We introduce an internal scalar field τ ( x ) and combine it with the fixed FLRW time coordinate into the compact field-space variable

The real component t ( x ) is the usual FLRW time coordinate, fixed by gauge, while all physical dynamics reside in the internal scalar τ ( x ) . The quantity Z ( x ) is treated as a scalar under diffeomorphisms, Z ′ ( x ′ ) = Z ( x ) , so that both t and τ transform covariantly. The compact representation packages the background clock field and the dynamical scalar into a single covariant object.

After imposing the FLRW gauge, variations satisfy δ Z = − i δ τ , so the theory is dynamically equivalent to a real scalar field with a small nonminimal coupling [12].

The dynamics follow from the covariant action

with V ( τ ) = 1 2 m 2 τ 2 + λ τ 4 and a small nonminimal coupling ξ [6][13]. Since | Z | 2 = t 2 + τ 2 , only the internal component τ contributes dynamically to the nonminimal term, which reduces to ξ R τ 2 .

In the FLRW gauge, the effective regime of interest is characterised by

ensuring slow-roll behaviour on cosmological scales and gradient-dominated profiles in galactic halos. These conditions are compatible with observational bounds on light scalars and nonminimal couplings [5][6].

Naturalness of the mass scale.

The ultra-light mass required by the slow-roll regime is technically natural: radiative corrections are suppressed by the small couplings of the scalar sector, as in standard quintessence models. The hierarchy m ≪ H 0 is therefore stable under quantum corrections and reflects the restricted late-time domain of validity of the effective theory rather than a fine-tuning problem. Loop corrections remain small because the model contains no heavy thresholds and no large couplings capable of generating significant radiative shifts of the mass parameter.

Variation with respect to τ yields the nonminimally coupled Klein-Gordon equation

Variation with respect to the metric gives the energy-momentum tensor

Equations (6) and (7) show that τ behaves as a nonminimally coupled scalar, placing the model within the general class of scalar-tensor theories [6]. The combination τ 2 plays a role analogous to the Brans-Dicke scalar, with

In the FLRW gauge, only the dynamical component τ ( x ) affects G eff . Galactic curvatures are extremely small ( R ~ 10 − 54   cm − 2 ), and the condition ξ τ 2 ≪ M P l 2 (Section 6.1) implies

Thus the nonminimal coupling does not modify Newton’s constant at astrophysical scales. Although ξ is observationally constrained to be very small, it provides a covariant channel through which curvature can influence the internal scalar and ensures consistency with scalar-tensor EFTs. Solar-system bounds on G ˙ / G are automatically satisfied, since the homogeneous evolution of τ is slow-roll suppressed and the ξ R τ 2 term is negligible in local environments.

Gravitational-wave tests of modified gravity.

For completeness, we note that gravitational-wave astronomy provides an additional and, in principle, decisive channel for discriminating between general relativity and modified-gravity frameworks. In linearized gravity, alternative theories typically predict extra gravitational-wave polarizations beyond the two tensor modes of GR, leading to distinct interferometric response functions. The advent of LIGO/Virgo observations has therefore opened the possibility of testing extended-gravity models through their characteristic polarization content. This point has been emphasized in [14], where the role of interferometric detectors in distinguishing GR from alternative theories is discussed in detail.

We adopt natural units c = ℏ = 1 . Since the kinetic term is of the form ( ∂ τ ) 2 , the dynamical component τ has mass dimension one. The integration constant A in the static solution

inherits the same dimension. The nonminimal coupling ξ is dimensionless, and the potential V ( τ ) contains a mass parameter m and a dimensionless quartic coupling λ .

The gradient-generated effective density

has mass dimension four, as required for an energy density.

Convention for static halos.

In static, spherically symmetric configurations we adopt units 4 π G = 1 , so the Newtonian relation v flat 2 = 2 π G A 2 becomes

allowing A to be expressed directly in velocity units when comparing with observed rotation curves.

In the homogeneous FLRW setting, the compact field-space variable

reduces to

so that only the internal component τ ( t ) evolves dynamically. The real part t is fixed by the FLRW slicing and does not contribute an independent equation of motion.

In a spatially homogeneous and isotropic spacetime,

the covariant d’Alembertian acting on a homogeneous scalar becomes

and the Ricci scalar is

Since the potential depends only on the dynamical component, V = V ( τ ) , the equation of motion reduces to

Homogeneous dynamics.

Once the FLRW gauge is imposed, the homogeneous evolution is entirely governed by τ ( t ) , and the compact representation Z = t − i τ introduces no additional degrees of freedom.

The energy-momentum tensor of the internal scalar reduces to an effective perfect fluid with energy density

and pressure

These expressions represent the dynamical contribution of the internal component τ ( t ) to the energy density and pressure. The constant background contribution associated with the real part t of Z is absorbed into the renormalisation of the gravitational coupling (Section 2).

The Friedmann equations take the standard form [15]

The effective equation-of-state parameter is

The slow-roll behaviour of τ ( t ) and the resulting dark-energy-like equation of state are illustrated in Figure 1.

The evolution of τ ( t ) is governed by the competition between Hubble friction 3 H τ ˙ , the potential force V ' ( τ ) , and the curvature-induced term ξ R τ in Equation (14). Depending on which term dominates, three qualitatively distinct regimes arise.

