Time Dilation Between the Quantum and Expansionary Epochs

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60605

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

We examine the discrete description proposed by Measurement Quantization for relations between the expansionary epoch and a pre-referenceable quantum epoch. The Internal Frame is a nondimensional configuration domain. The System Frame is the continuous relational description obtained from those counts through the Frames mapping. Dimensional descriptions of the quantum epoch are assigned from that observer perspective. The dilation considered here is the geometric scaling between those descriptions. We combine classical kinematic and radiation relations with the count and mapping assumptions of this framework. No parameter is fitted to the age comparison reported here. The quantum-epoch construction has an observer-represented SI-equivalent extent of 363,312 years and, under the adopted unit-leg closure, places the referenceability threshold at a realized radius of √3 fundamental lengths. Combining its mapped extent with the realized mass accretion rate gives the cosmic microwave background mass allocation. Using the measured cosmic microwave background temperature with that allocation gives a universe age of 13.7943(38) billion years within this construction. The corresponding calculated universal volume recovers the temperature input as a consistency check. The epoch ratio can be expressed as an algebraic speed parameter equal to 84.4755% of the speed of light. This parameter does not represent relative motion between physical clocks in the two epochs.

Keywords: Time Dilation, Epochs, Quantum Epoch, Universe, Inflation, Expansion

1. INTRODUCTION

We present expressions that describe the geometric scaling of the Cosmic Microwave Background (CMB) allocation between quantum-epoch and expansionary-epoch descriptions. Inflation was proposed to address the horizon and flatness problems [1, 2]. The Measurement Quantization (MQ) interpretation instead assigns a pre-referenceable quantum epoch ([3], Appx. BF–BH). Here pre-referenceable means that the external relations do not yet admit a globally consistent realized description under the adopted closure. Its nondimensional count structure is described from the observer’s perspective, without an intrinsic quantum-epoch clock or an independently measurable internal velocity. In the observer’s description, its temporal extent is represented by 363,312 years and its referenceability boundary by √3 fundamental lengths.

The fundamental measures l_f, m_f and t_f are length, mass and time references in m, kg and s ([3], Appx. W). They are distinct MQ references compared with conventional Planck units [4, 5]. Their numerical uncertainties are conditional on the stated calibration and do not establish a more precise direct measurement of the Planck scale. Their counts are nondimensional. We write c for the speed of light, 299,792,458 m s⁻¹. The interepoch comparison uses dimensional quantities realized through these references.

Within this interpretation, pre-referenceable mass allocation is represented by an observer’s dimensional construction. After realization, mass accretion is expressed as a steady rate ([3], Appx. BG, BE). These descriptions are related through the epoch scaling, from which the CMB mass equivalent and, for a calculated universal volume, its present density and temperature are obtained ([3], Appx. BH). Fixsen reports a CMB temperature of 2.72548 ± 0.00057 K [6]. Its use in a calculation must be distinguished from its use as an independent comparison.

The present calculation does not model inflation, recombination or the full CMB anisotropy spectrum. Its numerical comparisons therefore do not by themselves replace those physical descriptions [7, 2].

The MQ construction adds discrete counts and a Frames mapping to descriptions of length, mass and time. Conventional Planck units provide a dimensional comparison [4, 5]. The coefficient is calibrated using the inverse fine-structure input, while the CMB temperature enters the age inference. Holding those inputs fixed avoids fitting a parameter to the quoted age benchmark, but does not remove the framework assumptions. We do not develop the loop-quantum-gravity approaches discussed in [8, 9] or the string-theory framework of [10]. Earlier MQ work discusses applications to physical constants [11]. Those applications supply background rather than independent proof of every assumption used here.

We denote the SI-equivalent quantum-epoch extent by A_qe and the associated expansionary-epoch elapsed-time representation by A_ee, both expressed in s or y. The invariant dimensionless MQ coefficient is θ_si. The expansion scalar 2θ_si gives the cube-root factor in the three-dimensional epoch mapping. Ages expressed in billions of years are converted to seconds for SI representation using 1 Gyr = 10⁹ × 31,557,600 s. This is an ordinary unit conversion. Equations (1)–(2) describe the relation between representations and do not assign physical duration to the nondimensional configuration domain. In the exponential expressions below, e is the dimensionless base of the natural logarithm.

The factor

scales the two epoch descriptions. Calling this relation a dilation refers to that scaling. It does not assert special-relativistic clock dilation between the epochs. Numerical results are MQ calculations using the 2022 CODATA source inputs [5]. Parenthetical uncertainties give one standard uncertainty in the final printed digits, propagated through the shared input dependencies. They quantify propagated source-input uncertainty within these MQ calculations and do not include an independently estimated model-discrepancy contribution.

