In a classic setting, an amount of gas is held in a sealed vessel, which means the gas is trapped inside a physical boundary: The walls of the vessel. The boundary of a sealed gas can change, like in a piston. All bound gases are sealed. However, not all boundaries are seals. There’s imaginary boundaries, which don’t really exist. They’re used for constant amounts of gas. An unsealed gas is unbound, despite any imaginary boundary we may apply. The math terms, bound and boundary, are common to textbooks over the range of disciplines we use in this paper, so we’ll describe gas behavior in sealed vessels this way.

There are two kinds of gas expansion: reversible and free. Reversible expansion is isoentropic by definition: ΔS = 0, where S is the entropy of the gas (kg-m²/s²-K or J/K). A classic, perfectly reversible expansion must also be adiabatic, which means there is no heat transferred into or out of the vessel. When a bound gas expands both adiabatically and isoentropically, its pressure P (kg/m-s²), internal kinetic energy U_i (J), and temperature T (K) decrease. Energy -ΔU_i is lost, leaves the vessel, and converted into work Δ(PV) as the boundary moves. This PV work from e.g. a piston can be stored and reused.

An adiabatic bound gas can also undergo free, Joule expansion, which is entropic (ΔS > 0). No work is performed. As the bound volume V (m³) increases, U_i does not decrease and only P drops. Adiabatic, freely expanding bound gases do convert energy, it’s just not through loss of U_i. It’s referred to as entropic energy TS and its gain TdS, or more generally d(TS), measures the bound gas’s reduced ability to convert U_i into a storable form. Inside an adiabatic bound vessel, the total energy [U_i - PdV + TdS] of a freely expanding gas remains constant. For any bound gas, its density ρ is a primary metric for how much of its U_i can be harnessed.

The first and second laws of thermodynamics are held inviolate, by engineers at least, and can be expressed at scale. The first law of thermodynamics, in its broadest definition, says that energy is neither created nor destroyed: Where E is the sum of mass and energy in an at-scale sphere. We note here that the terms E and dE do double duty in this paper. They have the meaning given by (1) in our discussion of the fluid and acceleration equations (21)-(23). They also refer to adiabatic thermal loss from work: E = -ΔU_i, and when isoentropic, dE = -PdV. There is conflation of these two meanings in derivation of the fluid equation.

The second law of thermodynamics is more subtle in meaning than the first law, and has had several descriptions over the years. The broadest of these says that entropy at scale is always increasing over time:

This links time and entropy. If one assumes an isoentropic process then (2) requires that no time shall elapse. Equation (2) can’t be compared directly to (1) because they have different units of measurement. To make direct comparison possible, I will restate (2) in terms of energy:

Note that (3) only applies to an unbound system. “System” usually means e.g. a bound vessel’s contents. In this paper it also refers to constant amounts of gas that aren’t bound. It’s possible to have d(TS)/dV < 0 and d(S)/dV > 0 in a bound system if heat transfer to or from its surroundings is neglected. However, for both the system and surroundings combined, (3) is always true. At scale, the system is the surroundings. There is no scenario at scale where the entropy of the system is increasing and its entropic energy isn’t. The converse applies: If d(TS)/dV > 0 at scale then d(S)/dV > 0 as well.

We conduct three different “thought experiments” in which gravity is unimportant. These will help us better understand the nature of free expansion within the GCDM model, where gravity plays a central role.

Take a spherical helium balloon, of radius r₁ = 10 cm, at a temperature T = 300K and pressure P = 1 atmosphere, and place it in the center of a perfectly rigid, insulated, spherical vacuum chamber of radius r₂ = 50 cm. Gravitational effects are infinitesimal. The insulation and rigidity of the chamber means any gas expansion from r₁ to r₂ will be adiabatic. The gas in the balloon is monatomic, and its internal kinetic energy U_i is 100% thermal. For a monatomic gas, this is given by:Where R is the gas constant (8.314 J/mole-K), Ж is the atomic weight of the gas (kg/mole), n is the number of moles of gas, and M is the thermodynamic mass of the gas (kg). Other forms of mass are important at scale and discussed later. The thermal energy U_i in the balloon is defined as the instant sum of its atoms’ individual kinetic energies:

“Instant” means time stands still. The tensor v is the atom’s instant kinetic energy, m is the mass of the helium atom (6.6 x 10^-27 kg), r is the distance from the center, θ is the conic angle of latitude, 𝜑 is the angle of longitude, and ${ϴ}^{\prime }$ is the conic angle of v’s deviance from radial. These are shown in two dimensions in Figure 1. If you spin Figure 1 around its polar axis you get 𝜑; this is omitted in the graphic for simplicity. The void between r₁ and r₂ makes no contribution to U_i as long as the balloon is intact. The balloon is an idle sphere, having a constant radius r₁.

We pop the balloon. The thermal energy U_i is temporarily and partly transformed into radial kinetic energy E_k:

Which is the scalar difference in energy between the outward and inward radial components of the atoms’ tensors. Implementation of (6) isn’t as sequential as (5). We have to determine if the atom is moving in or out before assigning it. Another definition for $U_i$ can now be given:

For an idle sphere, the inward and outward radial scalars of v in (6) are equal, so we can just double the inward scalar and replace the radial term in (5). This gives (7), which yields the same result as (5) for an idle sphere but can also be used to get U_i for an expanding sphere. Note that that (6) and (7) always have precise instant values.

