# Constraints on the time variation of the speed of light using Strong lensing

Seokcheon Lee<EMAIL_ADDRESS>Department of Physics, Institute of Basic
Science, Sungkyunkwan University, Suwon 16419, Korea

###### Abstract

Due to the recent growth of discoveries of strong gravitational lensing (SGL)
systems, one can statistically study both lens properties and cosmological
parameters from 161 galactic scale SGL systems. We analyze meVSL model with
the velocity dispersion of lenses by adopting the redshift and surface mass
density depending power-law mass model. Analysis shows that meVSL models with
various dark energy models including $\Lambda$CDM, $\omega$CDM, and CPL
provide the negative values of meVSL parameter, $b$ when we put the prior to
the $\Omega_{\text{m}0}$ value from Planck. These indicate the faster speed of
light and the stronger gravitational force in the past. However, if we adopt
the WMAP prior on $\Omega_{\text{m}0}$, then we obtain the null results on $b$
within 1-$\sigma$ CL for the different dark energy models.

##  I Introduction

Gravitational lensing represents the deflection of light rays owing to
intermediate inhomogeneous distributions of matter between the source and the
observer. Most sources seem slightly distorted compared to the way they would
appear in a perfectly homogeneous and isotropic universe. This is called weak
gravitational lensing (WGL) Refregier:2003ct ; Hoekstra:2008db ;
Bartelmann:2010fz .

On the other hand, strong gravitational lensing (SGL) implies that the
deflection angle is sufficiently large to produce multiple images, arcs, or
Einstein rings of a distant background source. SGL systems require massive
foreground objects that act as lenses and a background source to be almost
perfectly aligned along the line of sight (LOS) from the observer. Any lens
with a surface mass density above the so-called critical density can generate
multiple images. Hence, the SGL of stars is quite rare. However, SGL is more
commonly observed among galaxies and clusters of galaxies. They are massive
enough to split multiple images by more than an arcsecond to be resolved by
astronomical observations in various wavelengths Treu:2010 .

SGL can be used as a powerful method to probe the properties of various
astrophysical and cosmological objects.

  * •

Mass distribution : SGL provides an accurate measurement of the total mass
(including dark matter, DM) of galaxies and galaxy clusters at cosmological
scales. After accounting for the luminous matter in these systems through
observations, we can infer the fraction of dark matter and put constraints on
its mass distribution.

  * •

Statistical probes : Statistical investigation of lenses their abundances,
image separations, and the orientation of their lensed arcs, are useful probes
of the foreground lens population. These can be used to constrain physical
properties such as shapes of their mass distribution and ellipticities. They
can be used to constrain parameters of cosmological models of the Universe.

  * •

Cosmology : Intrinsically variable lensed sources (such as supernovae and
quasars) exhibit variabilities among the multiple images at different times
due to the different paths taken by light rays to the observer. These time
delays can be combined with a lens mass model and lens stellar kinematics to
measure both the time-delay distance and the angular diameter distance to the
lens, which can constrain cosmological parameters. One can investigate the
equation of state (e.o.s) of dark energy (DE) and/or its cosmological
evolution by measuring the angular diameter distances of background sources at
multiple redshifts behind a lens galaxy or cluster.

SGL by elliptical galaxies has been used to probe both cosmology and galaxies
Chae:2002uf ; Oguri:2006qp ; Paraficz:2009 ; Paraficz:2010 ; Cao:2012 ;
Oquri:2012 ; Chen:2018jcf ; Tu:2019vcj ; Wong:2019kwg ; Oguri:2019fix . We
summarize three main approaches to these methods.

  * •

Image separation : SGL systems have been used as an alternative way to probe
the property of DE over the years by using measurements for the velocity
dispersion of the lens DM halo $\sigma_{\text{DM}}$ and the angular Einstein
radius $\theta_{\text{E}}$ to constrain cosmological parameters Biesiada:2006
; Jullo:2010 ; Biesiada:2010 ; Cao:2012 ; Cao:2015qja ; Magana:2015wra ;
Magana:2017gfs ; Amante:2019xao . It has been demonstrated that the lensing
mass distribution of early-type galaxies (ETGs) is close to a singular
isothermal sphere (SIS) and thus the separation among the multiple images
depends on the mass of the deflector and the angular diameter distances to the
lens and to the source when a galaxy acts as a lens. The comoving distance
$D_{\text{C}}$, the transverse comoving distance $D_{\text{M}}$, and the
angular diameter distance $D_{\text{A}}$ in meVSL model are given by
Lee:2020zts

