Hubble’s Constant

A single expansion geometry with two realizations. In MQ, the Frames mapping relates a common expansion geometry to two physically realized comparison rates under a shared age normalization.

System Frame Realization

H_{sys} = 68.259(19) km s⁻¹ Mpc⁻¹

H_sys denotes the System Frame comparison rate. MQ associates this calculated rate with radiation-anchored comparisons, including cosmic-microwave-background determinations of the present expansion rate. The number is an MQ model calculation, not the quoted Planck observational estimate. A CMB-based Hubble determination is an inference under a specified cosmological model, not a direct measurement of the present rate at last scattering.

Internal Frame Realization

H_{int} = 73.508(20) km s⁻¹ Mpc⁻¹

H_int denotes the Internal Frame-resolved comparison rate. MQ associates it with matter-anchored distance-ladder observations, including Cepheids and Type Ia supernovae. Both H_sys and H_int are physically realized frequencies, with SI units inverse seconds; their conventional astronomical units here are kilometres per second per megaparsec. They are distinct from the nonphysical, vectorless, dimensionless pre-realization configuration domain. This graphic does not assign an SI expansion rate to that primitive domain or identify either model result as an independent observation.

Frames mapping and dimensionless offset

Δ_{H} = θ_{si} - π

The dimensionless invariant MQ coefficient theta_si is distinct from its momentum realization p_si. The suffix si is italic once. Pi is the upright dimensionless circle constant. Delta_H is the dimensionless circle-coefficient offset in MQ3325; it is not itself the physical frequency difference. The MQ rate definitions are

H_{sys} = \frac{π}{θ_{si} A_{U}}, H_{int} = \frac{2 θ_{si} - π}{θ_{si} A_{U}}

Here A_U is the realized System Frame age in seconds. These formulas give rates in inverse seconds before conversion to astronomical units. The age is reconstructed from the measured CMB temperature under the stated MQ normalization; it is not an additional independent input. The model’s absolute time-normalization premise remains unverified. Both results share the age and input dependencies, so their conditional uncertainties and comparison must retain that covariance. They do not constitute two independent combined pieces of evidence.

Physical rate difference

H_{int} - H_{sys} = 5.249 km s⁻¹ Mpc⁻¹

The value 5.249 is the arithmetic difference of the displayed rounded central values, 73.508 minus 68.259. It is a dimensionful separation, distinct from Delta_H. No independent uncertainty for the difference is inferred by treating the two correlated rates as independent. Parentheses on the two rates denote one standard propagated input uncertainty in their final displayed digits, conditional on the MQ age normalization and relations. They do not include unquantified model discrepancy. The current H_int uncertainty is (20), replacing the older source (21), following MQ3383 and DEF1689.

Scientific and observational scope

The association between measurement classes and the two MQ realizations is the framework interpretation under test. Neither the midpoint nor the existence of observational tension alone validates that interpretation. This illustration does not establish a resolution of the Hubble tension, a completed fit of the CMB anisotropy spectrum, or agreement with every local expansion estimator. DEF1949 distinguishes model outputs from the separately inferred observational H_0. The relevant MQ records are MQ3325, MQ3381–3383, DEF1502, DEF1688–1689, DEF2251 and NAR3775.

Conceptual illustration

The CMB sphere at redshift about 1100, cosmic mesh, galaxy field, universal-expansion arrow and small magenta distance-ladder schematic are illustrative. They are not an authenticated CMB sky map, measured spatial shape of the universe, fitted expansion history or plotted observational dataset. Planck 2018 parameter tables give the photon-last-scattering redshift near 1090, consistent with the rounded era label. Riess et al. (2022), Astrophysical Journal Letters 934 L7, doi:10.3847/2041-8213/ac5c5b, describe the Cepheid and Type Ia-supernova distance ladder. The generated cleanup preserves source artwork in character, with local texture changes, and restores all labels and arrows as native SVG. The two original structural text panels remain, with no added solid backings behind their labels or equations.

