The Inevitable Expansion of the Universe in ECM: Empirical Alignment

Soumendra Nath Thakur

April 02, 2025

The expansion of the universe is an inherent consequence of the dynamics described by the Extended Classical Mechanics (ECM) framework. According to ECM, the interaction between effective mass (Mᵉᶠᶠ) and negative apparent mass (Mᵃᵖᵖ) results in a repulsive gravitational effect, driving the expansion of the universe.

In the ECM model, the effective mass (Mᵉᶠᶠ) is expressed as:

Mᵉᶠᶠ = (Mᴍ - Mᵃᵖᵖ) when |- Mᵃᵖᵖ| > Mᴍ ⇒ Mᵉᶠᶠ < 0

where:

• Mᴍ: represents the matter mass, including the mass of dark matter.
• Mᵃᵖᵖ represents the negative apparent mass Mᵃᵖᵖ<0.

This relationship directly corresponds with the observed cosmological equation by A. D. Chernin et al., where:

Mɢ = Mᴍ + (-Mᵃᵖᵖ)

This implies that the gravitational dynamics, represented by Mɢ, can be understood as the sum of matter mass Mᴍ and the negative apparent mass component Mᵃᵖᵖ.

In ECM, the repulsive gravitational effect caused by Mᵃᵖᵖ leads to cosmic acceleration, and this is captured by the equation:

Fᴇᴄᴍ = (Mᴍ - Mᵃᵖᵖ)aᵉᶠᶠ

where:

• Fᴇᴄᴍ is the effective force acting on the system,
• aᵉᶠᶠ is the effective acceleration.

Furthermore, ECM aligns with the standard Friedmann equation, where the negative apparent mass replaces the cosmological constant, leading to the equation:

H² = (8πG/3) × (ρᴍ - ρᵃᵖᵖ)

where:

• H is the Hubble parameter,
• ρᴍ is the matter energy density,
• ρᵃᵖᵖ is the effective energy density associated with negative apparent mass.

Finally, to address the contribution of different mass components, we observe that:

Mᴍ = Mᴏʀᴅ + Mᴅᴍ

where:

• Mᴏʀᴅ represents ordinary matter mass,
• Mᴅᴍ represents dark matter mass.

This equation encapsulates the total matter mass (Mᴍ) in the universe, which, combined with the negative apparent mass (Mᵃᵖᵖ), determines the effective mass and drives the accelerated expansion of the universe.

Thus, through ECM's framework, we see that the interplay between ordinary matter, dark matter, and negative apparent mass directly contributes to the observed cosmic acceleration, providing a more consistent and empirically grounded explanation for the expansion of the universe.

List of mathematical terms in alphabetical order:

• aᵉᶠᶠ: Effective acceleration
• Fᴇᴄᴍ: ECM force equation
• G: Gravitational constant
• H²: Hubble parameter squared
• Mᴍ: Matter mass including mass of dark matter
• Mᴏʀᴅ: Mass of ordinary (baryonic) matter  
• Mᴅᴍ: Effective mass of dark matter
• Mᵃᵖᵖ: Negative apparent mass component
• ρₘ: Mass-energy density of matter
• ρʌ: Vacuum energy density associated with Λ
• ρᵃᵖᵖ: Density contribution of negative apparent mass (-Mᵃᵖᵖ)