$\displaystyle D_{\text{C}}(z)$
$\displaystyle\equiv\int_{0}^{r}\frac{dr^{\prime}}{\sqrt{1-kr^{2}}}=\frac{\tilde{c}_{0}}{H_{0}}\int_{0}^{z}\frac{dz^{\prime}}{E^{(\text{GR})}(z^{\prime})}\quad\text{where}\quad
E^{2}\equiv\left(\sum_{i}\Omega_{i0}a^{-3(1+\omega_{i})}\right)\,,$ (1)
$\displaystyle D_{\text{M}}(z)$
$\displaystyle=\begin{cases}\frac{\tilde{c}_{0}}{H_{0}}\frac{1}{\sqrt{\Omega_{K0}}}\sinh\left(\sqrt{\Omega_{k0}}\frac{H_{0}}{\tilde{c}_{0}}D_{\text{C}}\right)&\Omega_{k0}>0\\\
D_{\text{C}}&\Omega_{k0}=0\\\
\frac{\tilde{c}_{0}}{H_{0}}\frac{1}{\sqrt{|\Omega_{K0}|}}\sin\left(\sqrt{|\Omega_{k0}|}\frac{H_{0}}{\tilde{c}_{0}}D_{\text{C}}\right)&\Omega_{k0}<0\end{cases}\,,\quad
D_{\text{A}}(z)=\frac{1}{(1+z)}D_{\text{M}}(z)$ (2)

where $H_{0}$ is the present value of the Hubble parameter, $\tilde{c}_{0}$ is
the present value of the speed of light,
$\Omega_{i0}\equiv\rho_{i0}/\rho_{\text{cr}0}$ is the present mass density
contrasts of i-component, and $\omega_{i}$ is its equation of state (e.o.s).

One defines $\mathcal{R}^{\text{th}}$ as the theoretical angular diameter
distance ratio of the angular diameter distance between the lens and source
$D_{\text{A}}(z_{l},z_{s})$ and that between observer and source
$D_{\text{A}}(0,z_{s})$. Also, its observable counterpart
$\mathcal{R}^{\text{obs}}$ can be defined as

$\displaystyle\mathcal{R}^{\text{th}}\left(z_{l},z_{s};{\bf p}\right)$
$\displaystyle=\frac{D_{\text{A}}\left(z_{l},z_{s};{\bf
p}\right)}{D_{\text{A}}\left(0,z_{s};{\bf
p}\right)}=\frac{D_{\text{M}}\left(z_{l},z_{s};{\bf
p}\right)}{D_{\text{M}}\left(0,z_{s};{\bf p}\right)}\,,$ (3)
$\displaystyle\mathcal{R}^{\text{obs}}\left(\sigma_{\text{DM}},\theta_{\text{E}}\right)$
$\displaystyle=\frac{\tilde{c}_{0}^{2}\theta_{\text{E}}}{4\pi\sigma_{\text{DM}}^{2}}\,.$
(4)

where $\textbf{p}=\sum_{i}(\Omega_{i0},\omega_{i})$ denote cosmological
parameters, $z_{l}$ and $z_{s}$ are the redshifts to the lens and source
respectively. Thus, one can estimate cosmological parameters by minimizing the
$\chi^{2}$ function by using observed values of $z_{l}$, $z_{s}$,
$\theta_{\text{DM}}$, and $\theta_{\text{E}}$ from SGL systems

$\displaystyle\chi^{2}(\textbf{p})=\sum_{i=1}^{N_{\text{SGL}}}\left(\frac{\mathcal{R}_{i}^{\text{th}}\left(z_{l},z_{s};{\bf
p}\right)-\mathcal{R}_{i}^{\text{obs}}\left(\sigma_{\text{DM}},\theta_{\text{E}}\right)}{\Delta\mathcal{R}_{i}^{\text{obs}}}\right)^{2}\quad,\quad\Delta\mathcal{R}_{i}^{\text{obs}}\equiv\mathcal{R}_{i}^{\text{obs}}\sqrt{\left(\frac{\Delta\theta_{\text{E}}}{\theta_{\text{E}}}\right)_{i}^{2}+\left(2\frac{\Delta\sigma_{\text{DM}}}{\sigma_{\text{DM}}}\right)_{i}^{2}}\,,$
(5)

where $\Delta\theta_{\text{E}}$ and $\Delta\sigma_{\text{DM}}$ are the errors
for the angular Einstein distance and for the DM velocity dispersion
respectively.

