Finite-Range Scalar–Tensor Gravity: Constraints from Cosmology and Galaxy Dynamics

Abstract

Objective: We examine whether a finite-range scalar–tensor modification of gravity can be simultaneously compatible with cosmological background data, galaxy rotation curves, and local/astrophysical consistency tests, while satisfying the luminal gravitational-wave propagation constraint ($c_T=1$) implied by GW170817 at low redshifts. Methods: We formulate the model at the level of an explicit covariant action and derive the corresponding field equations; for cosmological inferences, we adopt an effective background closure in which the late-time dark-energy density is modulated by a smooth activation function characterized by a length scale $\lambda$ and amplitude $ϵ$. We constrain this background model using Pantheon+, DESI Gaussian Baryon Acoustic Oscillations (BAOs), and a Planck acoustic-scale prior, including an explicit $\Lambda$CDM comparison. We then propagate the inferred characteristic length by fixing $\lambda$ in the weak-field Yukawa kernel used to model 175 SPARC galaxy rotation curves with standard baryonic components and a controlled spherical approximation for the scalar response. Results: The joint background fit yields ${\Omega }_m=0.293\pm 0.007$, $\lambda =7.{69}_{-1.71}^{+1.85}$$\mathrm{Mpc}$, and $H_0=72.33\pm 0.50$$\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$. With $\lambda$ fixed, the baryons + scalar model describes the SPARC sample with a median reduced chi-square of ${\chi }_{\nu }^2=1.07$; for a 14-galaxy subset, this model is moderately preferred over the standard baryons + NFW halo description in the finite-sample information criteria, with a mean $\Delta$AICc outcome in favor of the baryons + scalar model (≈2.8). A Vainshtein-type screening completion with $\Lambda =1.3\times {10}^{-8}$ eV satisfies Cassini, Lunar Laser Ranging, and binary pulsar bounds while keeping the kpc scales effectively unscreened. For linear growth observables, we adopt a conservative General Relativity-like baseline (${\mu }_0=0$) and show that current $f{\sigma }_8$ data are consistent with ${\mu }_0\simeq 0$ for our best-fit background; the model predicts $S_8=0.791$, consistent with representative cosmic-shear constraints. Conclusions: Within the present scope (action-level weak-field dynamics for galaxy modeling plus an explicitly stated effective closure for background inference), the results support a mutually compatible characteristic length at the Mpc scale; however, a full perturbation-level implementation of the covariant theory remains an issue for future work, and the role of cold dark matter beyond galaxy scales is not ruled out.

1. Introduction

The success of General Relativity (GR) across laboratory, solar system, and binary pulsar tests coexists with persistent large-scale phenomenology, including the late-time acceleration of the cosmic expansion [1,2] and the apparent need for cold dark matter (CDM) in structural formation and galaxy dynamics [3,4]. $\Lambda$CDM considers this through a cosmological constant and a non-baryonic matter component [3]. An alternative program explores whether this phenomenology can partly arise from additional gravitational degrees of freedom, provided they are screened in high-density environments and are compatible with gravitational-wave (GW) constraints [5,6].

Scalar–tensor theories offer a well-controlled setting for these types of explorations. Among them, the subclass with derivative couplings to curvature (notably couplings to the Einstein tensor) is notable because it remains within the Horndeski/Lovelock framework of second-order field equations and can satisfy $c_T=1$ [7,8]. For the scalar with a small mass, the associated fifth force has a finite range $\lambda =m^{-1}$, potentially allowing gravity to interpolate between GR-like behavior at short distances and modified behavior on astrophysical/cosmological scales.

This study explores a finite-range scalar–tensor theory within a single, consistent framework across two regimes: (i) cosmology, where we constrain a single length scale $\lambda$ using background-expansion data, and (ii) galaxy dynamics, where we test whether the same $\lambda$ can capture part of the observed rotation-curve phenomenology in the SPARC sample. To avoid ambiguity, we distinguish explicitly between results derived from the covariant action and those introduced as an effective phenomenological closure. Accordingly, we (a) express the full-field equations explicitly, (b) present the Friedmann–Lemaître–Robertson–Walker (FLRW) specialization, and (c) derive the weak-field limit and circular velocity expressions used in the SPARC analysis. Furthermore, we posit that the background expansion function used in the cosmological Markov Chain Monte Carlo (MCMC) is a phenomenological, coarse-grained closure motivated by void-percolation arguments and not a closed-form solution of the homogeneous scalar background.

