We start from the classical potential per unit mass inside a uniform-density bulge and want to adjust this for the Hubble expansion, putting a limit to the effective Newtonian reach:

$\frac{V\left(r\right)}{m}=-\frac{GM}{2R}\left(3-\frac{r^2}{R^2}\right)$

First, we convert this to a radial inflow velocity of the metric, using the escape kinetic energy equivalence:

$\frac{1}{2}{v}_{\text{rad}}^2\left(r\right)=-\frac{V\left(r\right)}{m}\Rightarrow v_{\text{rad}}\left(r\right)=\sqrt{\frac{2GM}{2R}\left(3-\frac{r^2}{R^2}\right)}=\sqrt{\frac{GM}{R}\left(3-\frac{r^2}{R^2}\right)}$

We then subtract the Hubble expansion velocity, which is a metric effect too, and define the effective intrinsic radial velocity of space as

$v_{\text{rad},\text{eff}}\left(r,Hz\right)=v_{\text{rad}}\left(r\right)-v_{Hz}=v_{\text{rad}}\left(r\right)-H_{z}r.$

We then convert this back to a modified potential energy and call the result the effective Newtonian potential, as corrected for the Hubble expansion. For this, we again use the escape kinetic energy to potential relation through the escape velocity procedure:

$\frac{1}{2}{v}_{\text{rad},\text{eff}}^2\left(r,Hz\right)=-\frac{V_{\text{eff}}\left(r,Hz\right)}{m}\Rightarrow \frac{V_{\text{eff}}\left(r,Hz\right)}{m}=-\frac{1}{2}{\left(v_{\text{rad}}\left(r\right)-H_{z}r\right)}^2$

The final result is:

$\frac{V_{\text{eff}}\left(r,Hz\right)}{m}=-\frac{1}{2}{\left(\sqrt{\frac{GM}{R}\left(3-\frac{r^2}{R^2}\right)}-H_{z}r\right)}^2=\frac{1}{2}{v}_{\text{rad},\text{eff}}^2\left(r,Hz\right)$

so we also have

$v_{\text{rad},\text{eff}}^2\left(r,Hz\right)={\left(\sqrt{\frac{GM}{R}\left(3-\frac{r^2}{R^2}\right)}-H_{z}r\right)}^2.$

From this, we can arrive at the bulge-disk boundary condition, for which we set $r=R$, as:

$\frac{V_{\text{eff}}\left(r,Hz\right)}{m}=-\frac{1}{2}{\left(\sqrt{\frac{2GM}{R}}-H_{z}R\right)}^2=\frac{1}{2}{v}_{\text{rad},\text{eff}}^2\left(R,Hz\right)$

so

$v_{\text{rad},\text{eff}}^2\left(R,Hz\right)={\left(\sqrt{\frac{2GM}{R}}-R{H}_z\right)}^2.$

And for $r=0$, we get

$v_{\text{rad},\text{eff}}^2\left(0,Hz\right)=\frac{3GM}{R}=\frac{3}{2}{v}_{\text{esc}}^2.$

We define outer effective potential per unit mass in the same way through the intermediary of the velocities of space:

$\frac{V_{\text{eff}}\left(r,Hz\right)}{m}=-\frac{1}{2}{\left(v_{\text{esc}}-v_H\right)}^2=-\frac{1}{2}{\left(\sqrt{\frac{2GM}{r}}-H_{z}r\right)}^2$

which at $r=R$ can be valuated as:

$\frac{V_{\text{eff}}\left(R,Hz\right)}{m}=-\frac{1}{2}{\left(\sqrt{\frac{2GM}{R}}-H_{z}R\right)}^2$

and gives the same result as the inner effective potential energy. We define $r_c=r$ as the distance at which $v_{\text{esc}}=v_H$, so where $V_{\text{eff}}=0$.

