Supernova Hubble Diagram Constraints on a Unified-Flow Model with Redshift-Dependent Fundamental Constants

1. Introduction

Type Ia supernovae (SNe Ia) are luminous transients widely interpreted as thermonuclear disruptions of carbon–oxygen white dwarfs in binary systems. Their spectra and light-curve morphologies form a comparatively homogeneous class, and empirical correlations between peak brightness, post-maximum decline rate, and color enable their use as standardizable candles for distance determination [1,2,3,4,5]. For each object, a standardized distance modulus $\mu$ provides an estimate of the luminosity distance $d_L$, up to an overall calibration offset that absorbs the absolute magnitude and the present-day distance scale.

A SN Ia Hubble diagram is the empirical relation between standardized distance moduli and redshift. In practice, the Hubble diagram is constructed by plotting $\mu$ (or equivalent magnitude residuals) against a spectroscopic redshift corrected to an appropriate cosmological frame, and comparing the resulting distance–redshift trend to model predictions for $d_L\left(z\right)$. Because $d_L\left(z\right)$ depends on the line-of-sight integral of the light-propagation scale $c/H$ in homogeneous kinematics, the Hubble diagram probes the shape of the distance law and its redshift dependence. Early Hubble-diagram extensions to more distant SNe Ia [6] and subsequent studies have investigated potential systematics and wavelength dependence, including host-galaxy correlations [7] and near-infrared Hubble diagrams with reduced sensitivity to dust [8,9]. High-redshift Hubble diagrams provided the first direct evidence that the observed distances depart from a simple fixed-constant, matter-only prediction, motivating models that introduce a cosmological constant or other late-time component [10,11,12].

Although these phenomenological extensions describe a broad set of observations, the microphysical origin of an effective late-time acceleration remains uncertain, and the cosmological constant problem continues to motivate empirical tests of alternative distance–redshift parameterizations [13,14]. In this context, modifications that act directly on the distance scale—rather than on an explicit energy component—provide a complementary way to interrogate the information encoded in the SN Hubble diagram alone.

Redshift dependence of effective couplings is one class of such modifications. Theoretical motivations and phenomenological frameworks include scalar–tensor gravity [15], varying-speed-of-light (VSL) proposals [16,17,18,19], and broader observational reviews emphasizing that physical interpretation must ultimately be cast in terms of dimensionless quantities [20,21]. Even when interpreted purely as an effective ansatz, redshift-dependent scalings modify the combination $c\left(z\right)/H\left(z\right)$ that determines $d_L\left(z\right)$, and therefore admit direct confrontation with Hubble-diagram distances. Despite the extensive literature on varying-constant ideas, empirical, covariance-respecting fits that apply a redshift-dependent scaling directly to SN Ia distance moduli remain limited.

A one-parameter unified-flow distance law is therefore tested against the Pantheon+SH0ES compilation. The model encodes an effective redshift dependence through a single exponent $\beta$ in $\varphi \left(z\right)={(1+z)}^{\beta }$, adopts a matter-closure scaling for $H\left(z\right)$, and yields an analytic expression for the dimensionless luminosity distance. The parameter $\beta$ is estimated using the full Pantheon+SH0ES statistical+systematic covariance matrix with an exact analytic profiling of the distance-modulus offset, and the resulting fit is compared to a fixed-constants matter-only baseline and to a supernova-only flat $\Lambda$CDM benchmark.

2. Materials and Methods

2.1. Data

The analysis uses $N=1701$ SNe Ia from the Pantheon+SH0ES data products released with Pantheon+ [22]. The subset is labeled “Pantheon+SH0ES” because it overlaps the Cepheid-calibrated supernova sample employed by SH0ES in the distance-ladder analysis [23]. The Hubble-diagram redshift supplied with the data products, $z\equiv z_{\mathrm{HD}}$, is adopted throughout, giving $z\in [0.00122,2.26137]$, together with the corresponding observed distance moduli ${\mu }_{\mathrm{obs}}$ (column MU_SH0ES in the release files). The full covariance matrix $\mathbf{C}$ supplied with the data products (statistical and systematic components) is used without approximation.

