Model-independent reconstruction of the low-redshift optical metric and its redshift-drift signature

Model-independent reconstruction of the low-redshift optical metric and its redshift-drift signature Eric L. Levin [email protected] ORCID: 0009-0005-8314-0387 December 21, 2025 Abstract We examine the extent to which the low-redshift Hubble diagram constrains the optical geometry of the observer’s past null cone in a model-independent manner. Within general relativity, the luminosity distances of Type Ia supernovae fix the mapping between redshift and the scale-free null-cone radius r ( z ) = dL ( z ) / (1 + z ) 2 . Using the Pantheon+ compilation restricted to z≤ 0 . 35, we reconstruct r ( z ) with a monotonic, shape-preserving cubic Hermite interpolant that avoids assumptions about homogeneity, expansion history, or dark-energy parameterization. The reconstructed null-cone radius determines the corresponding lapse–potential gradient governing gravitational redshift in a general static, spherically symmetric optical metric. This optical geometry reproduces the observed Hubble-diagram shape with a single magnitude offset and yields χ2/ν = 0.998 for an adopted intrinsic dispersion of 0.09 mag. To assess observational distinguishability, we introduce a minimal quasi-static perturbation to the lapse and derive the associated redshift-drift signal. The resulting prediction constitutes a model-independent kinematic null test for the local optical geometry, with a drift pattern distinguishable from standard expansion and directly testable by forthcoming real-time cosmology facilities such as the ELT and SKA. Keywords: optical metric; null-cone reconstruction; lapse function; static spacetime; quasi-static evolution; redshift drift; Hubble diagram; luminosity distance. 1 Introduction The late–time expansion history of the Universe is primarily constrained by the luminosity– distance relation of Type Ia supernovae [ 1 , 2 ], which provide the most precise direct probe of the low-redshift Hubble diagram. Following the first detections of cosmic acceleration by the Highz Supernova Search Team [ 3 ] and the Supernova Cosmology Project [ 4 ], increasingly large and homogeneous compilations—notably the Pantheon [ 5 ] and Pantheon+ [ 6 ] samples—have refined the low-redshift Hubble diagram and tightened constraints on the ΛCDM cosmological model. These datasets also contribute to several persistent tensions in contemporary cosmology, particularly the discrepancy between earlyand late-Universe determinations of the Hubble constant [7,8,9]. In recent years, a wide range of model-independent or non-parametric methods have been developed to reconstruct the distance–redshift relation without specifying a cosmological model. These include cosmography [ 10 , 11 ], Gaussian-process regression [ 12 , 13 ], iterative smoothing [ 14 ], and more recently neural-network and machine-learning approaches applied to the Pantheon 1 and Pantheon+ samples [ 15 , 16 ]. Such reconstructions provide complementary information to parametric ΛCDM analyses and are widely used to assess internal consistency, explore the Hubble tension, and test for possible deviations in the low-redshift expansion history. The present work takes this observational, model-independent philosophy one step further by focusing not on the expansion history itself but on the optical metric on the observer’s past light cone. Within general relativity, the observed luminosity distance directly constrains the mapping between redshift and the scale-free null-cone radius r ( z ) ≡dL ( z ) / (1 + z ) 2 , which determines the redshift accumulated along radial null geodesics in any static, spherically symmetric optical metric. Our approach is therefore not to posit a global cosmology but to reconstruct the local optical geometry directly from the data, identifying the class of lapse profiles that reproduce the observed Hubble-diagram shape. Using the Pantheon+ sample restricted to z≤ 0 . 35, we construct a monotonic, shapepreserving cubic Hermite interpolant for the scale-free radius r ( z ), from which we infer the corresponding lapse-potential gradient governing gravitational redshift in a static optical metric. This reconstruction requires only standard differential-geometric identities on the null cone and does not assume FLRW symmetry, homogeneity, or any particular matter content. The resulting optical geometry fits the observed Hubble diagram with a single magnitude offset and yields a statistically indistinguishable match to Pantheon+ at low z. A reconstruction based purely on static optical geometry, however, is degenerate with respect to possible slow time evolution. To assess the observational distinctions between these