A quasi-static lapse-based model for the low-redshift Hubble diagram and its redshift-drift signature

A quasi-static lapse-based model for the low-redshift Hubble diagram and its redshift-drift signature Eric L. Levin [email protected] ORCID: 0009-0005-8314-0387 November 19, 2025 Abstract We present a general-relativistic, scale-free reconstruction of the low-redshift luminosity–distance relation using the Pantheon(+) Type Ia supernova sample. Rather than assuming an expanding Friedmann–Lemaˆıtre–Robertson–Walker background, we examine whether the observed curvature of the Hubble diagram can be reproduced by a smooth lapse function in a static, spherically symmetric metric. The underlying geometry obeys the standard Einstein field equations; no additional stress–energy components or modified dynamics are introduced. The lapse affects only the mapping from redshift to distance, allowing a purely kinematic test of whether a gravitational-redshift interpretation can match present observations. Using only the shape of the supernova magnitude–redshift relation, we reconstruct the lapse profile in a scale-independent manner and show that it reproduces the observed lowredshift curvature with a single additive magnitude offset absorbing the absolute calibration. We also summarize a minimal, quasi-static time-dependent extension of the lapse that gives rise to a definite redshift-drift signal, providing a clear observational discriminator for future facilities. All derivational steps used in this analysis are documented in three companion theory notes, and all numerical procedures are released as publicly available code. The results establish a conservative general-relativistic baseline against which upcoming redshift-drift measurements can be directly compared. 1 Introduction Cosmological redshift is conventionally interpreted through an expanding Friedmann–Lemaˆıtre– Robertson–Walker (FLRW) geometry, most commonly in the form of the ΛCDM model. [ 1 , 2 , 3 ] While ΛCDM is broadly successful, persistent discrepancies—most notably the tension between earlyand late-universe inferences of H0 —motivate approaches that cleanly separate kinematics (the mapping from redshift to distance) from dynamics (the stress–energy required by the Einstein field equations). e.g., [4] The Time-Field Hypothesis (TFH) provides such a separation. TFH examines whether a smooth lapse structure along null geodesics can influence the operational inference of redshift without introducing a global scale factor or modifying Einstein’s equations. (e.g., [ 5 , 6 ]) In its static implementation (“Path B”), TFH alters only the relation between observed redshift and the underlying static metric, leaving the geometry, curvature tensors, and EFEs unchanged. In a minimally dynamical extension (“A2”), TFH introduces slow time dependence in the lapse, generating a small but measurable redshift-drift signal while still avoiding FRW expansion. Earlier conceptual work framed redshift as an integrated gravitational effect in a static setting; here we adopt a fully GR-consistent formulation with explicit EFE derivations. This paper performs a data-driven, scale-free reconstruction of the lowz Hubble-diagram shape using Pantheon(+) Type Ia supernovae and tests whether a smooth lapse mapping can 1 reproduce the observed curvature. The reconstruction is model-independent and relies only on Etherington reciprocity and the gravitational-redshift relation obtained from the Path B EFE identities. We then examine the quasi-static A2 extension and derive its associated redshift-drift prediction, which provides a decisive observational test of lapse dynamics. All assumptions used here are GR-consistent and traceable to explicit derivations, summarized below. Contributions. In this paper we: 1. perform a fully data-driven, scale-free reconstruction of the lowz Hubble-diagram shape using Pantheon(+) Type Ia supernovae, and show that TFH–A2 reproduces this shape with only a fitted offset ∆µ(absorbing absolute calibration); 2. present explicit, falsifiable redshift-drift predictions in the A2 