The total kinetic energy U_k in the sphere is:

When idle, U_k = U_i. When expanding, U_k stays the same and E_k diminishes U_i. The instant E_k given by (6) is the differential entropic energy gain:

Equation (9) is the single most important concept of this entire paper. It links kinetic and entropic energies. Entropic energy gain can be expressed with (9) through kinetic energy calculations without resort to direct calculation of entropy. All conclusions herein arise from (9)’s basic premise: Unbound kinetic energy and entropic energy are two facets of the same phenomenon, the second law of thermodynamics (2)-(3).

In the special condition of uniform comoving density ρ, E_k is given as:Where $E_k^{\prime }$ is the entropic pressure:

At last scatter, the Universe’s ρ was uniform throughout its volume, so (10) and (11) apply. In the present bound example, uniform ρ only occurs at the instant the balloon is popped, giving $E_k^{\prime }$ = P. Since ρ is not uniform after the balloon is popped, (10) and (11) don’t describe the later behavior of the atoms in this example. However, (8) and (9) remain accurate, and since loss to gravity is negligible, E_k = -Δ(U_i) for the one-meter sphere as a whole during expansion. It lasts for maybe a second; the exact amount of time is unimportant. During the initial phase of expansion, U_i drops to a minimum value U_i’ and E_k reaches its maximum. The atoms quickly bounce off the wall and E_k drops. When equilibrium is reestablished, E_k→0, the terms of (6) again cancel, and U_i = U_k is unchanged for the enlarged idle sphere. The entropic energy gain E_S from volume increase is:

Take that same balloon, put it in the center of a large vacuum chamber (r₂ = 10⁸ m) and pop it. A helium atom at T = 300K has a root mean square speed v_rms = 1368 m/s. Those atoms will take about 20 hours to reach the wall of the chamber if their tensor of movement is perfectly radial. As they expand, they stop colliding with each other at any meaningful rate. After that happens, we can say that atomic movement is in a nonequilibrant “unbound free expansion” regime which is best considered at a time period when the atoms have stopped colliding, but haven’t hit the wall yet. During the regime, almost all of the kinetic energy is radial: U_i ≈ 0 and U_k ≈ E_k. The radial component of each outward atom’s speed, or radial velocity v_r, is proportional to its distance from the center:

Where t is the elapsed time. The atomic Hubble parameter H is simply expressed by (13). Once the atoms stop colliding, E_k remains unchanged until they start to hit the wall. Eventually the atoms bounce off the wall, the regime slowly comes to an end, thermal equilibrium is reestablished, and U_i rises back to its starting value. A classic Joule expansion has a similar U_i profile: Helium gas at 300K is allowed to pass unimpeded through a connecting tube from a small pressurized chamber into a much larger vacuum chamber. The gas cools while it passes through the tube as U_i partitions into E_k, which in the tube is linear kinetic energy, not radial. The thermal energy U_i inside the tube is well defined since the instant temperature is constant along short lengths of the tube and can be measured in situ. 4 The enlarged vessel’s boundary again eventually yields thermal equilibrium, with U_i unchanged from its starting value.5

What if there’s no boundary? There’s no equilibrium to be reached, so for a freely expanding gas, more and more of U_i is permanently converted to gain as time passes. One can approach Universal conditions by looking only at the comoving central core of a large popped sphere with r_inner = 10^-6 r_outer, or some similar small fraction of the total. That inner sphere would be nearly homogenous (dρ/dr ≈ 0), and as such, has a uniform instant value of U_i which obeys (10) and (11). The H value of the inner sphere’s surface would be more complex than (13) and perhaps similar to the Universe at its cosmic redshift z = 1089,6 the time of last scattering. That is, if the universe happens to be 100% helium, and denser. With proper parameters and a computer, this sort of treatment could be accurate at scales large enough to include gravity. I’m not suggesting that the Universe has finite mass, only that it can be so modeled.

The behavior of common mass in e.g. a rock closely follows the laws of gravity and motion discovered by Isaac Newton. Einstein’s laws of general relativity, a refinement of Newton’s laws, considers Newtonian mass as a form of energy through the well-known equation E = mc², a special case of:Where E_m is the total or Einsteinian energy of the mass. The rest mass m is when it stands still relative to its neighbors, and is exactly Newtonian. This m could mean a helium atom as before, or a larger mass. The term c is the speed of light (3 x 10⁸ m/s), and m’ is the relativistic mass, an increase m’/m at a relative speed v’.7 When v’ << c, (14) simplifies to:

The ΛCDM model discards $\frac{1}{2}m{v^{\prime }}^2$ as insignificant (22). The GCDM model discards $m{c}^2$ as unchanged between successive thermodynamic states (45). The Einsteinian energy of rest mass plays only a supporting role in the GCDM model, as a source of kinetic energy in the IGM arising from nuclear fusion in the cosmic web.

Newtonian laws operate in Euclidean or flat spacetime, a continuous array of infinitely large three-dimensional instant Cartesian grids x,y,z over linear time t. General relativity combines space and time into a single curved non-Euclidean description. All measurements to date support its conclusions, which led to the question of Universal curvature beyond scale. This does not preclude the idea that at scale, the Universe is well approximated by flat space and linear time if v' << c. This author is hardly an expert in general relativity, but will attempt to show why Newtonian laws in Euclidean space are adequate.

We consider two isolated massive objects with v' << c. Every point in spacetime around these objects has a set of ten gravity tensors and ten momentum tensors in a four-dimensional normalized coordinate system x, y, z, and t. All the tensors must be used to accurately describe the relative movement of the two masses. When the objects are far away from each other and moving at a substantial but not relativistic speed, the gravitational tensors of the system approach two isolated sets, and their relative movement can be fairly described with Newtonian momentum in Euclidean space. Exactly what “far away” means depends on the masses in question, and the distance between them.

