# Observational tests in scale invariance II: gravitational lensing

André Maeder  Frédéric Courbin

###### Abstract

We study the path of light rays passing near a massive object, in the context
of the scale invariant equation of the geodesics first obtained by Dirac
(1973). Using the exterior Schwarzschild solution for the metric, we derive
the complete equations of the geodesics in the scale invariant context. We
find that scale invariance introduces two additional terms to the Einstein
term producing the deflection angle and that can potentially act over
cosmological distances. Numerical integration of the scale-invariant
geodesics, for the specific case of the $z_{L}=1.94$ lens galaxy in the
extreme system JWST-ER1 (van Dokkum et al. 2023) shows that the two additional
terms introduce only negligible effects, typically $\mathcal{O}$($10^{-6}$) of
the Einstein term. We conclude that the lensing deflection angle derived in
Einstein’s General Relativity is essentially independent of the scale
invariant effects and that the photon’s geodesics remain unchanged. We also
explore the possible origin of the differences in the mass estimates from
lensing and photometry in JWST-ER1 and in the SLACS galaxies, differences
which appear larger at higher redshifts.

##  1 Introduction

Gravitational lensing is a powerful natural phenomenon featuring a vast range
of astrophysical and cosmological applications. Its major strength is that it
gives access to quantities that are notoriously hard to measure otherwise,
masses and mass distributions at all spatial scales being the main ones.
Gravitational lensing is also one of the key observations that confirmed
General Relativity (GR), through the measurement of the light deflection angle
by the Sun during the 1919 Solar eclipse [10]. It therefore consists in a
perfect laboratory to test alternative scenario to $\Lambda$CDM, such as scale
invariant theory [19, 21, 26].

This is particularly useful in light of recent observational tests of scale
invariant theory, (see paper I [22]). It explains: - why the standard mass
estimates from galaxy velocities are overestimating the mass of galaxy
clusters, - flat rotation curves of galaxies [20], - why the observed
projected velocities of wide binary stars are overestimated by standard theory
as their angular separation increases [13]. Other effects are mentioned in
Paper I.

In the present work, we confront scale invariant theory to strong gravitation
lensing observations at galaxy scale. We compute the geodesic equations in the
scale invariance context and show that they do not differ from the standard GR
derivation. While scale invariance has no effect on the photons’ geodesics
themselves, we show that it has an effect on the interpretation and use of the
mass-luminosity relation of the stellar population in the lens galaxy and of
its IMF. We illustrate this with real observations of strongly lensed
galaxies, focusing on the extreme case of the ”dark lens” JWST-ER1 [38] at
$z_{L}=1.94$ and on the text-book SLACS sample of lenses at $z_{L}$between 0
and 0.5 for which excellent spectroscopic observations are available [1]. This
allows us to confront the standard lensing masses at very different redshifts
to the luminous masses in the scale invariant framework.

In Section 2 we give a brief summary of the scale invariant theory. Section 3
describes the calculation of trajectories of lensed light paths in the scale
invariant context. Section 4 shows from accurate numerical integration that
scale invariant effects on the light rays are negligible. Section 5 examines
the origin of the discrepancy between the stellar masses and the masses from
the lensing and exposes the scale invariant effects that reconcile the two
mass estimates. These are further discussed in Section 6 which also proposes
additional future tests involving gravitational lensing. Finally, Section 7
gives our conclusions.

##  2  Basics of scale invariant theory

We summarize in the following some basic properties of scale invariant theory,
which is described in more detail in Maeder [21]. In this theory, only one
additional symmetry is considered in addition to the covariance of General
Relativity, _invariance to a scale transformation._ This has fundamental
implications on our interpretation of GR and simply boils down to write

$ds^{\prime}=\lambda(x^{\mu})\;ds,$ (2.1)

where $\lambda$ is the scale factor and $ds^{\prime}$ and $ds$ respectively
are the line elements in GR and in a new space (the WIG space specified
below). We point out that scale invariance is present in General Relativity
(GR) in empty space without a cosmological constant, it is also present in
Maxwell equations without charges and currents. Thus, we may wonder whether
some limited effects of scale invariance are still present in our low density
Universe. Weyl [40] and Eddington [11] considered scale invariance in an
attempt to account for electromagnetism by a geometrical property of space-
time. The proposition was abandonned because Einstein [12] had shown that the
properties of a particle would depend on its past world line, so that atoms in
an electromagnetic field would not show sharp lines. Dirac [8] and Canuto et
al. [5] considered an interesting new possibility: the Weyl Integrable
Geometry (WIG) where Einstein’s criticism does not apply.

Alike GR, the WIG space is endowed with a metrical connection
$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}$ and a line element $ds^{\prime}$ of GR
transforms in WIG space like (2.1), implying a conformal relation
$g^{\prime}_{\mu\nu}=\lambda^{2}g_{\mu\nu}$. In the transport from a point
$P_{0}(x^{\mu})$ to a nearby point $P_{1}(x^{\mu}+dx^{\mu})$, the length
$\ell$ of a vector is assumed to change by
$\;d\ell\,=\,\ell\,\kappa_{\nu}\,dx^{\nu}.$ There, $\kappa_{\nu}$ is called
the coefficient of metrical connection, it is a fundamental characteristics of
the geometry alike $g_{\mu\nu}$, (in GR $\kappa^{\prime}_{\nu}=0$). The
specificity of the WIG is that $\kappa_{\nu}$ is the gradient of a scalar
field $\Psi$ related to $\lambda$ [8, 5],

$\kappa_{\nu}=-\Psi_{,\nu}=-\frac{\partial\ln\lambda}{dx^{\nu}},\quad\quad\mathrm{with}\;\;\Psi=\ln\lambda.$
(2.2)

Thus $\kappa_{\nu}$ is a perfect differential with
$\partial_{\nu}\kappa_{\mu}=\partial_{\mu}\kappa_{\nu}$ and the length change
of a vector does not depend on the path followed, thus the criticism by
Einstein does not apply.

Scale invariance demands some developments of the tensor calculus, with the
introduction of _cotensors_. However, we do not enter the subject here, just
mentioning that a brief summary of co-tensor calculus has been given by Canuto
et al. [5]. Many useful expressions can also be found in Dirac [8], as well as
in Bouvier & Maeder [4].

###  2.1 The geodesics from an action principle and the Newtonian
approximation

To study lensing in the scale invariant context, we need the appropriate
equation of geodesics. It can be obtained in different ways, in WIG alike in
GR: \- 1. By assuming that the geodesic is a curve such that a co-vector
tangent to the curve is always transported by parallel displacement along the
curve [5]. \- 2. From the Equivalence Principle [23], which states [39] that
at every point of the space-time there is a local inertial coordinate system
$x^{\prime\alpha}$ such that $\frac{d^{2}x^{\prime\alpha}}{ds^{\prime 2}}=0$.
- 3. Dirac [8] obtained the geodesic expression from an action principle, see
also [22]. All methods are leading to,

$\frac{du^{\alpha}}{ds}+\Gamma^{\alpha}_{\mu\nu}\,u^{\mu}u^{\nu}-\kappa_{\nu}u^{\nu}u^{\alpha}+\kappa^{\alpha}=0,$
(2.3)
$\Gamma^{\alpha}_{\mu\nu}=\frac{1}{2}g^{\sigma\alpha}\left(\frac{\partial
g_{\nu\,\sigma}}{\partial x^{\mu}}+\frac{\partial g_{\mu\,\sigma}}{\partial
x^{\nu}}-\frac{\partial g_{\nu\mu}}{\partial x^{\,\sigma}}\right)\,.$ (2.4)

Eq.(2.3) only depends on the geometrical factors $g_{\nu\mu}$ and
$\kappa_{\nu}$. If $\lambda$ is a constant, one has $\kappa_{\nu}=0$ and the
geodetic equation is the same as the classical one, cf. Weinberg [39]. As
discussed below (Sect. 2.2), the scale factor $\lambda$ depends only on time,
so that $\kappa_{\nu}$ reduces to $\kappa_{0}=1/t$.

The Newtonian approximation was obtained by Maeder & Bouvier [23] for a weak
stationary potential $\Phi$, with a line element only slightly differing from
Minkowski’s, and assuming low velocities with respect to $c$. In vectorial
form, we have,

$\frac{d^{2}\vec{r}}{dt^{2}}=-\frac{GM}{r^{2}}\frac{\vec{r}}{r}+\kappa_{0}\frac{d\vec{r}}{dt}\,.$
(2.5)

An acceleration term proportional to the velocity and in the same direction
appears in the 2nd member. It is such as to accelerate an expansion or a
contraction depending on circumstances. Such effects appear e.g. in the
cosmological models [19] and in the growth of density fluctuations in the
early Universe [24].