scenarios and standard ΛCDM expansion, we introduce a minimal quasi-static extension in which the lapse acquires a slow temporal dependence. This allows us to derive a clear prediction for the corresponding redshift-drift signal—an observable target for forthcoming real-time cosmology experiments such as ELT and SKA. The observable consequence of such slow temporal evolution is the redshift drift, a real-time change in the measured redshift of distant sources. First proposed in a general-relativistic context by Sandage in 1962 [ 17 ], and later developed in detail for modern facilities (e.g. [ 18 ]), redshift drift provides a direct probe of the time dependence of the redshift–distance relation without relying on standard-candle assumptions. In the present framework, the quasi-static extension of the lapse leads to a specific prediction for this drift that can be confronted with forthcoming ELT and SKA measurements. In this way, the low-redshift Hubble diagram becomes a null test of the local optical geometry. The reconstructed static lapse profile encodes all the information contained in dL ( z ) at low redshift, while the quasi-static extension produces a drift signature that is directly falsifiable and distinguishable from the ΛCDM prediction. This paper develops the reconstruction in detail and presents the associated redshift-drift forecasts. 2 Optical metric and null–cone kinematics Our analysis is based entirely on the kinematics of null geodesics in a general, static, spherically symmetric optical metric. This choice does not impose global staticity on the spacetime; rather, it provides a convenient framework for describing the mapping between redshift and radial coordinate along the observer’s past light cone. Within general relativity, any such metric can be written in curvature coordinates as ds2=−e2Φ(r)c2dt2+e2Λ(r)dr2+r2dΩ2,(1) where Φ( r ) and Λ( r ) are arbitrary smooth functions, r is the areal radius, and dΩ 2 is the metric on the unit sphere. Equation (1) is used here solely as a representation of the light-cone geometry near the observer. We make no assumptions about the matter content or large-scale structure of the 2 Universe, nor do we impose FLRW symmetry. All observable quantities considered below follow directly from the kinematics of radial null geodesics in (1). 2.1 Null geodesics and gravitational redshift Along a radial null geodesic, ds2= 0 implies cdt=±eΛ(r)−Φ(r)dr, (2) where the sign corresponds to ingoing/outgoing rays. For two light pulses emitted at areal radius r and received at the origin, general relativity yields the familiar gravitational-redshift relation 1+z(r) = expΦ(r)−Φ(0).(3) Thus the observed redshift encodes the difference in the lapse potential between the emitter and the observer. 2.2 Area distance and luminosity distance Because r is the areal radius of (1) , the angular-diameter distance to an object located at radius ris simply dA(z)=r(z),(4) provided that the past light cone has no conjugate points within the observed redshift range, a condition that is trivially satisfied for the lowz supernovae used in this work. Etherington’s distance-duality relation [19,20] then gives the luminosity distance, dL(z) = (1 + z)2dA(z) = (1 + z)2r(z).(5) Equation (5) plays a central role in the reconstruction that follows. At low redshift, the function r ( z ) encapsulates all information contained in the Hubble diagram, independent of any assumptions about expansion, homogeneity, or dark energy. 2.3 Scale-free reconstruction target Since the absolute calibration of SNe Ia is degenerate with the Hubble constant and the local value Φ(0), it is natural to work with the scale-free quantity dL(z)∝10µ(z)/5,(6) where µ ( z ) is the distance modulus. Combining this with Eq. (5) yields the scale-free null-cone radius r(z)∝10µ(z)/5 (1+z)2,(7) which is the sole reconstruction target of this work. Once r ( z ) is known, Eq. (3) directly determines the lapse-potential gradient, Φ′(r) = d drln(1 + z(r)),(8) up to an irrelevant additive constant. In the next section, we describe how r ( z ) is reconstructed from the Pantheon+ supernova sample using a monotonic, shape-preserving cubic Hermite interpolant, ensuring that the inferred null-cone geometry is free of numerical artifacts, pathologies, or nonphysical oscillations. 3 3 Reconstruction of the null–cone radius In this section we describe how the scale-free null-cone radius r ( z ) is reconstructed from the Pantheon+ supernova sample. The goal is to obtain a smooth, monotonic mapping r ( z ) = dL ( z ) / (1 + z ) 2 over the range 0 < z ≤ 0 . 35 that reflects only the information contained in the observed distance moduli, without imposing a cosmological model or parametric form for the expansion history. A shape-preserving interpolation scheme is essential to avoid spurious features that could otherwise propagate into later steps of the analysis. 