Gravitational Self-Energy: A Unified Origin for Inflation and Dark Energy

The standard cosmological model, ΛCDM, introduces a mysterious component known as Dark Energy (Λ), which accounts for approximately 68% of the total energy, to explain the accelerated expansion of the universe. However, the physical nature of dark energy remains unknown. Furthermore, the model faces significant challenges, including the catastrophic discrepancy between theoretical predictions and observed values (the Cosmological Constant Problem), the recently highlighted Hubble Tension, and the problem of massive galaxies in the early universe.

This paper proposes the Matter-Only Cosmology (MOC) model, which argues that "dark energy is not a separate, mysterious component, but rather originates from the Gravitational Self-Energy (GSE) inherent to Matter itself." This model does not introduce new particles or fields but explains the history of the universe, from primordial inflation to late-time accelerated expansion, unifyingly through the interaction between matter and gravity alone. Here, "Matter-Only" does not imply the absence of radiation; rather, it signifies that dark energy is not a new fluid independent of matter, but a dependent energy arising directly from matter.

1. Derivation of the Complete Gravitational Self-Energy Equation

Since the existing equation for gravitational self-energy is incorrect, we must derive a complete expression for gravitational self-energy. (Please refer to the paper for the detailed derivation.)

Our fundamental postulate is that the source term M'(r) must be replaced by an equivalent mass M_{eq}(r), which includes not only the material mass but also the equivalent mass of its own gravitational self-energy, M'_{gs}(r). Because the mass inside the shell is not free state, but already bound.

M_{eq}(r) = M'(r) - M'_{gs}(r)

For a general mass distribution, we define the GPE of the inner sphere of radius r and mass M'(r) using a structural parameter β. This parameter encapsulates the geometric distribution of mass and relativistic corrections, ranging from β = 3/5 for a uniform sphere in Newtonian mechanics to values in the range of β ~1.0 - 2.0 for various astrophysical configurations in General Relativity.

By integrating this equation and replacing the Newtonian coefficient of 3/5 with β to reflect general relativistic effects and the structural evolution of the universe, we obtain the following final expression.

The first term (U_gs) corresponds to the conventional gravitational binding energy we are familiar with, while the second term (U_{m-gs}) represents the newly discovered interaction term between gravitational self-energy and matter.

2. Identifying the New Gravitational Self-Energy Equation with Dark Energy

The total mass density of a gravitational system consists of the mass density of matter plus the mass density term due to gravitational self-energy. This gravitational self-energy corresponds precisely to dark energy.

ρ_T = ρ_m + ρ_{m-gs} - ρ_{gs} = ρ_m + ρ_{Λ_m}

By dividing the potential energy terms derived above by the volume, we obtain the expression for mass density. The dark energy term ρ_{Λ_m} = ρ_{m-gs} - ρ_{gs} is given as follows

Examining the dark energy term, we can see that it is a function of the matter density ρ_m. Dark energy is not an independent entity but arises from matter itself.

3. Numerical Analysis of ρ_{Λ_m} Characteristics

To investigate whether this ρ_{Λ_m} equation exhibits characteristics similar to the current dark energy, we performed numerical calculations.

General Characteristics of the Data: 
Across various simulations, the dark energy density is negative in the early universe (t < 5 Gyr), transitions to positive values in the middle epoch to contribute to cosmic accelerated expansion, peaks at approximately 7 ~ 9 Gyr, and subsequently decreases. This demonstrates that ρ_{Λ_m} can explain the current value of dark energy density.

Furthermore, several characteristics align with recently published results regarding the properties of dark energy. Refer to BAO+CMB, BAO+CMB+SN or BAO+CMB+SN(corrected)

4. Interpretation of Numerical Results

1) Natural Resolution of the Hubble Tension

• Problem: The persistent discrepancy between the Hubble constant (H_0) measured in the early universe (CMB) and the late universe (SH0ES).

• Solution: MOC argues that the structural parameter β evolves as cosmic structures form. Since the physical state of the early, uniform universe (β ~ 1.39; Ω_m=0.346 model) differs from that of the current, clustered universe (β ~ 1.27; Ω_m=0.346 model), attempting to describe expansion with a single constant causes the tension. Thus, the Hubble Tension is not an error but evidence of the structural evolution of the universe.

2) Resolution of the Early Massive Galaxy Problem (JWST Observations)

• Problem: The James Webb Space Telescope (JWST) has discovered massive galaxies formed much earlier than expected.

• Solution: MOC provides a crucial prediction that differs from the standard ΛCDM model. In the early universe, there existed a phase where the dark energy density was negative, implying a period with a negative cosmological constant. According to MOC, in the early universe (z > 1.5), dark energy was negative energy. This acted to enhance gravity (attraction), allowing matter to clump together much faster than predicted by existing theories.

3) Weakening Dark Energy

• Observation: Recent observations from DESI and others suggest the possibility that dark energy is not constant but weakens over time.

• Solution: In MOC, dark energy is not a constant; it possesses dynamic properties where it gradually decreases after initiating accelerated expansion. This is in exact agreement with recent observational trends.

