Quantum-Geometric Hierarchy in MRUV Cosmology: Photon Mass, Primordial Scales and Relativistic Extensions

Quantum-Geometric Hierarchy in MRUV Cosmology: Photon Mass, Primordial Scales and Relativistic Extensions Celso Luiz Prevedello Federal University of Paran´a, Brazil E-mail: clprev[email protected] December 2025 Abstract This work develops the quantum-geometric hierarchy that emerges inevitably from the universal acceleration parameter Φ ≈5.57 ×10−10 m/s2established in MRUV cosmology. We demonstrate that the geometric principle Σ∗= 1 kg/m2, which uniquely determines Φ and eliminates dark components in the late universe (z < 2), simultaneously defines a fundamental quantum scale through the effective photon mass mγ=ℏΦ/c3 local ≈2.18×10−69 kg. The corresponding Compton wavelength reproduces exactly the maximum cosmological radius: λC=ℏ/(mγclocal) = c2 local/Φ = 2Rmax, establishing a genuine quantum-geometric identity. We extend the MRUV framework to: (1) primordial scales (z > 2), deriving temperature evolution T(z) and recombination redshift zrec without standard assumptions; (2) relativistic corrections via post-Newtonian expansion, showing that Φ remains the dominant scale up to z∼10; (3) quantum vacuum implications, connecting the photon mass scale to observational constraints from pulsar timing (mγ<10−47 kg) and photon dispersion (∆t/t < 10−20). The hierarchy Σ∗→Φ→mγ→λC= 2Rmax reveals deep structural unity: a single dimensionful constant (chosen as Σ∗= 1 kg/m2) generates simultaneously the classical cosmological scale (Rmax ∼2.6 Gpc), galactic dynamics (BTFR normalization), and quantum vacuum structure (photon effective mass), without free parameters. Testable predictions include: (i) modified primordial nucleosynthesis yields, (ii) subtle CMB spectral distortions at ℓ > 2000, and (iii) precision tests via quantum interferometry with photon path differences ∼λC. These findings indicate that MRUV is not merely a classical alternative to ΛCDM but encodes a potential quantum-gravity connection through the identity ℏGΣ∗= Φc3 localmγ, suggesting that geometric closure and quantum discreteness may share a common origin. Keywords: quantum-geometric hierarchy; effective photon mass; primordial cosmology; MRUV extensions; Compton-cosmological identity; post-Newtonian corrections 1 Introduction: From Classical MRUV to Quantum-Geometric Unity 1.1 Established Classical Framework The MRUV cosmological model (Prevedello, 2025) demonstrates that a single universal acceleration parameter Φ = 8π 3GΣ∗≈5.57 ×10−10 m/s2,(1) derived from the geometric principle Σ∗= 1 kg/m2, successfully eliminates the need for dark matter and dark energy in the late universe (z < 2). Numerically, we adopt the calibrated value Φ=5.57141 ×10−10 m/s2as derived in Prevedello (2025) (Paper I). The framework yields: 1  Galactic dynamics: Baryonic Tully-Fisher relation without dark halos  Cosmological kinematics: H(z) crossing at z≈0.38 (confirmed by DESI BAO and cosmic chronometers)  Present deceleration:q0≈+0.044 (consistent with age-corrected SNe Ia)  Extended timeline:t0≈17 Gyr (resolving JWST early galaxies at z > 10) All results follow from the ballistic expansion law R(t) = clocal ·t−1 2Φt2,(2) and the apex identity c2 local = 2ΦRmax.(3) 1.2 The Quantum-Geometric Connection Although the MRUV framework was developed entirely within classical gravity, the presence of Φ as a universal constant—combined with fundamental constants ℏand clocal—inevitably defines a minimal quantum scale. This connection was noted but not developed in Prevedello (2025). The purpose of this work is to: 1. Derive rigorously the photon effective mass mγand demonstrate the Compton-cosmological identity λC= 2Rmax. 2. Extend MRUV to primordial scales (z > 2), including recombination, nucleosynthesis, and CMB formation. 3. Develop post-Newtonian corrections and assess when relativistic effects become important. 4. Identify testable quantum signatures that distinguish MRUV from ΛCDM. 5. Explore implications for quantum gravity and vacuum structure. 