The orbital velocity of stars in galaxies and the gravitational lensing of galaxy clusters indicates that there is gravity (curving of space-time) that is not caused by ordinary matter. It is generally assumed that this gravity is caused by dark matter, which is a proposed type of particle that curves space-time, but that has not been directly detected in space or in particle colliders. We describe an alternative to dark matter, where the gravity attributed to dark matter is caused by curvature in space-time created during inflation.

We provide a framework for interpreting fundamental physics and we discuss the expansion of space-time, gravity, the relationship of particles to gravity, inflation, and gravitational waves in light of the framework. We use that discussion to elucidate how space-time structures generated during inflation curve space-time and cause gravity. We also show that pre-inflation stochastic gravitational waves provided the energy for such space-time structures.

Expansion

Space-time can be represented as a mesh of small cubes of space, where each cube comprises an X dimension, a Y dimension, and a Z dimension. The space-time cubes are also connected to time, which ticks at all locations in space. Time can be represented as a special type of dimension. Each of the six faces of a cube matches exactly to the corresponding face of an adjacent cube.

When a tick of time occurs, each space-time cube experiences a discrete period of time and the X, Y, Z, and T dimensions all expand by extremely small, but equivalent amounts. Accordingly, the volume of a space-time cube grows very slightly larger with each tick of time and the magnitude of a tick of time increases by the same proportion as one of the dimensions.

Since X, Y, and Z expand as a function of time, the space-time cubes must comprise a potential that gets actuated into the expansion as time progresses. The constant expansion of space-time is similar to a compressed elastic membrane that was compressed until X and Y got very small and Z got large, where the compression has stopped and the elastic membrane is in the process of decompressing. When the membrane decompresses, the X and Y dimensions of the membrane are expand, while the Z dimension contracts. The elastic potential energy is transformed into expansion of the X and Y dimensions of the membrane and into the contraction of the Z dimension, where a small thinning of the Z dimension enables a large expansion of X and Y.

Similarly, we can consider each cube of space to be a four dimensional membrane with X, Y, and Z being extended dimensions that are expanding with time and with each cube further including an additional small orthogonal dimension that provides a type of thickness for the membrane. The energy from the decompression of the space-time membrane causes X, Y, and Z to expand with time and it causes the additional dimension to contract or thin with time. The space-time membrane has an elastic potential energy which is actualized by the decompression of X, Y, and Z and by the thinning of the additional dimension.

The additional dimension is much smaller than X, Y, and Z. However, a small thinning of the additional dimension corresponds to a large increase in X, Y, and Z. Accordingly, the state of expansion of each cube can be defined by the magnitude of the extra dimension or by the magnitude of any one of X, Y, Z, and T. The space cube’s potential energy for expansion and that potential's actualization can be thought of as a physical manifestation of the positive cosmological constant. When the elastic potential energy exhausts and X, Y, and Z reach the end of their expansion, the cosmological constant will transition through zero to negative, which will cause the energy of expansion to transition into the potential energy of elasticity.

In accordance with Einstein’s theory of General Relativity (GR), the geometry of space time is altered by the presence of mass. Space-Time is represented as a four dimensional non-Euclidean space. Mass causes curvature in that 4d space. The magnitude of curvature caused by a mass increases as you approach that mass and as the magnitude of the mass increases.          The magnitude of the curvature of space caused by a distribution of mass can be calculated with the GR equations. Macroscopic objects follow a geodesic path through the curved four dimensional space. However, particles can’t follow geodesic paths through space time, since the path of a particle is defined by the evolution of the particle’s probability wave.

The probability of detecting a particle at a point in space is determined by the Schrodinger equation, where the solution to the Schrodinger equation provides vectors (kets) that define the probability of detecting a particle at an eigen value of position. [1], [2] The probability of detecting the particle at a particular Eigen value of position is determined by taking the inner product of the vector with itself.

The path of a charged particle interacting with the Earth’s magnetic field and traveling through the gravity of the Earth/Moon system can be numerically determined using the GR equations and the Schrodinger equation. We could define a base cell of a variable cell mesh small enough such that a finite element solution of the time dependent Schrodinger equation with a magnetic potential converges in a truncated Hilbert space. [1], [2], [3] The X, Y, Z, and T dimensions for each cell of the Hilbert space are contracted according to the GR equations and in accordance with the contraction factor calculated using the contraction of time. The volume of a cell in the mesh would be equal to the cube of the magnitude of a dimension of the base cell in non-curved space multiplied by the contraction factor. Each cell of the mesh would be a software object having the following elements: X, Y, and Z coordinates for each of the vertices; magnitude of the magnetic field for that cell; a magnitude of T; a contraction factor; a probability vector; a total probability for the cell; and the width matrix.