  * •

Time delay cosmography : In the SGL systems, multiple images appear to the
observer. If the emission intensity from the source is variable in time as an
active galactic nucleus (AGN), a supernova (SN), or a quasar, the difference
in arrival time is measurable. This arrival time of the images depends on both
the path of the light ray and also the gravitational potential of the lens
Suyu:2012ApJ . The light travel time $t\left(\bm{\theta}\,,\bm{\beta}\right)$
for each image at angular position $\bm{\theta}$ relative to a fiducial
unperturbed ray from the source position $\bm{\beta}$ through a single,
isolated, thin gravitational lens is given by

$\displaystyle t\left(\bm{\theta}\,,\bm{\beta}\right)$
$\displaystyle\equiv\frac{D_{\Delta
t}}{c}\phi\left(\bm{\theta}\,,\bm{\beta}\right)\,,$ (6)

where a so-called “time-delay distance” $D_{\Delta t}$ and the Fermat
potential $\phi\left(\bm{\theta}\,,\bm{\beta}\right)$ are defined as

$\displaystyle D_{\Delta t}$
$\displaystyle\equiv\frac{D_{\text{A}}(0,z_{l})D_{\text{A}}(0,z_{s})}{D_{\text{A}}(z_{l},z_{s})}\,,$
(7) $\displaystyle\phi\left(\bm{\theta}\,,\bm{\beta}\right)$
$\displaystyle\equiv\left[\frac{\left(\bm{\theta}-\bm{\beta}\right)^{2}}{2}-\psi\left(\bm{\theta}\right)\right]\,.$
(8)

The deflection angle of light rays is determined by the lens equation
$\bm{\theta}-\bm{\beta}\equiv\bm{\alpha}$ which is related to the lens
potential as $\bm{\alpha}=\bm{\nabla}\psi$. The first term of Eq. (8) comes
from the geometric path difference as a result of the strong lens deflection
and the second term is due to the gravitational delay described by the lens
potential $\psi$. Therefore, one can estimate cosmological parameters
including the present value of the Hubble parameters that appear in the time
delay distance by modeling the lens potential and the source position. There
have been quasar-galaxy strong lensing systems with time delay observations
Suyu:2013kha ; Chen:2016fwu ; Wong:2016dpo ; Bonvin:2018dcc ; Birrer:2018vtm ;
Bonvin:2019xvn ; Rusu:2019xrq ; Chen:2019ejq ; Wong:2019kwg . $H_{0}$ Lenses
in COSmological MOnitoring of GRAvItational Lenses (COSMOGRAIL)’s Wellspring
(H0LiCOW) collaboration reveals six lensed quasars and one from STRong lensing
Insights into the Dark Energy Survey (STRIDES) collaboration Shajib:2019toy .
The eight lenses with measured time delays consist of the following systems :
B1608+656 Myers:1995 ; Fassnacht:1996 ; Suyu:2009by ; Jee:2019hah ,
RXJ0911+0551 Bade:1997 ; Kneib:2000ty ; Hjorth:2002gs , RXJ1131-1231
Sluse:2003iy ; Sluse:2007cn , HE0435-1223 Wisotzki:2002dp ; Morgan:2004xu ;
Eigenbrod:2005ub ; Sluse:2012rg ; Millon:2020xab , SDSS1206+4332 Oguri:2004qu
; Agnello:2015ala , WFI2033-4723 Sluse:2012rg ; Morgan:2003bd ; Sluse:2019 ,
PG1115+080 Weymann:1980 ; Tonry:1997pc , and DES J0408-5354 Shajib:2019toy .
Recently, there are also time delays in 18 strongly lensed quasars from
optical monitoring Millon:2020xab .

  * •

Lens velocity dispersion : If we combine SGL observations with stellar
dynamics in elliptical galaxies, then we can use the spectroscopically
measured stellar velocity dispersion (VD) of the lens as statistical quantity
to constrain both cosmological parameters and lens model Futamase:2001ij ;
Biesiada:2006 ; Grillo:2007iv ; Schwab:2010 ; Cao:2017nnq ; Chen:2018jcf ;
Amante:2019xao . The main assumption is that the dynamical mass enclosed
within the Einstein radius and lensing mass of early-type lens galaxies should
be equal. Available SGL systems of VD have grown Koopmans:2002ic ; Treu:2002ee
; Koopmans:2002qh ; Treu:2004wt ; Bolton:2005nf ; Treu:2005aw ;
Koopmans:2006iu ; Grillo:2007iv ; Ruff:2011 ; Brownstein:2012 ; Gavazzi:2012 ;
Sonnenfeld:2013xga ; Sonnenfeld:2013cha ; Sonnenfeld:2014 ; Cao:2015qja ;
Zhou:2019vou . We use this method to analyze the cosmological parameters of
meVSL model in this manuscript. Thus, we will briefly review the methodology
of this approach in the next section.