The quantum-epoch extent is therefore an SI-equivalent description of a pre-referenceable allocation. It is not an intrinsic elapsed time and is not identified with a measured photon-decoupling age. The mapped expansionary extent is the quantity used with the realized accretion rate to express the CMB mass allocation.

2. METHODS

The calculation uses the MQ count framework with classical kinematic and radiation relations. Earlier MQ analyses discuss gravitational, uncertainty-principle and cosmological applications [11–16]. Heisenberg’s uncertainty principle [17] is part of that background; its use in a count construction does not alone establish that every physically realized length or time is discrete.

The MQ framework and its background derivations are presented in [3]. They provide the source assumptions and mathematical context for this calculation. Empirical comparisons must additionally identify the measured observable, its uncertainty and any input shared with the calculation. The cited paper and supplementary resources are available through Online Resources.

https://www.researchgate.net/publication/361148724_OnlineResources

The supplementary spreadsheets provide background calculations. The numerical results reported here use the fixed 2022 CODATA input edition [5] and the stated CMB temperature [6]. Earlier workbook values are not independent measurements or additional tests of the present results.

3. RESULTS

Within MQ, the construction distinguishes the observable expansionary epoch from an earlier pre-referenceable configuration domain. Its ordering is expressed through discrete relations. The epoch comparison changes the observer’s representation of those relations by the geometric expansion scalar. The argument does not require clocks or relative inertial motion within the quantum epoch.

We first describe the quantum-epoch count construction, its referenceability boundary and its SI-equivalent representation. Before global referenceability, external points cannot be assigned the realized relations used in the expansionary description. The mass allocation is then represented dimensionally using the observer’s reference measures. In this presentation that allocation supplies the CMB mass equivalent. Its scaling is governed by the epoch mapping rather than relative motion of two physical clocks.

The adopted realized right-triangle construction uses nondimensional orthogonal leg counts n_a and n_b. A triangle with two unit legs gives a diagonal of √2. Combining that diagonal with a further orthogonal unit leg gives the three-dimensional threshold. The repeated unit leg in the nested construction produces the single equation

Within this closure construction, multiplying the count by the fundamental length gives the realized referenceability boundary ([3], Appx. BF). In the same observer representation, v_i denotes the SI-equivalent expansion rate in m s⁻¹ ([3], Appx. BG). It is evaluated using the represented quantum-epoch extent.

The realized mass accretion rate M_acr has units kg s⁻¹ ([3], Appx. BE). Here p_si is the momentum realization, in kg m s⁻¹, of the coefficient introduced above. In the adopted SI realization their numerical coefficients agree. Its defining relation is

The numerical rate uses the 2022 CODATA inverse fine-structure input [5] and the exact SI constants c and h, where h is the Planck constant in J s. Multiplying the rate in Eq. (5) by the mapped extent in Eq. (7) gives the mass allocation in Eq. (8).

For use with the rate in Eq. (5), we repeat the two extents from Eqs. (1)–(2) ([3], Appx. BH). The cube-root mapping is applied before the expansionary extent is multiplied by the rate.

We denote the CMB mass equivalent by M_CMB, in kg, and combine the expansionary extent with the realized rate. In this allocation,

is the dimensionless count of the mapped extent. Multiplication by the fundamental mass converts the mass count to kg. The time-times-rate and count-times-mass forms in Eq. (8) express the same allocation ([3], Appx. BH).

The universal volume V_U has units m³ and is calculated from the universe age A_U, expressed in s, and the expansion scalar. It describes the calculated whole universe, distinct from the observable-universe volume. Equation (9) supplies the geometric dependence ([3], Appx. BI). We write ρ for radiation energy density in J m⁻³ and T_CMB for CMB temperature in K. The radiation constant a has units J m⁻³ K⁻⁴ and is obtained from

where k is the exact SI Boltzmann constant in J K⁻¹ [5]. Here π is the dimensionless circular constant. The age is obtained from Eq. (22) using the measured temperature. Equations (10)–(11) then check the resulting density and temperature.

In Eq. (22), the measured temperature reported above [6] determines the age and hence the volume in Eq. (9). Substituting this volume into Eqs. (10)–(11) recovers the input temperature. This agreement checks internal consistency. An independent temperature prediction would require an age fixed without that same temperature input.

