# Conformal theory of gravitation and cosmic expansion

R. K. Nesbet  IBM Almaden Research Center, 650 Harry Road, San Jose, CA
95120-6099, USA<EMAIL_ADDRESS>
###### 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 relations and dark galactic haloes of definite
large radius. Cosmological CHM parameters exclude a massive Higgs boson but
are consistent with a novel alternative particle of the observed mass.
Keywords: conformal gravity theory; conformal Higgs model; depleted galactic
halo model

###### pacs:

04.20.Cv,98.80.-k,98.62.Gq

##  I Introduction

Conformal gravity (CG), reviewed in detail(MAN06, ; MAN12, ) by Mannheim,
replaces the Einstein-Hilbert Lagrangian density by a quadratic contraction of
the conformal Weyl tensor(WEY18, ; WEY18a, ), invariant under Weyl scaling by
an arbitrary scalar field. The implied Schwarzschild equation, in spherical
geometry(MAK89, ; MAN91, ), fitted to observed galactic rotation(MAN06, ),
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 geometryJW18 , 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 $\Phi$ is postulated to
have Weyl scaling symmetry, its Lagrangian density acquires a gravitational
term(MAN06, ). The resulting conformal Higgs model (CHM) implies a modified
Friedmann equation consistent with observed Hubble expansion(NESM1, ).

In spherical geometry, the CHM Friedmann equation determines luminosity
distance as a function of redshift $z\leq 1$ for observed supernovae as
accurately as any alternative theory(NESM1, ; NES13, ). The Higgs symmetry-
breaking mechanism produces the gravitational effect of a dark energy source
cosmological constant(NESM1, ). Centrifugal cosmic acceleration persists in
present epoch $z\leq 1$. CG and CHM together have been found to determine
basic parameters of the Higgs model from observed galactic and cosmological
data. These results are reviewed here.

##  II Qualitative regularities of cosmological data

Radial centripetal acceleration $a$ implies $v^{2}=ra$ for 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 $ra_{N}$ 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(MIL83, ; FAM12, ). MOND implies flat $v(r)$ for large
$r$. For $a_{N}\ll a_{0}$, MOND $v^{4}=a^{2}r^{2}\to GMa_{0}$, the empirical
baryonic Tully-Fisher relation(TAF77, ; MCG05, ; MCG11, ; OCM18, ).

It has recently been shown(MLS16, ) 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 relation
(RAR) is a postulate of the empirical MOND model(MIL83, ; FAM12, ; MIL16, ),
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 $a=a_{N}+\Delta a$, a function of radius
in an assumed spherical galactic halo. DM $\Delta a$ is added to baryonic
Newtonian $a_{N}$. DM fits to galactic rotation (orbital velocity vs. circular
orbit radius) depend on model parameters central density $\rho_{0}$ and core
radius $r_{0}$ for a DM halo distribution. Observed data imply that the
surface density product $\rho_{0}r_{0}\simeq 100M_{\odot}pc^{-2}$ is nearly
constant for a large range of galaxies (KAF04, ; GEN09, ; DON09, ).

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

##  III 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 $\Lambda$CDM paradigm for cosmology. Dark energy $\Lambda$ remains
without an explanation. The search for tangible dark matter has continued for
many years with no conclusive results(SAN10, ).

Consideration of an alternative paradigm is motivated by this situation. The
postulate of universal conformal symmetry, requiring local Weyl scaling
covariance(WEY18, ) for all massless elementary physical fields(NES13, ) is a
falsifiable alternative. Conformal symmetry, already valid for fermion and
gauge boson fields (DEW64, ), is extended to both the metric tensor field of
general relativity and the Higgs scalar field of elementary-particle
theory(CAG98, ). This postulate is satisfied by conformal gravity(MAK89, ;
MAN91, ; MAN06, ; MAN12, ) and by the conformal Higgs model(NESM1, ; NES13, ),
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
$g_{\mu\nu}(x)\to g_{\mu\nu}(x)\Omega^{2}(x)$(WEY18, ) for arbitrary real
differentiable $\Omega(x)$. Conformal symmetry is defined by invariant action
integral $I=\int d^{4}x\sqrt{-g}{\cal L}$. For any Riemannian tensor $T(x)$,
$T(x)\to\Omega^{d}(x)T(x)+{\cal R}(x)$ defines weight $d[T]$ and residue
${\cal R}[T]$. For a scalar field, $\Phi(x)\to\Phi(x)\Omega^{-1}(x)$, so that
$d[\Phi]=-1$. Conformal Lagrangian density ${\cal L}$ must have weight
$d[{\cal L}]=-4$ and residue ${\cal R}[{\cal L}]=0$ up to a
4-divergence(MAN06, ).

Variational theory for fields in general relativity is a straightforward
generalization of classical field theory(NES03, ). Given scalar Lagrangian
density ${\cal L}=\sum_{a}{\cal L}_{a}$, action integral $I=\int
d^{4}x\sqrt{-g}{\cal L}$ 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
${\cal L}_{a}$ determines energy-momentum tensor
$\Theta_{a}^{\mu\nu}=-2X_{a}^{\mu\nu}$, evaluated for a solution of the
coupled field equations. Generalized Einstein equation
$\sum_{a}X_{a}^{\mu\nu}=0$ is expressed as
$X_{g}^{\mu\nu}=\frac{1}{2}\sum_{a\neq g}\Theta_{a}^{\mu\nu}$. Hence summed
trace $\sum_{a}g_{\mu\nu}\Theta_{a}^{\mu\nu}$ vanishes for exact field
solutions. Given $\delta{\cal L}=x^{\mu\nu}\delta g_{\mu\nu}$, metric
functional derivative $\frac{1}{\sqrt{-g}}\frac{\delta I}{\delta g_{\mu\nu}}$
is $X^{\mu\nu}=x^{\mu\nu}+\frac{1}{2}{\cal L}g^{\mu\nu}$, evaluated for a
solution of the field equations. Tensor $\Theta^{\mu\nu}=-2X^{\mu\nu}$ is
symmetric.

Conformal gravity was introduced by Weyl(WEY18, ; WEY18a, ) 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(MAN06, ; MAN12, ). The formidable Lagrangian density, a quadratic
contraction of the Weyl tensor, simplified by elimination of a
4-divergence(LAN38, ), reduces to two terms, quadratic respectively in the
Ricci tensor and Ricci scalar(MAN06, ): ${\cal
L}_{g}=-2\alpha_{g}(R^{\mu\nu}R_{\mu\nu}-\frac{1}{3}R^{2})$. Conformal
symmetry fixes the relative coefficient of the two quadratic terms.

In the conformal Higgs model, the unique Lagrangian density for scalar field
$\Phi$ adds a gravitational term(MAN06, ; NESM1, ; NESM2, ) to Higgs
$-V$(CAG98, ). The conformal Lagrangian density is
$(\partial_{\mu}\Phi)^{\dagger}\partial^{\mu}\Phi-\frac{1}{6}R\Phi^{\dagger}\Phi$,
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 $\Delta{\cal
L}_{\Phi}=(w^{2}-\lambda\Phi^{\dagger}\Phi)\Phi^{\dagger}\Phi$.