We use the virial theorem at $r=R$ for a circular orbit in an effective potential and use it for the metric instead of material particles and we only use it at the boundary between bulge and disk:

$K_{\text{orb}}=-\frac{1}{2}{V}_{\text{eff}}\left(R\right)\Rightarrow \frac{1}{2}m{v}_{\text{orb}}^2=-\frac{1}{2}{V}_{\text{eff}}\left(R\right)\Rightarrow v_{\text{orb}}^2=-\frac{V_{\text{eff}}\left(R\right)}{m}$

From the outside in, we arrive at $R$ from above. We use the earlier expression for the effective potential outside the bulge:

$\frac{V_{\text{eff}}\left(R\right)}{m}=-\frac{1}{2}{\left(\sqrt{\frac{2GM}{R}}-H_{z}R\right)}^2$

so

$v_{\text{orb}}^2=-\frac{V_{\text{eff}}\left(R\right)}{m}=\frac{1}{2}{\left(\sqrt{\frac{2GM}{R}}-H_{z}R\right)}^2$

We recall the Lagrangian definition to be:

$L=K_{\text{orb}}-V_{\text{eff}}=\frac{1}{2}m{v}_{\text{orb}}^2-V_{\text{eff}}\left(R\right)$

We now substitute $v_{\text{orb}}^2=-\frac{V_{\text{eff}}\left(R\right)}{m}$:

$L=\frac{1}{2}m\left(-\frac{V_{\text{eff}}\left(R\right)}{m}\right)-V_{\text{eff}}\left(R\right)=-\frac{1}{2}{V}_{\text{eff}}\left(R\right)-V_{\text{eff}}\left(R\right)=-\frac{3}{2}{V}_{\text{eff}}\left(R\right)$

In short, we have

$L\left(R,Hz\right)=-\frac{3}{2}{V}_{\text{eff}}\left(R\right)$

We use the earlier expression for the effective potential:

$\frac{V_{\text{eff}}\left(R\right)}{m}=-\frac{1}{2}{\left(\sqrt{\frac{2GM}{R}}-H_{z}R\right)}^2$

in terms of the velocity of space, the square of the Lagrangian velocity becomes:

$\frac{2L}{m}=v_L^2=-3\cdot \left(\frac{V_{\text{eff}}\left(R\right)}{m}\right)=-3\cdot \left[-\frac{1}{2}{\left(\sqrt{\frac{2GM}{R}}-H_{z}R\right)}^2\right]=\frac{3}{2}{\left(\sqrt{\frac{2GM}{R}}-H_{z}R\right)}^2$

This gives the new Lagrangian velocity at $R$, assuming:

In a previous version of the galactic rotation curves, I constructed a model galaxy with a model bulge with mass $M$ and radius $R$ and an empty space around it [1] . In such a mode galaxy, the Newtonian gravitational potential was fully determined by the bulge. The model bulge has constant density ${\rho }_0=\frac{M}{V}=\frac{3M}{4\pi R^3}$ and its composing stars rotate on geodetics in a quasi-solid way. So all those stars in the bulge have equal angular velocity $\omega$ on their geodetic orbits, with $v_{\text{orbit}}=\omega r$. So we had $v_{\text{orbit}}^2={\omega }^2{r}^2$ and

$\frac{K_{\text{orbit}}}{m}=\frac{{\omega }^2{r}^2}{2}$(1)

On the boundary between the quasi solid spherical bulge and the emptiness outside of it, the orbital velocities are behaving smoothly. So the last star in the bulge and the first star in the region outside of the bulge have equal velocities and potentials. We now apply the effective potential. This allows us to determine $\omega$ as in

${\omega }_{\text{orb}}^2=\frac{v_{\text{orb}}^2\left(R\right)}{R^2}=-\frac{V_{\text{eff}}\left(R\right)}{m{R}^2}=\frac{1}{2R^2}{\left(\sqrt{\frac{2GM}{R}}-H_{z}R\right)}^2$

So we get

$v_{\text{orb}}^2\left(r\right)={\omega }_{\text{orb}}^2{r}^2=\frac{v_{\text{orb}}^2\left(R\right)r^2}{R^2}=-\frac{V_{\text{eff}}\left(R\right)r^2}{m{R}^2}=\frac{1}{2}{\left(\sqrt{\frac{2GM}{R}}-H_{z}R\right)}^2\frac{r^2}{R^2}$

We can then determine $L\left(r\right)$ inside the bulge because $L\left(r\right)=K_{\text{orb}}\left(r\right)+K_{\text{rad},\text{eff}}\left(r\right)$, so

$\begin{array}{c}\frac{2L}{m}\left(r\right)=v_{\text{orb}}^2\left(r\right)+v_{\text{rad},\text{eff}}^2\left(r\right)\\ =\frac{1}{2}{\left(\sqrt{\frac{2GM}{R}}-H_{z}R\right)}^2\frac{r^2}{R^2}+{\left(\sqrt{\frac{GM}{R}\left(3-\frac{r^2}{R^2}\right)}-H_{z}r\right)}^2.\end{array}$

As a result, inside such a model bulge, $L$ is a not constant of the motion of the metric any more.

