Cosmic Gas Thermodynamics at z = 1089


 The Universe at z = 1089 is treated as an expanding ideal gas. Its internal kinetic energy loss exceeds the amount absorbed by gravity and drives further expansion. A Hubble relation (Hg) is derived and compared to the value found by the ΛCDM model (HΛ) over the range z = 1089 to 0. The results suggest that the adiabatic release of energy from cosmic gas accounts for about half of present-day Universal expansion.

The presently preferred description of Universal expansion is the flat-universe ΛCDM model, given in simple form by (1): Where ( ) is the Hubble parameter at scale factor = 1 (1 + ) ⁄ , z is the cosmic redshift ( − 0 )⁄ of an emitted photon of known 0 , and 0 is the present-day Hubble constant, 67.4 Km/sec/Mpc or 2.184 x 10 -18 sec -1 . The Ω values or density parameters add up to one and relate their energy density to the present-day critical density = 3 2 ( 0 ) 2 8 ⁄ : In eq. (1), Ωm = 0.0486 is baryonic matter, Ωcdm = 0.259 is cold dark matter, Ωrad = 9.00 x 10 -5 is relativistic energy (photons and neutrinos), and ΩΛ ≈ 0.69 is the "dark energy" parameter. The ΛCDM model treats Universal expansion as a function of the sum of the mass-energy components Ω, three of which (Ωrad, ΩM, and ΩCDM) have constant comoving energy densities εrad, εM, and εCDM. The fourth parameter ΩΛ has a density εΛ which is not comoving, but rather is the same for any volume of space at any time. The critical density εcrit is the total mass-energy density which gives an exact balance to the energy loss from gravity over time. The empirical accuracy of the ΛCDM model is high and we use it to calibrate our model. This paper explores the energy release associated with density reduction ("expansion") of cosmic gas and plasma, by treating those portions the Universe not bound by gravity as an ideal gas.
First, we select a time: z = 1089, just after recombination. Baryonic matter was almost all neutral gas and acoustic oscillation was minimal so the Universe had constant density. Baryons were then present as a mixture of 75% monatomic hydrogen (H1) : 25% helium (He) by weight, or about 92 mole % H1 : 8 mole % He. This gives a mean molecular weight Ж = 1.2475 x 10 -3 Kg/mol. Monatomic gas thermodynamic laws can be reasonably applied to this time period. 1 The 2018 Planck survey 2 gives a ΛCDM-based value for today's mean baryon density: ρ0 = 4.21 x 10 -28 Kg/m 3 . The baryon density ρ1089 was thus ( 0 / 3 ) = (4.21 x 10 -28 )(1090) 3 = 5.45 x 10 -19 Kg/m 3 , or 2.6 x 10 8 atoms per cubic meter. The background radiation had just decoupled so the baryon temperature was ≈ 2971 K (CMB = 2.7255 K)(1/a = 1090). 3 The Universe as a whole is an adiabatic system. 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, its internal kinetic energy 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, but with differences. In the first, cosmic isoentropic ("reversible") expansion, the internal kinetic energy (Ui) lost isn't equal to the energy stored by gravity. The excess becomes vectored kinetic energy. The second, cosmic free expansion, derives from the fact that the entropy of the Universe always increases over time. 4 On a cosmic scale, gas expansion can't be exclusively isoentropic as no time would have elapsed. There must also be an entropic volume increase. These two forms of gas expansion are linked: one can't happen without the other. We develop our model isoentropically ( only) and examine later on.
Consider a finite sphere around a single atom of H1, of radius about Earth size (1 au = 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: 5 Where G is the gravitational constant (6.67408 x 10 -11 m 3 kg -1 sec -2 ). The ΛCDM model also contains cold dark matter (CDM), Ωcdm/(ΩM + Ωcdm) = 0.842, ≈ 84% of all cold mass in the Universe, and doesn't act as a gas. Its only influence is gravitational. This is included by dividing the baryon mass by 0.158: The ideal gas law is: Where R is the universal gas constant (=8.31446 kg-m 2 sec -2 mole -1 K -1 ). The volume of a sphere is: When (6) and (7) are combined we get the internal pressure (P1): Where ρ is the mass density. Entering our values for M, T, and r, we obtain P1 =1.08 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 3 , giving a volume increase of one percent. 6 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 = 2.92 x 10 -13 J. The internal kinetic energy loss (-E) is, however, much greater than Ur: 7 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 = 1.16 x 10 8 J. This is 10 20 times as much energy released as absorbed. The excess (Ek) is now outward, radial kinetic energy: 6 We define the increment as Gravity loss is negligible and Ek = E ≈ 10 8 J. The internal pressure drops to a new value, P2: Eq. (12) gives P2 = 1.06 x 10 -11 Pa. Dividing Ek by V2 gives the increase in expansion pressure (ΔPE): Our sphere was static to start so 1 = 0. Our expanded sphere has ΔPE = 1.06 x 10 -13 Pa, or 1% of P2. It's important to emphasize that PE does not add to P2, 8 but is instead a vector quantity which results in radial increase only. Each atom is moving in a straight line away from the center, like a bunch of tiny rockets blasting away from their despoiled planet. It helps to ignore Ui to properly visualize this. Expansion 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 → √1.01 3 au. The gas is cooling faster than the photons. Eq. (14) is also used to verify the consistency of the calculations. 