Metric tensor $g_{\mu\nu}$ is determined by conformal field equations derived
from ${\cal L}_{g}+{\cal L}_{\Phi}$(NESM3, ), driven by energy-momentum tensor
$\Theta_{m}^{\mu\nu}$, where subscript $m$ refers to conventional matter and
radiation. The gravitational field equation within halo radius $r_{H}$ is

$\displaystyle
X_{g}^{\mu\nu}+X_{\Phi}^{\mu\nu}=\frac{1}{2}\Theta_{m}^{\mu\nu}.$ (1)

Defining mean galactic source density ${\bar{\rho}}_{G}$ and residual density
${\hat{\rho}}_{G}=\rho_{G}-{\bar{\rho}}_{G}$ and assuming
$\Theta_{m}^{\mu\nu}(\rho)\simeq\Theta_{m}^{\mu\nu}({\bar{\rho}})+\Theta_{m}^{\mu\nu}({\hat{\rho}})$,
solutions for $r\leq r_{G}$ of the two equations

$\displaystyle
X_{\Phi}^{\mu\nu}=\frac{1}{2}\Theta_{m}^{\mu\nu}({\bar{\rho}}_{G})$
$\displaystyle
X_{g}^{\mu\nu}=\frac{1}{2}\Theta_{m}^{\mu\nu}({\hat{\rho}}_{G})$ (2)

imply a solution of the full equation. The $X_{\Phi}$ equation has an exact
solution for source density ${\bar{\rho}}$ in uniform, isotropic geometry
(FLRW metric tensor)(NESM1, ; NES13, ). The two field equations are made
compatible by imposing appropriate boundary conditions. Fitting constants of
integration of source-free $X_{g}$ at $r_{G}$ (MAK89, ) extends a combined
solution out to halo radius $r_{H}$.

Conformal gravity is described in Schwarzschild metric(MAK89, )

$\displaystyle
ds_{ES}^{2}=-B(r)c^{2}dt^{2}+\frac{dr^{2}}{B(r)}+r^{2}d\omega^{2},$ (3)

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

$\displaystyle
ds_{\Phi}^{2}=-c^{2}dt^{2}+a^{2}(t)(\frac{dr^{2}}{1-kr^{2}}+r^{2}d\omega^{2}),$
(4)

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

$\displaystyle
ds^{2}=-B(r)c^{2}dt^{2}+a^{2}(t)(\frac{dr^{2}}{B(r)}+r^{2}d\omega^{2}),$ (5)

##  IV Depleted galactic halo model

In the $\Lambda$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 gravitational field that deflects photons in gravitational lensing and
increases the velocity of orbiting mass particles. Conformal theory(MAN06, ;
MAN12, ; NES13, ), 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(NESM3, ).

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(NES13, ) 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 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 15kpc would be accompanied by a halo of radius $8\times
15=120$kpc. Equivalence of galactic and deleted halo mass resolves the paradox
for $\Lambda$CDM 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(SAL07, ; MCG05a, ).

##  V Conformal gravity

Given spherical mass-energy density enclosed within $r\leq{\bar{r}}$, the
$X_{g}$ field equation in the ES metric is(MAK89, ; MAN91, )

$\displaystyle\partial_{r}^{4}(rB(r))=rf(r),$ (6)

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

An exact solution of the tensorial $X_{g}$ equation, for source-free
$r\geq{\bar{r}}$, derived by(MAK89, ; MAN91, ; MAN06, ), is

$\displaystyle y_{0}(r)=rB(r)=-2\beta+\alpha r+\gamma r^{2}-\kappa r^{3}.$ (7)

This adds two constants of integration to the classical external potential:
nonclassical radial acceleration $\gamma$ and halo cutoff parameter
$\kappa$(MAN06, ; NESM3, ). Derivative functions
$y_{i}(r)=\partial^{i}_{r}(rB(r)),0\leq i\leq 3$ satisfy differential
equations

$\displaystyle\partial_{r}y_{i}=y_{i+1},0\leq i\leq 2,$
$\displaystyle\partial_{r}y_{3}=rf(r).$ (8)

Independent constants $a_{i}=y_{i}(0)$ determine coefficients
$\beta,\alpha,\gamma,\kappa$ such that at endpoint ${\bar{r}}$

$\displaystyle
y_{0}({\bar{r}})=-2\beta+\alpha{\bar{r}}+\gamma{\bar{r}}^{2}-\kappa{\bar{r}}^{3},$
$\displaystyle y_{1}({\bar{r}})=\alpha+2\gamma{\bar{r}}-3\kappa{\bar{r}}^{2},$
$\displaystyle y_{2}({\bar{r}})=2\gamma-6\kappa{\bar{r}},$ $\displaystyle
y_{3}({\bar{r}})=-6\kappa.$ (9)

To avoid a singularity at the origin, $a_{0}=0$. $\gamma$ and $\kappa$ must be
determined consistently with the $X_{\Phi}$ equation. Specified values can be
fitted by adjusting $a_{1},a_{2},a_{3}$, subject to $a_{0}=0,\alpha=1$(MAN06,
). The solution for $B(r)$ remains regular if $\gamma/\beta\simeq 0$. If Ricci
scalar $R(r\to 0)$ is unconstrained, a particular solution for
$r\leq{\bar{r}}$ is given with specified constants
$a_{0}=0,\alpha,\gamma,\kappa$(NESM5, ):

$\displaystyle rB(r)=y_{0}(r)=\alpha
r-\frac{1}{6}\int_{0}^{r}q^{4}fdq-\frac{1}{2}r\int_{r}^{\bar{r}}q^{3}fdq$
$\displaystyle+\gamma r^{2}+\frac{1}{2}r^{2}\int_{r}^{\bar{r}}q^{2}fdq-\kappa
r^{3}-\frac{1}{6}r^{3}\int_{r}^{\bar{r}}qfdq.$ (10)

Integrated parameter $2\beta=\frac{1}{6}\int_{0}^{\bar{r}}q^{4}fdq$. This
particular solution differs from(MAK89, ; MAN06, ), replacing
$-\frac{1}{2}r^{2}\int_{0}^{r}q^{2}fdq$ by $\gamma
r^{2}+\frac{1}{2}r^{2}\int_{r}^{\bar{r}}q^{2}fdq$(NESM5, ). Constant $\gamma$
is determined independently of the CG equation. At the galactic center, $r\to
0$ implies regular $B(r)$ but singular Ricci $R(r)$ for this solution. This is
consistent with development of a massive black hole, recently observed for
many galaxies.

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

$\displaystyle B(r)=-2\beta/r+\alpha+\gamma r-\kappa r^{2},$ (11)

is exact for a source-free region, outside a spherical mass-energy source
density(MAK89, ; MAN91, ), in particular within an empty galactic halo.

For a test particle in a stable exterior circular orbit with velocity $v$ the
centripetal acceleration is $a=v^{2}(r)/r=\frac{1}{2}B^{\prime}(r)c^{2}$.
Newtonian $B(r)=1-2\beta/r$, where $\beta=GM/c^{2}$, so that classical
acceleration $a_{N}=\beta c^{2}/r^{2}=GM/r^{2}$. CG adds nonclassical $\Delta
a=\frac{1}{2}c^{2}\gamma-c^{2}\kappa r$, a universal constant when halo cutoff
$2\kappa r/\gamma\simeq r/r_{H}$ can be neglected (NESM5, ). Orbital velocity
squared is the sum of $v^{2}(a_{N};r)$ and $v^{2}(\Delta a;r)$, 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 $v(r)$ centered at stationary point
$r_{TF}$, without constraining behavior at large $r$.

MOND(MIL83, ; SAN10, ; FAM12, ) modifies the Newtonian force law for
acceleration below an empirical scale $a_{0}$. Using $y=a_{N}/a_{0}$ as
independent variable, for assumed universal constant $a_{0}\simeq
10^{-10}m/s^{2}$, MOND postulates an interpolation function $\nu(y)$ such that
observed radial acceleration $a=f(a_{N})=a_{N}\nu(y)$. 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 $v^{4}=a^{2}r^{2}\to GMa_{0}$, the empirical baryonic
Tully-Fisher relation(TAF77, ; MCG05, ; MCG11, ; OCM18, ).