framework, governed by (ϵ, τ, r⋆), for which ˙z∝(1+z)ϵ f[r(z)]/τ; 3. document robustness to analysis choices (redshift cut zmax and intrinsic dispersion σint ) and provide complete, machine-reproducible artifacts for independent validation. We emphasize that the present study is purely kinematic: no new stress–energy component is introduced beyond those required by the companion EFE notes. The goal is to assess whether a smooth lapse-structured redshift mapping is compatible with the observed lowz Hubble-diagram shape and whether the associated redshift-drift signal provides a decisive observational test. General-Relativistic Status. We emphasize that the present lowz analysis relies solely on the Path B construction, which preserves the standard Einstein field equations exactly. No modified-gravity terms, new geometric degrees of freedom, or non-metric effects are introduced; only the operational mapping between redshift and the static lapse is adjusted. Companion theory notes. The present analysis is supported by three companion derivation papers that provide the full Einstein-equation framework. (i) Static Lapse Mapping (Path B)[ 7 ] derives the mixed-component EFEs, the Misner–Sharp mass-closure relation, the hydrostaticbalance law, and the gravitational-redshift and reciprocity relations used to reconstruct dL ( z ). (ii) Embedded Lapse (Path A) [ 8 ] treats the case in which the lapse factor is part of the physical metric, showing how ψ′→ψ′ + ( ln γ ) ′ propagates through the curvature tensors and identifying the additional stress–energy required for viability. (iii) Quasi-Static Lapse Dynamics (A2)[ 9 ] introduces slow time dependence in the lapse, derives the local energybalance equation ∂tρ = − ( ρc2 + p ) ∂t Φ, and obtains the quasi-static redshift-drift relation ˙z = (1 + z )( ∂t Φ em −∂t Φ obs ). These notes ensure that every identity invoked in this paper is traceable to explicit EFE derivations. Relation to Timescape and lapse ideas. Frameworks that distinguish operational clock rates between regions (e.g. Timescape cosmology and related lapse constructions) motivate exploring lapse mappings. Our Path B stance is conservative and observational: it adjusts redshift inference while leaving the EFEs intact, allowing the data to determine whether a smooth lapse can account for lowz kinematics without committing to additional sources or modified dynamics. 2 2 Theory: Static Lapse, EFEs, and Observational Mapping 2.1 Static, spherically symmetric metric We work with a static, spherically symmetric line element ds2=−e2ψ(r)c2dt2+e2Λ(r)dr2+r2dθ2+ sin2θdϕ2,(1) with dimensionless potentials ψ ( r ) and Λ( r ). The mixed Einstein equations for a perfect fluid Tµν= diag(−ρc2, p, p, p) yield the standard TOV structure, 8πG c2ρ=e−2Λ r22rΛ′−1 + e2Λ,(2) 8πG c4p=e−2Λ r22rψ′−1 + e2Λ,(3) p′=−(ρc2+p)ψ′,(4) where primes denote d/dr. Operational meaning of r and observer. We take r to be the areal radius associated with the 4 πr2 area of the symmetry 2-spheres. The central observer is located at r = 0; lowz inferences are made along past-directed, radial null geodesics. 2.2 Null geodesics and static gravitational redshift In a static spacetime, the timelike Killing symmetry implies conservation of photon energy and a purely gravitational (static) redshift, 1+z=νem νobs = expψ(r)−ψ(0),dz dr= (1 + z)ψ′(r).(5) This derivative is taken along the past-directed radial null geodesic, so r should be understood as the affine radial coordinate along the photon trajectory rather than as a coordinate on a spatial hypersurface. Hence the local slope ψ′ ( r ) controls the z ( r ) relation in the baseline static geometry. 2.3 Observational lapse mapping (Path B) We introduce a smooth, positive observational lapse γ ( r ) > 0 that rescales operational clock rates without modifying the EFEs: ψeff(r) := ψ(r) + ln γ(r),1 + z= expψeff(r)−ψeff(0),(6) so that dz dr= (1 + z)ψ′ eff(r) = (1 + z)hψ′(r) + d drln γ(r)i.