The remainder of this paper is organized as follows. Section 2 introduces the action, defines parameters, and presents the field equations, conservation laws, GW speed, and FLRW specialization. Section 3 presents the effective background closure used for cosmological inference, data and likelihood, and the cosmological constraints (including an explicit $\Lambda$CDM comparison and cosmographic diagnostics). Section 4 derives the weak-field Yukawa limit, the modified potential, and the circular velocity used for SPARC fits and presents population-level results and morphology trends. Section 5 compares the scalar fits to standard Navarro–Frenk–White (NFW) halo fits for representative galaxies. Section 6 discusses additional consistency tests: screening across scales, growth-rate constraints, and peculiar-velocity bounds. Finally, Section 7 concludes the study. Supplementary tables are provided, and technical derivations are presented in appendices and explicitly cited in the main text.

2. Theoretical Framework and Field Equations

In this section, we establish the covariant formulation of the theory. We first present the action and define the coupling parameters, then derive the exact field equations and conservation laws. Finally, we establish the specific limits required for our analysis: the tensor propagation speed on cosmological backgrounds and the effective closure used for the expansion history.

2.1. Action and Parameter Definitions

We consider a scalar–tensor theory with a canonical scalar, a derivative coupling to the Einstein tensor, and a mass term that sets a finite interaction range [7,8]. The minimal action is

$$S_{\min }=\int d^{4}x\sqrt{-g}\left[{\displaystyle \frac{M_{\mathrm{Pl}}^2}{2}}R-{\displaystyle \frac{1}{2}}{(\nabla \varphi )}^2-{\displaystyle \frac{\alpha }{2}}{G}^{\mu \nu }{\nabla }_{\mu }\varphi {\nabla }_{\nu }\varphi -{\displaystyle \frac{m^2}{2}}{\varphi }^2\right]+S_m\left[{\tilde{g}}_{\mu \nu },{\psi }_m\right],$$

(1)

where $M_{\mathrm{Pl}}$ is the reduced Planck mass, $G^{\mu \nu }$ is Einstein’s tensor, and $\alpha$ is a coupling with dimensions of length². The scalar mass m defines a finite interaction range,

$$\lambda \equiv m^{-1},$$

(2)

which is the central scale constrained in this work. We assume the matter fields ${\psi }_m$ couple universally to a Jordan-frame metric ${\tilde{g}}_{\mu \nu }$. To ensure definiteness and to connect with the weak-field galaxy limit, we use a leading-order conformal coupling ${\tilde{g}}_{\mu \nu }=A^2\left(\varphi \right)g_{\mu \nu }$ with $A\left(\varphi \right)\simeq 1+g\varphi /M_{\mathrm{Pl}}$. This leading-order coupling generates an interaction of the form

$$S_{\mathrm{int}}\simeq -{\displaystyle \frac{g}{M_{\mathrm{Pl}}}}\int d^{4}x\sqrt{-g}\varphi T,$$

(3)

where T is the trace of the matter energy–momentum tensor.

In high-density environments, derivative self-interactions can provide Vainshtein-type screening. To discuss screening across scales, we also consider an extended action with a cubic Galileon operator [9],

$$S=S_{\min }+\int d^{4}x\sqrt{-g}\left[{\displaystyle \frac{c_3}{{\Lambda }^3}}{(\nabla \varphi )}^2\square \varphi \right],$$

(4)

with a dimensionless $c_3$ and a strong-coupling scale $\Lambda$. In the galaxy fits of Section 4 and Section 5, we operate in a Yukawa regime where the cubic term is subdominant, but we use it in Section 6 to assess screening.

2.2. Metric and Scalar Field Equations

Varying the action with respect to the metric yields the modified Einstein equations,

$$M_{\mathrm{Pl}}^2{G}_{\mu \nu }=T_{\mu \nu }^{\left(m\right)}+T_{\mu \nu }^{\left(\varphi \right)}+\alpha {\Theta }_{\mu \nu }+T_{\mu \nu }^{\left(3\right)},$$

(5)

where $T_{\mu \nu }^{\left(m\right)}$ is the matter stress tensor (defined with respect to $g_{\mu \nu }$),

$$T_{\mu \nu }^{\left(\varphi \right)}={\nabla }_{\mu }\varphi {\nabla }_{\nu }\varphi -{\displaystyle \frac{1}{2}}{g}_{\mu \nu }{(\nabla \varphi )}^2-{\displaystyle \frac{1}{2}}{m}^2{\varphi }^2{g}_{\mu \nu },$$

(6)

${\Theta }_{\mu \nu }$ is the contribution from the Einstein’s tensor coupling term, and $T_{\mu \nu }^{\left(3\right)}$ is the contribution from the cubic Galileon operator. As ${\Theta }_{\mu \nu }$ is lengthy, we introduce its explicit covariant form in Appendix A and use Equation (5) in the main text as the definition of our field equations.

