The GCDM Model

– A model for the reduction in Universal density over time, the “GCDM” model, is derived using gas thermodynamics with z = 1089 as the starting point. In the GCDM model, the Universe is pushing itself apart with internal gas pressure. A simple three-term Hubble expression H G is derived and found to be independent, or zero-order, in temperature and molecular weight of the gas. Isoentropic expansion of the gas at any z yields an entropic energy term which is modified to include energetic electrons, derived in turn from high-energy photons. These electrons are proposed as the source of the “dark energy” term found in the Λ CDM model. The presently preferred description of Universal expansion is the flat-universe Λ CDM model. Its empirical accuracy is accepted as high. The Λ CDM model treats Universal expansion as a function of the sum of the density parameters Ω , three of which ( Ω rad , Ω b , and Ω c ) have comoving mass densities ρ rad , ρ b , and ρ c . The fourth parameter Ω Λ , termed “dark energy”, has a density ρ Λ which is not comoving, but rather is the same for any volume of space at any time.

Where ( ) is the Hubble parameter at scale factor = 1 (1 + ) ⁄ , z is the cosmic redshift ( 9 − ' ) 9 ⁄ of an emitted photon of known ' , and ' is the present-day Hubble constant, 67.70 Km/sec/Mpc, 1 from the Planck 2018 survey. 2 The Ω values add up to one and relate their density to the present-day critical density 2*<= = 3 ' " 8 ⁄ = 8.6075 x 10 -27 Kg/m 3 , where G is Newton's constant, 6.67408 x 10 -11 m 3 /(Kg-sec 2 ): The critical density ρcrit is the total mass-energy density which, within the ΛCDM model, gives an exact balance to the energy loss from gravity over time. 3 By convention, the Ω parameters are expressed with the value h = H/100. Table 6 of ref. 2 gives the baryon Ωbh 2 = 0.022447, so at z = 0 Ωb = (10 4 x 0.022447)/H0 2 = 0.04898. Similarly for cold dark matter (CDM), Ωch 2 = 0.11923 and Ωc = 0.26014 right now. For Ωrad at z = 0, we treat neutrino energy as relativistic and combine its energy with CMB photons. 4 Ref. 2 gives the redshift of matter-radiation equality at z = 3387-3402, which corresponds at z = 0 to Ωrad ≈ 9.1 x 10 -5 . For ΩΛ, Ref. 2 gives ΩΛ = 0.6894. We get ΩΛ = 1-(0.04898+0.26014+9x10 -5 ) = 0.6908; we will use our value. An important variable in the GCDM model is the baryon mass density at z = 0. Its value is Ωbρcrit = 4.216 x 10 -28 Kg/m 3 by our calculation, close to Ryden and Pogge's value. 5 The sources of the ΛCDM model are the Friedmann and fluid equations, and the equation of state. We compare, in advance, the differences between the ΛCDM and GCDM models. The reader is referred to some more recent textbooks for a summary of the ΛCDM model. 6 We focus not on the equations themselves but their assumptions and conclusions, especially the fluid equation, where the differences between the models are most clear. Not all of these differences will be subject to quantitative analysis in the following discussion, but rather serve as postulates of the GCDM model for further quantitative treatment, analysis, and disproof.
The first difference is that the GCDM model considers only the internal kinetic energy < = K " ⊽ < " of gaseous baryons as contributing to expansion, whereas both the fluid and state equations view the baryon rest mass 0 = " as a component of internal energy, and therefore the overwhelmingly dominant constituent: Ub -Ui ≈ Ub.
The second difference is that in the fluid equation, the internal pressure P of the gas is attractive and retards expansion, whereas in the GCDM model, P is repulsive and the source of expansion.
The third difference is that the ΛCDM model makes no separate provision for entropic energy TΔS. The GCDM model includes TΔS.
The fourth difference is that in the fluid equation, Ub of a comoving volume is constant and there is no heat input Q. In the GCDM model, Ui is fed by Q from high-energy stellar photons and active galactic nuclei.
The fifth difference is that the dark energy component of the ΛCDM model is of unknown origin and nonconservative. In the GCDM model, this is caused by entropic energy gain TΔS from photoionization, and is conservative.
