\pi G}}.$$

(83)

When the Schwarzschild radius in the Haug–Wojnow black-hole luminosity calculation is identified with the Hubble radius,

$$R_s=R_H,$$

(84)

the luminosity obtained in their framework is algebraically the same as the luminosity obtained here. The difference is therefore mainly interpretational and organizational. Haug and Wojnow emphasize a black-hole-cosmology interpretation in which the luminosity is evaluated at the Schwarzschild radius and is the same for every Schwarzschild black hole. The present paper emphasizes the Hubble-sphere flux first: $F_{\mathrm{cmb}}$ decreases as $R_H^{-2}$, while the total luminosity becomes constant only after multiplying by the spherical area $4\pi R_H^2$.

This distinction is useful because it separates the local surface-like quantity, the radiant exitance or one-sided flux $F_{\mathrm{cmb}}$, from the global Hubble-sphere quantity $L_H$. The flux depends explicitly on the Hubble radius, whereas the luminosity does not.

10. Energy Inside the Hubble Sphere

The black-body-equivalent CMB energy inside the Hubble sphere can be written as

$$E_{\mathrm{cmb}}=u_{\mathrm{cmb}}{V}_H,$$

(85)

where

$$V_H={\displaystyle \frac{4\pi R_H^3}{3}}.$$

(86)

Using

$$u_{\mathrm{cmb}}={\displaystyle \frac{\hslash c}{15360{\pi }^2{R}_H^2{l}_p^2}},$$

(87)

one obtains

$$E_{\mathrm{cmb}}={\displaystyle \frac{\hslash c}{15360{\pi }^2{R}_H^2{l}_p^2}}{\displaystyle \frac{4\pi R_H^3}{3}}.$$

(88)

Hence

$$E_{\mathrm{cmb}}={\displaystyle \frac{\hslash c{R}_H}{11520\pi l_p^2}}.$$

(89)

Using $l_p^2=\hslash G/c^3$, this becomes

$$E_{\mathrm{cmb}}={\displaystyle \frac{c^4{R}_H}{11520\pi G}}.$$

(90)

The corresponding mass-equivalent is

$$M_{\mathrm{cmb}}={\displaystyle \frac{E_{\mathrm{cmb}}}{c^2}}={\displaystyle \frac{c^2{R}_H}{11520\pi G}}.$$

(91)

Thus, in this framework,

$$E_{\mathrm{cmb}}\propto R_H\propto t,$$

(92)

while the energy density and flux both scale as $R_H^{-2}$.

11. Discussion

The main result of this note is that the assumed CMB temperature relation

$$T_{\mathrm{cmb}}={\displaystyle \frac{\hslash c}{k_{B}4\pi \sqrt{2R_H{l}_p}}}$$

(93)

leads to the black-body-equivalent flux

$$F_{\mathrm{cmb}}={\displaystyle \frac{\hslash c^2}{61440{\pi }^2{R}_H^2{l}_p^2}}.$$

(94)

The same result follows from the geometric-mean form

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

(95)

In that representation,

$$F_{\mathrm{cmb}}=\sigma T_{max}^2{T}_{min}^2.$$

(96)

This provides a compact thermodynamic link between the Planck scale and the Hubble scale. The minimum temperature is Hubble-scale, the maximum temperature is Planck-scale, and the CMB temperature appears as their geometric mean. The associated flux then depends on the squared product of these limiting temperatures.

It is important to interpret the word “flux” carefully. The directly observed CMB is an almost isotropic radiation field; its net vector flux at a point is approximately zero. The quantity $F_{\mathrm{cmb}}=\sigma T_{\mathrm{cmb}}^4$ is instead the one-sided black-body-equivalent radiant exitance associated with the CMB temperature. It is the same flux one would assign to an ideal black-body surface at temperature $T_{\mathrm{cmb}}$, or equivalently the quantity related to the radiation energy density by

$$F_{\mathrm{cmb}}={\displaystyle \frac{c}{4}}{u}_{\mathrm{cmb}}.$$

(97)

Within the assumed $R_H=ct$ cosmological model, the results can be summarized as

$$F_{\mathrm{cmb}}\propto R_H^{-2},u_{\mathrm{cmb}}\propto R_H^{-2},n_{\gamma }\propto R_H^{-3/2},{\Omega }_{\gamma }={\displaystyle \frac{1}{5760\pi }},E_{\mathrm{cmb}}\propto R_H,L_H=\mathrm{constant}.$$

(98)

The luminosity expression agrees algebraically with the CMB/internal black-hole luminosity discussed by Haug and Wojnow when the Schwarzschild radius is identified with the Hubble radius. The present derivation, however, highlights the prior flux relation and its $R_H^{-2}$ scaling before forming the area-integrated luminosity.

