The Hubble diagram is an important empirical relation [1][2][5], but it is not by itself a direct measurement of the expansion of space. The interpretation that cosmological redshift is caused by the expansion of space is therefore an assumption of the FLRW framework. Although supported indirectly by a broad range of observations, a direct observational verification has not yet been achieved.

Occam’s Razor

Scientific theories should be based on the smallest possible number of assumptions. One way to reduce assumptions is to determine through observation whether an assumption corresponds to reality. Another is to show that an assumption is unnecessary. For example, in [9][10], we eliminated the assumption of the cosmological principle in our theory of cosmology. Here we demonstrate how, through astronomical observations, the Hubble diagram and redshift drift, we can determine if the assumption of the cosmological redshift is actually due to the expansion of space, which would make it a fact instead of an assumption, or if on the contrary it is not due to the expansion of space, in which case this interpretation would need to be eliminated from cosmological theory.

It is clear that a useful cosmological test should reduce the dependence on interpretive assumptions wherever possible. Here, we suggest that the redshift distance relationship and redshift drift could provide a test of the expansion of space. Redshift drift is more direct because it measures whether the redshift of a distant source changes with observer time [11]-[14]. In an expanding FLRW model, comoving sources generally exhibit a nonzero cosmological redshift drift. In a stationary metric, a source at fixed coordinate distance has no cosmological drift. This contrast motivates the present comparison.

Other cosmological probes, including the cosmic microwave background, baryon acoustic oscillations, weak lensing, number counts, and structure formation, are essential to modern precision cosmology. They are outside the scope of this paper, which is limited to redshift-based discriminants.

We consider three possible methods for distinguishing an expanding FLRW description from a stationary Taub-NUT description:

1) The supernova Hubble diagram;

2) The redshift drift;

3) Astrometric radial velocities.

The first two are practical or potentially practical observational tests. Astrometric radial velocities are included mainly for conceptual completeness because, as shown quantitatively in Appendix A.1., the perspective-acceleration signal is far too small to be useful at cosmological distances.

The observational supernova distance is inferred from the distance modulus,

where μ is the published corrected distance modulus. Figure 1 compares z HD with this inferred luminosity distance. For the Taub-NUT overlay, the practical comparison identifies the radial coordinate r with the empirical distance scale d L in Gpc. This identification is a phenomenological comparison, not a full derivation of the luminosity-distance relation in the Taub-NUT spacetime. A fully self-consistent treatment of photon propagation, flux dilution, and luminosity distance in this stationary metric remains an important extension of the present work.

Figure 1. Supernova redshift z HD versus luminosity distance inferred from the distance modulus. The solid curve is the best-fit Taub-NUT relation with α = 318.72 ± 53.40 Gpc and n = 37.61 ± 3.23 Gpc. The dashed curve is a reference flat FLRW model with Ω m = 0.30 , Ω Λ = 0.70 , and best-fit H 0 = 69.30   km ⋅ s − 1 ⋅ Mpc − 1 . Individual error bars are omitted for clarity.

In this analysis, the fit uses the Pantheon + SH0ES supernova compilation with columns z HD , z HDERR , μ , and σ μ . Only objects with positive z HD , positive μ , and finite positive σ μ were retained. This gives N = 1768 supernovae in the range

The distance d L was computed from Equation (6). The Taub-NUT parameters α and n were obtained by a nonlinear least-squares fit of Equation (4) to z HD ( d L ) . For the reference FLRW comparison, a spatially flat ΛCDM model with Ω m = 0.30 and Ω Λ = 0.70 was used, with H 0 fit to the same data.

For the goodness-of-fit calculation, the effective uncertainty in redshift was computed by combining the tabulated z HD uncertainty with the distance-modulus uncertainty propagated into redshift through the local slope of the fitted curve,

This treatment uses diagonal uncertainties only and is therefore an approximate model-comparison statistic. A full covariance-matrix analysis would be a natural next refinement.

Table 1. Approximate model comparison using the same supernova sample.

Table 1 shows that the present redshift-distance data do not provide a decisive separation between the two descriptions in this approximate comparison. The Taub-NUT fit gives a slightly smaller χ 2 and RMSE, while the FLRW model gives a smaller BIC because it uses one fitted parameter rather than two. Thus, the Hubble diagram by itself is not sufficient to establish the Taub-NUT interpretation; rather, it shows that a stationary gravitational-redshift curve can mimic much of the observed redshift-distance relation over the present supernova range.

High-redshift supernovae observed with JWST may eventually extend the Type Ia supernova Hubble diagram into the approximate range 2 ≲ z ≲ 4 . This is potentially important because extrapolations of the Taub-NUT and FLRW curves can separate at high redshift. This possible separation is illustrated in Figure 2. However, the current observational situation must be stated cautiously: Type Ia supernova constraints at z > 2 remain limited, and independent luminosity-distance measurements in this range are still developing [15]-[17].

Astrometric radial velocities can in principle be obtained from perspective acceleration rather than spectroscopic redshift [28]. If a source has transverse velocity v t , radial velocity v r , and distance d , its proper motion is

If the distance varies as d ( t ) = d 0 + v r t , then

or equivalently

Combining these relations gives

The inverse-square distance dependence makes the signal impractically small at cosmological distances.

For a representative source at d = 100 Mpc with v t ≃ 300   km ⋅ s − 1 and v r ≃ H 0 d ≃ 7000   km ⋅ s − 1 , using typical peculiar-velocity scales from the literature [29]-[32], the perspective acceleration is approximately

Over a 20 yr observing baseline, the angular curvature from this acceleration is only

The corresponding observational requirements are summarized in Table A1.

Table A1. Number of repeated observations required to detect proper motion and perspective acceleration for a source at d = 100 Mpc over a 20 yr baseline.

Assumptions: d = 100   Mpc , v t = 300   km ⋅ s − 1 , v r = 7000   km ⋅ s − 1 , and a 20 yr baseline. The estimates include statistical averaging only and neglect systematic errors, source-structure effects, calibration floors, spacecraft stability, and photon-counting limitations.

As shown in Table A1, the required number of observations becomes enormous for perspective-acceleration measurements at cosmological distances. If one assumes, unrealistically, one statistically independent observation every second, the required observing time is

where R = 3.16 × 10 7   yr − 1 . The resulting times remain decades to centuries even under highly optimistic assumptions. Thus astrometric radial velocities cannot be used as a practical cosmological expansion test at 50 - 100 Mpc.

Many stationary or non-expanding cosmological models have been proposed in the literature, including general-relativistic stationary models [33]-[37], gravitational-redshift interpretations [38]-[40], and tired-light mechanisms [41]-[44]. The purpose of mentioning these models is not to place them on the same empirical footing as standard cosmology. Rather, their existence motivates the value of direct observational tests, especially redshift drift, that can distinguish expanding from stationary interpretations.