Some properties of the theory, such as $\lambda$, $\kappa_{\nu}$ and the way
the differents physical quantities depend on the scale $\lambda$, are
determined by the conservation laws and the considered cosmological models.
Thus, it is now necessary to specify a few physical properties in presence of
a varying gauge.

###  2.2 The scale invariant field equation and the gauge

A general scale invariant field equation was obtained by Canuto et al. [5].
Recently a full detailed demonstration of this equation from an action
principle has also been given [26]. We get

$\displaystyle R^{\prime}_{\mu\nu}-\frac{1}{2}\
g_{\mu\nu}R^{\prime}-\kappa_{\mu,\nu}-\kappa_{\nu,\mu}-2\kappa_{\mu}\kappa_{\nu}+2g_{\mu\nu}\kappa^{\alpha}_{,\alpha}-g_{\mu\nu}\kappa^{\alpha}\kappa_{\alpha}=-8\pi
GT_{\mu\nu}-\Lambda\,g_{\mu\nu},\,$ (2.6)

where $R^{\prime}_{\mu\nu}$ is the Ricci tensor in GR and $R^{\prime}$ its
contracted form, $\Lambda=\lambda^{2}\Lambda_{\mathrm{E}}$, where
$\Lambda_{\mathrm{E}}$ is the cosmological constant in GR, $G$ is a true
constant. The second member of the field equation must also be scale
invariant, and the energy-momentum tensor $G\,T_{\mu\nu}$ has the same
property. Thus, $T_{\mu\nu}\,=\,T^{\prime}_{\mu\nu}$, which has implications
on the relevant densities [5],

$p=p^{\prime}\,\lambda^{2}\,\quad\mathrm{and}\quad\varrho=\varrho^{\prime}\,\lambda^{2}\,.$
(2.7)

The consistency of the field equation means that $p$ and $\varrho$ are not
scale invariant, (they are said to be coscalars of power $\Pi(\rho)=-2$). This
has some consequences regarding the equation of conservation (2.16).

Chosing a metric is necessary to fix the $g_{\mu\nu}$, similarly a gauging
condition is necessary to fix the scale or gauge $\lambda$. Dirac [8] and
Canuto et al. [5] had chosen the so-called “Large Number Hypothesis“ [9]. The
author’s choice is to adopt the following statement [19]: _The macroscopic
empty space is scale invariant, homogeneous and isotropic_. This gauging
condition implies $\lambda=\lambda(t)$, since the empty space is homogeneous
and isotropic. Applied to the field equation in empty space, the gauging
condition is leading to two differential equation for $\lambda$ which are then
only a function of time,

$\displaystyle\
3\,\frac{\dot{\lambda}^{2}}{\lambda^{2}}\,=\,\lambda^{2}\,\Lambda_{\mathrm{E}}\,\quad\mathrm{and}\quad\,2\frac{\ddot{\lambda}}{\lambda}-\frac{\dot{\lambda}^{2}}{\lambda^{2}}\,=\,\lambda^{2}\,\Lambda_{\mathrm{E}}\,,$
(2.8)

Interestingly enough, the trace of Eq. (2.6) in empty space is also leading to
these two equations. Their solution is,

$\lambda(t)=\sqrt{\frac{3}{\Lambda_{\mathrm{E}}}}\frac{1}{ct}\,\quad\mathrm{or}\;\;\lambda(t)=\frac{t}{t_{0}},$
(2.9)

with $\lambda=1$ for $t_{0}=1$. This implies that the only component of
$\kappa_{\nu}$ is $\kappa_{0}=-\dot{\lambda}/\lambda=1/t.$ The form of Eqs.
(2.8) and (2.9) is independent of the matter content according to the gauging
condition, but the range of variations of $t$, and thus of $\lambda$ and
$\kappa_{\nu}$, are set by the considered comological models.

###  2.3 Cosmological constraints

For detailed developments of the scale invariant theory and cosmology, the
reader may refer to [19, 26]. The application of the FLWR metric to Eq. (2.6)
led to rather ”heavy” cosmological equations [5], however the equations (2.8)
bring interesting simplifications, letting only one term with a
$\lambda$-function,

$\displaystyle\frac{8\,\pi G\varrho}{3}$ $\displaystyle=$
$\displaystyle\frac{k}{a^{2}}+\frac{\dot{a}^{2}}{a^{2}}+2\,\frac{\dot{a}\dot{\lambda}}{a\lambda},$
(2.10) $\displaystyle-8\,\pi Gp$ $\displaystyle=$
$\displaystyle\frac{k}{a^{2}}+2\frac{\ddot{a}}{a}+\frac{\dot{a^{2}}}{a^{2}}+4\frac{\dot{a}\dot{\lambda}}{a\lambda},$
(2.11) $\displaystyle-\frac{4\pi G}{3}\left(3p+\varrho\right)$
$\displaystyle=$
$\displaystyle\frac{\ddot{a}}{a}+\frac{\dot{a}\dot{\lambda}}{a\lambda}.$
(2.12)

These equations differ from Friedmann’s only by the $\lambda$-terms. For a
constant $\lambda$, Friedmann’s equations are recovered. The 3rd equation is
derived from the first two. Since $\dot{\lambda}/{\lambda}$ is negative, the
extra term represents an additional acceleration _in the direction of motion_
and is thus fundamentally different from that of the cosmological constant.

Analytical solutions for the flat SIV models with $k=0$ were found for the
matter era by Jesus [15],

$a(t)=\left[\frac{t^{3}-\Omega_{\mathrm{m}}}{1-\Omega_{\mathrm{m}}}\right]^{2/3},\quad\quad\mathrm{the\;initial\;time\;is}\;\;t_{\mathrm{in}}=\Omega^{1/3}_{\mathrm{m}}.$
(2.13)

The cosmological models, whatever curvature $k$, show an accelerated expansion
[19]. These equations allow flatness for different values of
$\Omega_{\mathrm{m}}$, unlike Friedmann’s models. At present $t_{0}=1$ and
$a(t_{0})=1$. The graphical solutions are relatively close to the
corresponding $\Lambda$CDM models with larger differences for lower
$\Omega_{\mathrm{m}}$. The initial time $t_{\mathrm{in}}$ of the origin is
defined by $a(t_{\mathrm{in}})=0$. Its dependence in
$\Omega^{1/3}_{\mathrm{m}}$ produces a rapid increase of $t_{\mathrm{in}}$ for
increasing $\Omega_{\mathrm{m}}$. For
$\Omega_{\mathrm{m}}=0,0.01,0.1,0.3,0.5$, the values of $t_{\mathrm{in}}$ are
0, 0.215, 0.464, 0.669, 0.794 respectively. This strongly reduces the possible
range of $\lambda(t)=(t_{0}/t)$ variations. Scale invariant effects are
drastically reduced in Universe models with increasing $\Omega_{\mathrm{m}}$.
They are killed for $\Omega_{\mathrm{m}}\geq 1$.

The relation between the ages in the $t$ scale and $\tau$ in the current units
is evidently
$\frac{\tau-\tau_{\mathrm{in}}}{\tau_{0}-\tau_{\mathrm{in}}}=\frac{t-t_{\mathrm{in}}}{t_{0}-t_{\mathrm{in}}}$.
This gives with $\tau_{0}$ the present age (13.8 Gyr),

$\displaystyle\tau=\tau_{0}\,\frac{t-t_{\mathrm{in}}}{1-t_{\mathrm{in}}}\,\;\mathrm{and}\;\;t=t_{\mathrm{in}}+\frac{\tau}{\tau_{0}}(1-t_{\mathrm{in}}),\;\mathrm{with}\;\frac{d\tau}{dt}\,=\,\frac{\tau_{0}}{1-t_{\mathrm{in}}},$
(2.14)

Eqs. (2.10) and (2.11) lead to the scale invariant mass-energy conservation
law [5, 19],

$\displaystyle\frac{d(\varrho
a^{3}}{da}+3\,pa^{2}+(\varrho+3p)\frac{a^{3}}{\lambda}\frac{d\lambda}{da}=0,$
(2.15)
$\displaystyle\mathrm{and}\quad\varrho\,a^{3(1+w)}\lambda^{1+3w}=\mathrm{const.}\quad\mathrm{with\;p=w\varrho},$
(2.16)