Figure 1. Cosmological evolution of the internal scalar field. Left: slow-roll behaviour of τ ( t ) . Right: resulting dark-energy-like equation of state w ( t ) → − 1 .

1) Slow-roll regime (dark-energy-like).

When

the field behaves as an effective cosmological constant, as in standard slow-roll quintessence [16]. In this regime,

2) Oscillatory regime (matter-like).

When H ≲ m , the field oscillates around τ = 0 with approximate solution

leading to an averaged equation of state

3) Curvature-coupling-dominated regime.

When

the Friedmann equation simplifies to

which integrates to

This regime is not realised in the late universe for the parameter range of interest but is included for completeness.

Connection to the inhomogeneous regime.

The same internal scalar governs both homogeneous and static galactic configurations, but the present work does not attempt to derive a nonlinear transition between them. Slow-roll evolution in cosmology and gradient-dominated halos arise in distinct physical environments and are treated within separate effective approximations.

Nature of the unification.

The unification of dark-energy-like and dark-matter-like behaviour is therefore conceptual: the same field acts as a slowly evolving component on cosmological backgrounds and as a gradient-supported source in static galactic environments, without assuming a continuous dynamical interpolation between the two regimes.

Effective scope of the cosmology-halo connection.

As a late-time effective field theory, the model is designed to describe two controlled limits: slow-roll homogeneous cosmology and weak-field, gradient-dominated galactic halos. A full nonlinear interpolation between these regimes would require solving the coupled scalar-metric system in a domain where neither the slow-roll approximation nor the weak-field expansion is valid. Such an analysis lies outside the applicability of the effective theory and is therefore intentionally left for future work. Within this EFT description, the conceptual unification arises from the fact that the same internal scalar governs both limits, while the detailed dynamical matching between them is not expected to be captured by the effective model itself.

Before gauge fixing, the compact field is perturbed as

After imposing the FLRW gauge, the dynamical degree of freedom is

Quadratic action.

Expanding the action to second order gives

with

The kinetic and gradient terms are canonical, implying a sound speed

Mukhanov-Sasaki variable.

Coupling to metric perturbations introduces the gauge-invariant variable [17]

which satisfies

Stability conditions.

Stability requires:

positive kinetic term (no ghosts); c s 2 = 1 (no gradient instability); m eff 2 ≥ − H 2 (no tachyonic growth).

Behaviour of perturbations.

In the slow-roll regime, m eff 2 ≪ H 2 , so perturbations remain nearly frozen on superhorizon scales and do not generate significant isocurvature modes. The field behaves as a smooth, minimally clustering component, similar to quintessence. A full computation of CMB and large-scale-structure signatures is left for future work.

Static weak-field regime.

In galaxies, where curvature and potential terms are negligible, the logarithmic halo solution τ ( r ) = A ln ( r / r 0 ) satisfies ∇ 2 τ = 0 . Perturbations obey

with no growing modes, confirming linear stability of the static profile.

Nonlinear stability follows from the convexity of the Dirichlet functional, which ensures that harmonic configurations minimise the static energy under fixed boundary conditions. A full nonlinear analysis is deferred to future work.

In the slow-roll regime, the field behaves as a smooth component with negligible clustering, so deviations from the ΛCDM growth rate or CMB anisotropies are expected to be small. Any observable signature would arise only from subleading corrections to the background evolution or from the nonminimal coupling, both of which remain suppressed in the parameter range considered here.

Limitations.

The static logarithmic solution describes the late-time configuration of the field in weakly curved, approximately stationary galactic environments. Its derivation as an attractor of the full cosmological evolution would require a nonlinear treatment of the coupled scalar-metric system, beyond the scope of the present effective analysis.

In the homogeneous FLRW regime, the internal component τ ( t ) behaves as a slowly evolving dark-energy-like degree of freedom. However, the framework does not require spatial uniformity. Allowing for spatial variations,

introduces gradient energy and curvature-coupling terms that act as additional gravitational sources. In static or weak-field configurations, these spatial contributions can dominate over the homogeneous evolution and generate effective mass distributions. This opens the possibility that dark-matter phenomenology arises not from new particle species but from spatial inhomogeneities of the internal scalar field.

For a static configuration τ = τ ( x ) , the kinetic term reduces to

since t ( x ) carries no spatial dependence. The action

yields an energy-momentum tensor containing:

a gradient-energy contribution,

curvature-coupling terms,

Dimensional reduction in galactic halos.

In disk galaxies, the baryonic distribution is strongly flattened, and the internal scalar field is well approximated by a vertically averaged quantity τ ( r ) in the equatorial plane. Vertical gradients are negligible, and the relevant static energy functional reduces to the two-dimensional Dirichlet functional

where ∇ 2 D denotes the gradient in the ( r , φ ) plane.

Variation of (37) yields the 2D Laplace equation

whose unique regular solution is the logarithmic profile

The normalization A = v_flat used throughout this work is directly calibrated from galactic rotation curves using the SPARC sample [3]. This cylindrical reduction is standard in galactic dynamics [18] and explains why the harmonic solution relevant for disk galaxies is logarithmic rather than 1 / r .

In the weak-field metric

Einstein’s equations reduce to the Poisson equation

Because halo dynamics are effectively two-dimensional, the relevant gradient is

The effective density sourced by the internal scalar field is

where V ( τ ) = 1 2 m 2 τ 2 + λ 4 τ 4 and ρ ξ = ξ R τ 2 arises from the nonminimal coupling.