We consider two isolated helium atoms. The distance beyond which Newtonian and Euclidian space becomes an accurate description is very low: maybe a micron, depending on the level of rigor one chooses to pursue. At last scattering, the mean distance between atoms was more than a millimeter, so Newtonian laws in Euclidean space give a good description of their relative movement.

We consider a large assembly of atoms, like a gas. If the atoms are evenly dispersed, the gravitational stress tensors in x,y,z between any two atoms remain near zero as density increases. There’s no lateral stress even at very high densities. This means that the instant volume occupied by the gas is Euclidean. In the Universe at scale, we make the approximation that the instant IGM is uniformly dense. This works well, as we will see later. Practically, the instant Euclidean approximation is accurate for any two atoms if they both lie in the same gravitationally unbound region of the Universe.

Time stress is a different story. There’s always some t stress in an unbound assembly of atoms, and there’s two tensors: gravitational and entropic. The gravity tensor changes monotonically with ρ. In Newtonian physics, attractive gravity stress in a model sphere can be expressed as dU/dρ, where U is the gravitational potential energy (33). We assume Newton’s G stays constant. A variable G gives Einstein’s time curvature. If G increases with time, so do the gravity tensors.

Repulsive entropic stress d(TS)/dρ has a different density dependence than U. At low ρ, d(TS)/dρ is strictly dominant over dU/dρ. At higher ρ, dU/dρ can wrest control, resulting in collapse into accreted bodies. This paper only considers low-density conditions in the IGM, where d(TS)/dρ rules.

Universal curvature is presently considered by astronomers be “very small” (Planck 2020). The present paper goes farther in that direction with two axioms:

These axioms consider the debate about Universal curvature as settled in favor of absolute flatness. This may be controversial, along with the neglect of non-Euclidean curvature near e.g. galaxies. At scale these bodies of accreted matter are only local perturbations in a much more voluminous and massive flat landscape. All available evidence suggests that the Universe has no curvature.

If all the energy in the Universe was bodies of accreted mass, its expansion could be fairly described with (16) and (17). However, as Einstein pointed out, CMB light has energy which also imparts mass density. CMB energy density drops off faster than that of accreted mass. To reconcile these differing rates of density drop, the fluid equation (21) was devised. Its derivation starts with (18), the engineer’s preferred expression of the first law of thermodynamics. This is not the same as (1). Engineers work with bound systems, and (18) describes the behavior of gas in e.g. a vessel:Where dE is the differential change of thermal energy U_i inside the vessel. Also in this vessel, Where dQ is the differential heat flow (J) to or from the vessel. A restriction is placed on (19), dQ = 0. So far, so good: The system is adiabatic,9 like the Universe. If dQ = 0 in (19), then dS = 0 as well. This precept is used to set dS in (18) to 0. However, a vessel’s boundary is required for heat to flow, or not, in (19). A vessel isn’t required for (18), but if dS in (18) is differentiated over time, a term TdS/dt arises which at scale cannot be set to zero since that is inconsistent with (2). This issue isn’t taken seriously enough in current theory. It’s instead skirted by removal of TdS prior to differentiation. The outcome is:

From (1), dE/dt should at scale be zero. Neglect of (2) leads to inconsistency with (1). Equation (20) is nonetheless used to derive the fluid equation (21):

By excising entropic gain, (21) inverts P into a gravity term. Pressure becomes a proxy for mass density ρ, or ϵ if you prefer. Gas “pressure” is now attractive, and trivially small.

The Newtonian expression of (21) is:

Equation (22) better shows why gas pressure is thought to be insignificant. The kinetic energy of any one atom is, from (15), dwarfed by its rest mass term. The fluid equation, however, isn’t accurate. It has a problem in its derivation: Application of a bound expression (18) to unbound conditions. This gives some odd results:

CMB entropic energy gain is unaccounted. This will be discussed shortly and isn’t especially important. It doesn’t affect H after last scatter and is only needed to balance the energy budget (45).

The Friedmann equation (16) is differentiated and combined with (21) to give the acceleration equation:

In (23), the expression dH/dt is governed only by G and energy (mass) density (ϵ + P). Entropic pressure $E_k^{\prime }$, which comprises part of P, is insignificant and even if significant would be an attractive term.

The acceleration equation (23) is inaccurate. It derives from the fluid equation (21) which is inconsistent with both laws of thermodynamics (1) and (2). These laws cannot be simply ignored. The attempt by the fluid equation to adhere to general relativity as the sole source of Universal behavior improperly conflates E between bound (18) and unbound (1) systems. Setting dS = 0 in (19) is conditionally allowed in a perfectly adiabatic bound system. However, transfer of dS = 0 from (19) to (18) is inconsistent with (2) at scale. This results in (20) which is again conditionally allowed when bound, but inconsistent with (1) at scale, so (20) is inaccurate. Equation (20)’s inaccuracy is then incorporated into (21), followed by (23).

The Jeans resonance model of star formation (Owen and Villumsen 1997) is relevant to our discussion here. At last scatter, both (23) and the Jeans model operated concurrently within any given volume. The Jeans model treats P as repulsive, an offset against gravitational collapse. The acceleration equation (23) treats P as attractive and is inconsistent with the Jeans model. The GCDM model treats P as repulsive, which is consistent with the Jeans model’s treatment of P.