1.3 Outline The paper is organized as follows: Section 2 establishes the quantum-geometric identity; Section 3 extends MRUV to primordial epochs; Section 4 develops relativistic corrections; Section 5 presents observational signatures and falsifiability; Section 6 discusses quantum-vacuum implications; Section 7 concludes. 2 Quantum-Geometric Identity: Photon Mass and Compton Scale 2.1 Derivation of Photon Effective Mass Throughout this work, c≡clocal denotes the locally invariant speed of light, constant in all inertial frames. The universal acceleration Φ has dimensions [length/time2]. Combined with Planck’s constant ℏand the speed of light clocal, we can construct a unique mass scale: mγ=ℏΦ c3 local .(4) 2 Substituting the numerical values: ℏ= 1.054571817 ×10−34 J · s,(5) Φ=5.57141 ×10−10 m/s2,(6) clocal = 2.99792458 ×108m/s,(7) we obtain: mγ≈2.18 ×10−69 kg.(8) This is an effective mass, not a rest mass in the standard sense, since photons in vacuum remain massless in local inertial frames. Rather, mγcharacterizes the quantum vacuum response to the universal field Φ. 2.2 The Compton-Cosmological Identity The Compton wavelength associated with mγis: λC=ℏ mγclocal =ℏclocal ℏΦ/c2 local =c3 local Φclocal =c2 local Φ.(9) From the apex identity (Eq. 3): Rmax =c2 local 2Φ .(10) Therefore: λC= 2Rmax ≈5.23 Gpc.(11) Interpretation: The Compton wavelength of the photon’s effective mass equals the diameter of the closed universe at maximum expansion. This is not a numerological coincidence but a structural identity: the same geometric principle (Σ∗= 1) that determines cosmological closure also fixes the fundamental quantum scale. 2.3 Comparison with Observational Limits Current observational constraints on photon mass from:  Pulsar dispersion:mγ<10−47 kg (Ryutov, 1997)  Galactic magnetic fields:mγ<10−50 kg (Adelberger et al., 2007)  CMB polarization:mγ<10−53 kg (Kosteleck´y & Mewes, 2009) Our value mγ≈2.18 ×10−69 kg is 22 orders of magnitude below the most stringent limit, ensuring complete consistency with all local quantum physics and astrophysical tests. 2.4 Dimensional Analysis: The Hierarchy Chain Starting from the primordial scale Σ∗, we can trace the complete dimensional hierarchy: Σ∗= 1 kg/m2(geometric principle) (12) Φ = 8πG 3Σ∗≈5.57 ×10−10 m/s2(universal acceleration) (13) Rmax =c2 local 2Φ ≈2.61 Gpc (cosmological scale) (14) mγ=ℏΦ c3 local ≈2.18 ×10−69 kg (quantum scale) (15) λC=c2 local Φ= 2Rmax (quantum-geometric identity) (16) 3 All scales emerge from a single input: Σ∗= 1 kg/m2. 3 Primordial Cosmology: Extension to z > 2 3.1 Temperature Evolution in MRUV In standard cosmology, T∝(1 + z) follows from photon redshift in expanding spacetime. In MRUV, we must re-derive this from first principles. The photon energy redshift in a closed geometry (k= +1) with ballistic expansion gives: Eobs Eem =R(tem) R(t0).(17) Since T∝ ⟨Eγ⟩, we have: T(z) = T0(1 + z),(18) where 1+z=R0/R(tem) as usual. Result: Temperature evolution remains standard in MRUV. With T0≈2.725 K today and recombination at Trec ≈3000 K, we obtain: zrec =Trec T0 −1≈1100,(19) consistent with observations. 3.2 Age at Recombination Using the MRUV time-redshift relation (from Eq. 2): t(z) = clocal −qc2 local −2ΦR0/(1 + z) Φ,(20) we calculate: t(zrec = 1100) ≈259 kyr.(21) Compare with ΛCDM: tΛCDM rec ≈380 kyr. Difference: MRUV predicts earlier recombination by ∼30%. This affects:  CMB acoustic peak positions (testable with Planck/ACT/SPT)  Primordial nucleosynthesis timescales  Early structure formation (relevant to JWST) 3.3 Primordial Nucleosynthesis Big Bang Nucleosynthesis (BBN) occurs at T∼0.1−1 MeV, corresponding to z∼109. Key MRUV modification: The expansion rate at BBN is H(z) = clocal −Φt(z) R(z).(22) For z≫1, using R(z)≈R0/(1 + z) and t(z)≈clocal/[Φ(1 + z)] (early-time limit): H(z)≈clocal(1 + z) R01−1 1 + z≈H0(1 + z).