5. Applicability to Inflation and Black Hole Singularity Problems

1)Inflation: To explain inflation, we don't introduce new elements, such as inflaton fields or false vacuums. The previously derived equation for the dark energy density ρ_{Λ_m} also applies to inflation.
Even in the Planck era of the early universe, ρ_{Λ_m} was approximately 40 times larger than the matter density ρ_m. Since ρ_{Λ_m} had a positive value during this period, it drove the accelerated expansion (inflation) of the universe via negative pressure. Furthermore, the ρ_{Λ_m} equation contains a natural self-termination mechanism for inflation.

2)Black Hole: In the case of black holes, it can be mathematically verified that when R is smaller than a critical radius R_gs, the dark energy density generates a repulsive force, which prevents the formation of a singularity.

6. Conclusion

Gravitational self-energy resolves the problems of dark energy and inflation through a single equation within the framework of existing physics, without introducing new fields or particles.

When deriving the equation for gravitational self-energy, the mass inside a shell must be the equivalent mass that includes negative binding energy. However, by using the free-state mass M_fr instead of the equivalent mass, we have been led down the wrong path. Consequently, this oversight has given rise to various problems related to gravity, such as inflation, dark energy, singularities, and divergences.

#Paper:

Matter-Only Cosmology: A Unified Origin for Inflation and Dark Energy

Recently, several more simulations were performed, and modifications were made to the comoving horizon, resulting in more complete results.

General Characteristics of the Data:
Across various simulations, the dark energy density is negative in the early universe (t < 6 Gyr, z>1.0), transitions to positive values in the middle epoch to contribute to cosmic accelerated expansion, peaks at approximately 11.8 Gyr, and subsequently decreases. This demonstrates that ρ_{Λ_m} can explain the current value of dark energy density. The dark energy equations suggest damped oscillations, and the cycles of decelerating and accelerating expansion are predicted to become longer and longer.

Assuming w=-1 for dark energy, as in the ΛCDM model, the condition for accelerated expansion in the acceleration equation is ρ_m - 2ρ_{Λ_m}<0, which occurs when the dark energy density exceeds 50% of the matter density. From the data avobe, we can see that the accelerated expansion of the universe occurs around 8.8 Gyr (approximately 5 billion years ago).

The total gravitational self-energy has properties consistent with the core properties of dark energy.
1) Current dark energy sign and density values
2) The point at which the universe transitioned to accelerated expansion: Around t=8.8 Gyr (approximately 5 billion years ago)
3) Recent decrease in dark energy: All simulations show a peak around t=11.8 Gyr and then a decrease.
4) The problem of massive galaxies in the early universe: In the early universe, dark energy was negative, contributing to the decelerating expansion and thus promoting the formation of galaxy structures.

In particular, the prediction of a sign change in dark energy is remarkable.

Gravitational self-energy resolves the problems of dark energy and inflation through a single equation within the framework of existing physics, without introducing new fields, new particles, or any free parameters.

The structure parameter β is a coefficient derived from the calculation of gravitational self-energy and is not a free parameter.
Because the function of dark energy is clearly defined, countless verification methods exist.

Matter-Only Cosmology: A Unified Origin for Inflation and Dark Energy

Edited December 9Dec 9 by icarus2

On 11/26/2025 at 7:10 PM, icarus2 said:

Gravitational Self-Energy: A Unified Origin for Inflation and Dark Energy

The standard cosmological model, ΛCDM, introduces a mysterious component known as Dark Energy (Λ), which accounts for approximately 68% of the total energy, to explain the accelerated expansion of the universe. However, the physical nature of dark energy remains unknown. Furthermore, the model faces significant challenges, including the catastrophic discrepancy between theoretical predictions and observed values (the Cosmological Constant Problem), the recently highlighted Hubble Tension, and the problem of massive galaxies in the early universe.

This paper proposes the Matter-Only Cosmology (MOC) model, which argues that "dark energy is not a separate, mysterious component, but rather originates from the Gravitational Self-Energy (GSE) inherent to Matter itself." This model does not introduce new particles or fields but explains the history of the universe, from primordial inflation to late-time accelerated expansion, unifyingly through the interaction between matter and gravity alone. Here, "Matter-Only" does not imply the absence of radiation; rather, it signifies that dark energy is not a new fluid independent of matter, but a dependent energy arising directly from matter.

1. Derivation of the Complete Gravitational Self-Energy Equation

Since the existing equation for gravitational self-energy is incorrect, we must derive a complete expression for gravitational self-energy. (Please refer to the paper for the detailed derivation.)

Our fundamental postulate is that the source term M'(r) must be replaced by an equivalent mass M_{eq}(r), which includes not only the material mass but also the equivalent mass of its own gravitational self-energy, M'_{gs}(r). Because the mass inside the shell is not free state, but already bound.

