Conformal Theory of Gravitation and Cosmic Expansion

Abstract

The postulate of universal Weyl conformal symmetry for all elementary physical fields introduces nonclassical gravitational effects in both conformal gravitation (CG) and the conformal Higgs model (CHM). The resulting theory is found to explain major observed phenomena, including excessive galactic rotation velocities and accelerating Hubble expansion, without invoking dark matter (DM). The recent history of this development is surveyed here. The argument is confined to implications of classical field theory, which include galactic baryonic Tully–Fisher relationships and dark galactic haloes of a definite large radius. Cosmological CHM parameters exclude a massive Higgs boson but are consistent with a novel alternative particle of the observed mass.

1. Introduction

Conformal gravity (CG), reviewed in detail [1,2] by Mannheim, replaces the Einstein–Hilbert Lagrangian density by a quadratic contraction of the conformal Weyl tensor [3,4], invariant under Weyl scaling by an arbitrary scalar field. The implied Schwarzschild equation, in spherical geometry [5,6], fitted to observed galactic rotation [1], determines parameters with no need for dark matter (DM).

Conformal theory is limited here to symmetry defined by invariance of the coupled field action integral under Weyl scaling, defined below, without altering the postulated Riemannian geometry of Einstein–Hilbert general relativity. The further development of Weyl geometry [7], possibly relevant to the intense gravitation of black holes and the Big Bang model, is not considered here.

The Weyl tensor vanishes identically in the isotropic uniform geometry postulated to describe cosmic Hubble expansion, precluding explanation by CG. However, if the Higgs scalar field of particle theory $\mathsf{\Phi }$ is postulated to have Weyl scaling symmetry, its Lagrangian density acquires a gravitational term [1]. The resulting conformal Higgs model (CHM) implies a modified Friedmann equation consistent with observed Hubble expansion [8].

In spherical geometry, the CHM Friedmann equation determines luminosity distance as a function of redshift $z\le 1$ for observed supernovae as accurately as any alternative theory [8,9]. The Higgs symmetry-breaking mechanism produces the gravitational effect of a dark-energy-source cosmological constant [8]. Centrifugal cosmic acceleration persists in the present epoch $z\le 1$. CG and CHM together have been found to determine the basic parameters of the Higgs model from observed galactic and cosmological data. These results are reviewed here.

2. Qualitative Regularities of Cosmological Data

Radial centripetal acceleration a implies $v^2=ra$ for the velocity of circular geodesics. When it became possible to observe and measure velocities of matter in stable circular galactic orbits, $v^2$ was found to exceed Newtonian $r{a}_N$ of observed baryonic matter by quite large factors and to remain nearly constant (flat) as r increases. This was well before the observation of galactic haloes with implied fixed cutoff boundaries. This motivated the MOND model [10,11]. MOND implies flat $v\left(r\right)$ for large r. For $a_N\ll a_0$, MOND $v^4=a^2{r}^2\to GM{a}_0$, the empirical baryonic Tully–Fisher relationship [12,13,14,15].

It has recently been shown [16] for 153 disc galaxies that observed acceleration a is effectively a universal function of Newtonian acceleration $a_N$, computed for the observed baryonic distribution. The empirical function has negligible observed scatter. Such a radial acceleration relationship (RAR) is a postulate of the empirical MOND model [10,11,17], which does not invoke dark matter (DM). It implies some very simple natural law.

Observed deviations from Newton/Einstein galactic gravitation have been modeled by distributed but unobserved dark matter. Typical DM halo models imply centripetal radial acceleration $a=a_N+\mathsf{\Delta }a$, a function of radius in an assumed spherical galactic halo. DM $\mathsf{\Delta }a$ is added to baryonic Newtonian $a_N$. DM fits galactic rotation (orbital velocity vs. circular orbit radius) depending on model parameter central density ${\rho }_0$ and core radius $r_0$ for a DM halo distribution. Observed data imply that the surface density product ${\rho }_0{r}_0\simeq 100M_{\odot }p{c}^{-2}$ is nearly constant for a large range of galaxies [18,19,20].

It will be shown here that the conformal postulate for gravitation and the Higgs scalar field implies all these regularities and derives values of the required parameters without dark matter.

3. Formalism for Unified Conformal Theory

Gravitational phenomena that cannot be explained by general relativity as formulated by Einstein are attributed to cold dark matter in the currently accepted $\mathsf{\Lambda }$CDM paradigm for cosmology. Dark energy $\mathsf{\Lambda }$ remains without an explanation. The search for tangible dark matter has continued for many years with no conclusive results [21].

Consideration of an alternative paradigm is motivated by this situation. The postulate of universal conformal symmetry, requiring local Weyl scaling covariance [3] for all massless elementary physical fields [9], is a falsifiable alternative. Conformal symmetry, already valid for fermion and gauge boson fields [22], is extended to both the metric tensor field of general relativity and the Higgs scalar field of elementary-particle theory [23]. This postulate is satisfied by conformal gravity [1,2,5,6] and by the conformal Higgs model [8,9], eliminating the need for dark matter, with no novel elementary fields. The Higgs symmetry-breaking mechanism produces an effective cosmological constant driving Hubble expansion.

For fixed coordinates $x^{\mu }$, local Weyl scaling is defined by $g_{\mu \nu }\left(x\right)\to g_{\mu \nu }\left(x\right){\mathsf{\Omega }}^2\left(x\right)$ [3] for arbitrary real differentiable $\mathsf{\Omega }\left(x\right)$. Conformal symmetry is defined by invariant action integral $I=\int d^{4}x\sqrt{-g}\mathcal{L}$. For any Riemannian tensor $T\left(x\right)$, $T\left(x\right)\to {\mathsf{\Omega }}^d\left(x\right)T\left(x\right)+\mathcal{R}\left(x\right)$ defines weight $d\left[T\right]$ and residue $\mathcal{R}\left[T\right]$. For a scalar field, $\mathsf{\Phi }\left(x\right)\to \mathsf{\Phi }\left(x\right){\mathsf{\Omega }}^{-1}\left(x\right)$, so that $d\left[\mathsf{\Phi }\right]=-1$. Conformal Lagrangian density $\mathcal{L}$ must have weight $d\left[\mathcal{L}\right]=-4$ and residue $\mathcal{R}\left[\mathcal{L}\right]=0$ up to a 4-divergence [1].

Variational theory for fields in general relativity is a straightforward generalization of classical field theory [24]. Given scalar Lagrangian density $\mathcal{L}={\sum }_a{\mathcal{L}}_a$, action integral $I=\int d^{4}x\sqrt{-g}\mathcal{L}$ is required to be stationary for all differentiable field variations, subject to appropriate boundary conditions. g here is the determinant of metric tensor $g_{\mu \nu }$. Standard conservation laws follow from the variational principle.