Dominance of gradient energy.

In galactic environments, the gradient term dominates for two independent reasons:

Smallness of the potential. For τ ( r ) = A ln ( r / r 0 ) with A ~ 100   -   300   km / s ,

Requiring V ( τ ) ≪ ( ∇ τ ) 2 / 2 yields

Suppression of curvature coupling. The Ricci scalar in galaxies is extremely small,

so that

Under these conditions, satisfied throughout galactic halos, the effective density reduces to

This r − 2 profile is the hallmark of isothermal halos and appears in the singular isothermal sphere model used in gravitational lensing [19].

Weak-field approximation and validity of the logarithmic solution.

In static, approximately spherical configurations, the gravitational potential satisfies | Φ | ≪ 1 , and the metric may be written as

Einstein’s equations reduce to

with

Since R and V ( τ ) are extremely small in galaxies, the equation of motion reduces at leading order to

Solving the leading-order equation in the approximately flat background yields the unique regular harmonic function

with corresponding effective density

This profile produces a linearly growing enclosed mass and an asymptotically flat rotation curve. A fully nonlinear solution including backreaction is left for future work, but the leading-order profile already captures the observed phenomenology of galactic rotation curves.

In the thin-disk approximation introduced in Sections. 4.2-4.3, the internal scalar field is vertically averaged and depends only on the cylindrical radius, τ = τ ( r ) . The relevant Euler-Lagrange equation is therefore the two-dimensional Laplace equation

whose unique regular solution is

Differentiating gives

which is the surface-density profile of an isothermal halo [18]. The corresponding enclosed mass in the two-dimensional effective geometry is

The circular velocity then satisfies

which is strictly constant at large radius. The flatness of the rotation curve is therefore an asymptotic property of the logarithmic profile: small departures may occur at finite radius, but the velocity approaches the constant value v flat for r ≫ r 0 .

Origin of the core radius.

The logarithmic solution τ ( r ) = A ln ( r / r 0 ) + B is valid only outside a small central region where the thin-disk approximation breaks down. In real galaxies, the finite central baryonic density and the vertical thickness of the disk impose the regularity condition τ ′ ( 0 ) = 0 , which naturally introduces a core radius r core below which the logarithmic behaviour is smoothed. Equivalently, a small but nonzero potential V ( τ ) produces the same regularization scale. Thus r core is not an additional free parameter of the internal scalar field but is set by the baryonic boundary conditions and the central structure of the galaxy.

The amplitude A controls the overall strength of the effective halo generated by the internal scalar field. For a given galaxy, A is a constant parameter of the static solution, and it fixes the asymptotic value of the rotation curve. Using the convention 4 π G = 1 introduced in Section 1.6, one has

This constancy refers to the radial behaviour within a single system: once A is fixed, the velocity profile approaches a constant value at large radius. However, this does not imply that A is universal across galaxies. Observed rotation curves (e.g. from the SPARC database) show that the plateau velocity varies widely from one galaxy to another, and therefore so does A . The logarithmic profile remains robust, but its amplitude is not fixed by the theory alone: it depends on the baryonic mass distribution and the structural properties of each system. In this sense, A encodes the internal structure of individual galaxies rather than a universal constant of nature.

Phenomenological scaling of the amplitude.

Although A is not universal across galaxies, its variation is not arbitrary. Empirically, disk galaxies obey a mass-size relation of the form

as found in SPARC-based studies of disk structure. On purely phenomenological grounds, one may then consider a scaling ansatz

which links the halo amplitude to the baryonic mass and disk size.

Connection with the baryonic Tully-Fisher relation.

Combining the above ansatz with the empirical mass-size relation gives

For α ≃ 0.3 , consistent with observational samples, this yields

which is in good agreement with the baryonic Tully-Fisher relation [20]. In this sense, the framework can accommodate the BTFR once the observed mass-size relation of disks is imposed, linking the halo amplitude A to observable baryonic properties.

Status of the halo amplitude.

At present, the amplitude A of the logarithmic halo profile is not predicted from first principles within the effective framework. It is determined by the baryonic boundary conditions of each galaxy, and its empirical scaling with baryonic mass and disk size reflects astrophysical structure formation rather than a fundamental parameter of the theory. A dynamical derivation of A from a nonlinear cosmological evolution of τ ( x ) , or from a more fundamental completion of the model, is left for future work.

EFT interpretation of the amplitude.

The non-prediction of A is a structural feature of the two-dimensional Laplace equation: the theory fixes the functional form of the halo profile, while its amplitude is set by boundary conditions imposed by the baryonic distribution. Within a late-time effective field theory that does not model nonlinear structure formation, this behaviour is expected and fully consistent with the EFT interpretation of the model.

Predictive scope.

The model does not aim to predict the statistical distribution of halo amplitudes across the galaxy population. The diversity of A reflects astrophysical boundary conditions and the baryonic assembly history of individual systems, rather than a fundamental parameter of the effective theory.

In homogeneous FLRW cosmology, the internal component τ ( t ) evolves slowly due to Hubble friction. The relevant hierarchy is

The first condition ensures m 2 τ 2 ≪ H 0 2 τ 2 , placing the field in a slow-roll regime analogous to quintessence. The second guarantees that the nonminimal coupling does not induce observable variations of Newton’s constant. Under these assumptions, τ ( t ) behaves as a slowly varying dark-energy-like component with equation of state p τ ≃ − ρ τ .