We construct the GCDM model using Newtonian laws, and include relativistic energy (mass) and entropic effects later on. First, we select a time: z = 1089, just after recombination. This is the earliest at which monatomic gas thermodynamic laws can be reasonably applied. Baryonic matter was almost all neutral gas and acoustic oscillation was minimal so the Universe had constant density. We use the convention that baryons were then present as a mixture of 75% monatomic hydrogen (H1) : 25% helium (He) by weight, or 92.256 mole % H1 : 7.744 mole % He. This gives a mean molecular weight Ж = 1.2399 x 10 -3 Kg/mol. 7 The baryon density ρ1089 was ( ' / 1 ) = (4.216 x 10 -28 )(1090) 3 = 5.46 x 10 -19 Kg/m 3 , or 2.6 x 10 8 atoms per cubic meter. The background radiation had just decoupled and the baryon temperature at z = 1089 will be set to the proposed value, 2971 K (CMB = 2.7255 K)(1/a = 1090).
Absent high-energy photons, a gaseous Universe is adiabatic insofar as Ui is concerned, and the Universe is a "closed" system, so if isoentropically expanding, Ui does all the work: -ΔUi = W.
In a classic setting, there are two kinds of adiabatic gas expansion: reversible and free. Reversible expansion is isoentropic by definition. When a gas expands reversibly, Ui decreases, the gas performs work, and the temperature and pressure drop. When a gas expands freely, Ui does not decrease and only the pressure drops. The temperature stays the same: Both happen cosmically. In cosmic isoentropic expansion ( R ), not all of the internal kinetic energy lost (-ΔUi) performs work against gravity. The excess, entropic energy, may give additional volume increase. This cosmic entropic expansion differs from classic free expansion in that entropic energy is lost to gravity. As with classic free expansion, the internal kinetic energy of the gas, and so its temperature, remains unchanged ( S ).
We start with isoentropic expansion. Consider a finite sphere around a single atom of H1, of radius r about Earth size (6.3781 x 10 6 m), at 2971 K, which at ρ1089 has baryon mass M = 593 Kg. This sphere is still in thermal equilibrium, a major but necessary departure from reality. Nonequilibrium thermodynamics must be set aside so that the underlying transfer of conserved energy is more clearly described. The sphere's gravitational potential energy (U) is: 8 The ΛCDM model contains cold dark matter (CDM), Ωc/(Ωb + Ωc) ≈ 84% of all cold mass in the Universe, and doesn't act as a gas. Its only influence is gravitational. This is also needed in the GCDM model, and included as a mass multiplier ɱ = (Ωb + Ωc)/Ωb = 6.3111: 7 Atomic weights: H1 = 0.00100797 kg/m 3 ; He = 0.0040260 kg/m 3 . Heavier elements and diatomic hydrogen are treated as negligible. Hydrogen is in its isotopic distribution. 8 The terms U1, U2, Ur, and Us refer to gravitational potential energy. The term Ui is used to denote the internal kinetic energy of a gas or plasma, and Ub, the rest mass of the baryons, is used briefly in an energy context.
The multiplier ɱ is adjusted to include photon and neutrino mass later on (eq. 29). The ideal gas law is: Where R is the universal gas constant (=8.31446 J-mole -1 K -1 ). The volume of a sphere is: When (6) and (7) are combined we get the internal pressure (P1): Entering our values for M, T, and r, we obtain P1 =1.09 x 10 -11 Pa. We will also suppose that the sphere isn't getting any bigger over time. It is but for now we'll say it isn't. We increase the sphere's radius by √1.01 h , giving a volume increase of one percent. 9 Work is performed against gravity: Where U1 and U2 are the gravitational potential energies at radii r1 and r2 respectively. Entering the values for M, r1 and r2 we find that Ur = 2.916 x 10 -13 J. The internal kinetic energy loss (-ΔUi = W = E) is, however, much greater than Ur: 10 Where W has the classic meaning of work performed by the gas, P1 is the internal pressure before expansion, and V1 and V2 are the before and after volumes of the sphere respectively. V and P can be calculated from (7) and (8). Entering these into (10) gives E = 1.17 x 10 8 J. This is 10 20 times as much energy released as absorbed. The excess (Ek) is now outward, radial kinetic energy: The radial kinetic energy r is exactly the entropic energy. They are the same thing: At r = 6 x 10 6 m, gravity loss is negligible and r = ≈ 10 8 J. The internal pressure drops to a new value, P2: Eq. (13) gives P2 = 1.06 x 10 -11 Pa. Dividing r by V2 gives the increase in entropic pressure ( R ): Our sphere was static to start so r g = 0. Our expanded sphere has ΔPS = 1.07 x 10 -13 Pa, or 1% of P2. It's important to emphasize that PS does not add to P2, but is instead a vector quantity which results in radial increase only. If we ignore or "freeze" Ui, each atom can then be seen as moving in a straight line away from the center, like a bunch of tiny rockets blasting away from their despoiled planet, or a bomb going off. 11 Entropic pressure already existed in the sphere since the Universe has been expanding all along.
