How a Thermodynamic Version of the Friedmann Equation Appears to Solve the Early Galaxy Formation Problem

1. The Cosmic Age in $R_h=ct$ Cosmology

It is well known that, in $R_h=ct$ cosmology, the age of the universe at any point in time must simply be $t_t=\frac{1}{H_t}$, and then the current age of the universe becomes $t_0=\frac{1}{H_0}$, as pointed out, for example, by Melia and Shevchuk [1]. However, this formula still results in a very wide uncertainty in cosmic age, due to the significant uncertainty in $H_0$ present in both Melia’s $R_h=ct$ cosmology and in $\mathsf{\Lambda }$-CDM cosmology. The uncertainty in $H_0$ arises both from measurement difficulties and from the Hubble tension uncertainty. Considering the Hubble tension, $H_0$ within one standard deviation ranges from $63.3$ km/s/Mpc to $79.2$ km/s/Mpc, as calculated from seven recent measurement studies of $H_0$ which we will show. This wide range of $H_0$ values leads to a cosmic age estimate range of about $12.3$ to $15.4$ billion years.

In the $\mathsf{\Lambda }$-CDM model, the estimated cosmic age is further adjusted according to the assumption that the universe has expanded faster than the speed of light. Of interest, in the special case where ${\mathsf{\Omega }}_{\mathrm{matter}}=0.26$ and ${\mathsf{\Omega }}_{\mathsf{\Lambda }}=0.74$, the $\mathsf{\Lambda }$-CDM estimated cosmic age coincidentally matches that predicted by $R_h=ct$ models, namely $t_0=\frac{1}{H_0}$, as recently pointed out by Kutschera and Dyrda [2]. So, it is clear that large uncertainties in cosmic age estimates due to measurement errors in $H_0$, as well as the Hubble tension itself [3], result when using only $H_0$ in isolation. Moreover, in the $\mathsf{\Lambda }$-CDM model, one also uses inputs other than $H_0$, including cosmological redshifts, to estimate the cosmic age; their best current age estimate is $13.8\pm 0.02$ Gyr [4]. Their small uncertainty in this number is surprising, to say the least.

Recently, Haug and Tatum demonstrated [5] that the Friedmann equation [6] can be written in thermodynamic form:

where (called “Upsilon”) is a composite coupling constant given by Tatum et al [7,8] equal to:

The only uncertainty in this constant comes from the uncertainty in G, as the other constants are exact according to the NIST CODATA 2018 standard. When working with our particular sub-class of $R_h=ct$ cosmology model, one can set $k=\mathsf{\Lambda }=0$.

Our thermodynamic formulation of the Friedmann equation is a result based on many years of work by multiple researchers, starting with Tatum et. al [9] in 2015. Their cosmic temperature formula linked the CMB temperature, given as:

$$T_{cmb,t}=\frac{\hslash c^3}{k_{b}8\pi G\sqrt{M_{c,t}{m}_p}}=\frac{\hslash c}{k_{b}4\pi \sqrt{R_{h,t}2l_p}}$$

(3)

with the Hubble constant at any cosmic time, $H_t$, according to $R_{h,t}=\frac{c}{H_t}$. So, $R_{h,t}$ is the Hubble radius at cosmic time t. The same formula was later derived from the Stefan-Boltzmann law; see Haug and Wojnow [10,11]. As our sub-class of $R_h=ct$ cosmology must have $t_t=c{R}_{h,t}$ and also $c{t}_p=l_p$, equation (3) can naturally also be re-written as:

$$T_{cmb,t}=\frac{\hslash }{k_{b}4\pi \sqrt{t_{t}2t_p}}$$

(4)

where $t_t$ is the cosmic time as it evolves. Solving for $t_t$ gives:

We also must have:

$$\begin{array}{ccc}\hfill \frac{1}{H_t^2}& =& \frac{3}{8\pi G{\rho }_t+\mathsf{\Lambda }{c}^2-k^2{c}^2}\hfill \\ \hfill t_t^2& =& \frac{3}{8\pi G{\rho }_t+\mathsf{\Lambda }{c}^2-k^2{c}^2}\hfill \end{array}$$

