MASS ACCRETION IN THE UNIVERSE

An assessment of the total mass-equivalent content of the universe is challenging because cosmology ordinarily infers its contents from gravitational dynamics, expansion history, the cosmic microwave background, and other observations rather than through a direct global measurement. Measurement Quantization (MQ) approaches the problem differently. Beginning with discrete relations among length, mass, and time, MQ develops a count-based description in which the size, age, mass-equivalent content, and geometric domains of the universe arise from a common construction.

The distinction between the Internal Frame and System Frame is central to this description. The Internal Frame is the discrete, count-based configuration domain in which the fundamental measures $l_f$, $m_f$, and $t_f$ are defined. Relational structure is encoded through integer counts such as $n_L$, $n_M$, and $n_T$. The System Frame is the non-discrete relational realization of that count structure. Observable quantities arise through the Frames mapping from discrete count configurations in the Internal Frame to continuous relational quantities in the System Frame.

For the universe-level construction, MQ begins with the dimensionally explicit fundamental relation.

Here $p_{\mathrm{si}}$ is the momentum realization of the invariant MQ scalar ${\theta }_{\mathrm{si}}$. Extending the count structure to the universe gives the realized-diameter relation $n_{L\_U}=2{\theta }_{\mathrm{si}}{n}_{T\_U}$. The corresponding total mass-equivalent count is central to the accretion argument.

The two relations give $\frac{n_{M\_U}}{n_{L\_U}}=\frac{{{\theta }_{\mathrm{si}}}^2}{4}$. The first relation connects realized diameter count to elapsed time count. The second connects total mass-equivalent count to elapsed time count. Within MQ, accumulated mass-equivalent content therefore increases linearly with elapsed count after physical realization.

The same construction relates dimensionless counts to the realized dimensions of the universe. With $A_U$ denoting universe age, $D_U$ its diameter, and $c$ the speed of light, the relation is $\frac{n_{L\_U}}{2n_{T\_U}}=\frac{D_U}{2A_{U}c}={\theta }_{\mathrm{si}}$. The invariant scalar ${\theta }_{\mathrm{si}}$ therefore relates elapsed count, spatial count, and their corresponding realized quantities. In the MQ construction this relation follows from the count geometry rather than from fitting the present diameter or age of the universe.

GEOMETRIC DOMAINS

The same geometry partitions the expanding universe into five fixed referenceability-domain constants. These should not be confused with empirically fitted cosmological density parameters. The fundamental domain is ${\mathrm{\Omega }}_{\mathrm{f}}=\frac{2}{{{\theta }_{\mathrm{si}}}^2}=\mathrm{18.79135772(58)}\%$. The dark domain is ${\mathrm{\Omega }}_{\mathrm{dk}}=\frac{{{\theta }_{\mathrm{si}}}^2-2}{{{\theta }_{\mathrm{si}}}^2+2}=\mathrm{68.36241612(59)}\%$. The observable domain is ${\mathrm{\Omega }}_{\mathrm{obs}}=\frac{4}{{{\theta }_{\mathrm{si}}}^2+2}=\mathrm{31.63758388(59)}\%$. Its presently visible portion is ${\mathrm{\Omega }}_{\mathrm{vis}}=\frac{2}{{\theta }_{\mathrm{si}}({{\theta }_{\mathrm{si}}}^2+2)}=\mathrm{4.84883489(10)}\%$, leaving ${\mathrm{\Omega }}_{\mathrm{uobs}}={\mathrm{\Omega }}_{\mathrm{obs}}-{\mathrm{\Omega }}_{\mathrm{vis}}=\mathrm{26.78874899(59)}\%$ observable in principle but not presently visible.

The numerical correspondence is nevertheless notable. Planck 2018 reports a matter-density parameter near 0.315 in base spatially flat ΛCDM, leaving approximately 0.685 for the cosmological-constant contribution. The corresponding MQ constants are ${\mathrm{\Omega }}_{\mathrm{obs}}=0.3163758388$ and ${\mathrm{\Omega }}_{\mathrm{dk}}=0.6836241612$. The MQ visible-domain fraction, 0.0484883489, is likewise close to the baryonic fraction inferred from the Planck baryon-density result, while the unobserved-domain fraction, 0.2678874899, is numerically close to the non-baryonic matter fraction. Planck 2018 Results VI provides the observational ΛCDM comparison.

The related MQ interpretations are discussed further in Dark Energy, Dark Matter, and Mass in the Universe. The domain fractions also normalize their associated mass-equivalent allocations through $\frac{M_f}{M_{\mathrm{tot}}}={\mathrm{\Omega }}_{\mathrm{f}}$ and $\frac{M_{\mathrm{obs}}}{M_{\mathrm{tot}}}={\mathrm{\Omega }}_{\mathrm{obs}}$. Thus, the fundamental-domain mass-equivalent allocation is one geometrically determined fraction of the total rather than the total mass-equivalent content itself.