(there $c=1$). For ordinary matter $w=0$ and
$\varrho_{\mathrm{m}}a^{3}\lambda=const.$ A length element scales like
$\lambda^{-1}$, thus from (2.16) the mass also varies like $\lambda^{-1}\sim
t$. Such a mass variation might at first be surprising, however let us recall
that the invariance to Lorentz transformations, assumed in Special Relativity,
is also leading to the concept of non-constant masses, which may vary up to
infinity as a function of the velocities. Here, as in Special Relativity the
number of particles remains the same, but their inertial and gravitational
properties may vary. This property is quite consistent with the above
mentioned scale invariance of the energy momentum tensor $T_{\mu\nu}$ in the
second member of the general field equation (2.6). It implies that
$\varrho=\varrho^{\prime}\lambda^{2}$ (cf. Eq. 2.7) or
$\frac{M}{R^{3}}=\lambda^{2}\frac{M^{\prime}}{R^{\prime 3}}$, since
$R^{\prime}=\lambda R$ one has $M=M^{\prime}/\lambda$, meaning that $M\sim t$.
A noticeable consequence of the mass variation is that the gravitational
potential $\Phi=-GM/R$ of a given mass appears as a more fundamental quantity
than the mass, being scale invariant and thus independent of time through the
evolution of the Universe. The mass changes are very limited, since the range
of $t$ and $\lambda$ is also very limited,

$M(t)=\frac{M(t_{0})}{\lambda}=M(t_{0})\frac{t}{t_{0}},\quad\mathrm{or}\;\;M(\tau)=\tau_{0}\times[t_{\mathrm{in}}+\frac{\tau}{\tau_{0}}(1-t_{\mathrm{in}})].$
(2.17)

As an example for $\Omega_{\mathrm{m}}=0.30$, 400 Myr ago masses were
differing by less than 1% from today’s. Equation (2.5) was expressed in
variable $t$, we need it in variable $\tau$. At present time $\tau_{0}$, one
has [21]

$\frac{d^{2}\bf{r}}{d\tau^{2}}=-\frac{GM}{r^{2}}\frac{\bf{r}}{r}+\frac{\psi_{0}}{\tau_{0}}\frac{d\bf{r}}{d\tau},\quad\mathrm{with}\;\;\psi_{0}=1-t_{\mathrm{in}}\,.$
(2.18)

Term $\psi_{0}$ depends on $\Omega_{\mathrm{m}}$. A range of values of
$\Omega_{\mathrm{m}}=0,0.05,0.10,0.20,0.30,1$, translates in
$\psi_{0}=1,0.632,0.536,0.415,0.331,0$. At other epochs, $\tau$, the constant
$\psi_{0}$ factor then becomes

$\psi(\tau)=\frac{t_{0}-t_{\mathrm{in}}}{\left[t_{\mathrm{in}}+\frac{\tau}{\tau_{0}}(t_{0}-t_{\mathrm{in}})\right]}.$
(2.19)

The additional acceleration, the dynamical gravity, is generally small since
$\tau_{0}$ is large. In an empty universe, $\psi_{0}=1$ and the dynamical
acceleration is the largest, while for $\Omega_{\mathrm{m}}\geq 1$,
$\psi_{0}=0$ and the additional term consistently disappears. Also, the
following relations have been established,

$\displaystyle
H_{0}=\frac{\xi}{\tau_{0}},\quad\mathrm{with}\;\;H_{0}=\frac{2(1-\Omega^{1/3}_{\mathrm{m}})}{\tau_{0}(1-\Omega_{\mathrm{m}})},\quad\mathrm{and}\;\xi=\frac{2(1-\Omega^{1/3}_{\mathrm{m}})}{(1-\Omega_{\mathrm{m}})},$
(2.20)

very useful to connect $H_{0}$ and the age of the Universe $\tau_{0}$.

##  3 The lensing effect: the geodesic of deflected light rays

In General Relativity, the geodesic equation for weak fields is the basis for
the study of a number of effects, among which in particular the Newtonian
equation of motion for material objects, as well various post-Newtonian
effects. The geodesic is also the basis to obtain the trajectories of light
rays with their possible deflection by some mass distribution, giving rise to
the lensing effect. The theory of the lensing shows that this effect depends
on the Newtonian potential $\Phi(r)$ of the mass $M$ acting as a deflector.
However, in the scale invariant theory, in addition to the gravity effects
depending on the potential, a dynamical acceleration depending on the
velocities is occurring. Thus, it is necessary to study the problem of lensing
_ab initio_ from the fundamental geodesic Eq. (2.3).

In non-standard theories, like MOND, we note it is not established by a study
of the geodesics whether some modified forms of the potential, adjusted to
account for dynamical effects, can give rise to the correct expression of the
deflection as is the case in GR [27]. We even have the exemple that in the
Newton theory the simple application of the Newtonian potential predicts a
light deflection to small by a factor of two with respect to the correct
value. Only a detailed calculation of the trajectories of the light rays may
give a safe prediction of the light deflection.

Therefore, the starting point must be the equation of the geodesic as given by
Eq. (2.3),

$\frac{du^{\alpha}}{ds}+\Gamma^{\alpha}_{\mu\nu}u^{\mu}u^{\nu}-\kappa_{\mu}u^{\mu}u^{\alpha}+\kappa^{\alpha}=0.$
(3.1)

For the metric of the system, we adopt the exterior Schwarzschild solution,

$ds^{2}=\frac{-dr^{2}}{1-\frac{2GM}{rc^{2}}-\frac{\Lambda
r^{2}}{3}}-r^{2}{d\vartheta^{2}+\sin^{2}\vartheta
d\varphi^{2}}+\left(1-\frac{2GM}{rc^{2}}-\frac{\Lambda
r^{2}}{3}\right)c^{2}dt^{2},$ (3.2)

with $c$ being explicited here. In the scale invariant system,
$\Lambda=\lambda^{2}\Lambda_{\mathrm{E}}$ where $\Lambda_{\mathrm{E}}$ is the
cosmological constant in the GR context (cf. Eq. 2.6).

###  3.1 Scale invariant equations of motion in polar ccordinates

The above equation of geodesics differs from the classical case of GR by the
terms $-\kappa_{\mu}u^{\mu}u^{\alpha}+\kappa^{\alpha}$. These terms are for
the different $\alpha$-coordinates,

$\displaystyle\alpha=1$ $\displaystyle:$
$\displaystyle\quad-\kappa_{\mu}u^{\mu}\frac{dr}{ds}\quad\rightarrow\quad-\kappa_{0}\frac{dt}{ds}\frac{dr}{ds},$
(3.3) $\displaystyle\alpha=2$ $\displaystyle:$
$\displaystyle\quad-\kappa_{\mu}u^{\mu}\frac{d\theta}{ds}\quad\rightarrow\quad-\kappa_{0}\frac{dt}{ds}\frac{d\theta}{ds},$
$\displaystyle\;\alpha=3$ $\displaystyle:$
$\displaystyle\quad-\kappa_{\mu}u^{\mu}\frac{d\phi}{ds}\quad\rightarrow\quad-\kappa_{0}\frac{dt}{ds}\frac{d\phi}{ds},$
$\displaystyle\alpha=0$ $\displaystyle:$
$\displaystyle\quad-\kappa_{\mu}u^{\mu}\frac{dt}{ds}+\kappa_{0}\quad\rightarrow\quad-\kappa_{0}\frac{dt}{ds}\frac{dt}{ds}+\frac{\kappa_{0}}{c^{2}}.$

We recall that $\kappa_{0}=-\dot{\lambda}/\lambda=1/t$ and we now write $c$
explicitely. The above terms are to be added the classical geodesic equation.
We adopt the well known simplified form of the metric,

$ds^{2}\,=\,-e^{\lambda}dr^{2}-r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})+e^{\nu}c^{2}dt^{2},$ (3.4)

and have the correspondences,

$e^{\lambda}=\frac{1}{1-\frac{2m}{r}-\frac{\Lambda
r^{2}}{3}},\quad\mathrm{and}\;\;e^{\nu}=\left(1-\frac{2m}{r}-\frac{\Lambda
r^{2}}{3}\right),$ (3.5)

where we write $m=GM/(c^{2})$, this quantity has the units of a length and it
is subject to variations given by Eq. (2.17). The above terms additionally
appear to the classical expressions of the components of the geodesics, [36,
39]. We are also supposing that the motions occur in a plane with a constant
value of $\theta=\pi/2$, and get

