RESONANT RELATIVITY
BACK REF

INVESTIGATION: Cosmic Distance SWAGS

Purpose

This study records a set of numerical comparisons concerning the estimated size of the observable universe. The purpose is not to propose a cosmological model, but to place several independently calculated numbers beside one another and examine their relationships.

The calculations deliberately use simple relationships and published observational parameters. They are intended as working data: measurements are converted into numerical information from which further questions may be developed.

Conventional Observable-Universe Scale

A commonly quoted value for the diameter of the observable universe is approximately 92–93 billion light-years. NASA currently describes the observable universe as approximately 92 billion light-years across. The exact value depends upon the cosmological parameters used in the calculation.

For the calculations in this study, the working value is therefore:

\[ D_{\mathrm{current}} \approx 93\ \mathrm{billion\ ly} \]

This number represents the existing reference scale against which the other calculations below are compared.

Standard-Candle Luminosity Scale

Type-Ia supernovae provide a useful standardizable luminosity reference. The inverse-square relationship between luminosity, observed flux, and distance is

\[ F = \frac{L}{4\pi d^2} \]

Rearranging gives the luminosity distance:

\[ d = \sqrt{\frac{L}{4\pi F}} \]

Using the available calibrated Type-Ia supernova distance range as the working luminosity scale produces a characteristic diameter of approximately:

\[ D_{\mathrm{candle}} \approx 112.5\ \mathrm{billion\ ly} \]

This is retained here as a working observational number, not as a claim that the physical universe necessarily has this diameter.

Difference Between the Two Size Estimates

Comparing the standard-candle working scale with the conventional observable-universe scale:

\[ \Delta D = D_{\mathrm{candle}}-D_{\mathrm{current}} \]
\[ \Delta D = 112.5-93.0 \approx 19.5\ \mathrm{billion\ ly} \]

The corresponding ratio is

\[ \frac{D_{\mathrm{candle}}}{D_{\mathrm{current}}} = \frac{112.5}{93.0} \approx 1.21 \]

Thus the standard-candle working scale is approximately 21% larger in diameter than the conventional 93-billion-light-year reference.

Volume Consequence

If the two diameters are treated as spherical scales, volume varies as the cube of diameter:

\[ V \propto D^3 \]

Therefore the relative volume represented by the standard-candle scale is

\[ \frac{V_{\mathrm{candle}}}{V_{\mathrm{current}}} = \left( \frac{112.5}{93.0} \right)^3 \approx 1.77 \]

The larger diameter therefore corresponds to approximately 1.77 times the volume, or approximately 77% more volume.

Dark-Energy Density Comparison

Planck 2018 base-\(\Lambda\)CDM parameters give a matter density parameter of approximately \(\Omega_m=0.315\). The corresponding dark-energy component is approximately \(\Omega_\Lambda=0.685\).

For the purposes of this study, these values are used only as numerical reference points.

\[ \Omega_m \approx 0.315 \]
\[ \Omega_\Lambda \approx 0.685 \]

Their ratio is approximately

\[ \frac{\Omega_\Lambda}{\Omega_m} = \frac{0.685}{0.315} \approx 2.17 \]

In this numerical comparison, the presently inferred dark-energy density is therefore about 2.17 times the inferred matter density.

Size Required for Equal Average Densities

A separate question can be asked without changing the presently estimated total amount of dark energy:

What volume would be required to dilute that fixed amount of dark energy until its average density equaled the presently estimated matter density?

Since density is amount divided by volume, the required volume ratio is approximately

\[ \frac{V_{\mathrm{balance}}}{V_{\mathrm{current}}} = \frac{\Omega_\Lambda}{\Omega_m} \approx 2.17 \]

Because volume scales as the cube of diameter,

\[ \frac{D_{\mathrm{balance}}}{D_{\mathrm{current}}} = \sqrt[3]{2.17} \approx 1.295 \]

Using the 93-billion-light-year working reference:

\[ D_{\mathrm{balance}} = 93.0\times1.295 \approx 120.4\ \mathrm{billion\ ly} \]

The resulting working number is therefore:

\[ \boxed{ D_{\mathrm{balance}} \approx 120.4\ \mathrm{billion\ light\ years} } \]

This is a deliberately simplified calculation. It asks only what diameter would result from holding the present inferred dark-energy amount fixed while increasing the volume until the average density matched the present matter-density reference.

The Three Working Scales

Numerical comparison of universe-size scales
Reference Diameter Relative to 93 B ly
Conventional observable-universe estimate 93.0 B ly 1.000
Standard-candle working scale 112.5 B ly 1.210
Equal-density balance scale 120.4 B ly 1.295

Numerical Relationships

Derived differences
Comparison Difference Relative Difference
Standard candle vs. conventional +19.5 B ly +20.9%
Balance scale vs. conventional +27.4 B ly +29.5%
Balance scale vs. candle scale +7.9 B ly +7.0%

The last comparison is particularly useful as a numerical observation:

\[ 112.5\ \mathrm{B\,ly} \;<\; 120.4\ \mathrm{B\,ly} \]

The two independently constructed working scales differ by only approximately 7.9 billion light-years, with the standard-candle scale being approximately 93.4% of the equal-density balance scale.

Workbench Summary

Three numbers are retained from this study:

  1. 93.0 billion light-years — conventional observable-universe diameter.
  2. 112.5 billion light-years — working standard-candle luminosity scale.
  3. 120.4 billion light-years — working diameter obtained by holding the presently inferred dark-energy amount fixed and increasing the volume until the average density matches the matter-density reference.

No physical interpretation is assigned to the numerical proximity of the latter two values in this study. The purpose of recording the relationship is simply to preserve the information for subsequent investigation.

The useful observation is that the two independently constructed working scales are numerically close compared with their difference from the conventional 93-billion-light-year reference.

References