(6)

In the flat space $R_h=ct$ critical Friedmann solution, one sets $k=0$ and $\mathsf{\Lambda }=0$ and we then have:

$$\begin{array}{ccc}\hfill H_t^2& =& \frac{8\pi G{\rho }_{c,t}}{3}\hfill \\ \hfill \frac{1}{H_t^2}& =& \frac{3}{8\pi G{\rho }_{c,t}}\hfill \\ \hfill t_t& =& \sqrt{\frac{3}{8\pi G{\rho }_{c,t}}}\hfill \\ \hfill t_t& =& \sqrt{\frac{3}{8\pi G\frac{3H_t^2}{8\pi G}}}\hfill \\ \hfill t_t& =& \frac{1}{H_t}\hfill \end{array}$$

(7)

This is as expected, and a well-known result. From the work of Tatum et al. [7,8], we also must have:

So, our $R_h=ct$ critical Friedmann solution can also be expressed as:

And, for the current spatially-flat epoch of the universe, we also have:

Since $T_0$ has been measured much more precisely than $H_0$, this leads to a much more accurate estimate of current cosmic age, as shown in Table 1. If $H_0$ had been measured with the same precision as the current CMB temperature, one could have also obtained the same precision in $t_0$ (when one ignores the Hubble tension, which we claim to have solved; see [12,13]). The justification for such precision is explained in [7]. Given this background, please see Table 1 and Table 2.

In $R_h=ct$ models, one can now estimate the current cosmic age much more precisely by adopting our thermodynamic Friedmann equation; see [5]. In the Melia sub-class of $R_h=ct$ model [1], without our thermodynamic Friedmann equation, there is greater uncertainty in its estimate of the cosmic age, ranging from about 13.2 to 14.8 billion years. In sharp contrast, based on four recent CMB studies, we can now constrain the cosmic age to slightly greater than 14.6 billion years, with very high precision. See again Table 1. This is largely due to the current high precision (i.e., low uncertainty) in current CMB temperature measurements.

Tatum and Haug [25] were first to derive a 14.6 billion year cosmic age estimate from their new cosmological model, when calibrating their redshift prediction formula using all 580 type Ia supernova redshifts in the Union2 database. This shows that our particular sub-class of $R_h=ct$ model is internally consistent, because it is also consistent with our thermodynamic form of the Friedmann equation.

It is worth mentioning here that Haug and Tatum [12] also solved the Hubble tension by using the Union2 supernova redshift database. We initially started with a mere guess for $H_0$, but then employed intelligent algorithms, such as the well-known bisection method and the Newton-Raphson method, in order to minimize errors between predicted and observed redshift values. When applying this new approach to predicted redshifts using the formula: $z=\sqrt{\frac{R_h}{R_t}}-1$, we found: $H_0=66.8711\pm 0.0019\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$. This level of $H_0$ precision surpasses that of any other study, and yields a cosmological time of $t_0=\frac{1}{H_0}=14,622,044,{825}_{-415,442}^{+415,466}$ years. Our particular solution to the Hubble tension, and its alignment with our thermodynamic form of the Friedmann equation, produces the same cosmic time estimate when using $H_0$ as that when using the CMB temperature measurement directly. It is important to note that the original Friedmann equation, and now its thermodynamic counterpart, are, at a deeper level, essentially the same equation. They are based on two different observations, either current CMB temperature or $H_0$, but are now coupled by an exact mathematical relationship (using our Upsilon composite constant) that has been overlooked in other cosmological models.