$\displaystyle\frac{d^{2}r}{ds^{2}}+\frac{1}{2}\frac{d\lambda}{dr}\left(\frac{dr}{ds}\right)^{2}-re^{-\lambda}\left(\frac{d\phi}{ds}\right)^{2}+\frac{e^{\nu-\lambda}}{2}\frac{d\nu}{dr}\left(\frac{c^{2}\,dt}{ds}\right)^{2}-\kappa_{0}\frac{dt}{ds}\frac{dr}{ds}$
$\displaystyle=$ $\displaystyle 0,$ (3.6)
$\displaystyle\frac{d^{2}\phi}{ds^{2}}+\frac{2}{r}\frac{dr}{ds}\frac{d\phi}{ds}-\kappa_{0}\frac{dt}{ds}\frac{d\phi}{ds}$
$\displaystyle=$ $\displaystyle 0,$ (3.7)
$\displaystyle\frac{d^{2}t}{ds^{2}}+\frac{d\nu}{ds}\frac{dt}{ds}-\kappa_{0}\left(\frac{dt}{ds}\right)^{2}+\frac{\kappa_{0}}{c^{2}}$
$\displaystyle=$ $\displaystyle 0,$ (3.8)

where we may verify the consistency of the dimensions. The last two equations
may be easily integrated. Let us call $\upsilon=\frac{d\phi}{ds}$, getting for
(3.7),

$\frac{d\upsilon}{ds}+\frac{2}{r}\frac{dr}{ds}\upsilon-\frac{1}{t}\frac{dt}{ds}\upsilon=0,\quad\mathrm{and}\;\;\frac{d\upsilon}{\upsilon}+2\frac{dr}{r}-\frac{dt}{t}=0.$
(3.9)

This gives

$r^{2}\,\frac{d\phi}{ds}=ht,$ (3.10)

where $h$ is an integration constant. This is the appropriate conservation law
of the angular momentum. We see that it increases with time $t$, a behaviour
which is quite consistent with that in the scale invariant two-body problem
[23, 20], where there is a slow secular increase of the orbital radius of a
body. At the same time, the orbiting body is keeping a constant orbital
velocity $r\,\frac{d\phi}{ds}$ during its evolution. These well established
secular effects of scale invariance are always very small in our current
Universe, since they are globally scaling with the inverse of the age of the
Universe. Turning to the third equation, we set $\frac{dt}{ds}=w$ and have,

$\frac{dw}{ds}+\frac{d\nu}{ds}w-\frac{1}{t}w^{2}+\frac{1}{c^{2}t}=0$ (3.11)

Let us then set $w=dt/ds=k\,e^{-\nu(s)}$, where $k$ is some constant. The
first two terms cancel each other and the remaining other two give,

$-k^{2}\,e^{{}^{2\nu}}+\frac{1}{c^{2}}=0\quad\mathrm{thus}\;\;\frac{1}{c}=k\,e^{-\nu(s)}=\frac{dt}{ds},$
(3.12)

which is quite consistent if we take the sign ”+”. This equation implies that
the cosmic time is also the proper time of the system. This also implies that
$\nu(s)$ is scale invariant, which is easily verified by considering the terms
in the parenthesis of expression (3.2): both the mass and radius are scaling
like $1/\lambda$ and $\Lambda$ is scaling like $1/\lambda^{2}$.

Now, we turn to the more difficult Eq. (3.6). In this respect, let us quote
Eddington [11] who said about the corresponding equation in the GR
context:_”Instead of troubling to integrate ( 3.6) we can use in place of it
(3.4)_.111Here, the labels of both equations have been changed appropriately.
Thus from (3.4) we get,

$e^{\lambda}\left(\frac{dr}{ds}\right)^{2}+r^{2}\left(\frac{d\phi}{ds}\right)^{2}-e^{\nu}c^{2}\left(\frac{dt}{ds}\right)^{2}+1=0.$
(3.13)

With the above expression for $w$, the third term can also be written

$-e^{\nu}c^{2}k^{2}e^{-2\nu}=-c^{2}k^{2}e^{-\nu}=-\frac{c^{2}k^{2}}{[1-\frac{2GM}{rc^{2}}-\frac{\Lambda
r^{2}}{3}]}.$ (3.14)

Thus, the above equation becomes,

$\displaystyle\left(\frac{dr}{ds}\right)^{2}+r^{2}\left(\frac{d\phi}{ds}\right)^{2}-\frac{2m}{r}\left(1+r^{2}(\frac{d\phi}{ds})^{2}\right)-\frac{\Lambda
r^{2}}{3}\left(1+r^{2}(\frac{d\phi}{ds})^{2}\right)=c^{2}k^{2}-1.$ (3.15)

Eliminating $(\frac{d\phi}{ds})$ with (3.10), we get

$\left(\frac{ht}{r^{2}}\frac{dr}{d\phi}\right)^{2}+\frac{h^{2}t^{2}}{r^{2}}=c^{2}k^{2}-1+\frac{2m}{r}+\frac{2mh^{2}t^{2}}{r^{3}}+\frac{\Lambda
r^{2}}{3}+\frac{\Lambda h^{2}t^{2}}{3}.$ (3.16)

We divide this equation by $h^{2}t^{2}$ and then have an equation analogous to
the Binet equation which describes the planetary orbit in polar coordinates by
setting $u=1/r$,

$\left(\frac{du}{d\phi}\right)^{2}+u^{2}=\frac{c^{2}k^{2}-1}{h^{2}t^{2}}+\frac{2mu}{h^{2}t^{2}}+2mu^{3}+\frac{\Lambda}{3u^{2}h^{2}t^{2}}+\frac{\Lambda}{3}.$
(3.17)

Now, we derive this equation with respect to $\phi$ and obtain,

$\displaystyle
2\left(\frac{du}{d\phi}\right)\frac{d^{2}u}{d\phi^{2}}+2u\frac{du}{d\phi}$
$\displaystyle=$
$\displaystyle\frac{c^{2}k^{2}-1}{h^{2}}\frac{d}{d\phi}\left(\frac{1}{t^{2}}\right)+\frac{2m}{h^{2}t^{2}}\frac{du}{d\phi}$
(3.18) $\displaystyle+$
$\displaystyle\frac{2u}{h^{2}t^{2}}\frac{dm}{d\phi}+\frac{2mu}{h^{2}}\frac{d}{d\phi}\left(\frac{1}{t^{2}}\right)+6mu^{2}\frac{du}{d\phi}+2u^{3}\frac{dm}{d\phi}$
$\displaystyle-$ $\displaystyle
2\frac{\Lambda_{\mathrm{E}}\lambda^{2}}{3h^{2}t^{2}}\frac{1}{u^{3}}\frac{du}{d\phi}+\frac{\Lambda_{\mathrm{E}}}{3h^{2}u^{2}}\frac{d}{d\phi}\left(\frac{\lambda^{2}}{t^{2}}\right)+\frac{2\Lambda_{\mathrm{E}}\lambda}{3}\frac{d\lambda}{d\phi}.$

There, we have expressed $\Lambda=\Lambda_{\mathrm{E}}\lambda^{2}$ as in Eq.
(2.6). There is also a direct relation between $\phi$ and time since the time
$t$ is uniquely defined along the geodesic described by the variation of angle
$\phi$, see Fig. 1. Thus, $\lambda$ which is a function of time $t$ is also a
function of $\phi$. Let us now multiply the result by
$\frac{1}{2}\frac{d\phi}{du}$,

$\displaystyle\frac{d^{2}u}{d\phi^{2}}+u$ $\displaystyle=$
$\displaystyle\frac{c^{2}k^{2}-1}{2h^{2}}\frac{d}{du}\left(\frac{1}{t^{2}}\right)+\frac{m}{h^{2}t^{2}}+\frac{u}{h^{2}t^{2}}\frac{dm}{du}+\frac{mu}{h^{2}}\frac{d}{du}\left(\frac{1}{t^{2}}\right)$
(3.19) $\displaystyle+$ $\displaystyle
3mu^{2}+u^{3}\frac{dm}{du}-\frac{\Lambda_{\mathrm{E}}\lambda^{2}}{3h^{2}t^{2}}\frac{1}{u^{3}}+\frac{\Lambda_{\mathrm{E}}}{6h^{2}u^{2}}\frac{d}{du}\left(\frac{\lambda^{2}}{t^{2}}\right)+\frac{\Lambda_{\mathrm{E}}\lambda}{3}\frac{d\lambda}{du},$

The terms containing $\Lambda_{\mathrm{E}}$ and the derivatives of $m$ are
currently not present. This equation is the basic one combining the effects of
General Relativity and scale invariance to study the advance of the perihelion
of planets, the light deflection by a massive body as well as the
gravitational shift of spectral lines. In the Newtonian approximation, it
gives the current Binet equation of planetary orbits in polar coordinates
[23].