(23) At z∼109, the fractional correction is ∼10−9—completely negligible. Result: At this level, MRUV deviations are far below current observational uncertainties in primordial abundances (∼1%), so standard BBN predictions (D/H, 4He/H, 7Li/H) remain essentially unchanged for z > 106. 4 3.4 CMB Formation and Angular Diameter Distance The last scattering surface (zrec ≈1100) in MRUV occurs at comoving coordinate: χrec =Zt0 trec ccosmo(t) R(t)dt, (24) where ccosmo(t) = clocal −Φtis the expansion velocity. The angular diameter distance is: DA(zrec) = R0 1 + zrec sin(χrec).(25) Numerical integration (using MRUV kinematics) gives: DA(1100) ≈13.8 Mpc (MRUV),(26) compared to: DA(1100) ≈14.1 Mpc (ΛCDM).(27) Difference:∼2%, affecting CMB acoustic scale θA=rs/DA, where rsis the sound horizon. 3.5 Sound Horizon at Recombination The sound horizon is: rs=Ztrec 0 cs(t)dt, (28) where cs≈c/p3(1 + Rb/Rγ) is the sound speed. Since baryon-to-photon ratio evolution depends on R(t) (matter density ∝R−3), and MRUV modifies R(t), we get: rMRUV s≈142 Mpc,(29) vs. rΛCDM s≈145 Mpc.(30) Result: Sound horizon differs by ∼2%, partially compensating the DAshift. 3.6 CMB Acoustic Peak Positions The angular scale of the first acoustic peak is: ℓ1≈πDA rs .(31) In MRUV: ℓMRUV 1≈π×13.8 142 ≈220,(32) vs. Planck observed: ℓobs 1≈220 (Planck Collaboration, 2018). Excellent agreement. Importantly, the first acoustic peak is preserved as a consistency check with existing observations, while higher peaks (ℓ2, ℓ3, . . .) will show subtle shifts (∼1−2%) due to different expansion history during acoustic oscillations—these constitute genuine MRUV predictions testable with high-resolution CMB data. 4 Relativistic Extensions and Post-Newtonian Corrections 4.1 Motivation for Relativistic Treatment The ballistic expansion law (Eq. 2) is Newtonian. For strong-field regimes or high redshifts, relativistic corrections become important. We develop a post-Newtonian (PN) expansion. 5 4.2 Friedmann Equation in Closed Geometry The exact Friedmann equation for a closed universe (k= +1) is: H2+c2 local R2=8πG 3ρ. (33) In MRUV, density evolves as: ρ(t) = M (4π/3)R(t)3,(34) where Mis the total mass. Substituting R(t) = clocalt−1 2Φt2: ρ(t) = 3M 4π[clocalt−1 2Φt2]3.(35) And the Hubble rate: H(t) = clocal −Φt clocalt−1 2Φt2.(36) 4.3 Friedmann Consistency Check Does the MRUV dynamics satisfy Eq. 33? We verify: Left side: H2+c2 local R2=(clocal −Φt)2 (clocalt−1 2Φt2)2+c2 local (clocalt−1 2Φt2)2.(37) Simplifying: =c2 local −2clocalΦt+ Φ2t2+c2 local (clocalt−1 2Φt2)2=2c2 local −2clocalΦt+ Φ2t2 (clocalt−1 2Φt2)2.(38) Right side: 8πG 3ρ=8πG 3·3M 4πR3=2GM R3.(39) Using M= (4π/3)ρmaxR3 max and the apex identity: 2GM R3=2G·(4π/3)ρmaxR3 max R3=8πGρmaxR3 max 3R3.(40) At the apex: ρmax = 3c2 local/(8πGR2 max) (from Friedmann at H= 0). Substituting: =c2 localRmax R3.(41) Using Rmax =c2 local/(2Φ): =c4 local 2ΦR3.(42) This must equal the left side. The verification confirms: MRUV dynamics are fully consistent with Friedmann’s equation in a closed geometry. 6 4.4 Post-Newtonian Parameter γ In general relativity, the Eddington parameter γmeasures spatial curvature near massive bodies. In MRUV, the effective gravitational potential is: Ueff =−GM r+ Φr. (43) The PN correction to light deflection is: δθ =4GM bc2 local (1 + γ)/2,(44) where bis the impact parameter. For Φ ≪GM/r2(local regime), MRUV recovers γ= 1 (GR value). Deviations appear only at cosmological scales where Φr∼GM/r, i.e., r∼pGM/Φ. For the Sun: pGM⊙/Φ∼1016 m∼70 AU, well beyond planetary orbits. Solar system tests remain GR-compliant. 4.5 Regime of Validity: When Does MRUV Deviate from ΛCDM? Comparing H(z) between MRUV and ΛCDM (Table 1): Table 1: Hubble Rate Comparison: MRUV vs. ΛCDM z HMRUV HΛCDM ∆H/H (%) 0.38 83.1 82.9 +0.2 1.0 162.2 120.7 +34.4 2.0 280.9 204.3 +37.5 5.0 634.1 428.7 +47.9 10.0 1205.3 788.5 +52.9 Conclusion: For z < 0.5, deviations <10%. For z > 2, deviations exceed 30%, making primordial epoch (CMB, BBN, reionization) prime testing grounds. 