The theoretical framework presented in this paper evolved through a sequence of modeling refinements, each driven by the goal of improving the physical realism and explanatory power of galactic rotation curve fits. This progression not only deepens our understanding of galactic structure but ultimately enables a novel, physically grounded estimation of the local expansion rate H z in the early universe.

We begin with a single-region model for individual galaxies, assuming a constant Lagrangian metric inflow within a symmetric baryonic bulge. In this regime, spiral inflow velocities can be expressed as a combination of tangential metric motion and radial metric components, encoded via a fixed effective potential. The effective potential incorporates Newtonian gravity and Hubble expansion through a superposition of respective metric velocities. This initial framework offers a good match for certain galaxies where the baryonic structure is dominated by a central bulge, but proves insufficient when confronted with more complex rotation curves.

To address this, the model is extended to incorporate multi-region virial transitions. By defining physically distinct domains—such as the bulge interior, a first pure metric spiral disk region, and an outer region with additional virial motion—the rotation curve can be decomposed into segments with different dynamical behaviors. This segmentation leads to more accurate fits and reveals structural transitions not visible in the single-region model.

However, some galaxies exhibit rotation curves that cannot be reproduced by a single galaxy wide Lagrangian, even with multiple regions, but can be fit using two constant Lagrangian curves. We now interpret these systems as displaying clear signatures of nested spirals: an inner, bar-like structure (interpreted as a relic spiral) and an outer disk spiral, each with distinct dynamical histories and possibly formed at different cosmic epochs [1]. To capture this, the model is generalized to include two distinct Lagrangians—one for the inner bulge-bar spiral, one for the outer disk spiral—each characterized by its own radius, mass, and expansion parameter H z [2]. The outer disk spiral appears like a reset of the original bulge by including the mass of the bar and starting at the outer realm of the bulge-bar system, its visible or invisible ring with its specific critical radius.

Finally, by treating the inner bulge-bar as a fossil remnant of an earlier structural phase, i.e., a proto-galaxy, and assuming it constituted a small fraction (e.g., 0.2% - 5%) of today’s bulge mass, we derive an independent estimate of the early-universe expansion rate H z via the local condition v H = v esc at the bar-ring-disk transition radius. This condition depends solely on the measured M and r c of an individual galaxy and is therefore independent of cosmological assumptions. Only after mapping H z to cosmic time using the global ΛCDM H ( t ) relation—based on chosen cosmological parameters—do the local measurements converge with the large-scale matter-era scaling ρ ∝ t − 2 .

This hierarchical modeling—from single-region fits [3] to nested spirals, to the locally anchored v H = v esc condition, and finally to its cosmological interpretation—forms the backbone of the present analysis and underpins the results discussed in the sections that follow.

In our original constant Lagrandian fit of the SPARC database galaxies, thirteen stood out because we needed two constant Lagrangians to fit the rotation curve. Each Lagrangian fit had its own radius and mass. It was as if the accumulation of mass forced a reset of the bulge. For the time, we didn’t have an interpretation for this occurrence. Now we interpret it as nested spirals, ie bars, inside a larger spiral, ie the disk. So the previous three to four region fit, with bulge to disk to disk-with-virial-regions has to be enlarged with a bulge-bar to bar-as-new-bulge to disk to disk-with-virial-regions. The bulge-bar Lagrangian has its M, R set and so does the bar-as-new-bulge to disk to disk-with-virial-regions.

Inside the bulge ( r ≤ R ):

Outside the bulge, inside the bar until the onset of the disk ( R b u l g e < r < R b a r ):

Outside the bar, so on the disk, until the first virial region ( R b a r < r < r v i r i a l 1 ):

Outside the bar for the first virial region ( r v i r i a l 1 < r < r v i r i a l 2 ):

Outside the bulge for the second virial region ( r v i r i a l 2 < r < r c ):

Figure 9. Original, pre-Hz, two Lagrangian fit for galaxy NGC 3741 [2][4]. The RMS of the rel. residuals was 13%.

Figure 10. Observed squared rotation velocities v obs 2 (blue circles) with error bars from SPARC data, compared to the model velocity squared v model 2 (red squares) for NGC 3741 as a function of galactocentric radius. R bulge = 0.537   kpc , R bar = 2.31   kpc and R virial = 5.0   kpc , deviding the fit in four regions: bulge, bar, disk and virial disk.