We again postulate that the Lagrangian is constant in the whole disk. For $r>R$, the effective potential is:

$\frac{V_{\text{eff}}\left(r\right)}{m}=-\frac{1}{2}{\left(\sqrt{\frac{2GM}{r}}-H_{z}r\right)}^2$

We define the Lagrangian per unit mass outside the bulge as:

$\frac{L}{m}=\frac{K_{\text{orb}}}{m}+\frac{K_{\text{rad},\text{eff}}}{m}=\frac{1}{2}{v}_{\text{orb}}^2+\frac{1}{2}{\left(\sqrt{\frac{2GM}{r}}-H_{z}r\right)}^2$

Substitute the conserved value of $\frac{L}{m}$:

$\frac{1}{2}{v}_{\text{orb}}^2+\frac{1}{2}{\left(\sqrt{\frac{2GM}{r}}-H_{z}r\right)}^2=\frac{3}{4}{\left(\sqrt{\frac{2GM}{R}}-H_{z}R\right)}^2$

Solve for $v_{\text{orb}}^2$:

$v_{\text{orb}}^2=\frac{3}{2}{\left(\sqrt{\frac{2GM}{R}}-H_{z}R\right)}^2-{\left(\sqrt{\frac{2GM}{r}}-H_{z}r\right)}^2.$

In brief, we have

$v_{\text{orb}}^2=\frac{3}{2}{v}_{\text{rad},\text{eff}}^2\left(R\right)-v_{\text{rad},\text{eff}}^2\left(r\right).$

And for $v_{\text{final},\text{eff}}=v_L$ we get

$v_{\text{final},\text{eff}}^2=\frac{3}{2}{v}_{\text{rad},\text{eff}}^2\left(R\right)=\frac{3}{2}{\left(\sqrt{\frac{2GM}{R}}-H_{z}R\right)}^2$

$v_{\text{orb}}^2\left(r\right)=\frac{1}{2}{\left(\sqrt{\frac{2GM}{R}}-H_{z}R\right)}^2\frac{r^2}{R^2}$

$v_{\text{rad},\text{eff}}^2\left(r,Hz\right)={\left(\sqrt{\frac{GM}{R}\left(3-\frac{r^2}{R^2}\right)}-H_{z}r\right)}^2.$

$v_L^2\left(r\right)=\frac{1}{2}{\left(\sqrt{\frac{2GM}{R}}-H_{z}R\right)}^2\frac{r^2}{R^2}+{\left(\sqrt{\frac{GM}{R}\left(3-\frac{r^2}{R^2}\right)}-H_{z}r\right)}^2.$

$v_{\text{orb}}^2=\frac{3}{2}{\left(\sqrt{\frac{2GM}{R}}-H_{z}R\right)}^2-{\left(\sqrt{\frac{2GM}{r}}-H_{z}r\right)}^2.$

$v_{\text{rad},\text{eff}}^2={\left(\sqrt{\frac{2GM}{r}}-H_{z}r\right)}^2.$

$v_L^2=v_{\text{final}}^2=\frac{3}{2}{\left(\sqrt{\frac{2GM}{R}}-H_{z}R\right)}^2$

