Thermodynamic pressure is caused by particles with velocity. Jean’s analysis indicates that accumulation begins when gravitational pressure overwhelms resisting thermodynamic pressure.

According to Wiki, the approximate value of the Jeans radius may be derived through a simple physical argument. One begins with a spherical gaseous region of radius, mass, and gaseous sound speed. Perturbations compress the gas slightly, and it takes time (called crossing time) for sound waves to cross the region, push back against gravity and re-establish the system’s pressure balance. Gravity will contract the system if not stopped and will do so based on free-fall time. When the sound crossing time is higher than the gravity free fall time, the gas cloud is unstable and will collapse (accumulate).

Gravity free fall time is derived below.

If a particle starts falling it will accelerate and fall a distance R in free fall time t.

Values from the proton model are key to understanding temperature related events during expansion. The initial kinetic energy for expansion = 10.15 MeV/proton. When expansion decreases the kinetic energy to 0.11 MeV, fusion to He4 dE = 2.0 (Equation (14)) increases temperature by a factor of 65.5. These values allow expansion temperature history be determined (Table 3 below).

Table 3. Temperature history of straight-line expansion.

Original energy 10.15 MeV/proton and nucleosynthesis energy have been depleted in the last row of Table 3 and the final temperature = 0.33 K. WMAP measured the current sky temperature = 2.73 K. The ratio 2.73/0.33 = 8.3 is important because it indicates that primordial CMB sky temperature variations have been overwritten. This changes the interpretation of WMAP measurements.

Equality is the point in expansion where mass density is equal to photon mass density. After equality mass dominates and waves are free to propagate. The conditions at equality for straight-line expansion are in Table 4.

Table 4. Conditions at equality.

Waves and large-scale perturbations exist in normal matter surrounding the developing black holes. Perturbations, called pancakes, were predicted and analyzed by Zel’dovich [9]. This caused dark matter and developing black holes to form large structures we call the cosmic web. Perturbations are normally transitory, but the black holes swept into the structure become stable. Expansion of the universe continues and the space between black holes expands. Table 5 below details the accumulation process based on a “Rapid cosmic evolution” [12]. The calculations below start with a single dark matter particle in line 12 and end when the dark matter is depleted at mass 3.6e39 kg. This establishes the number of galaxies at 3.4e50/3.6e39 = 1e11.

Table 5. Black hole accumulation.

3.2.1. Interpretation of the Angle 0.0104 Radians

At expansion ratio Z = 80300, the Jeans length = 5.43e19 meters is visualized as a spot that forms a small angle against the sky radius = 1.57e21 meters (values in red from Table 5).

The angle is conserved throughout expansion and can be used to estimate current large R.

The angle 0.0104 radians was observed by WMAP. According to their documentation, the value 0.0104 radians was the size of the peak micro-degree variation in CMB [4][11]. Figure 2 presents the geometry that allows overall radius (large R) to be determined from the 0.0104 radian distance between “spots” that this document identifies as galaxies. They were formed around black holes at Z = 9300 but the distance between them expanded with the universe.

Figure 2. Geometry that determines R from Jeans.

WMAP documented CMB variations with multi-pole moments. This analysis describes volumes within other volumes and can be converted to the radius of each volume. Multipole moment 220 is radius 4.12e24 meters. The angle 0.0104 is interpreted as large-scale structure. Table 6, line 1 below is the size of features (peaks and valleys) in the CMB documented by WMAP [4]. The features are labelled as cosmic web features in Table 6. The large-scale structure is associated with Zel’dovich pancakes. The column labelled galaxies is the distance between galaxies identified in Table 7 below. This structure forms around the black holes. The remainder of the features are described in reference 13 using cosmic web data.

Table 6. Cosmic web structure.

3.2.2. Mass and Energy Balance

Equation (2) number of particles = 1.49e78 is proportioned into two parts by the mass of dark matter particles and normal matter in Table 7. Each mass is accounted for down through the table as it is partitioned into black holes and normal matter galaxies that form around the black holes. The number and distance between features are determined from current density. The goal of the table is to account for all mass and energy as it changes form.

Table 7. Distance between galaxies.

There is evidence that galaxies are represented in the CMB at multipole moment 2000. The distance 5.7e22 meters on the left side of Table 7 is the key. The center column of Table 7 shows 1e11 galaxies after accumulation around the central black hole seeds in the left column. Volumes = mass/density allows the distance between galaxies to be calculated at the current time. In the center column of Table 7, the distance is 5.3e22 meters between galaxies.

This is comparable to the Table 7 left column distance 5.7e22 meters and an important clue that the small CMB observations are galaxies. The left column of Table 7 has the same spacing. This is consistent with normal matter galaxies forming around dark holes.

Energy balance: The proton model provided an initial value of 10.15 MeV/proton. Nucleosynthesis added energy but at the end of expansion the temperature was only 0.33 K. A calculation in topic 3.6 “Number of stars,” adds energy from 6.6e20 stars to achieve the measured sky temperature 2.73K.