The outline of this manuscript is as follows. In Section II, we briefly review
the method of using stellar VD as statistical quantity and derive the
modifications of equations in meVSL model. We analyze data to obtain
cosmological parameters and time evolutions of the speed of light and the
gravitational constant in Section III. In Section IV, we conclude and
summarize.

##  II Review on VD method

A typical configuration considered in SGL is depicted in Fig. 1, where a lens
at distance $D_{l}$ deflects the light rays from a source at distance $D_{s}$.
The source and lens planes are perpendicular to the optical axis (line $NO$)
from the source to the observer. $\vec{\zeta}$ denotes the two-dimensional
position of the source on the source plane and it is related to the true
angular position $\vec{\beta}=\vec{\zeta}/D_{s}$ in the absence of the
deflector. If there are no other lenses close to the LOS and the extent of the
lens mass along it is much smaller than both $D_{l}$ and the distance from the
deflector to the source $D_{ls}$, the actual light rays which are smoothly
curved in the vicinity of the deflector can be replaced by two straight rays
with the deflection angle $\hat{\alpha}$. This is also related with the scaled
deflection angle $\vec{\alpha}=(D_{ls}/D_{s})\hat{\alpha}$. The lens equation
relates the true position of the source to its observed position,
$\vec{\beta}=\vec{\theta}-\vec{\alpha}$ as shown in Fig. 1.

Figure 1: Configuration of strong gravitational lensing system.

Among several methods of using SLG to probe cosmology, the method of using
stellar VD has merit since its theoretical prediction depends only on the lens
mass model without relying on the luminosity function. We adopt the method in
Chen:2018jcf  and thus briefly review it in this section and also extend
results of GR to those of the meVSL model.

The main assumption of this method is that the projected surface mass due to
the lens $M^{\text{E}}_{\text{gr}}$ equals the projected dynamical mass
$M^{\text{E}}_{\text{dyn}}$ with the Einstein radius
$R_{\text{E}}=D_{l}\theta_{\text{E}}$. The projected gravitational mass is
related with the critical surface mass density by
$\pi\Sigma_{\text{cr}}R_{\text{E}}^{2}=M^{\text{E}}_{\text{gr}}$ where
$\Sigma_{\text{cr}}=c^{2}/(4\pi G)\cdot D_{s}/(D_{s}D_{ls})$. Thus, the
projected gravitational surface mass of meVSL model is generalized as

$\displaystyle
M^{\text{E}}_{\text{gr}}=\frac{\tilde{c}_{0}^{2}}{4\pi\tilde{G}_{0}}(1+z_{l})^{-\frac{b}{2}}\frac{D_{s}D_{l}}{D_{ls}}\theta_{\text{E}}^{2}\,,$
(9)

where $\tilde{c}_{0}$ and $\tilde{G}_{0}$ correspond to the present values of
the speed of light and of the gravitational constant, respectively. We use the
consequences of meVSL model $\tilde{c}=\tilde{c}_{0}(1+z)^{-b/4}$ and
$\tilde{G}=\tilde{G}_{0}(1+z)^{b}$. One should make an assumption on the
galaxy lens mass distribution in order to estimate the projected dynamical
mass of the lens galaxy from its VD. The density distribution of total mass
$\rho(r)$ assumed to follow that of early-type galaxies (ETGs) and the
luminosity density of stars $\nu(r)$ is given by

$\displaystyle\rho(r)$
$\displaystyle=\rho_{0}\left(\frac{r}{r_{0}}\right)^{-\gamma(z)}\quad\text{where}\quad\gamma=\gamma_{0}+\gamma_{z}\cdot
z_{l}+\gamma_{s}\cdot\log\tilde{\Sigma}\,,$ (10) $\displaystyle\nu(r)$
$\displaystyle=\nu_{0}\left(\frac{r}{r_{0}}\right)^{\delta}\quad,\quad\beta(r)=1-\frac{\sigma_{\theta}^{2}}{\sigma_{r}^{2}}\,,$
(11)

where $\tilde{\Sigma}$ is the normalized surface mass density of the lens
galaxy given by
$(\sigma/100\,\text{km}\,\text{s}^{-1})^{2}/(R_{\text{E}}/10h^{-1}\,\text{kpc})$
and $\beta(r)$ denotes the stellar orbital anisotropy with $\sigma_{\theta}$
and $\sigma_{r}$ represent the tangential and radial VDs, respectively.

