Soumendra Nath Thakur

April 02, 2025

In the standard ΛCDM model, lambda (Λ) acts as a form of dark energy, providing an outward pressure that explains the observed accelerated expansion of the universe.

From the Extended Classical Mechanics (ECM) perspective, however, Λ can be replaced by Negative Apparent Mass (-Mᵃᵖᵖ), eliminating the need for a cosmological constant. ECM attributes cosmic acceleration to antigravity effects associated with -Mᵃᵖᵖ, offering a dynamic explanation rather than an imposed constant.

1. ECM Interpretation of Cosmological Expansion

The ΛCDM model treats Λ as a uniform vacuum energy density that causes accelerated expansion. However, in ECM, this acceleration is a consequence of negative apparent mass (-Mᵃᵖᵖ) dynamically interacting with gravitational systems. The effective force equation in ECM is:

Fᴇᴄᴍ = (Mᴍ - Mᵃᵖᵖ) aᵉᶠᶠ

where:

• Mᴍ: is the matter mass,
• Mᵃᵖᵖ: is the negative apparent mass component,
• aᵉᶠᶠ: is the effective acceleration.

This equation shows that as Mᵃᵖᵖ increases in magnitude (negative), it effectively induces an antigravitational effect, leading to the observed acceleration of cosmic expansion.

2. Replacing the Cosmological Constant Λ with -Mᵃᵖᵖ:

The standard Friedmann equation in the ΛCDM model is:

H² = (8πG/3) × (ρₘ + ρʌ)  - (k/a²)

where:

• ρₘ: is the mass-energy density of matter,
• ρʌ: is the vacuum energy density associated with Λ,
• k: represents spatial curvature.

In ECM, instead of using ρʌ, we define an effective mass density that includes the negative apparent mass component:

H² = (8πG/3) × (ρᴍ - ρᵃᵖᵖ)

where:ρᵃᵖᵖ dynamically replaces ρʌ as a function of cosmic evolution.

Thus, rather than introducing an artificial Λ-term, ECM interprets accelerated expansion as an emergent effect due to the natural presence of -Mᵃᵖᵖ.

3. Effective Gravitational Acceleration in ECM:

The gravitational acceleration due to matter mass alone follows:

a𝑔ᵣₐᵥ = GM/r²

However, when incorporating -Mᵃᵖᵖ, the net acceleration becomes:

aᵉᶠᶠ = G(Mᴍ - Mᵃᵖᵖ)/r²

Since Mᵃᵖᵖ is negative, the term -Mᵃᵖᵖ contributes positively to the acceleration, leading to a repulsive effect that drives cosmic expansion.

4. Cosmological Redshift and -Mᵃᵖᵖ:

Cosmological redshift is naturally explained by the evolution of -Mᵃᵖᵖ. As the universe expands:

Mᵃᵖᵖ(t) ∝ -1/aⁿ

where n depends on the cosmic epoch. This dynamic scaling modifies the expansion rate without requiring a static Λ.

Conclusion:

By integrating -Mᵃᵖᵖ into ECM’s gravitational framework, we can eliminate the need for the cosmological constant Λ. The accelerated expansion is not an imposed effect but a natural outcome of how negative apparent mass dynamically interacts with matter and gravity.

List of mathematical terms in alphabetical order:

• aᵉᶠᶠ: Effective acceleration
• a𝑔ᵣₐᵥ: Gravitational acceleration due to matter mass alone
• c: Speed of light (implicitly mentioned in conversions)
• Fᴇᴄᴍ: ECM force equation
• G: Gravitational constant
• H²: Hubble parameter squared
• k: Spatial curvature
• Mᴍ: Matter mass
• 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)
• t: Time (in cosmological redshift context)
• a: Scale factor (used in redshift equation)
• n: Scaling exponent (depends on the cosmic epoch)
• ℓP: Planck length (implicitly mentioned in some of the constants)