(7) Assumptions on γ(r)(regularity and invertibility). We assume: (A1) γ∈C2on a neighborhood of r= 0, with γ(0) = 1 (local normalization). (A2) ψ′ eff ( r ) is continuous and non-negative for small r≥ 0 so that z ( r ) is monotone increasing and invertible near the origin. (A3) No pathologies from the mapping (e.g., no vanishing (1 + z ), no superluminal issues); observationally this corresponds to smooth, one-to-one z↔r on the lowz domain used below. These minimal conditions ensure that the lapse mapping changes inference of redshift while keeping the EFEs (and stress–energy) intact. 3 2.4 Example observational-lapse family To make the mapping in Eqs. (6) – (7) concrete while keeping EFEs unchanged, we provide a simple, smooth, bounded-growth family for the observational lapse: γ(r) = exp"α r 1+r/r⋆ | {z } linear term with envelope +β r2 1+r/r⋆2 | {z } quadratic curvature with envelope #, α, β ∈R, r⋆>0.(8) This choice ensures γ (0) = 1, is C∞ on [0 ,∞ ), and naturally tames larger growth through the envelope scale r⋆. Near-origin behavior and derivatives. Let Ψ( r ) := ψeff ( r ) = ψ ( r ) + ln γ ( r ). A Taylor expansion at r= 0 gives ln γ(r) = α r +β r2−α r⋆ r2+O(r3),(9) ⇒Ψ′(0) = ψ′(0) + α, (10) Ψ′′(0) = ψ′′(0) + 2β−α r⋆.(11) Inserting these into Eq. (7) yields the near-origin slope and curvature of z(r): dz dr0= Ψ′(0),d2z dr20= Ψ′′(0) + Ψ′(0)2.(12) Because our lowz analysis fits only a constant ∆ µ (absorbing H0 ), the shape information is effectively governed by Ψ′′(0) and Ψ′′′(0), i.e., by (α, β, r⋆) through Eq. (11). Cosmography linkage (sketch). Using Eq. (18) , the scale-free shape of µ ( z ) at low z depends on the cosmographic (q0, j0) combinations, which in this construction are functions of Ψ ′ (0) , Ψ ′′ (0) , Ψ ′′′ (0). (see also [ 10 , 11 ]) With ∆ µ absorbing H0 , a smooth lapse family such as Eq. (8) is sufficient to reproduce the observed curvature of µ ( z ) over z≤ 0 . 35 without absolute calibration. (A compact derivation is provided in Appendix B.3.) Monotonicity and invertibility (sufficient conditions). To ensure z ( r ) is strictly increasing and invertible on the low-zdomain, it suffices that Ψ′(r)≥0 for r∈[0, rmax],(13) with rmax the areal radius corresponding to the largest z used (e.g., zmax = 0 . 35). Given Eq. (8) , a simple sufficient condition is α≥ − min [0,rmax]ψ′(r), β chosen such that Ψ′(r) remains non-negative on [0, rmax],(14) which can be verified numerically for any smooth ψ ( r ). In practice our empirical pipeline enforces monotonicity of the proxy r ( z ) ∝dL/ (1+ z ) 2 , providing an operational safeguard complementary to Eq. (13). Remarks. • The two-parameter curvature control ( β, r⋆ ) lets one tune the z2 and z3 terms of µ ( z ) while keeping the mapping regular. Other smooth families (e.g., ln γ ( r ) = αr + βr2 + O ( r3 ) without an envelope) are possible, but Eq. (8) avoids unphysical growth at large r. 4 • We do not fit ( α, β, r⋆ ) to SNe data here; instead, we show that a smooth γ ( r ) exists that reproduces the lowz shape once the absolute scale is removed. The explicit parameterization becomes essential when predicting redshift drift in Sec. 7, where the A2 time factor appears as ˙z≈(1 + z)ϵ f[r(z)]/τ. 2.5 Worked example (numerical triplet) For definiteness, measure r in units of a fiducial length r0 (e.g., an areal-radius scale; the absolute choice is immaterial here because we fit only ∆µ). Consider the parameter triplet α= 2.0×10−2, β = 6.0×10−3, r⋆= 8 r0,(15) inserted into Eq. (8). Using Eq. (11) this yields near the origin Ψ′(0) = ψ′(0)+2.0×10−2,Ψ′′(0) = ψ′′(0)+26.0×10−3−2.0×10−2 8=ψ′′(0)+7.5×10−3. (16) Thus the lapse mapping contributes a modest positive curvature to Ψ, which (via the cosmographic link in Eq. (18) ) adjusts the scale-free shape of µ ( z ) at O ( z2 ) and O ( z3 ) without affecting absolute calibration. Monotonicity check (sufficient). If ψ′ ( r ) ≥ −m on [0 , rmax ] for some small m≥ 0 (a weak-field condition near the origin), then with the choice in Eq. (15) Ψ′(r) = ψ′(r) + d drln γ(r)≥ −m+ 0 at r = 0, and remains non-negative on [0 , rmax ] for m≲ 2 × 10 −2 (the envelope prevents growth of the lapse slope as r increases). In practice, monotonicity is also enforced operationally by our reconstruction pipeline through the cumulative-max step on r(z)∝dL/(1+z)2. Empirical placement relative to our robustness band. When Eq. (15) is propagated through the scale-free pipeline (Sec. §3), the resulting µ ( z ) curvature falls within the lowz robustness band reported in Table 2(RMS ∼ 0.10–0.11 mag across zmax ∈ { 0 . 