Gravitational field equations are determined by metric functional derivative $X^{\mu \nu }=\frac{1}{\sqrt{-g}}\frac{\delta I}{\delta g_{\mu \nu }}$. Any scalar ${\mathcal{L}}_a$ determines energy-momentum tensor ${\mathsf{\Theta }}_a^{\mu \nu }=-2X_a^{\mu \nu }$, evaluated for a solution of the coupled field equations. Generalized Einstein equation ${\sum }_a{X}_a^{\mu \nu }=0$ is expressed as $X_g^{\mu \nu }=\frac{1}{2}{\sum }_{a\ne g}{\mathsf{\Theta }}_a^{\mu \nu }$. Hence summed trace ${\sum }_a{g}_{\mu \nu }{\mathsf{\Theta }}_a^{\mu \nu }$ vanishes for exact field solutions. Given $\delta \mathcal{L}=x^{\mu \nu }\delta g_{\mu \nu }$, metric functional derivative $\frac{1}{\sqrt{-g}}\frac{\delta I}{\delta g_{\mu \nu }}$ is $X^{\mu \nu }=x^{\mu \nu }+\frac{1}{2}\mathcal{L}{g}^{\mu \nu }$, evaluated for a solution of the field equations. Tensor ${\mathsf{\Theta }}^{\mu \nu }=-2X^{\mu \nu }$ is symmetric.

Conformal gravity was introduced by Weyl [3,4] with the prospect of uniting electromagnetic theory and gravitation. This was shown to fail by Einstein, which side-tracked CG for many years. It was reconsidered by Mannheim as a possible source of observed deviations from general relativity [1,2]. The formidable Lagrangian density, a quadratic contraction of the Weyl tensor, simplified by elimination of a 4-divergence [25], reduces to two terms, quadratic, respectively, in the Ricci tensor and Ricci scalar [1]: ${\mathcal{L}}_g=-2{\alpha }_g(R^{\mu \nu }{R}_{\mu \nu }-\frac{1}{3}{R}^2)$. Conformal symmetry fixes the relative coefficient of the two quadratic terms.

In the conformal Higgs model, the unique Lagrangian density for scalar field $\mathsf{\Phi }$ adds a gravitational term [1,8,26] to Higgs $-V$ [23]. The conformal Lagrangian density is ${\left({\partial }_{\mu }\mathsf{\Phi }\right)}^{†}{\partial }^{\mu }\mathsf{\Phi }-\frac{1}{6}R{\mathsf{\Phi }}^{†}\mathsf{\Phi }$, where Ricci scalar $R=g_{\mu \nu }{R}^{\mu \nu }$ is the trace of gravitational Ricci tensor $R^{\mu \nu }$. This is augmented by Higgs $\mathsf{\Delta }{\mathcal{L}}_{\mathsf{\Phi }}=(w^2-\lambda {\mathsf{\Phi }}^{†}\mathsf{\Phi }){\mathsf{\Phi }}^{†}\mathsf{\Phi }$.

Metric tensor $g_{\mu \nu }$ is determined by conformal field equations derived from ${\mathcal{L}}_g+{\mathcal{L}}_{\mathsf{\Phi }}$ [27], driven by energy-momentum tensor ${\mathsf{\Theta }}_m^{\mu \nu }$, where subscript m refers to conventional matter and radiation. The gravitational field equation within halo radius $r_H$ is

$$\begin{array}{c}\hfill X_g^{\mu \nu }+X_{\mathsf{\Phi }}^{\mu \nu }=\frac{1}{2}{\mathsf{\Theta }}_m^{\mu \nu }.\end{array}$$

(1)

Defining mean galactic source density ${\overline{\rho }}_G$ and residual density ${\widehat{\rho }}_G={\rho }_G-{\overline{\rho }}_G$ and assuming ${\mathsf{\Theta }}_m^{\mu \nu }\left(\rho \right)\simeq {\mathsf{\Theta }}_m^{\mu \nu }\left(\overline{\rho }\right)+{\mathsf{\Theta }}_m^{\mu \nu }\left(\widehat{\rho }\right)$, solutions for $r\le r_G$ of the two equations

$$\begin{array}{c}\hfill X_{\mathsf{\Phi }}^{\mu \nu }=\frac{1}{2}{\mathsf{\Theta }}_m^{\mu \nu }\left({\overline{\rho }}_G\right)\\ \hfill X_g^{\mu \nu }=\frac{1}{2}{\mathsf{\Theta }}_m^{\mu \nu }\left({\widehat{\rho }}_G\right)\end{array}$$

(2)

imply a solution to the full equation. The $X_{\mathsf{\Phi }}$ equation has an exact solution for source density $\overline{\rho }$ in uniform, isotropic geometry (FLRW metric tensor) [8,9]. The two field equations are made compatible by imposing appropriate boundary conditions. Fitting constants of integration of source-free $X_g$ at $r_G$ [5] extends a combined solution out to halo radius $r_H$.

Conformal gravity is described in Schwarzschild metric [5]

$$\begin{array}{c}\hfill d{s}_{ES}^2=-B\left(r\right)c^{2}d{t}^2+\frac{d{r}^2}{B\left(r\right)}+r^{2}d{\omega }^2,\end{array}$$

(3)

which defines Schwarzschild potential $B\left(r\right)$, assuming spherical symmetry. The conformal Higgs model is described in the FLRW metric

$$\begin{array}{c}\hfill d{s}_{\mathsf{\Phi }}^2=-c^{2}d{t}^2+a^2\left(t\right)(\frac{d{r}^2}{1-k{r}^2}+r^{2}d{\omega }^2),\end{array}$$

(4)

which defines Friedmann scaling factor $a\left(t\right)$. The Higgs model breaks both gauge and conformal symmetry. Compatibility requires a hybrid metric with $k=0$, such as

$$\begin{array}{c}\hfill d{s}^2=-B\left(r\right)c^{2}d{t}^2+a^2\left(t\right)(\frac{d{r}^2}{B\left(r\right)}+r^{2}d{\omega }^2),\end{array}$$

(5)

4. Depleted Galactic Halo Model

In the $\mathsf{\Lambda }$CDM model, an isolated galaxy is considered to be surrounded by a much larger spherical dark matter halo. What is actually observed is a halo of the gravitational field that deflects photons in gravitational lensing and increases the velocity of orbiting mass particles. Conformal theory [1,2,9], which modifies both Einstein–Hilbert general relativity and the Higgs scalar field model, supports an alternative interpretation of lensing and anomalous rotation as gravitational effects due to depletion of the cosmic background by concentration of diffuse primordial mass into an observed galaxy [27].

In standard theory, Poisson’s equation determines a distribution of dark matter for any unexplained gravitational field. If dark matter interacts only through gravity, the concept of dark matter provides a compact description of observed phenomena, not a falsifiable explanation. Postulated universal conformal symmetry [9] promises a falsifiable alternative free of novel elementary fields.

As a galaxy forms, background matter density ${\rho }_m$ condenses into observed galactic density ${\rho }_g$. Conservation of mass and energy requires total galactic mass M to be missing from a depleted background. Since the primordial density is uniform and isotropic, the depleted background can be modeled by an empty sphere of radius $r_H$, such that $4\pi {\rho }_m{r}_H^3/3=M$. In particular, the integral of ${\rho }_g-{\rho }_m$ must vanish. The resulting gravitational effect, derived from conformal theory, describes a dark-depleted halo. This halo model accounts for the otherwise remarkable fact that galaxies of all shapes are embedded in essentially spherical haloes. Conformal theory relates the acceleration parameter of anomalous galactic rotation to the observed acceleration of Hubble expansion.