1. Calculations

The dimensional fundamental relation l_fm_f=2p_sit_f ([3], Appx. V) separates the momentum realization from the dimensionless coefficient. Cubing Eq. (2) gives Eq. (12). Its numerator uses the expansionary extent and its denominator the quantum-epoch extent, with the common geometric factors cancelling. This is a ratio of the two cubic scale constructions, rather than a comparison of independently measured whole-universe volumes. Equation (13) repeats the quantum-epoch extent used in the temporal-count ratio.

Let n_Tl and n_To denote the dimensionless temporal counts in the expansionary and SI-equivalent quantum-epoch descriptions. With A_ee=n_Tlt_f and A_qe=n_Tot_f, the common time reference gives

This is a relation between two representations of the temporal count. To express the same scalar ratio in the conventional length-contraction form, n_L denotes a displacement count and n_Lc the distance count for light over the same represented interval. We use the following algebraic identification ([3], Appx. AU, BK). It introduces a comparison parameter and does not assert physical special-relativistic dilation between epochs.

Equating the scalar count ratio with that algebraic form gives the following sequence ([3], Appx. BK).

The conventional speed parameter in [18], Secs. 3–4, gives an algebraic comparison for the same dimensionless ratio. Let n_T be the common dimensionless time count, t its represented time in s, and l and l_c the displacement and light-distance measures in m. The comparison speed v is

, in m s⁻¹. Converting the counts to those measures gives Eq. (20), using the same represented interval in both velocity quotients.

The resulting ratio v/c is an algebraic comparison with the conventional speed parameter ([3], Appx. BK). Its value is

expressed as a fraction of the speed of light. Its numerical value is an MQ calculation using the 2022 CODATA input [5]. The dilation in this paper remains the geometric rescaling between epoch descriptions. It does not require the physical relative velocity of two clocks. Laboratory clock tests compare actual clock rates under specified motion or gravitational conditions [19]. No such pair of clocks is identified between the MQ descriptions here.

2. Observational Comparison

Combining Eq. (8) with the volume in Eq. (9) gives the density in Eq. (10). Equating that density to the radiation density associated with Eq. (11) and solving for the positive age yields Eq. (22) ([3], Appx. BI). Substitution of Eq. (5) cancels the common cubic factor in θ_si. The quotient inside the cube root has units s³, so the result is a time. The exponential remains in the expansionary extent defined earlier. The global volume is then calculated from that age through Eq. (9); no observed whole-universe boundary is required.

The age in Eq. (22) can be compared with 13.787 ± 0.020 billion years from the Planck collaboration’s combined temperature, polarization, lensing and baryon-acoustic-oscillation data, evaluated within its standard cosmological model [7]. The MQ central value lies within that quoted interval. This numerical consistency does not establish the MQ epoch interpretation or provide a comparison independent of cosmological assumptions and possible shared inputs. The calculated whole-universe volume remains the quantity defined in Eq. (9).

4. DISCUSSION

MQ distinguishes nondimensional count structure from its dimensional realization through the Frames mapping. In this presentation, the quantum epoch is described from the observer’s expansionary perspective. The dilation factor rescales the descriptions as a function of the expansion scalar. It does not describe an intrinsic clock in the configuration domain or special-relativistic motion between the epochs. The CMB allocation and calculated universal volume are evaluated within that same realization. Their numerical consistency checks use shared inputs and do not by themselves establish a complete early-universe dynamics. Independent empirical tests must distinguish an inferred quantity from a reused calibration input.

The mass allocation and epoch mapping describe the expanding construction used here. They do not supply a contracting branch or a dynamical bounce condition. Extending them to a bouncing cosmology would require an account of how the allocation changes during contraction, how an initiation event relates to the subsequent expansion, and what produces a reversal. The equations in this paper therefore leave those questions outside the present derivation.

5. APPENDICES

APPENDIX A: TERMS

Additional MQ definitions and derivations are given in [3].

• The angular realization Θ_si is expressed in radians and remains distinct from the dimensionless coefficient and momentum realization. The value 3.26239 rad in Shwartz and Harris is a calculated signal/idler propagation angle relative to crystal planes for a 25-keV pump in diamond (111), not a measured polarization rotation or MQ coefficient [20]. We retain 2 μrad as an adopted comparison tolerance, not a published standard uncertainty. The different down-conversion geometry in [21] and the wave-mixing widths in [22] do not establish an uncertainty for that calculated angle. The rounded tabulation therefore supplies a numerical comparison rather than an independent measurement of the coefficient.