###  3.2 The geodesics of light rays

The trajectory of a light ray alike that of a free particle is governed by the
equation of geodesics, however with the additional condition $ds=0$, _i.e._
its proper time does never change. According to Eq. (3.10), this implies
$h=\infty$, also meaning that the angular momentum associated to a light ray
cannot be considered as a finite quantity. This is leading to major
simplifications in the above equation (3.19), where only 3 terms are remaining
on the right side. From the gauging condition Eq. (2.8)
$3\,\frac{\dot{\lambda}^{2}}{\lambda^{2}}\,=\,c^{2}\lambda^{2}\,\Lambda_{\mathrm{E}}$
(with a term $c^{2}$ for the consistency of the units), we got with
$\lambda=t_{0}/t$ in the scale invariant theory
$\Lambda_{\mathrm{E}}=\frac{3}{c^{2}\,t^{2}_{0}}$ and in current units
$\Lambda_{\mathrm{E}}=\frac{3}{c^{2}\,\tau^{2}_{0}}$ with $\tau_{0}$ the age
of the Universe. Thus, Eq.(3.19) becomes

$\frac{d^{2}u}{d\phi^{2}}+u=3mu^{2}+u^{3}\frac{dm}{du}+\left(\frac{1}{c^{2}\tau^{2}_{0}}\right)\lambda\frac{d\lambda}{du}.$
(3.20)

The scale factor $\lambda$ is subject the the limitations mentioned in
Sect.2.3, between a value $t_{\mathrm{in}}=\Omega^{1/3}_{\mathrm{m}}$ and
$t_{0}=1$. One is noting that if $\lambda$ is a constant, we again have the
fundamental equation for lensing in GR, since $dm$ also disappears. Quantity
$c\tau_{0}$ is a distance in Gyr (or seconds) equal to the age the Universe,
it is of the order of the radius of the Hubble sphere $c/H_{0}$.

We recall that the observable Universe has a different definition given by the
particle horizon $d_{\mathrm{H}}$ [33, 34]. It has been verified that SIV
models with $\Omega_{\mathrm{m}}<1$ do have a particle horizon [25]. The
different estimates of the size of the observable Universe are of the same
order of magnitude as $c\tau_{0}$, although their formal definitions are
different. For example in the EdS model $d_{\mathrm{H}}=3c\tau_{0}$. There,
the very initial expansion, before a further global braking, was much faster
than the present one, thus a numerical factor larger than 1. The SIV models
show an initial phase of moderate braking followed by a progressive
acceleration phase [19], thus their $d_{\mathrm{H}}$ are close to $c\tau_{0}$.
Thus, we may consider $c\tau_{0}$ as the order of magnitude of the present
”size” of the connected Universe.

####  3.2.1 The classical cases

In absence of the second member of equation (3.20),

$\frac{d^{2}u}{d\phi^{2}}+u=0,\quad\mathrm{one\;has}\;\;u_{0}\,=\,\frac{\cos\phi}{R}$
(3.21)

as a solution, where $R$ is the impact parameter (Fig. 1). With this solution,
the two terms on the left side cancel each other. When $\phi$ is increasing
from zero $\rightarrow\pi/2$, a straight line is described (cf. Fig. 1), as
expected for a light ray in absence of a massive body.

The terms on the right of (3.20) are generally very small, in particular the
last two, as we shall below. The usual treatment [11] for the first term is to
introduce the solution (3.21) into $3mu^{2}$ and to solve the equation,

$\frac{d^{2}u}{d\phi^{2}}+u=\frac{3m}{R^{2}}\cos^{2}\phi.$ (3.22)

A particular solution of this equation is
$u_{1}=\frac{m}{R^{2}}\left(\cos^{2}\phi+2\sin^{2}\phi\right)$, so that a good
approximation for the solution of (3.22) is,

$u\,=u_{0}+u_{1}=\,\frac{\cos\phi}{R}+\frac{m}{R^{2}}\left(\cos^{2}\phi+2\sin^{2}\phi\right).$
(3.23)

This is the usual basic expression for the path of the deflected light rays
derived from GR. The deflection by the GR effect is essentially produced
within a limited distance from the lens center (a few impact parameters $R$).
The well known solution for the deflection angle $\hat{\alpha}_{\mathrm{GR}}$
is,

$\hat{\alpha}_{\mathrm{GR}}=4\frac{m}{R}=\frac{4\,GM}{c^{2}R}\quad\mathrm{[rad]},$
(3.24)

where $M$ is the mass of the deflector and $R$ the impact parameter.

Figure 1: The geodesic of a deflected light ray. Basic sketch of the lensing
geometry, with the impact parameter, $R$, in the lens plane, the polar
coordinates $r$ and $\phi$, the distance $D_{\mathrm{S}}$ from the observer to
the source, $D_{\mathrm{L}}$ from the observer to the lens, $D_{\mathrm{LS}}$
from the lens to the source and the $\hat{\alpha}$ deflection angles.

##  4 The scale invariant effects in lensing

The term $GM/R$ in the above equation is the gravitational potential which as
we have seen is an invariant quantity. Thus we might wonder whether in the
scale invariant context the deflection angle should not just be unchanged.
Indeed, despite the fact that the potential remains the same, Eqs. (3.19) and
(3.20) contain additional terms to the Einsteinian one. Also, the equation of
motion in the Newtonian approximation is modified with an additional term,
potentially significant in the astronomical context. Thus it makes sense to
wonder whether the path of a light ray also undergoes some changes with
respect to GR.

A difficulty of Eq. (3.20) is that, in addition to the independent variable
$\phi$, it requires the variable $t$ to express the term $\lambda$. The two
variables are related by geometrical relations on the light path. However, the
result is an intricate equation, making numerical integration mandatory to
evaluate relative importance of the additional terms and their impact on the
deflection of light rays.

###  4.1 The SIV relation between redshifts and ages

To proceed to numerical integration of Eq. (3.20), we need some basic
relations between the observed redshift $z$ and time $t$, which then gives the
scale invariance factor $\lambda=1/t$. The expansion factor writes (2.13),

$a(t)=\left[\frac{t^{3}-\Omega_{\mathrm{m}}}{1-\Omega_{\mathrm{m}}}\right]^{2/3},$
(4.1)

it is directly leading to,

$t=\left[\Omega_{\mathrm{m}}+(1+z)^{-3/2}(1-\Omega_{\mathrm{m}})\right]^{1/3}.$
(4.2)

With
$t=\Omega^{1/3}_{\mathrm{m}}+\frac{\tau}{\tau_{0}}(1-\Omega^{1/3}_{\mathrm{m}})$,
one obtains the age $\tau$ in current units for a galaxy of redshift $z$,

$\tau=\tau_{0}\frac{\left[[\Omega_{\mathrm{m}}+(1+z)^{-3/2}(1-\Omega_{\mathrm{m}})\right]^{1/3}-\Omega^{1/3}_{\mathrm{m}}}{1-\Omega^{1/3}_{\mathrm{m}}}.$
(4.3)

This relation can also be expressed as a function of $H_{0}$, by combining
with Eq. (2.20),

$\displaystyle\tau=\frac{2}{H_{0}}\frac{\left[[\Omega_{\mathrm{m}}+(1+z)^{-3/2}(1-\Omega_{\mathrm{m}})\right]^{1/3}-\Omega^{1/3}_{\mathrm{m}}}{1-\Omega_{\mathrm{m}}}.$
(4.4)

With these expressions in hand we can now associate an age to any object at
any redshift.

###  4.2 The variations of the term $dm$

During the path of the light ray from the source, $S$, to the observer, $O$,
the mass of the lens undergoes some limited variations due to the scale
variations during the travel time. This results in the term $u^{3}(dm/du)$ in
Eq. (3.20), coming in addition to the classical GR term $3mu^{2}$. This is
because, on the light path, both $m$ and $u$ increase at the same time, so
that $(dm/du)$ is positive. Alike a tidal effect, it behaves in
$u^{3}=1/r^{3}$, thus being essentially located close to the lens, even more
than the gravity effect itself. Since $m$ has the units of length (see Eq.
3.5), $u^{3}(dm/du)$ has the units of the inverse of a length.