Assuming that galactic mass is concentrated within an average radius $r_g$, the ratio of radii $r_H/r_g$ should be large, the cube root of the mass–density ratio ${\rho }_g/{\rho }_m$. Thus, if the latter ratio is $8^3=512$, a galaxy of radius 15 kpc would be accompanied by a halo of radius $8\times 15=120$ kpc. Equivalence of galactic and deleted halo mass resolves the paradox for $\mathsf{\Lambda }$CDM such that, despite any interaction other than gravity, the amount of dark matter inferred for a galactic halo is strongly correlated with the galactic luminosity or baryonic mass [28,29].

5. Conformal Gravity

Given spherical mass-energy density enclosed within $r\le \overline{r}$, the $X_g$ field equation in the ES metric is [5,6]

$$\begin{array}{c}\hfill {\partial }_r^4\left(rB\left(r\right)\right)=rf\left(r\right),\end{array}$$

(6)

where $f\left(r\right)$ is determined by the source energy-momentum tensor.

An exact solution of the tensorial $X_g$ equation, for source-free $r\ge \overline{r}$, derived by [1,5,6], is

$$\begin{array}{c}\hfill y_0\left(r\right)=rB\left(r\right)=-2\beta +\alpha r+\gamma r^2-\kappa r^3.\end{array}$$

(7)

This adds two constants of integration to the classical external potential: nonclassical radial acceleration $\gamma$ and halo cutoff parameter $\kappa$ [1,27]. Derivative functions $y_i\left(r\right)={\partial }_r^i\left(rB\left(r\right)\right), 0\le i\le 3$ satisfy differential equations

$$\begin{array}{c}\hfill {\partial }_r{y}_i=y_{i+1}, 0\le i\le 2,\\ \hfill {\partial }_r{y}_3=rf\left(r\right).\end{array}$$

(8)

Independent constants $a_i=y_i\left(0\right)$ determine coefficients $\beta ,\alpha ,\gamma ,\kappa$ such that at endpoint $\overline{r}$

$$\begin{array}{c}\hfill y_0\left(\overline{r}\right)=-2\beta +\alpha \overline{r}+\gamma {\overline{r}}^2-\kappa {\overline{r}}^3,\\ \hfill y_1\left(\overline{r}\right)=\alpha +2\gamma \overline{r}-3\kappa {\overline{r}}^2,\\ \hfill y_2\left(\overline{r}\right)=2\gamma -6\kappa \overline{r},\\ \hfill y_3\left(\overline{r}\right)=-6\kappa .\end{array}$$

(9)

To avoid a singularity at the origin, $a_0=0$. $\gamma$ and $\kappa$ must be determined consistently with the $X_{\mathsf{\Phi }}$ equation. Specified values can be fitted by adjusting $a_1,a_2,a_3$, subject to $a_0=0,\alpha =1$ [1]. The solution for $B\left(r\right)$ remains regular if $\gamma /\beta \simeq 0$. If Ricci scalar $R(r\to 0)$ is unconstrained, a particular solution for $r\le \overline{r}$ is given with specified constants $a_0=0,\alpha ,\gamma ,\kappa$ [30]:

$$\begin{array}{c}\hfill rB\left(r\right)=y_0\left(r\right)=\alpha r-\frac{1}{6}{\int }_0^r{q}^{4}fdq-\frac{1}{2}r{\int }_r^{\overline{r}}{q}^{3}fdq\\ \hfill +\gamma r^2+\frac{1}{2}{r}^2{\int }_r^{\overline{r}}{q}^{2}fdq-\kappa r^3-\frac{1}{6}{r}^3{\int }_r^{\overline{r}}qfdq.\end{array}$$

(10)

Integrated parameter $2\beta =\frac{1}{6}{\int }_0^{\overline{r}}{q}^{4}fdq$. This particular solution differs from [1,5], replacing $-\frac{1}{2}{r}^2{\int }_0^r{q}^{2}fdq$ by $\gamma r^2+\frac{1}{2}{r}^2{\int }_r^{\overline{r}}{q}^{2}fdq$ [30]. Constant $\gamma$ is determined independently of the CG equation. At the galactic center, $r\to 0$ implies regular $B\left(r\right)$ but singular Ricci $R\left(r\right)$ for this solution. This is consistent with the development of a massive black hole, recently observed for many galaxies.

For uniform density $\overline{\rho }$ the Weyl tensor vanishes, so that $X_g^{\mu \nu }\equiv 0$ for a uniform, isotropic cosmos. Schwarzschild potential,

$$\begin{array}{c}\hfill B\left(r\right)=-2\beta /r+\alpha +\gamma r-\kappa r^2,\end{array}$$

(11)

is exact for a source-free region outside a spherical mass-energy source density [5,6], in particular, within an empty galactic halo.

For a test particle in a stable exterior circular orbit with velocity v, the centripetal acceleration is $a=v^2\left(r\right)/r=\frac{1}{2}{B}^{\prime }\left(r\right)c^2$. Newtonian $B\left(r\right)=1-2\beta /r$, where $\beta =GM/c^2$, so that classical acceleration $a_N=\beta c^2/r^2=GM/r^2$. CG adds nonclassical $\mathsf{\Delta }a=\frac{1}{2}{c}^2\gamma -c^2\kappa r$, a universal constant when halo cutoff $2\kappa r/\gamma \simeq r/r_H$ can be neglected [30]. Orbital velocity squared is the sum of $v^2(a_N;r)$ and $v^2(\mathsf{\Delta }a;r)$, which cross with equal and opposite slope at some $r=r_{TF}$, if $2\kappa r/\gamma$ can be neglected. This defines a flat range of $v\left(r\right)$ centered at stationary point $r_{TF}$, without constraining behavior at large r.

MOND [10,11,21] modifies the Newtonian force law for acceleration below an empirical scale $a_0$. Using $y=a_N/a_0$ as an independent variable, for assumed universal constant $a_0\simeq {10}^{-10}$ m/s${}^2$, MOND postulates an interpolation function $\nu \left(y\right)$ such that observed radial acceleration $a=f\left(a_N\right)=a_N\nu \left(y\right)$. A flat velocity range approached asymptotically requires $a^2\to a_N{a}_0$ as $a_N\to 0$. For $a_N\ll a_0$, MOND $v^4=a^2{r}^2\to GM{a}_0$, the empirical baryonic Tully–Fisher relationship [12,13,14,15].

In conformal gravity (CG), centripetal acceleration $a=v^2/r$ determines the exterior orbital velocity $v^2/c^2=ra/c^2=\beta /r+\frac{1}{2}\gamma r-\kappa r^2$, compared with asymptotic $r{a}_N/c^2=\beta /r$. Assuming Newtonian constant $\beta ={\beta }_N$ and neglecting $2\kappa r/\gamma$, the slope of $v^2\left(r\right)$ vanishes at $r_{TF}^2=2\beta /\gamma$. This implies that $v^4\left(r_{TF}\right)/c^4={(\beta /r_{TF}+\frac{1}{2}\gamma r_{TF})}^2=2\beta \gamma$ [15,31]. This is the Tully–Fisher relationship, exactly at stationary point $r_{TF}$ of the $v\left(r\right)$ function. The implied MOND acceleration constant is $a_0=2\gamma c^2=1.14\times {10}^{-10}$ m/s${}^2$ [30].