9 The linear rate of expansion, or radial velocity (vs) of the sphere is: Direct use of (15) ignores the fact that the sphere is already expanding and 1 ≫ 0, so vs is inaccurate and quite low. We can get corrected values for vs at an instant in time by modeling (11), using a small increment Δri/r = 10 -9 and increasing r independently. 8 Since one of our assumptions is thermal equilibrium, P 2 remains a state function. 9 The internal energy of an ideal monatomic gas is given as = First we define the gravity ratio (X): With our above T, ρ, and Δri/r held constant, we increase r1 stepwise. The mass rises and vs falls until the adiabatic radius, or endpoint (re = 1.2732 x 10 17 m), 10 is reached, where X = 1 and = 0. This adiabatic sphere conserves energy around the central atom.
The cutoff ratio ( ′ ) is defined as: Below the cutoff radius (rc), gravity can be neglected and (11) simplifies to (10). The choice of X' < 10 -5 is best seen graphically. Figure 1 shows a semilog map of vs vs. r from 10 12 to 10 16 m at 2971K and ρ1089. 11 Below rc (= 4 x 10 14 m) vs is constant to 2 ppm, giving the same (E/M): 12 This gives the initial radial velocity (vi): Above rc, gravity takes its toll, and vs drops, reaching zero at re. The radial velocity (v) of the adiabatic sphere is the sum of the contained shells: Where is the step, ′ = � 0 �, and 0 is the constant value of vs at r << rc. For all r < rc, v = 19.8 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 = 1 is 4973.2 m/sec. Adding 19.8 to this gives 4993 m/sec, or (0.7934±0.0001) vi. This proportion K shows little change with input values; v/vi is constant to nearly the 4 th decimal place.
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 vr is: 10 The endpoint r e is found on a spreadsheet by convergence of r around X = 1. 11 Both r c and r e are independent of Δr i /r over a wide range. 12 For isoentropic adiabatic expansion, And the comoving gas-derived Hubble value ( ) is: Which at z = 1089 and K = 0.7934 gives = 3.87 x 10 -14 /sec, or 17955H0. This is 78% of the value found from (1). If we set K = 1, we get = 4.94 x 10 -14 /sec, or 22630H0. This is 99% of the value found from (1). It appears that isoentropic treatment of the gas at vi is more consistent with the ΛCDM model than treatment at v/vi: Use of (23) 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 that (23) has no temperature dependence. The model is also independent of molecular weight. A Universe made of xenon atoms (0.131 Kg/mole) returns the same result as our primordial mix. Other than the CDM adjustment, the mass density ρ is the only remaining variable in the model, and it's a function of the cosmic redshift z. This makes z the sole independent variable. 13 We now examine entropy, by looking at . To do this, we use a two-increment model at r << rc, where the expansion is first performed reversibly and then freely, giving r1-r3 and U1-U3. The first increment r2 -r1 generates vs. Free expansion of the sphere r << rc has no gravity loss and occurs simultaneously with reversible expansion, so in the second increment r3 -r2 the atoms coast along at vs for the same time. The two increments are equal. From (3): And from (18): Which again gives (18).
higher increment Δ = 0.001, U3-U2 = 0.006 Ei and � ′ = 0.93, so there is some model-based dependency best addressed by keeping the increment low. We can more accurately see the effect of entropic expansion by examining a linear map of /H Λ vs. z from z = 999 to 19 using equations (23) and (1) (Figure 3). The entropic expansion at z = 1089 is arbitrarily set to zero. The influence of ΩΛ is negligible in this z range, and the variance kz is well fit by (26) We include kz in (23), giving (27): This new density ρ" = 0.85ρ gives an adjustment of 0.92 and a new ratio Hg"/H0 = (0.92)(Hg'/H0) = 0.51. This is more than the sum of Ωm + Ωcdm + Ωrad = 0.31, which means that Hg" is responsible for some of H0 ascribed to ΩΛ in (1). The model outlined herein does not further address that issue. It does, however, demonstrate that a known source of energy, Ui, can be used to account for much of Universal expansion. While a dark energy field cannot be excluded as a source, the results of this paper suggest that more mundane sources, e.g., kinetic energy stored in gases or plasma, make a substantial contribution.
The presented model only superficially addresses the issue of entropic expansion. Proper treatment of entropy using classic gas thermodynamic principles, and applied to cosmic conditions in more recent times, may yield meaningful results.
The author declares no competing interest. Figure 1 Uncorrected sphere radial velocity vs. radius at z = 1089

Figure 2
Corrected sphere radial velocity vs. r/re at z = 1089 Uncorrected and corrected Hubble ratios at z = 0 to 10

Supplementary Files
This is a list of supplementary les associated with this preprint. Click to download. cosmicgasworkbook20211012.xlsx