2.2. Unified-Flow Factor and Redshift Scalings

A single flow factor

$$\varphi \left(z\right)={(1+z)}^{\beta }$$

(1)

where z is redshift. The flow factor parameterizes redshift dependence of the effective constants through the canonical scalings

$$G\left(z\right)=G_0\varphi \left(z\right),c\left(z\right)={\displaystyle \frac{c_0}{\varphi \left(z\right)}},\mathrm{d}t\left(z\right)=\mathrm{d}{t}_0\varphi \left(z\right),$$

(2)

where subscript “0” denotes the $z=0$ reference values. Here $G\left(z\right)$ denotes an effective gravitational coupling, $c\left(z\right)$ an effective light-propagation scale, and $\mathrm{d}t\left(z\right)$ an effective proper-time interval; $G_0$, $c_0$, and $\mathrm{d}{t}_0$ denote the corresponding $z=0$ reference values. The exponent $\beta$ is assumed constant across the dataset (i.e., a single parameter describing the full redshift range of Pantheon+SH0ES) and is not promoted to a function $\beta \left(z\right)$ in this formulation.

In this supernova-only application, Equation (2) is treated as an effective scaling ansatz: the Hubble-diagram fit depends only on the derived combination $c\left(z\right)/H\left(z\right)$ that enters the luminosity-distance integral (Equation (5)), while a complete dynamical theory would be required to connect $\varphi \left(z\right)$ to the evolution of dimensionless couplings and to local tests.

2.3. Background Expansion and Luminosity Distance

A matter-dominated background with the Friedmann scaling $H^2\propto G{\rho }_m$ (where ${\rho }_m$ is the matter density) and ${\rho }_m\propto {(1+z)}^3$ implies

$$H\left(z\right)=H_0{(1+z)}^{(3+\beta )/2}.$$

(3)

In Equation (3), $H\left(z\right)$ is the Hubble expansion rate at redshift z and $H_0$ is its value at $z=0$. Combining Equations (2) and (3) yields a power-law form for the light-travel factor

$${\displaystyle \frac{c\left(z\right)}{H\left(z\right)}}={\displaystyle \frac{c_0}{H_0}}{(1+z)}^{-\gamma \left(\beta \right)},\gamma \left(\beta \right)={\displaystyle \frac{3}{2}}(\beta +1).$$

(4)

Assuming spatial flatness for the purpose of the distance integral, the luminosity distance can be written as

$$d_L(z;\beta )=(1+z){\int }_0^z{\displaystyle \frac{c\left(z^{\prime }\right)}{H\left(z^{\prime }\right)}}\mathrm{d}{z}^{\prime }.$$

(5)

where $z^{\prime }$ is a dummy integration variable. The overall scale $c_0/H_0$ is degenerate with the SN absolute magnitude and is absorbed into the offset parameter M (Section 2.5). A dimensionless distance proxy is defined by factoring out $c_0/H_0$:

$${\tilde{d}}_L(z;\beta )\equiv {\displaystyle \frac{H_0}{c_0}}{d}_L(z;\beta ).$$

(6)

Evaluating Equation (5) gives the closed form

$${\tilde{d}}_L(z;\beta )=(1+z)\left\{\begin{array}{cc}{\displaystyle \frac{{(1+z)}^{1-\gamma }-1}{1-\gamma },}\hfill & \gamma \ne 1,\hfill \\ {\displaystyle ln(1+z),}\hfill & \gamma =1(\beta =-1/3),\hfill \end{array}\right.$$

(7)

with $\gamma \equiv \gamma \left(\beta \right)$.

2.3.1. Continuity and Baseline Reduction Checks

Two identities are required for mathematical continuity and for verification of limiting behavior.

Baseline reduction ($\beta =0$).

At $\beta =0$, Equation (7) must reproduce the matter-only baseline

$${\tilde{d}}_L(z;0)=(1+z)2\left(1-{\displaystyle \frac{1}{\sqrt{1+z}}}\right).$$

(8)

Continuous limit at $\gamma \to 1$ ($\beta \to -1/3$).