For each time step, the magnitude of time, the contraction factor for each cube, and the coordinates of its vertices would be numerically calculated using the GR equations.

Second, the magnitude of the magnetic field for each cell would be determined. Third, the particle would be evolved for the time step, where the evolution involves numerically solving the partial differential equations of the Schrodinger equation using an iterative finite element technique. Each cell evolves by the magnitude of time associated with that cell. Also, the width matrix is multiplied by the scalar contraction factor for the specific cell and thereby concentrates the probability into the more contracted cells.

Since energy in the form of mass causes X, Y, and Z to contract and since it causes the additional dimension to expand, that energy associated with the mass must exist within the additional dimension. However, the energy of mass comes from three distinct forces. Accordingly, there must also be three distinct additional dimensions. Each force is associated with a distinct dimension that is connected to X, Y, Z, and T, but that is not connected to the other additional dimensions. Energy, in the form of oscillations of space-time, in any one of the additional dimensions, will cause that dimension to expand equivalently to the total expansion of all the additional dimensions caused by increasing the elastic potential energy by the same amount. The energy of mass will cause X, Y, and Z to contract in accordance with the contraction factor calculated using the contraction of time. Accordingly, with each tick of time, X, Y, Z, and T get a little bit bigger and all of the additional dimensions associated with forces get a little bit smaller.

Particles consist of oscillations of space in the additional dimensions. Massless particles would be oscillations similar to traveling waves and they would have no stationary reference frame. Massive particles would be similar to a standing waves and have a reference frame in which they have a fixed location. However, these particles, which consist of oscillations or groups of oscillations, would have to follow the rules of quantum mechanics. Accordingly, each particle would be located at a superposition. The oscillations in the additional dimensions would be located at all allowable eigen states of position in X, Y, and Z, except when that particle undergoes a detection and it collapses to an oscillation at a particular eigen state of X, Y, and Z. The magnitudes of the superposition of oscillations would be described by a Gaussian and that magnitude would correspond to the magnitude of the particle’s probability wave, where a greater magnitude of oscillation corresponds to a greater probability of detection.

Instead of a single distinct dimension, each force could be associated with a distinct set of dimensions. The sets associated with each force contain multiple connected dimensions that are also connected to X, Y, and Z. However, the dimensions form one of the sets are not connected to the dimensions from the other two sets. Having multiple dimensions would provide additional degrees of freedom to match up with all of the known properties of particles. Accordingly, the electromagnetic force, the weak nuclear force, and the strong nuclear force would each comprise a set of dimensions that are connected to X, Y, and Z, but that are not connected to the additional dimensions of the other forces. For symmetry considerations, we assume that space dimensions always come in matched triplets, such that each force will comprise a set of three identical connected dimensions.

Some particles, like electrons, would consist of a single standing wave that oscillates in only one set of the additional dimensions. Other particles, like protons, are composite particles that consist of a group of waves that have oscillations in multiple additional dimensions. Since the dimensions of the individual forces aren’t connected to each other, composite particles would also include an oscillation within X, Y, or Z to maintain resonant relationship among the oscillations. Such an oscillation would occur entirely in X, Y, and Z, unlike a gravitational wave which, as discussed below, is an oscillation between X, Y, and Z and the additional dimensions. Because of their stationary position in the overall resonant wave, the oscillations entirely in X, Y, and Z would form standing waves. However, in the instance of nuclear decay, these oscillations wholly in X, Y, and Z can obtain an independent existence as neutrinos.

The additional dimensions would function similar similar to a wave guide. In a wave guide, the magnitudes of small perpendicular dimensions determine the allowable modes of oscillation for the waves propagating through the extended dimension. Similarly, the magnitudes of the additional dimensions determine the allowable modes of oscillation for the particles moving through X, Y, and Z. The magnitude of the dimensions of the strong nuclear force will determine the oscillation modes and wavelength of a quark, where the different types of quarks would correspond to different modes of oscillation.

The magnitude of the dimensions of the electromagnetic force will determine the oscillation modes and wavelengths of an electron and of a muon. The magnitude of the weak nuclear force dimensions would determine the oscillation modes and wavelengths for the W and Z bosons and the Higgs boson.

When a particle enters more contracted space, it enters space with larger additional dimensions. Accordingly, the particles will have larger wavelengths, which means the particle will have less internal energy. Accordingly, a particle’s kinetic energy will have to increase as the particle goes into contracted space.