The projected dynamical mass contained within a cylinder with a radius equals
to $R_{\text{E}}$ with the density distribution given in Eq. (10) is

$\displaystyle M^{\text{E}}_{\text{dyn}}$
$\displaystyle=2\sqrt{\pi^{3}}\frac{R_{\text{E}}^{3-\gamma}}{3-\gamma}\frac{\Gamma\left[\left(\gamma-1\right)/2\right]}{\Gamma\left[\gamma/2\right]}\rho_{0}r_{0}^{\gamma}\,.$
(12)

The total mass contained within a sphere of radius $r$ related to
$M^{\text{E}}_{\text{dyn}}$ can be written as

$\displaystyle
M(r)=4\pi\frac{r^{3-\gamma}}{3-\gamma}\rho_{0}r_{0}^{\gamma}=\frac{2}{\sqrt{\pi}}\frac{\Gamma\left[\gamma/2\right]}{\Gamma\left[\left(\gamma-1\right)/2\right]}\left(\frac{r}{R_{\text{E}}}\right)^{3-\gamma}M^{\text{E}}_{\text{dyn}}$
(13)

One can obtain the radial component of VD, $\sigma_{r}$ by using the radial
Jeans equation in spherical coordinate under the assumption that the stellar
number density is spatially constant compared to stellar luminosity density
$\nu(r)$

$\displaystyle\sigma_{r}^{2}(r)$
$\displaystyle=\tilde{G}_{0}\left(1+z_{l}\right)^{-b}\int_{r}^{\infty}dr^{\prime}r^{{}^{\prime}2\left(\beta-1\right)}\nu\left(r^{\prime}\right)M(r^{\prime})/\left(r^{2\beta}\nu(r)\right)\,.$
(14)

The final expression of radial VD is obtained by inserting $M(r)$ in Eq. (13)
into Eq. (14)

$\displaystyle\sigma_{r}^{2}(r)$
$\displaystyle=\frac{2}{\sqrt{\pi}}\frac{GM^{\text{E}}_{\text{dyn}}}{R_{\text{E}}}\frac{1}{\eta-2\beta}\frac{\Gamma\left[\gamma/2\right]}{\Gamma\left[\left(\gamma-1\right)/2\right]}\left(\frac{r}{R_{\text{E}}}\right)^{2-\gamma}\,,$
(15)

where $\eta=\gamma+\delta-2$. The actually measured VD of the lens galaxy is
the component of luminosity weighted average along the LOS and over the
effective spectroscopic aperture $R_{\text{A}}$ and the final expression for
this is given by

$\displaystyle\sigma_{\parallel}^{2}\left(\leq R_{\text{A}}\right)$
$\displaystyle=\frac{2}{\sqrt{\pi}}\frac{\tilde{G}_{0}M^{\text{E}}_{\text{dyn}}}{R_{\text{E}}}f\left(\gamma\,,\beta\,,\delta\right)\left(\frac{R_{\text{A}}}{R_{\text{E}}}\right)^{2-\gamma}(1+z_{l})^{-b}$
$\displaystyle=\frac{\tilde{c}_{0}^{2}}{2\sqrt{\pi}}\frac{D_{s}}{D_{ls}}\theta_{\text{E}}f\left(\gamma\,,\beta\,,\delta\right)\left(\frac{\theta_{\text{eff}}}{2\theta_{\text{E}}}\right)^{2-\gamma}(1+z_{l})^{-\frac{b}{2}}\,,\quad\text{where}$
(16) $\displaystyle f\left(\gamma\,,\beta\,,\delta\right)$
$\displaystyle\equiv\frac{3-\delta}{\left(\eta-2\beta\right)\left(3-\eta\right)}\left[\frac{\Gamma\left[\left(\eta-1\right)/2\right]}{\Gamma\left[\eta/2\right]}-\beta\frac{\Gamma\left[\left(\eta+1\right)/2\right]}{\Gamma\left[\left(\eta+2\right)/2\right]}\right]\frac{\Gamma\left[\gamma/2\right]\Gamma\left[\delta/2\right]}{\Gamma\left[\left(\gamma-1\right)/2\right]\Gamma\left[\left(\delta-1\right)/2\right]}\,,$

where we also use $\theta_{\text{A}}=\theta_{\text{eff}}/2$. Thus, Eq. (16) is
the theoretical value of radial VD. The corresponding observational value of
VD is obtained from the spectroscopic data $\sigma_{\text{ap}}$ inside the
circular aperture with the angular radius $\theta_{\text{ap}}$. From the
aperture correction formula, one can estimate the observational radial VD as