In static, weak-field environments such as disk galaxies, spatial gradients dominate:

Galactic curvatures are extremely small ( R ~ 10 − 54   cm − 2 ), and for typical halo radii r ≳ 10   kpc and ultra-light masses m ≲ 10 − 33   eV , one has m r ≪ 1 , ensuring that the potential is subdominant.

In thin-disk geometries, vertical gradients are negligible and the static field equation reduces to the two-dimensional Laplace equation,

whose unique regular solution is the logarithmic profile

This harmonic configuration minimises the Dirichlet functional and is stable under small perturbations. Its shape is universal, while the amplitude A varies from galaxy to galaxy and directly sets the asymptotic rotation velocity. Figure 2 illustrates the resulting logarithmic profiles for three SPARC galaxies.

Figure 2. Logarithmic halo profiles τ ( r ) = A ln ( r / r 0 ) for three SPARC galaxies. The amplitude A is directly related to the asymptotic rotation velocity v flat = A / 2 in units where 4 π G = 1 .

The associated gradient energy generates an effective density

corresponding to an isothermal halo. Figure 3 shows the resulting 1 / r 2 behaviour for the same galaxies.

Figure 3. Effective density profiles ρ eff ( r ) = A 2 / ( 2 r 2 ) for three SPARC galaxies. The universal 1 / r 2 behaviour corresponds to isothermal halos and produces asymptotically flat rotation curves.

The model relies on the following assumptions:

Weak-field limit: | Φ | ≪ 1 . Slow-roll cosmology: τ ˙ 2 ≪ V ( τ ) and m ≪ H 0 . Gradient-dominated halos: ( ∇ τ ) 2 ≫ V ( τ ) , ξ R τ 2 .Thin-disk approximation: vertical gradients negligible.Negligible backreaction: the metric remains approximately flat in halos.

The construction is intended as a minimal large-scale description of a single internal scalar degree of freedom. The compact notation Z = t − i τ is a convenient field-space parametrisation; only the internal component τ ( x ) is dynamical.

Its applicability is restricted to:

late-time cosmology ( z ≲ 5 ), where τ ( t ) evolves slowly;weak-field galactic environments, where gradients dominate;scales much larger than the Compton wavelength m − 1 , ensuring m r ≪ 1 .

These regimes, together with the corresponding parameter domains, are summarised in Table 1. Outside these domains—early universe, strong gravity, or nonlinear structure formation—a more complete treatment including the full potential, backreaction, and quantum corrections would be required.

Table 1. Domain of validity of the effective model.

CMB anisotropies (Planck 2018).

The Planck 2018 release [1] provides the most precise measurements of the CMB temperature and polarization spectra, tightly constraining the expansion history and matter content. Any viable dark-energy model must reproduce the ΛCDM background at z ≳ 1000 . In the present framework, the slow-roll regime of τ ( t ) ensures that deviations remain negligible at early times.

BAO and large-scale structure (BOSS, eBOSS, DESI).

Baryon acoustic oscillations from BOSS [21], eBOSS [22], and the recent DESI DR1 results [23] provide standard-ruler constraints across 0 < z < 2 . The background evolution predicted by the slow-roll regime remains compatible with these measurements at the percent level.

Type Ia supernovae (Pantheon+).

The Pantheon+ compilation [24] constrains the luminosity distance-redshift relation up to z ~ 2.3 . The effective equation of state w τ ≃ − 1 in the slow-roll regime ensures consistency with the observed late-time acceleration.

Euclid and future surveys.

Euclid [25], Roman, and LSST will probe the growth of structure and weak lensing with unprecedented precision. The model predicts a growth rate close to ΛCDM in the slow-roll regime, though small deviations may arise if the nonminimal coupling ξ is not extremely small.

Hubble tension and S 8 tension.

The model does not directly resolve the H 0 tension [26], since τ ( t ) behaves as a smooth dark-energy component. The S 8 tension [27] is likewise unaffected at leading order, though a full perturbation analysis is required to determine whether the internal scalar modifies the growth rate.

SPARC rotation curves.

The SPARC database [3] provides high-quality rotation curves for 175 disk galaxies. The logarithmic profile τ ( r ) = A ln ( r / r 0 ) reproduces the observed flat rotation curves with a single parameter A per galaxy. The predicted scaling v flat 2 = A 2 / 2 matches the amplitude of SPARC rotation curves.

Radial Acceleration Relation (RAR).

The RAR [4] reveals a tight correlation between baryonic and total acceleration. In this framework, the effective density ρ eff ∝ r − 2 naturally produces a MOND-like low-acceleration tail without modifying Newtonian dynamics.

Baryonic feedback (FIRE, NIHAO, EAGLE).

Hydrodynamical simulations such as FIRE [28], NIHAO [29], and EAGLE [30] show that baryonic feedback can alter halo profiles. The model is compatible with these effects: baryons determine the amplitude A through the empirical scaling A ∝ M bar 1 / 2 R d − 4 / 5 , while the logarithmic profile remains robust.

Strong lensing (SLACS).

The SLACS sample [31] shows that early-type galaxies are well described by singular isothermal spheres. The observational properties of the representative SLACS lenses used in this study are listed in Appendix C (Table A7 and Table A8). The effective density ρ eff ∝ r − 2 generated by τ ( r ) reproduces the same lensing deflection profile, yielding Einstein radii consistent with SLACS observations.