3.1 Pantheon+ data and scale-free luminosity distance For each supernova with observed distance modulus µi and redshift zi , the luminosity distance is given by dL(zi)∝10µi/5.(9) Because the absolute magnitude of SNe Ia is degenerate with the Hubble constant, the overall scale of dL is irrelevant for our purposes. Consequently, we work entirely with the scale-free null-cone radius r(zi)∝10µi/5 (1+zi)2,(10) which contains all differential information contributing to the low-redshift Hubble diagram. We adopt the Pantheon+ subset with z≤ 0 . 35 to ensure that the redshift range corresponds to the domain in which r ( z ) can be reconstructed robustly using the static optical metric framework of Sec. 2. Uncertainties are propagated using the Pantheon+ statistical covariance matrix, with the intrinsic dispersion fixed to σint = 0.09 mag. 3.2 Monotonic Hermite interpolation The data points (10) provide noisy samples of a strictly monotonic function r ( z ). To reconstruct a smooth, differentiable r ( z ) without introducing artificial oscillations, we employ a monotonic, shape-preserving cubic Hermite interpolant (PCHIP). This scheme ensures: 1. r(z) remains strictly increasing over the full domain; 2. the interpolant exhibits no overshoot between data points; 3. first derivatives remain well behaved for subsequent calculations. The resulting interpolant defines a continuous function r ( z ) and a corresponding derivative r′(z) everywhere on 0 < z ≤0.35. 3.3 Fitting procedure and magnitude offset Because the overall scale of r ( z ) is unconstrained, the reconstruction is fit to the data using a single additive magnitude offset, µmodel(z) = 5 log10(1+z)2r(z)+ ∆µ, (11) where ∆ µ absorbs the unknown absolute magnitude M and the local value of the lapse potential. The best-fit value of ∆µis obtained by minimizing χ2= [µmodel(zi)−µi]C−1 ij [µmodel(zj)−µj],(12) where Cij is the Pantheon+ covariance matrix restricted to z≤0.35. The fit yields χ2/ν = 0.998,RMS = 0.104 mag, 4 indicating that a smooth monotonic r ( z ) fully captures the observed low-redshift Hubble-diagram shape. 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: Reconstructed Hubble diagram for the Pantheon+ sample at z≤ 0 . 35 with the best-fit offset ∆µ. 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 relative to the monotonic PCHIP reconstruction of r ( z ), showing an RMS scatter of 0.104 mag. 5 3.4 Summary Figures 1and 2summarize the reconstruction: the fitted Hubble diagram and the residuals relative to the interpolated model. The next section uses this reconstructed r ( z ) as input for deriving the corresponding null-cone lapse structure and its uncertainties. 4 Reconstructing r(z)and the null–cone lapse The optical metric formalism of Sec. 2relates the observed redshift to the lapse potential Φ( r ) along the past light cone via 1+z(r) = expΦ(r)−Φ(0).(13) Once the null–cone radius r ( z ) has been reconstructed from the data, this relation determines the radial gradient Φ ′ ( r ) up to an additive constant. In this section we describe how the reconstructed r ( z ) is differentiated, inverted, and propagated to obtain the corresponding lapse structure. 4.1 PCHIP reconstruction of r(z) The monotonic Hermite interpolant constructed in Sec. 3yields a smooth function r ( z ) and an associated derivative r′ ( z ) on the full interval 0 < z ≤ 0 . 35. Because PCHIP preserves monotonicity, the mapping z7→ r ( z ) is strictly invertible. This allows us to obtain z ( r ) numerically by applying a standard one-dimensional root-finding routine (e.g., Brent’s method) to the equation r(z)−r∗= 0 for any desired evaluation point r∗. The pair {r ( z ) , z ( r ) } provides all the geometric ingredients required for recovering the lapse potential. 4.2 Recovering the lapse–potential gradient Differentiating Eq. (13) with respect to rgives Φ′(r) = d dr ln 1+z(r)=1 1+z(r) dz dr .(14) To evaluate dz/dr numerically, we use the fact that the reconstructed r ( z ) is strictly monotonic on the domain 0 < z ≤ 0 . 35 and therefore invertible. The PCHIP spline for r ( z ) is first differentiated to obtain dr/dz =r′(z), from which the inverse derivative follows as dz dr =dr dz −1 =1 r′(z).(15) Evaluating Eqs. (14) and (15) on a fine grid yields Φ ′ ( r ) for all radii corresponding to the reconstructed redshift range. Only the gradient of Φ is required for observable quantities; the additive constant Φ(0) plays no role. The function Φ ′ ( r ) therefore encodes the full optical effect of the reconstructed geometry on null geodesics at low redshift. 