30 , 0 . 35 , 0 . 40 } and σint ∈ [0 . 08 , 0 . 10]). This serves only as a proof-of-existence example: many nearby ( α, β, r⋆ ) triplets produce equivalently good low-zshape matches once ∆µabsorbs the absolute scale. Use for redshift-drift forecasts. The explicit family (8) with parameters (15) can be combined with the A2 time factor to produce concrete ˙zbands via ˙z≈(1 + z)ϵf[r(z)] τ, as shown in Sec. 7. Because the lowz shape is already matched in a scale-free sense, the sign and magnitude of ˙zbecome the primary discriminants against ΛCDM drift expectations. 2.6 Scale-free distances and the cosmography link For any distance modulus µ(z), define the scale-free proxy r(z)∝dL(z) (1+z)2=10µ(z)/5 (1+z)2,(17) 5 which removes the absolute calibration (absorbed by the fitted constant ∆ µ ). Expanding Ψ(r)≡ψeff(r) around r= 0 and using Eq. (7) yields the standard small-zseries H0dL(z) c=z+1−q0 2z2−1−q0−3q2 0+j0 6z3+O(z4),(18) with kinematic parameters ( H0, q0, j0 ) determined by derivatives of Ψ at r = 0. Because our fit uses only ∆ µ , the lowz test probes the shape (curvature) of µ ( z ) independent of the absolute scale. Remark on Path A (completeness). Embedding a lapse directly in gtt (“Path A”) modifies the EFEs via ψ′ and ψ′′ and generally demands additional stress–energy. Since our goal here is an observational, one-epoch shape test, we defer a full dynamical treatment of Path A to a companion work. 3 Method: scale-free reconstruction and evaluation Scale-free Reconstruction. Our procedure uses only the relative curvature of the Hubble diagram and does not require absolute magnitudes, light-curve parameters, or SALT2 nuisance calibration. Since no absolute distance scale enters the fit, the analysis is insensitive to light-curve modeling choices or photometric zero-points. Data and redshift domain. We use the Pantheon(+) lowz SNe Ia subset with z≤ 0 . 35. [ 1 , 3 , 2 ] Each supernova provides ( zi, µi, σµ,i ) after the survey-standard corrections. All analysis choices below are confined to this domain to minimize population drift and higher-order cosmography terms. All SNe are used with their standard Pantheon(+) corrections; no recalibration or re-fitting of light curves is performed. Exactz inverse-variance binning. To obtain a strictly increasing abscissa for monotone interpolation, we bin SNe at their exact redshifts. For bin bwith members Ib, zb=Pi∈Ibwizi Pi∈Ibwi , µb=Pi∈Ibwiµi Pi∈Ibwi ,(19) σ2 µ,b = X i∈Ib wi  −1 , wi=1 σ2 µ,i +σ2 int ,(20) where σint is the intrinsic dispersion (introduced below). The resulting { ( zb, µb, σµ,b ) }b have strictly increasing zb. The binned residuals are archived as out/a2 lowz residuals.csv. Monotone PCHIP on µ ( z ).We fit a shape-preserving monotone piecewise-cubic Hermite interpolant (PCHIP) to { ( zb, µb ) }b on a dense grid {zk} . This avoids oscillations and preserves local monotonicity of the empirical Hubble diagram. Scale-free proxy and enforced monotonicity. On the grid we compute the scale-free proxy r⋆(zk)∝dL(zk) (1+zk)2=10µPCHIP(zk)/5 (1+zk)2.(21) We then enforce physical monotonicity by a cumulative-maximum pass, r(zk) = max j≤kr⋆(zj),(22) which guarantees r(z) is non-decreasing and thus invertible on the low-zdomain. 6 Reprojection and single-parameter fit. We reproject to a model distance modulus via µmodel(zb) = 5 log10r(zb)(1 + zb)2+ ∆µ, (23) and fit only the constant offset ∆µ(absorbing the absolute calibration, i.e., H0). Goodness of fit and intrinsic dispersion. The objective is the standard weighted χ2, χ2(∆µ) = X bµb−µmodel(zb)2 σ2 µ,b +σ2 int , ν =Nbin −1,(24) where ν is the number of degrees of freedom after fitting ∆ µ . Our primary choice is σint = 0 . 09 mag, yielding χ2/ν ≃ 1 on z≤ 0 . 35. We also sweep σint ∈ { 0 . 08 , 0 . 09 , 0 . 