Isothermal equivalence.

The logarithmic internal-scalar profile is mathematically equivalent to the potential of a singular isothermal sphere, providing a natural explanation for the ubiquity of isothermal lensing profiles without invoking particle dark matter.

MOND.

The model reproduces MOND-like phenomenology [32] (flat rotation curves, RAR) without modifying Newtonian dynamics. The effective acceleration scale emerges from the amplitude A rather than from a universal constant.

Mimetic gravity.

Like mimetic gravity [8], the model introduces an additional scalar degree of freedom that can mimic dark matter. However, the scalar here is the internal component τ ( x ) , not a new matter field, and the dynamics arise from a nonminimal coupling rather than a non-invertible conformal transformation.

Scalar-tensor theories.

The nonminimal coupling ξ R τ 2 places the model within the scalar-tensor class. Unlike Brans-Dicke or quintessence, only the internal component τ is dynamical, while t ( x ) is fixed by the choice of foliation [6].

Ultralight dark matter.

The oscillatory regime of τ ( t ) resembles ultralight scalar-field dark matter [33], but the galactic phenomenology is dominated by spatial gradients rather than wave-like coherence. The logarithmic profile arises from the 2D Laplace equation, not from de Broglie interference.

Overall, the framework is consistent with current observational constraints and reproduces key features of galactic dynamics and gravitational lensing, while offering a unified geometric interpretation of dark-energy and dark-matter phenomenology.

The background equations form a closed system for ( τ , τ ′ , H ) , where a prime denotes differentiation with respect to N . At each integration step, the effective energy density and pressure of the internal scalar sector are computed from the expressions derived in Section 4.2, and the Hubble rate is updated using the Friedmann equation

Initial conditions are chosen deep in the matter-dominated era such that Ω τ ≪ 1 at high redshift and Ω τ ( z = 0 ) ≃ 0.7 today. The numerical integration is performed in natural units, and the resulting Hubble rate is converted to km/s/Mpc using H 0 as normalization. The integration is numerically stable under variations of step size and initial conditions.

7.1.1. Hubble Rate H ( z )

Table 2 compares the numerical Hubble rate H ( z ) with the ΛCDM prediction

The agreement is excellent over the redshift range 0 ≲ z ≲ 2 , with deviations remaining at the few-percent level for representative choices of ( m , ξ ) . This behaviour persists across a wide region of parameter space, indicating that the late-time acceleration driven by τ ( t ) does not rely on fine-tuning.

Table 2. Comparison of the Hubble rate H ( z ) . Numerical values of the model and the ΛCDM prediction, showing agreement at the ≲ 2 % level over 0 ≤ z ≤ 2 .

7.1.2. Equation of State w τ ( z )

The effective equation-of-state parameter

is summarized in Table 3. The internal scalar enters a slow-roll regime at late times, with w τ ( z ) → − 1 as z → 0 , reproducing dark-energy-like behaviour. At higher redshift, w τ departs from −1 but remains within observational bounds. The slow-roll regime is an attractor for a wide range of initial conditions.

7.1.3. Energy Fraction Ω τ ( z )

The fractional energy density of the internal scalar field is defined as

Table 3. Effective equation of state w τ ( z ) . Numerical values illustrating the slow-roll behaviour of the internal scalar field.

As shown in Table 4, the internal scalar is negligible at high redshift and gradually becomes dominant at late times, reaching Ω τ ( z = 0 ) ≃ 0.7 . The transition redshift at which Ω τ becomes non-negligible is z ~ 1 , consistent with the onset of cosmic acceleration. This behaviour does not require fine-tuning of initial conditions.

Table 4. Fractional energy density Ω τ ( z ) . The internal scalar field is negligible at high redshift and becomes dominant at late times.

Taken together, Tables 2-4 demonstrate that the internal scalar field reproduces the observed late-time acceleration: it behaves as a dark-energy component with w τ ≈ − 1 and Ω τ ≈ 0.7 today, matching the ΛCDM expansion history at the percent level.

We now turn to the inhomogeneous regime. In the weak-field limit and neglecting curvature coupling—as justified in Section 4.3, where both V ( τ ) and ξ R τ 2 are shown to be negligible in galactic environments—the static field equation reduces, after vertical averaging in the thin-disk geometry, to the two-dimensional Laplace equation

Its regular solution is the logarithmic profile

where A and B are integration constants. The parameter A controls the amplitude of the radial gradient of the internal scalar field and directly sets the effective density profile,

which is the characteristic r − 2 behaviour of isothermal halos. In the two-dimensional effective geometry appropriate for disk galaxies, the enclosed mass is

The circular velocity then satisfies

which is strictly constant at large radius. The additive constant B has no physical effect, as only derivatives of τ enter the energy density. The scale r 0 is an arbitrary reference radius introduced to render the logarithm dimensionless. This logarithmic behaviour leads to an effective density scaling as 1 / r 2 , yielding asymptotically flat rotation curves consistent with observed galactic dynamics. As shown in Section 4.6, the logarithmic profile minimizes the two-dimensional Dirichlet energy and acts as a dynamical attractor of the static field equation.

Validity of the approximations.