6. The Conformal Higgs Model

The Newton-Einstein model of galactic growth and interaction is substantially altered by conformal theory [8,27]. A gravitational halo is treated as a natural consequence of galaxy formation. Condensation of matter from the primordial uniform mass-energy distribution leaves a depleted sphere that has an explicit gravitational effect. The implied subtracted mass, which integrates into the total galactic mass, cannot be ignored. If this mass were simply removed, the analogy to vacancy scattering in solids implies a lensing effect. A background geodesic is no longer a geodesic in the empty sphere [27].

The conformal gravitational field equation is

$$\begin{array}{c}\hfill X_g^{\mu \nu }+X_{\mathsf{\Phi }}^{\mu \nu }=\frac{1}{2}{\mathsf{\Theta }}_m^{\mu \nu },\end{array}$$

(12)

where index m refers to matter and radiation. An exact solution inside the depleted halo radius is given by

$$\begin{array}{c}\hfill X_g^{\mu \nu }=\frac{1}{2}{\mathsf{\Theta }}_m^{\mu \nu }\left({\rho }_g\right),X_{\mathsf{\Phi }}^{\mu \nu }=0, r\le r_H.\end{array}$$

(13)

The exact source-free solution [5] of the $X_g$ equation is valid in the external halo, $r_g\le r\le r_H$, because ${\rho }_g$ vanishes. Constants of integration fitted at $r_g$ and $r_H$.

In uniform, isotropic geometry with uniform mass/energy density $\overline{\rho }$, ${\mathcal{L}}_{\mathsf{\Phi }}$ implies a modified Friedmann equation [8,9,32] for cosmic distance scale factor $a\left(t\right)$, in FLRW metric

$$\begin{array}{c}\hfill d{s}_{\mathsf{\Phi }}^2=-c^{2}d{t}^2+a^2\left(t\right)(\frac{d{r}^2}{1-k{r}^2}+r^{2}d{\omega }^2),\end{array}$$

(14)

which determines Friedmann scaling factor $a\left(t\right)$. With $a\left(t_0\right)=1$ at present time $t_0$:

$$\begin{array}{c}\hfill \frac{{\dot{a}}^2}{a^2}+\frac{k}{a^2}-\frac{\ddot{a}}{a}=\frac{2}{3}(\overline{\mathsf{\Lambda }}+\overline{\tau }{c}^2\overline{\rho }\left(t\right)).\end{array}$$

(15)

Consistency with the $X_g$ equation requires $k=0$. $\overline{\mathsf{\Lambda }}=\frac{3}{2}{w}^2\ge 0$ and $\overline{\tau }\sim -3/{\varphi }_0^2\le 0$ are determined by parameters of the Higgs model [8]. For uniform $\rho =0$ within the halo radius, this solves the second of Equation (13).

Dividing the modified equation by ${(\dot{a}/a)}^2$ determines dimensionless sum rule ${\mathsf{\Omega }}_m+{\mathsf{\Omega }}_{\mathsf{\Lambda }}+{\mathsf{\Omega }}_k+{\mathsf{\Omega }}_q=1$. Scale factor $a\left(t\right)$ determines dimensionless Friedmann acceleration weight ${\mathsf{\Omega }}_q\left(t\right)=\frac{\ddot{a}a}{{\dot{a}}^2}$ [8]. Radiation energy density is included in ${\mathsf{\Omega }}_m$ here. The dimensionless weights are ${\mathsf{\Omega }}_m\left(t\right)$ for mass–density, ${\mathsf{\Omega }}_k\left(t\right)$ for curvature, and ${\mathsf{\Omega }}_{\mathsf{\Lambda }}\left(t\right)$ for dark energy, augmented by acceleration weight ${\mathsf{\Omega }}_q\left(t\right)$. Galactic rotation velocities, observed at relatively small redshifts, determine radial acceleration values, which can be compared with acceleration weights inferred from Hubble expansion data. Omitting ${\mathsf{\Omega }}_m$ completely, with $k=0$, conformal sum rule ${\mathsf{\Omega }}_{\mathsf{\Lambda }}\left(t\right)+{\mathsf{\Omega }}_q\left(t\right)=1$ fits observed data accurately for redshifts $z\le 1$ (7.33 Gyr) [8,26,30]. This eliminates any need for dark matter to explain Hubble expansion.

The $X_{\mathsf{\Phi }}$ equation includes dark energy, present regardless of density $\rho$. This produces a centrifugal acceleration of background Hubble expansion [8,9], which must be subtracted to compute observable radial acceleration. Observed geodesics, whose bending determines the extragalactic centripetal acceleration responsible for lensing and anomalous orbital rotation velocities, are defined relative to the cosmic background.

Defining $\frac{\dot{a}}{a}=h\left(t\right)H_0$, for Hubble constant $H_0$, it is convenient to use Hubble units, such that $c=\hslash =1$, and dimensionless $a\left(t_0\right)=1$, $h\left(t_0\right)=1$. The units for frequency, energy, and acceleration are $H_0,\hslash H_0,c{H}_0$, respectively. Setting ${\mathsf{\Omega }}_k={\mathsf{\Omega }}_m=0$, for $\alpha =w^2={\mathsf{\Omega }}_{\mathsf{\Lambda }}\left(t_0\right)$, the CHM Friedmann equation [8,9] is

$$\begin{array}{c}\hfill \frac{{\dot{a}}^2}{a^2}-\frac{\ddot{a}}{a}=\frac{2}{3}\overline{\mathsf{\Lambda }}=\alpha .\end{array}$$

(16)

In Hubble units, Equation (16) reduces to $\frac{d}{dt}h\left(t\right)=-\alpha$. The explicit solution for $t\le t_0$ is

$$\begin{array}{c}\hfill h\left(t\right)=\frac{\dot{a}}{a}\left(t\right)=\frac{d}{dt}lna\left(t\right)=1+\alpha (t_0-t),\\ \hfill lna\left(t\right)=-(t_0-t)-\frac{1}{2}\alpha {(t_0-t)}^2,\\ \hfill a\left(t\right)=exp[-(t_0-t)-\frac{1}{2}\alpha {(t_0-t)}^2].\end{array}$$

(17)

From the definition of redshift $z\left(t\right)$:

$1+z\left(t\right)=\frac{1}{a\left(t\right)}=exp[(t_0-t)+\frac{1}{2}\alpha {(t_0-t)}^2]$,

${(t_0-t)}^2+\frac{2}{\alpha }(t_0-t)=\frac{2}{\alpha }ln(1+z)$.

In Hubble units with $\frac{\dot{a}}{a}\left(t_0\right)=1$, this implies

$$\begin{array}{c}\hfill t_0-t\left(z\right)=(\sqrt{2\alpha ln(1+z)+1}-1)/\alpha ,\\ \hfill \frac{dt}{dz}=\frac{-1}{(1+z)\sqrt{2\alpha ln(1+z)+1}}.\end{array}$$

(18)

If $\alpha =0.732$ and $z=1$, $(t_0-t)/H_0=7.33 \mathrm{Gyr}$, for $H_0=67.8\pm 0.9$ km/s/Mpc $=2.197\times {10}^{-18}$/s [33].