The ΛCDM equation of state describes the relation between pressure P and density ϵ. It treats P as attractive, and has three terms: baryonic10, relativistic, and Λ:

The ϵ terms are the Einsteinian energy densities of baryons (ϵ_b), photons (ϵ_rel), and Λ (ϵ_Λ). The w terms are dimensionless numbers: w_b << 1, w_rel = 1/3, and w_Λ = -1. Equation (24) is combined with (23) to complete the ΛCDM model (36).

The mass of baryonic, everyday matter is nonrelativistic, which means it moves much slower than light: v’ << c. Its Einsteinian energy content is given by (15). Baryons comprise stars, cars, and helium balloons. Baryonic mass is considered attractive in the ΛCDM model. This might disturb a vendor watching his balloons implode. It’s repulsive in the GCDM model; the balloon vendor feels better. At last scatter, baryon mass was 100% elastically colliding atoms, i.e. a repulsive atomic gas: Helium and monatomic hydrogen.11 Presently, the repulsive : attractive ratio of baryon mass in the Universe is about 5:1.

The ΛCDM term $w_b{ϵ}_b$ is expressed as:Where μ is the mean atomic mass (kg), ρ_b is the mean baryon density (kg/m³), and k is Boltzmann’s constant, 1.38 x 10^-23 (m²kg)/(s²K). Without ado, (25) gives 1.088 x 10^-11 Pa at z = 1089, the same value obtained from the GCDM model’s equation of state (31). They are, for slow neutral atoms, equivalent expressions. Equation (25), however, treats baryon rest mass as part of the internal energy density ϵ. The baryons in the IGM are considered a perfect fluid or “dust” with almost all of ϵ contained in their rest mass. Thermal energy U_i is relegated to the status of a rounding error (shown in (25) as ≈), and -ΔU_i = E is unaddressed. In the GCDM model, -ΔU_i is the source of repulsion, and the Einsteinian energy density of rest mass is irrelevant (45).

Relativistic mass, expressed as $w_{rel}{ϵ}_{rel}$ in the ΛCDM model, is attractive in both models and arises from photon and neutrino energy (37)-(38).12 We digress briefly into photon energy. An expanding sphere of CMB light has an r^-4 dependence of energy density (Ryden 2017). Volume increases as r³, so there appears to be a 1/r loss of CMB energy upon expansion. During the “dark age” from last scattering until reionization began (Miralda-Escude 2003; Natatajan and Yoshida 2014), there was no coupling of the CMB with free electrons or stripped protons because there weren’t any. There was no mechanism through which that lost energy could perform work. It vanished and the energy budget became unbalanced. We get inconsistency with (1). I see no escape from this conundrum except to apply (3): CMB light yields entropic energy gain $E_{S_{\Delta \lambda }}$ through wavelength stretch Δλ. Any one CMB photon’s wavelength increases with time and their combined lost energy is the entropic gain at scale: where $E_{{CMB}_1}$ and $E_{{CMB}_2}$ are the before and after CMB energies, h is Planck’s constant (6.6 x 10^-34 J/Hz), λ₁ and λ₂ are the before and after wavelengths of the stretched photon (m), and $n_{\lambda }$ is the number of photons at a wavelength λ₁. The distribution $n_{\lambda }$ vs. λ for any z in the dark age is given by the Boltzmann curve at last scatter, T = 2971K. The ratio λ₂/λ₁ between two CMB states is akin to the scale factor a (34).

The above analysis of the CMB gives an individual photon’s entropy $S_{\lambda }$ as equal to Planck’s constant:

Its entropic energy $E_{S_{\lambda }}$ is the photon energy:Where $ʄ$ is the frequency of the photon (Hz; s^-1). Entropy is expressed as J/Hz rather than the more conventional J/K.13

Photon energy is 100% entropic. This makes sense, given that entropic energy gain is linked to E_k (9), hence volume increase (12). The rate of volume increase of radial light $d{V}_{\lambda }/dt$ in an unbound model sphere is: which far outpaces other radial energy in the sphere.

Current treatment of CMB energy is isoentropic, also begins with (18), and concludes that radiation expands more slowly than baryonic matter (Liddle 2015). A balanced budget may give a different result, if $E_{S_{\Delta \lambda }}$ is included in an ab initio derivation. In the observable Universe, $E_{S_{\Delta \lambda }}$ may affect E_k and H at z ≈ 1089, as free electrons were still present at z > 1089, and photons were more strongly coupled to baryon movement.

Electrons’ low rest mass makes them much more wavelike than nucleons. Electron entropic energy, like that of photons, is a function of wavelength (28).

The remaining term, $w_{\Lambda }{ϵ}_{\Lambda }$, describes repulsion. In the ΛCDM model, Einstein’s time-invariant Λ is used to account for the behavior of distant stars (Perlmutter et al. 1999). The term w_Λ = -1 arises because (23) treats P as attractive, so w_Λϵ_Λ has to have negative pressure. In the GCDM model the behavior described by w_Λϵ_Λ arises from suprathermal electrons, whose pressure P is repulsive. These electrons do create a scalar field, but unlike Λ its value changes with time. A time-invariant Λ field has a constant ϵ. This creates more and more energy at comoving scale, which is inconsistent with (1). If (1) is obeyed, Λ must change with time.14