The $\gamma \ne 1$ form in Equation (7) must approach $(1+z)ln(1+z)$ as $\gamma \to 1$, ensuring continuity at $\beta =-1/3$.

The numerical verification of these identities (Figures 7 and 8) is reported in Section 2.9.

2.4. Distance Modulus Model

The theoretical distance modulus is defined as

$${\mu }_{\mathrm{th}}(z;\beta ,M)=5{log}_{10}\left[{\tilde{d}}_L(z;\beta )\right]+M,$$

(9)

where M is a nuisance offset absorbing the absolute magnitude calibration and the overall distance normalization (Equation (6)).

2.5. Likelihood, Covariance Weighting, and Analytic Profiling of M

Assuming a multivariate normal likelihood with covariance $\mathbf{C}$, the chi-squared function is

$${\chi }^2(\beta ,M)={\left({\mu }_{\mathrm{obs}}-{\mu }_{\mathrm{th}}(\beta ,M)\right)}^T{\mathbf{C}}^{-1}\left({\mu }_{\mathrm{obs}}-{\mu }_{\mathrm{th}}(\beta ,M)\right).$$

(10)

A Cholesky factorization $\mathbf{C}=\mathbf{L}{\mathbf{L}}^T$ is used to avoid explicit matrix inversion. Defining the residual vector

$$\mathbf{r}(\beta ,M)={\mu }_{\mathrm{obs}}-{\mu }_{\mathrm{no}M}\left(\beta \right)-M\mathbf{1},$$

(11)

where ${\mu }_{\mathrm{no}M}\left(\beta \right)=5{log}_{10}\left[{\tilde{d}}_L(z_i;\beta )\right]$ and $\mathbf{1}$ is the N-vector of ones, the chi-squared becomes

$${\chi }^2(\beta ,M)={∥{\mathbf{L}}^{-1}\mathbf{r}(\beta ,M)∥}^2.$$

(12)

Because M enters linearly, it is profiled out analytically at each $\beta$. Defining ${\mathbf{y}}_1={\mathbf{L}}^{-1}\mathbf{1}$ and ${\mathbf{y}}_{\mathrm{obs}}={\mathbf{L}}^{-1}{\mu }_{\mathrm{obs}}$ and ${\mathbf{y}}_{\mathrm{no}M}\left(\beta \right)={\mathbf{L}}^{-1}{\mu }_{\mathrm{no}M}\left(\beta \right)$, the profiled offset is

$$M_{*}\left(\beta \right)={\displaystyle \frac{{\left({\mathbf{y}}_{\mathrm{obs}}-{\mathbf{y}}_{\mathrm{no}M}\left(\beta \right)\right)}^T{\mathbf{y}}_1}{{\mathbf{y}}_1^T{\mathbf{y}}_1}}.$$

(13)

Substituting $M_{*}\left(\beta \right)$ into Equation (12) yields the profile chi-squared ${\chi }^2\left(\beta \right)\equiv {\chi }^2(\beta ,M_{*}\left(\beta \right))$.

2.6. Parameter Estimation, Confidence Intervals, and Model Comparison

The best-fit exponent $\widehat{\beta }$ minimizes the profile chi-squared ${\chi }^2\left(\beta \right)$. Profile-likelihood intervals for a single parameter use

$$\Delta {\chi }^2\left(\beta \right)={\chi }^2\left(\beta \right)-{\chi }^2\left(\widehat{\beta }\right),$$

(14)

with thresholds $\Delta {\chi }^2=1$ (68% confidence) and $\Delta {\chi }^2\approx 3.84$ (95% confidence) [24].

Model comparison between the unified-flow model (free $\beta$) and the fixed-constants baseline ($\beta =0$) uses the likelihood-ratio statistic $\Delta {\chi }^2$ and the Akaike information criterion (AIC) [25,26]:

$$\mathrm{AIC}={\chi }_{min}^2+2k,$$

(15)

where k is the number of fitted parameters ($k=2$ for $(\beta ,M)$ and $k=1$ for $\left(M\right)$ at $\beta =0$).