Neglecting both curvature weight ${\mathsf{\Omega }}_k$ and cosmic mass/energy weight ${\mathsf{\Omega }}_m$, conformal sum rule ${\mathsf{\Omega }}_{\mathsf{\Lambda }}+{\mathsf{\Omega }}_q=1$ holds for acceleration weight ${\mathsf{\Omega }}_q=\frac{\ddot{a}a}{{\dot{a}}^2}$. For ${\mathsf{\Omega }}_k=0$, luminosity distance $d_L\left(z\right)=(1+z)\chi \left(z\right)$ [26], where

$$\begin{array}{c}\hfill \chi \left(z\right)={\int }_{t\left(z\right)}^{t_0}\frac{dt}{a\left(t\right)}={\int }_z^{0}d\overline{z}(1+\overline{z})\frac{dt}{dz}\left(\overline{z}\right)=\\ \hfill {\int }_0^z\frac{d\overline{z}}{\sqrt{2\alpha ln(1+\overline{z})+1}}.\end{array}$$

(19)

Evaluated for $\alpha ={\mathsf{\Omega }}_{\mathsf{\Lambda }}\left(t_0\right)=0.732$, the fit to scaled luminosity distances $H_0{d}_L/c$ from Hubble expansion data [8,34] is given in

Table 1. Scaled luminosity distance fit to Hubble data.

			Theory	Observed
z	${\mathsf{\Omega }}_{\mathsf{\Lambda }}$	${\mathsf{\Omega }}_{\mathit{q}}$	${\mathit{H}}_{\mathbf{0}}{\mathit{d}}_{\mathit{L}}/\mathit{c}$ Equation (19)	${\mathit{H}}_{\mathbf{0}}{\mathit{d}}_{\mathit{L}}/\mathit{c}$ [34]
0.0	0.732	0.268	0.0000	0.0000
0.2	0.578	0.422	0.2254	0.2265
0.4	0.490	0.510	0.5013	0.5039
0.6	0.434	0.566	0.8267	0.8297
0.8	0.393	0.607	1.2003	1.2026
1.0	0.363	0.637	1.6209	1.6216

.

Observed redshifts have been fitted to an analytic function [34] with statistical accuracy comparable to the best standard $\mathsf{\Lambda }$CDM fit, with ${\mathsf{\Omega }}_m=0$. Table 1 compares CHM $d_L\left(z\right)$ to this Mannheim function. Because $\overline{\tau }$ is negative [1,8], cosmic acceleration ${\mathsf{\Omega }}_q$ remains positive (centrifugal) back to the earliest time [8].

Conformal Friedmann Equation (15) [8,26] determines cosmic acceleration weight ${\mathsf{\Omega }}_q$. With both weight parameters ${\mathsf{\Omega }}_k$ and ${\mathsf{\Omega }}_m$ set to zero, Equation (15) fits scaled Hubble function $h\left(t\right)=H\left(t\right)/H_0$ for redshifts $z\le 1$, Table 1, as accurately as standard $\mathsf{\Lambda }$CDM, with only one free constant. This determines Friedmann weights, at the present time $t_0$, ${\mathsf{\Omega }}_{\mathsf{\Lambda }}=0.732,{\mathsf{\Omega }}_q=0.268$ [8]. Hubble constant $H\left(t_0\right)=H_0=2.197\times {10}^{-18}/s$ [33] is independent of these data. For uniform primordial energy-momentum source density ${\rho }_m$, the dimensionless sum rule [8] with ${\mathsf{\Omega }}_k=0$ determines ${\mathsf{\Omega }}_q\left({\rho }_m\right)=1-{\mathsf{\Omega }}_{\mathsf{\Lambda }}-{\mathsf{\Omega }}_m$ in the cosmic background, and ${\mathsf{\Omega }}_q\left(0\right)=1-{\mathsf{\Omega }}_{\mathsf{\Lambda }}$ in the depleted halo [27].

Schwarzschild $B\left(r\right)$ parameter $\gamma >0$, independent of galactic mass and structure, implies centripetal acceleration due to an isotropic cosmological source [31]. The parametrized gravitational field forms a spherical halo [9]. The depleted halo model removes a particular conceptual problem in fitting $B\left(r\right)$ parameters $\gamma ,\kappa$ to galactic rotation data [1,31,35]. In empirical parameter $\gamma ={\gamma }^{*}{N}^{*}+{\gamma }_0$, ${\gamma }_0$ does not depend on galactic mass, so it must be due to the surrounding cosmos [31]. Since the interior term, coefficient ${\gamma }^{*}{N}^{*}$, is centripetal, one might expect the term in ${\gamma }_0$ to be centrifugal, describing an attraction to exterior sources. However, coefficient $\gamma$, derived here for a depleted halo, determines net centripetal acceleration, in agreement with observation [30]. Recent galactic rotation results for galaxies with independently measured mass [16] imply that $\gamma$ cannot depend on galactic mass [36].

The conformal Friedmann cosmic evolution equation implies dimensionless cosmic acceleration parameters ${\mathsf{\Omega }}_q\left(\rho \right)$ [27] which are locally constant but differ across the halo boundary $r_H$. Smooth evolution of the cosmos implies observable particle acceleration $\gamma$ within $r_H$ proportional to ${\mathsf{\Omega }}_q\left(in\right)-{\mathsf{\Omega }}_q\left(out\right)={\mathsf{\Omega }}_q\left(0\right)-{\mathsf{\Omega }}_q\left({\rho }_m\right)$. Uniform cosmological ${\rho }_m$ implies constant $\gamma$ for $r\le r_H$, independent of galactic mass [30]. This result is consistent with recent observations of galactic rotational velocities for galaxies with directly measured mass [16,36].

From the CHM, observed nonclassical gravitational acceleration $\frac{1}{2}\gamma c^2$ in the halo is proportional to $\mathsf{\Delta }{\mathsf{\Omega }}_q={\mathsf{\Omega }}_q\left(0\right)-{\mathsf{\Omega }}_q\left({\rho }_m\right)={\mathsf{\Omega }}_m\left({\rho }_m\right)$ [27], where, given ${\rho }_m$ and $H_0$, ${\mathsf{\Omega }}_m\left({\rho }_m\right)=\frac{2}{3}\frac{\overline{\tau }{c}^2{\rho }_m}{H_0^2}$ [8]. Positive ${\rho }_m$ implies ${\mathsf{\Omega }}_m<0$ because coefficient $\overline{\tau }<0$ [1,8]. Therefore, the depleted halo model determines constant $\gamma$ from uniform, universal cosmic baryonic mass–density ${\rho }_m/c^2$, which includes radiation energy density here.

${\mathsf{\Omega }}_m<0$ implies centripetal acceleration, converted from Hubble units, $\frac{1}{2}\gamma c^2=-c{H}_0{\mathsf{\Omega }}_m\left({\rho }_m\right)$ [27]. Hence, $\mathsf{\Delta }{\mathsf{\Omega }}_q={\mathsf{\Omega }}_m<0$ is consistent with nonclassical centripetal acceleration, confirmed by inward deflection of photon geodesics observed in gravitational lensing [27]. This logic is equivalent to requiring radial acceleration to be continuous across halo boundary $r_H$:

$$\begin{array}{c}\hfill \frac{1}{2}\gamma c^2-c{H}_0{\mathsf{\Omega }}_q\left(0\right)=-c{H}_0{\mathsf{\Omega }}_q\left({\rho }_m\right).\end{array}$$

(20)

Signs here follow from the definition of ${\mathsf{\Omega }}_q$ as centrifugal acceleration weight.

