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CMB Power Spectrum

The Cosmic Microwave Background (CMB) is one of the most information-rich observations in cosmology. Its temperature and polarization anisotropies are conventionally represented by angular power spectra whose structure constrains cosmological parameters. Within the standard Lambda-CDM framework, those spectra are modeled through the evolution of radiation, baryonic matter, cold dark matter, primordial perturbations, and the geometry and expansion history of the universe. Measurement Quantization (MQ) approaches the same observational record from a different starting point. It asks whether characteristic CMB quantities can arise from the same count-constrained geometry and Frames mapping used elsewhere in the theory.

That distinction is important. MQ does not identify its visible, unobserved, observable, and dark domains with four physical substances. They are geometric referenceability domains. Their numerical relationships to quantities inferred in conventional cosmology are therefore comparisons to observation rather than identities assumed at the outset.

The construction begins with the MQ quantum epoch. In the current formulation, this is a pre-referenceable count regime rather than an interval during which an already realized physical universe simply expands at an unusually low rate. Before a valid System Frame exists, ordinary physical length, velocity, density, and elapsed System Frame time cannot be assigned to the configuration in the usual classical sense. The Internal Frame instead contains the underlying count relations from which physically realized measure subsequently emerges.

The minimum three-dimensional count geometry establishes the referenceability condition below, where $R_{\mathrm{qe}}$ is the quantum-epoch radial measure and $l_f$ is the fundamental measure of length.

The square-root-of-three relation follows from the minimum orthogonal three-dimensional construction. This threshold closes the quantum epoch and permits realization of a physical System Frame. Consequently, the quantum epoch should not be described as an already realized universe possessing an ordinary physical radius smaller than one fundamental length. Before realization, the MQ construction is not yet an ordinary physical spacetime to which such a measured radius can be assigned.

The SI-equivalent temporal extent associated with the quantum-epoch count construction is described by $A_{\mathrm{qe}}$, with ${\theta }_{\mathrm{si}}$ denoting the MQ dimensionless scalar that governs the relevant frame relations:

The 363,312-year value is therefore an SI-equivalent mapping of the quantum-epoch count construction. It should not be interpreted as 363,312 years of ordinary expansionary System Frame time, nor should it be identified directly with the conventional photon-decoupling age inferred within Lambda-CDM.

The Frames mapping between the quantum and expansionary descriptions produces the corresponding expansionary-epoch age marker

which evaluates to 2.14242727551(18) × 10¹³ s, or 678,894.236415(88) y. This factor is not an independently fitted cosmological parameter. It follows from the same MQ Frames mapping between the Internal Frame and System Frame.

This early-universe construction can be propagated forward to the present CMB. The standard radiation constant is

where $a$ is the radiation constant, $\sigma$ is the Stefan-Boltzmann constant, $c$ is the speed of light, $k_B$ is the Boltzmann constant, and $\hslash$ is the reduced Planck constant.

The CMB radiation energy density is related to temperature by the standard blackbody relation, where $E_{\rho :\mathrm{CMB}}$ is the CMB radiation energy density and $T_{\mathrm{CMB}}$ is the CMB temperature.

The total CMB energy $E_{\mathrm{CMB}}$ and corresponding mass equivalent $M_{\mathrm{CMB}}$ then follow from

where $V_U$ is the present universe volume.

The significance of this calculation is methodological. The observed CMB monopole temperature is not fitted to obtain the MQ result. Instead, the calculation propagates the expanding-frame count construction through the quantum-epoch allocation, present universe volume, radiation density, and standard blackbody relation. The adopted cosmic age remains an observational normalization and is the dominant source of uncertainty in this forward calculation. The resulting agreement with the measured CMB temperature is therefore properly characterized as a cross-scale consistency test rather than as a parameter-free determination independent of cosmological measurement.

The relationship can also be inverted. Given $T_{\mathrm{CMB}}$, MQ obtains a corresponding cosmic age $A_U$:

This inverse calculation supplies a separate internal consistency test connecting the early-universe construction to the observed CMB temperature. Related calculations are developed in Diameter & Age of the Universe and the discussion of the quantum epoch.

Geometric Domains

The second part of the CMB argument concerns the large-scale partition of the universe. MQ distinguishes four referenceability domains. ${\Omega }_{\mathrm{vis}}$ is the fixed fraction presently visible; ${\Omega }_{\mathrm{uobs}}$ is the fixed fraction not presently visible but ultimately referenceable; ${\Omega }_{\mathrm{obs}}$ is their combined observable domain; and ${\Omega }_{\mathrm{dk}}$ is the complementary domain that remains permanently non-referenceable.