For completeness, the Bayesian information criterion (BIC) is also reported:

$$\mathrm{BIC}={\chi }_{min}^2+klnN.$$

(16)

2.7. Residual Diagnostics and Goodness-of-Fit Tests

Whitened residuals are defined by

$${\mathbf{y}}_r={\mathbf{L}}^{-1}\mathbf{r}(\widehat{\beta },M_{*}\left(\widehat{\beta }\right)).$$

(17)

A covariance-weighted root-mean-square statistic is reported as

$$\mathrm{WRMS}=\sqrt{{\displaystyle \frac{{\chi }^2\left(\widehat{\beta }\right)}{N}}}.$$

(18)

To assess possible redshift-dependent structure in the marginal residual distribution, a two-sample Kolmogorov–Smirnov (KS) statistic is computed for the unwhitened residuals after splitting the sample at $z=0.5$; the associated p-value is reported as a descriptive diagnostic [27].

2.8. Low-Redshift Sensitivity Bound (no Low-z-Only Fit)

The exponent $\beta$ affects ${\tilde{d}}_L(z;\beta )$ only beyond the linear Hubble-law term. Expanding Equation (7) for $z\ll 1$ gives

$${\tilde{d}}_L(z;\beta )=z+\left(1-{\displaystyle \frac{\gamma \left(\beta \right)}{2}}\right)z^2+\mathcal{O}\left(z^3\right),$$

(19)

so that the corresponding distance modulus satisfies

$${\displaystyle \frac{\partial {\mu }_{\mathrm{th}}}{\partial \beta }}=-{\displaystyle \frac{15}{4ln10}}z+\mathcal{O}\left(z^2\right).$$

(20)

Because the nuisance offset M is fitted simultaneously, the effective sensitivity to $\beta$ in a low-z subsample is further reduced by partial degeneracy with the intercept.

Let the parameter vector be $\theta =(\beta ,M)$. Under the Gaussian likelihood of Equation (10), the Fisher matrix for a subset with covariance $\mathbf{C}$ is

$$F_{ij}={\left({\displaystyle \frac{\partial {\mu }_{\mathrm{th}}}{\partial {\theta }_i}}\right)}^T{\mathbf{C}}^{-1}\left({\displaystyle \frac{\partial {\mu }_{\mathrm{th}}}{\partial {\theta }_j}}\right),\theta =(\beta ,M).$$

(21)

Profiling over the linear offset M yields an effective Fisher information for $\beta$,

$${\tilde{F}}_{\beta \beta }=F_{\beta \beta }-{\displaystyle \frac{F_{\beta M}^2}{F_{MM}}},{\sigma }_{\beta }\simeq {\tilde{F}}_{\beta \beta }^{-1/2}.$$

(22)

For the 500 lowest-redshift objects in Pantheon+SH0ES ($z\le 0.03143$), evaluation with the supplied covariance submatrix gives ${\sigma }_{\beta }\approx 0.73$, consistent with weak constraining power from this redshift range once the intercept is profiled. Accordingly, no low-z-only best-fit $\beta$ is reported, and the full-sample inference is adopted as the definitive constraint within the assumed constant-$\beta$ model.

2.9. Numerical Verification of Limiting Cases

Two numerical checks validate the implementation of Equation (7). Figure 7 reports the absolute relative error between Equation (7) evaluated at $\beta =0$ and the baseline expression in Equation (8). Figure 8 reports the absolute relative error between the $\gamma \ne 1$ form near $\beta =-1/3$ and the logarithmic limit, validating continuity.

3. Results

3.1. Best-Fit Parameters and Goodness of Fit

The unified-flow model provides an excellent match to the Pantheon+SH0ES Hubble diagram (Figure 1). Minimization of the profiled chi-squared yields

$$\widehat{\beta }=-0.49752,{\sigma }_{\beta }\approx 0.0190,M_{*}\left(\widehat{\beta }\right)=43.0902.$$

(23)

The minimum chi-squared is ${\chi }_{min}^2=1751.82$ for $\nu =N-2=1699$, giving ${\chi }_{\nu }^2=1.031$ (Table 1).