For a single spherical solar mass isolated in a galactic halo, mean internal mass–density ${\overline{\rho }}_{\odot }$ within $r_{\odot }$ determines an exact solution of the conformal Higgs gravitational equation, giving internal acceleration ${\mathsf{\Omega }}_q\left({\overline{\rho }}_{\odot }\right)$. Given $\gamma$ outside $r_{\odot }$, continuous acceleration across boundary $r_{\odot }$,

$$\begin{array}{c}\hfill \frac{1}{2}{\gamma }_{\odot ,in}{c}^2-c{H}_0{\mathsf{\Omega }}_q\left({\overline{\rho }}_{\odot }\right)=\frac{1}{2}\gamma c^2-c{H}_0{\mathsf{\Omega }}_q\left(0\right),\end{array}$$

(21)

determines constant ${\gamma }_{\odot ,in}$ valid inside $r_{\odot }$. ${\gamma }_{\odot ,in}$ is determined by local mean source density ${\overline{\rho }}_{\odot }$. $\gamma$ in the halo is not changed. Its value is a constant of integration that cannot vary in the source-free halo [27,30]. Hence, there is no way to determine a mass-dependent increment to $\gamma$. This replaces the usually assumed ${\gamma }_0+N^{*}{\gamma }^{*}$ by ${\gamma }_H$ determined at halo boundary $r_H$.

7. Values of Relevant Parameters

Anomalous rotation velocities for 138 galaxies are fitted using only four universal CG parameters ${\beta }^{*},{\gamma }^{*},{\gamma }_0,\kappa$ [1,2,35,37,38] such that $\beta =N^{*}{\beta }^{*}=GM/c^2,\gamma ={\gamma }_0+N^{*}{\gamma }^{*}$. $N^{*}$ is galactic baryonic mass M in solar mass units. Inferred parameter values [1,37],

$$\begin{array}{c}\hfill {\beta }^{*}=1.475\times {10}^3 \mathrm{m},{\gamma }_0=3.06\times {10}^{-28}/\mathrm{m},\\ \hfill {\gamma }^{*}=5.42\times {10}^{-39}/\mathrm{m},\kappa =9.54\times {10}^{-50}/{\mathrm{m}}^2,\end{array}$$

(22)

fit conformal gravity to galactic rotation velocities. The observed radial acceleration relationship (RAR) [16] is consistent with the CHM conclusion that ${\gamma }^{*}$ must vanish. Then $\gamma$ is a universal constant [36]. Using data for the Milky Way galaxy, $\gamma ={\gamma }_0+N^{*}{\gamma }^{*}=6.35\times {10}^{-28}$/m [38,39].

$\zeta >0$ for computed $R\left(t\right)$ [8] implies $\lambda <0$. $\hslash \varphi \left(t_0\right)=174 \mathrm{GeV}$ [40]$=1.203\times {10}^{44}\hslash H_0$ in Hubble units. For ${\mathsf{\Omega }}_m=0$, $\zeta \left(t_0\right)=2{\mathsf{\Omega }}_q\left(t_0\right)=0.536$. For $\lambda \left(t\right)=\zeta /(-2{\varphi }^2)$ and $\varphi \left(t_0\right)={\varphi }_0$, dimensionless $\lambda \left(t_0\right)=-0.185\times {10}^{-88}$.

It is widely assumed that negative Higgs $\lambda$ would imply an unstable physical vacuum, but the present analysis does not support this conclusion. The conformal Higgs scalar field does not have a well-defined mass, instead inducing dynamical $w^2$, which acts as a cosmological constant determining dark energy. Nearly constant $\dot{\varphi }/\varphi$ for redshifts $z\le 1$ is shown here to imply Higgs $\lambda <0$, consistent with a dynamical origin. Negative empirical $\lambda$ implies finite but ‘tachyonic’ mass for a scalar field fluctuation. This does not support the conventional concept of a massive Higgs particle [41].

Scalar field $ZZ=g_{\mu \nu }{Z}^{\mu *}{Z}^{\nu }$ interacts with neutral scalar field $WW=g_{\mu \nu }{W}_{-}^{\mu }{W}_{+}^{\nu }$ through the exchange of quarks and leptons. There is no contradiction in treating resulting scalar diboson $W_2$ as an independent field or particle, in analogy to atoms, molecules, and nuclei. Neutral $W_2$ dresses scalar field $\mathsf{\Phi }$ to produce biquadratic Lagrangian term $\lambda {\left({\mathsf{\Phi }}^{†}\mathsf{\Phi }\right)}^2$ [41]. If the mass of $W_2$ is $125 \mathrm{GeV}$, and the conformal Higgs $\lambda$ assumes its empirical value. This implies an alternative identification of the recently observed LHC resonance [42,43,44]. The interacting bare fields produce relatively stable $W_2=WWcos{\theta }_x+ZZsin{\theta }_x$ and complementary resonance $Z_2=-WWsin{\theta }_x+ZZcos{\theta }_x$ [41]. $W_2$ can dress the bare $\mathsf{\Phi }$ field while maintaining charge neutrality. Assumed $W_2$ mass 125 GeV implies $\lambda =-0.455\times {10}^{-88}$ [41], consistent with empirical value $\lambda \left(t_0\right)\simeq -{10}^{-88}$.

Parameters $w^2$ and $\lambda$ are defined as positive constants in Higgs [23,45]

$$\begin{array}{c}\hfill \mathsf{\Delta }{\mathcal{L}}_{\mathsf{\Phi }}=-V\left({\mathsf{\Phi }}^{†}\mathsf{\Phi }\right)=w^2{\mathsf{\Phi }}^{†}\mathsf{\Phi }-\lambda {\left({\mathsf{\Phi }}^{†}\mathsf{\Phi }\right)}^2.\end{array}$$

(23)

Stationary action implies constant finite ${\mathsf{\Phi }}^{†}\mathsf{\Phi }={\varphi }_0^2=w^2/2\lambda$, where $\hslash \mathsf{\Phi }$ and $\hslash w$ are energies. The conformal Higgs model [8] acquires an additional term in $\mathsf{\Delta }{\mathcal{L}}_{\mathsf{\Phi }}$: $-\frac{1}{6}R{\mathsf{\Phi }}^{†}\mathsf{\Phi }$ [1] for gravitational Ricci scalar $R=g_{\mu \nu }{R}^{\mu \nu }$. The modified Lagrangian, defined for neutral gauge field $Z_{\mu }$ [26], determines $w^2$, which becomes a cosmological constant in the conformal Friedmann cosmic evolution equation [8]. Lagrangian term $w^2{\mathsf{\Phi }}^{†}\mathsf{\Phi }$ is due to induced neutral $Z_{\mu }$ field amplitude [26], which dresses the bare scalar field. Finite $w^2$ breaks conformal symmetry but does not determine the Higgs mass.