2. Two Cosmological Redshift Formulas and Their Implications

In general, for multiple different models, we have $T_t=T_0{(1+z)}^{1-\beta }$, as pointed out by Lima et al. [26]. In our thermodynamic Friedmann equation, it is clear that one must have $\beta =0$, which is consistent with observational studies [27] indicating that $\beta$ must be close to zero. So, we now assume $T_t=T_0(1+z)$. This leads to the cosmological redshift formula in our particular sub-class of $R_h=ct$ cosmology, when setting $k=\mathsf{\Lambda }=0$ (i.e., flat space cosmology); one must have:

$$z=\sqrt{\frac{R_h}{R_t}}-1=\frac{T_t}{T_0}-1$$

(11)

Which, when solved for $T_t$, brings us back to the well-tested $T_t=T_0(1+z)$ relation. If, instead, we want to have $z=\frac{R_h}{R_t}-1$, then our thermodynamic Friedmann equation tells us that we also must have:

$$z=\sqrt{\frac{R_h}{R_t}}-1=\frac{T_t^2}{T_0^2}-1$$

(12)

which leads to $T_t=T_0{(1+z)}^{1-\beta }=T_0{(1+z)}^{1-\frac{1}{2}}=T_0{(1+z)}^{\frac{1}{2}}$, which we have recently shown does not appear to be consistent with observations. In making this discovery, we claim to have resolved the Hubble tension; see again [12]. It is important to note here that we now have both a new redshift formula and a new cosmological framework providing exact inter-relationships between $T_0$, $H_0$, z, and even the Planck scale [9,11,13].

3. What Modern Telescopes Have Revealed About the Earliest Galaxies and Their Supermassive Black Holes

Modern telescopes, most notably the James Webb Space Telescope (JWST), have revealed numerous early galaxy surprises with respect to expectations based on the Lambda-CDM cosmology model and our current understanding of the physics of galaxy formation. The most distant galaxies, particularly those with redshifts of $z>7$, appear shockingly different from previous expectations. Their combined unexpected features have given rise to what has been called the “early galaxy problem.” They present a severe challenge to Lambda-CDM cosmology and our previous theories and simulations of early galaxy formation.

Before the most modern space telescopes became available, the prevailing theory of galaxy evolution was the “Hierarchical Model.” The idea was that modern galaxy formation could only occur as a piecemeal process of gradually-accumulating evolutionary building blocks over many billions of years. However, in recent years, large galaxies with modern features could already be seen by 3-6 billion years of cosmic age. This discovery was one of the first indications that the Hierarchical Model theory must be incorrect. In addition, it has gradually become apparent that supermassive black holes (SMBHs) are more ubiquitous and fundamental as “seeds” for galaxy formation than previously thought.

The JWST confirmed these growing suspicions about the early universe and elucidated a number of surprising features of the highest redshift galaxies. These galaxies are found to be far more mature in appearance than previously expected for age. Their bulk mass and brightness is already much greater than expected [28,29,30]. Interestingly, they also tend to be ultracompact, having diameters roughly several hundred to one thousand times smaller than that of the Milky Way [31,32]. The oldest among them, having redshifts greater than 9, also have much lower ratios of their star mass to that of their SMBH. In mature galaxies, such as the Milky Way, this ratio is usually about 1000 to 1. However, very early galaxies tend to have much smaller ratios, less than 100 to perhaps as low as roughly 1 to 1. The very early SMBHs of one study [33] were typically at least 10-100 times smaller than the active galactic nuclei of quasars. Such findings suggest that these considerably smaller early SMBHs might act as initial “seeds” for galaxy star formation and bursty activity by some sort of positive feedback loop. It has also been proposed that larger SMBH “seeds” might even form by “direct collapse” from gargantuan clouds of primordial hydrogen gas [34].

Given the surprising degree of apparent maturity of the earliest visible galaxies, one must ask to what degree the early galaxy problem could be the result of some yet-undiscovered flaw in the Lambda-CDM cosmic age estimate of roughly 13.8 billion years, as opposed to an incomplete understanding of the physics of early galaxy and/or SMBH formation. While the discussion so far has focused on new concepts of early galaxy and SMBH formation (which can be “tested" in sophisticated computer simulations), the question with respect to cosmic age must also be recognized as highly model-dependent. A different cosmology model will naturally calculate a different cosmic age. Could it be that 13.8 billion years is in need of correction in favor of allowing for a longer time interval between the Big Bang and the appearance of the first galaxies?