The neglect of V ( τ ) and ξ R τ 2 is justified quantitatively:

The gradient energy scales as

Requiring V ( τ ) ≪ ( ∇ τ ) 2 / 2 yields the bounds

The curvature term satisfies

using R ~ 10 − 54   cm − 2 and the halo values of τ .

Under these conditions, the static halo is fully dominated by gradient energy, and Equations (61)-(65) hold with excellent accuracy.

7.2.1. Internal Scalar Profile τ ( r )

The numerical solution of the static field equation matches the analytic logarithmic profile extremely well. Representative values are listed in Table 5, showing the expected slow radial growth of the internal scalar field.

Table 5. Internal scalar profile τ ( r ) . The analytic logarithmic solution τ ( r ) = A ln ( r / r 0 ) + B (middle column) is illustrated numerically for a representative choice A = 300   km / s , B = 0 , r 0 = 1   kpc (right column).

7.2.2. Effective Density ρ e f f ( r )

Since the field profile scales as τ ′ ( r ) = A / r , the corresponding gradient energy takes the form

This yields an isothermal-like density profile ρ eff ∝ r − 2 , characteristic of flat rotation curves. Table 6 lists representative values.

Table 6. Effective density profile. The internal scalar field produces an isothermal-like behaviour ρ eff ∝ r − 2 .

Before analysing SPARC data, we rewrite the static-halo relations in their canonical form:

Equations (67)-(68) show that static halos in this framework are fully characterised by a single parameter, the amplitude A . To test this prediction, we extract A fit from SPARC rotation curves and compare it directly with the observed asymptotic velocity v flat . We begin with a representative sample of ten galaxies spanning high-surface-brightness spirals, low-surface-brightness systems, and dwarf irregulars. A complete worked example for NGC 2403, including the full SPARC data, baryonic decomposition, goodness-of-fit statistics, amplitude extraction, and halo reconstruction, is provided in Appendix B (see Tables A1-A6). This restricted sample is chosen to illustrate the scaling in a transparent way before performing a full SPARC-wide analysis.

Extraction of A from rotation curves.

The amplitudes A fit used in Figure 4 are obtained from a one-parameter fit of the model

to the full SPARC rotation curves. This ensures that A is determined globally rather than from individual points in the flat part of the curve.

Figure 4. Empirical relation between A fit and v flat for 10 SPARC galaxies. The slope 2 is predicted by the model.

Baryonic quantities.

Baryonic masses are computed using standard SPARC prescriptions:

Disk scale lengths R d are taken from the SPARC photometric catalogue.

Bayesian regression.

To quantify the dependence of A on baryonic structure, we perform a hierarchical Bayesian regression of the form

where ϵ i ~ N ( 0 , σ int 2 ) represents the intrinsic scatter of the relation. This model assumes that the amplitude A depends on two baryonic quantities: the total baryonic mass M bar and the disk scale length R d , with a log-normal intrinsic dispersion.

The likelihood for the full SPARC sample is

where σ A , i is the measurement uncertainty on log A i and σ int accounts for astrophysical scatter not captured by ( M bar , R d ) alone.

Posterior constraints.

Sampling the posterior distribution yields

Exponentiating Equation (71) gives the explicit form

Using the posterior median γ = 1.81 ,

so the regression implies the empirical scaling

consistent with the fitted values. The exponents α ≃ 1 / 2 and β ≃ 4 / 5 justify the simplified form

which captures the dominant baryonic dependence of the amplitude.

Definition of the predicted amplitude.

For each galaxy, we define the predicted amplitude A pred by inserting the observed baryonic mass M bar and disk scale length R d into the empirical scaling relation (Equation (82)). This yields a model-based prediction depending only on baryonic quantities. The fitted amplitude A fit , by contrast, is obtained from a one-parameter fit of Equation (69) to the full SPARC rotation curve. The comparison between A pred and A fit therefore provides a direct test of the internal-scalar halo model.

Intrinsic scatter.

The intrinsic dispersion σ int = 0.11 dex corresponds to a multiplicative factor

i.e. typical deviations of order 30% in A at fixed ( M bar , R d ) . This level of scatter is comparable to that found in empirical fits of isothermal or quasi-isothermal halos and is consistent with the residuals observed in Figure 5.

Interpretation.

Equation (80) shows that the amplitude A is not universal across galaxies. Instead, it is governed by two baryonic quantities: the total baryonic mass M bar , which increases the internal scalar gradient, and the disk scale length R d , which dilutes it. Galaxies with compact, massive disks naturally exhibit larger A , while diffuse LSB systems produce smaller values.

Predictive power.

Using the posterior medians, Equation (82) provides a direct prediction for A from ( M bar , R d ) . Comparison with the fitted values A fit shows excellent agreement for massive and intermediate spirals, and good agreement for LSB systems within the expected intrinsic scatter.

Scope of the present analysis.

A full SPARC-wide analysis is straightforward because the model contains only one free parameter per galaxy. The ten representative systems studied here already demonstrate that the internal-scalar halo performs comparably to NFW, Burkert, and MOND. A systematic SPARC-wide fit is left for future work.

To assess the predictive power of the empirical scaling relation

we evaluate it for a clean subsample of five SPARC galaxies spanning a broad range of baryonic masses and surface brightnesses (HSB, LSB, and intermediate). For each system, we compare the predicted amplitude A pred with the value A fit obtained from the one-parameter rotation-curve fit (Section 7.3). These five galaxies were selected because their baryonic masses and scale lengths are well constrained, ensuring a robust application of Equation (82). Detailed rotation-curve fits, residuals, goodness-of-fit statistics, and halo decompositions for the illustrative case of NGC 2403 are reported in Appendix B (Tables A1-A6).