We have

${\stackrel{\to }{v}}_L={\stackrel{\to }{v}}_{\text{orb}}+{\stackrel{\to }{v}}_{\text{rad},\text{eff}}$

with

${\stackrel{\to }{v}}_{\text{rad},\text{eff}}=-v_{\text{rad},\text{eff}}\hat{r}=-\left(\sqrt{\frac{2GM}{r}}-H_{z}r\right)\hat{r}$

and

${\stackrel{\to }{v}}_{\text{orb}}=v_{\text{orb}}\hat{\phi }=\sqrt{\left(\frac{3}{2}{\left(\sqrt{\frac{2GM}{R}}-H_{z}R\right)}^2-\left(\sqrt{\frac{2GM}{r}}-H_{z}r\right)\right)}\hat{\phi }$

so

$\tan \left({\alpha }_L\right)=\frac{\left|v_{\text{rad},\text{eff}}\right|}{\left|v_{\text{orb}}\right|}=\frac{\sqrt{\frac{2GM}{r}}-H_{z}r}{\sqrt{\frac{3}{2}{\left(\sqrt{\frac{2GM}{R}}-H_{z}R\right)}^2-{\left(\sqrt{\frac{2GM}{r}}-H_{z}r\right)}^2}}$

and

$v_L^2=v_{\text{orb}}^2+v_{\text{rad},\text{eff}}^2$

Inside the bulge:

$v_{\text{orb}}^2\left(r\right)=\frac{1}{2}{\left(\sqrt{\frac{2GM}{R}}-H_{z}R\right)}^2\frac{r^2}{R^2}$

and outside the bulge:

$v_{\text{orb}}^2=\frac{3}{2}{\left(\sqrt{\frac{2GM}{R}}-H_{z}R\right)}^2-{\left(\sqrt{\frac{2GM}{r}}-H_{z}r\right)}^2.$

In Figure 1 we present the specific Lagrangian $L/m$ as a function of radius $r$, separated into its orbital and radial effective contributions. For radii $r\le R$ the expressions follow directly from the Lagrangian formulation in Eq. (X), where the

quadratic terms in $v_{\text{orb}}$ and $v_{\text{rad},\text{eff}}$ combine to produce the Lagrangian density. For radii $r>R$ the continuation is given by the effective outside-bulge relations [Eqs. (IIF-2)-(X)], in which the orbital and radial terms exchange roles while the total value remains a constant $L/m=\frac{1}{2}{v}_L^2$. This asymptotic constant is split by construction between the orbital and radial channels. The figure further highlights the horizontal fractions $\frac{1}{3}L/m$ and $\frac{2}{3}L/m$, which provide natural reference levels for the partitioning of the constant value in the exterior domain. The vertical dashed line marks $r=R$, i.e. the bulge boundary. The shaded region indicates the continuation to radii $r>R$.

In order to assess how the balance between orbital and radial effective contributions evolves with redshift and stellar mass, we also evaluated the model for intermediate parameters, namely $M=5\times {10}^8{M}_{\odot }$ at $z=1$ with the same bulge radius $R=2$ kpc. Here the Hubble rate is $H\left(z=1\right)=123$ km∙s⁻¹∙kpc⁻¹, corresponding to 0.123 km∙s⁻¹∙kpc⁻¹. Figure 2 compares the resulting specific Lagrangian and contributions with the baseline case ( $M={10}^9{M}_{\odot }$, $H_0=70$ km∙s⁻¹∙kpc⁻¹, $R=2$ kpc). The asymptotic values are reduced from $\simeq 3.2\times {10}^3$ (km/s)² in the baseline to $\simeq 1.6\times {10}^3$ (km/s)² at $z=1$, so both curves remain visible on the same vertical scale. This illustrates that for realistic cosmological values the interior partitioning of the Lagrangian remains qualitatively unchanged, but the overall scale shifts systematically with mass and Hubble parameter.

We parameterize $M_{\text{bulge}}\left(z\right)$ with a two-channel model in which ~60% of the final mass is built in situ around the cosmic star formation peak ( $z~2$) and the remainder accrues ex situ through mergers toward lower redshift. This parameterization yields a steep rise in mass between $z~6$ and $z~1$ and a flattening thereafter. Figure 3 illustrates this track.

The asymptotic value of the specific Lagrangian is

${\frac{L}{m}|}_{\infty }\left(z\right)=\frac{3}{4}{\left(\sqrt{\frac{2G{M}_{\text{bulge}}\left(z\right)}{R}}-H\left(z\right)R\right)}^2.$(2)

For the adopted bulge masses and $R=2$ kpc, the gravitational term $\sqrt{2GM/R}$ is tens to hundreds of km∙s⁻¹, whereas the Hubble term $H\left(z\right)R$ remains of order a few km∙s⁻¹ even at $z=10$. As a result, ${L/m|}_{\infty }$ is governed almost entirely by the bulge mass, with $H\left(z\right)$ providing only a percent-level correction. This is evident in Figure 3 , where the redshift dependence of ${L/m|}_{\infty }$ directly tracks the growth of $M_{\text{bulge}}\left(z\right)$.