Precise calculation of cosmic age has everything to do with having the correct redshift formulae with respect to time-dependent cosmic temperature, radius and cosmological distance. That having the correct redshift formula is crucial to the calculation of cosmic age, is already well-known in the literature. A recent example is offered by Gupta [35]. By adopting a different (“hybrid”) redshift formula, Gupta stretches the cosmic age to 26.7 billion years, thus allowing “enough time to form massive galaxies” (his words).

In Section 2, we demonstrated the importance of having the correct redshift formula and methodology. In particular, we showed how the standard cosmological formula could be underestimating cosmic age by a significant degree. In our previous publications wherein we have offered a Hubble tension solution [12,13], we have already shown how such a formula can, conversely, overestimate a local Hubble constant value derived from a database of supernova redshifts.

4. How the Stefan-Boltzmann Law is Inextricably Linked with Our Thermodynamic Friedmann Equation

The Stefan-Boltzmann luminosity from a black body is directly linked to the Stefan-Boltzmann law and is given by:

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

(13)

where $\sigma$ is the Stefan-Boltzmann constant, defined as $\sigma =\frac{2{\pi }^5{k}_b^4}{15c^2{h}^3}$, and T is the temperature of the black body.

Here, we will demonstrate that we can recover the Stefan-Boltzmann law from the relationship between the past and current Friedmann critical density of the universe. The critical Friedmann density of the universe (which also plays a central role in Melia’s $R_h=ct$ cosmology as well as in the $\mathsf{\Lambda }$-CDM model) is given by the classical Friedmann equation:

$${\rho }_c=\frac{3H_0^2}{8\pi G}$$

(14)

Since the Haug-Tatum model largely relies upon the mathematical coupling between the current CMB temperature and the $H_0$ value, namely, by , we can substitute $H_0$ in the above equation:

In addition, using $T_t=T_0(1+z)$ as discussed in Section 2, and the same critical density formula for past epochs of the universe, within $R_h=ct$ cosmology, we must have:

Furthermore, in a growing black hole model, we must also have:

$${\rho }_{c,t}=\frac{3H_0^2}{8\pi G}=\frac{M_{c,t}}{\frac{4}{3}\pi R_t^3}$$

(17)

so we must have :

The Upsilon constant is given by so, by squaring, and remembering $\sigma =\frac{2{\pi }^5{k}_b^4}{15c^3{h}^3}$, we have:

Where $4\pi R_{s,p}\sigma T_{BH,p}^4$ is the internal luminosity of a Planck mass Schwarzschild black hole, and $R_{s,p}=\frac{2G{m}_p}{c^2}$ and $T_{BH,p}=\frac{T_p}{8\pi }$. This means that we must always have:

$$\begin{array}{ccc}\hfill L_{BH}& =& 4\pi R_t^2\sigma T_t^4\hfill \end{array}$$

(20)

The above results show that we have recovered the Stefan-Boltzmann luminosity from our thermodynamic Friedmann equation. This is to be expected, since the thermodynamic Friedmann equation relies upon the CMB temperature coupling to the Hubble constant, as first indicated by Tatum et al. [9] in 2015. Later, in 2023, Haug and Wojnow [10,11] used the Stefan-Boltzmann law to derive the same basic relationships. Thus, we believe that our thermodynamic Friedmann formula is always consistent with the Stefan-Boltzmann law. Can any other cosmological model make a similar claim? We invite proponents of other $R_h=ct$ models and proponents of the $\mathsf{\Lambda }$-CDM model to discuss our results and compare them with their own models.