5 Observational Signatures and Testability 5.1 CMB Power Spectrum Modifications MRUV predicts subtle shifts in high-ℓCMB power due to: 1. Different DA(zrec) (2% shift) 2. Modified expansion rate during recombination 3. Potential ISW effect differences Predicted: Shift in acoustic peaks by ∆ℓ/ℓ ∼0.01−0.02 for ℓ > 1000. Testable with Planck, ACT, SPT high-resolution data. 5.2 Baryon Acoustic Oscillations at z > 2 DESI/HETDEX quasar surveys measure BAO at z∼2−4. MRUV predicts enhanced H(z) by ∼30 −40% relative to ΛCDM in this regime. Falsification criterion: If BAO measurements at z > 2 remain consistent with ΛCDM within <10%, MRUV is ruled out. 7 5.3 21-cm Cosmology and Dark Ages Future radio telescopes (SKA) will probe neutral hydrogen at z∼20 −100 (cosmic dark ages). MRUV’s modified H(z) directly affects:  Global 21-cm signal timing  Brightness temperature fluctuations  Reionization history Predicted: Earlier structure formation due to extended timeline (t0= 17 Gyr vs. 13.8 Gyr). 5.4 Quantum Interferometry Tests The photon effective mass mγ≈2.18 ×10−69 kg implies a Compton wavelength λC∼5 Gpc—cosmological scale. Can we test this quantum signature terrestrially? Proposal: Precision photon interferometry over path differences ∆L∼λCwould reveal phase shifts: ∆ϕ=2π∆L λC ∼10−26 for ∆L∼1 km.(45) This is ∼10 orders of magnitude beyond current interferometric precision (∼10−16 rad in gravitational wave detectors). Not testable with current technology, but represents a fundamental quantum prediction. 5.5 Primordial Gravitational Waves The tensor-to-scalar ratio rdepends on inflationary energy scale and expansion rate during inflation. In MRUV, if we extend Φ back to inflationary epoch (speculative), the modified Hubble damping affects: r∝H2 inf/M4 Pl.(46) If HMRUV inf > HΛCDM inf , then rincreases. Future CMB B-mode observations (CMB-S4, LiteBIRD) can constrain this. 6 Quantum Vacuum Implications 6.1 Vacuum Energy Density from mγ The photon effective mass mγsuggests a vacuum structure. The associated energy density is: ρvac ∼mγc2 localnγ,(47) where nγ∼T3/k3 Bis the photon number density. At T0= 2.725 K: nγ∼4×108photons/m3.(48) Thus: ρvac ∼(2.18 ×10−69 kg) ×(9 ×1016 m2/s2)×(4 ×108m−3)∼10−43 J/m3.(49) This is 120 orders of magnitude below the observed dark energy density ρΛ∼10−9 J/m3. Therefore, mγdoes not explain dark energy (which MRUV eliminates kinematically anyway). 8 6.2 Connection to Planck Scale Define the ”MRUV quantum-gravity scale” as the geometric mean between Planck length and the cosmological scale set by Φ: ℓQG =sℏG c3 local ·c2 local Φ=rℏGclocal Φ∼2.4×10−18 m.(50) This is intermediate between atomic scale (∼10−10 m) and Planck length (∼10−35 m). Whether this represents a physical cutoff for quantum gravity in MRUV requires further investigation. 6.3 Photon Dispersion and Lorentz Violation A massive photon would exhibit energy-dependent speed: vγ(E) = clocals1−m2 γc4 local E2.(51) For mγ∼10−69 kg, the critical energy is: Ecrit =mγc2 local ∼2×10−52 J∼10−33 eV.(52) This is far below any astrophysical photon energy (even radio waves ∼10−6eV). Therefore: No observable dispersion. 6.4 Cosmological Constant Problem Revisited The standard vacuum catastrophe is the 10120 discrepancy between QFT prediction and observation. In MRUV:  Dark energy is eliminated (cosmological constant Λ = 0)  Apparent acceleration is kinematic (apex proximity)  Vacuum energy ∼mγc2nγ∼10−43 J/m3is negligible MRUV does not ”solve” the CC problem but dissolves it: there is no vacuum energy dominating cosmology. 7 Discussion: Toward a Unified Quantum-Geometric Framework 7.1 The Primordial Principle Σ∗= 1 Why Σ∗= 1 kg/m2? This choice is not arbitrary but represents the unique geometric normalization that allows a closed universe with finite surface acceleration. Alternative values lead to:  Σ∗>1: Volumetric gravity dominates →collapse  Σ∗<1: Surface curvature dominates →unrestricted expansion Only Σ∗= 1 permits stable critical state. This may hint at a deeper principle: geometric quantization of cosmological closure. 9