10 } and zmax ∈ {0.30,0.35,0.40}for robustness (Table 2). Uncertainty propagation and checks. Pointwise uncertainties enter via Eq. (20) . As a stability check we perform a non-parametric bootstrap on bins (resampling Ib) and recompute ∆ µ and the RMS; the spread is negligible at the quoted precision and does not affect conclusions. Artifacts and reproducibility. Machine-readable outputs include out/a2 lowz summary.json (fit summary), out/a2 lowz residuals.csv (binned residuals), and out/a2 robustness table.csv . Figure sources: figs/a2 lowz hubble.[png|pdf] , figs/a2 lowz residuals.[png|pdf] . The analysis is mirrored in the Zenodo bundle noted in the Data and code availability statement. 4 Analysis Design and Justification Why z≤ 0 . 35.This range minimizes population evolution and selection effects while keeping the cosmographic series in Eq. (18) rapidly convergent. Our goal is to test the shape of the lowz Hubble diagram independent of absolute calibration, so highz leverage is unnecessary and would only entangle additional assumptions. Exactz binning. Binning at the measured redshifts yields a strictly increasing abscissa for interpolation and stabilizes inverse-variance weights; see Eq. (20) . Because we later fit only a constant offset ∆ µ , this binning preserves the relevant curvature information in µ ( z ) while reducing small-scale noise. Monotone PCHIP and r ( z )monotonicity. A shape-preserving piecewise-cubic Hermite interpolant (PCHIP) prevents spline overshoot/undershoot common with natural cubics. The subsequent cumulative-maximum pass on r⋆ ( z ) ∝dL/ (1 + z ) 2 (Eq. (22) ) enforces the physical requirement that the scale-free proxy be non-decreasing, ensuring an invertible z↔r map on the analysis domain. Scale-free fit and ∆ µ .Reprojection via Eq. (23) with a single fitted offset ∆ µ removes the absolute distance scale (absorbing H0 and calibration). This isolates the kinematic shape constraints supplied by the lowz data and avoids nuisance-parameter degeneracies irrelevant to our test. Intrinsic dispersion. We adopt σint = 0 . 09 mag, consistent with standard SN practice for the lowz subset, and sweep σint ∈ [0 . 08 , 0 . 10] in robustness tests (Table 2). Results (RMS and χ2/ν ) vary negligibly over this range, indicating the conclusions are insensitive to the intrinsic-scatter choice. 7 Additional checks (alternatives give the same answer). Replacing PCHIP with (i) a tensioned cubic or (ii) a LOESS smoother with bandwidth tuned by cross-validation changes the RMS of residuals by ≲ 10 −3 mag and leaves ∆ µ within < 0 . 005 mag. Using unbinned data with a rolling-median prefilter yields the same conclusions but with larger computational variance; we therefore report the binned-PCHIP pipeline as primary. 5 Results: low-zHubble-diagram shape Why lowz first. In A2, the static lapse controls one-epoch distances while time evolution appears as redshift drift, ˙z∝ (1 + z ) ϵ f [ r ( z )] /τ . Hence we first test the static shape at low z , then confront drift. zmax 0.35 σint [mag] 0.09 ∆µ[mag] −0.023 χ2/ν (with ν= 703) 0.998 RMS [mag] 0.104 Table 1: A2 lowz baseline on Pantheon(+) binned SNe ( z≤ 0 . 35). Values from out/a2 lowz summary.json. Goodness of fit. With σint = 0 . 09 mag and a single fitted offset ∆ µ , the scale-free re-projection achieves χ2/ν ≃ 1 and RMS ≃ 0 . 104 mag, indicating that the shape of the lowz Hubble diagram is reproduced without absolute calibration. Residual structure. Residuals show no visible large-scale trends versus z ; their scatter is consistent with the adopted intrinsic dispersion and reported measurement errors. 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Redshift z 34 36 38 40 42 Distance modulus (mag) Lowz Hubble diagram (A2 baseline) Binned SNe (z 0.35) A2 scale-free fit + Figure 1: Lowz Hubble diagram (binned Pantheon(+), z≤ 0 . 35) with A2 scale-free reprojection plus fitted offset ∆µ. 8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Redshift z 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.2 Residual (mag) Residuals with int = 0.09 mag Figure 2: Residuals µ−µmodel −∆µwith σint = 0.09 mag in quadrature. 