In conformal gravity (CG), centripetal acceleration $a=v^{2}/r$ determines
exterior orbital velocity $v^{2}/c^{2}=ra/c^{2}=\beta/r+\frac{1}{2}\gamma
r-\kappa r^{2}$, compared with asymptotic $ra_{N}/c^{2}=\beta/r$. Assuming
Newtonian constant $\beta=\beta_{N}$ and neglecting $2\kappa r/\gamma$, the
slope of $v^{2}(r)$ vanishes at $r^{2}_{TF}=2\beta/\gamma$. This implies that
$v^{4}(r_{TF})/c^{4}=(\beta/r_{TF}+\frac{1}{2}\gamma r_{TF})^{2}=2\beta\gamma$
(MAN97, ; OCM18, ). This is the Tully-Fisher relation, exact at stationary
point $r_{TF}$ of the $v(r)$ function. The implied MOND acceleration constant
is $a_{0}=2\gamma c^{2}=1.14\times 10^{-10}m/s^{2}$(NESM5, ).

##  VI The conformal Higgs model

The Newton-Einstein model of galactic growth and interaction is substantially
altered by conformal theory(NESM1, ; NESM3, ). 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 to 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(NESM3, ).

The conformal gravitational field equation is

$\displaystyle
X_{g}^{\mu\nu}+X_{\Phi}^{\mu\nu}=\frac{1}{2}\Theta_{m}^{\mu\nu},$ (12)

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

$\displaystyle
X_{g}^{\mu\nu}=\frac{1}{2}\Theta_{m}^{\mu\nu}(\rho_{g}),X_{\Phi}^{\mu\nu}=0,r\leq
r_{H}.$ (13)

The exact source-free solution(MAK89, ) of the $X_{g}$ equation is valid in
the external halo, $r_{g}\leq r\leq r_{H}$, because $\rho_{g}$ vanishes.
Constants of integration fitted at $r_{g}$ and $r_{H}$ determine radial
acceleration in this external halo.

In uniform, isotropic geometry with uniform mass/energy density
${\bar{\rho}}$, ${\cal L}_{\Phi}$ implies a modified Friedmann equation
(FRI22, ; NESM1, ; NES13, ) for cosmic distance scale factor $a(t)$, in FLRW
metric

$\displaystyle
ds_{\Phi}^{2}=-c^{2}dt^{2}+a^{2}(t)(\frac{dr^{2}}{1-kr^{2}}+r^{2}d\omega^{2}),$
(14)

which determines Friedmann scaling factor $a(t)$. With $a(t_{0})=1$ at present
time $t_{0}$:

$\displaystyle\frac{{\dot{a}}^{2}}{a^{2}}+\frac{k}{a^{2}}-\frac{\ddot{a}}{a}=\frac{2}{3}({\bar{\Lambda}}+{\bar{\tau}}c^{2}{\bar{\rho}}(t)).$
(15)

Consistency with the $X_{g}$ equation requires $k=0$.
${\bar{\Lambda}}=\frac{3}{2}w^{2}\geq 0$ and
${\bar{\tau}}\sim-3/\phi_{0}^{2}\leq 0$ are determined by parameters of the
Higgs model(NESM1, ). For uniform $\rho=0$ within the halo radius this solves
the second of Eqs.(13).

Dividing the modified equation by $({\dot{a}}/a)^{2}$ determines dimensionless
sum rule $\Omega_{m}+\Omega_{\Lambda}+\Omega_{k}+\Omega_{q}=1$. Scale factor
$a(t)$ determines dimensionless Friedmann acceleration weight
$\Omega_{q}(t)=\frac{{\ddot{a}}a}{{\dot{a}}^{2}}$(NESM1, ). Radiation energy
density is included in $\Omega_{m}$ here. The dimensionless weights are
$\Omega_{m}(t)$ for mass density, $\Omega_{k}(t)$ for curvature, and
$\Omega_{\Lambda}(t)$ for dark energy, augmented by acceleration weight
$\Omega_{q}(t)$. 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
$\Omega_{m}$ completely, with $k=0$, conformal sum rule
$\Omega_{\Lambda}(t)+\Omega_{q}(t)=1$ fits observed data accurately for
redshifts $z\leq 1$ (7.33Gyr) (NESM1, ; NESM2, ; NESM5, ). This eliminates any
need for dark matter to explain Hubble expansion.

The $X_{\Phi}$ equation includes dark energy, present regardless of density
$\rho$. This produces centrifugal acceleration of background Hubble
expansion(NESM1, ; NES13, ), which must be subtracted off in order 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 $\frac{{\dot{a}}}{a}=h(t)H_{0}$, for Hubble constant $H_{0}$, it is
convenient to use Hubble units, such that $c=\hbar=1$, and dimensionless
$a(t_{0})=1$, $h(t_{0})=1$. The units for frequency, energy, and acceleration
are $H_{0},\hbar H_{0},cH_{0}$, respectively. Setting
$\Omega_{k}=\Omega_{m}=0$, for $\alpha=w^{2}=\Omega_{\Lambda}(t_{0})$, the CHM
Friedmann equation(NESM1, ; NES13, ) is

$\displaystyle\frac{{\dot{a}}^{2}}{a^{2}}-\frac{\ddot{a}}{a}=\frac{2}{3}{\bar{\Lambda}}=\alpha.$
(16)

In Hubble units, Eq.(16) reduces to $\frac{d}{dt}h(t)=-\alpha$. The explicit
solution for $t\leq t_{0}$ is

$\displaystyle h(t)=\frac{{\dot{a}}}{a}(t)=\frac{d}{dt}\ln
a(t)=1+\alpha(t_{0}-t),$ $\displaystyle\ln
a(t)=-(t_{0}-t)-\frac{1}{2}\alpha(t_{0}-t)^{2},$ $\displaystyle
a(t)=\exp[-(t_{0}-t)-\frac{1}{2}\alpha(t_{0}-t)^{2}].$ (17)

From the definition of redshift $z(t)$:
$1+z(t)=\frac{1}{a(t)}=\exp[(t_{0}-t)+\frac{1}{2}\alpha(t_{0}-t)^{2}],\\\
(t_{0}-t)^{2}+\frac{2}{\alpha}(t_{0}-t)=\frac{2}{\alpha}\ln(1+z)$.

In Hubble units with $\frac{{\dot{a}}}{a}(t_{0})=1$, this implies

$\displaystyle t_{0}-t(z)=(\sqrt{2\alpha\ln(1+z)+1}-1)/\alpha,$
$\displaystyle\frac{dt}{dz}=\frac{-1}{(1+z)\sqrt{2\alpha\ln(1+z)+1}}.$ (18)

If $\alpha=0.732$ and $z=1$, $(t_{0}-t)/H_{0}=7.33Gyr$, for $H_{0}=67.8\pm
0.9km/s/Mpc=2.197\times 10^{-18}/s$(PLC15, ).

Neglecting both curvature weight $\Omega_{k}$ and cosmic mass/energy weight
$\Omega_{m}$, conformal sum rule $\Omega_{\Lambda}+\Omega_{q}=1$ holds for
acceleration weight $\Omega_{q}=\frac{{\ddot{a}}a}{{\dot{a}}^{2}}$. For
$\Omega_{k}=0$, luminosity distance $d_{L}(z)=(1+z)\chi(z)$(NESM2, ), where

$\displaystyle\chi(z)=\int_{t(z)}^{t_{0}}\frac{dt}{a(t)}=\int_{z}^{0}d{\bar{z}}(1+{\bar{z}})\frac{dt}{dz}({\bar{z}})=$
$\displaystyle\int_{0}^{z}\frac{d{\bar{z}}}{\sqrt{2\alpha\ln(1+{\bar{z}})+1}}.$
(19)

Evaluated for $\alpha=\Omega_{\Lambda}(t_{0})=0.732$, the fit to scaled
luminosity distances $H_{0}d_{L}/c$ from Hubble expansion data(MAN03, ; NESM1,
) is given in Table 1.