The critical radius is defined by

$r_c\left(M,z\right)={\left(\frac{2GM}{H{\left(z\right)}^2}\right)}^{1/3}.$(3)

Unlike ${L/m|}_{\infty }$, here the Hubble rate enters with a strong exponent, $r_c\propto H{\left(z\right)}^{-2/3}$. Across $z=10$ to $z=0$, $H\left(z\right)$ decreases by roughly a factor of twenty, driving an order-of-magnitude increase in $r_c$. Mass growth contributes more weakly, as $r_c\propto M^{1/3}$: a thirty-fold bulge mass increase corresponds to only a factor ~3 rise in $r_c$. Consequently, the overall evolution of $r_c\left(z\right)$ is dominated by cosmology rather than bulge mass. Figure 4 compares the critical radius for the evolving $M_{\text{bulge}}\left(z\right)$ track to the constant-mass case $M={10}^{10}{M}_{\odot }$, highlighting the relative contributions.

The relative impact of bulge mass and Hubble expansion on the two quantities of interest is summarized in Table 1 . For bulges of realistic galactic mass scale ( $M\gtrsim {10}^8{M}_{\odot }$) at $R=2$ kpc, the asymptotic Lagrangian evolution is predominantly mass-driven, while the critical radius evolution is cosmology-driven. This separation of roles clarifies the relative impact of galactic growth versus the Hubble flow in shaping the dynamical scales. For rotation curves, this implies that the impact of $Hz$ on the size of $v_{\text{final}}^2$ is in the few percentage range. But the impact of higher $Hz$, through its impact on $r_c$, should be observable as a faster approach

of $v_{\text{orb}}^2$ towards the asymptotic $v_{\text{final}}^2=v_L^2$. And the shape of spirals should also be impacted by the evolution of $r_c$ because a smaller $r_c$ squeezes the spiral inwards and forces its faster towards the final circle at $r_c$.

For nearby galaxies ( $z\simeq 0$) the Hubble rate is nearly constant and the bulge radius $R$ can be treated as fixed on galactic scales. In this regime the only remaining free parameter controlling the morphology of the inflow spirals is the bulge mass $M$. To isolate this dependence, we computed dual-arm solutions for $R=2$ kpc and four different masses, $M={10}^7$, ${10}^8$, ${10}^9$, and ${10}^{10}{M}_{\odot }$, truncated at $r=20$ kpc. The integration employed the local Hubble parameter $H\left(z=0\right)=0.07$ km∙s⁻¹∙kpc⁻¹.

Figure 5 shows the resulting spiral loci in the $\left(x,y\right)$ plane. At the lowest mass ( $M={10}^7{M}_{\odot }$) the arms remain tightly wound near the bulge, with small pitch angles. As the mass increases, the stronger gravitational potential steepens the orbital component and opens the spiral morphology. By $M={10}^{10}{M}_{\odot }$ the arms extend rapidly to large radii, producing a much more open two-armed pattern. Thus, in the local universe the bulge mass alone sets the geometry of the inflow spiral, while cosmological effects encoded in $H\left(z\right)$ are negligible.

To illustrate the evolution of inflow patterns over cosmic time we computed the single-arm spirals for bulges of fixed physical radius $R=2$ kpc at redshifts $z=0$, 2, 3, and 6. The bulge mass at each epoch was taken from the fiducial mass-growth track normalized to $M_{\text{bulge}}\left(z=0\right)={10}^{10}{M}_{\odot }$. This yields masses $M\left(z=0\right)=1.0\times {10}^{10}{M}_{\odot }$, $M\left(z=2\right)=4.2\times {10}^9{M}_{\odot }$, $M\left(z=3\right)=2.6\times {10}^9{M}_{\odot }$, and $M\left(z=6\right)=8.4\times {10}^8{M}_{\odot }$. For each case the critical radius

$r_c\left(M\left(z\right),H\left(z\right)\right)={\left(\frac{2GM\left(z\right)}{H{\left(z\right)}^2}\right)}^{1/3}$(4)

was evaluated using a flat ΛCDM cosmology with $H_0=70$ km∙s⁻¹∙kpc⁻¹,

${\Omega }_m=0.3$, ${\Omega }_{\Lambda }=0.7$.