In Figure 9, the original, pre-Hz fit is given. In Figure 10 one can see the result with two Lagrangians and one virial region with the effective potentials that include Hz expansion correction to the original Lagrangian. The values of the respective radii and masses are: M bulge = 7.36 × 10 37   kg , R bulge = 1.543 × 10 19   m , M bar = 1.09 × 10 39   kg , R bar = 7.129 × 10 19   m , and H z = 2.2 × 10 − 18   s − 1 . The relative residuals are given in Figure 11 and the RMS of the relative residual was 6.8%. Around r = 2   kpc , two data-points couldn’t be fitted appropriately, with the interpretation that they marked the boundary between bar and disk and represent the transition from the one to the other.

Figure 11. Relative residuals ( v model 2 − v obs 2 ) / v obs 2 for NGC 3741. The model closely matches the observed SPARC rotation curve, with a root mean square deviation of approximately 6.8%.

In a previous paper, we introduced the postulate that bars are nested spirals, being the proto-galaxy before its Lagrangian reset to a new ‘bulge’ to start evolving the spiral of the disk. We put forward the simple idea that the starting point of the spiral of the disk should be the critical radius of the bulge and the largest possible extend of the bar. This nested spiral, ending at its critical radius, thus ends as a ring, see Figure 12. This ring might or might not be a visible aspect of such a galaxies morphology. This should allow to find the H z of the moment when the second spiral started to evolve, through the simple formula:

But to really find the correct epoch, we should realize that the mass of the bulge continues to grow, due to the spiral inflow of space functioning as a conveyor belt for inertial mass like hydrogen gas clouds. So, the present day mass of the bulge needs to be corrected to get the right estimate of the formation epoch. We worked with the assumption that the formation epoch was at least 12Gyr in the past and that the bulge in that time-span acquired 90% of its mass. So the original mass was estimated at 10% of the modelled mass. This constraint on the mass of the bulge was the only additional assumption to calculate the formative epoch of the spiral of the disk.

Figure 12. Nested spiral model. The straight lines through the figure are failures of the plotting algorithm at the ring part of the spiral. Usual spiral disks do not stretch out until the outer ring.

So, to constrain the epoch of formation for the nested bar (interpreted here as a relic spiral), we adopt two physically motivated inputs:

1) The **initial mass** of the bulge of the proto-bar is taken as 10% of the present bulge mass of the galaxy. Based on rotation curve analysis, this corresponds to:

2) The **critical radius** R c is chosen as the radius in the galaxy’s rotation curve where the outer spiral disk begins, estimated from the SPARC data as:

Using the gravitational inflow-based metric framework, the local expansion rate H z associated with this configuration is given by:

Inverting this local H z using the standard ΛCDM cosmology, we find the corresponding redshift and cosmic time to be:

This time falls at the very beginning of the “First Galaxies Form” epoch ( z ~ 15   -   6 , t ~ 300   -   950 Myr). The match is particularly significant because, besides estimating the original mass as 10% of today’s mass, no cosmological parameters were used in the derivation: the calculation is entirely local and metric-based. The predictive success of the model—achieved with just one realistic parameter guess and minimal assumptions—strongly suggests that it captures a real physical feature of early galaxy evolution, rather than merely reproducing known outcomes. This alignment supports the interpretation of bars or nested spirals as relics of early metric imprinting rather than late-time dynamical products.

We then applied the same approach to the rest of the thirteen galaxies with the same clear dual Lagrangian fit from our first attempt to fit the rotation curves of the 177 galaxies of the SPARC database [2], under the section “C. ABRUPT LAGRANGIAN TRANSITION CROSSOVER DUAL FIT GALAXIES”. To this we added four galaxies that in the renewes Hz-fit were also given a dual Lagrangian. The result is given in Table 1.

Table 1.Bar mass M bulge , critical radius R c , estimated local expansion rate H z , redshift z , and cosmic time t for 17 galaxies. The value of H z is computed from the relation H z = 2 G M / R c 3 , where M is assumed to be 10% of the present-day bulge mass. Redshift and cosmic time corresponding to each H z are derived assuming a flat ΛCDM cosmology with H 0 = 70 km∙s⁻¹∙Mpc⁻¹, Ω m = 0.3 , and Ω Λ = 0.7 .