• The exact SI value of c used above is also listed in the 2018 CODATA compilation [23] and the 2022 source [5]. Its count representation is

  $$\mathit{c}=\frac{{\mathit{n}}_{\mathit{L}}{\mathit{l}}_{\mathit{f}}}{{\mathit{n}}_{\mathit{T}}{\mathit{t}}_{\mathit{f}}}\text{.}$$

  (22a)
  Equal unit length and time counts recover the fundamental speed relation.

• For the gravitational coupling G, upright type denotes the MQ upper-count value,

  $$6.6740779383(10)\times {10}^{-11}\mathrm{m}\text{³}\mathrm{kg}\text{⁻}\text{¹}\mathrm{s}\text{⁻}\text{²}\text{.}$$

  (22b)

  The finite-count value G at the electromagnetic demarcation, the fixed MQ electromagnetic interaction-count distance, is

  $$6.6738448317(10)\times {10}^{-11}\mathrm{m}\text{³}\mathrm{kg}\text{⁻}\text{¹}\mathrm{s}\text{⁻}\text{²}\text{.}$$

  (22c)
  These are MQ calculations using the same 2022 input chain [5], with their difference governed by the Informativity differential, the count-dependent non-Lorentz mapping correction. These model-specific realizations are not new direct measurements or replacement CODATA recommendations.
• The fundamental measures introduced in the Introduction also provide the reference factors used to convert the dimensionless counts in Eqs. (8), (14)–(15) and (20).
• l_p, m_p and t_p are Planck’s units for length, mass, and time. Their SI units are m, kg and s, respectively.
• The count notation extends to n_M, the dimensionless count of the fundamental mass. The length and time counts used above retain their stated meanings in each application.
• In Eqs. (16)–(20), n_L and n_Lc retain the displacement and light-distance roles assigned before Eq. (16).
• For a center-of-mass description, the length count instead specifies separation from the center; its reference remains the fundamental length. This contextual use is distinct from the displacement assignment in the speed comparison.
• The common time interval in Eq. (20) is essential when converting the displacement-to-light-distance count ratio into the speed ratio.

• For the electromagnetic demarcation, the length count satisfies

  $${\mathit{n}}_{\mathit{L}}=\frac{276}{{\mathit{\theta }}_{\mathit{si}}}\text{.}$$

  (22d)

  The metric differential expresses the discrete–continuous count mismatch in the Frames mapping. Its adjusted System Frame count of the MQ coefficient is n_θ. Mixed left-floor and right-ceiling brackets denote the nearest-whole-unit operation, giving

  $${\mathit{n}}_{\mathit{\theta }}=\lfloor \frac{{\mathit{n}}_{\mathit{L}}}{2}\rceil =42\text{.}$$

  (22e)
• The argument in this nearest-whole-unit operation lies strictly between 41.5 and 42.5. No half-unit tie occurs in the present calculation.
• Q_L is the dimensionless fractional residual of a fundamental-length count used in the more precise count construction.
• Q_Ln_L enters the normalization associated with the Informativity differential introduced above. This residual-count product approaches one half at the upper count limit, where the length-normalization factor tends to unity and the associated contraction vanishes.

• The conventional reduced Planck constant ℏ is defined exactly by

  $$\frac{\mathit{h}}{2\mathrm{\pi }}$$

  (22f)

  in J s, with decimal value 1.054571817646156… × 10⁻³⁴ J s [5]. The distinct MQ upper-count action is written upright,

  $$\mathrm{ħ}=2{\mathit{p}}_{\mathit{si}}{\mathit{l}}_{\mathit{f}}=1.0545349844417293(56)\times {10}^{-34}\mathrm{J}\mathrm{s}\text{.}$$

  (22g)
  The latter is calculated from the same 2022 source input, with shared dependencies retained.
• m_e is the electron mass, in kg.
• a₀ is the Bohr radius, a length in m, used in the fine-structure calibration.
• α_f^-1, α_p^-1, and α_c^-1 are the dimensionless fundamental, Planck-like, and classical forms of the inverse fine structure constant. α_f^-1=42θ_si is defined with respect to the Internal Frame; the remaining two against the System Frame such that α_c^-1 differs from α_p^-1, accounting for the length contraction associated with discrete measure.
• γ denotes the historical electromagnetic transformation parameter, with SI units m⁶ C⁻² s⁻⁴ [3], Appx. AQ, AR. It is not used in the epoch equations.

STATEMENTS AND DECLARATIONS

The author has no relevant financial or non-financial interests to disclose.

The author has no competing interests to declare that are relevant to the content of this article.

DATA AVAILABILITY

The equations and numerical inputs are given in the manuscript. The accompanying calculation record provides source provenance, numerical dependencies and reproducibility details.

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