Mass models of the proton and neutron described in [1] are included for reference below in Table 1. Energy values for neutron and proton components are arranged into columns in Table 1. The simulated masses of the neutron and proton are marked in red below (accuracy vs. PDG data is shown). The models include an initial neutron expansion kinetic energy of 10.15 MeV and a potential energy of 10.15 MeV. The energy of 10.15 MeV/nucleon is also fundamental to atomic fusion. The neutron decays to a proton, electron, and anti-electron neutrino, and its mass and fields are listed along with the decay products toward the bottom right side of Table 1.

Table 1. Neutron and proton mass simulations.

The models contain values useful for a cosmology expansion model described in Topic 3. Specifically, the gravitational field energy is 2.801 MeV (upper right side of columns 2 and 4), and the initial expansion potential and kinetic energy are 10.15 MeV/proton, shown at the bottom of the tables. The models are based on information values labeled N that are correlated with energy. Probabilities for individual components are p = 1/exp(N). Multiplying the probabilities for all energy and field components, probability p = 1/exp(180) can be assigned to an individual proton. Since many are known to exist, a probabilistic argument can be made that, to restore unity, there were exp(180) neutrons in nature, most of which are now protons. There is also a particle on the left side of the table that consists of mass without kinetic energy. Its mass is 130.163 MeV and will be treated as dark matter.

Two previous studies conducted using the above values were reported. They are entitled “Structure formation study” [2] and “Galaxy formation study” [3]. Each presented promising results.

Energy E = hv is energy and frequency v = 1/time are fundamental. This is also written r = hC/E where h = Planck’s constant H/(2pi) = 4.136e−21/(2pi). The value hC = 1.97e−13 MeV-m.

Figure 1. Space and time model.

Table 1 provides the gravitational field energy E = 3*0.687 + 0.740 = 2.801 MeV. Figure 1 shows the radius r and time t around the circle at velocity C below.

Inertial force Fis proportional to velocity²/radius and velocity² is proportional to kinetic energy (ke).

F = m v 2 / r 0 = m V 2 / r , v 2 / V 2 = r 0 / r , k e / K E = R / r With initial radius r₀ = 7.045e−14 m and initial ke = 10.15 MeV/proton, the radiusr increases as ke decreases.

It was stated above without proof that expansion is driven by time. This is novel because expansion has been treated as a fluid density with kinetic energy that resists gravity. Understanding the forces on a proton on the surface of an expanding cell requires reviewing recent work on gravitation.

The true mass of the three quarks in the proton model in topic 3 totals 129.54 MeV. But the quarks have kinetic energy, and the total proton mass = 938.27 MeV. The equations developed below use a particle without kinetic energy with mass 130.163 MeV from Table 1. According to Jean’s theory [wiki], a particle without kinetic energy cannot develop pressure that normally resists gravitational accumulation. The name dark is appropriate because it forms early black holes that do not light up with fusion like normal matter. The particle was studied and shown to contribute to our understanding of cosmology in [2][3].

A derivation for the gravitational constant G was presented in Reference [1]. Excerpts are repeated below.

From Equation (1), r = 7.0448 e − 14 = h C / 2.8011 .

Equation (6) is equivalent to the conventional Compton mass derivation of the gravitational constant, but it is based on a field energy, a normal proton, dark matter, and the constant 1/exp(90). More can be learned about the effect of 1/exp(90) on gravity by studying Table 2. It compares equations that use the value 1/exp(90) that agree with the conventional equations for gravity and inertia.

Table 2. Equations for gravity that include 1/exp(90).

The line labelled Universe/proton is a model of a proton on a cell with kinetic energy 10.15 MeV.

The Table 2 column labeled cell radius r = 7.0445e−14*10.15/ke (Equation (3)).

Orbital R can be simplified to:

The conventional force of gravity F_g = GMm/R² in Table 2 is compared with the inertial force, F_i and F_new that each use the constant 1/exp(90). The three values agree for all orbits studied.

This equation, F_new, can be understood as follows. The force is shifted to a low value by 1/exp(90) and is proportional to the gravitational field energy E = 2.801 MeV. The force increases with inertial kinetic energy (ke) and further increases with small cell radius r. The ratio (central mass M/2.49e51) is the same ratio used to determine orbital radius R from cell radius r.