Varying the action with respect to $\varphi$ yields

$$\square \varphi -m^2\varphi +\alpha G^{\mu \nu }{\nabla }_{\mu }{\nabla }_{\nu }\varphi +{\displaystyle \frac{c_3}{{\Lambda }^3}}\left[{(\square \varphi )}^2-{\nabla }_{\mu }{\nabla }_{\nu }\varphi {\nabla }^{\mu }{\nabla }^{\nu }\varphi -R_{\mu \nu }{\nabla }^{\mu }\varphi {\nabla }^{\nu }\varphi \right]={\displaystyle \frac{g}{M_{\mathrm{Pl}}}}T.$$

(7)

In the weak-field and quasi-static regimes relevant to galaxies, the curvature-dependent terms are subleading, and Equation (7) reduces to a Yukawa-type equation, as shown in Section 4.

2.3. Conservation Laws and Frames

The Bianchi identity ${\nabla }_{\mu }{G}^{\mu \nu }=0$ implies covariant conservation of the total stress tensor on the right-hand side of Equation (5). If matter is minimally coupled in the Jordan frame ${\tilde{g}}_{\mu \nu }$, then ${\tilde{\nabla }}_{\mu }{\tilde{T}}_{\left(m\right)}^{\mu \nu }=0$ holds with respect to ${\tilde{g}}_{\mu \nu }$. In the Einstein frame $g_{\mu \nu }$, and for $A\left(\varphi \right)\ne 1$, the matter stress tensor is not separately conserved; at the leading order,

$${\nabla }_{\mu }{T}_{\left(m\right)}^{\mu \nu }\simeq {\displaystyle \frac{g}{M_{\mathrm{Pl}}}}T{\nabla }^{\nu }\varphi ,$$

(8)

which corresponds to an exchange of energy–momentum between matter and the scalar. In this study, the baryonic mass models used in galaxy fits are treated as stationary sources in the quasi-static limit, and the scalar-mediated force is encoded in the modified potential derived from Equation (7).

2.4. Gravitational-Wave Speed

In Horndeski-class scalar–tensor theories, the observation of GW170817/GRB170817A [10] constrains the low-redshift tensor speed to be extremely close to the speed of light. For the derivative coupling to the Einstein tensor in Equation (1), tensor perturbations around an FLRW background propagate with

$$c_T^2=1,$$

(9)

so that the theory is compatible with the GW170817 constraint at low redshifts (refer to the references [11,12]). We emphasize that GW170817 is a low-z event, and future, higher redshift multimessenger observations can test possible temporal dependencies of GW propagation in broader effective-field-theory frameworks [13,14].

2.5. FLRW Specialization Derived from the Action

For a spatially flat FLRW metric with a scale factor $a\left(t\right)$ and a homogeneous scalar $\varphi \left(t\right)$, Equations (5)–(7) reduce to generalized Friedmann equations. Based on the expression

$$3M_{\mathrm{Pl}}^2{H}^2={\rho }_m+{\rho }_r+{\rho }_{\varphi },-2M_{\mathrm{Pl}}^2\dot{H}=({\rho }_m+p_m)+({\rho }_r+p_r)+({\rho }_{\varphi }+p_{\varphi }),$$

(10)

the effective scalar energy density and pressure for the Einstein tensor coupling (with $c_3=0$) take the standard form [8]

$$\begin{array}{cc}\hfill {\rho }_{\varphi }& ={\displaystyle \frac{1}{2}}{\dot{\varphi }}^2\left(1+9\alpha H^2\right)+{\displaystyle \frac{1}{2}}{m}^2{\varphi }^2,\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill p_{\varphi }& ={\displaystyle \frac{1}{2}}{\dot{\varphi }}^2\left(1-3\alpha H^2-2\alpha \dot{H}\right)-{\displaystyle \frac{1}{2}}{m}^2{\varphi }^2-2\alpha H\dot{\varphi }\ddot{\varphi },\hfill \end{array}$$

(12)

and the homogeneous scalar equation becomes

$$\left(1+3\alpha H^2\right)\ddot{\varphi }+3H\left(1+3\alpha H^2+2\alpha \dot{H}\right)\dot{\varphi }+m^2\varphi ={\displaystyle \frac{g}{M_{\mathrm{Pl}}}}T.$$

(13)

These equations provide the action-level cosmological dynamics of the model.