Table 1: Scaled luminosity distance fit to Hubble data |  |  | Theory | Observed
---|---|---|---|---
z | $\Omega_{\Lambda}$ | $\Omega_{q}$ |  $H_{0}d_{L}/c$Eq.(VI) |  $H_{0}d_{L}/c$(MAN03, )
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(MAN03, ) with
statistical accuracy comparable to the best standard $\Lambda$CDM fit, with
$\Omega_{m}=0$. Table(1) compares CHM $d_{L}(z)$ to this Mannheim function.
Because ${\bar{\tau}}$ is negative(MAN06, ; NESM1, ), cosmic acceleration
$\Omega_{q}$ remains positive (centrifugal) back to the earliest time(NESM1,
).

Conformal Friedmann Eq.(15)(NESM1, ; NESM2, ) determines cosmic acceleration
weight $\Omega_{q}$. With both weight parameters $\Omega_{k}$ and $\Omega_{m}$
set to zero, Eq.(15) fits scaled Hubble function $h(t)=H(t)/H_{0}$ for
redshifts $z\leq 1$, Table(1), as accurately as standard $\Lambda$CDM, with
only one free constant. This determines Friedmann weights, at present time
$t_{0}$, $\Omega_{\Lambda}=0.732,\Omega_{q}=0.268$ (NESM1, ). Hubble constant
$H(t_{0})=H_{0}=2.197\times 10^{-18}/s$ (PLC15, ) is independent of these
data. For uniform primordial energy-momentum source density $\rho_{m}$, the
dimensionless sum rule(NESM1, ) with $\Omega_{k}=0$ determines
$\Omega_{q}(\rho_{m})=1-\Omega_{\Lambda}-\Omega_{m}$ in the cosmic background,
and $\Omega_{q}(0)=1-\Omega_{\Lambda}$ in the depleted halo(NESM3, ).

Schwarzschild $B(r)$ parameter $\gamma>0$, independent of galactic mass and
structure, implies centripetal acceleration due to an isotropic cosmological
source(MAN97, ). The parametrized gravitational field forms a spherical
halo(NES13, ). The depleted halo model removes a particular conceptual problem
in fitting $B(r)$ parameters $\gamma,\kappa$ to galactic rotation data(MAN97,
; MAN06, ; MAO11, ). In empirical parameter
$\gamma=\gamma^{*}N^{*}+\gamma_{0}$, $\gamma_{0}$ does not depend on galactic
mass, so must be due to the surrounding cosmos(MAN97, ). Since the interior
term, coefficient $\gamma^{*}N^{*}$, is centripetal, one might expect the term
in $\gamma_{0}$ to be centrifugal, describing attraction to exterior sources.
However, coefficient $\gamma$, derived here for a depleted halo, determines
net centripetal acceleration, in agreement with observation(NESM5, ). Recent
galactic rotation results for galaxies with independently measured mass(MLS16,
) imply that $\gamma$ cannot depend on galactic mass(NESM7, ).

The conformal Friedmann cosmic evolution equation implies dimensionless cosmic
acceleration parameters $\Omega_{q}(\rho)$(NESM3, ) 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 $\Omega_{q}(in)-\Omega_{q}(out)=\Omega_{q}(0)-\Omega_{q}(\rho_{m})$.
Uniform cosmological $\rho_{m}$ implies constant $\gamma$ for $r\leq r_{H}$,
independent of galactic mass (NESM5, ). This result is consistent with recent
observations of galactic rotational velocities for galaxies with directly
measured mass(MLS16, ; NESM7, ).

From the CHM, observed nonclassical gravitational acceleration
$\frac{1}{2}\gamma c^{2}$ in the halo is proportional to
$\Delta\Omega_{q}=\Omega_{q}(0)-\Omega_{q}(\rho_{m})=\Omega_{m}(\rho_{m})$(NESM3,
), where, given $\rho_{m}$ and $H_{0}$,
$\Omega_{m}(\rho_{m})=\frac{2}{3}\frac{{\bar{\tau}}c^{2}\rho_{m}}{H^{2}_{0}}$(NESM1,
). Positive $\rho_{m}$ implies $\Omega_{m}<0$ because coefficient
${\bar{\tau}}<0$ (MAN06, ; NESM1, ). Thus the depleted halo model determines
constant $\gamma$ from uniform universal cosmic baryonic mass density
$\rho_{m}/c^{2}$, which includes radiation energy density here.

$\Omega_{m}<0$ implies centripetal acceleration, converted from Hubble units,
$\frac{1}{2}\gamma c^{2}=-cH_{0}\Omega_{m}(\rho_{m})$(NESM3, ). Hence
$\Delta\Omega_{q}=\Omega_{m}<0$ is consistent with nonclassical centripetal
acceleration, confirmed by inward deflection of photon geodesics observed in
gravitational lensing (NESM3, ). This logic is equivalent to requiring radial
acceleration to be continuous across halo boundary $r_{H}$:

$\displaystyle\frac{1}{2}\gamma
c^{2}-cH_{0}\Omega_{q}(0)=-cH_{0}\Omega_{q}(\rho_{m}).$ (20)

Signs here follow from the definition of $\Omega_{q}$ as centrifugal
acceleration weight.

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

$\displaystyle\frac{1}{2}\gamma_{\odot,in}c^{2}-cH_{0}\Omega_{q}({\bar{\rho}}_{\odot})=\frac{1}{2}\gamma
c^{2}-cH_{0}\Omega_{q}(0),$ (21)

determines constant $\gamma_{\odot,in}$ valid inside $r_{\odot}$.
$\gamma_{\odot,in}$ is determined by local mean source density
${\bar{\rho}}_{\odot}$. $\gamma$ in the halo is not changed. Its value is a
constant of integration that cannot vary in the source-free halo(NESM3, ;
NESM5, ). 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 $r_{H}$.

##  VII Values of relevant parameters

Anomalous rotation velocities for 138 galaxies are fitted using only four
universal CG parameters $\beta^{*},\gamma^{*},\gamma_{0},\kappa$ (MAN06, ;
MAN12, ; MAO11, ; MAO12, ; OAM15, ) 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
(MAN06, ; MAO12, ),

$\displaystyle\beta^{*}=1.475\times 10^{3}m,\gamma_{0}=3.06\times 10^{-28}/m,$
$\displaystyle\gamma^{*}=5.42\times 10^{-39}/m,\kappa=9.54\times
10^{-50}/m^{2},$ (22)

fit conformal gravity to galactic rotation velocities. The observed radial
acceleration relation (RAR)MLS16  is consistent with the CHM conclusion that
$\gamma^{*}$ must vanish. Then $\gamma$ is a universal constant(NESM7, ).
Using data for the Milky Way galaxy,
$\gamma=\gamma_{0}+N^{*}\gamma^{*}=6.35\times 10^{-28}/m$(OAM15, ; MCG08, ).

$\zeta>0$ for computed $R(t)$(NESM1, ) implies $\lambda<0$.
$\hbar\phi(t_{0})=174GeV$(AMS08, )$=1.203\times 10^{44}\hbar H_{0}$ in Hubble
units. For $\Omega_{m}=0$, $\zeta(t_{0})=2\Omega_{q}(t_{0})=0.536$. For
$\lambda(t)=\zeta/(-2\phi^{2})$ and $\phi(t_{0})=\phi_{0}$, dimensionless
$\lambda(t_{0})=-0.185\times 10^{-88}$.

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 ${\dot{\phi}}/\phi$ for redshifts $z\leq 1$ 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(NESM4, ).