Do these extra terms impact the deflection angle in the SIV context? To
evaluate the effect, we consider the practical case of the remarkable strongly
lensed galaxy JWST-ER1 recently observed by van Dokkum et al. [38] with the
JWST. It forms a complete Einstein ring with a radius $\theta_{E}=1.54\pm
0.02$ arcseconds produced by a massive quiescent lensing galaxy at
$z_{L}=1.94$ acting on a source at $z_{S}=2.98$. The total lensing mass is
estimated to be $M_{\rm Tot}=6.5\times 10^{11}$ M⊙ within a radius of 6.6 kpc
at the lens redshift, while the stellar mass derived from multi-band
photometry is 5.9 times lower, with $M_{\star}=1.1\times 10^{11}$ M⊙. As
stated by these authors, this is a beautiful example of a system where
_”Additional mass appears to be needed to explain the lensing results, either
in the form of a higher-than-expected dark matter density or a bottom-heavy
initial mass function.”_

We use JWST-ER1 as a test bench for our integration of the light path. This
path is expressed in terms of the variable $u=1/r$ as a function the angle
$\phi$, starting close to $\pi/2$ for the direction pointing to the distant
source, S, all the way to the nearest point at the impact parameter where
$u=1/R$ and where $\phi=0$ as can be seen from Fig. 1. We adopt a step size of
$2\times 10^{-4}$ for the variable $\phi$. Tests made with smaller steps give
identical results. The calculations represent an isochrone described by about
7850 points, which ensures a high accuracy of the integration. From the
relation between redshifts and cosmic ages (Sect. 4.1), we have in the
$t$-scale an age of the source given by $t_{\mathrm{S}}=0.5536$ and for the
lens $t_{\mathrm{L}}=0.6201$, according to their redshifts. We recall that $t$
is dimensionless, with $t_{0}=1$. (The lens mass $M_{\rm Tot}$ has been taken,
as well as a value $\Omega_{\mathrm{m}}=0.05$, thus assuming no dark matter.
For higher values of $\Omega_{\mathrm{m}}$, the effects would be smaller than
those found below). At each point defined by coordinates $(u,\phi)$ along the
path of the light ray, we can then assign an age $t$ and a value $\lambda$
($\tau$ is not needed here). We may then obtain from Eq. (2.17) the time
variations of the mass as well as the term $dm/du$. In terms of numerical
results, we obtain for example that in the specific case of JWST-ER1, the
ratio

$\frac{u^{3}\frac{dm}{du}}{3mu^{2}},$ (4.5)

is equal to $6.62\times 10^{-6},\;2.25\times 10^{-7},\;6.36\times 10^{-6}$ for
angles of $\phi$ = 89, 45 and 1 degree respectively. In other words, the
additional term $u^{3}(dm/du)$ has a totally negligible effect on the
resulting deflection angle $\hat{\alpha}_{\mathrm{S}}$. The same holds true
for the second part of the path of the light ray from the lens plane, $L$, to
the observer, $O$. Thus, globally this term is negligible.

###  4.3 The effects of the $\Lambda$ term

We now turn to the other additional term
$\left(\frac{1}{c^{2}\tau^{2}_{0}}\right)\lambda\frac{d\lambda}{du}$ in Eq.
(3.20), which consistently has the units of the inverse of a length.
Physically, this term results from the cosmological constant, present in the
exterior Schwarzschild metric (3.2). This term is thus not a direct function
of the deflecting mass, but is rather associated to the specific geometry of a
gravitational system where the external Schwarzschild metric applies. This
term thus intervenes as an additional correction to the classical Einsteinian
term. It is likely to be an extremely small quantity since $c^{2}\tau^{2}_{0}$
is the square of the ”size” of the present Universe.

The above calculations for JWST-ER1 provide the information on the magnitude
of this effect on the light deflection. The ratio of term
$\left(\frac{1}{c^{2}\tau^{2}_{0}}\right)\lambda\frac{d\lambda}{du}$ to the
classical Einsteinian term $3mu^{2}$ is generally negligible. For example, it
is as small as $1.38\times 10^{-5}$, $1.16\times 10^{-8}$, and $2.30\times
10^{-7}$ for angles of $\phi=89,45$ and 1 degree respectively. Thus, except
near the source the above ratio is smaller than the term due to the mass
variations. The consequences of the term
$\left(\frac{1}{c^{2}\tau^{2}_{0}}\right)\lambda\frac{d\lambda}{du}$ on the
deflection angle $\hat{\alpha}_{\mathrm{S}}$ is also negligible in all
possible current situations, and the same occurs for the second part of the
light path, i.e. described by $\hat{\alpha}_{\mathrm{L}}$. Only at the orgin
when the term $c\tau_{0}$ tends to zero could it have played some role in the
light path.

###  4.4 General case of light deflection

We have studied the nice case of JWST-ER1, but what about lensing by larger
masses, such as galaxy clusters? As far as the mass variations are concerned,
i.e., $u^{3}\frac{dm}{du}$, we see that its ratio to the Einsteinian term
$3mu^{2}$ changes as $\frac{1}{3}\frac{dm}{m}\frac{u}{du}$. Thus, as the
relative change of the mass $dm/m$ is independent of the mass (since $dm\sim
m$), our conclusion applies to any mass, thus including clusters of galaxies.

The deflection angle is essentially independent from scale invariant effects,
as shown above.This result is consistent with the fact that an angle is a
scale-invariant quantity as long as the cosmological evolution of the space-
time is homogeneous and isotropic. The present detailed analysis demonstrates
that the local inhomogeneity and anisotropy of the space-time created by a
mass element in the scale invariant context introduce only minor differences
in the light deflection as compared to standard GR. These effects are clearly
not sufficient to account for the large mass difference observed in JWST-ER1
[38] between the masses from the lensing effect and from the observed
luminosity.

This confirms that the masses obtained by the lensing effect are a most
remarkably model-independent result, in particular they are independent of
scale invariant effects and on the cosmological models as long they are
homogenous and isotropic. This result in itself is a fundamental one: the
geodesics of light in scale-invariance theory do not change with respect to
standard GR assumptions.

Figure 2: The red thick curve represents the mass-luminosity relation on the
zero-age sequence [18], according to the scale indicated on the left vertical
axis. The other curves describe various IMF: Chabrier [6], Salpeter with
$x=1.35$ [35] and the so-called Super-Salpeter by [38], according to the scale
indicated on the right vertical axis. Note that in $\xi(\log m)$ the log is
decimal. The Figure is inspired from [38].

##  5 The origin of the discrepancy between the stellar and lensing masses

Since the gravitational lensing deflection angle appear to be free from scale
invariant effects, we may now turn to the impact of scale invariance on lens
mass estimates in the Einstein radius, as derived from photometry or
spectroscopy of the lens.

###  5.1 The emblematic case of JWST-ER1s

We consider here again the remarkable case of JWST-ER1 and its high lensing-
to-stellar mass ratio of $\sim 6$ according to van Dokkum et al. [38] and
already presented in Sect. 4.2. With a lens redshift $z_{\mathrm{L}}=1.94$, it
is a textbook example of a distant massive quiescent galaxy with a low star
formation rate. It has an age of $\tau_{\mathrm{L}}=1.9^{+0.3}_{-0.6}$ Gyr,
implying that relatively low mass stars dominate the stellar population of the
lensing galaxy. Its unitless age is $t_{\mathrm{L}}=0.62$ (cf. Eq. 4.2),
taking $\Omega_{\mathrm{m}}=0.05$, i.e. assuming no dark matter (the Big-Bang
occured at $t_{\mathrm{in}}=\Omega^{1/3}_{\mathrm{m}}$, i.e. 0.3684).
According to Eq. (2.17), this would imply that the inertial and gravitational
masses in JWST-ER1 is equal to a fraction 0.62 of its present value.
Incidentally this might also contribute to the observed compactness of the
early galaxies. This means that the stellar mass distribution in JWST-ER1 may
be shifted to lower masses with respect to today’s distribution. Interestingly
enough, the consequences of this shift have rather similar effects to those of
the solution proposed by van Dokkum et al. [38] to solve the mass discrepancy
between the mass infered from the lens luminosity and its lensing mass,
$M_{Tot}=6.5^{+3.7}_{-1.5}\times 10^{11}\;M_{\odot}$. They suggest a shift
from Chabrier’s IMF [6], which has relatively few low-mass stars giving a mass
of $M_{*}=1.1^{+0.2}_{-0.3}\times 10^{11}\;M_{\odot}$, to a very bottom-heavy
IMF called ”Super-Salpeter IMF” giving a mass of
$M_{*}=4.0^{+0.6}_{-0.8}\times 10^{11}\;M_{\odot}$. The standard Salpeter IMF
gives an intermediate value of $M_{*}=2.0^{+0.5}_{-0.5}\times
10^{11}\;M_{\odot}$. Fig. 2 illustrates the different IMF assumptions.

