Cosmic Gas Thermodynamics at z = 1090 and z = 0


 Dark energy is shown to be the internal kinetic energy of intergalactic plasma.

In all the cosmos, density reduction ("expansion") must be adiabatic. In cosmology, adiabatic expansion is treated as isoentropic. We use that here but it's not really true. 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, internal energy is not lost and only the pressure drops. The temperature stays the same: Both happen cosmically, but with differences. In the first, cosmic isoentropic expansion, the internal energy lost isn't equal to the energy stored by gravity. The excess becomes vectored kinetic energy. The second, cosmic free expansion, must also be included. There's an additional entropic component of expansion, from the second law. This gives more volume to accommodate the same amount of gas, just like classic free expansion. Unlike classic free expansion, internal energy is lost, to gravity.
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 ρ1090 and Ж has mass M = 455 Kg, or 2.02 x 10 8 atoms/m 3 . 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: 2 Where G is the gravitational constant (6.67408 x 10 -11 m 3 kg -1 sec -2 ). Cold dark matter, with Ωcdm = 0.259, has ≈ 84% of all cold mass in the Universe, and doesn't act as a gas. Its only influence is gravitational. 3 This is included by dividing the baryon mass by 0.16: 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): Entering our values for M, T, and r, we obtain P1 =1.07 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 i , giving a volume increase of one percent. 4 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 = 8.53 x 10 -11 J. The internal energy loss (-E) is, however, much greater than Ur: 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: 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 expansion pressure (PE): Our sphere was static to start so r^= 0. Our expanded sphere has PE = 1.06 x 10 -13 Pa, or 1% of P2. It's important to emphasize that PE does not add to P2, 5 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 linear rate of expansion, or radial velocity (vs) of the sphere is: which is addressed below. 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 ⁄ ƒ= 2961 K, or 0.3% for = 1 → √1.01 i au. The gas is cooling faster than the photons.
We can get a value for r^a nd vs at an instant in time by modeling (11), using a small increment Δri/r = 10 -10 and increasing r independently.
First we define the gravity ratio (X): As we increase r1, the mass rises and vs falls until the endpoint (re = 1.409 x 10 17 m), or adiabatic radius, is reached, where X = 1 and ‰t u ‰* = 0. The sphere at re is the adiabatic sphere, and its surface is the adiabatic surface.
The cutoff ratio ( Š ) 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  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 y is the shell thickness or step, y ' = ? --" -" \˜, and y \ is the constant vs < rc. For all r < rc, v = 24.3 m/sec. That leaves the remaining 99.6% 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 4956 m/sec, or 0.79vi. 8 For any two atoms separated by a distance r, their recession rate vr is: And the comoving Hubble value is: Which at z = 1090 gives H = 3.83 x 10 -14 /sec. While expansion pressure PE may be universal, it's very much a local phenomenon, dependent on local density, temperature, massive objects, etc. Furthermore, for all z post-nucleosynthesis, most Universal volume had a positive PE. When z > 1090 or z < 100, Ui was replenished by irradiation. By z < 8, the Universe was fully reionized, resulting in steady-state Ui in regions of low ρ. These ionized, low-density regions dominate Universal volume today, and define its expansion as a volume-proportional scalar energy field Ui of protons and ionized helium nuclei. 9 For example, we can define a region as having ρcrit/10 = 8.62 x 10 -28 Kg/m 3 at 10 4 K. For this region, vi = 11,545.5 m/sec, rc = 1 x 10 19 m, re = 5.95 x 10 21 m, v/vi = 0.790, and H = 1.5 x 10 -18 /sec, or 0.70H0. Raising the temperature to 10 5 K gives vi = 36,510 m/sec and H = 0.69H0. Raising the density to ρcrit gives H = 0.70H0.
To summarize: For large gaseous regions of low and homogenous density, every proton in the region is surrounded by a finite sphere of radius re, with the internal kinetic energy of the protons in that sphere exerting outward radial force on those protons.
For inhomogenous regions, the adiabatic surface around a proton isn't spherical. Of interest are the regions near the plane of the spiral arms of galaxies. The endpoint in these regions varies, can exceed the interstellar distance, and the regional composition of baryonic matter defies a simple expression of re.
Entropic volume gain, or cosmic free expansion, occurs simultaneously with cosmic reversible expansion and is more difficult to treat. Internal energy values of the 92/8 plasma are needed for calculation at z = 0. We can, however, get an expression for vi with added entropy, by looking at the increase in . Free expansion of the sphere r < rc has no energy cost and occurs simultaneously with reversible expansion. From (3): And from (19): So with entropic expansion added: This suggests a version of (21): Where K ≈ 0.7 -0.8 is a constant to be found. We can set K = 0.79 as before, while keeping the same endpoint value. This gives H = 0.49H0, about half. We know the endpoint will shrink from entropic gravity loss; this would appear to be around 50% in order to give H = H0. A more precise determination of both K and re is in order.
In summary, we have found a result consistent with what is seen through the telescope. The recession of galaxies is mostly caused by expanding hot intergalactic plasma. If there is some other form of dark energy, the conclusions of this paper should be taken into account.