Cosmic Acceleration from MQ’s Geometric Fractions

The universe’s expansion is accelerating. Measurement Quantization (MQ) supplies a striking present-epoch consistency relation for that behavior. Its geometric domain fractions, interpreted through the spatially flat general-relativistic matter-and-vacuum acceleration law, give $q_0$ of −0.5254. A published supernova and distance-ladder analysis reports −0.51 ± 0.024.^[1]

The connection begins with the same dimensionless coefficient that enters MQ’s physical-constant construction. The calculation below follows the fractional domains into the acceleration equation, then constructs a candidate metric mapping that carries the result into changing cosmic distances. Each physical identification is stated at the step where it enters.

Begin with the Fractional Domains

MQ organizes its cosmological domains by physical referenceability. ${\mathrm{\Omega }}_{\text{vis}}$ denotes the presently visible fraction, ${\mathrm{\Omega }}_{\text{uobs}}$ the unobserved but ultimately referenceable fraction, and ${\mathrm{\Omega }}_{\text{obs}}$ their combined observable domain. The complementary dark-domain fraction is ${\mathrm{\Omega }}_{\text{dk}}$. The fundamental fraction ${\mathrm{\Omega }}_{\text{f}}$ supplies a normalization pivot. These definitions and their public context are developed in Cosmological Domain Partitions.^[2]

Let ${\theta }_{\text{si}}$ be MQ’s dimensionless invariant coefficient. The starting normalization and visibility relation are ${\mathrm{\Omega }}_{\text{f}}=\frac{2}{{{\theta }_{\text{si}}}^2}$ and ${\mathrm{\Omega }}_{\text{obs}}=2{\theta }_{\text{si}}{\mathrm{\Omega }}_{\text{vis}}$. The normalized total satisfies ${\mathrm{\Omega }}_{\text{obs}}+{\mathrm{\Omega }}_{\text{dk}}=1$. MQ closes this construction with the additional domain relation below.

The second equality follows by replacing the visible fraction with $\frac{{\mathrm{\Omega }}_{\text{obs}}}{2{\theta }_{\text{si}}}$. Combining the closure relation with the complementary observable and dark domains yields $1-{\mathrm{\Omega }}_{\text{obs}}=\frac{{\mathrm{\Omega }}_{\text{obs}}}{{\mathrm{\Omega }}_{\text{f}}}-1$. Adding one and collecting the observable-domain terms gives $2={\mathrm{\Omega }}_{\text{obs}}\left(1+\frac{1}{{\mathrm{\Omega }}_{\text{f}}}\right)$. Therefore,

Subtracting this result from the normalized total gives the dark-domain expression. Both fractions are fixed once the coefficient and the MQ closure premise are specified.

The remaining domains follow directly. The visible fraction is ${\mathrm{\Omega }}_{\text{vis}}=\frac{{\mathrm{\Omega }}_{\text{obs}}}{2{\theta }_{\text{si}}}$, and the unobserved fraction is ${\mathrm{\Omega }}_{\text{uobs}}={\mathrm{\Omega }}_{\text{obs}}-{\mathrm{\Omega }}_{\text{vis}}$. Table 1 evaluates the construction using the working value ${\theta }_{\text{si}}=3.26239030504$, calibrated within MQ from the 2022 inverse-fine-structure input. All displayed percentages are rounded.

The visible and unobserved fractions sum to the observable fraction; the observable and dark fractions sum to one. The fundamental fraction appears in the normalization algebra rather than in that additive sum.

Translate Geometry into Acceleration

Astronomers express acceleration through the scale factor $a$, which relates the proper separation of comoving locations at different cosmic times. Define $H=\frac{\dot{a}}{a}$ and the dimensionless deceleration parameter $q$. Dots denote derivatives with respect to cosmic proper time $t$.

An expanding universe accelerates when $q<0$. The sign follows immediately from the first definition because $a$ and ${\dot{a}}^2$ are positive.

In this setting, the Friedmann and acceleration equations take the following forms. Writing densities as mass per unit volume keeps the factors of $c^2$ in the pressure term explicit before the vacuum equation of state is inserted.

Dividing the second expression by the first gives $q=\frac{{\rho }_{\text{m}}-2{\rho }_{\text{v}}}{2\left({\rho }_{\text{m}}+{\rho }_{\text{v}}\right)}$. With ${\Omega }_{\text{m}}=\frac{{\rho }_{\text{m}}}{{\rho }_{\text{m}}+{\rho }_{\text{v}}}$ and ${\Omega }_{\text{v}}=\frac{{\rho }_{\text{v}}}{{\rho }_{\text{m}}+{\rho }_{\text{v}}}$, this becomes $q=\frac{{\Omega }_{\text{m}}}{2}-{\Omega }_{\text{v}}$.