2.6. Effective Background Closure Used for Cosmological Inference

In this study, we do not claim to integrate the coupled FLRW system presented above for $\varphi \left(t\right)$ and $a\left(t\right)$. Instead, following a coarse-graining/void-percolation motivation, we close the background expansion phenomenologically by introducing an effective activation function $F\left(a\right)$ that modulates the dark-energy density at later times. This key phenomenological component in our cosmological inference is introduced explicitly as an effective closure.

We parameterize the activation as

$$F\left(a\right)\equiv {\displaystyle \frac{f\left(a\right)}{f\left(1\right)}},f\left(a\right)=ϵ{\displaystyle \frac{1}{2}}\left[1+tanh\left(2\left({\displaystyle \frac{R_{0}a}{\lambda }}-1\right)\right)\right],$$

(14)

where $R_0$ is a characteristic co-moving scale associated with the present-day void network (we adopt $R_0=35\mathrm{Mpc}$ as a representative value), and $ϵ$ is an effective normalization parameter. The ratio $f\left(a\right)/f\left(1\right)$ ensures that $F\left(1\right)=1$ so that ${\Omega }_{\mathrm{DE}}$ remains the present-day dark-energy fraction. Importantly, Equation (14) is not derived as a closed-form homogeneous solution of the scalar background in Section 2; it is an effective closure used only for the background expansion. This choice of scope is explicitly discussed again in Section 3 and Section 7.

With this closure, the effective Hubble function used in the cosmological likelihood is

$$H^2\left(z\right)=H_0^2\left[{\Omega }_m{(1+z)}^3+{\Omega }_r{(1+z)}^4+{\Omega }_{\mathrm{DE}}F\left({\displaystyle \frac{1}{1+z}}\right)\right],$$

(15)

where ${\Omega }_{\mathrm{DE}}=1-{\Omega }_m-{\Omega }_r$ and ${\Omega }_r$ is fixed by $T_{\mathrm{CMB}}$ and $N_{\mathrm{eff}}$. Equation (15) is the background model used in our MCMC; it must be considered an effective description rather than the exact solution of the homogeneous scalar dynamics.

3. Cosmological Constraints from Background Data

3.1. Data and Likelihood

We constrain the effective background model (15) using (i) the Pantheon+ supernova compilation [15,16] (with analytic marginalization over the SN absolute magnitude), (ii) BAO distance measurements, including DESI Gaussian BAO constraints [17] (used in the standard $D_M/r_d$, $D_H/r_d$, and $D_V/r_d$ combinations), and (iii) the Planck acoustic-scale prior ${\ell }_A$ [3]. The total likelihood is the product of the individual likelihoods; for independent datasets, we add their ${\chi }^2$ contributions,

$${\chi }_{\mathrm{tot}}^2={\chi }_{\mathrm{SN}}^2+{\chi }_{\mathrm{BAO}}^2+{\chi }_{{\ell }_A}^2.$$

(16)

For supernovae, we use the standard, analytically marginalized form (see Appendix C for details and notation),

$${\chi }_{\mathrm{SN}}^2=A-{\displaystyle \frac{B^2}{C}},$$

(17)

with A, B, and C defined from the Pantheon+ covariance matrix. For BAO, we use

$$\begin{array}{cc}\hfill D_M\left(z\right)& =c{\int }_0^z{\displaystyle \frac{d{z}^{\prime }}{H\left(z^{\prime }\right)}},\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill D_H\left(z\right)& ={\displaystyle \frac{c}{H\left(z\right)}},\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill D_V\left(z\right)& ={\left[z{D}_M^2\left(z\right)D_H\left(z\right)\right]}^{1/3},\hfill \end{array}$$

(20)

and compare the outcomes to the published $r_d$ scaled measurements. For the acoustic scale, we adopted a Gaussian prior on ${\ell }_A$ with mean and variance values obtained from Planck [3].