TOTAL MASS-EQUIVALENT CONTENT

The elapsed fundamental-time count of the universe is $n_{T\_U}=\frac{A_U}{t_f}$. After System Frame realization, the accumulated total mass-equivalent content is

This relation makes the time dependence explicit. Because ${\theta }_{\mathrm{si}}$ and the fundamental measures are fixed within the MQ construction while $n_{T\_U}$ increases with elapsed count, the realized mass-equivalent content increases linearly with elapsed time. For the adopted universe age, the count interpretation gives $n_{T\_U}=8.0763\times {10}^{60}$, $n_{L\_U}=5.2696\times {10}^{61}$, and $n_{M\_U}=1.4021\times {10}^{62}$. These values expose the scale of the discrete count description rather than introducing additional fitted parameters.

They also satisfy the defining ratios $\frac{n_{L\_U}}{2n_{T\_U}}={\theta }_{\mathrm{si}}=3.262390305$ and $\frac{n_{M\_U}}{n_{T\_U}}=\frac{{{\theta }_{\mathrm{si}}}^3}{2}=17.36112075$. The MQ result is therefore more specific than the statement that the universe contains a particular mass. It establishes a fixed count relation between elapsed time and accumulated mass-equivalent content.

THE MASS-ACCRETION RATE

Dividing accumulated mass-equivalent content by elapsed time removes the elapsed count. Using the dimensionally explicit fundamental relation then gives the principal expansion-epoch prediction.

The dimensions are mass per unit time. $M_{\mathrm{acr}}$ is therefore a rate and must be multiplied by a duration to obtain an accumulated mass-equivalent quantity.

Following realization, continued allocation manifests as net global mass accretion within an increasing cosmic volume. Gravity subsequently redistributes realized mass. This distinction is important to the modern MQ formulation. The increasing mass-equivalent content is not inferred merely because universe age appears in an equation. It follows directly from the count relation $n_{M\_U}=\frac{{{\theta }_{\mathrm{si}}}^3}{2}{n}_{T\_U}$, which fixes the increase in mass-equivalent count for each increase in elapsed fundamental-time count.

CONNECTION TO THE COSMIC MICROWAVE BACKGROUND

MQ provides an observational cross-scale test of this construction through the cosmic microwave background. The derivation propagates the underlying count geometry through early-universe referenceability, mass-energy allocation, realized volume, radiation energy density, and the blackbody relation to obtain the present CMB temperature.

An important distinction is required. The 363,312-year quantity is not simply identified with the conventional CMB formation age. It is the SI-equivalent quantum-epoch temporal scale, $A_{\mathrm{qe}}=e^{\sqrt{3}{{\theta }_{\mathrm{si}}}^3/2}=\mathrm{1.1465266(11)}\times {10}^{13}\mathrm{s}=\mathrm{363,312.35}\mathrm{y}$. The subsequent expansionary-epoch elapsed time scale is $A_{\mathrm{ee}}=A_{\mathrm{qe}}{\left(2{\theta }_{\mathrm{si}}\right)}^{\frac{1}{3}}=2.1424273\times {10}^{13}\mathrm{s}=\mathrm{678,894.26}\mathrm{y}$.

Propagating the associated mass-energy allocation through the realized universe gives $M_{\mathrm{CMB}}=1.5016\times {10}^{50}\mathrm{kg}$. The corresponding present CMB radiation energy density is ${\rho }_{\mathrm{CMB}}=4.1723\times {10}^{-14}\mathrm{J}{\mathrm{m}}^{-3}$. Using the blackbody radiation constant $a=\frac{4\sigma }{c}=\frac{{\pi }^2{k_B}^4}{15c^3{\hslash }^3}=7.565733250\times {10}^{-16}\mathrm{J}{\mathrm{m}}^{-3}{\mathrm{K}}^{-4}$, the predicted present-day monopole temperature is

The peer-reviewed observational determination compiled by Fixsen is 2.72548 ± 0.00057 K. The MQ value lies within the reported 1 $\sigma$ observational uncertainty. See D. J. Fixsen, The Temperature of the Cosmic Microwave Background, The Astrophysical Journal 707, 916-920 (2009).

The modern MQ description of cosmic mass is consequently more precise than the older statement that "the mass of the universe must be increasing." MQ distinguishes pre-realization count allocation, physical realization, expansion-epoch mass accretion, and subsequent gravitational redistribution. After realization, mass-equivalent count scales linearly with elapsed universe count, producing the fixed global rate $M_{\mathrm{acr}}=7.0088795\times {10}^{36}\mathrm{kg}{\mathrm{s}}^{-1}$. The same underlying geometry determines the five referenceability-domain constants and propagates into the MQ calculation of the present cosmic microwave background temperature. Mass accretion is therefore not introduced as an isolated cosmological assumption. Within MQ, it is a consequence of the universe-level count relations and the Frames mapping through which that count structure becomes physically realized.