M_{eq}(r) = M'(r) - M'_{gs}(r)

For a general mass distribution, we define the GPE of the inner sphere of radius r and mass M'(r) using a structural parameter β. This parameter encapsulates the geometric distribution of mass and relativistic corrections, ranging from β = 3/5 for a uniform sphere in Newtonian mechanics to values in the range of β ~1.0 - 2.0 for various astrophysical configurations in General Relativity.

By integrating this equation and replacing the Newtonian coefficient of 3/5 with β to reflect general relativistic effects and the structural evolution of the universe, we obtain the following final expression.

The first term (U_gs) corresponds to the conventional gravitational binding energy we are familiar with, while the second term (U_{m-gs}) represents the newly discovered interaction term between gravitational self-energy and matter.

2. Identifying the New Gravitational Self-Energy Equation with Dark Energy

The total mass density of a gravitational system consists of the mass density of matter plus the mass density term due to gravitational self-energy. This gravitational self-energy corresponds precisely to dark energy.

ρ_T = ρ_m + ρ_{m-gs} - ρ_{gs} = ρ_m + ρ_{Λ_m}

By dividing the potential energy terms derived above by the volume, we obtain the expression for mass density. The dark energy term ρ_{Λ_m} = ρ_{m-gs} - ρ_{gs} is given as follows

Examining the dark energy term, we can see that it is a function of the matter density ρ_m. Dark energy is not an independent entity but arises from matter itself.

3. Numerical Analysis of ρ_{Λ_m} Characteristics

To investigate whether this ρ_{Λ_m} equation exhibits characteristics similar to the current dark energy, we performed numerical calculations.

General Characteristics of the Data: 
Across various simulations, the dark energy density is negative in the early universe (t < 5 Gyr), transitions to positive values in the middle epoch to contribute to cosmic accelerated expansion, peaks at approximately 7 ~ 9 Gyr, and subsequently decreases. This demonstrates that ρ_{Λ_m} can explain the current value of dark energy density.

Furthermore, several characteristics align with recently published results regarding the properties of dark energy. Refer to BAO+CMB, BAO+CMB+SN or BAO+CMB+SN(corrected)

4. Interpretation of Numerical Results

1) Natural Resolution of the Hubble Tension

• Problem: The persistent discrepancy between the Hubble constant (H_0) measured in the early universe (CMB) and the late universe (SH0ES).

• Solution: MOC argues that the structural parameter β evolves as cosmic structures form. Since the physical state of the early, uniform universe (β ~ 1.39; Ω_m=0.346 model) differs from that of the current, clustered universe (β ~ 1.27; Ω_m=0.346 model), attempting to describe expansion with a single constant causes the tension. Thus, the Hubble Tension is not an error but evidence of the structural evolution of the universe.

2) Resolution of the Early Massive Galaxy Problem (JWST Observations)

• Problem: The James Webb Space Telescope (JWST) has discovered massive galaxies formed much earlier than expected.

• Solution: MOC provides a crucial prediction that differs from the standard ΛCDM model. In the early universe, there existed a phase where the dark energy density was negative, implying a period with a negative cosmological constant. According to MOC, in the early universe (z > 1.5), dark energy was negative energy. This acted to enhance gravity (attraction), allowing matter to clump together much faster than predicted by existing theories.

3) Weakening Dark Energy

• Observation: Recent observations from DESI and others suggest the possibility that dark energy is not constant but weakens over time.

• Solution: In MOC, dark energy is not a constant; it possesses dynamic properties where it gradually decreases after initiating accelerated expansion. This is in exact agreement with recent observational trends.

5. Applicability to Inflation and Black Hole Singularity Problems

1)Inflation: To explain inflation, we don't introduce new elements, such as inflaton fields or false vacuums. The previously derived equation for the dark energy density ρ_{Λ_m} also applies to inflation.
Even in the Planck era of the early universe, ρ_{Λ_m} was approximately 40 times larger than the matter density ρ_m. Since ρ_{Λ_m} had a positive value during this period, it drove the accelerated expansion (inflation) of the universe via negative pressure. Furthermore, the ρ_{Λ_m} equation contains a natural self-termination mechanism for inflation.

2)Black Hole: In the case of black holes, it can be mathematically verified that when R is smaller than a critical radius R_gs, the dark energy density generates a repulsive force, which prevents the formation of a singularity.

6. Conclusion

Gravitational self-energy resolves the problems of dark energy and inflation through a single equation within the framework of existing physics, without introducing new fields or particles.

When deriving the equation for gravitational self-energy, the mass inside a shell must be the equivalent mass that includes negative binding energy. However, by using the free-state mass M_fr instead of the equivalent mass, we have been led down the wrong path. Consequently, this oversight has given rise to various problems related to gravity, such as inflation, dark energy, singularities, and divergences.

#Paper:

Matter-Only Cosmology: A Unified Origin for Inflation and Dark Energy

I'm curious as to your rationale for saying the formula for binding energy of a sphere is wrong.

I'm far from expert on this, but doesn't the classical derivation of the binding energy make use of Newton's Shell Theorem, thereby avoiding the need to worry about the effect of the mass inside each infinitesimal shell on that shell, when doing the integration? But no doubt I am misunderstanding.