HUBBLE'S CONSTANT AND THE TWO REALIZATIONS OF COSMIC EXPANSION

The Hubble constant, $H_{0}$ , describes the present relation between recession velocity and distance in the expanding universe. Observationally, its determination has produced one of the most persistent discrepancies in modern cosmology. Measurements inferred from the cosmic microwave background under a cosmological model generally yield a lower value than locally calibrated distance-ladder measurements based on Cepheid variables and Type Ia supernovae.

Measurement Quantization (MQ) proposes a specific physical interpretation of this difference. The framework does not assign two independent expansion rates to the universe. Instead, it describes a single underlying expansion geometry whose realized value depends on the frame through which the measurement is expressed.

The distinction follows from MQ's fundamental measures of length $l_{f}$ , mass $m_{f}$ , and time $t_{f}$ . The Internal Frame contains the discrete count relations underlying a physical system. The System Frame is the continuous physical realization of those relations. The Frames mapping connects the two descriptions.

At cosmological scale, the mapping becomes directly relevant to the measured expansion rate. The MQ parameter $θ_{si}$ establishes the scalar geometry relating the fundamental measures across the two frames. Importantly, $θ_{si}$ is not obtained by fitting the observed Hubble tension. It is independently constrained within MQ through relations involving the fundamental constant and the fine structure constant.

The cosmological age scale then converts this underlying geometry into the conventional units used for the Hubble constant.

A Common Expansion Geometry

MQ first establishes a common expansion geometry and then resolves that geometry through the Frames mapping. The difference between the baseline geometry and its frame-resolved realizations is governed by $θ_{si}$ and $π$ . The associated differential is $ΔH = θ_{si} - π$ .

PHYSICAL INTERPRETATION

The two Hubble values are not introduced as separate empirical fit parameters. MQ assigns them to two realizations of one underlying expansion geometry. The observational question is therefore not only which value is measured, but whether different measurement classes continue to resolve the frame-dependent structure predicted by the model.

Within the MQ construction, the lower realization is associated with the System Frame, while the upper realization is associated with the Internal Frame. The resulting values are

H_{sys} = 68.259 (19) km s^{- 1} {Mpc}^{- 1}

H_{int} = 73.508 (21) km s^{- 1} {Mpc}^{- 1}

Here $H_{sys}$ denotes the System Frame realization and $H_{int}$ denotes the Internal Frame realization. Their separation is therefore

H_{int} - H_{sys} = 5.249 km s^{- 1} {Mpc}^{- 1}

This separation is central to the current MQ interpretation. The framework does not require the observational determinations to converge upon the same numerical value. It proposes that different classes of measurement can retain observable evidence of the frame through which the common expansion geometry is realized.

The System Frame Realization

The lower MQ value, $H_{sys} = 68.259 (19)$ km s⁻¹ Mpc⁻¹, is associated in the MQ model with globally resolved, radiation-anchored determinations such as those inferred from the cosmic microwave background.

Historically, this value showed particularly close numerical agreement with the CMB benchmark used in the original MQ Hubble analysis, 68.3 ± 1.5 km s⁻¹ Mpc⁻¹. That comparison remains part of the development history of the prediction, but it should not be confused with the present observational situation.

More precise CMB analyses now provide a stronger test. Planck 2018, assuming base-ΛCDM, reported $H_{0}$ = 67.4 ± 0.5 km s⁻¹ Mpc⁻¹. A newer SPT-3G analysis, using SPT polarization and lensing data with specified priors, obtained $H_{0}$ = 66.81 ± 0.81 km s⁻¹ Mpc⁻¹.

INTERPRETING THE HISTORICAL CONCORDANCE

The close agreement between the earlier MQ lower-branch value and the observational benchmark used in the original analysis is properly treated as historical numerical concordance. Newer CMB central values lie lower and therefore provide a more stringent test of the fixed MQ System Frame prediction. Historical residuals should not be interpreted as an independent discovery significance.

The Internal Frame Realization

The upper MQ value, $H_{int} = 73.508 (21)$ km s⁻¹ Mpc⁻¹, is associated with the Internal Frame realization and with locally calibrated, matter-anchored observations such as Cepheid and Type Ia supernova distance ladders.