The particle properties are determined by wave mechanics, the details of which still need to be worked out. However, the electromagnetic force likely derives from a type of constructive/destructive interference of the probability waves, which correlate to the superposition of oscillations in the additional dimensions. Protons and neutrons and quarks likely transmit gluons to maintain the appropriate distance for resonance.

Oscillations of energy in the additional dimensions will cause those dimensions to increase in size, which cause will X, Y, Z, and T to decrease in size. Accordingly, the more massive particles that get collected into a volume, the more contracted that X, Y, and Z become in the vicinity of the volume. This increase in contraction further concentrates the particles probability waves in the direction of the collected particles, which increases the contraction of X, Y, and Z and the increased contraction causes curvature in four dimensional space-time.

Because there is no parallel postulate, the distinct sets of dimensions merge into connected sets of dimensions, if the additional dimensions are sufficiently expanded by the presence of mass. First, the dimensions for the electromagnetic force and the dimensions for the weak nuclear force would merge into a single set of six connected dimensions. As the additional dimensions continue to expand and as X, Y, Z continued to contract, the dimensions for the strong nuclear force would connect to the previously connected sets to provide a nine dimensional space (thirteen including X, Y, Z, and T). Accordingly, even if X, Y, and Z contracted to to disappearance, there would still be 9 space dimensions for the universe to exist in.

In the first few moments of the expansion of X, Y, and Z, X, Y, and Z were very small and the additional dimensions were very large. Accordingly, the sets of dimensions for the strong nuclear force, the weak nuclear force and the electromagnetic force were all connected. The 13 dimensional space with the nine additional dimensions can be thought of as a type of resonance chamber for determining the allowed modes of oscillation for waves traveling through X, Y, and Z. The connected nine dimensions provided all of the oscillation modes associated with the dimensions for the strong nuclear force, all the oscillation modes associated with the dimensions for the weak nuclear force, and all of the oscillation modes associated with the dimensions of the electromagnetic force. However, the connected nine dimensions would also have provided cross modes of oscillation that included oscillations in dimensions of two or more sets. A large percentage of the energy in the early Universe would have existed in these cross modes.

When X, Y, and Z expanded sufficiently, the set of dimensions associated with the strong nuclear force disconnected form the electroweak dimensions. When that disconnection occurred, the cross modes involving the strong nuclear force dimensions would have become impossible. The energy associated with those cross modes immediately transitioned into the fabric of space time causing X, Y, and Z to expand by a very large amount almost instantaneously. This expansion caused the weak nuclear force dimensions to disconnect from the electromagnetic force dimensions, which caused the energy in the electroweak cross dimensions to also be absorbed by the fabric of space time and cause further expansion. This expansion of X, Y, and Z caused by the cross mode energy would have have caused what is known as “Inflation”.

If you have two large masses orbiting each other (orbiting a bari-center), then each of the masses will move through space. As a mass moves through space, it contracts X, Y, and Z for the space it moves into and it correspondingly increases the magnitude of the additional dimensions. When the mass moves out of the space, X, Y, and Z expand back out and the additional dimensions contract. This pumping of space-time generates a pulse wave that propagates through space-time in accordance with a differential equation.

When the gravitational wave passes through a volume of space, it contracts X, Y, Z, and T in that volume and it expands the additional dimensions in that volume. When there is a tick of time, the space-time cubes containing the wave will expand less than the adjacent space-time cubes. Accordingly, the space-time cubes that contained the waves at the click of time will have slightly more contracted X, Y, and Z dimensions and slightly larger additional dimensions than the adjacent space-time cubes. Accordingly, a small amount of the energy of the gravitational wave was incorporated into the elastic potential energy of the space-time cubes that contained the waves. This loss of energy will slowly decrease the energy of the gravitational wave, but the effect is appear small compared to the inverse square drop in amplitude caused by the wave translating through 3d space.

Figure 1: LIGO image of stochastic gravitational wave.

The pre-inflation Universe is presumed to have had stochastic gravitational waves.[4] Such waves are similar to the gravitational waves discussed above, but have a random distribution as shown if Fig 1.[4] Accordingly, at any given moment, pre-inflation space time would have had a pattern of stochastic gravitational waves over its entire surface.

The waves would have modulated the X, Y, Z, and T dimensions of our space-time cube. Accordingly, at any given moment, pre-inflation space time would have had volumes of space-time where X, Y, and Z were more contracted and time was slower and it would have had volumes of space-time where X, Y, and Z were less contracted and time was faster.