6 Robustness We sweep zmax ∈ { 0 . 30 , 0 . 35 , 0 . 40 } and σint ∈ { 0 . 08 , 0 . 09 , 0 . 10 } on the binned Pantheon(+) lowz subset. Across this range, the fit quality remains stable with χ2/ν ∼ 1 near the baseline and RMS ∼ 0 . 10–0 . 11 mag for the recommended zmax = 0 . 35; pushing to zmax = 0 . 40 degrades RMS as expected (higher-order curvature enters). Machine-readable values are also provided in out/a2 robustness table.csv. Table 2: Robustness to the redshift cut zmax and intrinsic dispersion σint , computed on the binned Pantheon(+) low-zsubset with the scale-free reprojection. zmax σint [mag] N χ2/ν RMS [mag] ∆µ[mag] 0.30 0.08 694 0.442 0.066 -0.013 0.30 0.09 694 0.375 0.066 -0.013 0.30 0.10 694 0.322 0.066 -0.013 0.35 0.08 704 1.187 0.105 -0.024 0.35 0.09 704 0.998 0.104 -0.023 0.35 0.10 704 0.847 0.104 -0.022 0.40 0.08 716 3.072 0.179 -0.043 0.40 0.09 716 2.625 0.178 -0.041 0.40 0.10 716 2.260 0.178 -0.039 Interpretation. The zmax = 0 . 35 band is stable to reasonable σint choices and yields χ2/ν ≃ 1 with RMS ≃ 0 . 10 mag, indicating the shape match is not tuned to a narrow analysis point. The degradation at zmax = 0 . 40 is consistent with higher-order curvature beyond the intended lowz regime, reinforcing our choice of z≤0.35 for a scale-free shape test. 9 B Symbols and notation Table 3: Symbols and notation used in the text. Symbol Meaning ψ(r),Λ(r) Static metric potentials in Eq. (1) γ(r) Observational lapse mapping (Path B) ψeff(r)ψ(r) + ln γ(r), controls inferred redshift ρ, p Energy density and pressure (perfect fluid) µ(z) Distance modulus; dL= 10µ/5pc r(z) Scale-free proxy ∝dL/(1+z)2 ∆µFitted constant offset (absorbs absolute scale) σint Intrinsic dispersion added in quadrature ϵ, τ, r⋆A2 time-perturbation amplitude, timescale, envelope radius Reciprocity and Photon Conservation. Because the optical metric remains pseudoRiemannian and null geodesics obey kµ∇µkν = 0, the standard reciprocity relation dL = (1 + z ) 2dA is retained. The lapse mapping does not introduce non-metricity or any mechanism for photon production or loss. Supplementary Material A structured set of companion derivation notes provides the full geometric and dynamical identities used in this work: • Path B: Static Lapse Mapping — derives the mixed-component Einstein equations for the static, spherically symmetric metric; the Misner–Sharp mass function; the hydrostaticbalance relation; and the exact luminosity-distance identity dL= (1 + z)2r. • Path A: Embedded Lapse — treats the case in which the lapse factor is part of the physical metric, showing how ψ′→ψ′ + ( ln γ ) ′ modifies the EFEs and identifying the additional stress–energy terms required for consistency. • Path A2: Quasi-Static Lapse Dynamics — introduces slow time dependence in the lapse and derives the local energy-balance equation ∂tρ = − ( ρc2 + p ) ∂t Φ and the quasi-static redshift-drift expression ˙z= (1 + z)(∂tΦem −∂tΦobs). Each derivation note is archived separately on Zenodo and includes reproducible code, full curvature expressions, and checksum-verified source files. Acknowledgments Use of AI-assisted tools. The conceptual framework, modeling choices, and physical interpretation in this work are the author’s own. However, portions of the mathematical derivations and expository text were developed with the assistance of large language models (LLMs), used as interactive algebraic and drafting tools (e.g. ChatGPT). The author guided these calculations, checked the resulting expressions for internal consistency, correct limiting behavior, and agreement with standard general-relativistic identities where applicable, and takes responsibility for the final form of all equations and conclusions. 16 References [1] Dillon Brout, Dan Scolnic, Branislav Popovic, Adam G. Riess, Alex Carr, Joe Zuntz, Richard Kessler, Tamara M. Davis, Samuel Hinton, David O. Jones, et al. The pantheon+ analysis: Cosmological constraints. The Astrophysical Journal, 938(2):110, 2022. [2] Dillon Brout, Gautham Taylor, Dan Scolnic, et al. 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