3.2. Comparison to the Fixed-Constants Baseline

The fixed-constants baseline corresponds to $\beta =0$ (Equation (8)) with a single profiled offset parameter M. For this baseline, ${\chi }^2(\beta =0)=2392.02$ for $\nu =N-1=1700$, giving ${\chi }_{\nu }^2=1.407$ (Table 1). The likelihood-ratio improvement is $\Delta {\chi }^2=640.20$ for one additional parameter. The AIC values are ${\mathrm{AIC}}_{\mathrm{var}}=1755.82$ and ${\mathrm{AIC}}_{\mathrm{fixed}}=2394.02$, yielding $\Delta \mathrm{AIC}=638.20$.

3.3. Residual diagnostics

Figure 2 displays residuals $\Delta \mu ={\mu }_{\mathrm{obs}}-{\mu }_{\mathrm{th}}(\widehat{\beta },M_{*})$ versus redshift with a split marker at $z=0.5$. A two-sample KS statistic computed on the marginal residual distributions for $z<0.5$ and $z>0.5$ yields $p=0.19$ (Table 2), providing no evidence for a pronounced change in the residual distribution across this split. The covariance-weighted WRMS (Equation (18)) is close to unity (Table 2).

Figure 2. Residuals $\Delta \mu ={\mu }_{\mathrm{obs}}-{\mu }_{\mathrm{th}}(\widehat{\beta },M_{*})$ versus redshift. The vertical dashed line marks the diagnostic split at $z=0.5$ used for the KS test. The redshift axis is logarithmic.

Figure 2. Residuals $\Delta \mu ={\mu }_{\mathrm{obs}}-{\mu }_{\mathrm{th}}(\widehat{\beta },M_{*})$ versus redshift. The vertical dashed line marks the diagnostic split at $z=0.5$ used for the KS test. The redshift axis is logarithmic.

Figure 3. Residual distributions for $z<0.5$ and $z>0.5$ corresponding to the diagnostic split. The Kolmogorov–Smirnov statistic and p-value for this split are reported in Table 2.

Figure 3. Residual distributions for $z<0.5$ and $z>0.5$ corresponding to the diagnostic split. The Kolmogorov–Smirnov statistic and p-value for this split are reported in Table 2.

Figure 4. Q–Q plot of the whitened residuals ${\mathbf{y}}_r={\mathbf{L}}^{-1}\mathbf{r}$ (Equation (17)) against standard-normal quantiles. The near-linear trend and WRMS close to unity support proper covariance weighting.

Figure 4. Q–Q plot of the whitened residuals ${\mathbf{y}}_r={\mathbf{L}}^{-1}\mathbf{r}$ (Equation (17)) against standard-normal quantiles. The near-linear trend and WRMS close to unity support proper covariance weighting.

3.4. Profile Likelihood for $\beta$ and Confidence Intervals

The profile likelihood $\Delta {\chi }^2\left(\beta \right)$ is shown in Figure 5. One-parameter confidence intervals derived from the profile likelihood are summarized in Table 3.

Figure 5. Profile likelihood for the unified-flow exponent $\beta$ computed as $\Delta {\chi }^2\left(\beta \right)={\chi }^2\left(\beta \right)-{\chi }^2\left(\widehat{\beta }\right)$. The shaded regions indicate the 68% and 95% confidence intervals for one parameter ($\Delta {\chi }^2=1$ and $3.84$).

Figure 5. Profile likelihood for the unified-flow exponent $\beta$ computed as $\Delta {\chi }^2\left(\beta \right)={\chi }^2\left(\beta \right)-{\chi }^2\left(\widehat{\beta }\right)$. The shaded regions indicate the 68% and 95% confidence intervals for one parameter ($\Delta {\chi }^2=1$ and $3.84$).

3.5. Profiled Nuisance Parameter $M_{*}\left(\beta \right)$

The nuisance offset profiled by Equation (13) varies smoothly with $\beta$ (Figure 6), with $M_{*}\left(\widehat{\beta }\right)=43.0902$ at the best-fit exponent.