The GCDM model follows a balanced energy budget. Energy is conserved through inclusion of E_k and $E_{{\lambda }_S}$ in the budget. We construct the model with a finite element method using the radius r of a sphere as the finite variable. A spreadsheet is used for the calculations. This is less satisfactory than an analytic derivation, but it does give solace in that the equilibrium expressions (30), (31), (49), and (50) are exact, as they describe changes in U_i which has a precise instant value (7). It is only in the partition of -ΔU_i between gain (44) and work (43) where error accrues. We must find a time period when the Universe was 100% gaseous and as homogenous as possible. That happened at z = 1089, the time of last scattering. Baryonic matter was all unaccreted atoms. We use the BBN estimate for baryons (Weinberg, 1988) as a mixture of about 75% hydrogen (H₁) : 25% helium (He) by weight, giving a mean atomic weight Ж = 1.24 x 10^-3 kg/mol. Hydrogen was monatomic and nonrecombinant to diatomic form, absent catalysis through aggregation. The isotropy in the CMB appears to indicate that the Universe at z = 1089 had a constant density, only minimally perturbed by the observed nascent Jeans resonance wiggles in the power spectra (Planck 2020). There is a metric not fully understood by this author, η_slip, found from the wiggles. It describes the accord between Einsteinian and Newtonian physics in a presumptively homogenous and isotropic Universe, and may be conversely used to estimate variation in mass density. At z = 1089, if η_slip = 1, then there was no variance, and this atomic Universe would have been homogenous and isotropic. The value of η_slip was found to be 1.004 ± 0.007. How exactly this translates to spatial density variation is unclear to me, but the text in Planck proclaims agreement between Einstein’s and Newton’s models for the presumed uniform gravitational potential. We proceed as follows: There was no accreted matter at z = 1089, and gas density variations from e.g. Jeans resonance were either averaged out or insignificant relative to the volumes used in the GCDM model, on the order of a sphere with r ≈ 10¹⁷ meters, or V ≈ 140 cubic parsecs. The wiggles tell us the atoms were dense enough to support the Jeans resonance, which is sonic pressure transmission vs. gravitational free fall. Since these atoms could transmit sound, they behaved like a gas back then so we can safely assume they had all the same properties we associate with gases today. The baryon density ρ_b(z=1089) was (Ω_bρ_crit)(1+z)³ = 5.46 x 10^-19 kg/m³. This is very low and we can say the gas behaved ideally in a thermodynamic sense. The critical density ρ_crit and the Ω values are given in Table 1 and are derived from table 6 of Planck. The CMB had decoupled right around then so the baryon temperature T at z = 1089 will be set to the extrapolated value (T_{CMB, z=0})(1 + z) = (2.726K)(1090) = 2971K.

The dark model, described immediately below, is constructed using equilibrium monatomic gas thermodynamic expressions, found in many introductory engineering textbooks and Wikipedia. Its z range, 1089→10, includes the entire “dark age” of the universe, hence the name. Its expression (58) is valid at z = 1089 as there was no high-energy light to perturb the model.

The light model, discussed later, has a range z = 10→0. Equation (58) is still used but one of its terms is adjusted to include suprathermal energy from cosmic and ß rays. The ß energy dominates and comes from impact of light upon electrons. One additional adjustment is made, to ρ_crit. This is constant (61), and precise in result.

Consider a comoving sphere of initial radius r₁ around a single atom of H₁, at 2971K and ρ = 5.46 x 10^-19 kg/m³. There are similar spheres around all the other atoms. Nonequilibrium conditions besides expansion, e.g. turbulence, Jeans resonance, etc. will be set aside so that the underlying transformation of conserved energy is more clearly described. There are two competing forces acting on the sphere: Repulsive entropic push, and attractive gravity pull. We are using a finite element method, so we define an increment: $\frac{\left(r_2-r_1\right)}{r_1}=\frac{\mathsf{\Delta }{r}_i}{r}$, which must be kept below 10^-4 for most purposes to minimize the partition error. I will use 10^-9, as low as the spreadsheet will tolerate. When the gas in the sphere expands, it must do so adiabatically, and there’s no void outside the sphere into which free expansion can occur. Under classic bound conditions, the comoving sphere would then have to lose U_i (through work). We postulate that applies in a cosmic setting as well. For monatomic gases this is:Where the numeric subscripts refer to the before and after U_i and V values. Volumes V₁ and V₂ are readily found (4πr³/3). The starting pressure P₁ is found from the equation of state for ideal gases:

If work against gravity is negligible, there is no alternative to free expansion within the sphere that I can find, so the released energy E from (30) is 100% entropic E_k. From (11), the finite differential E_k gives the entropic energy gain E_S:Where the subscripts refer to the before and after values on a T-S diagram. Volume increase is strictly local to the sphere. In an infinitely large Universe, it all just gets less dense.

An exception to the low-increment rule is that any size increment gives zero error in the calculation of -ΔU_i. You can get the temperature at any dark redshift just from the increment. This is discussed later (62)-(63).