The mean local expansion rate is 〈 H z 〉 = 8.18 × 10 − 17   s − 1 , corresponding to a mean redshift of 〈 z 〉 = 15.02 and a mean cosmic time of 〈 t 〉 = 331.0   Myr . These averages place the inferred start of the formation of disk spirals from proto-bars or nested spirals squarely in the transition from the Cosmic Dawn epoch to the onset of the First Galaxies Form period [5][6]. This temporal alignment, based solely on internal morphological parameters and minimal assumptions, strongly supports the hypothesis that present-day inner bars encode the memory of early gravitational structure formation.

We didn’t use empirical values for the bulge masses. But the used bulge masses, inferred from the rotation curves, fit span the range M bulge ~ 10 36   -   10 38   kg , consistent with observationally derived bulge masses in late-type spiral and dwarf galaxies. This supports the model’s assumption that present-day bulge mass grew over time, and that the early mass—estimated here as 10% of the current bulge—is representative of a plausible proto-bar structure. These values fall well within the typical bulge-to-total (B/T) ratios observed across Hubble types, especially for systems with pseudo-bulges or bar-driven secular evolution.

The same goes for the fitted critical radii R c , interpreted as the radial extent of the inner (nested) spiral or bar relic. They ranged from approximately 2.1 to 11.4 kpc. These values are consistent with bar lengths observed in both low- and intermediate-mass spiral galaxies. Specifically, the model’s R c values match the expected bar length distributions for late-type systems, where bars typically occupy 20% - 60% of the optical disk. This alignment with empirical data further supports the interpretation that the nested structures recovered from the kinematic fits correspond to real morphological features.

A key outcome of this work is that a purely local, data-anchored dynamical condition at the bar-ring-to-disk radius, when combined with the global ΛCDM matter-era scaling, inevitably reproduces the same ρ c ∝ t − 2 law that describes the Universe’s average matter density. In the framework adopted here, the onset condition at the bar-ring transition radius r c is

which can be rewritten as

Thus, in the v H = v esc formulation, the proportionality

is not an empirical correlation but a direct algebraic consequence of the local dynamical condition. For any galaxy with measured ( M , r c ) -both extracted from rotation-curve fitting anchored in observational data—this prescription yields a unique H z satisfying the above relation. The v H = v esc condition is therefore entirely local and data-inserted: it makes no assumptions about the global cosmology.

When this locally derived H z is then mapped to a cosmic time t using the global matter-era ΛCDM scaling

the density-time relation

follows inevitably. Here, the convergence between the v H = v esc approach and the Friedmann matter-era scaling is driven by the combination of two independent ingredients: 1) the local kinematic balance at r c , constrained by galaxy data, fixes ρ c ∝ H z 2 ; and 2) the global cosmological model fixes H ( t ) ∝ t − 1 during matter domination.

Figure 13. Local critical density ρ c versus cosmic time t for galaxy four cases considered. Data from both the v H = v esc model and the constant Lagrangian morphological fits collapse onto the same t − 2 scaling (solid line), matching the prediction from the ΛCDM matter-dominated epoch.

The slope in ρ c ( t ) , see Figure 13 is therefore not tuned but determined by these independent assumptions, and must be −2 in log-log space during the matter-dominated epoch. This means the agreement between the local and global scalings is a robust physical outcome of combining galaxy-scale dynamics with large-scale cosmology, rather than a trivial circularity.

The galaxies with available morphological classifications (see Table 2) confirm the physical structures predicted by the nested Lagrangian fit approach: an early-formed bar-like structure (interpreted as a relic spiral) and a secondary spiral disk. Strong agreement is found in several cases, including NGC 3741, IC 2574, NGC 247, and UGC 5005, where both a central bar and extended spiral disk are observed. For other galaxies such as UGC A444 and F574-1, the morphological features are consistent with the model, though further high-resolution imaging could strengthen the classification. In cases where the classification remains uncertain (e.g., UGC 12732), targeted follow-up via deep imaging surveys or citizen science projects (e.g., Galaxy Zoo) may provide additional validation of the proposed two-Lagrangian framework.

Table 2. Morphological support for nested spiral interpretation based on rotation curves.