Scalar field $ZZ=g_{\mu\nu}Z^{\mu*}Z^{\nu}$ interacts with neutral scalar
field $WW=g_{\mu\nu}W_{-}^{\mu}W_{+}^{\nu}$ through 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 $\Phi$ to produce biquadratic
Lagrangian term $\lambda(\Phi^{\dagger}\Phi)^{2}$(NESM4, ). If the mass of
$W_{2}$ is $125GeV$ the conformal Higgs $\lambda$ assumes its empirical value.
This implies an alternative identification of the recently observed LHC
resonance (DJV12, ; ATL12, ; CMS12, ). The interacting bare fields produce
relatively stable $W_{2}=WW\cos\theta_{x}+ZZ\sin\theta_{x}$ and complementary
resonance $Z_{2}=-WW\sin\theta_{x}+ZZ\cos\theta_{x}$(NESM4, ). $W_{2}$ can
dress the bare $\Phi$ field while maintaining charge neutrality. Assumed
$W_{2}$ mass 125GeV implies $\lambda=-0.455\times 10^{-88}$(NESM4, ),
consistent with empirical value $\lambda(t_{0})\simeq-10^{-88}$.

Parameters $w^{2}$ and $\lambda$ are defined as positive constants in
Higgs(HIG64, ; CAG98, )

$\displaystyle\Delta{\cal
L}_{\Phi}=-V(\Phi^{\dagger}\Phi)=w^{2}\Phi^{\dagger}\Phi-\lambda(\Phi^{\dagger}\Phi)^{2}.$
(23)

Stationary action implies constant finite
$\Phi^{\dagger}\Phi=\phi_{0}^{2}=w^{2}/2\lambda$, where $\hbar\Phi$ and $\hbar
w$ are energies. The conformal Higgs model(NESM1, ) acquires an additional
term in $\Delta{\cal L}_{\Phi}$: $-\frac{1}{6}R\Phi^{\dagger}\Phi$(MAN06, )
for gravitational Ricci scalar $R=g_{\mu\nu}R^{\mu\nu}$. The modified
Lagrangian, defined for neutral gauge field $Z_{\mu}$(NESM2, ), determines
$w^{2}$, which becomes a cosmological constant in the conformal Friedmann
cosmic evolution equation(NESM1, ). Lagrangian term $w^{2}\Phi^{\dagger}\Phi$
is due to induced neutral $Z_{\mu}$ field amplitude(NESM2, ), which dresses
the bare scalar field. Finite $w^{2}$ breaks conformal symmetry, but does not
determine a Higgs mass.

The conformal scalar field equation including parametrized $\Delta{\cal
L}_{\Phi}$ is(MAN06, ; NESM1, )

$\displaystyle\frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}\partial^{\mu}\Phi)=(-\frac{1}{6}R+w^{2}-2\lambda\Phi^{\dagger}\Phi)\Phi.$
(24)

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

$\displaystyle\frac{{\ddot{\phi}}}{\phi}+3\frac{{\dot{a}}}{a}\frac{{\dot{\phi}}}{\phi}=-(\zeta(t)+2\lambda\phi^{2}).$
(25)

Omitting $R$ and assuming constant $\lambda>0$ and $w^{2}$, Higgs solution
$\phi_{0}^{2}=w^{2}/2\lambda$(HIG64, ) is exact. All time derivatives drop
out. In the conformal scalar field equation, cosmological time dependence of
Ricci scalar $R(t)$, determined by the CHM Friedmann cosmic evolution
equation, introduces nonvanishing time derivatives and implies
$\lambda<0$(NESM1, ). An exact solution for $\phi(t)$ should include the time
derivative terms in $\zeta(t)$. Neglecting them here limits the argument to
redshift $z\leq 1$.

Ricci scalar $R(t)$ drops out of the scalar field equations if time
derivatives are neglected. This justifies an estimate of Higgs parameter
$w^{2}$(NESM1, ). Dressing scalar field $\Phi$ by neutral gauge field
$Z^{\mu}$ produces Lagrangian term $w^{2}\Phi^{\dagger}\Phi$. Defining $\Delta
I_{\Phi}=\int d^{4}x\sqrt{-g}\Delta{\cal L}_{\Phi}$ from Eq.(23), the
parametrized effective potential term in the scalar field equation is given by

$\displaystyle\frac{1}{\sqrt{-g}}\frac{\delta\Delta
I_{\Phi}}{\delta\Phi^{\dagger}}=(w^{2}-2\lambda\Phi^{\dagger}\Phi)\Phi.$ (26)

Gauge invariance replaces bare derivative $\partial_{\mu}$ by gauge covariant
derivative(CAG98, )

$\displaystyle D_{\mu}=\partial_{\mu}-\frac{i}{2}g_{z}Z_{\mu}.$ (27)

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

$\displaystyle{\cal
L}^{0}_{\Phi}=(\partial_{\mu}\Phi)^{\dagger}\partial^{\mu}\Phi-\frac{1}{6}R\Phi^{\dagger}\Phi$
(28)

by coupling term

$\displaystyle\Delta{\cal
L}=(D_{\mu}\Phi)^{\dagger}D^{\mu}\Phi-(\partial_{\mu}\Phi)^{\dagger}\partial^{\mu}\Phi=$
$\displaystyle\frac{i}{2}g_{z}Z_{\mu}^{*}\Phi^{\dagger}\partial^{\mu}\Phi-\frac{i}{2}g_{z}Z^{\mu}(\partial_{\mu}\Phi)^{\dagger}\Phi+\frac{1}{4}g_{z}^{2}\Phi^{\dagger}Z_{\mu}^{*}Z^{\mu}\Phi.$
(29)

Parametrized for a generic complex vector field(CAG98, ),

$\displaystyle\Delta{\cal
L}_{Z}=\frac{1}{2}m^{2}_{Z}Z^{*}_{\mu}Z^{\mu}-\frac{1}{2}(Z^{*}_{\mu}J_{Z}^{\mu}+Z^{\mu}J^{*}_{Z\mu}),$
(30)

given mass parameter $m_{Z}$ and source current density $J_{Z}^{\mu}$. The
field equation for parametrized $Z^{\mu}$ is(CAG98, )

$\displaystyle\partial_{\nu}Z^{\mu\nu}=\frac{2}{\sqrt{-g}}\frac{\delta\Delta
I_{Z}}{\delta Z_{\mu}^{*}}=m_{Z}^{2}Z^{\mu}-J_{Z}^{\mu}.$ (31)

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

$\displaystyle\frac{2}{\sqrt{-g}}\frac{\delta\Delta I}{\delta
Z_{\mu}^{*}}=\frac{1}{2}g_{z}^{2}\Phi^{\dagger}\Phi
Z^{\mu}+ig_{z}\Phi^{\dagger}\partial^{\mu}\Phi.$ (32)

This implies Higgs mass formula
$m_{Z}^{2}=\frac{1}{2}g_{z}^{2}\Phi^{\dagger}\Phi$ and field source density
$J_{Z}^{\mu}=-ig_{z}\Phi^{\dagger}\partial^{\mu}\Phi$. Using $\Delta{\cal L}$
derived from the covariant derivative,

$\displaystyle\frac{1}{\sqrt{-g}}\frac{\delta\Delta
I_{\Phi}}{\delta\Phi^{\dagger}}=\frac{1}{4}g_{z}^{2}Z_{\mu}^{*}Z^{\mu}\Phi+\frac{i}{2}g_{z}(Z_{\mu}^{*}+Z_{\mu})\partial^{\mu}\Phi.$
(33)

Comparison with Eq.(26) implies
$w^{2}=\frac{1}{4}g_{z}^{2}Z_{\mu}^{*}Z^{\mu}$. Neglecting derivatives of
$Z^{\mu}$, Eq.(31) reduces to

$\displaystyle
Z^{\mu}=J_{Z}^{\mu}/m_{Z}^{2}=-\frac{2i}{g_{z}}\Phi^{\dagger}\partial^{\mu}\Phi/\Phi^{\dagger}\Phi.$
(34)

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

##  VIII 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 $100M_{\odot}pc^{-2}$ for a large range of galaxies (KAF04, ; GEN09, ;
DON09, ). Conformal theory implies nonclassical centripetal acceleration
$\Delta a$, for Newtonian $a_{N}$ due to observable baryonic matter.
Neglecting the halo cutoff for $r\ll r_{H}$, conformal $\Delta a$ is constant
over the halo and $a=a_{N}+\Delta a$ is a universal function, consistent with
a recent study of galaxies with independently measured mass(MLS16, ). An
equivalent dark matter source would be a pure cusp distribution with cutoff
parameter determined by halo boundary radius. This is shown here to imply
universal central surface density for any dark matter core model.