Now make the stated present-epoch domain-to-density identification. Substitution gives a closed expression for the conditional acceleration value $q_{\theta }$.

The algebra exposes the sign before any numerical calculation. For positive ${\theta }_{\text{si}}$, acceleration follows when ${\theta }_{\text{si}}>2$. At the working MQ coefficient, the matter term contributes +0.1581879194 and the vacuum-form term contributes −0.6836241612. Their sum is −0.5254362418. The coefficient’s geometric partition therefore selects an accelerating present-epoch value under the stated density interpretation.

Place the Result beside the Published Measurement

In “A Comprehensive Measurement of the Local Value of the Hubble Constant with 1 km s⁻¹ Mpc⁻¹ Uncertainty from the Hubble Space Telescope and the SH0ES Team,” Adam Riess and colleagues report a joint inference from their distance ladder and higher-redshift supernovae. Section 5.2 and Figure 17 give the comparison shown in Table 2.^[1]

The conditional geometric value lies inside the published quoted interval, from −0.534 to −0.486. The calculation fixes its coefficient through the MQ fine-structure calibration and derives the value through the displayed domain and density relations. Its evidential content is a present-epoch numerical consistency result under those premises.

Input sensitivity can be followed directly. Differentiating the closed expression gives $\frac{d{q}_{\theta }}{d{\theta }_{\text{si}}}=-\frac{12{\theta }_{\text{si}}}{{\left({{\theta }_{\text{si}}}^2+2\right)}^2}$. The magnitude of this derivative is 0.244908. Propagating only the quoted coefficient uncertainty of 0.00000000025 gives an uncertainty in $q_{\theta }$ of 6.12e-11. That small calibration contribution is distinct from the unquantified uncertainty of the physical density assignment and the metric completion. The observational error and that model uncertainty serve different roles in assessing the comparison.

A Constant Domain Radius Rate and Accelerating Distances

MQ’s cosmic size relation is $D_{\text{U}}=2{\theta }_{\text{si}}c{A}_{\text{U}}$, where $D_{\text{U}}$ is the adopted universe-scale diameter and $A_{\text{U}}$ is the associated age parameter. Consequently $R_{\text{U}}={\theta }_{\text{si}}c{A}_{\text{U}}$. Its public derivation is developed in Diameter and Age of the Universe.^[3]

For the mapping calculation, identify the age parameter with cosmic proper time $t$ and let $t_0$ denote its present adopted value. Comoving separations evolve as $D=a\chi$, with fixed comoving coordinate $\chi$. A domain boundary can move relative to those locations. This makes the domain’s radius and the metric scale separate quantities with a physical relation to determine. The distance definitions are discussed in Tamara Davis and Charles Lineweaver’s “Expanding Confusion: Common Misconceptions of Cosmological Horizons and the Superluminal Expansion of the Universe.”^[4]

Represent that relation by a dimensionless function $F$ and normalize the metric scale to one today. Then

The time dependence of $F$ supplies the additional response of internal separations. For a present rate normalization $H_0{t}_0=k$ and $F\left(t_0\right)=1$, direct differentiation fixes the first two required derivatives.

Adopting the present System Frame normalization $k=\frac{\pi }{{\theta }_{\text{si}}}$ gives 0.96297265497. With $q_0=q_{\theta }$, the required scaled derivatives are -0.03702734503 and 0.56130029971, respectively. These numbers specify precisely how the mapping must bend the metric history while the domain radius retains its constant radial rate. The present rate relation is documented on Hubble’s Constant and the Two Realizations of Cosmic Expansion.^[5]

Construct a Present-Epoch Metric Completion

An explicit candidate makes the required response visible. Define $x=\frac{t}{t_0}$ and the temporary dimensionless coefficients $A=k-1$ and $B=1-\left(1+q_{\theta }\right)k^2$. Choose the following positive mapping function.

The working values are $A$ = -0.03702734503 and $B$ = 0.55992927543. This exponential-polynomial shape is a new mapping ansatz. Its coefficients enforce the selected present $H_0$ and $q_{\theta }$. MQ’s primitive dynamics still has to select a physical mapping and source history; this construction supplies an explicit local candidate for that task.