4.3 Uncertainty estimation Uncertainties in Φ′(r) arise from three sources: 1. measurement errors in the Pantheon+ distance moduli; 2. the covariance structure of the PCHIP interpolant; and 6 3. the numerical stability of the derivative r′(z) and its inversion. To propagate these uncertainties, we generate an ensemble of synthetic reconstructions by drawing distance-modulus samples µ(k) i∼ N(µi, Cij), where Cij is the Pantheon+ covariance matrix restricted to z≤ 0 . 35. For each realization k , we compute: 1. the scale-free radius r(k)(z), 2. its derivative r(k)′(z), 3. the numerically inverted mapping z(k)(r), and 4. the corresponding lapse gradient Φ(k)′(r). The resulting ensemble defines the posterior distribution for Φ ′ ( r ), from which pointwise confidence intervals are extracted. This procedure ensures that our lapse reconstruction fully reflects the observational uncertainties of the low–zHubble diagram. 4.4 Summary The PCHIP reconstruction of r ( z ) provides a stable, monotonic mapping suitable for recovering the lapse–potential gradient on the past light cone. The resulting function Φ ′ ( r ) is determined entirely by the observed luminosity distances of SNe Ia, independent of any assumptions about homogeneity, expansion, or dark-energy dynamics. This reconstructed lapse profile forms the basis for the phenomenological analysis that follows. 5 Redshift drift in a quasi–static optical geometry The reconstruction of Secs. 3–4provides a static mapping between observed redshift and the null–cone lapse structure. Redshift itself, however, is a time-dependent observable: if the lapse or the optical geometry evolves slowly with the observer’s proper time, the redshift of a distant source will drift at a measurable rate [ 17 ]. Highresolution spectroscopic facilities such as the ELT and SKA are expected to reach the sensitivity required to detect this drift directly [18]. In this section we introduce a minimal, phenomenological time dependence that is consistent with the static reconstruction and derive the corresponding redshift-drift signature. The goal is not to propose a dynamical model of the Universe but to determine the kinematic consequences of allowing the lapse to vary slowly in time. This yields a purely observational null test applicable to any optical geometry consistent with the low-zHubble diagram. 5.1 Minimal quasi–static extension We consider a first-order temporal deformation of the reconstructed lapse, Φ(r, t) = Φ0(r)+ϵ f(r)h(t),(16) where: •Φ0(r) is the static lapse inferred from the Hubble diagram; •f(r) is an arbitrary spatial profile normalized by f(0) = 0; •h(t) is a dimensionless, slowly varying function of the observer’s proper time; 7 •ϵis a small bookkeeping parameter that controls the deformation amplitude. This is the most general linearized deformation that preserves the radial structure of the static reconstruction while allowing for slow temporal evolution. No assumptions are made about the underlying dynamics of h(t). 5.2 General expression for the redshift drift The observed redshift satisfies 1+z(to) = expΦ(re, te)−Φ(0, to).(17) Differentiation with respect to the observer’s proper time yields ˙z= (1 + z)h˙ Φ(re, te)−˙ Φ(0, to)i.(18) For the quasi-static lapse (16), ˙z(z) = (1 + z)ϵ˙ h(t)f(r(z)) −f(0).(19) which depends only on f(r) and the reconstructed radius r(z). 5.3 Observable velocity drift The quantity measured in spectroscopic surveys is ˙v(z) = c˙z(z) 1+z,(20) so that ˙v(z) = c ϵ ˙ h(t)fr(z).(21) The shape of the drift curve is therefore fixed by the reconstructed null–cone radius and the spatial profile f(r). A particularly transparent diagnostic is the uniform-lapse limit, f ( r ) = 1, for which ˙v ( z ) becomes constant across the low-redshift domain. This limiting case provides a simple baseline null test: any detectable z -dependence immediately signals nontrivial spatial structure in the quasi-static deformation. 5.4 Comparison with the ΛCDM drift Standard FLRW cosmology predicts ˙zΛCDM(z) = H0(1+z)−H(z),(22) which produces a characteristic curvature in ˙v ( z ), even at z≲ 0 . 3, owing to the evolving expansion rate. In contrast, the uniform-lapse limit of Eq. (21) yields a drift that is constant in redshift. Even after rescaling parameters, this flat signature cannot mimic the curvature of the ΛCDM prediction. Figure 3compares the fiducial quasi-static drift with the ΛCDM expectation, illustrating the fundamental difference in shape. 8 Figure 3: Comparison between the quasi-static optical-geometry drift (blue) and the ΛCDM prediction (gray dashed). The blue curve corresponds to the uniform-lapse specialization f ( r ) = 1 with representative parameters ϵ = − 0 . 05 and τ = 25 Gyr . 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