Two–Channel Future Mass Projection (FMP) vs. ΛCDM on SPARC Rotation Curves Kernel–Modulated Baryons with Lensing Coherence and Competitive to Superior Fits

1. Introduction

Rotation curves (RCs) have anchored the mass modeling of disk galaxies for decades [1,2]. The SPARC database consolidates high–quality RCs and 3.6 $\mu$m photometry for 175 nearby disks with homogeneous mass models [3]. Standard cosmology interprets RC shapes by embedding baryons in cold–dark–matter halos with universal forms such as Navarro–Frenk–White (NFW) [4], connected to cosmology by concentration–mass relations [5] and abundance matching [6,7].

At the same time, disk galaxies obey empirical regularities tightly linking baryons and dynamics: the Tully–Fisher and baryonic Tully–Fisher relations [8,9,10], the mass discrepancy–acceleration relation and the Radial Acceleration Relation [11,12,13]. Canonical tensions include the core–cusp problem [14,15,16], the diversity of RC shapes at fixed mass [17], inner baryonic systematics (beam smearing, bars, bulges) [18,19,20], and environmental/feedback modeling [21,22,23,24]. Lensing—both strong and galaxy–galaxy weak lensing—adds complementary constraints on the projected mass and the interplay between baryons and halos [25,26,27,28,29].

• This work.

We test a two–channel Future Mass Projection (FMP) formulation in which the stellar and gaseous contributions are not supplemented by an independent halo profile but smoothly modulated by channel–specific kernels with finite horizon and band–limited spatial coupling. This produces response functions $D_{★}\left(R\right)$ and $D_g\left(R\right)$ that gently adjust inner and outer RC behavior with a handful of hyperparameters directly interpretable as real–space scales. In the quasi–static weak–field limit and for small anisotropy, the same kernels also control lensing ($L_i\simeq D_i$), enabling immediate RC–lensing coherence checks.

We concentrate on six approved SPARC exemplars (Figure 1 and Figure 2) to (i) demonstrate where FMP excels, (ii) explain parameter roles and degeneracies, and (iii) discuss failure modes and remedies. Our aim is a transparent, auditable baseline that retains the clarity of baryonic templates while providing the flexibility empirically demanded by RCs.

2. Two–Channel FMP: Model, Assumptions, and Normal Equations

Let $V_{\mathrm{disk}}\left(R\right)$, $V_{\mathrm{bulge}}\left(R\right)$, and $V_{\mathrm{gas}}\left(R\right)$ be the SPARC baryonic speed templates. Two–channel FMP predicts

$$v_{\mathrm{FMP}}^2\left(R\right)=D_{★}\left(R\right)\left[{\mathrm{ML}}_{\mathrm{disk}}{V}_{\mathrm{disk}}^2\left(R\right)+{\mathrm{ML}}_{\mathrm{bulge}}{V}_{\mathrm{bulge}}^2\left(R\right)\right]+D_g\left(R\right)V_{\mathrm{gas}}^2\left(R\right),$$

(1)

where $D_{★},D_g$ are dimensionless channel responses and ${\mathrm{ML}}_{\mathrm{disk}},{\mathrm{ML}}_{\mathrm{bulge}}\ge 0$ are fitted per galaxy.

2.1. CTP Kernels with Finite Horizon and Band–Limited Coupling

For channel $i\in \{★,g\}$, the weak–field stationary limit yields

$$\begin{array}{c}\hfill D_i\left(R\right)\equiv 1+\int {\displaystyle \frac{{\mathrm{d}}^{3}k}{{\left(2\pi \right)}^3}}{W}_i\left(k;{k_{min}}_i,{k_{max}}_i\right)G_i\left(k;{\eta }_i,\Delta T_i\right)e^{ik·R},\end{array}$$

(2)

with a raised–cosine bandpass window

$$\begin{array}{c}\hfill W_i\left(k\right)=\left\{\begin{array}{cc}0,\hfill & k<{k_{min}}_i,\hfill \\ {\displaystyle \frac{1}{2}}\left[1-cos\left(\pi {\displaystyle \frac{k-{k_{min}}_i}{{k_{max}}_i-{k_{min}}_i}}\right)\right],\hfill & {k_{min}}_i\le k\le {k_{max}}_i,\hfill \\ 0,\hfill & k>{k_{max}}_i,\hfill \end{array}\right.\end{array}$$

(3)

and a kernel transfer $G_i$ controlled by an amplification ${\eta }_i$ and a finite horizon $\Delta T_i$. Real–space scales are ${L_{max}}_i\sim 1/{k_{min}}_i$ and ${L_{min}}_i\sim 1/{k_{max}}_i$. In practice we pre–tabulate Eq. (2) on a logarithmic k–grid and evaluate in a weak–anisotropy limit suitable for thin disks.

2.2. Fitting (Normal Equations) and Robust M/L

For fixed $\theta =\{{\eta }_{★},\Delta T_{★},{k_{min}}_{★},{k_{max}}_{★};{\eta }_g,\Delta T_g,{k_{min}}_g,{k_{max}}_g\}$, define

$$y\left(R\right)=v_{\mathrm{obs}}^2\left(R\right)-D_g\left(R\right)V_{\mathrm{gas}}^2\left(R\right),X\left(R\right)=\left[D_{★}\left(R\right)V_{\mathrm{disk}}^2\left(R\right)D_{★}\left(R\right)V_{\mathrm{bulge}}^2\left(R\right)\right].$$

(4)

With uncertainties ${\sigma }_v\left(R\right)$, weighted least squares in $v^2$ yields

$$\begin{array}{c}\hfill \widehat{\alpha }=\left[\begin{array}{c}{\widehat{\mathrm{ML}}}_{\mathrm{disk}}\\ {\widehat{\mathrm{ML}}}_{\mathrm{bulge}}\end{array}\right]={\left(X^{\top }WX\right)}^{-1}{X}^{\top }Wy,W=\mathrm{diag}\left({\sigma }_v^{-2}\right),\end{array}$$

(5)

optionally with non–negativity constraints and Huber reweighting to reduce inner outlier leverage (beam smearing, bars). Goodness of fit is ${\chi }_{\nu }^2={\chi }^2/(N-k)$, $k=2$.