Since we must have:

$$L_{BH}=4\pi R_t^2\sigma T_t^4=4\pi R_{s,p}\sigma T_{BH,p}^4$$

(21)

The above results can also be proven in a much simpler way. From Section 2 we know we must have:

$$\begin{array}{c}\hfill z=\sqrt{\frac{R_h}{R_t}}-1=\frac{T_t}{T_0}-1\end{array}$$

(22)

If we want to satisfy $T_t=T_0(1+z)$, which basically has been confirmed by observations, this means that we must also have:

$$\begin{array}{ccc}\hfill \sqrt{\frac{R_h}{R_t}}-1& =& \frac{T_t}{T_0}-1\hfill \\ \hfill T_t& =& T_0\sqrt{\frac{R_h}{R_t}}\hfill \end{array}$$

(23)

This would imply that we must have:

$$\begin{array}{ccc}\hfill L_{BH}& =& 4\pi R_t^2\sigma T_t^4\hfill \\ \hfill L_{BH}& =& 4\pi R_t^2\sigma {\left(T_0\sqrt{\frac{R_h}{R_t}}\right)}^4\hfill \\ \hfill L_{BH}& =& 4\pi R_t^2\sigma T_0^4\frac{R_h^2}{R_t^2}\hfill \\ \hfill L_{BH}& =& 4\pi R_h^2\sigma T_0^4\hfill \end{array}$$

(24)

So, mathematically, we have determined that the internal black hole luminosity must remain constant across all Schwarzschild black holes (and possibly other types of black holes). This naturally does not imply that the internal temperature (CMB) is the same during the growth of the black hole Hubble sphere. A smaller radius corresponds to an earlier epoch of the universe when the CMB temperature was higher. However, the internal luminosity remains constant when measured at $R_t$ because the shorter radius is precisely offset by the higher temperature. This finding aligns with Haug and Wojnow’s [36] derivation, which began from a different angle since the thermodynamic Friedmann equation had not yet been invented at the time of their investigation. They also found that what they termed CMB luminosity surprisingly remains constant in growing black hole $R_h=ct$ cosmology. Thus, while the temperature and radius change as the universe expands, the internal luminosity remains unchanged (at radius $R_t=ct$). If we, instead, were to assume:

$$z=\frac{R_h}{R_t}-1=\frac{T_0}{T_t}-1$$

(25)

This would lead to :

$$\begin{array}{ccc}\hfill L_{BH}& =& 4\pi R_t^2\sigma T_t^4\hfill \\ \hfill L_{BH}& =& 4\pi R_t^2\sigma {\left(T_0\frac{R_h}{R_t}\right)}^4\hfill \\ \hfill L_{BH}& =& 4\pi R_t^2\sigma T_0^4\frac{R_h^4}{R_t^4}\hfill \\ \hfill L_{BH}& =& 4\pi \frac{R_h^4}{R_t^2}\sigma T_0^4\hfill \end{array}$$

(26)

which would require a modification of the Stefan-Boltzmann law, which we think would be highly unlikely to hold against observations. It would also require that we have $T_t=T_0{(1+z)}^{1-\beta }=T_0{(1+z)}^{\frac{1}{2}}$, which does not seem to fit current observations in favor of $\beta =0$.

Again, we invite proponents of other $R_h=ct$ models and proponents of the $\mathsf{\Lambda }$-CDM model to discuss our results and compare them with their own models.

5. Summary and Conclusions

We have demonstrated how our particular sub-class of $R_h=ct$ model, in conjunction with our thermodynamic form of the Friedmann equation, leads to an estimated cosmic age of approximately 14.6 billion years. This suggests that the universe is about 800 million years older than estimated by standard cosmology model proponents. Thus, this cosmic age extension potentially solves the early galaxy formation problem related to recent observations made by the James Webb Space Telescope. Furthermore, use of our thermodynamic Friedmann equation also results in significantly higher precision in cosmic age estimates, in comparison to those given using only the standard Friedmann equation and $H_0$ in isolation. In our view, this new approach to calculating cosmic age with high precision is revolutionary and has potentially important implications for the early universe. Nevertheless, we recommend that the astrophysics community scrutinize our model over time and form their own opinion.