The temperature drop is given as: The temperature drops from 2971 to 2951 K or 0.7%. This can be compared to the CMBderived temperature: (2.7255)•1090 ∛1.01 ⁄ "= 2961 K, or 0.3% for " = √1.01 h K . The gas is cooling slightly faster than the photons. Eq. (15) is also used to verify the calculations. The internal kinetic energy < of a monatomic ideal gas is given by: The new internal energy < X can be found with the T2 value from (15). The residual error < − ( < X + ) is exactly zero to the limit of the spreadsheet. We can also get temperature drops over large z changes with eq. (16). Starting at z = 1089 and T = 2971 K, we set the increment to 10, basically one large step, and get z = 108.9. This gives T108.9 = 24.6 K, and Ui,108.9 = 0.0083Ui,1089. Before star formation around z = 100, the Universe was colder than liquid nitrogen, having exhausted 99+% of its internal kinetic energy. 12 The linear rate of expansion, or increment radial velocity ( † X ) of the sphere is: 11 ). Some say that "all bombs are entropic" in the sense that there is an instant rise in Ui within the bomb, followed by a large entropic increase when the bomb casing ruptures. This analogy may help the reader to better understand eq. (12). 12 Calculations at this large of an increment are unreliable, but -ΔUi remains accurate regardless of increment size. † X = ‡ Direct use of (17) from (11) ignores the fact that the sphere is already expanding, so it's inaccurate and quite low. It's also increment-dependent; larger y* H * gives larger † X values. We can get correct values ( † ) at an instant in time with (11). We set y* H * = 10 -9 and increase r independently. First we define the gravity ratio (X): With our above T, ρ, and Δri/r held constant, we increase r stepwise. The mass rises and † X falls until the adiabatic radius, or endpoint (re) is reached, where X = 1 and ‹{ | ‹* = 0. This adiabatic sphere conserves energy around the central atom; its surface is the adiabatic surface. Below the cutoff radius (rc = 0.003re), gravity can be neglected and (11) This gives the initial radial velocity (vi): We can compare this with the PV change (E) upon incremental increase and see if it's consistent.
We increment a small sphere, giving T2, and examine the residual error (21): By use of (21) we find that in fact, vi is better expressed by rearranging (16): which gives: By use of (23) instead of (20), the residual error (21) is minimal (2 x 10 -8 ). Again using (23), the residual error of vi for a small sphere is: Which is about as good as we are going to get with an increment this large. 14 Now that we have a proper value of < , we can suggest an expression for † : † = Where † X • is the constant value of † X at r < rc. Eq. (25) gives a zero value at the endpoint, and gives vi at low r. The endpoint is found by convergence of r around X = 1, which at ρ(z =1089) = 5.46 x 10 -19 Kg/m 3 and T = 2971 K, gives re = 1.28 x 10 17 meters (4.1 parsecs).
The choice of rc is best seen graphically. Figure 1 shows a semilog map of vs/vs0 vs. r/re from 10 -5 re to 0.1re. 15 Below rc/re = 0.003, vs is constant to 4 ppm and drops at higher r/re as gravity takes its toll. The radial velocity (v) of the adiabatic sphere is the sum: At our chosen T and ρ, for all r < rc, v = 23.2 m/sec. That leaves the remaining 99.7% of v to be found. Integration of (11) is problematic so we resort to a map ( Figure 2) whose cumulative value at r = re is 6103 m/sec. Adding 23.2 to this gives 6126 m/sec, or 0.79245±0.00005 vi. If rc/re is kept constant, the proportion of vi not lost to gravity, or curvature K, shows little change with either ρ, T, or Ж; K is constant to the 4th decimal place. 16 In the special case of atoms separated by 2re, their adiabatic spheres are joined at a tangent point and they are moving apart at 2v. More generally, for any two atoms separated by a distance r, their recession rate * is: Rearrangement of (27) gives the fundamental equation of this paper: 14 Spreadsheet errors begin to creep in at Δri/r < 10 -9 . At Δri/r = 10 -8 , the residual vi error is 1.4 x 10 -4 . 15 The temperature and density were found to be irrelevant. Also, both rc and re are independent of Δri/r over a wide range; Δri/r = 10 -8 was used for Fig. 1. 16 Initial HG values reported herein were derived starting with eq. (20) using 998-999 steps of linearly increasing r/re, beginning at rc and ending at r/re = 0.999 or 1. The integrals were calculated with the plotting program, Dplot, giving third-order correlation > 0.9999 in all cases. Replacement of the integral constant with vi(rc/re) gave the reported K = v/vi, 0.79245. Using 499 data points gave the same result.