The structural sequence proposed by our model—a central bulge, a relic spiral or bar-like structure, a critical radius defining a transition ring, and an outer spiral disk—finds strong support in observed galactic morphologies, particularly among barred spiral galaxies. Many systems classified as SB or SAB types in the de Vaucouleurs classification scheme display precisely this architecture. Galaxies such as NGC 4314, NGC 1291, and NGC 1433 are well-known examples of systems exhibiting a bulge-bar-ring-disk hierarchy. In these galaxies, a prominent central bulge is followed by an elongated bar, which terminates at a bright ring structure. Beyond this ring, spiral arms emerge, forming a large-scale disk. This pattern mirrors our theoretical model, wherein the inner bar-like spiral dominates until the critical radius R c , beyond which the spiral disk begins to form [7].

From a dynamical perspective, such ring features are frequently interpreted as signatures of resonances, particularly the inner Lindblad resonance (ILR) and the corotation radius (CR). In our approach, however, the ring is reinterpreted as the unavoidable end of the Lagrangian spiral at the critical distance r c . In a nested spiral situation, its ring at r c is where the proto-galactic inflow terminates and a new Lagrangian regime begins. This reinterpretation not only fits the morphological data but also allows for a physical estimation of the local expansion rate H z based on the mass and extent of the bar structure.

Thus, the presence of a ring at the boundary between bar and disk is not merely a morphological coincidence but a dynamic signature of nested gravitational metric regimes. This reinforces the claim that the bulge-bar-ring-disk configuration reflects a real and layered gravitational-metric history, one that can be used to extract cosmologically relevant quantities such as H z , redshift z , and cosmic time t . In this light, the nested spiral structure is more than a visual motif: it encodes the evolutionary layering of galaxies and provides a bridge between internal galactic dynamics and the cosmic expansion history.

Using the v esc = v H equilibrium model to estimate the cosmic time of the bar-risk transition for a sample of 17 galaxies, we tested a range of initial bulge mass fractions relative to the present-day bulge mass, M transition = f   M bulge , today , with f spanning from 0.2% to 20%. For each scenario we computed the corresponding local expansion rate H z from the measured critical radius R c , and converted H z into redshift z and cosmic time t via the ΛCDM integral ( H 0 = 70 km∙s⁻¹∙Mpc⁻¹, Ω m = 0.3 , Ω Λ = 0.7 ) [8]. The resulting timelines were compared with observational constraints on the earliest known spiral galaxies and the global ΛCDM epochs of Cosmic Dawn, early galaxy formation, and mature disk emergence, see Figure 14 [9]-[11].

Scenarios with very large early mass fractions ( f ≈ 20 % ) produce bar-disk transition times that are too early ( t ≲ 0.3 Gyr), preceding the observed epoch of the earliest spirals (e.g., BRI 1335-0417, Zhú lóng at t ≈ 1.1   -   1.3 Gyr [9]-[11]) and leaving too time for the remaining ~80% of bulge mass to assemble over the next ~13 Gyr, creating an unrealistic slow mass accretion. Conversely, scenarios with very small initial bulge masses ( f ≲ 0.2 % ) delay the transition well beyond t ≳ 2 Gyr for many systems, providing an unrealistically short ≳ 99 % mass-assembly window.

The optimal agreement with both the ΛCDM cosmic timeline and realistic bulge growth histories is obtained for f in the range 0.5 % ≲ f ≲ 1 % . In this “sweet spot,” bar-disk transitions occur predominantly in t ≈ 0.5   -   2.5 Gyr, overlapping the early galaxy formation epoch and the half of middle galaxy formation epoch, while leaving ~99.0% - 99.5% of the present-day bulge mass to be assembled over the subsequent 12 - 13 Gyr. This range minimizes conflicts with both early-onset and late-onset extremes, and provides a physically plausible pathway for our bar-to-disk transition within the ΛCDM cosmological framework, see Figure 15.

Figure 14. Timeline of disk onset for seven bulge growth scenarios ( M transition = 0.2%, 0.6%, 1%, 5%, 10%, 15%, and 20% of today’s bulge mass). Bars show the minimum—maximum cosmic time for the galaxies in each scenario, with the black vertical tick indicating the median. Background shaded regions denote approximate ΛCDM cosmic epochs: Cosmic Dawn (100 - 500 Myr), Early formation (500 - 1500 Myr), Middle formation (1.5 - 4 Gyr), and Late formation (>4 Gyr). Cosmic times are computed by converting the local expansion rate H z from the v esc = v H equilibrium model into redshift and cosmic time using a flat ΛCDM cosmology with H 0 = 70 km∙s⁻¹∙Mpc⁻¹, Ω m = 0.3 , and Ω Λ = 0.7 .