Figure 6. Profiled nuisance parameter $M_{*}\left(\beta \right)$ obtained from Equation (13). The dashed line marks $M_{*}\left(\widehat{\beta }\right)=43.0902$ at the best-fit $\widehat{\beta }=-0.4975$.

Figure 6. Profiled nuisance parameter $M_{*}\left(\beta \right)$ obtained from Equation (13). The dashed line marks $M_{*}\left(\widehat{\beta }\right)=43.0902$ at the best-fit $\widehat{\beta }=-0.4975$.

Figure 7. Verification of the baseline reduction identity at $\beta =0$: absolute relative error between the general closed form (Equation (7)) and the matter-only baseline (Equation (8)) across the dataset redshifts. The redshift and error axes are logarithmic.

Figure 7. Verification of the baseline reduction identity at $\beta =0$: absolute relative error between the general closed form (Equation (7)) and the matter-only baseline (Equation (8)) across the dataset redshifts. The redshift and error axes are logarithmic.

Figure 8. Verification of continuity at $\gamma \to 1$ ($\beta \to -1/3$): absolute relative error between the $\gamma \ne 1$ form of Equation (7) evaluated at $\beta =-1/3\pm {10}^{-4}$ and the logarithmic limit $(1+z)ln(1+z)$. The redshift and error axes are logarithmic.

Figure 8. Verification of continuity at $\gamma \to 1$ ($\beta \to -1/3$): absolute relative error between the $\gamma \ne 1$ form of Equation (7) evaluated at $\beta =-1/3\pm {10}^{-4}$ and the logarithmic limit $(1+z)ln(1+z)$. The redshift and error axes are logarithmic.

3.6. Implementation Verification Figures

4. Discussion

The Pantheon+SH0ES Hubble diagram admits a high-quality fit within a one-parameter unified-flow model in which a constant exponent $\beta$ modifies the distance–redshift relation through Equations (1)–(7). The model comparison metrics in Table 1 indicate strong preference for the variable-$\beta$ fit relative to the fixed-constants baseline under the AIC. Residual diagnostics show no pronounced redshift-dependent structure under a split at $z=0.5$ and give a whitened WRMS close to unity (Table 2), consistent with a statistically adequate covariance-weighted fit.

4.1. Interpretation of Information-Criterion Preference for Variable $\beta$

The fixed-constants baseline at $\beta =0$ contains only the offset M and can therefore adjust the vertical normalization of the Hubble diagram but not its curvature with redshift. Allowing $\beta$ to vary changes the exponent $\gamma \left(\beta \right)$ in the light-propagation factor $c\left(z\right)/H\left(z\right)$ (Equation (4)), which directly controls the redshift dependence of the integrated distance scale in Equation (5). The large decrease in chi-squared relative to the baseline indicates that the Pantheon+SH0ES distance moduli require a redshift-dependent departure from the matter-only distance law beyond an intercept shift. Under the Akaike information criterion, the penalty for one additional parameter is 2, which is negligible compared with the observed improvement $\Delta {\chi }^2=640.20$; consequently, $\Delta \mathrm{AIC}$ is dominated by the change in fit quality.

Empirically, the fitted value $\widehat{\beta }<0$ primarily captures the smooth, monotonic deviation of $\mu \left(z\right)$ from the $\beta =0$ prediction at intermediate and high redshift, while the low-z normalization remains absorbed by the profiled offset M. Within the adopted ansatz, preference for $\beta \ne 0$ is therefore a quantitative constraint on the redshift scaling required to reproduce the SN Ia Hubble diagram, rather than a standalone demonstration of variation in any particular dimensional constant.