3.2. Sampling and Priors

We explore the posterior of $({\Omega }_m,\lambda ,ϵ,H_0)$ with an affine-invariant ensemble sampler. Priors and convergence diagnostics are summarized in Appendix C and Table S3 (Supplementary Materials).

3.3. Cosmological Posteriors and Best-Fit Values

The posterior constraints are shown in Figure 1. We obtained (posterior medians and 68% credible intervals)

$${\Omega }_m=0.{293}_{-0.007}^{+0.007},\lambda =7.{69}_{-1.71}^{+1.85}\mathrm{Mpc},H_0=72.{33}_{-0.49}^{+0.50}\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1},$$

(21)

with $ϵ$ weakly constrained because of the normalization $F\left(1\right)=1$ in Equation (14). We emphasize that the cosmological inference constrains the activation scale $\lambda$ that enters the effective closure, which is propagated to the galaxy sector as a fixed interaction range.

3.4. $\Lambda$CDM Comparison and Cosmographic Diagnostics

To address model comparison explicitly, we fitted the same datasets with a $\Lambda$CDM baseline (constant dark-energy density, i.e., $F\left(a\right)=1$) and compared the minimum ${\chi }^2$ and Akaike information criterion (AIC) outcomes.

Table 1. Comparison to a $\Lambda$CDM baseline using the same background datasets.

Model	Parameters	${\mathit{\chi }}_{min}^2$	$\mathbf{\Delta }\mathbf{AIC}$
$\Lambda$CDM	$({\Omega }_m,H_0)$	$15,455.45$	0
Effective activation ($F\left(a\right)$)	$({\Omega }_m,\lambda ,ϵ,H_0)$	$15,455.54$	$4.09$

summarizes the results. The improvement in ${\chi }^2$ relative to $\Lambda$CDM is negligible, and the extra parameters of the effective activation model are therefore disfavored by AIC, as expected for a minimal background data comparison.

We also report basic cosmographic diagnostics derived from the best-fitted $H\left(z\right)$; the present-day deceleration parameter $q_0\simeq -0.56$ and jerk $j_0\simeq 1.00$, and the $Om\left(z\right)$ diagnostic remains almost constant within the range of $z\lesssim 2.5$ (see Appendix D).

4. Galaxy Dynamics and SPARC Rotation Curves

4.1. Weak-Field Limit and Modified Potential

In the weak-field regime around Minkowski’s space, curvature terms are negligible ($G_{\mu \nu }\approx 0$) and the scalar Equation (7) reduces to a Yukawa equation,

$$\left({\nabla }^2-m^2\right)\varphi \simeq {\displaystyle \frac{g}{M_{\mathrm{Pl}}}}{\rho }_b,$$

(22)

where ${\rho }_b$ is the (nonrelativistic) baryonic density. The solution to Green’s function is

$$\varphi \left(\mathbf{r}\right)=-{\displaystyle \frac{g}{4\pi M_{\mathrm{Pl}}}}\int d^3{\mathbf{r}}^{\prime }{\displaystyle \frac{{\rho }_b\left({\mathbf{r}}^{\prime }\right)e^{-|\mathbf{r}-{\mathbf{r}}^{\prime }|/\lambda }}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}}.$$

(23)

For conformal matter coupling $A\left(\varphi \right)\simeq 1+g\varphi /M_{\mathrm{Pl}}$, the scalar-mediated acceleration of a test particle is ${\mathbf{a}}_{\varphi }\simeq -(g/M_{\mathrm{Pl}})\nabla \varphi$. Defining an effective scalar potential ${\Phi }_{\varphi }\equiv g\varphi /M_{\mathrm{Pl}}$, we have ${\mathbf{a}}_{\varphi }=-\nabla {\Phi }_{\varphi }$. In spherical symmetry, Equation (22) can be reduced to one-dimensional integrals; the explicit kernel and the form used in the numerical fits are derived in Appendix B.