The conformal scalar field equation including parametrized $\mathsf{\Delta }{\mathcal{L}}_{\mathsf{\Phi }}$ is [1,8]

$$\begin{array}{c}\hfill \frac{1}{\sqrt{-g}}{\partial }_{\mu }\left(\sqrt{-g}{\partial }^{\mu }\mathsf{\Phi }\right)=(-\frac{1}{6}R+w^2-2\lambda {\mathsf{\Phi }}^{†}\mathsf{\Phi })\mathsf{\Phi }.\end{array}$$

(24)

Ricci scalar R introduces gravitational effects. Only real-valued solution $\varphi \left(t\right)$ is relevant in uniform, isotropic geometry. Defining $V\left(\varphi \right)=\frac{1}{2}(\zeta +\lambda {\varphi }^2){\varphi }^2$, where $\zeta \left(t\right)=\frac{1}{6}R-w^2$, the field equation is

$$\begin{array}{c}\hfill \frac{\ddot{\varphi }}{\varphi }+3\frac{\dot{a}}{a}\frac{\dot{\varphi }}{\varphi }=-(\zeta \left(t\right)+2\lambda {\varphi }^2).\end{array}$$

(25)

Omitting R and assuming constant $\lambda >0$ and $w^2$, Higgs solution ${\varphi }_0^2=w^2/2\lambda$ [45] is exact. All-time derivatives drop out. In the conformal scalar field equation, the cosmological time dependence of Ricci scalar $R\left(t\right)$, determined by the CHM Friedmann cosmic evolution equation introduces nonvanishing time derivatives and implies $\lambda <0$ [8]. An exact solution for $\varphi \left(t\right)$ should include the time derivative terms in $\zeta \left(t\right)$. Neglecting them here limits the argument to redshift $z\le 1$.

Ricci scalar $R\left(t\right)$ drops out of the scalar field equations if time derivatives are neglected. This justifies an estimate of Higgs parameter $w^2$ [8]. Dressing scalar field $\mathsf{\Phi }$ by neutral gauge field $Z^{\mu }$ produces Lagrangian term $w^2{\mathsf{\Phi }}^{†}\mathsf{\Phi }$. Defining $\mathsf{\Delta }{I}_{\mathsf{\Phi }}=\int d^{4}x\sqrt{-g}\mathsf{\Delta }{\mathcal{L}}_{\mathsf{\Phi }}$ from Equation (23), the parametrized effective potential term in the scalar field equation is given by

$$\begin{array}{c}\hfill \frac{1}{\sqrt{-g}}\frac{\delta \mathsf{\Delta }{I}_{\mathsf{\Phi }}}{\delta {\mathsf{\Phi }}^{†}}=(w^2-2\lambda {\mathsf{\Phi }}^{†}\mathsf{\Phi })\mathsf{\Phi }.\end{array}$$

(26)

Gauge invariance replaces bare derivative ${\partial }_{\mu }$ by gauge covariant derivative [23]

$$\begin{array}{c}\hfill D_{\mu }={\partial }_{\mu }-\frac{i}{2}{g}_z{Z}_{\mu }.\end{array}$$

(27)

This retains ${\mathcal{L}}_Z$ in terms of $Z_{\mu \nu }={\partial }_{\mu }{Z}_{\nu }-{\partial }_{\nu }{Z}_{\mu }$ and augments conformal

$$\begin{array}{c}\hfill {\mathcal{L}}_{\mathsf{\Phi }}^0={\left({\partial }_{\mu }\mathsf{\Phi }\right)}^{†}{\partial }^{\mu }\mathsf{\Phi }-\frac{1}{6}R{\mathsf{\Phi }}^{†}\mathsf{\Phi }\end{array}$$

(28)

by coupling term

$$\begin{array}{c}\hfill \mathsf{\Delta }\mathcal{L}={\left(D_{\mu }\mathsf{\Phi }\right)}^{†}{D}^{\mu }\mathsf{\Phi }-{\left({\partial }_{\mu }\mathsf{\Phi }\right)}^{†}{\partial }^{\mu }\mathsf{\Phi }=\\ \hfill \frac{i}{2}{g}_z{Z}_{\mu }^{*}{\mathsf{\Phi }}^{†}{\partial }^{\mu }\mathsf{\Phi }-\frac{i}{2}{g}_z{Z}^{\mu }{\left({\partial }_{\mu }\mathsf{\Phi }\right)}^{†}\mathsf{\Phi }+\frac{1}{4}{g}_z^2{\mathsf{\Phi }}^{†}{Z}_{\mu }^{*}{Z}^{\mu }\mathsf{\Phi }.\end{array}$$

(29)

Parametrized for a generic complex vector field [23],

$$\begin{array}{c}\hfill \mathsf{\Delta }{\mathcal{L}}_Z=\frac{1}{2}{m}_Z^2{Z}_{\mu }^{*}{Z}^{\mu }-\frac{1}{2}(Z_{\mu }^{*}{J}_Z^{\mu }+Z^{\mu }{J}_{Z\mu }^{*}),\end{array}$$

(30)

given mass parameter $m_Z$ and source current density $J_Z^{\mu }$. The field equation for parametrized $Z^{\mu }$ is [23]

$$\begin{array}{c}\hfill {\partial }_{\nu }{Z}^{\mu \nu }=\frac{2}{\sqrt{-g}}\frac{\delta \mathsf{\Delta }{I}_Z}{\delta Z_{\mu }^{*}}=m_Z^2{Z}^{\mu }-J_Z^{\mu }.\end{array}$$

(31)

$\mathsf{\Delta }\mathcal{L}$ determines parameters for field $Z^{\mu }$:

$$\begin{array}{c}\hfill \frac{2}{\sqrt{-g}}\frac{\delta \mathsf{\Delta }I}{\delta Z_{\mu }^{*}}=\frac{1}{2}{g}_z^2{\mathsf{\Phi }}^{†}\mathsf{\Phi }{Z}^{\mu }+i{g}_z{\mathsf{\Phi }}^{†}{\partial }^{\mu }\mathsf{\Phi }.\end{array}$$

(32)

This implies Higgs mass formula $m_Z^2=\frac{1}{2}{g}_z^2{\mathsf{\Phi }}^{†}\mathsf{\Phi }$ and field source density $J_Z^{\mu }=-i{g}_z{\mathsf{\Phi }}^{†}{\partial }^{\mu }\mathsf{\Phi }$. Using $\mathsf{\Delta }\mathcal{L}$ derived from the covariant derivative,

$$\begin{array}{c}\hfill \frac{1}{\sqrt{-g}}\frac{\delta \mathsf{\Delta }{I}_{\mathsf{\Phi }}}{\delta {\mathsf{\Phi }}^{†}}=\frac{1}{4}{g}_z^2{Z}_{\mu }^{*}{Z}^{\mu }\mathsf{\Phi }+\frac{i}{2}{g}_z(Z_{\mu }^{*}+Z_{\mu }){\partial }^{\mu }\mathsf{\Phi }.\end{array}$$

(33)

Comparison with Equation (26) implies $w^2=\frac{1}{4}{g}_z^2{Z}_{\mu }^{*}{Z}^{\mu }$. Neglecting derivatives of $Z^{\mu }$, Equation (31) reduces to

$$\begin{array}{c}\hfill Z^{\mu }=J_Z^{\mu }/m_Z^2=-\frac{2i}{g_z}{\mathsf{\Phi }}^{†}{\partial }^{\mu }\mathsf{\Phi }/{\mathsf{\Phi }}^{†}\mathsf{\Phi }.\end{array}$$

(34)

Then $|Z^0{|}^2=\frac{4}{g_z^2}{\left(\frac{\dot{\varphi }}{\varphi }\right)}^2$, so that the scalar field equation implies nonvanishing $w^2=\frac{1}{4}{g}_z^2{\left|Z^0\right|}^2={\left(\frac{\dot{\varphi }}{\varphi }\right)}^2$. Implied pure imaginary $Z^{\mu }$ does not affect parameter $\lambda$.