4.2. Directionality and Example Scalings

Because $\widehat{\beta }<0$, the flow factor satisfies $\varphi \left(z\right)<1$ for $z>0$ and therefore implies $G\left(z\right)<G_0$, $c\left(z\right)>c_0$, and $\mathrm{d}t\left(z\right)<\mathrm{d}{t}_0$ toward the past (higher redshift), directly from Equation (2). For illustration at $z=1$ (so $1+z=2$), the best fit gives

$$G\left(1\right)=G_0{2}^{-0.498}\approx 0.71G_0,c\left(1\right)=c_0{2}^{+0.498}\approx 1.41c_0,\mathrm{d}t\left(1\right)=\mathrm{d}{t}_0{2}^{-0.498}\approx 0.71\mathrm{d}{t}_0.$$

(24)

The interpretation of changes in dimensional constants must ultimately be expressed in terms of dimensionless observables [20,21]. In this parameterization, the observational content is fully encoded by the modified luminosity-distance relation and the fitted exponent $\beta$; the scalings in Equation (2) should be read as an effective parameterization that reproduces the SN Hubble diagram within the assumed framework.

4.3. Scope Limitation: Constant $\beta$ for One Dataset

The parameter $\beta$ is treated as constant across the Pantheon+SH0ES sample, and the inference $\widehat{\beta }=-0.4975\pm 0.019$ should be interpreted strictly within that modeling choice and dataset. The results establish that a constant-$\beta$ unified-flow parameterization fits this dataset and is strongly preferred to the fixed-constants baseline under AIC. No claim is made that a single exponent must describe all cosmological datasets or that $\beta$ cannot vary with redshift; testing $\beta \left(z\right)$, alternative functional forms for $\varphi \left(z\right)$, or joint fits including other probes constitutes distinct empirical hypotheses.

4.4. Kinematic Interpretation from the Fitted Exponent

The fitted exponent $\widehat{\beta }$ fixes a simple kinematic interpretation for the background expansion implied by Equation (3). Writing $a={(1+z)}^{-1}$ gives $H\left(a\right)=H_0{a}^{-(3+\beta )/2}$, which corresponds to a constant deceleration parameter

$$q\equiv -{\displaystyle \frac{\ddot{a}}{a{H}^2}}={\displaystyle \frac{1+\beta }{2}}.$$

(25)

At the best fit, $q\left(\widehat{\beta }\right)=0.251\pm 0.010$, which is strictly decelerating ($q>0$) despite reproducing the observed SN Ia Hubble diagram. Formally, if the same power-law $H\left(a\right)$ were re-expressed within a constant-w FLRW parameterization with fixed constants, $H\left(a\right)\propto a^{-3(1+w)/2}$ implies an effective equation-of-state mapping

$$w_{\mathrm{eff}}={\displaystyle \frac{\beta }{3}},$$

(26)

so that $w_{\mathrm{eff}}\left(\widehat{\beta }\right)=-0.166\pm 0.006$ for the present fit. This mapping is an algebraic equivalence for $H\left(a\right)$ only; a complete dynamical theory is required to interpret $w_{\mathrm{eff}}$ in terms of microphysics.

4.5. Sensitivity to the Covariance Structure

The main results use the full statistical+systematic covariance matrix supplied with Pantheon+SH0ES. As a robustness check, a diagonal-only fit was computed by replacing $\mathbf{C}$ with $\mathrm{diag}\left(\mathbf{C}\right)$ while retaining the same distance model and analytic profiling of M. The diagonal-only best fit is ${\widehat{\beta }}_{\mathrm{diag}}=-0.5015$ with a 68% profile interval $[-0.5153,-0.4876]$. The agreement between ${\widehat{\beta }}_{\mathrm{diag}}$ and $\widehat{\beta }$ indicates that the preferred exponent is not primarily driven by off-diagonal correlations, while the wider full-covariance uncertainty reflects the information loss associated with correlated systematics.

4.6. Standard-Model Benchmark and Discriminating Predictions

For context, a flat $\Lambda$CDM benchmark was fitted to the same Pantheon+SH0ES distance moduli using the same full covariance matrix and the same analytic profiling of the offset M. The dimensionless luminosity distance proxy is

$${\tilde{d}}_L^{\Lambda \mathrm{CDM}}(z;{\Omega }_m)=(1+z){\int }_0^z{\displaystyle \frac{\mathrm{d}{z}^{\prime }}{E(z^{\prime };{\Omega }_m)}},E(z;{\Omega }_m)=\sqrt{{\Omega }_m{(1+z)}^3+(1-{\Omega }_m)}.$$

(27)

The integral in Equation (27) was evaluated by trapezoidal quadrature on a dense redshift grid and linearly interpolated to the observed redshifts; convergence was verified by grid refinement.