$\displaystyle\sigma_{\parallel}^{\text{obs}}=\sigma_{\text{ap}}\left(\frac{\theta_{\text{eff}}}{2\theta_{\text{ap}}}\right)^{\tau}\quad\text{where}\quad\tau=-0.066\pm
0.035\,,$ (17)

where the value of $\tau$ is from Cappellari:2005ux . There are 161 galaxy
scale SGL systems with given observed values of $z_{s}$, $z_{l}$,
$\theta_{\text{E}}$, $\theta_{\text{eff}}$, $\theta_{\text{ap}}$, and
$\sigma_{\text{ap}}$ with its error. Thus, we can perform the likelihood
analysis by doing the minimum $\chi^{2}$ for VD as

$\displaystyle\chi^{2}=\sum_{i=1}^{N}\left(\frac{\sigma_{\parallel\,,i}^{\text{th}}-\sigma_{\parallel\,,i}^{\text{obs}}}{\Delta\sigma_{\parallel\,,i}^{\text{obs}}}\right)^{2}\,,$
(18)

where $N$ is the number of the data points,
$\sigma_{\parallel\,,i}^{\text{th}}$ is given in Eq. (16),
$\sigma_{\parallel\,,i}^{\text{obs}}$ is the observed value of VD given in Eq.
(17), and $\Delta\sigma_{\parallel\,,i}^{\text{obs}}$ is its uncertainty.

##  III Analysis and results

We analyze three different DE models for two different given values of
$\Omega_{\text{m}0}$ by using the data given in Chen:2018jcf . We investigate
the cosmological constant ($\omega_{0}=-1$), the constant $\omega_{0}$ (i.e.
$\omega_{a}=0$), and the Chevallier-Polarski-Linder (CPL) parametrization
($\omega=\omega_{0}+\omega_{a}(1-a)$) for the DE equation of state (e.o.s) as
viable DE models under the meVSL model. We adopt $\Omega_{\text{m}0}=0.27$
from the Wilkinson Microwave Anisotropy Probe (WMAP) seven-year results
Jarosik:2010  and $\Omega_{\text{m}0}=0.32$ from 2018 Planck data release
Aghanim:2018eyx . The overall results for different models are given in Table.
1. Overall results show that the speed of light is slower than that of present
and the gravitational constant is smaller than that of the present ( i.e.
$b>0$) when we adopt the WMAP value of the matter density
$\Omega_{\text{m}0}=0.27$. Within 1-$\sigma$ CL those variations are null.
However, if we adopt the matter density from the Planck
$\Omega_{\text{m}0}=0.32$, then we obtain the larger values both of the speed
of light and of the gravitational constant in the past ( i.e. $b>0$). Usually,
there is no dependence on $h$ based on the VD analysis method because it
depends on the ratio of angular diameter distances. However, $h$ dependence
appears in the $\gamma$ due to its dependence on $\tilde{\Gamma}$ as shown in
Eq. (10).

###  III.1 Cosmological constant : $\Lambda$

In this subsection, we constrain the meVSL model parameter $b$ based on the
cosmological constant, $\omega_{0}=-1$ and $\omega_{a}=0$ by using the minimum
chi-square analysis. We first fix the best fit values of lens model parameters
for the different values of $\Omega_{\text{m}0}$.

  * •

$\Omega_{\text{m}0}=0.27$ : We obtain the best-fit values of lens model
parameters as
$\left(\gamma_{0}\,,\gamma_{z}\,,\gamma_{s}\,,\beta\,,\delta\right)=\left(1.201\,,-0.568\,,0.815\,,2.349\,,-0.022\right)$
with $h=0.717$ and $\chi^{2}=344.8$ for $b=0$. If we consider this model for
meVSL model, then we obtain the other best-fit values
$\left(\gamma_{0}\,,\gamma_{z}\,,\gamma_{s}\,,\beta\,,\delta\right)=\left(1.182\,,-0.575\,,0.818\,,2.370\,,-0.019\right)$
with $h=0.738$. For the 1-$\sigma$ confidence level (CL), $b=0.032\pm 0.092$
and $\chi^{2}=359.6$ for this model. Thus, we obtain the null results for the
time variation of the physical constants like the speed of light and the
gravitational constant in this model.