Figure 15. Bar-disk transition timeline for the 0.6 % × M bulge scenario. The horizontal bar spans the minimum to maximum cosmic times t derived for the 17-galaxy sample, with the black vertical tick marking the median. Individual galaxies are shown as dots, plotted slightly above the bar for clarity. Cosmic times are computed from the local expansion rate H z via the v esc = v H equilibrium model and converted to redshift z and t using a flat ΛCDM cosmology ( H 0 = 70 km∙s⁻¹∙Mpc⁻¹, Ω m = 0.3 , Ω Λ = 0.7 ). Shaded background regions indicate approximate ΛCDM epochs: Cosmic Dawn (100 - 500 Myr), Early formation (500 - 1500 Myr), Middle formation (1.5 - 4 Gyr), and Late formation (>4 Gyr).

But in order to connect observed present-day galaxy morphology and bulge mass to their likely formation epochs in a more flexible way, we propose to adopt a 0.5 - 4 Gyr cosmic time band for the onset of spiral disk formation from a pre-existing bar structure. This time window encompasses the expected ΛCDM epoch for large-scale disk emergence while allowing for diversity in individual growth histories. For each galaxy, we take its present-day bulge mass and critical radius R c (from morphology and rotation curve fitting) and apply the v H = v esc criterion to infer the onset H z and corresponding cosmic time for a given early bulge fraction. Instead of fixing this fraction universally, we choose it individually for each galaxy from a physically motivated 0.2% - 5% range of today’s bulge mass. This bracket captures the plausible early mass fraction in ΛCDM, excludes unreasonably large proto-bulges at early times, and allows for realistic late growth [12][13]. The chosen fraction is the one that places the galaxy’s inferred spiral onset within the 0.5 - 4 Gyr band, ensuring consistency with both the cosmological timeline and the observed morphological scale. This approach thus links local structural parameters directly to cosmological formation epochs while retaining flexibility to account for galaxy-to-galaxy variations. In the next section, we will delve deeper in the “sweet spot” selection on individual galaxic level.

In this stage of the analysis, we reverse the usual forward-modelling procedure. Rather than starting from an assumed early bulge fraction and evolving the system forward to match present-day properties, we anchor the calculation at the current epoch ( t 0 , M today , H 0 ) and work backward in time to determine the onset of the outer disk spiral at the bar-ring interface. We adopt a power-law bulge growth law motivated by cosmological merger and secular evolution studies, and test a discrete set of candidate onset fractions f of the present-day bulge mass. For each trial f , the system is re-evolved forward from the inferred onset time to t 0 and checked against the observed M today . The f value yielding the closest mass closure, M final / M today ≈ 1 , is selected as the optimal onset fraction f opt , from which the onset time t on , opt and redshift z on , opt follow directly.

The resulting distribution of onset times for the galaxy sample is shown in Fig. 16, overlaid on the Cosmic Dawn (0 - 0.5 Gyr) and Early Formation (0.5 - 1.5 Gyr) epochs. This backward-time minimization thus links the observed bar-ring-disk morphology to a physically plausible growth history, while ensuring consistency with the v H = v esc condition and the ΛCDM expansion history.

To estimate the original bulge fraction at the bar-to-disk spiral onset, we solved a backward-time optimization anchored at today’s state ( t 0 , M today , H 0 ) . The onset is defined at the critical radius r c by

with f the (unknown) onset fraction of the present-day bulge mass.

a) Bulge growth law. Between t on and t 0 we model secular + merger-driven buildup with a power law,

motivated by cosmological assembly histories. In the idealized case where Equation (20) holds strictly from t on to t 0 , one obtains the schematic closed form

Operationally, in our pipeline we still evaluate t on and z on self-consistently via the ΛCDM mapping H ( z ) and the equality in Equation (19); Equation (21) simply clarifies how f and α set the onset time in the pure power-law limit.

b) Mismatch functional and selection rule. For each trial f ∈ { 0.002 , 0.006 , 0.01 , 0.02 , 0.05 } we evolve M ( t ) forward from t on to t 0 and compute the dynamical consistency metric

with H ( t ) from flat ΛCDM. Our primary selection criterion is the mass-closure ratio

choosing the f opt that minimizes | R f − 1 | . If R f = 1 lies between two discrete f values, we linearly interpolate to obtain f opt and read off the corresponding t on , opt and z on , opt from Equations (19)-(20) (Figure 16).