The sphere has to get quite large before gravity begins to play any kind of role. To find out just how large, we now look at the gravitational potential energy U of the sphere:

The potential energy U must take into account the total mass $M^{\prime }$, not just the thermodynamic mass M of the baryons. In addition to baryon mass there’s cold dark matter (CDM) which is about five times as abundant as baryon mass. Its only interaction with baryons, electrons, or light, is through gravity. CDM does move relative to accreted baryons like stars, but all that occurs within the cosmic web, and at scale, does not affect H. A consistent description of CDM’s composition and origin remains to be found (Bertone and Hooper 2018). There’s widespread belief that CDM’s mass density evolution over time is inverse third-order in r, like baryons. We use this convention. Due to η_slip ≈ 1, its density at z = 1089 can be kept constant with respect to baryon density. Both follow 1/r³, as expressed by the scale factor a: where r₀ is the comoving radius of a sphere today, and z is the cosmic redshift used throughout this paper:Where λ_ob is the observed wavelength of light of known laboratory value, λ_em. There’s also relativistic mass from the CMB, whose comoving density follows 1/r⁴. This is addressed by the minimum flat-universe ΛCDM model:Where H_Λ is the ΛCDM Hubble parameter, H₀ is today’s Hubble constant (z = 0), and the Ω values are energy density ratios at z = 0. These are listed in Table 1. The Ω values are dimensionless and in theory, always add up to one at any given z. They share a common denominator ϵ_crit, and have identical values when expressed as mass density ratios using the common denominator ρ_crit. To get the relative density, and therefore mass $M^{\prime }$ for a given volume, we divide them by each other; the denominators cancel. This gives an Einsteinian density multiplier ɱ_E: which we use to get the total Einstein mass:

In an Einsteinian Universe, $M_E^{\prime }$ = 6.313M at z = 0 and increases to ${M^{\prime }}_E$= 8.336M at z = 1089. The reader may be curious as to why Ω_Λ wasn’t included in the calculation of $M_E^{\prime }$. It’s a repulsive energy term generated by the ΛCDM model and unrelated to the gravitational effect of mass.

In a Newtonian Universe, the mass equivalent of Ω_rel doesn’t exist. This gives a Newtonian density multiplier ɱ_N: which we use to get the total Newton mass:

Which is $M_N^{\prime }$ = 6.3111M for all z.

Throughout this paper, we assume that the ΛCDM model is an empirically perfect description of H vs. z. Neither ${ɱ}_N$ nor ${ɱ}_E$ matches H_Λ at z = 1089. The Einstein mass $M_E^{\prime }$ overshoots and the Newton mass $M_N^{\prime }$ undershoots. I will add a third multiplier, the J density multiplier ${ɱ}_J$, whose relativistic contribution ${\Omega }_{\lambda }$ is treated as an inverse j power: giving the total J mass:

Unlike ${ɱ}_E$ and ${ɱ}_N$ which derive from known theory, ${ɱ}_J$ is ad hoc. We proceed using a single term $M^{\prime }$ which may be any of $M_E^{\prime }$, $M_N^{\prime }$, or $M_J^{\prime }$, depending on context. The energy lost to gravity upon expansion of the sphere is:

Where U₁ and U₂ are the before and after gravitational potential energies, respectively.

We’ve seen that atoms can freely expand without colliding. Less obvious to the engineer reader is the fact that they can also perform work against gravity without colliding. How is this possible? Well, the Friedmann equation (16) does the same thing, but with stars instead of atoms. Any volume of space containing evenly dispersed atoms, however dilute, has a uniform gravitational potential energy at scale. When the atoms all move away from the central atom of a comoving sphere, they are climbing out of a gravity well caused by the reduced density resulting from their movement, and E_k diminishes accordingly. It is this loss of radial kinetic energy to gravity, not PV work, which is responsible for the “isoentropic” portion of -ΔU_i.

Combining (30) and (43) gives a finite differential E_k value which includes loss to gravity:

An expression of conserved Einsteinian energy upon expansion is given by (45):Where E₁ and E₂ are the total Einsteinian energies of the before and after sphere. The CMB gain $E_{S_{\Delta \lambda }}$ is decoupled from E_k at z < 1089 and separately expressed. Energy for nonrelativistic mass is given by (15). This is accurate: Relativistic mass increase [(m’/m) -1)] for an atom of H₁ at 2971K is only around 10^-9. Furthermore, the baryon rest mass M_b, electron mass M_e, and CDM mass M_c are unchanged so their rest mass energies Mc² cancel. It is only their Newtonian properties which are relevant for (44).

At any instant, as the radius r increases isotemporally, the mass density ρ remains constant and the subsumed mass in the sphere increases as r³. The loss and gain terms in (44) shift toward loss. When r reaches the adiabatic radius or endpoint r_e, they cancel, giving an adiabatic sphere: E_k = 0. The adiabatic sphere is a principal construct of the GCDM model. Energy is conserved within any one such sphere, indeed all of them, as they expand over time. The comoving imaginary boundary of a sphere with r = r_e is the adiabatic surface. This isn’t adiabatic in the classical sense. Energy can flow freely in both directions across the boundary, but any such net transfer between many spheres would always be zero. The term “adiabatic” is apt and so repurposed. In today’s Universe, the adiabatic surface around a central atom isn’t always spherical due to anisotropic stress from accreted baryons, e.g. stars. This happens near the cosmic web. Density variation also occurs locally in the IGM. These are inconsequential at scale. The cosmic web’s mass in this context is addressed later (60)-(61). At z = 1089, there’s no anisotropy, let alone a web, so it’s all spheres. The endpoint r_e is found from (44) by convergence of r around -U_r/E = 1. If we use the Einsteinian ${ɱ}_E$, we get r_e = 9.69 x 10¹⁶ meters, about 20 ly in diameter. If we use the Newtonian ${ɱ}_N$, a bigger sphere results: r_e = 1.28 x 10¹⁷ m. Either way we get an adiabatic sphere about 20 ly across.

For an adiabatic sphere, the postulate connecting classic to cosmic gas behavior in (30) is clearly seen. The thermal loss in the sphere just balances gravity, like a piston’s expansion just holding up a weight. The postulate holds for lesser, medium spheres, as differential work and E_k combined.