  * •

$\Omega_{\text{m}0}=0.32$ : In this model, we obtain
$\left(\gamma_{0}\,,\gamma_{z}\,,\gamma_{s}\,,\beta\,,\delta\right)=\left(1.184\,,-0.589\,,0.882\,,2.362\,,-0.018\right)$
with $h=0.721$ and $\chi^{2}=431.1$ for $b=0$. If we allow $b$ to vary, then
the best fit values become
$\left(\gamma_{0}\,,\gamma_{z}\,,\gamma_{s}\,,\beta\,,\delta\right)=\left(1.183\,,-0.588\,,0.822\,,2.363\,,-0.018\right)$
, $h=0.731$, and $b=-0.339\pm 0.091$ with $\chi^{2}=421.4$. In this model, the
lens model parameters are almost independent of the value of $b$. We obtain
the cosmological evolutions of the speed of light and that of the
gravitational constant in this model. This result is shown in Fig. 2. In the
left panel of this figure, we depict the cosmological evolution of
$\tilde{c}/\tilde{c}_{0}$ as a function of the redshift $z$. The central
dashed line indicates the best fit value of $\tilde{c}/\tilde{c}_{0}$. Its
value increases as a function of $z$. The shaded region enveloped by the solid
lines represents the 1-$\sigma$ CL region. The cosmological evolution of
$\tilde{G}/\tilde{G}_{0}$ within 1-$\sigma$ CL is shown in the right panel of
Fig. 2.

|
---|---
Figure 2: The time variations of $\tilde{c}$ and $\tilde{G}$ of $\Lambda$CDM
meVSL model as a function of $z$ when $\Omega_{\text{m}0}=0.32$. a) The
cosmological evolution of the ratio of $\tilde{c}$ to its present value
$\tilde{c}_{0}$ within 1-$\sigma$ CL. b) The cosmological evolution of the
value of $\tilde{G}/\tilde{G}_{0}$ for 1-$\sigma$ CL.

###  III.2 Constant DE e.o.s : $\omega$

We investigate the so-called $\omega$CDM model (i.e. constant $\omega_{0}$ and
$\omega_{a}=0$) in this subsection. Again, we obtain the best fit values of
lens model parameters first and then do the likelihood analysis for the
$\omega_{0}$ and $b$ values for the different values of $\Omega_{\text{m}0}$.

  * •

$\Omega_{\text{m}0}=0.27$ : For $b=0$, the best fit values of
$\left(\gamma_{0}\,,\gamma_{z}\,,\gamma_{s}\,,\beta\,,\delta\right)=\left(1.197\,,-0.570\,,0.816\,,2.352\,,-0.021\right)$
with $h=0.717$. The 1-$\sigma$ CL of $\omega_{0}$ is $-0.953\pm 0.044$ with
the minimum chi-square value is $345.8$. We also obtain the other set of the
lens model parameters best fit values
$\left(\gamma_{0}\,,\gamma_{z}\,,\gamma_{s}\,,\beta\,,\delta\right)=\left(1.198\,,-0.574\,,0.817\,,2.367\,,-0.006\right)$.
The value of $h=0.710$ with $\chi^{2}=357.0$. Within the 1-$\sigma$ CL,
$\omega_{0}=-1.006\pm 0.062$ and $b=0.127\pm 0.120$. Interestingly,
$\omega_{0}$ is the same as the $\Lambda$CDM model but this model provides the
time-varying speed of light within 1-$\sigma$ CL. We also investigate the
1-$\sigma$ confidence level values of $\omega_{0}$ and $b$ when we adopt the
same best fit values of lens model parameters
$\left(\gamma_{0}\,,\gamma_{z}\,,\gamma_{s}\,,\beta\,,\delta\right)=\left(1.18\,,-0.49\,,0.81\,,2.48\,,0\right)$
for the different values of $\Omega_{\text{m}0}=0.32$. In this model, we
obtain $(\omega_{0}\,,h\,,b)=(-0.944\pm 0.054\,,0.743\pm 0.009\,,0.331\pm
0.171)$ with $\chi^{2}=557.4$.