The best-fit flat $\Lambda$CDM parameter is ${\widehat{\Omega }}_m=0.3612\pm 0.0187$ (68% profile interval), with ${\chi }_{min}^2=1752.51$ for $\nu =1699$ (Table 4). Because both unified flow and flat $\Lambda$CDM employ one shape parameter plus the same offset M, their information criteria differ only by $\Delta {\chi }^2$. The unified-flow fit improves the SN-only ${\chi }^2$ by $0.69$ relative to the flat $\Lambda$CDM benchmark, indicating near-degeneracy at current SN precision.

Despite comparable fit quality, the two distance laws separate at the highest redshifts. After profiling M in both models, the predicted difference in distance modulus is $\Delta \mu \left(z\right)={\mu }_{\mathrm{flow}}\left(z\right)-{\mu }_{\Lambda \mathrm{CDM}}\left(z\right)$, which reaches $\Delta \mu \left(2.26137\right)\approx 0.259\mathrm{mag}$ at the highest-redshift Pantheon+SH0ES object and grows to $\Delta \mu \left(3\right)\approx 0.372\mathrm{mag}$ and $\Delta \mu \left(5\right)\approx 0.621\mathrm{mag}$ under direct extrapolation (Figure 9). High-redshift standard candles or standard sirens can provide a direct discriminant between the two distance laws, because the predicted separation grows with redshift even when both models fit existing SN data well.

4.7. Relation to Other Constraints and External Observations

Laboratory and Solar-System tests, including lunar laser ranging and binary-pulsar timing, constrain departures from standard gravitational dynamics in the contemporary epoch [28,29,30]. Geophysical and spectroscopic measurements provide complementary limits on the variation of dimensionless parameters [31,32,33,34,35]. Early-Universe physics also constrains variations through nucleosynthesis and related processes [36,37]. Mapping the present phenomenological $\beta$ constraint to these bounds requires a complete dynamical theory specifying how dimensionless couplings evolve and how local time derivatives relate to redshift, which is not attempted here.

The unified-flow fit provides an observationally grounded alternative explanation of SN dimming without introducing an explicit dark-energy component. Contextual motivations for exploring such alternatives include empirical tensions and open questions in early structure formation. Several JWST analyses have reported candidate galaxies with high inferred stellar masses at $z\gtrsim 10$ [38,39], while subsequent studies have emphasized the role of selection effects, dust, and interloper contamination [40]. No quantitative analysis of these external datasets is performed; the SN result is presented as an internally consistent, covariance-respecting fit that motivates broader multi-probe tests.

5. Conclusions

A one-parameter unified-flow model with flow factor $\varphi \left(z\right)={(1+z)}^{\beta }$ provides a statistically strong description of the Pantheon+SH0ES SN Ia Hubble diagram when the full covariance matrix is used. With $N=1701$ SNe over $z\in [0.00122,2.26137]$, the best fit is $\widehat{\beta }=-0.4975\pm 0.019$ under the modeling choice that $\beta$ is constant across this dataset. Residual diagnostics provide no indication of pronounced redshift-dependent structure under the adopted tests (Table 2). Within the adopted matter-closure kinematics, the fitted exponent corresponds to a constant deceleration parameter $q=(1+\beta )/2>0$ while reproducing the SN Ia Hubble diagram, indicating that the Pantheon+SH0ES distances can be fit without introducing an explicit dark-energy term within this phenomenological distance law. A supernova-only flat $\Lambda$CDM benchmark fit yields a statistically indistinguishable goodness of fit (Table 4), but the two distance laws predict increasing separation at $z\gtrsim 2$ (Figure 9), enabling direct falsification with higher-redshift standard candles or standard sirens. Extending the test to additional probes and to more general $\varphi \left(z\right)$ forms remains a direct next step for empirical evaluation.