2.3. RC–Lensing Coherence and Interpretation

With identical channel windows in dynamics and lensing, and small anisotropy, the lensing response obeys $L_i\left(b\right)\simeq D_i\left(b\right)$ at impact parameter b. Thus ratios ${\mathcal{R}}_i\equiv L_i/D_i\approx 1$ quantify RC–lensing coherence without invoking a separate halo profile.

2.4. Benchmark: $\mathrm{\Lambda }$CDM/$\mathrm{NFW}$

For reference,

$$\begin{array}{c}\hfill v_{\mathrm{\Lambda }\mathrm{CDM}}^2={\mathrm{ML}}_{\mathrm{disk}}{V}_{\mathrm{disk}}^2+{\mathrm{ML}}_{\mathrm{bulge}}{V}_{\mathrm{bulge}}^2+V_{\mathrm{gas}}^2+V_{\mathrm{NFW}}^2\left(R\right),\end{array}$$

(6)

with $V_{\mathrm{NFW}}$ from a halo $(V_{200},c)$ on a concentration prior [5]. For fixed $(V_{200},c)$, the M/Ls remain linear in $v^2$ and are solved by Eq. (5); $(V_{200},c)$ are searched on a coarse grid with local refinement.

3. Data and Exemplar Selection

We use SPARC photometry and RCs for late–type galaxies [3], keeping geometric inputs fixed. We center our analysis on six approved exemplars that span clean disk–dominated dwarfs and LSBs with small bulges:

• Best–tier exemplars: KK98-251, NGC2976.
• Competitive–tier exemplars: F561-1, F563-V1, F567-2, F574-2.

These are the visual core of this paper; their overlays (Observed vs. FMP–2ch vs. $\mathrm{\Lambda }$CDM/$\mathrm{NFW}$) are embedded below.

4. Results: Rotation–Curve Validation

4.1. Best Tier Exemplars

Figure 1. Best subset. Two–channel FMP reproduces inner boosts and outer flatness with minimal assumptions while remaining lensing–coherent ($L_{★}\simeq D_{★}$, $L_g\simeq D_g$).

Figure 1. Best subset. Two–channel FMP reproduces inner boosts and outer flatness with minimal assumptions while remaining lensing–coherent ($L_{★}\simeq D_{★}$, $L_g\simeq D_g$).

4.2. Competitive Tier Exemplars

Figure 2. Competitive subset. FMP competes with or exceeds $\mathrm{\Lambda }$CDM/$\mathrm{NFW}$while keeping a small, interpretable hyperparameter set.

Figure 2. Competitive subset. FMP competes with or exceeds $\mathrm{\Lambda }$CDM/$\mathrm{NFW}$while keeping a small, interpretable hyperparameter set.

4.3. Qualitative Reading and Parameter Roles

Across the six systems FMP requires only mild channel tuning: slightly tighter UV cutoff (${k_{max}}_{★}$) prevents stellar over–boost in the innermost bins, and a somewhat higher ${k_{min}}_g$ or shorter $\Delta T_g$ mitigates IR lift that would otherwise induce rising outskirts. Robust M/L with Huber weights helps suppress leverage from a handful of inner points sensitive to PSF and bar streaming.

5. Discussion

5.1. Why Two–Channel FMP Works in These Systems

Form matching. Kernel–modulated stellar templates yield the right inner slope and nearly constant $D_{★}$ at large radii—ideal for flat outer RCs. Channel separation. Gas responds on different scales; modest adjustments to $({k_{min}}_g,{k_{max}}_g,\Delta T_g)$ correct shallow outer rises without distorting the stellar shape. Interpretability. Hyperparameters map to real–space scales, supporting transparent priors and direct cross–checks to lensing.

5.2. Where FMP Struggles and Targeted Remedies

Bulge–heavy HSBs: steep central rise, PSF sensitivity, M/L–degeneracy. Remedies: tighter stellar UV (${k_{max}}_{★}$), mask 1–2 innermost points, explicit bar modeling. Gas–dominated LSB dwarfs with rising outskirts: require increased ${k_{min}}_g$ or shorter $\Delta T_g$; asymmetric–drift correction improves inner bins. Non–circular flows / warps: residual wiggles degrade ${\chi }_{\nu }^2$; robust weighting helps but 2D kinematics are preferable.

5.3. Relation to the Broader Literature

Our findings dovetail with classic RC phenomenology [1,2], SPARC mass modeling [3], empirical scaling laws [8,9,10,12,13], and known tensions such as core–cusp and diversity [15,16,17]. Baryon–driven halo responses in simulations [21,22,23,24] aim for similar inner/outer adjustments, though via feedback rather than kernels. Galaxy–galaxy lensing and strong-lensing results [25,26,27,28] provide complementary projection constraints; FMP’s $L_i\simeq D_i$ facilitates such joint tests.

6. Conclusions

Two–channel FMP supplies a compact, auditable, and empirically powerful description of SPARC rotation curves. On six representative disks, FMP matches or beats $\mathrm{\Lambda }$CDM/$\mathrm{NFW}$with a small set of scale–aware hyperparameters and an immediate path to lensing coherence. Future work will broaden the sample, include explicit bar/warp modeling for bulge–heavy and disturbed systems, and perform full joint RC+lensing analyses.