Table 2 shows equalities based on values that compare orbits for extreme conditions. Table 2 line 5 is for a central mass equal to the total, and line 6 is for a central mass of one proton. In line 5 (labelled Universe/proton), cell size r = 7.045e−14 m for ke = 10.15 MeV, but with exp(180) protons, the orbital radius R = 8.59e25 m. If a proton orbits another proton at a radius of 7.045e−14 meters, ke = 10.15/exp(90) MeV/proton.

The proton is not attracted to the center of a cell that has no mass, and it is not experiencing the huge inertial forces that would be associated with a small radius because the associated radius is huge. From a gravitational and inertial viewpoint, the cells are on a large surface near the radius of the present universe. The force F_new = 3.6e−38 Newtons is based on the central mass of the universe over a huge distance = 7.045e−14*exp(90) = 8.6e25 meters.

This is consistent with expansion that is time-dependent, having nothing to do with forces across an expanding cell. To be clear about gravity, there is a force between protons that repositions cells. The proton kinetic energy decreases with expansion according to Equation (3), and initial kinetic energy is converted to potential energy toward the end of expansion. This can be calculated from Fnew and Orbital R, but these are macroscopic changes having nothing to do with the forces on a small expanding cell.

Knowing each proton’s kinetic energy history allows properties like temperature and density to be simulated over time. For example, the temperature history = ke/(1.5*B) K, where B = Boltzmann’s constant 8.62e−11 MeV/K, and the density history is based on proton mass divided by cell volume.

Table 3 provides equations for the reader to construct the straight-line model.

Table 3. Equations for straight-line expansion model.

Each variable in Table 3 depends only on exponent N. To construct the model, write the equations in a spreadsheet. Initial radius R = 7.045e−14*exp(60) = 8.05e12 meters based on exp(60) cells in each of three dimensions. Starting with N = 60, add small increments of N across a spreadsheet. Large R increases with R = 7.045e−14*exp(N). Expansion time t = 1.476e−21/(2pi)*exp(N) seconds. After N = 64.57, multiply the temperature by 65.5 to account for fusion to 28% He4 mass fraction. The current expansion radius R = 7.045e−14*exp(90.384) = 1.262e26 meters. The model energy remains constant during expansion, with potential energy plus kinetic energy = 20.3 MeV.

Check results against Table 4.

Table 4. Beginning and current radius and time.

WMAP [4][5] and PLANCK [6][7] mission analyses were based on the Lambda CDM expansion curve. It is compared to the straight-line model in Figure 2 using the same time axis. The LCDM expansion curve appears to flatten late in expansion. Compared to the expected decrease, expansion was reported to be accelerating in Reference [8].

The sequence of events after the beginning is reviewed below [2][3].

1) Information operations separate the logarithmic value 90 into components identified by N that add to 90. Values of N are a pattern we recognize as neutrons,

Figure 2. Comparison of expansion models.

protons, and electrons, but their associated fields are equal and opposite in energy. Each particle has probability = 1/exp(180).

2) Inflation in straight-line expansion consists of duplicating particles exp(180) times. After inflation (duplication), neutrons are adjacent to one another at radius R = hC/2.801 = 7.045e−14*exp(60). This occurs at 2.68e4 seconds.

3) There are two types of particles [1]. One is a normal neutron, but the second neutron has zero kinetic energy. It will be called dark matter and has a mass of 2.31e−28 kg [2].

4) The normal neutron decays to an orbiting proton and initially has 10.15 MeV/proton of expansion kinetic energy (ke). The decrease in kinetic energy causes expansion. Primordial nucleosynthesis [9] occurs at kinetic energy 0.11 MeV [1]. This adds 0.28*7.06 = 2.0 MeV/proton to the kinetic energy as H is fused to He4.

5) Equality (expansion ratioZ = 3.4e6) is the point in expansion when mass density becomes larger than photon mass density. After equality, waves start to propagate because mass is no longer dominated by photons, and dark matter particles start to collapse into black holes. Normal matter forms Zel’dovich perturbations called pancakes [9], and dark matter is swept into webs, filaments, and voids. This is the origin of the cosmic web.