Spheres larger than adiabatic result in gravitational contraction. Although quite interesting, treatment of these large spheres as a description of gravitational collapse lies outside the scope of the present paper.

One might suppose that because E_k = 0 at r_e, the adiabatic sphere isn’t comoving: d(r_e)/dt = v = 0. That’s not true. It is, just very slowly: v > 0.15 The adiabatic sphere contains medium spheres: For r < r_e, E_k > 0. To find v, we have to figure out how fast these are expanding (46)-(55), and add up their combined radial speeds (56).

The finite differential E_k gives the increment radial velocity $v_s^{\prime }$:

This is best visualized as each and every atom in the sphere moving away from the center at $v_s^{\prime }$. The true picture is messier (6). Note that $v_s^{\prime }$ is increment-dependent: A larger $\frac{\mathsf{\Delta }{r}_i}{r}$ gives more $v_s^{\prime }$. This state of affairs can be sorted by following $v_s^{\prime }$ as a function of r. The cutoff radius r_c = 0.003r_e is important. Below r_c, loss to gravity is negligible and all these small spheres have the same E_k/M value to within 5 ppm (Figure 2):

For an adiabatic system with an imaginary boundary, dE =-PdV at the instant isoentropic limit. The minus sign is omitted. Combining (46) and (47) gives the initial radial velocity v_i:

We can compare this with E and see if energy is conserved (51). We expand a small sphere (r = r₁ = 1 x 10¹² m) by $\frac{\mathsf{\Delta }{r}_i}{r}$ =10^-9, giving P₂ and T₂. The pressure drop of an adiabatically expanding bound monatomic gas is given by:

And the temperature drop by:

We examine the partition error:

Development in (47) gets dV/dM from the ideal gas law (31) and not from the thermal energy (4). By use of (51) we find that v_i is better expressed by rearranging (4):

Which gives:

By use of (53) instead of (48), the partition error (51) is at its minimum (2 x 10^-8). Note that v_i is the fastest rate at which any sphere can expand. Small spheres are all expanding this fast, so an increment should not affect v_i. Again using (53), the partition error of v_i for our small sphere is:

Which is about as good as we are going to get with an increment, and sphere, this large. Now that we have a proper value of $v_i$, I propose that the radial velocity $v_s$ of any one medium sphere is given as:Where ${v_s^{\prime }}_0$ is the constant value of $v_s^{\prime }$ at r < r_c. Equation (55) gives a zero value at the endpoint, and gives v_i at low r. My guess is $v_s/v_s^{\prime }$ remains constant. The radial velocity v of the adiabatic sphere is the sum of the radial speeds of the contained medium shells, plus the small core:

At our chosen T, ρ, and Ж, for all r < r_c, v = 23.2 m/s. That leaves the remaining 99.7% of v to be found. Numerical integration of the sigma portion of (56) (Figure 3)16 gives 6103 m/s. Adding 23.2 to this gives 6126 m/s, or 0.7925 v_i. If r_c/r_e is kept constant, the proportion of v_i not lost to gravity, termed K, shows little change with any of $M^{\prime }$, ρ, T, or Ж. The term K is constant to the 4th decimal place. About 63%, (v/v_i)² = K², of E is converted to entropic E_k. Only 37% is stored by gravity. The term K² is the conversion ratio. In the special case of atoms separated by 2r_e, their adiabatic spheres are joined at a tangent point and they are moving apart at 2v. A line of adiabatic spheres, connected at their tangent points, can be constructed in the instant Euclidean space. Anywhere along this line, for any two atoms separated by a distance r, their recession rate $v_r$ is:

Rearrangement of (57) gives the fundamental equation:Where $H_G=v_r/r$ is the Hubble parameter of the GCDM model.

For any given $M^{\prime }$, deployment of (58) at varying T from 10K to 50,000K at z = 1089, or any other dark z value, gives the same H_G to five decimal places every time. The dark model is zero-order in temperature. It’s also zero-order in Ж. A universe made of xenon atoms (0.131 Kg/mole) at the same ρ and T returns 100% of our primordial mix. The mass density ρ is the only remaining independent variable in the dark model. This fact is hidden inside of (58) and not obvious from cursory inspection.

The differences between the models arising from the partition of mass into gravitationally bound and unbound domains evolves over the range z = 1089 to 10, and is unchanged from z = 10 to 0. “Dark energy” interferes with accurate visualization of this process at low z values. We remove the Ω_Λ term in (36), giving:Where $H_{\Lambda }^{\prime }$ is the ΛCDM H parameter, without dark energy. Both H_G (58) and $H_{\Lambda }^{\prime }$ (59) are purely density-dependent functions and we can look at their evolution without interference from extraneous repulsive effects. Figure 4 is a plot of H_G/H_Λ and H_G/$H_{\Lambda }^{\prime }$ vs. (z + 1) using data derived from each of the three density multipliers ${ɱ}_N$, ${ɱ}_E$ and ${ɱ}_J$. There’s a total of six curves, but it looks like just two or three due to overlap. There are two separate ranges of overlap: z > 10, where H_G/H_Λ = H_G/$H_{\Lambda }^{\prime }$ for each of the three $ɱ$’s, giving three sets of two curves, and z < 10, where the six curves converge to two sets, each set having the same H_G/H_Λ or H_G/$H_{\Lambda }^{\prime }$. Maximum convergence between all six curves occurs at z = 10, where the values are within 0.2% of each other.