Figure 16. Spiral onset times t on for the sample (points), shown over the Cosmic Dawn (0 - 0.5 Gyr; amber) and Early Formation (0.5 - 2 Gyr; teal) epochs. Each point is placed at the onset time inferred from the backward-time procedure with the power-law growth model, after selecting f opt such that M final / M today ≈ 1 .

With this scheme, each galaxy yields an optimal onset fraction f opt and timeline placement t on , opt consistent with both the v H = v esc condition and a realistic bulge growth history as it shown in Table 3 for galaxy UGC 5005. In Table 4, the selected f for each galaxy is given, still leaving enough room for further improving the growth fitting towards M final / M today ≈ 1 .

Table 3. Backward-time onset solutions for UGC 5005 under different assumed onset fractions f of present-day bulge mass. Columns give the onset redshift z on from ΛCDM, onset cosmic time t on , the mismatch metric from Equation (22) in arbitrary units, and the mass-closure ratio M final / M today . The f = 0.02 onset M bulge was then selected as optimal. A more refined analysis process should be able to fit the f leading to M final / M today = 1 .

Table 4. Backward-time minimization results: onset fraction f , onset redshift z on (flat ΛCDM), onset cosmic time t on , mismatch metric ℰ (arbitrary units), and mass-closure ratio M final / M today .

c) Conclusion on Onset Epochs.The derived onset times for disk spiral formation, ranging from 0.4 Gyr to 1.8 Gyr after the Big Bang, fall entirely within the early galaxy formation epoch of ΛCDM cosmology. The lower bound coincides with the immediate post-Cosmic Dawn era, when the first massive, dynamically organized structures could plausibly emerge, while the upper bound remains well before the bulk of morphological settling in the later universe. This range is consistent with both high-resolution cosmological simulations [14][15] and recent JWST detections of massive disks at z ≈ 3   -   6 (∼0.9 - 2.1 Gyr) [16][17], supporting the physical plausibility of the fitted onset times.

In the Appendix, we present the result of the fitting method developed thus far to eight other galaxies from the SPARC database. We then used the eleven fits of this paper with other rotation curves fitting approaches as DM and MOND. The Lagrangian-metric model used in this study fits galaxy rotation curves without invoking exotic matter or modifying Newtonian dynamics. To assess its empirical validity, we compare its performance—measured by the root-mean-square (RMS) of the relative residuals—with two widely used paradigms: the dark matter (DM) halo model and modified Newtonian dynamics (MOND).

Table 5 presents the RMS residuals for our fits to 11 galaxies, alongside representative RMS values typically reported for DM and MOND fits from the literature. These comparative values are approximate, reflecting typical performance ranges rather than system-specific optimizations.

Table 5. Relative residual RMS values for our metric model fits compared with typical ranges for dark matter and MOND fits [18].

This study introduces and applies a novel approach to probing the early universe by linking internal galactic structure—specifically, the dynamics of nested bar and disk spirals—to the local expansion rate H z . Through a stepwise refinement of rotation curve modeling, we moved from single-region constant Lagrangian fits, to multi-region virial decompositions, and ultimately to a dual-Lagrangian framework capturing the physical transition between a relic inner bar and a later-formed spiral disk.

We have shown that the onset of spiral disk formation at the bar—ring transition radius can be used to estimate a local Hubble parameter H z for individual galaxies. This is achieved by equating the radial Hubble expansion velocity to the local escape velocity, v H ( r c ) = v esc ( r c ) , defining a critical radius r c that encapsulates the interplay between cosmic expansion and local gravitational binding. This v H = v esc condition is entirely local and data-driven—depending only on a galaxy’s measured mass and structural radius—yet, when mapped to cosmic time using the global ΛCDM H ( t ) relation, it reproduces the same scaling behavior as the matter-era universe.