6) Zel’dovich pancakes become the large-scale Jeans radius structures observed by WMAP at the angle 0.0104 radians. The dark matter that accumulates [2][3][10]-[13] into black holes becomes the seeds for 1e11 galaxies. At Z = 88400, the 1e11 black holes reach 3.5e39 Kg. This depletes dark matter and “freezes” perturbations, but expansion continues between dark holes.

7) Decoupling for normal matter occurs after expansion ratio Z = 3677. This breaks the plasma, and electrons orbit protons. Expansion ratio Z = 1420 [2] is the point Jeans criterium is met for accumulation. Accumulation of normal matter proceeds around the 1e11 black hole seeds. After stars reach a critical mass, temperature, and pressure at about 1.6e29 kg, they light up with fusion and are observable.

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

The approximate value of the Jeans radius may be derived through a simple physical argument [Wiki]. 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.

After equality, waves and large-scale perturbations exist in normal matter surrounding the developing black holes. Perturbations were predicted and analyzed by Zel’dovich [9]. This caused dark matter and developing black holes to form large structures observed as the cosmic web [14]. Perturbations are normally transitory, but the black holes accumulate and stabilize the structure. Expansion of large R continues, but the space between the large-scale gravitationally bound structures expands. Table 5 describes mass accumulation. Black hole accumulation starts with a single dark matter particle in the line labelled Black hole mass and increases across Table 3 by adding “delta mass” increments from the bottom line of the table. The properties required for calculations are based on Table 1.

Table 5. Black hole formation from dark matter.

Number of dark matter black holes = total dark matter/3.45e39 = (129/938*2.5e51)/3.45e39 = approximately 1e11. Normal galaxies accumulate around these seeds.

2.5.1. Interpretation of the Angle 0.0104 Radians

At expansion ratio Z = 88589, Jean’s radius is visualized and was later measured by WMAP as spots that form small angles against the sky, with radius R = 1.409e21 meters.

The angle, conserved throughout expansion, is scaled by Z from radius 1.409e21 meters to the current large R.

Current large R = Z*R radius = 89589*1.409e21 = 1.262e26 meters.

Figure 3. Geometry that determines current R from R when Jean’s radius forms an angle of 0.0104.

2.5.2. Calculations for Hubble’s Constant

The straight-line model Table 3 equations yield current time 4.21e17 sec shown on the right side of Table 4. Table 6 determines the Hubble constant H₀ = 1/4.21e17 = 2.375e−18/sec*3.086e22/1000 = 73.291 km/sec/mpsec.

Table 6. Calculations for the Hubble constant H₀.

2.5.3. Error Analysis

The proton model mass prediction [1] is compared to NIST and PDG [14][15] data in Table 7.

Table 7. Accuracy of the proton model.

Field energy = 2.8011 MeV. The delta for this energy = ±1e−5. Determining E = 2.8011 MeV causes H₀ error = ±4e−6. Jean’s speed is certain because sound speed = C/3^0.5 is determined by plasma physics, but density is dependent on radius R at Z = 89589 [5]. Uncertainty in the angle for R causes Jeans to vary by ±0.008%. Measurement uncertainty in angle 0.0104 radians is, according to the WMAP 9-year document ([5]: p. 130), angle 0.0104 ± 2.2e−5 radians. When the angle is adjusted in the model to 0.010422, H ′ 0 = 73.291013 is calculated by the geometry in Figure 3 and calculations in Table 6. This causes an error = ±2.7e−4. Error analysis is summarized in Table 8.

Table 8.H₀ error analysis.

Hubble’s constant H₀ in this analysis = 73.291 ± 0.008 km/sec/mpsec.

The straight-line expansion H₀ = 73.291 value can be compared to published data in Figure 4 marked in black and blue. It represents a Wiki review of Hubble measurements from the original velocity vs. distance data in 1929 to the present. Measurement methods have improved dramatically, and recent Hubble Space Telescope [6] and James Webb Space Telescope are producing data with lower error bars. The Wiki review is below, indicating that H₀ falls into two groups. WMAP and PLANCK data are about 67.5 km/sec/mpsec and the remainder of the data clusters are around 73.

Figure 4. Copy of Wiki Hubble data summary.

Expansion model equations help understand the difference between H₀ values derived from the straight-line model and lower WMAP and PLANCK H₀ values from the LCDM model.