8. Alternative Models of Observed Data

Numerous dark matter studies of galactic halo gravitation depend on models with core radius $r_0$ and central density ${\rho }_0$. Central surface density product ${\rho }_0{r}_0$ is found to be nearly a universal constant of order $100M_{\odot }p{c}^{-2}$ for a large range of galaxies [18,19,20]. Conformal theory implies nonclassical centripetal acceleration $\mathsf{\Delta }a$, for Newtonian $a_N$ due to observable baryonic matter. Neglecting the halo cutoff for $r\ll r_H$, conformal $\mathsf{\Delta }a$ is constant over the halo, and $a=a_N+\mathsf{\Delta }a$ is a universal function, consistent with a recent study of galaxies with independently measured mass [16]. An equivalent dark matter source would be a pure cusp distribution with a cutoff parameter determined by the halo boundary radius. This is shown here to imply universal central surface density for any dark matter core model.

CG implies $\mathsf{\Delta }a=\frac{1}{2}\gamma c^2(1-r/r_H)$ in a depleted halo. For $r\ll r_H$, CG acceleration constant $\gamma$ [1,36] predicts $\mathsf{\Delta }a=\frac{1}{2}\gamma c^2$ [27,36] for all DM models.

Uniform constant $\mathsf{\Delta }a$ puts a severe constraint on any DM model. The source density must be of the form $\xi /r$, a pure radial cusp [30,36], where constant $\xi =\mathsf{\Delta }a/2\pi G=0.06797$ kg/m${}^2=32.535M_{\odot }/p{c}^2$ is determined by CG parameter $\gamma$, with CODATA $G=6.67384\times {10}^{-11}$ m${}^3$s${}^{-2}$kg${}^{-1}$ [46].

A DM galactic model equivalent to conformal theory would imply uniform DM radial acceleration $\mathsf{\Delta }a=2\pi G\xi$, attributed to radial DM density $\xi /r$ for universal constant $\xi$, modified at large galactic radius by a halo cutoff function. Enclosed mass $M_r=2\pi \xi r^2$ implies $r\mathsf{\Delta }a/c^2=G{M}_r/r$. DM models avoid a distribution cusp by assuming finite central core density.

The distribution of source matter within a sphere cannot affect asymptotic gravitational acceleration. For arbitrary $r_0$, asymptotic radial acceleration is unchanged if mass within $r_0$ is redistributed to uniform density $\rho \left(r\right)$ within a sphere of this radius. Conformal density $\xi /r$ implies mass $M_0=2\pi \xi r_0^2$ in volume $V_0=\frac{4\pi }{3}{r}_0^3$. For a DM spherical model core that replaces a central cusp density, conformal theory implies constant $\rho \left(r_0\right)r_0=r_0{M}_0/V_0=3\xi /2$. For assumed PI core DM density [18] $\rho \left(r\right)={\rho }_0{r}_0^2/(r^2+r_0^2)$, central ${\rho }_0=2\rho \left(r_0\right)$. Hence, for a PI core, ${\rho }_0{r}_0=3\xi =\frac{3\mathsf{\Delta }a}{2\pi G}=0.204\times {10}^{-2}$ kg/m${}^2=97.6M_{\odot }p{c}^{-2}$, independent of $r_0$.

A radial acceleration relationship RAR [16] is postulated by MOND [10]. Paradigms $\mathsf{\Lambda }$CDM and CG/CHM both represent galactic radial acceleration by $a_N+\mathsf{\Delta }a$, for baryonic Newtonian $a_N$. The RAR requires $\mathsf{\Delta }a$ to be a universal constant [36], for $r\ll r_H$. Given $a_N$, the three paradigms differ only in the behavior of $\mathsf{\Delta }a$ for large galactic radius.

Milky Way Tully–Fisher radius $r_{TF}=17.2$ kpc. Halo radius $r_H=107.8$ kpc [27,30], for $r_G\simeq 15.0$ kpc. Implied MOND constant $a_0=2\gamma c^2=1.14\times {10}^{-10}$ m/s${}^2$. Outside $r_G$, $a_N\simeq \beta c^2/r^2$. Then a(CDM)$=a_N+\frac{1}{2}\gamma c^2$, using empirical CG $\mathsf{\Delta }a$ but omitting halo cutoff; a(CG) $=a_N+\frac{1}{2}\gamma c^2(1-r/r_H)$, including halo cutoff; and a(MOND [16])$\simeq a_N/(1-e^{-\sqrt{a_N/a_0}})$, just MOND with a particular interpolation function and $a_0=1.20\times {10}^{-10}$ m/s${}^2$. $a\left(CDM\right)$ is generic for any model with universal constant $\mathsf{\Delta }a$.

Table 2. Milky Way: radial acceleration a (${10}^{-10}$ m/s${}^2$).

r		CDM		CG		MOND
kpc	${\mathit{a}}_{\mathit{N}}$	a	${\mathbf{10}}^{\mathbf{3}}\frac{\mathit{v}}{\mathit{c}}$	a	${\mathbf{10}}^{\mathbf{3}}\frac{\mathit{v}}{\mathit{c}}$	a	${\mathbf{10}}^{\mathbf{3}}\frac{\mathit{v}}{\mathit{c}}$
15	0.376	0.661	0.584	0.621	0.566	0.877	0.672
20	0.212	0.497	0.584	0.444	0.552	0.617	0.650
25	0.135	0.420	0.601	0.354	0.551	0.475	0.638
30	0.094	0.379	0.625	0.300	0.556	0.385	0.630
35	0.069	0.354	0.652	0.261	0.560	0.324	0.624
40	0.053	0.338	0.682	0.232	0.565	0.279	0.619
45	0.042	0.327	0.717	0.208	0.567	0.246	0.616
50	0.034	0.319	0.740	0.187	0.566	0.219	0.613

[36] compares detailed predictions for the implied external orbital velocity curve of the Milky Way galaxy. The CDM curve rises gradually, and the CG curve remains remarkably flat, while the MOND [16] curve falls gradually toward asymptotic velocity.

9. Conclusions

The postulate of universal conformal symmetry is shown to be in good agreement with gravitational phenomena wherever tested. Implied values of Higgs scalar field parameters are also found to be valid. The resulting gravitational paradigm can be considered a valid alternative to $\mathsf{\Lambda }$CDM and MOND, justifying wider study and application. Selection among theoretical models depends on the improvement of the precision of observations of galactic rotation at large galactic radius, which is currently scarce and of limited accuracy.