The original MQ comparison used a local observational benchmark near 73.5 ± 1.1 km s⁻¹ Mpc⁻¹. The correspondence was notable because the MQ value arose from the frame construction rather than from fitting the Hubble measurement itself.

The continuing significance of this result lies not simply in the proximity of one calculated number to one observational central value. MQ makes the stronger structural claim that the higher and lower determinations arise as different realizations of the same underlying geometry.

From the Hubble Tension to a Frames Test

In conventional cosmology, the Hubble tension concerns the disagreement between determinations of $H_{0}$ obtained by different methods. Within MQ, that disagreement becomes a proposed test of the physical significance of the frames of reference.

The System Frame and Internal Frame are not introduced to reproduce the Hubble measurements. They belong to the broader architecture of MQ and appear in its treatment of the fundamental measures, physical constants, gravitation, quantum phenomena, and cosmological observables. The Hubble problem is therefore a cross-domain application of an independently used structure.

This also changes what constitutes the most informative observational test. Agreement with a single historical Hubble determination is less important than whether increasingly precise independent measurements preserve a separation compatible with the fixed MQ frame-resolved values.

If future independent measurements converge upon a single common value with uncertainties incompatible with the two MQ realizations, the present frame interpretation of the Hubble tension would fail that empirical test. Likewise, persistent lower and upper observational branches would not by themselves establish MQ. Their numerical values and separation would also have to remain consistent with the values obtained from the MQ construction.

WHAT THE OBSERVATION MUST TEST

A persistent Hubble tension alone is not sufficient evidence for the MQ interpretation. The stronger test is quantitative. Independent measurements must remain compatible with the predicted lower realization, upper realization, and their fixed 5.249 km s⁻¹ Mpc⁻¹ separation.

Connection to the Fundamental Measures

The Hubble construction is not isolated from the rest of Measurement Quantization. Expansion is related to the same fundamental measures that establish MQ's description of length, mass, and time. The cosmological geometry consequently shares mathematical structure with the framework's treatment of physical constants.

This connection extends to gravitation. MQ relates $θ_{si}$ and the fundamental measures to the gravitational constant, although the current Hubble calculation does not depend on selecting a particular experimental determination of $G$ . The modern MQ treatment of $G$ is maintained separately because gravitational measurements present their own experimentally resolved structure.

The Hubble calculation also connects to MQ's description of the cosmic microwave background and the transition between its early count-resolved cosmological description and the later expansionary realization. These are not independent additions to the Hubble model. They arise from the same frame architecture.

A Falsifiable Cosmological Prediction

The present MQ interpretation is therefore more specific than the claim that different observational methods happen to produce different values of $H_{0}$ . It assigns definite values to two frame-resolved realizations and a definite separation between them.

Improved CMB, distance-ladder, standard-siren, and other independent determinations can progressively test whether the observational structure remains compatible with $H_{sys} = 68.259 (19)$ km s⁻¹ Mpc⁻¹, $H_{int} = 73.508 (21)$ km s⁻¹ Mpc⁻¹, and their 5.249 km s⁻¹ Mpc⁻¹ separation.

The larger significance of Hubble's constant within Measurement Quantization is therefore not the reproduction of one observational value. It is the possibility that a cosmological measurement retains observable evidence of the frame through which physical measure is realized.

For the historical development of this calculation, see Resolving the Hubble Tension and the earlier ResearchGate Hubble analysis.

AIP Advances
8: Resolving the Hubble tension 68.261 and 73.510 km s Mpc

International Journal of Geometric Methods in Modern Physics
7: Measurement Quantization

Journal of High Energy Physics, Gravitation and Cosmology
6: Physically Significant Units of Measure
5: Measurement Quantization Unites Classical and Quantum Physics
4: Quantum Model of Gravity Unifies Relativistic Effects, Describes Inflation/Expansion Transition, Matches CMB Data
3: Measurement Quantization Accounts for Galactic Rotational Velocities and Obviates Dark Matter
2: Measurement Quantization Describes History of Universe -
Quantum Inflation, Transition to Expansion, CMB Power Spectrum

International Journal of Theoretical and Mathematical Physics
1: Measurement Quantization Describes the Physical Constants