Figure 6 shows the resulting spiral loci in the $\left(x,y\right)$ plane together with their $r_c$ circles. The present-day case ( $z=0$) exhibits an extended spiral with $r_c~260$ kpc, while at higher redshift the combination of reduced bulge mass and larger $H\left(z\right)$ compresses the pattern to smaller scales ( $r_c\simeq 24$ kpc at $z=6$). Thus the spiral inflow pattern contracts strongly with redshift, tracking the cosmological evolution of $H\left(z\right)$ modulated by the concurrent bulge mass growth.

This numerical trend is consistent with the analytic scaling relation $r_c\propto M^{1/3}H{\left(z\right)}^{-2/3}$. Over the interval $z=0$ to $z=6$, $H\left(z\right)$ increases by nearly an order of magnitude, while $M\left(z\right)$ decreases by a factor of ~10. The net effect is a contraction of $r_c$ by an order of magnitude, in good agreement with the values shown in the figure. Hence the plotted spirals provide a direct visualization of how the competing dependencies on bulge mass and the Hubble rate shape the inflow geometry.

To illustrate the morphology of inflow patterns in the early universe, we computed dual-arm spirals (a single solution and its 180˚ rotation) at redshifts $z=3$, 10, 15, and 20, as shown in Figure 7 and Figure 8 . The bulge radius $R$ was set to 2 kpc for $z=3$ and 10, and to 1 kpc for $z=15$ and 20, while the bulge mass was chosen to match the fiducial growth track at each epoch. The resulting parameters are $M\left(z=3\right)=2.6\times {10}^9{M}_{\odot }$, $M\left(z=10\right)=3.0\times {10}^8{M}_{\odot }$, $M\left(z=15\right)=1.0\times {10}^8{M}_{\odot }$, and $M\left(z=20\right)=5.0\times {10}^7{M}_{\odot }$.

The spirals were integrated outward from $R$ to the critical radius where the radial inflow velocity vanishes. At $z=20$ we find $H\left(z\right)=3.69$ km∙s⁻¹∙kpc⁻¹

and $r_c=3.16$ kpc, at $z=15$ we find $H\left(z\right)=2.45$ km∙s⁻¹∙kpc⁻¹ and $r_c=5.23$ kpc, at $z=10$ the pattern extends to $r_c=10.96$ kpc, and at $z=3$ the present observational limit for galaxy imaging is reached, with $H\left(z\right)=0.31$ km∙s⁻¹∙kpc⁻¹ and $r_c=61.2$ kpc. Figure 7 and Figure 8 show both spiral arms, the bulge radius, and the critical radius as dashed and dash-dotted circles.

The feedback process described in Sec. 6 can produce galaxies in which two distinct inflow spirals coexist: an older spiral that has expanded to larger radii, and a younger spiral formed after a bulge reset event. In such cases the outer spiral traces the previous inflow morphology, while the newly formed inner spiral develops within the reset bulge region. The result is a nested pattern in which two sets of arms overlap, potentially evolving into bar-like features embedded inside larger disks.

Figure 9 shows an example with an outer spiral representing $z=3$ ( $M=2.6\times {10}^9{M}_{\odot }$, $R=3$ kpc) truncated at 15 kpc, and a younger nested spiral at $z=20$ ( $M=5.0\times {10}^7{M}_{\odot }$, $R=1$ kpc). The inner structure retains its own critical radius $r_c=3.16$ kpc, well separated from the larger-scale $z=3$ pattern.

A second case is shown in Figure 9 , right, where the outer spiral corresponds to $z=0$ ( $M={10}^{10}{M}_{\odot }$, $R=3$ kpc) and is truncated at 15 kpc, while the nested inner spiral again represents the $z=20$ configuration. This illustrates how an early inner spiral can survive as a bar-like or ring-like feature embedded within the much more extended present-day spiral structure.