CG implies $\Delta a=\frac{1}{2}\gamma c^{2}(1-r/r_{H})$ in a depleted halo.
For $r\ll r_{H}$, CG acceleration constant $\gamma$ (MAN06, ; NESM7, )
predicts $\Delta a=\frac{1}{2}\gamma c^{2}$ (NESM3, ; NESM7, ) for all DM
models.

Uniform constant $\Delta a$ puts a severe constraint on any DM model. The
source density must be of the form $\xi/r$, a pure radial cusp (NESM7, ;
NESM5, ), where constant $\xi={\Delta a}/{2\pi
G}=0.06797kg/m^{2}=32.535M_{\odot}/pc^{2}$ is determined by CG parameter
$\gamma$, with CODATA $G=6.67384\times 10^{-11}m^{3}s^{-2}kg^{-1}$ (COD12, ).

A DM galactic model equivalent to conformal theory would imply uniform DM
radial acceleration $\Delta a=2\pi G\xi$, 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 $r\Delta
a/c^{2}=GM_{r}/r$. 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 $\rho(r)$ 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
$\rho(r_{0})r_{0}=r_{0}M_{0}/V_{0}=3\xi/2$. For assumed PI core DM density
(KAF04, ) $\rho(r)=\rho_{0}r_{0}^{2}/(r^{2}+r_{0}^{2})$ , central
$\rho_{0}=2\rho(r_{0})$. Hence for a PI core,
$\rho_{0}r_{0}=3\xi=\frac{3\Delta a}{2\pi G}=0.204\times
10^{-2}kg/m^{2}=97.6M_{\odot}pc^{-2}$, independent of $r_{0}$.

A radial acceleration relation RAR(MLS16, ) is postulated by MOND(MIL83, ).
Paradigms $\Lambda$CDM and CG/CHM both represent galactic radial acceleration
by $a_{N}+\Delta a$, for baryonic Newtonian $a_{N}$. The RAR requires $\Delta
a$ to be a universal constant(NESM7, ), for $r\ll r_{H}$. Given $a_{N}$, the
three paradigms differ only by the behavior of $\Delta a$ for large galactic
radius.

Milky Way Tully-Fisher radius $r_{TF}=17.2kpc$. Halo radius
$r_{H}=107.8kpc$(NESM3, ; NESM5, ), for $r_{G}\simeq 15.0kpc$. 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 $\Delta a$ but omitting halo cutoff;
a(CG)$=a_{N}+\frac{1}{2}\gamma c^{2}(1-r/r_{H})$, including halo cutoff; and
a(MOND(MLS16, ))$\simeq a_{N}/(1-e^{-\sqrt{a_{N}/a_{0}}})$, just MOND with a
particular interpolation function and $a_{0}=1.20\times 10^{-10}m/s^{2}$.
$a(CDM)$ is generic for any model with universal constant $\Delta a$.

Table 2: Milky Way: radial acceleration a ($10^{-10}m/s^{2}$) r |  | CDM |  | CG |  | MOND |
---|---|---|---|---|---|---|---
kpc | $a_{N}$ | a | $10^{3}\frac{v}{c}$ | a | $10^{3}\frac{v}{c}$ | a | $10^{3}\frac{v}{c}$
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

Table 2(NESM7, ) compares detailed predictions for the implied external
orbital velocity curve of the Milky Way galaxy. The CDM curve rises gradually,
the CG curve remains remarkably flat, while the MOND(MLS16, ) curve falls
gradually toward asymptotic velocity.

##  IX 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 $\Lambda$CDM and MOND,
justifying wider study and application. Selection among theoretical models
depends on improvement of the precision of observations of galactic rotation
at large galactic radius, currently scarce and of limited accuracy.

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††thanks: co-first author††thanks: co-first author††thanks<EMAIL_ADDRESS>correspondence author

# Do we need to use regularization on the thermal part in the NJL model?

Kai Xue  School of Automotive and Traffic Engineering , Jiangsu University,
Zhenjiang 212013 P.R. China  Xiaozhu Yu  Xinyang Wang  Department of Physics,
Jiangsu University, Zhenjiang 212013 P.R. China

###### Abstract

The Nambu–Jona-Lasinio (NJL) model is one of the most useful tools to study
non-perturbative strong interaction matter. Because it is a nonrenormalizable
model, the choosing of regularization is a subtle issue. In this paper, we
discuss one of the general things of the regularization in the NJL model,
which is whether we need to use the regularization on the thermal part by
evaluating the quark chiral condensate and thermal properties in the two-
flavor NJL model. The calculations in this work include three regularization
schemes that contain both gauge covariant and invariant schemes. We found no
matter which regularization scheme we choose, it is necessary to use the
regularization on the thermal part when calculating the chiral condensate
related physics quantities and do not use the regularization on the thermal
part when calculating the grand potential related physical quantities.

##  I Introduction

The understanding of QCD matter with finite temperature and chemical potential
is one of the main topics in theoretical physics. In recent decades, the
heavy-ion collision experiment especially RHIC and LHC create a ground
experiment environment to study such matter. In the theoretical viewpoint,
because the strong interaction coupling constant $\alpha_{s}$ is not small in
most of the regimes, the perturbative method failed. Therefore, the non-
perturbative methods are needed to understand the physics in the large
$\alpha_{s}$ regime. One of the useful tools is the Nambu-Jona-Lasinio(NJL)
model Nambu:1961tp ; Nambu:1961fr . The NJL model has a long history back to
1961 (for reviews, see Refs. Klevansky:1992qe ; Buballa:2003qv ;
Hatsuda:1994pi ), it incorporates the chiral symmetry and its spontaneous
breaking, the gluonic degrees of freedom are replaced by a local four-point
interaction of color currents. The essential of this model is that it keeps
the important symmetry in QCD, it is trivial to represent the symmetry
breaking in QCD which is very useful for helping us to understand the
properties of QCD matter.

Because of the point interaction for quarks in the NJL model, the model is
nonrenormalizable. Therefore, regularization should be used during the
calculation. One of the fundamental problems is that when the temperature is
finite, the contribution from the thermal part is convergent. It is
problematic whether we need to use the regularization on this part the same as
the divergent vacuum part. The different choice is not only to change the
numerical values of the results but also to change the physical behaviors. A
good example is given in Ref. Yu:2015hym , where they were discussed by
using/not using the regularization on the thermal part, the dependence of the
critical temperature on the chiral chemical potential gave two opposite
results. On the one hand, the thermal part does not need a regularization
since it is convergent, to get the full contribution from the thermal part, we
should not use the regularization on this part. On the other hand,
regularization should be applied to all parts of the equation to get correct
physics quantities under the same energy limit for physics consistency.
Therefore, to answer this regularization of the thermal part problem, a
systematic study is needed.

In this paper, we calculate the quark chiral condensate/quark constituent mass
and grand potential related thermodynamics by using the framework of the two-
flavor NJL model. We evaluate the physics quantities by using three different
regularization schemes and introducing two different treatments on the thermal
part for each scheme. We will investigate the simple and fundamental question:
should we use the regularization on the thermal part when we are using the NJL
model to calculate the physics quantities?