A central result is the discovery that the local critical density ρ c ∝ M / r c 3 inferred from this model obeys a robust ρ c ∝ t − 2 scaling with cosmic time t . This scaling emerges directly from the v H = v esc framework—where ρ c ∝ H z 2 combined with the ΛCDM matter-era relation H ( t ) ∝ t − 1 leads to t − 2 —and is anchored in rotation curve fits that independently yield the same density—time trend.

In the second part of this work, we introduced a backward-time minimization scheme to refine the onset estimates. Anchored at today’s bulge mass M today and H 0 , and assuming a power-law bulge growth model M ( t ) ∝ t α with α ≃ 1.5 , we searched over discrete onset fractions f to find the f opt that minimizes | M final / M today − 1 | . This yields a self-consistent pair ( f opt , t on ) for each galaxy, with all inferred onsets lying between 0.4 and 1.8 Gyr after the Big Bang—squarely within the early formation epoch of ΛCDM cosmology.

Importantly, this method yields internal, independent estimates of H z without requiring direct redshift observations. While not cosmological measurements in the strictest sense, the convergence of results across a diverse sample suggests that early disk formation may have been more systematic—and occurred earlier—than standard models imply. The approach is scalable to large galaxy samples, provided sufficient rotation curve and morphological resolution.

This work therefore links observable bar—ring morphology directly to both local dynamical conditions and the global cosmological timeline, providing a new tool for constraining galaxy formation scenarios. The agreement between independent modelling approaches indicates that the ρ c ∝ t − 2 scaling is not an artefact of any single assumption, but an intrinsic property of the galaxy—cosmos connection. Future work should extend this method to higher-redshift galaxies and marginalize over cosmological parameter sets to test the robustness of the onset timeline against uncertainties in H ( z ) .

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Figure A1.Top: Observed squared orbital velocity v 2 (with error bars) versus radius R for galaxy UGC 12632, compared to the model prediction based on a constant Lagrangian approach. Bottom: Relative residuals ( v obs 2 − v model 2 ) / v model 2 with corresponding propagated uncertainties. The RMS of the relative residuals is 0.0158.

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Figure A2.Top: Observed squared orbital velocity v 2 (with error bars) versus radius R for galaxy F574-1, compared to the model prediction based on a constant Lagrangian approach. Bottom: Relative residuals ( v obs 2 − v model 2 ) / v model 2 with corresponding propagated uncertainties. The RMS of the relative residuals is 0.0449.

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Figure A3.Top: Observed squared orbital velocity v 2 (with error bars) versus radius R for galaxy UGC A444, compared to the model prediction based on a constant Lagrangian approach. Bottom: Relative residuals ( v obs 2 − v model 2 ) / v model 2 with corresponding propagated uncertainties. The RMS of the relative residuals is 0.0610.

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Figure A4.Top: Observed squared orbital velocity v 2 (with error bars) versus radius R for galaxy NGC 24, compared to the model prediction based on a constant Lagrangian approach. Bottom: Relative residuals ( v obs 2 − v model 2 ) / v model 2 with corresponding propagated uncertainties. The RMS of the relative residuals is 0.0676.

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Figure A5.Top: Observed squared orbital velocity v 2 (with error bars) versus radius R for galaxy NGC 55, compared to the model prediction based on a constant Lagrangian approach. Bottom: Relative residuals ( v obs 2 − v model 2 ) / v model 2 with corresponding propagated uncertainties. The RMS of the relative residuals is 0.0368.

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Figure A6.Top: Observed squared orbital velocity v 2 (with error bars) versus radius R for galaxy NGC 247, compared to the model prediction based on a constant Lagrangian approach. Bottom: Relative residuals ( v obs 2 − v model 2 ) / v model 2 with corresponding propagated uncertainties. The RMS of the relative residuals is 0.0246.

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Figure A7.Top: Observed squared orbital velocity v 2 (with error bars) versus radius R for galaxy NGC 3877, compared to the model prediction based on a constant Lagrangian approach. Bottom: Relative residuals ( v obs 2 − v model 2 ) / v model 2 with corresponding propagated uncertainties. The RMS of the relative residuals is 0.0425.

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Figure A8.Top: Observed squared orbital velocity v 2 (with error bars) versus radius R for galaxy UGC 5005, compared to the model prediction based on a constant Lagrangian approach. Bottom: Relative residuals ( v obs 2 − v model 2 ) / v model 2 with corresponding propagated uncertainties. The RMS of the relative residuals is 0.067.