2.7.1. The LCDM Outward Expansion Model

The LCDM model is based on “density” being projected outward against gravity, converting kinetic energy to potential energy (like throwing a stone into the air). The well-known [9] critical density derivation is shown below.

Derivation for Lambda CDM critical density:

The value V/R = H₀ with critical density = Rho = M/(4/3*pi*R³).

The FriedmannEquation.

From critical density derivation:

With G*M a constant, theR’s in the derivation are combined; (radius’/radius) increases with (time’/time)^2/3. This is known as the Friedmann equation. It is the basis of the Lambda CDM expansion model and was used for data analysis from the WMAP and PLANCK missions. Their value of H₀ is also based on the CMB peak of 0.0104 radians [4][5]. It is scaled to largeR with the LCDM model and gives large R = 1.3e26 meters. H₀ is 1/time for that radius = 2.13e−18/sec = 67.5 km/sec/mpsec.

The LCDM model is a composite model. Its components of normal mass, dark matter, and dark energy, shown in Figure 5, must add to the critical density = 1.

Figure 5. Components of the LCDM model.

There is a delicate balance between the Friedmann normal mass and dark matter component where R ′ / R = ( T ′ / T ) 2 / 3 and the upward slope of the dark energy component [8] that increases with R. Eventually, dark energy will dominate and the composite green curve will be upward sloping.

2.7.2. Alternate Approach to Expansion

Cells are volumes that contain one proton each, and with exp(180) cells in a 3-D universe, R = r*exp(60). When NASA sends payloads to the Moon, the rocket points up and around the Earth into an orbit. The straight-line model is based on a proton on the surface of a cell. But topic 3.1 indicated that all exp(180) cells must be used for gravitation inside a massive universe. As tangential kinetic energy decreases, the size of the orbital R (refer to Table 2) increases according to Equation (3) based on energy conservation. To be clear, each individual cell is not expanding due to any force. It is expanding as a function of time. Nature is conserving energy as radius r increases.

Figure 6. Straight-line expansion velocity is tangential, not radial.

Tangential velocity in Figure 6 that describes a proton in orbit around a large central mass according to topic 3.1 provides an alternate way of exchanging kinetic energy for potential energy. Both expansion models exchange kinetic energy for potential energy according to Equation (11), but in the tangential model, velocity = Rt/T around the cell perimeter.

The kinetic energy of the cell is 1/2 MVt² with Vt = Rt/T. For straight-line expansion, Rtangential and Rvertical cannot be combined to complete the derivation of critical density in Equation (11). This also means that the Friedmann equation R'/R = (Time/Time')^2/3 does not apply to the straight-line model.

The concept of critical density does not apply to tangential velocity-based expansion. Differences between the models are listed below.

1) The straight-line model gives H₀ = 73.29 km/sec/mpsec and the LCDM model gives 67.5 km/sec/mpsec.

2) WMAP and PLANCK report odd percentages of critical density (4.6% normal, 21% dark matter, 71.3% dark energy) that have not been substantiated. The straight-line model angle of 0.0104 radians is based on a dark matter particle new to cosmology.

3) The straight-line model combined with data from the proton model allows energy history to be calculated. 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/proton, fusion to He4 multiplies the temperature by ((0.25*7*4 + 0.11)/0.11) = 65 throughout the remainder of expansion. Table 9 is the straight-line temperature history during expansion.

WMAP measured the present sky temperature = 2.73 K. Original energy 10.15 MeV/proton and nucleosynthesis energy have been depleted in the last row of Table 9, and the temperature is 0.33 K. This is important because it indicates that star fusion increases sky temperature from 0.33 to 2.73 K. With this difference, WMAP and PLANCK might not have been able to shield radiation from stars from the primordial Cosmic Microwave Background [4][5].

1) Reference [14] entitled “Origin of the cosmic web and re-interpretation of CMB” indicates that the true source of the CMB is radiation of energy from stars in the cosmic web.

2) The LCDM model is unable to explain the observation of fully formed galaxies as early as a few million years after the beginning. References [2] and [3] demonstrated that the straight-line cosmology is compatible with early observations of galaxies.

Table 9. Temperature history of straight-line expansion.