##  II model

The Lagrangian density of the two-flavor NJL model is given by

$\displaystyle\mathcal{L}=\bar{\psi}(i\gamma_{\mu}\partial^{\mu}-m)\psi+G_{S}\left[(\bar{\psi}\psi)^{2}+(\bar{\psi}i\gamma_{5}\mathbf{\tau}\psi)^{2}\right].$
(1)

Here $m$ is the current quark mass,
$\mathbf{\tau}=(\tau^{1},\tau^{2},\tau^{3})$ is the isospin Pauli matrices and
$G_{S}$ is the coupling constant with respect to the (pseudo)scalar channels.
By using mean-field (Hartree) approximation the Lagrangian density could be
given by Klevansky:1992qe ,

$\displaystyle\mathcal{L}=-\frac{\sigma^{2}}{4G_{S}}+\bar{\psi}(i\gamma_{\mu}\partial^{\mu}-M)\psi,$
(2)

where the dynamical quark mass $\sigma=-2G_{S}\left<\bar{\psi}\psi\right>$ and
$\left<\bar{\psi}\psi\right>$ is the chiral condensate. Then the constituent
quark mass is given by $M=m+\sigma$. The general grand potential can be
written in vacuum part and thermal part,

$\displaystyle\Omega$ $\displaystyle=$
$\displaystyle\Omega_{vac}+\Omega_{th},$ (3)

where the vacuum and the thermal part are given by

$\displaystyle\Omega_{vac}=\frac{\sigma^{2}}{4G_{S}}-2N_{f}N_{c}\int\frac{d^{3}p}{(2\pi)^{3}}E_{p}(\mathbf{p},M),$
(4a) $\displaystyle\Omega_{th}$ $\displaystyle=$
$\displaystyle-2N_{f}N_{c}\int\frac{d^{3}p}{(2\pi)^{3}}T$ (4b)
$\displaystyle\times$
$\displaystyle\left\\{\ln\left[1+\exp\left(-\frac{E_{p}(\mathbf{p},M)-\mu}{T}\right)\right]\right.$
$\displaystyle+$
$\displaystyle\left.\ln\left[1+\exp\left(-\frac{E_{p}(\mathbf{p},M)+\mu}{T}\right)\right]\right\\}.$

Here, the on shell energy of quark is given by
$E_{p}(\mathbf{p},M)=\sqrt{\mathbf{p}^{2}+M^{2}}$. $N_{f}$ and $N_{c}$ are the
number of flavours and colors, which are 2 and 3, respectively, in this work.
The constituent mass/chiral condensate can be solved by the corresponding gap
equation $\frac{\partial\Omega}{\partial\sigma}=0$ and
$\frac{\partial^{2}\Omega}{\partial\sigma^{2}}>0$. It is easy to see that the
integral for the vacuum part in Eq. (3) is divergent and the integral in the
thermal part is convergent by integrating over the three momenta from 0 to
infinity.

In this study, we will use three different regularization schemes, (1) the
three-momentum hard cutoff, (2) the three-momentum soft cutoff, and (3) Pauli-
Villas regularization. One should be mentioned here, the first two schemes are
gauge covariant and the third one is gauge invariant.

Then the vacuum part of grand potential with using regularizations could be
written as

$\displaystyle\Omega^{H}_{vac}$ $\displaystyle=$
$\displaystyle\frac{\sigma^{2}}{4G_{S}}-N_{f}N_{c}\int_{0}^{\Lambda}\frac{dp}{\pi^{2}}p^{2}E_{p}(p,M),$
(5a) $\displaystyle\Omega^{S}_{vac}$ $\displaystyle=$
$\displaystyle\frac{\sigma^{2}}{4G_{S}}-N_{f}N_{c}\int_{0}^{\infty}\frac{dp}{\pi^{2}}f_{\Lambda}^{2}p^{2}E_{p}(p,M),$
$\displaystyle\Omega^{PV}_{vac}$ $\displaystyle=$
$\displaystyle\frac{\sigma^{2}}{4G_{S}}-\sum_{a=0}^{2}C_{a}\frac{N_{f}N_{c}}{\pi^{2}}\frac{M_{a}^{4}\ln
M_{a}}{8}.$ (5c)

And the thermal part with using regularizations is given by

$\displaystyle\Omega^{H}_{th}$ $\displaystyle=$ $\displaystyle-
N_{f}N_{c}\int_{0}^{\Lambda}\frac{dp}{\pi^{2}}p^{2}T$ (6a)
$\displaystyle\times$
$\displaystyle\left\\{\ln\left(1+\exp\left(-\frac{E_{p}(p,M)-\mu}{T}\right)\right)\right.$
$\displaystyle+$
$\displaystyle\left.\ln\left(1+\exp\left(-\frac{E_{p}(p,M)+\mu}{T}\right)\right)\right\\},$
$\displaystyle\Omega^{S}_{th}$ $\displaystyle=$ $\displaystyle-
N_{f}N_{c}\int_{0}^{\infty}\frac{dp}{\pi^{2}}p^{2}f_{\Lambda}^{2}T$ (6b)
$\displaystyle\times$
$\displaystyle\left\\{\ln\left(1+\exp\left(-\frac{E_{p}(p,M)-\mu}{T}\right)\right)\right.$
$\displaystyle+$
$\displaystyle\left.\ln\left(1+\exp\left(-\frac{E_{p}(p,M)+\mu}{T}\right)\right)\right\\},$
$\displaystyle\Omega^{PV}_{th}$ $\displaystyle=$
$\displaystyle-\sum_{a=0}^{2}C_{a}\frac{N_{f}N_{c}}{\pi^{2}}\int_{0}^{\infty}\frac{dp}{\pi^{2}}p^{2}T$
(6c) $\displaystyle\times$
$\displaystyle\left[\ln\left(1+\exp\left(-\frac{E_{p}(p,M_{a})-\mu}{T}\right)\right)\right.$
$\displaystyle+$
$\displaystyle\left.\ln\left(1+\exp\left(-\frac{E_{p}(p,M_{a})+\mu}{T}\right)\right)\right].$

Where, the upper index H, S, PV represent hard cutoff, soft cutoff and Pauli-
Villas, respectively. The overall grand potential with different
regularizations and with/without applying the regularizations on the thermal
part can be written by

$\displaystyle\Omega^{H}$ $\displaystyle=$
$\displaystyle\Omega^{H}_{vac}+\Omega^{H}_{th},~{}~{}\Omega^{H^{\prime}}=\Omega^{H}_{vac}+\Omega_{th},$
$\displaystyle\Omega^{S}$ $\displaystyle=$
$\displaystyle\Omega^{S}_{vac}+\Omega^{S}_{th},~{}~{}\Omega^{S^{\prime}}=\Omega^{S}_{vac}+\Omega_{th},$
$\displaystyle\Omega^{PV}$ $\displaystyle=$
$\displaystyle\Omega^{PV}_{vac}+\Omega^{PV}_{th},~{}~{}\Omega^{PV^{\prime}}=\Omega^{PV}_{vac}+\Omega_{th},$
(7)

where the primes represent regularization-free on thermal part. The general
parameters are given in Table. 1. In addition, the soft cutoff weight function
is chosen as

$\displaystyle
f_{\Lambda}(p)=\sqrt{\frac{\Lambda^{2N}}{\Lambda^{2N}+|\mathbf{p}|^{2N}}},$
(8)

here we use $N=5$. For Pauli-Villas regularization, we set $C_{a}=(1,1,-2)$,
$\alpha_{a}=(0,2,1)$ and $M_{a}^{2}=M^{2}+\alpha_{a}\Lambda^{2}$ for
$a=(0,1,2)$, respectively.

Regularization Scheme |  $\Lambda$(MeV) |  m(MeV) |  GS
---|---|---|---
three-momentum hard cutoff | 653 | 6 | $2.14/\Lambda^{2}$
three-momentum soft cutoff | 626.76 | 6 | $2.02/\Lambda^{2}$
Pauli-Villas regularization | 859 | 6 | $2.84/\Lambda^{2}$
Table 1: The parameters of three different regularization schemes
Klevansky:1992qe ; Fukushima:2010fe .