At the moment inflation some volumes of space would have had faster time and some volumes of space would have had slower time. Since inflation occured as a function of time, the slower time areas would have expanded less than the faster time areas. By experiencing less time during the inflationary epoch, the X, Y, Z, and T dimensions of the slower time space-time cubes would have expanded less than the X, Y, Z, and T dimensions of faster time space time cubes. This relative reduction of expansion for the cubes corresponding to slower time would, would result in an increased elastic potential energy for the less expanded space-time cubes

Since each tick of the clock provides more time for the less contracted space-time cubes than it provides for the more contracted space-time cubes, the less contracted space-time cubes would have expanded much more during the first few clicks of time of inflation than the more contracted space-time cubes would have expanded during the same clicks. This large variation in the amount of expansion of space during the first few clicks of time during inflation would have caused the space time to have a large variation in the size of of X, Y, and Z, and T for the space-time cubes. These variations in in the size of of X, Y, and Z, and T would cause 4D space-time to curve.

The shape of the curvature would have been set very early in the inflationary epoch, since the magnitudes of the gravitational waves would have quickly dropped to near zero before the wave could move to an adjacent volume. Even moving at the speed of light, the gravitational wave could not reach the adjacent volume before being absorbed by the fabric of space-time. Accordingly, all of the energy of the gravitational wave would have gone into the elastic potential energy of the fabric of space-time. The relative increase in the amount of elastic potential energy causes X, Y, and Z to be more contracted in the volumes of space that correspond to a smaller magnitude of time at the onset of inflation.

The shape of gravity provided by this curvature/contraction would have been self-sustaining throughout the remainder of inflation and thereafter because the less expanded (more contracted) space would continue to experience less time. Accordingly, the shape of the gravity caused by less expanded space-time would have be very similar to the shape of the stochastic gravitational waves present at the first moment of inflation.

Figure 2: Image of the proposed distribution of dark matter in a galaxy cluster EVRARD, A. E. NATURE 394, 122–123 (09 JULY 1998).

Hubble data has been used to plot a proposed distribution of dark matter in a galaxy cluster.[5] The shape of the plotted dark matter in the galaxy cluster is shown ion Fig 2 and that proposed shape of dark matter corresponds to the shape of gravity.[5] The shape of the gravity, from Fig 2, is essentially the same as the shape of stochastic gravitational waves from Fig 1, which indicates that the pre-inflation stochastic gravitational waves caused the gravity.

Figure 3

The Dark Energy Survey has determined that dark matter is distributed as a web-like structure with dense clumps of matter separated by large empty voids. [5]   This is what you would expect from curvature and contraction resulting from pre-inflation gravitational waves because the slower time pre-inflation volumes would have expanded into relatively smaller volumes of post inflation space and the faster volumes would have expanded into relatively larger volumes. The pre-inflation space-time cubes with smaller X, Y, and Z would map into space-time cubes with smaller post inflation X, Y, and Z. The pre-inflation space-time cubes with larger X, Y, and Z would map into space-time cubes with larger post inflation X, Y, and Z. The post-inflation volumes of space with larger X, Y, and Z would have smaller amplitudes of curvature and smaller variations in contraction and with smaller first and second derivatives. The combination of the lower volume, higher curvature/higher contraction space-time with the higher volume, lower curvature, lower expansion volumes would have yielded a structure described by the Dark Energy Survey similar to that shown in Fig 3. [3]

KiDs’ data indicates that dark matter is smoother than expected and that the Standard Model with dark matter may not be able to recreate observations. [4] The described alternative to dark matter should be able to correct for such discrepancies.

Accordingly, space-time curvature created during inflation by stochastic gravitational waves is the likely cause of the gravity normally associated with dark matter.

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2. Jørgensen L, Cardozo DL, Thibierge E, Numerical Resolution of  The Schrödinger Equation, École Normale Supérieure de Lyon Master Sciences de la Matière 2011, Numerical Analysis
3. Burkhard S, Hochbruck M, Dörich B, Lasser C, Variational Gaussian Approximation For The Magnetic Schrodinger Equation, 2023.
4. LSC LIGO Scientific Collaboration, introduction to ligo & gravitational waves, stochastic gravitational waves.
5. EVRARD, E. NATURE 394, 1998; 122-123.
6. Jeffrey, Dark Energy Survey.
7. Catherine Heymans, KiDS-1000 Cosmology: Multi-probe weak gravitational lensing and spectroscopic galaxy clustering constraints, arXiv; 2020.