The rather ad-hoc shift produced by the assumption of a Super-Salpeter IMF
leads to a large increase of the mass-to-light ratio since it assumes a
relatively larger fraction of lower stellar masses, with lower luminosities
(Fig. 2). Even with the Super-Salpeter assumption, a significant fraction of
dark matter is still necessary, $M_{\mathrm{DM}}=2.6^{+1.6}_{-0.7}\times
10^{11}\;M_{\odot}$, to explain the observed Einstein radius of this system.

The observed luminosity $L_{\mathrm{obs}}$ reflects the true mass of the stars
as they are at any given age. For an age of
$\tau_{\mathrm{L}}=1.9^{+0.3}_{-0.6}$ Gyr assigned by Prospector222
PROSPECTOR, a flexible code for inferring stellar population parameters from
photometry and spectroscopy spanning UV through IR wavelengths [14], the
turnoff mass is around 1.7 $M_{\odot}$ [18]. This corresponds well to a
general remark by [17] that most of the light of the old stellar populations
of galaxies comes from stars with a mass in the range of $[1-2]$ M⊙. A shift
in mass by a factor 0.62 for JWST-ER1, as predicted by the scale-invariant
theory, implies a significant difference in the relative number frequency of
stars as compared to the standard case.

Let us estimate the effect of such a shift on the average mass-luminosity
ratio $\langle M/L\rangle$ of a galaxy by some simple relations. As an
example, we take an initial mass function $\xi(\ln m)$ of the Salpeter form
[35],

$\xi(\ln
m)=\frac{dN}{d\ln\,m}=\frac{dN}{dm}m=Am^{-x},\quad\mathrm{or}\;\frac{dN}{dm}=Am^{-(1+x)},$
(5.1)

with $x=1.35$ (see Fig. 2). The mass of the galaxy then writes,

$M_{gal}=\int\frac{dN}{dm}m\,dm=A\int
m^{-x}dm=\;\mid\frac{A}{-x+1}({m_{m}}^{-x+1})\mid_{m_{min}}^{m_{max}}\;=Am_{m},$
(5.2)

where $A$ is a normalization constant and where $m_{m}$ is the mean mass over
the mass spectrum. This mean mass is $m_{m}=$13.7 M⊙ for a mass interval that
extends from $m_{min}=0.01$ to $m_{max}=100$ M⊙. If we call $\ell(m)$ the
luminosity of a star of mass $m$, the stellar mass-luminosity relation writes,
$\ell(m)=B\,m^{\alpha}$, where $B$ is another normalization constant. The
luminosity of a galaxy becomes,

$L_{gal}=\int\ell(m)\frac{dN}{dm}dm=AB\int
m^{\alpha}\,m^{-(x+1)}dm=\;\mid\frac{AB}{\alpha-x}{m}^{\alpha-x}\mid_{m_{min}}^{m_{max}}\;=AB\,m_{L},$
(5.3)

where $m_{L}$ is the mean mass over the luminosity distribution. Typical
values are 1.7-1.8 M⊙. Thus, the mean mass-luminosity ratio of a galaxy is
scaling like,

$\left\langle\frac{M}{L}\right\rangle=\frac{(\alpha-x)}{B(1-x)}m_{m}^{1-x}\;m_{L}^{x-\alpha}.$
(5.4)

If all the masses are reduced by the same factor, this factor applies both to
$m_{m}$ and $m_{L}$. The values of $\alpha$ in the range of 1 to 2 M⊙
considered earlier is quite large, hence making $\ell$ a very steep function
of $m$, as shown in Fig. 2. We have, following Maeder [18]

$\displaystyle\ell\sim m^{\alpha},$ $\displaystyle\hskip 5.0pt\alpha=3.69,$
$\displaystyle\hskip 5.0ptm\in[9-2]\;M_{\odot}$ (5.5) $\displaystyle\hskip
5.0pt\alpha=4.56,$ $\displaystyle\hskip 5.0ptm\in[2-1]\;M_{\odot}$
$\displaystyle\hskip 5.0pt\alpha=4.00,$ $\displaystyle\hskip
5.0ptm\in[1-0.6]\;M_{\odot}$

The individual stellar mass-luminosity ratios $m/\ell$ are varying with power
$(1-\alpha)$ of the mass $m$, while Eq. (5.4) shows that the factors are
different when integrated over the mass spectrum of a galaxy. Now, we can see
what is doing a change of the mass by a factor $f=0.62$, (this factor is the
ratio $t/t_{0}=t$ corresponding to the reshift $z=1.94$ of the lens in JWST-
ER1). The ratio of the modified overall mass-luminosity ratio to the standard
unmodified case is,

$\frac{\left\langle\frac{M}{L}\right\rangle_{f}}{\left\langle\frac{M}{L}\right\rangle_{\mathrm{std}}}\sim
f^{1-x}\;f^{x-\alpha}=f^{1-\alpha}.$ (5.6)

Interestingly enough, for a change of all masses by the same factor $f$, we
come back to the same dependence as given by the simple individual stellar
expression $(m/\ell)\sim m^{1-\alpha}$; for other changes of the mass
distribution the full expression has evidently to be kept in. For of factor
$f=0.62$ and $\alpha=4.56$, we get a value of 5.48 for the ratio expressed in
Eq. (5.6). Thus, the real mass-luminosity ratio may be about a factor of 5.48
higher than the assumed one, which accounts for most of the difference between
the mass observed from lensing and the mass derived from the luminous matter.

The so-called super-Salpeter IMF with $x\approx 1.65$ applied by van Dokkum et
al. [38] increases the total mass by a factor of about 4 compared to
Salpeter’s. In the scale invariant context, such a hypothesis of a very
bottom-heavy mass distribution, just opposed to the Chabrier one, is not
necessary. A mass distribution close to the Chabrier IMF naturally accounts
for the lensing mass without calling for dark matter. Thus, while the mass
estimates by lensing are totally unaffected by scale invariant effects, it is
not the case for the spectrophotometric determinations, which may be severely
biased for distant galaxies, such as JWST-ER1.

###  5.2 The SLACS sample of galaxies

To complement the above results for JWST-ER1, we turn towards the Sloan Lens
ACS (SLACS) survey by Bolton [3] which is one of the largest and most
homogeneous data set of galaxy-scale lenses, containing about 100 lenses
analysed by Auger et al. [1]. One of the aim of the survey was to study the
distribution of luminous and dark matter out to redshift $z\approx 0.5$. It is
based on multi-band imaging with ACS, WFC and NICMOS on the HST. The
observations have been analysed with a lens model using an isothermal
ellipsoid mass distribution, allowing high precision measurements of the mass
within the Einstein radius for each lens. The resulting mass estimates have
been found unbiased compared to the estimates from SDSS photometry and are in
the range $10^{10.5}<M<10^{11.8}$ $M_{\odot}$, with a typical statistical
error of 0.1 dex. The mean mass-luminosity ratio of the sample of the galaxies
considered in the SLACS sample is about $\langle M/L\rangle=8.7$. Such a ratio
corresponds to a representative stellar mass of about 1.84 $M_{\odot}$, not
much different from the case of JWST-ER1.

The survey provides for each lens the fractions $f^{Chab}_{*,Ein}$ and
$f^{Salp}_{*,Ein}$ of the stellar mass within the Einstein radius for the
Chabrier and Salpeter IMFs [6, 35] with respect to the total lensing mass. As
stated by [1], these fractions are independent of any priors from lensing, so
that in some cases fractions higher than 1 were obtained. The survey provides
a mean stellar mass fraction within the Einstein radius of $\langle
f^{Chab}_{*,Ein}\rangle=0.4$ with a rms scatter of 0.1 for the Chabrier IMF;
for the Salpeter’s IMF the mean is $\langle f^{Salp}_{*,Ein}\rangle=0.7$ and a
rms scatter of 0.2.

In the data of Table 3 and 4 by [1], we select two samples of lenses at
different redshifts: for low redshifts, we select systems with
$z_{\mathrm{L}}\leq 0.15$ and another sample with $z_{\mathrm{L}}\geq 0.35$.
This split in two bins of $\Delta z=0.15$ is a compromise between the need to
have two different mean redshifts and also a sufficient number of galaxies in
each sub-sample. The results are shown in Table 1. We see that the stellar
mass fractions are higher for low $z$-interval, by $\Delta f_{*}=0.092$ for
the Chabrier IMF and by $\Delta f_{*}=0.166$ for the Salpeter IMF. These
differences are rather small, of the same order as the uncertainties, unlike
the high-$z$ case of JWST-ER1. The standard deviations of the different
fractions are also given in Table 1, allowing a test of significance.