Where T = * ⁄ is the comoving Hubble constant of the GCDM model. At z = 1089 eq. (28) gives T = 4.79 x 10 -14 /sec, or 21816HΛ0. This is 0.950 or 95% of the HΛ,1089 value found from (1). If we set K = 1, we get T = 6.04 x 10 -14 /sec, or 27531HΛ0. This is 120% of H Λ,1089. Use of (28) at varying T from 100 to 4000K at z = 1089 gives the same result to five decimal places every time. More extensive input change reveals no temperature dependence. The model is also zero-order in Ж. A Universe made of xenon atoms (0.131 Kg/mole) at the same density returns 100% (HG,1089 = 21816 HΛ,0) of our primordial mix. The mass density ρ is the only remaining thermodynamic variable in the model, and it's a function of z. This relation between HG and z is strictly monotonic and exclusive of other variables.
We now include relativistic mass. In general relativity, energy and mass are equivalent. Relativistic energy from photons and neutrinos 17 makes a contribution to gravity through its equivalent mass density ρrad, and is incorporated into the M' term of (5) through the mass multiplier ɱ: which at z = 1089 gives ɱ = 8.3363, about a 1/3 increase. Consequently re shrinks to 9.69 x 10 16 m with a resulting increase in HG: 6.32 x 10 -14 sec -1 . This is 125% of the HΛ value found from (1).
We now examine the GCDM model at z = 1 to 10, where non-Planck astronomers actually look. This gives substantial negative deviance from the ΛCDM model at z = 0 to 2. We suppose this is due to dark energy. A more useful comparison of the two models is thus performed by removing ΩΛ from (1): Values of HG/HΛ and HG/ ′ 3 vs. z are shown in Figure 3 for z = 10 to 0. 18 While HG/HΛ drops off sharply at low z, HG/ ′ 3 remains constant. This linear relation suggests that the two models are simply connected by the slope. We suppose that the slope deviance (1.09-1 = 0.09) of the line (HG/ ′ 3 vs. z) may be from a missing entropic term. We've seen that isoentropic treatment of an expanding sphere of monatomic gas gives entropic energy r inside the sphere. This entropic energy may give hitherto unaccounted radial increase. We add an entropic increment R = 1 − " : 17 As before, this considers neutrino energy to be entirely relativistic for all z. 18 These numbers were calculated at T = 2971K; any T may be used and gives the same result.
(<"R) = < + R Both increments occur at the same time and so the same speed, but additional distance is traveled with R . This is found by iteration. At r < rc, the gravity terms are negligible, so we will say the increments are equal: (<"R) = 2 < . 19 When r > rc, gravity kicks in and R shrinks, 20 analogous to an ascending rocket that's run out of fuel. The entropic gravity loss R • is: Where R • is the starting value for iteration of the free increment. We recalculate r : Where the initial values are r • = r and r g is the kinetic energy gain adjusted for entropic gravity loss. We resize R : The new R is entered back into eqs. (32) -(34) to get new values R ª and , repeating until R (ª«g) = R (ª) . 21 The obtained ¦* § ¦* H is used for the next step, and the steps carried out as for the isoentropic model, to the same endpoint. The resulting entropic curvature KS, also shown in Fig. 2, is only minimally less and has a similar invariance with ρ, T, and Ж: KS = 0.9935 K. Entropic expansion doesn't seem to be very important in the overall scheme of things so we continue to use K and not KS. 22 Our slope deviance issue, HG / ′ 3 vs. z (= 1.09), wasn't solved by an entropic term, so we look elsewhere: the critical density ρcrit. In the ΛCDM model, ρcrit is found from H0. With GCDM, "ρcrit" is not dependent on H0 so we can choose a best fit value: £ = 0.84 2*<= . This adjustment, also shown in Figure 3, gives a result within 0.05% of the ΛCDM ′ 3 values from z = 0 to 5. The gravitationally unbound mass fraction in the Universe is presently estimated at about 0.85 and its volume fraction at around 0.9, giving an unbound density of 0.94 2*<= . Our 19 The entropic increment R • can be made larger but the end result is the same. 20 Since both r1 and r2 are independent variables, the isoentropic increment < is unchanged. 21 These iterations were performed sequentially and not recursively. Ten is enough for convergence to six decimal places. 22 There may be some dependence of KS on the number of data points. More steps may give KS→K, but this hasn't been thoroughly investigated.
finding suggests that the volume fraction of unbound matter is closer to 1 than present estimates, or the mass fraction is lower than 0.85, perhaps as low as 0.76. We are tempted to conclude from Figure 3 that the partitioning of Universal mass into gravitationally bound and unbound domains happened very early on, but such conjecture is best left to the reader.