The total domain is normalized by the principal relation

The observable domain is related to the visible domain by ${\Omega }_{\mathrm{obs}}=2{\theta }_{\mathrm{si}}{\Omega }_{\mathrm{vis}}$, while the unobserved domain is ${\Omega }_{\mathrm{uobs}}={\Omega }_{\mathrm{obs}}-{\Omega }_{\mathrm{vis}}$. The fixed frame-domain fraction is given below.

The remaining fixed domain constants follow directly:

The unobserved constant is the difference between the observable and visible domains, ${\Omega }_{\mathrm{uobs}}={\Omega }_{\mathrm{obs}}-{\Omega }_{\mathrm{vis}}=\mathrm{26.7887490004(13)}\%$. Consequently, ${\Omega }_{\mathrm{obs}}={\Omega }_{\mathrm{vis}}+{\Omega }_{\mathrm{uobs}}$ and ${\Omega }_{\mathrm{obs}}+{\Omega }_{\mathrm{dk}}=1$.

The distinction from Lambda-CDM is conceptual as well as notational. ${\Omega }_{\mathrm{dk}}$ is not defined in MQ as the density of a dark-energy substance or cosmological fluid. Likewise, ${\Omega }_{\mathrm{uobs}}$ is not defined as a population of non-baryonic dark-matter particles. They are fixed geometric domain partitions arising from referenceability and the Frames mapping. Any correspondence between these quantities and observations conventionally interpreted through dark matter or dark energy is therefore an empirical comparison rather than an identification built into the definitions.

The CMB power-spectrum figure on this page summarizes these fixed domain relations alongside the angular-power structure. Its domain labels and numerical values should therefore be read as MQ geometric constants rather than as fitted cosmological density parameters.

The CMB Power Spectrum

MQ next asks whether the same fixed geometric partition is reflected in the CMB angular power spectrum. Conventionally, the angular power spectrum is expressed through the multipole coefficients $C_{\ell }$. The commonly plotted rescaled angular-power quantity is

Here $\ell$ is the angular multipole number, $C_{\ell }$ is the angular power at that multipole, and $D_{\ell }$ is the corresponding rescaled quantity.

Within MQ, the CMB is associated with the non-discrete System Frame. Its CMB-resolved expansion-rate realization is therefore $H_{\mathrm{sys}}$. Distance-ladder and Cepheid-like determinations are associated in the MQ Hubble analysis with the discrete Internal Frame realization $H_{\mathrm{int}}$. These are frame-resolved realizations of a common underlying expansion description rather than two independent primitive expansion parameters. This construction is developed separately in Hubble's Constant.

The power-spectrum construction associates the fixed geometric domains with corresponding components of the angular-power structure. In compact form, the mappings are

Modern CMB analyses infer cosmological parameters by fitting temperature and polarization power spectra within an explicit cosmological model. MQ instead derives fixed geometric domain fractions from ${\theta }_{\mathrm{si}}$ and tests whether the observational CMB structure is consistent with those fractions. Numerical proximity between an MQ domain fraction and a conventionally inferred cosmological density fraction does not establish that the two quantities have the same physical meaning.

The CMB calculation nevertheless connects several parts of MQ that would otherwise appear independent. The quantum-epoch count geometry determines the transition to a realizable System Frame. The Frames mapping determines how the count construction is represented in the expansionary epoch. The resulting mass-energy allocation can be related to the present CMB temperature through the standard blackbody radiation law. Large-scale referenceability geometry independently produces the fixed visible, unobserved, observable, and dark domain partitions. CMB-based expansion measurements are assigned to the System Frame realization of the expansion rate. Within MQ these results arise from a common count-and-mapping framework rather than from unrelated mechanisms introduced separately for each cosmological observation.

MQ consequently does not introduce a separate inflationary stage as a primitive component of its own early-universe construction. Instead, its formulation contains a finite pre-referenceable quantum epoch terminating at the minimum realizable three-dimensional geometry. This should not be described as ordinary physical expansion lasting 363,312 years. The 363,312.34127(22)-year value is the SI-equivalent temporal extent assigned to the pre-referenceable quantum-epoch count construction, whereas ordinary elapsed physical time is associated with the realized System Frame.

The CMB therefore provides several logically distinct tests of Measurement Quantization. These include the quantum-to-expansionary transition, the relation between the early count construction and present CMB temperature, the fixed geometric domain partitions, the frame assignment of CMB-based expansion measurements, and the proposed correspondence between those geometric domains and CMB angular-power structure. The power-spectrum relations describe a mapping between MQ geometric fractions and spectrum components rather than a complete replacement for a full CMB anisotropy calculation. Maintaining that distinction keeps the empirical claims commensurate with the mathematical development presently established in MQ.