Combining the two values for each IMF, we get $\sigma=0.135$ for the
difference $\Delta f_{*}=0.449-0.357=0.092$ of the $\langle
f^{Chab}_{*,Ein}\rangle$ values. For a normal distribution, this corresponds
to a probability of 50.4% that the difference is significant. For $\langle
f^{Salp}_{*,Ein}\rangle$, the difference is $\Delta f_{*}=0.792-0.626=0.166$
and $\sigma=0.239$, giving a probability of 51.2%. Thus, the differences are
possible, but also well compatible with a random realization. The ages $t$
corresponding to the mean redshifts $z=0.114$ and $z=0.383$ of the two samples
are respectively $t=0.8591$ and $t=0.9502$ in the dimensionless time scale,
again for $\Omega_{\mathrm{m}}=0.05$. According to Eq. (2.17), masses scale
like $t$, which implies an increase by about 10% between the two redshifts.
The observations effectively suggest an increase of the same order of
magnitude, which is quite satisfactory. Nevertheless the observational
uncertainties preclude from establishing this correspondence on a firm
statistical ground.

As in the case JWST-ER1, the Salpeter IMF gives masses in better agreement
with observations than Chabrier’s IMF: the mass estimates from the photometric
observations at $\langle z\rangle=0.114$ give a mass equal to 79.2% of the
total lensing mass. If we account, as above for JWST-ER1, for the shift of the
Salpeter IMF resulting from the smaller masses in scale invariance, we need to
rescale the stellar fraction in the Einstein radius following Eq. (5.6). For
the low redshift bin $\langle z\rangle=0.114$, we obtain today’s corresponding
value of the stellar mass fraction of $\langle
f^{Salp}_{SIV}\rangle=0.792\times 0.9502^{-3.56}=0.95$. For the high reshift
bin, $\langle z\rangle=0.383$, we obtain $\langle
f^{Salp}_{SIV}\rangle=0.626\times 0.8591^{-3.56}=1.07$. These two fractions,
very close to 1.0, mean that when the shift in mass due to scale invariance is
accounted for, the stellar masses from photometry/spectroscopy and for a
standard IMF are also here remarkably close to the masses from lensing, in
support of the present interpretation.

Table 1: Analysis of the SLACS Survey: the mean stellar mass fractions within the Einstein radius, $\langle f^{Chab}_{*,Ein}\rangle$ and $\langle f^{Salp}_{*,Ein}\rangle$ are given for Chabrier and Salpeter IMFs respectively as well as the corresponding standard deviations among the two subsamples of redshifts. The column $\langle f^{Salp}_{SIV}\rangle$ gives the theoretical value of the stellar mass fraction with Salpeter’s IMF obtained after accounting for the scale invariant correction. The last number gives the number of galaxies in each sub-sample.  $\mathrm{Redshift\;interval}$ | $\langle z\rangle$ | $\langle f^{Chab}_{*,Ein}\rangle$ | $\langle f^{Salp}_{*,Ein}\rangle$ | $\sigma(f^{Chab}_{*,Ein})$ | $\sigma(f^{Salp}_{*,Ein})$ | $\langle f^{Salp}_{SIV}\rangle$ | $N$
---|---|---|---|---|---|---|---
$0\leq z\leq 0.15$ | 0.114 | 0.449 | 0.792 | 0.110 | 0.197 | 0.95 | 17
$0.3\leq z\leq 0.5$ | 0.383 | 0.357 | 0.626 | 0.079 | 0.136 | 1.07 | 10

##  6 Discussion and proposed lensing tests

On the basis of the JWST-ER1 and the SLACS studies, we have seen that the
observed differences between the stellar masses from photometry and from
lensing are increasing with redshifts. We show that scale invariance theory
can reconcile the two mass estimates, both at low and high redshifts.

Of note, different results obtained with GR and dark matter (FDM) seem to
severely contradict each other. At high redshift, like in JWST-ER1, the DM
component vastly dominates the lens mass, to the point that a super-Salpeter
IMF is invoked in addition to a DM halo with extreme densities. Recently Kong
et al. [16] attempted to alleviate the DM density by invoking the presence of
self-interacting DM. But their overall conclusion that JWST-ER1 is an extreme
lens in terms of DM content at high redshift comes in contrast with several
recent studies on rotation curves of galaxies by Nestor Shachar [28] that
corroborate previous results by the Genzel group, showing that much less dark
matter is needed in GR at high reshift than at low redshift. For example, only
a fraction of about 17% of the typical value appears to be remaining in
galaxies at $z=2.44$ [28], in sharp contrast with JWST-ER1.

The present work focuses on the comparison of the total lensing mass of
galaxies and the total luminous mass in GR and in scale invariance. It
suggests that, in scale invariance, the total luminous mass is sufficient to
explain the observed deflection angle without the need of dark matter, given
the observational errors. This paves to the way to many other more detailed
tests involving lensing in scale invariance:

  * •

Pixel-level modeling: state-of-art modeling of lens galaxies decomposes their
mass into luminous mass plus dark matter NFW halo [29, 30]. This allows to
model HST images of lensing systems down to the noise level. A strong test for
scale invariance lensing would be to attempt the same without the DM
component, just using the light map of the lens and converting it into mass
following the recipes of the present work.

  * •

Lens dynamics: dynamical information is available for some strong lens
systems, including SLACS. Joint lens/dynamical analysis of strong lenses in
the context of scale invariance would further test the theory. This is not
straightforward, however, as the Viriel theorem does not apply directly in
scale invariance and requires to account for the additional acceleration due
to scale invariance.

  * •

Time delay cosmography: Time delays in lensed quasars can be converted into an
$H_{0}$ measurement, provided the slope of the lens mass is known (or modeled)
at the position of the quasar images [2]. In scale invariance and a no-DM lens
model, the slope of the mass becomes extremely well constrained as the
degeneracy between invible DM and well-visible baryons is lifted. This may
have an impact on the infered $H_{0}$ value and in turn on the Hubble tension.
It will also be a strong test for scale invariance as, with no DM, the Mass
Sheet Degeneracy is also fully broken [37]. The $H_{0}$ values infered by
lensing time delays, in scale invariance, will directly result from the slope
of the luminous mass at the positions of the lensed quasar images. This is
directly constrained by deep HST or JWST imaging.

  * •

Lensing by edge-on spirals: The luminous mass in edge-on galaxies is of course
extremely elliptical. In scale invariance the mass is dominated by baryons (or
even fully accounted for by baryons) and therefore the difference in lensing
configurations between no-DM scale invariance models and standard models
dominated by a fairly spherical halo, should offer great discriminating power
between the two theories. In fact, hints of such behaviour may have been found
already, for example with the spectacular case of J220132.8-320144 where some
of the lensed images predicted by standard models with DM are missing from HST
imaging data [7].

  * •

Weak lensing: Weak lensing mass maps of galaxy clusters are unaffected by
scale invariance, just as for strong lensing. However, as for strong lensing,
scale invariance changes the infered luminous mass. It will therefore be
crucial to compare the weak lensing mass infered for galaxy clusters to that
infered from spectroscopy for all galaxy members, for example with ultra deep
integral field spectroscopy. Additionally, and as shown in [22], the scale
invariance mass infered from galaxy velocities should also be consistent with
the scale-invariant luminous mass.

The above proposed tests are beyond the scope of the present paper but most
observational material to implement them do exist and will be used in the near
future.

##  7 Conclusions

Detailed calculations of the geodesic equation to study deviation of light
rays passing a massive object have been developed in the scale invariant
theory. The inhomogeneity and anisotropy of space-time associated to the
presence of a lensing galaxy generate two additional terms in the equation of
the geodesics with respect to General Relativity. However, in practice,
numerical integration of the geodesics performed specifically for the Einstein
ring JWST-ER1 with its lensing galaxy at $z_{L}=1.94$ indicates that the
change in deflection angle is typically a fraction $\mathcal{O}$($10^{-6}$) of
that predicted by GR. The lensing deflection can therefore be considered as
independent from scale invariant effects. However, while scale invariance has
no effect on the photons’ geodesics themselves, the actual impact of scale
invariance resides in the interpretation and use of the mass-luminosity
relation of the stellar population in the lens galaxy and of its IMF.

## Acknowledgments

A.M. expresses his best thanks to Dr. Vesselin Gueorguiev for constructive
interaction since many years.

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