The constancy of $L_H$ is particularly interesting:

$$L_H={\displaystyle \frac{c^5}{15360\pi G}}={\displaystyle \frac{1}{15360\pi }}{P}_{\mathrm{Pl}}.$$

(99)

This is a fixed fraction of the Planck power. Thus, in the assumed framework, the flux dilution with the growing Hubble radius is exactly compensated by the increasing Hubble-sphere area. The photon number density behaves differently, because it counts photons rather than energy; with the same temperature relation, it scales as $R_H^{-3/2}$ rather than $R_H^{-2}$. Haug’s exact result ${\Omega }_{\gamma }=1/\left(5760\pi \right)$ is a dimensionless density parameter, not a dimensional photon count or energy density. This difference is important when comparing the present flux relation with photon-density and radiation-density results such as those derived by Haug [10].

12. Conclusion

Starting from Planck’s black-body law, integration over wavelength and the outward hemisphere gives the Stefan–Boltzmann law,

$$F=\sigma T^4.$$

(100)

This law gives the radiant exitance of a black body and is also commonly called the emitted radiative flux in astrophysical contexts.

Assuming the CMB temperature relation

$$T_{\mathrm{cmb}}={\displaystyle \frac{\hslash c}{k_{B}4\pi \sqrt{2R_H{l}_p}}},$$

(101)

the CMB Hubble-sphere flux becomes

$$F_{\mathrm{cmb}}=\sigma T_{\mathrm{cmb}}^4={\displaystyle \frac{\hslash c^2}{61440{\pi }^2{R}_H^2{l}_p^2}}={\displaystyle \frac{\hslash {\overline{\lambda }}_c^2}{245760{\pi }^2{l}_p^6}}.$$

(102)

Using the geometric-mean temperature representation

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

(103)

with

$$T_{max}={\displaystyle \frac{\hslash c}{k_{B}8\pi l_p}},T_{min}={\displaystyle \frac{\hslash c}{k_{B}4\pi R_H}},$$

(104)

the same flux can be written as

$$F_{\mathrm{cmb}}=\sigma T_{max}^2{T}_{min}^2.$$

(105)

The corresponding photon number density is not the same as this radiation energy density or flux. It is

$$n_{\gamma }={\displaystyle \frac{\zeta \left(3\right)}{64\sqrt{2}{\pi }^5{\left(R_H{l}_p\right)}^{3/2}}},$$

(106)

which scales as $R_H^{-3/2}$, in contrast to $F_{\mathrm{cmb}}\propto R_H^{-2}$ and $u_{\mathrm{cmb}}\propto R_H^{-2}$. Haug’s exact result for this model is instead the dimensionless photon radiation-density parameter

$${\Omega }_{\gamma }={\displaystyle \frac{1}{5760\pi }},$$

(107)

which is the ratio $u_{\mathrm{cmb}}/\left({\rho }_c{c}^2\right)$, not the dimensional energy density and not the photon number density [10].

The resulting Hubble-sphere luminosity is

$$L_H=4\pi R_H^2{F}_{\mathrm{cmb}}={\displaystyle \frac{c^5}{15360\pi G}}.$$

(108)

This is algebraically the same constant luminosity discussed by Haug and Wojnow for the CMB/internal black-body luminosity of a Schwarzschild black hole when the Schwarzschild radius is identified with the Hubble radius, $R_s=R_H$ [6]. Thus, in the assumed $R_H=ct$ framework, the CMB black-body-equivalent flux decreases as $R_H^{-2}$, while the corresponding Hubble-sphere luminosity is constant and equal to a fixed fraction of the Planck power.