Mass balance: The proton model identifies total mass = 2.42e51 kg (Equation (3)). Data indicates that the average mass of stars = 2e30 kg. Data for the number of galaxies is 1e11 to 2e11. The left-hand column of Table 7 has mass fractions 0.135 and 0.855. Densities is lower in the dark matter column because particles have less mass. There are several numbers that affect one another but the table accounts for 95% of 2.2e51 kg in the proton model [2]. The remaining 5% is normal matter inside galaxies but between stars. The galaxies and stars are gravitationally bound but the distance between them expands. Volumes are mass/current density. This allows distance between features to be calculated and compared to current observations. Observations indicate large variations in star mass, galaxy size and type. The average galaxy mass in Table 7 is 2e40 kg. The table is based on averages, but the overall mass balance supports the proton model total mass.

The SAHA criterion (Equation (21)) predicts the point where expansion has reduced temperature to the point electrons settle into orbits and become atoms. This is called decoupling and occurs when the SAHA value exceeds unity. Table 8 below shows the straight-line expansion model mass density and photon mass density as a function of time. Decoupling occurs at Z = 9300.

Table 8. Decoupling.

Before decoupling (Z = 9300), the normal matter universe consisted of plasma. At decoupling the speed of sound drops from V = 3e8/3^0.5 to a much lower value. This is the start of what cosmologists call the dark ages. Hydrogen gas absorbs light at the Lyman-alpha wavelength (Wiki) and obscures observations.

The Jeans radius analysis is shown below for the start of normal matter accumulation. Density, temperature, and radius from the straight-line expansion model are in Table 9.

Table 9. Jeans analysis for star accumulation.

At straight-line expansion ratio Z = 1421 the Jeans criteria for accumulation of H and He4 mass is satisfied. The sound crossing time analysis is based on this radius. For this radius gravity free fall time dominates sound speed. This means the volume is unstable and will collapse into stars. The volume that collapses has a Jeans radius of 2.5e19 meters. The volume*(density at Z 1420) = 6e40 kg is the mass enclosed by this radius. This volume has the potential of becoming stars. The initial radius between galaxies that form will be about 2.6e19 meters.

Rapid Cosmic Evolution

An equation called the touch down equation is derived below.

Mass must travel inwardly distance L to reach the accumulation surface. The derivation indicates that as mass gets closer, its acceleration increases because gravity is an inverse square phenomenon. The acceleration causes mass M to traverse R toward the center in delta time t. Mass inside distance R (volume 4/3*pi*L³) will accumulate to a point if not stopped.

1) Low-pressure volume around the Jeans collapse.

The touch down equation incrementally crunches mass to the density of a star after mass becomes unstable. A low-pressure volume develops around the collapsed mass.

2) Backfilling the low-pressure volume.

The flow calculated below backfills the low-pressure volume [12]. Table 10 below details the accumulation process. The accumulation calculation starts with a single proton (line 3). The volume of each incremental mass = mass/density (density from the straight-line expansion model). Its associated radius R is labelled R. This radius is the basis for the surface area that mass flows through at velocity V to accumulate with the starting mass. The flow velocity is derived from expansion kinetic energy. Its initial value is 71 meters/sec. Mass/density is converted to volume, and its associated radius is used to determine the flow surface area (4*pi*R²). Incremental mass = rho*area*velocity*delta time where rho is the density at this point in expansion. Delta time is the interval between step wise mass accumulation calculations. Delta mass in the bottom line is added to line 3 mass for the next increment.

Galaxy mass accumulation can be early (Z = 1420 to present) but this analysis is low since dark matter black holes accelerate accumulation.

Table 10. Accumulation simulation.

The accumulation calculated above is for a star, but stars are forming throughout the 1e11 galaxies (Table 8). Figure 3 and Figure 4 are plots of accumulation from Table 10. Figure 4 indicates that stars can easily form during this early period, but do not light up until around 1.6e29 kg (Z = 40).

Figure 3. Star accumulation from Z = 1400.

Figure 4. End of Figure 3, star accumulation curve.

Based on the proton model, the total mass of the universe = 2.2e51 kg. Table 8 indicates that 95% of the universe mass 2.2e51 kg has been converted to galaxies of stars surrounding black hole seeds. The number of stars was estimated based on the current temperature = 2.725 K. In the straight-line model, the temperature at 1.26e26 meters radius would be only 0.33 K, not the measured 2.725 K. The calculation for sky temperature in Table 11 is based on multiplying the energy radiated from the 6e18 m² surface area of stars at surface temperature 5778 K. In the example below, the energy for each star is estimated. The Stefan Boltzmann constant = 3.54e5 MeV/(m²*K⁴) is used below.

For 6.6e20 stars, the total energy is 4.5e54 MeV. This energy is radiated to the surface area of the universe at radius R = 1.26e26 meters.

Table 11 is for the end of expansion, where the sky temperature is 2.73K. The energy radiated from stars is calculated showing that 6.6e22 stars raises the sky temperature. The fourth root of 32 K⁴ is the required star fusion to raise the temperature from 0.33K to 2.728 K.

Table 11. Star energy required to achieve 2.75 K.