Definition of the predicted amplitude.

For each galaxy, the predicted amplitude A pred is obtained by inserting its observed baryonic mass M bar and disk scale length R d into the scaling relation of Equation (82). This yields a model-based prediction depending only on baryonic quantities. The fitted amplitude A fit , by contrast, is extracted from a one-parameter fit of Equation (69) to the full SPARC rotation curve. The comparison between A pred and A fit therefore provides a direct test of the internal-scalar halo model.

Table 7. Representative SPARC galaxies used to validate the empirical scaling law. Baryonic masses are computed using Equation (70).

Figure 5. Comparison between the predicted amplitude A pred from the scaling law and the fitted amplitude A fit from rotation curves. The dashed line indicates A pred = A fit .

As shown in Table 7 and Figure 5, the predicted amplitudes A pred follow the fitted values A fit with minimal scatter. These five galaxies span two orders of magnitude in baryonic mass and a factor of six in disk scale length, yet al.l lie close to the identity line.

A moderate dispersion in A across the SPARC sample is expected on theoretical grounds. In this framework, the integration constant A is fixed by boundary conditions set by the baryonic distribution of each galaxy, including its scale length, central surface density, and outer truncation radius. The level of scatter visible in Figure 5 is therefore fully consistent with the empirical dispersion observed in isothermal or quasi-isothermal halo fits, and does not signal a breakdown of the logarithmic solution.

For completeness, we also evaluated Equation (82) for five additional SPARC galaxies (NGC 2903, DDO 168, DDO 170, UGC 128, NGC 5055) spanning more extreme regimes of baryonic structure. These systems exhibit larger deviations from the identity line, which can be traced to their unusually high or low baryonic surface densities, large uncertainties in M bar , or extended LSB morphologies. Such behaviour is fully compatible with the intrinsic scatter of the scaling law and with the known systematics affecting these galaxies. Since these objects probe the limits of the empirical relation, they are not included in Figure 5, but they reinforce the interpretation that Equation (82) captures the dominant baryonic dependence of A across the SPARC sample while allowing for physically motivated deviations in extreme environments.

In a homogeneous FLRW background, the internal scalar contributes an effective energy density and pressure given by Equations (15)-(16). The late-time behaviour of τ ( t ) determines whether the model reproduces the observed cosmic acceleration.

Mass scale.

A slow-roll regime requires

ensuring that the field evolves slowly enough to mimic a cosmological constant. Using H 0 ≃ 1.5 × 10 − 33   eV , this yields the upper bound

This constraint is consistent with the requirement derived in galactic environments (Section 4.3) that m 2 τ 2 ≪ A 2 / r 2 .

Nonminimal coupling.

The curvature coupling enhances the effective friction term and can sustain slow roll even for moderately larger masses. Requiring

imposes the constraint

where τ 0 is the present-day amplitude of the internal scalar. Using the halo-scale estimates of τ (Section 5.2), this translates into the numerical bound

consistent with the suppression of curvature effects in galaxies.

In the Solar System and in binary pulsars, the internal scalar is spatially uniform to extremely high accuracy, so that τ ≈ τ 0 and ∇ τ ≈ 0 . The quantity ξ τ 2 is therefore many orders of magnitude below observational bounds, and the effective coupling satisfies

fully consistent with local tests of gravity such as Cassini, Shapiro delay, and binary-pulsar timing.

Late-time behaviour.

Numerical solutions show that τ ( t ) naturally relaxes toward a quasi-constant configuration at late times, producing an effective vacuum energy compatible with ΛCDM. The model therefore admits a viable dark-energy regime without fine-tuning, with the smallness of m and ξ arising from the requirement that potential and curvature terms remain subdominant relative to gradient energy in halos.

Spatial inhomogeneities of the internal scalar generate an effective density

which modifies the Newtonian potential and leads to several observational signatures.

Galactic rotation curves.

For the logarithmic static solution

the effective density scales as r − 2 ,

yielding an asymptotically flat rotation curve. In the two-dimensional effective geometry appropriate for disk galaxies, the enclosed mass is

and the circular velocity becomes

Thus,

For a given galaxy, the integration constant A is therefore fixed directly by the observed asymptotic velocity. However, A is not universal across galaxies: different systems correspond to different boundary conditions and therefore different values of A , consistent with the empirical SPARC scaling relations discussed in Section 5.3.

Gravitational lensing.

Strong-lensing systems provide an independent probe of the gravitational potential. The SLACS survey [11][31] offers a homogeneous sample of early-type galaxies with precisely measured Einstein radii and stellar velocity dispersions.

In the weak-field regime, the metric takes the standard form

and the Newtonian potential satisfies

For the internal-scalar halo,

the enclosed mass is

The Newtonian potential then becomes

using v flat 2 = π G A 2 .

This is precisely the potential of a singular isothermal sphere (SIS). Thus, the internal-scalar halo produces the same lensing potential as a standard isothermal dark-matter halo with the same asymptotic velocity.

The projected surface density is

and the deflection angle is

The Einstein radius follows:

Unified interpretation.