Also, we want to introduce the thermodynamics quantities which depend on the
grand potential directly in this work.The pressure is just equal to the
opposite sign of the grand potential, i.e., $P=-\Omega$. By introducing the
normalized grand potential we will normalize the pressure at zero temperature
and chemical potential equal to zero, then we have

$\displaystyle P(\mu,T)=\Omega(0,0)-\Omega(\mu,T),$ (9)

where $\Omega(0,0)$ is the grand potential in the vacuum. The energy density
$\epsilon$ is given by

$\displaystyle\epsilon=-T^{2}\left.\frac{\partial(\Omega/T)}{\partial
T}\right|_{V}=-T\left.\frac{\partial\Omega}{\partial
T}\right|_{V}+\Omega-\Omega(0,0),$ (10)

and the corresponding specific heat

$\displaystyle C_{V}=\left.\frac{\partial\epsilon}{\partial
T}\right|_{V}=-T\left.\frac{\partial^{2}\Omega}{\partial T^{2}}\right|_{V}.$
(11)

The square of velocity of sound at constant entropy $S$ is given by

$\displaystyle v_{s}^{2}=\left.\frac{\partial
P}{\partial\epsilon}\right|_{S}=\left.\frac{\partial\Omega}{\partial
T}\right|_{V}\bigg{/}\left.T\frac{\partial^{2}\Omega}{\partial
T^{2}}\right|_{V}.$ (12)

##  III Numerical Result

Figure 1: Constituent quark mass as a function of temperature with zero
chemical potential. From panel (a)-(c), the three momentum hard cut off, three
momentum soft cut off and Pauli-Villas regularization are used, respectively.
In all the panels, the red dashed lines represent we are using the
regularization on thermal part, the black solids line represent we are not
using the regularization on the thermal part and the purple dotted lines are
the current quark mass $m=6$ MeV.

For simplicity, in the numerical calculations, we only consider the zero
chemical potential case. It is obvious that the results should be equivalent
to finite chemical potential cases. By solving the gap equation we calculated
the constituent mass of quarks and several thermodynamical properties with
zero chemical potential by using different regularizations. The constituent
mass varies with temperature with different regularization schemes are showing
in Fig. 1. It is showing, by using all three different regularization schemes,
the constituent quark mass is smaller than the current quark mass at high
temperature when we don’t use the regularizations on the temperature part.
Therefore, the chiral condensate is positive in this case, this is definitely
an incorrect physics result. When we use the regularizations on the
temperature part, the chiral condensate goes towards zero when the temperature
is increasing, this means the chiral symmetry is restoring at high temperature
which is physically correct. By definition, the fact of constituent mass is
always larger than the current mass in the framework of NJL model has been
mentioned in Ref. Buballa:2003qv . In fact, this is not a model dependent
result. In the chiral limit, the chiral symmetry is an exactly symmetry. The
chiral condensate is negative in vacuum and increases to zero at some critical
temperature. Since the occurrence of chiral condensation is due to dynamical
effect, similar feature should follow for nonzero current quark mass, that is,
chiral condensate is always nonpositive at high temperature. Furthermore, the
change of chiral condensate from negative value through zero to positive one
would indicate that chiral symmetry is firstly restored and then broken again
by only increasing temperature. This of course does not make sense.

Figure 2: Dimensionless thermal properties as a function of normalized
temperature with zero chemical potential by using three momentum hard cutoff.
In all the panels, the red dashed lines represent we are using the
regularization on thermal part, the black solid lines represent we are not
using the regularization on the thermal part.

Figure 3: Dimensionless thermal properties as a function of normalized
temperature with zero chemical potential by using three momentum soft cutoff.
In all the panels, the red dashed lines represent we are using the
regularization on thermal part, the black solid lines represent we are not
using the regularization on the thermal part.

Figure 4: Dimensionless thermal properties as a function of normalized temperature with zero chemical potential by using Pauli-Villas regularization. In all the panels, the red dashed lines represent we are using the regularization on thermal part, the black solid lines represent we are not using the regularization on the thermal part. Regularization Scheme | With(MeV) | Without(MeV)
---|---|---
three momentum hard cutoff | 195 | 178
three momentum soft cutoff | 202 | 179
Pauli-Villas regularization | 172 | 138
Table 2: The value of $T_{0}$ from different regularization schemes.
“With”(“Without”) is standing for using(not using) the regularization on the
thermal part.

In Figs. 2$-$4, we show the dimensionless quantities of the pressure, energy
density, specific heat, and the square of sound velocity as a function of
normalized temperature $T/T_{0}$ with different regularization schemes(hard
cutoff, soft cutoff, Pauli-Villas regularization, respectively) at zero
chemical potential in the NJL model. $T_{0}$ is the critical temperature at
zero chemical potential, the values by using different regularization schemes
are given in Table 2. By applying all three different regularization schemes,
it is obvious to show the behavior with using and without using the
regularizations on the thermal part are quite different for these quantities.
It is well known when the temperature is increasing, the thermal quantities
should be approaching the results of the free Fermi gas. The most trivial
quantity is the speed of the sound, i.e., $v_{s}^{2}=1/3$ at high temperature.
It is easy to see when we are not using the regularizations on the thermal
part, the square of the speed of sound is approaching this value at high
temperature, and it is incorrect when we are using the regularizations on the
thermal part because it is larger than the speed of light when the temperature
is high. We have the same issue for other thermal quantities, no matter which
regularization we are using.

##  IV Conclusion

In this paper, we discussed a fundamental question, whether we need to apply
the regularization on the thermal part when we evaluate the physics quantities
in the NJL model by considering two sets of physics qualities: the chiral
condensate and thermodynamical quantities which is basically from the grand
potential. By selecting a wide range of temperature and three popular
regularization methods (include both gauge covariant and gauge invariant
schemes) in the NJL model, we find that to get a physical result, we need to
use(not use) the regularization on the thermal part when we are evaluating the
chiral condensate(grand potential) related physics quantities. A good example
is the net baryon fluctuation in QCD matter which is very useful for studying
the QCD phase transition, in the theoretical viewpoint, it purely depends on
orders of derivative of the grand potential which we should not use the
regularizations on the thermal part, e.g., in one of the author’s previous
papers Li:2018ygx ; Wang:2018sur . On the other hand, for some quantities such
as meson masses, which are directly related to the chiral condensate of
quarks, we need to use the regularizations on the thermal part, e.g., Refs.
Liu:2018zag ; Sheng:2020hge . We could also come back to the question left in
Refs. Yu:2015hym  which we mentioned in the introduction section. For
calculating the chiral condensate and related chiral phase transition
temperature which are evaluated by the derivative of the chiral condensate, we
need to use the regularization on the thermal part. Then the behavior of the
critical temperature and chiral condensate will consistent with the results
from lattice QCD Braguta:2014ira  and Dyson-Schwinger Equations Xu:2015vna . A
similar discussion about this specific problem is given in Ref. Cui:2016zqp .

Another thing we want to mention here is, although the numerical result is
from the simplest two flavor NJL model, this should be correct for all NJL
based models because of the model consistency. Also, we have covered the
results for both gauge covariant and gauge invariant regularization schemes,
our conclusion should be good for using all other regularization schemes.
Therefore, the results should help understand the physics qualities of QCD
matter, especially in the non-perturbation regime. Also, this work should be
very helpful for beginners who are working on NJL based model projects.

###### Acknowledgements.

The work of X. Yu was supported by the Natural Science foundation of Jiangsu
province under Grants No.BK20140523 and the start-up funding No. 15JDG042 of
Jiangsu University. The work of X.W. was supported by the start-up funding No.
4111190010 of Jiangsu University and NSFC under Grant No. 11735007.

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