The total potential is

$$\Phi \left(r\right)={\Phi }_N\left(r\right)+{\Phi }_{\varphi }\left(r\right),$$

(24)

where ${\Phi }_N$ is the Newtonian potential sourced by baryons. The circular velocity of particles in stable circular orbits is then

$$v_c^2\left(r\right)=r{\displaystyle \frac{d\Phi }{dr}}=v_b^2\left(r\right)+v_{\varphi }^2\left(r\right),$$

(25)

with $v_b\left(r\right)$ being the standard baryonic contribution and $v_{\varphi }^2\left(r\right)\equiv r\left|a_{\varphi }\left(r\right)\right|$.

4.2. Practical Implementation for SPARC Fits

We use the SPARC database of rotation curves and baryonic components herein. For each galaxy, SPARC provides the gas, disk, and bulge contributions to the rotation curve under a fiducial mass-to-light ratio. We express the baryonic contribution as

$$v_b^2\left(r\right)=v_{\mathrm{gas}}^2\left(r\right)+{\mathrm{Y}}_{★}\left[v_{\mathrm{disk}}^2\left(r\right)+v_{\mathrm{bulge}}^2\left(r\right)\right],$$

(26)

where ${\mathrm{Y}}_{★}$ is a free stellar mass-to-light parameter. Given $v_b\left(r\right)$, we reconstruct an effective spherical baryonic density profile via

$${\rho }_b\left(r\right)={\displaystyle \frac{1}{4\pi r^2}}{\displaystyle \frac{d}{dr}}\left({\displaystyle \frac{r{v}_b^2\left(r\right)}{G}}\right),$$

(27)

which is exact for spherical mass distributions and provides an effective radial source for the Yukawa kernel.

4.3. Spherical Symmetry Assumption and Robustness

The SPARC baryonic components are disk-like rather than spherical. Our scalar-force kernel, however, is evaluated by assuming spherical symmetry for tractable analysis across the full sample and because the scalar field depends primarily on the enclosed mass profile in the quasi-static regime. We therefore adopted spherical symmetry as a controlled approximation for the population study. Appendix E summarizes its scope and limitations for disk-like baryonic components, and an explicitly axisymmetric implementation is deferred to future work.

4.4. Galaxy-Level Inference and Fixed-$\lambda$ Analysis

Herein, we adopt the cosmological posterior median $\lambda =7.69\mathrm{Mpc}$ (i.e., $\lambda =7.69\times {10}^3\mathrm{kpc}$) and maintain it for all SPARC galaxies, interpreting it as the Yukawa range parameter appearing in the weak-field solution of Equation (22) (see Appendix B for the explicit spherical kernel used in the pipeline). For each galaxy, we sample the posterior over the parameter set $(g,{\mathrm{Y}}_{★},{\sigma }_{\mathrm{sys}},{\sigma }_{\mathrm{turb}})$, where ${\sigma }_{\mathrm{sys}}$ and ${\sigma }_{\mathrm{turb}}$ are non-negative nuisance dispersions added in quadrature to the reported SPARC velocity uncertainties, ${\sigma }_i^2={\sigma }_{i,\mathrm{SPARC}}^2+{\sigma }_{\mathrm{sys}}^2+{\sigma }_{\mathrm{turb}}^2$, to account for residual systematics and noncircular motions. We report the posterior median model and a 68% credible band; unless stated otherwise, quoted constraints refer to g and ${\mathrm{Y}}_{★}$ with $({\sigma }_{\mathrm{sys}},{\sigma }_{\mathrm{turb}})$ marginalized.

Figure 2 and Figure 3 show representative fits for NGC7814 (high surface brightness) and PGC51017 (low surface brightness), illustrating the range of rotation-curve morphologies captured by the baryons + scalar model with a fixed $\lambda$ value.

4.5. Population Results, Goodness of Fit, and Morphology

We found a median reduced chi-square value of ${\chi }_{\nu }^2\simeq 1.07$ (Supplementary Table S2) across the full sample with a tail at higher ${\chi }_{\nu }^2$ values. Figure 4 shows the distribution. We report representative galaxies in

Table 2. Representative SPARC galaxies spanning different fit qualities and morphologies (see Table S1 for the full sample). Values are posterior medians, and ${\chi }_{\nu }^2$ is the reduced chi-square.

Galaxy	Morphology	g	${\mathbf{{\rm Y}}}_{\mathbf{★}}$	${\mathit{\chi }}_{\mathit{\nu }}^2$
UGC05005	Irr/LSB	0.0403	0.567	0.072
F574-2	Sa–Sb	0.00062	0.547	0.277

and provide full-sample fit summaries in Table S1.