To verify it, define $h\left(x\right)=t_{0}H$. A prime denotes differentiation with respect to $x$. The logarithmic derivative of the scale factor gives

At $x=1$, these expressions give $h\left(1\right)=k$ and $h^{\prime }\left(1\right)=-\left(1+q_{\theta }\right)k^2$. Inserting them into $q=-1-\frac{h^{\prime }}{h^2}$ recovers $q_{\theta }$ exactly. The construction therefore reproduces the desired present acceleration and rate normalization together.

The chosen shape carries physical commitments. It replaces an all-time inverse-age metric history with the $h\left(x\right)$ just derived. If instead one imposes $H=\frac{k}{t}$ throughout proper time, direct differentiation gives $q=\frac{1}{k}-1$, which evaluates to +0.03845109 for the System Frame coefficient. The accelerating candidate uses the new mapping history above.

For a numerical rate example, adopt $t_0$ of 13.7943 billion years. The candidate then has $H_0$ of 68.25915 km s⁻¹ Mpc⁻¹. The joint observational benchmark also infers $H_0$ of 73.30 ± 1.04 in those units. This rate comparison remains a separate model requirement alongside the favorable $q_0$ comparison. Here the age is an adopted normalization; its physical time calibration and radiation-evolution derivation remain part of the cosmological completion.

Follow the Source Densities through the Mapping

Flat GR with the same matter and vacuum-form components determines their densities from the candidate $H$ and $\dot{H}$. The two equations used earlier give

At the present epoch, these reconstructed densities return the domain fractions assigned above. Across 1,000 sampled past points $0<x\le 1$, both reconstructed densities are positive. Their ratio evolves with the candidate metric history. The fixed MQ domain fractions supply the present density identification in this calculation.

The same equations also fix the candidate’s future range under this source interpretation. The reconstructed matter density reaches zero at $x=\frac{1}{\sqrt{B}}$, which evaluates to 1.33639060, and becomes negative beyond that point. Extending the physical cosmology past this boundary requires a changed mapping or source description. Likewise, if MQ’s enclosed mass is assigned the meaning of homogeneous GR total mass inside the linear domain radius, the candidate produces $\frac{M}{M_{\text{linear}}}={\left(\frac{xh\left(x\right)}{k}\right)}^2$, with present normalization matched. This quantity is 0.77278 at redshift 0.5 and 0.76952 at redshift 1. These explicit source constraints accompany the present-epoch acceleration result.

A further consistency condition applies if the geometric fraction is instead assigned to a constant density fraction at every epoch. In the same flat-GR dust/vacuum setting, $\dot{H}=-\frac{3}{2}\left(1-{\Omega }_{\text{v}}\right)H^2$. With fixed ${\Omega }_{\text{v}}$ and proper age measured from the power-law origin, integration gives $Ht=\frac{2}{3\left(1-{\Omega }_{\text{v}}\right)}$. Substituting the dark-domain fraction gives 2.10719842. The candidate preserves the present fraction while allowing the reconstructed density fraction to evolve, consistently separating those two choices of source history.

Carry the Result into an Observable Distance Curve

The next observational quantity is luminosity distance as a function of redshift $z$. For a flat metric, standard photon propagation and $1+z=\frac{1}{a}$, define the luminosity distance $d_L$ by

For the candidate, determine the emission time $x_z$ from $a\left(x_z\right)=\frac{1}{1+z}$. The same integral can then be evaluated in the normalized time variable. Its dimensionless form is $\frac{H_0{d}_L}{c}=k\left(1+z\right){\int }_{x_z}^1\frac{dx}{a\left(x\right)}$.

Table 3 compares that calculated shape with a flat ΛCDM reference using a matter fraction of 0.334, the supernova-only result reported by Dillon Brout and colleagues in “The Pantheon+ Analysis: Cosmological Constraints.”^[6] Both curves use a common present $H_0$ normalization. The listed difference is ${\mathrm{\Delta \mu }}_{\text{shape}}=5{\log }_{10}\left(\frac{d_{\text{L,candidate}}}{d_{\text{L,reference}}}\right)$.

These are differences between calculated curves. Their assessment against individual supernovae requires the observed standardized magnitudes, redshifts, calibration covariance and an explicit likelihood. The distance curve tests the mapping’s history as well as its present acceleration. A complete cosmology must also supply compatible source dynamics, clock normalization and early-universe evolution.

The central connection is now explicit. MQ’s fractional-domain construction fixes a geometric ratio; the stated density interpretation converts that ratio into a present acceleration value; a candidate mapping realizes that value in a metric history with calculable distances and source constraints. This chain is the precise content of the present-epoch consistency result.