We now focus on H_G/$H_{\Lambda }^{\prime }$ in the z ≤ 10 range (Figure 5). Relativistic mass from the CMB had largely disappeared by then. The ratio H_G/$H_{\Lambda }^{\prime }$ converged to a constant value 1.09 and remains so all the way to z = 0. The transformation of H_G/$H_{\Lambda }^{\prime }$ from 1.09 to 1 is achieved through a single precise adjustment, to ρ_crit. We multiply ρ_crit by a best-fitting mass partition ratio ${\rho }^{\prime }$: giving the repulsive mass density $\rho$: The two models H_G and $H_{\Lambda }^{\prime }$ now give nearly identical results from z = 10 to 0 for any of the three $M^{\prime }$. The substitution $\rho =0.84{\rho }_{crit}$ is shown in Figure 5, using $M_E^{\prime }$ for total mass. It gives a result of -0.05% of the ΛCDM $H_{\Lambda }^{\prime }$ values at z = 0, increasing to +0.1% at z = 10. Its variance is positive with z. The J mass $M_J^{\prime }$ gives the most uniform variance, -0.045% ± 0.004%, not shown. The Newton mass $M_N^{\prime }$ gives negative variance, decreasing from -0.08% at z = 0 to -0.22% at z = 10, not shown.

There is a physical event which underlies ${\rho }_{crit}$ → $\rho$, namely the accretion of mass into the cosmic web of galaxies. The repulsive mass density $\rho$ is defined as the IGM mass alone divided by the total volume, web included. There’s a reason for this definition: It’s the IGM that’s fed by U_i. Web mass isn’t repulsive like a gas and can be seen as having Friedmann behavior (16). Since $\rho =0.84{\rho }_{crit}$, we can conclude there’s 5¼ times as much mass in the IGM which separates the tendrils of the cosmic web as there is in the tendrils themselves. The IGM is estimated to occupy up to nine times as much volume as the web at z = 0. Any given tendril or node of the web isn’t expanding as much as the IGM surrounding it. That is different from two tendrils separated by an intervening IGM volume; they are moving apart at a rate comparable to the IGM’s rate of expansion. Density and distribution of mass within the cosmic web changes over time along with its structure, but any reduction in web density is far exceeded by the reduction in density of the IGM over the same time period. A scalar $\rho$ is an oversimplification of any local IGM variance which might better describe the development of partitioned domains, but ${\rho }^{\prime }$ gives an astonishingly good fit with $H_{\Lambda }^{\prime }$ at scale, and it tells us roughly when the mass partition into IGM and web was complete. Introduction of an ad hoc partition ratio does raise questions about its accuracy as a physical interpretation. For one thing, the cosmic web’s mass isn’t included in ${\rho }^{\prime }$. The entropic energy gain arising from expansion of accreted matter in the web is treated as negligible this way, and if its rest mass doesn’t change a constant partition ratio ${\rho }^{\prime }$ may result. I see no way at present to quantify the effect of accreted matter on entropic pressure except through ${\rho }^{\prime }$. Other questions might arise, but the fit of ${\rho }^{\prime }$ with $H_{\Lambda }^{\prime }$ is good and we’ll use these terms as dark baselines later.

Two new issues emerge.

The redshift z₂ can be obtained from any starting value z₁ by changing the increment $\frac{\Delta r_i}{r}$:

The dark temperature change $T_z$ can be found via (62) from (49) and (50). We let $z^{\prime }=z+1=1/a$. Given T_(z’=1090) = 2971K, we find via spreadsheet that $T_{z^{\prime }}$ is exactly:17Where $T^{\prime }$ = 0.002500631 is expressed as degrees K. At z = 60, T = 10K; at z = 10, T = 0.3K. The Universe was a very cold place just before stars appeared.18

The value of v_i in (58) is found by inserting (63) into (53):

The value of r_e in (58) must now be adjusted for both T and ρ.

For the T adjustment, the dark radius change $r_{e_2}/r_{e_1}$ vs. T at constant ρ and Ж is found exactly:

We use (63) to get:

For the ρ adjustment, the dark radius change $r_{e_2}/r_{e_1}$ vs. ρ at constant T and Ж is found exactly:

For nonrelativistic mass,

Combining (66), (67), and (68) gives:

Inserting (64) and (69) into the dark model (58) gives:

We adjust for relativistic mass using:

Where:

And:

A linear dependence on ${\Omega }_2^{\prime }/{\Omega }_1^{\prime }$ is found:When ${\rho }^{\prime }$ is set to 1, equation (74) matches the manually calculated values of H_G from (58) to within 0.0004% for all z = 1089 to 0. In this ideal case, one only needs to calculate $r_e$, T and $z^{\prime }$ at last scatter to get dark H values at lower z. However, mass accretion is untreated in (74) so a term ${\rho }_z^{\prime }$ must be inserted, from (67) as the square root:

In (75), the constant ${\rho }^{\prime }$ = 0.84 for z < 10 is now a variable ${\rho }_z^{\prime }$ for z > 10. Rearrangement of (75) gives:

To estimate ${\rho }_z^{\prime }$ from (76), stars or galaxies with a known emission profile are required, along with an estimate of their distance via their luminosity. These earliest stars’ light energy won’t perturb the dark model (Figure 5). To the extent we get better at seeing them, use of (76) to estimate ${\rho }_z^{\prime }$ will allow us to better understand the progress of accretion in the dimly lit Universe.

The variance of H_G/H_Λ due to added repulsive energy remains, as shown by the circles in Figure 5.