Inferring the Global Topology of the Universe from Local Temporal Metrology

J. Rogers, SE Ohio

Abstract

The total size and mass of the universe remain among the most fundamental unknowns in cosmology, as direct observation is limited by the cosmological horizon. This paper proposes a novel method for constraining these global properties by reinterpreting the observed cosmic acceleration. We build upon the hypothesis that the local rate of proper time is a function of the integrated scalar gravitational potential from the entire mass-energy content of the universe, Φ_U. The apparent acceleration attributed to Dark Energy is modeled as a direct manifestation of the secular evolution of this potential, dΦ_U/dt, as the universe expands. By formalizing the relationship between Φ_U and cosmological parameters within a simplified topological model (a finite, homogeneous sphere), we derive an expression that relates the observable parameter Ω_Λ to the total radius of the universe, R_total. This transforms the problem of the universe's total size from one of speculative extrapolation to one of precision measurement, positing that local atomic clocks function as sensors for the global structure of the cosmos.

1. Introduction: The Horizon Problem and the Metrological Hypothesis

The standard cosmological model, ΛCDM, provides a remarkable description of the observable universe. However, it offers no definitive answer to the question of the universe's total size, which may be finite or infinite. Our causal contact is limited to the Hubble sphere, leaving the global topology unconstrained.

This paper explores the proposition that information about the global structure is locally accessible, not through a propagated signal, but through its influence on the static spacetime metric. Our central hypothesis is that the local rate of time is not an intrinsic constant but is set by the summed scalar gravitational potential Φ_U from all mass-energy in the universe. The observed cosmic "acceleration" is then a measure of the rate of change of our own clocks, d(dτ/dt)/dt, driven by the evolution of Φ_U in an expanding cosmos.

This approach is predicated on two principles:

1. Scalar Accumulation: Unlike vector forces that can cancel by symmetry, scalar potentials are strictly accumulative.

2. Acausal Influence of Static Fields: As evidenced by a black hole, the source of a static gravitational field does not need to be in causal contact for its external field to exist and affect the metric. The cosmological horizon is one of observation, not necessarily of gravitational influence on the static metric.

2. A Simplified Cosmological Model

To formalize the calculation, we model the universe as a finite, non-rotating, spherically symmetric entity with a total radius R_total and a uniform average mass-energy density ρ(t). The total mass is M_total = ρ(t) * (4/3)πR_total(t)³. We assume that M_total is conserved.

The gravitational potential inside a uniform sphere of mass is given by standard Newtonian physics (a valid weak-field approximation for this purpose):

Φ(r) = -2πGρ(R_total² - r²/3)

For any observer like us, our radial position r from the center is negligible compared to the total radius R_total. Therefore, the dominant potential we experience is effectively constant throughout our local region:

Φ_U(t) ≈ -2πGρ(t)R_total(t)² (Eq. 1)

This Φ_U(t) is the global potential that, we hypothesize, sets the baseline for our local rate of time.

3. Linking Global Potential to Local Metrology

From General Relativity, the relationship between proper time τ and coordinate time t in a weak, static potential is dτ/dt = sqrt(-g_00) ≈ 1 + Φ_U/c².

The rate of change of our clock's speed is what we seek:

d/dt(dτ/dt) ≈ (1/c²) * dΦ_U/dt (Eq. 2)

We must now express dΦ_U/dt in terms of observable cosmological parameters. From Eq. 1, using ρ(t) = M_total / ((4/3)πR_total(t)³):

Φ_U(t) ≈ -2πG * [M_total / ((4/3)πR_total³)] * R_total² = - (3/2) * G * M_total / R_total(t) (Eq. 3)

Differentiating with respect to time:

dΦ_U/dt = (3/2) * (G * M_total / R_total²) * dR_total/dt

We define the expansion of the total universe by a global Hubble-like parameter, dR_total/dt = H(t)R_total(t). Substituting this in:

dΦ_U/dt = (3/2) * (G * M_total / R_total) * H(t) (Eq. 4)

This equation links the rate of change of the global potential to the total mass, total radius, and expansion rate.

4. Deriving the Total Radius from Cosmological Observables

The observed cosmic acceleration is parameterized by the Dark Energy density parameter, Ω_Λ. The acceleration term in the Friedmann equations is proportional to H₀²Ω_Λ. In our model, this term is not a real force but the signature of our changing clock rate. We therefore set our derived effect proportional to the observed effect:

d/dt(dτ/dt) ∝ H₀²Ω_Λ

Using Eq. 2 and Eq. 4, we get:

(1/c²) * [(3/2) * (G * M_total / R_total) * H₀] ≈ H₀²Ω_Λ (Eq. 5)

(Here, we use the present-day value of the Hubble parameter, H₀, as this is where the measurement is made).

Now we can solve for R_total. First, express M_total in terms of the measurable critical density ρ_crit = 3H₀²/8πG and the matter density parameter Ω_m. The matter we see inside our Hubble volume V_H is M_H = Ω_m ρ_crit V_H. Extrapolating the same density to the whole universe gives M_total = Ω_m ρ_crit V_total = Ω_m (3H₀²/8πG) * (4/3)πR_total³.

Substitute this expression for M_total into Eq. 5:

(1/c²) * [(3/2) * (G / R_total) * H₀] * [Ω_m (3H₀²/8πG) * (4/3)πR_total³] ≈ H₀²Ω_Λ

Many terms cancel and simplify:

(1/c²) * (3/2) * (G/R_total) * H₀ * Ω_m * (H₀²/2G) * R_total³ ≈ H₀²Ω_Λ

(3/4) * (H₀³Ω_m/c²) * R_total² ≈ H₀²Ω_Λ

Now, we isolate R_total:

R_total² ≈ (4/3) * (c² * H₀²Ω_Λ) / (H₀³Ω_m)

R_total ≈ (2/√3) * (c/H₀) * sqrt(Ω_Λ / (H₀Ω_m)) (Eq. 6)

5. Discussion and Implications

Equation 6 is a remarkable result. It provides a direct formula to calculate the total radius of the universe, R_total, using five measurable quantities:

• c: The speed of light.

• H₀: The Hubble constant.

• Ω_Λ: The Dark Energy density parameter (reinterpreted as the magnitude of our temporal acceleration).

• Ω_m: The matter density parameter.

• √3/2: A geometric factor from our model.

This formula takes the most mysterious number in cosmology (Ω_Λ) and identifies it as the key that unlocks one of the most fundamental questions (R_total). It implies the universe is finite and its size is not only knowable but calculable.

The model is, of course, a first-order approximation. A full treatment would require a general relativistic analysis of the potential within a finite FLRW metric. However, this derivation serves as a proof-of-concept for the methodology. It demonstrates that if the premise of a globally-determined temporal metric is accepted, the universe's total size ceases to be an unanswerable question.

6. Conclusion

By reinterpreting the physical meaning of the constants G and c and following the logic of accumulative scalar potentials to its cosmological conclusion, we have outlined a method to measure the entire universe. This framework proposes that every high-precision clock is a local probe of global topology. The phenomenon we call "Dark Energy" is not the influence of a mysterious substance, but the readout of our cosmic speedometer's own accelerating calibration. The further refinement of this model and its testing against observational data (such as BAO and quasar variability) represents a potentially revolutionary path forward in cosmology.