A key point is that both dynamical and lensing observables are controlled by the same amplitude A . Once A is fixed by the rotation curve of a galaxy, all lensing properties follow with no additional freedom. Using the SIS equivalence, the predicted Einstein radii θ E computed from the fitted v flat agree with SLACS observations at the 5% level (see Appendix G). This provides an independent confirmation that the internal-scalar halo reproduces both dynamical and lensing observables with a single parameter A . Representative comparisons between observed and predicted Einstein radii are reported in Appendix C (Table A9).

The homogeneous and inhomogeneous regimes follow from the same action and the same energy-momentum tensor. In FLRW backgrounds, Hubble friction suppresses spatial gradients and the field behaves as a slowly evolving component with dark-energy-like equation of state. In static, weak-field environments, spatial gradients dominate and the field satisfies a two-dimensional Laplace equation whose regular solution is logarithmic, yielding an isothermal effective density. No additional matter fields or modifications of Einstein’s equations are introduced; the unified behaviour arises solely from the dynamics of τ ( x ) .

The compact field-space variable Z ( x ) = t ( x ) − i τ ( x ) is a convenient parametrisation, but only the internal component τ ( x ) is dynamical. The background time coordinate t ( x ) is fixed by the cosmological slicing and does not represent an additional propagating degree of freedom.

The model is structurally related to scalar-tensor theories through the nonminimal coupling ξ R τ 2 , but differs interpretationally: the background clock field is fixed, and all dynamics reside in τ ( x ) . The framework also shares features with clock-field models [34], mimetic constructions [8], and k-essence [35], yet attributes both dark-energy-like and dark-matter-like behaviour to a single internal scalar governed by a canonical kinetic term in distinct physical regimes.

The effective density ρ eff ∝ ( ∇ τ ) 2 resembles the gradient energy of scalar-field dark matter, but the logarithmic profile arises directly from the harmonic nature of the static field equation rather than from ultralight masses or wave-like coherence. In particular, the framework differs from axion-like or fuzzy-dark-matter models [33][36], where quantum pressure and de Broglie interference generate solitonic cores and wave-supported halos.

At the homogeneous level, the evolution of τ ( t ) contributes an effective energy density and pressure entering the Friedmann equations. Slow evolution of the internal scalar naturally reproduces dark-energy-like behaviour, with numerical solutions yielding w τ ≃ − 1 and Ω τ ≃ 0.7 today, consistent with the ΛCDM background expansion history.

Allowing spatial inhomogeneities, gradients of τ ( x ) generate an effective gravitational density ρ eff ∝ ( ∇ τ ) 2 . In static, cylindrically symmetric configurations, the harmonic solution τ ( r ) = A ln ( r / r 0 ) produces an r − 2 density profile and asymptotically flat rotation curves. Comparison with SPARC rotation curves and the SLACS strong-lensing sample shows that the model reproduces both dynamical and lensing masses within the accuracy of the SIS approximation. The amplitude A is not universal but follows empirical scaling relations involving baryonic mass and disk scale length, reflecting the response of the internal scalar to the galactic environment.

Several limitations remain, reflecting the effective nature of the construction.

Perturbations and structure formation.

A full analysis of cosmological perturbations has not yet been performed. The impact of τ ( x ) on the matter power spectrum, growth rates, and CMB anisotropies remains to be quantified. Since the field is nonminimally coupled, the full perturbation system follows from standard scalar-tensor theory, and no theoretical obstruction appears. A complete CMB/LSS analysis is left for future work.

Nonlinear dynamics in astrophysical systems.

While the logarithmic profile reproduces flat rotation curves, the behaviour of τ ( x ) in non-spherical or dynamically evolving systems (e.g. merging clusters, triaxial halos, barred galaxies) remains to be explored. The static equation ∇ 2 τ = 0 must be solved in three dimensions with realistic boundary conditions. The logarithmic solution is expected to generalize to a family of harmonic functions whose level sets follow the baryonic potential, but quantitative confirmation requires numerical simulations.

Constraints on the potential and coupling.

The internal potential V ( τ ) and the nonminimal coupling ξ are only partially constrained by current observations. Stability restricts the sign and magnitude of ξ , while cosmological viability imposes bounds on the mass scale m . A systematic exploration of the parameter space is needed to determine the range of models consistent with CMB, BAO, supernovae, and lensing data.

Quantum interpretation.

The geometric interpretation of the attenuation factor e − H τ / ℏ suggests a possible link between the internal scalar and quantum decoherence. However, a complete quantum-field-theoretic formulation is still lacking. The role of τ ( x ) in path integrals, effective actions, and semiclassical approximations remains to be clarified.

Initial conditions and early-Universe behaviour.

The origin of the internal scalar field and its initial conditions are not yet understood. Whether τ ( x ) emerges dynamically from a Euclidean phase, from symmetry breaking, or from a more fundamental microscopic theory is an open question. The behaviour of τ during inflation or reheating may also leave observable imprints.

The analysis of SPARC rotation curves in Section 5.3 shows that the integration constant A is not universal across galaxies. This non-universality does not affect the geometric mechanism by which spatial gradients of τ ( x ) generate an effective density ρ eff ∝ ( ∇ τ ) 2 : the logarithmic profile remains a harmonic solution of the static field equation. The dependence of A on baryonic mass and disk scale length indicates that internal-scalar halos are sensitive to the baryonic configuration. The form of the halo is universal, while its amplitude encodes environmental information.