We also examined the trends in morphology using a coarse two-bin classification scheme (early-type spirals Sa–Sb versus irregular/LSB systems). Figure 5 shows that the inferred coupling g exhibits a mild morphology dependence in this binning, with Sa–Sb systems tending to slightly higher median g values than Irr/LSB galaxies. We emphasize that this is an empirical trend within the model and does not imply a unique causal interpretation.

5. Halo-Model Comparison

A key question is whether the baryons + scalar fits effectively replace a standard dark-matter halo or whether a halo component is still preferred in some systems. To illustrate this, we performed fits to representative galaxies under three models with a fixed $\lambda$ value: (i) baryons + scalar, (ii) baryons + NFW, and (iii) a combined model (baryons + scalar + NFW). Figure 6 shows the comparison for three illustrative galaxies. In some cases, the baryons + scalar and baryons + NFW models achieve comparable ${\chi }_{\nu }^2$, while the combined model can reduce the residual structure at the cost of additional parameters.

6. Additional Consistency Tests and Constraints

6.1. Screening Across Scales

To connect the finite-range interaction to local tests, we examined screening using the cubic Galileon operator in Equation (4). Figure 7 summarizes the screening factor across the solar system, galaxy, and cluster scales for parameter choices consistent with our inferred $\lambda$. The solar system panel includes the Cassini scale based on [18].

6.2. Growth of Structure and ${\sigma }_8$

Our cosmological inference uses background observables. In scalar–tensor theories, perturbations can induce time- and scale-dependent modifications to the effective gravitational coupling $\mu (z,k)$ and the gravitational slip. We adopted a conservative baseline with ${\mu }_0=0$ in the growth analysis. Evolving dark-energy perturbations can modify $\mu$ and hence affect $f{\sigma }_8$ and ${\sigma }_8$, as emphasized recently by [19]. We therefore treat ${\mu }_0$ as a possible nuisance degree of freedom and show in Appendix F that current $f{\sigma }_8$ data are consistent with ${\mu }_0\simeq 0$ (within the uncertainty range) for our background best fit. A full Boltzmann-code implementation of the covariant action (1) at the perturbation level is beyond the scope of this study and constitutes future work.

6.3. Peculiar-Velocity Floor

A finite-range modification can also affect large-scale peculiar velocities. We compile peculiar velocity measurements and compare them to an envelope implied by the best-fit activation scale. Figure 8 shows the resulting constraint, which places a conservative upper bound on the impact of the finite-range interaction on low-redshift velocities.

7. Conclusions

We enforce conceptual consistency between the covariant scalar–tensor theory and the phenomenological components used in cosmology and galaxy dynamics by explicitly separating action-derived results from effective closures. Our main results are the following:

• We formulated the theory at the explicit action level and provided the full-field equations from metric and scalar variations (Section 2, Appendix A). The theory satisfies $c_T=1$ and is compatible with GW170817 at low redshifts, while higher z multimessenger tests remain important (Section 2).
• We derived the FLRW specialization from the action and formulated the generalized Friedmann and scalar background equations. We then made a deliberate scope choice: the background expansion used for cosmological inference was considered an effective closure (Equation (14)), motivated by coarse-graining/void-percolation arguments and not claimed to be a closed-form solution of the homogeneous scalar dynamics.
• Using Pantheon+, BAO, and the acoustic scale, we constrained the activation length to $\lambda \simeq 7.7\mathrm{Mpc}$ and obtained a background expansion close to $\Lambda$CDM in cosmographic diagnostics. We also included an explicit $\Lambda$CDM comparison via information criteria (Section 3).
• Fixing $\lambda$ to the cosmological posterior median, we derived the weak-field Yukawa limit and used it to fit SPARC rotation curves. We obtained statistically acceptable fits for a large fraction of the sample (median ${\chi }_{\nu }^2\simeq 1.07$) and documented population statistics, morphology trends, and representative examples (Section 4).
• We compared scalar fits to standard NFW halo fits for representative galaxies and discussed screening, growth, and peculiar-velocity constraints (Section 5 and Section 6).

The logical structure is explicit, indicating that the covariant action defines the modified gravity theory and its local/weak-field behavior, while the late-time background expansion used in the cosmological likelihood is an effective phenomenological closure. Once this separation is made explicit, the paper provides a coherent and testable set of constraints on a finite interaction range across cosmology and galaxy dynamics.