None of the above discussion comes any closer to explaining the phenomenon of dark energy. The GCDM model gives us the means to do so from known and conserved sources, rather than from the proposed (and nonconservative) "vacuum energy field" of the ΛCDM model. The GCDM model has three terms: vi, K, and re. If we want to increase HG, we need to increase vi or K, decrease re, or some combination. This boils down to finding a source of entropic energy that does not derive from the internal kinetic energy Ui of the adiabatic sphere. We use the expression: Where R is the added entropic energy and ß is the change in entropy from a source ß. We modify (23) by adding R : As yet we have no a priori means of evaluating ß and therefore ES, but we can find values for (Ui+ES) which give a best fit with HΛ. Since £ = 0.84 2*<= gives an excellent fit to £ over the range z = 0 to 5, we will use £ ( ). We use the same temperature, 4000K, for all calculations. The curvature K is kept at 0.79245; re is left unchanged. 23 The best fit value is found by varying ES/Ui to convergence around HG/HΛ = 1. A plot of best-fit ES/Ui values vs. (z+1) is shown in Figure 4. At z = 0, ES/Ui = 2.24, and drops steadily to 0.01 at z = 5. Whatever the source, R is clearly the most important contributor to HG presently, and played little role in earlier times.
We now propose that high-energy photons are the source of R . Star formation is believed to have commenced around z = 100, maybe earlier; new old stars are being found at this is written. Some give off light with photon energy sufficient to ionize H1, yielding a proton and a free electron. This photon energy is partitioned into potential energy gain < , electron energy ß , and added proton kinetic energy ¯« : Where n is the number of impact photons, Eλ is the mean photon energy, h is Planck's constant = 2 x 10 -25 J-m, λ is a suitable averaged wavelength of the impact photon, c is the speed of light = 3 x 10 8 m/sec, Ei is the work function of the impacted particle, Eß is the mean energy of the emitted electron, and ¯« is the mean kinetic energy added to the proton. For H1 the work function Ei is its ionization potential (13.6 eV or 1.6 x 10 -19 J). A plasma is created, which at this density has thermodynamic behavior much like any other monatomic gas. 24 We focus on the latter two terms of (37), Eß and ¯« , and return to our sphere upon which the GCDM model is based. The direction of motion of the protons and electrons after photon impact is random. In order for them to make any meaningful contribution to (26) through (36) they have to be moving faster than v, or they will never cross the adiabatic surface of the sphere, and their movement simply adds to Ui. We calculate v at T = 4000K and Ж = 0.00124 Kg/m 3 , which from (23) gives vi = 8971 m/sec and v = Kvi = 7109 m/sec. For an impact photon of hc/λ = 14 eV, 13.6 eV is absorbed and the remaining 0.4 eV is partitioned into proton (¯«) and electron ( 9 ) kinetic energies; the electron carries away 99.945% of the energy. 25 The particle speeds £ 9 and £¯« are, at hc/λ = 14 eV, respectively: which isn't enough to escape the sphere. It appears that the principal source of ES is photoionized electrons. These do not necessarily have to come from hydrogen. There's plenty of, e.g., helium around, albeit requiring a much higher hc/λ, 24.6 eV.
This added entropic energy, which corresponds to dark energy in the ΛCDM model, is a function of the impact photon flux across the surface of the adiabatic sphere. A complex subject, it's well beyond the scope of our discussion. 26 Briefly, the high-energy photon content of the Universe is estimated from the Wein tail of the Boltzmann distribution function at the surface of short-lived type O giant stars, 27 and their rate of formation. 28 One thing we can conclude from Fig. 4 is that the amount of high-energy photons is up in recent times, and their sources, such as type O stars and the like, are present in increasing number.
The author declares no competing interest. 24 Actually, plasma at low density is much more kinematically responsive than the corresponding neutral gas. Protons, for example, are charged and repel according to an inverse-square law. This causes path deflection at distances many times that of the van der Waals radius of the neutral atom. 25 Momentum is conserved. 26 For some details of photon production, the reader is referred to chapters 4 and 9 of ref. 5. 27 In the Harvard system (OBAFGKM), "O" is the hottest. 28 There is also a contribution from active galactic nuclei, subdwarf O stars, and probably other sources as well. Normalized sphere radial velocity vs. normalized sphere radius Best-t values of E S /U I for H G = H Λ

Supplementary Files
This is a list of supplementary les associated with this preprint. Click to download. GCDM20210106.xlsx