  * •

$\Omega_{\text{m}0}=0.32$ : The best fit values of
$\left(\gamma_{0}\,,\gamma_{z}\,,\gamma_{s}\,,\beta\,,\delta\right)=\left(1.181\,,-0.591\,,0.822\,,2.364\,,-0.017\right)$
with $h=0.721$ for $b=0$. The 1-$\sigma$ CL of $\omega_{0}$ is $-0.992\pm
0.042$ with $\chi^{2}=431.1$. For $b\neq 0$, we obtain the best fit values of
the lens model parameters as
$\left(\gamma_{0}\,,\gamma_{z}\,,\gamma_{s}\,,\beta\,,\delta\right)=\left(1.205\,,-0.590\,,0.821\,,2.359\,,-0.017\right)$.
The best value of $h$ is $0.700$ and $\chi^{2}=418.1$. Within the 1-$\sigma$
CL, $\omega_{0}=-0.996\pm 0.052$ and $b=-0.246\pm 0.108$. The 1-$\sigma$
confidence level values of $\omega_{0}$, $h$, and $b$ for the same best fit
values of lens model parameters of the different values of
$\Omega_{\text{m}0}=0.27$ become $(-0.914\pm 0.046\,,0.749\pm
0.009\,,-0.214\pm 0.166)$ with $\chi^{2}=649.7$. We show this result in Fig.
3. In the left panel of this figure, we show the constraint contours for
$b$-$\omega_{0}$. The best fit value of $(b\,,\omega_{0})=(-0.214\,,-0.914)$
and the different color shaded regions represent the 1-$\sigma$, 2-$\sigma$,
and 3-$\sigma$ constraints. These two parameters show a stronger negative
correlation than those with $\omega_{0}$. We depict the contours of $b$-$h$
constraints for the different confidence levels in the middle panel of Fig. 3.
$h$-$\omega_{0}$ constraint contours are shown in the right panel of this
figure.

|  |
---|---|---
Figure 3: $\omega_{0}$, $h$, and $b$ constraints for $\omega$CDM model when
$\Omega_{\text{m}0}=0.32$. a) The contours for $b$-$\omega_{0}$ constraint.
The contour levels represent the 1-$\sigma$, 2-$\sigma$, and 3-$\sigma$
constraints. b) The constraint contours for $b$ and $h$. c) The contours for
$h$-$\omega_{0}$ constraint.

###  III.3 CPL : $\omega_{0}+\omega_{a}(1-a)$

In this subsection, we investigate the so-called CPL model. The best fit
values of the lens model parameters are almost the same for all models as
shown in other subsections. We consider the different values of
$\Omega_{\text{m}0}$ models for the different choices of $\omega_{0}$.
Unfortunately, we are not able to obtain the converge values of cosmological
parameters when we also allow $\omega_{0}$ to vary. Thus, we investigate
cosmological parameters for the given values of $\omega_{0}=0.95,-1.0$, and
$-1.05$ with fixing $h=0.74$. For the lower values of $h$, these models suffer
from converging of the chi-square analysis.

  * •

$\Omega_{\text{m}0}=0.27$ : First, we investigate $\omega_{0}=-0.95$.
$\left(\gamma_{0}\,,\gamma_{z}\,,\gamma_{s}\,,\beta\,,\delta\right)=\left(1.198\,,-0.560\,,0.813\,,2.354\,,-0.023\right)$.
Within 1-$\sigma$ CL, we obtain $(b\,,\omega_{a})=(-0.037\pm 0.113\,,-1.115\pm
0.301)$ with $\chi^{2}=326.3$. The best-fit values of lens model parameters
are almost the same as each other for the different values of $\omega_{0}$ in
this model. The 1-$\sigma$ CL values of $(b\,,\omega_{a})=(-0.053\pm
0.103\,,-0.992\pm 0.247)$ with $\chi^{2}=350.2$ for $\omega_{0}=-1.0$. Also,
$b=-0.057\pm 0.090$ with $\chi^{2}=338.4$ for
$(\omega_{0}\,,\omega_{a})=(-1.5\,,-0.88)$. The analysis does not converge for
the general $\omega_{a}$ case in the last model and thus we fix its value in
this model. For these models, the data show no time variation of the speed of
light because of the null results of $b$.

  * •

$\Omega_{\text{m}0}=0.32$ :
$\left(\gamma_{0}\,,\gamma_{z}\,,\gamma_{s}\,,\beta\,,\delta\right)=\left(1.185\,,-0.582\,,0.820\,,2.363\,,-0.019\right)$
for $\omega_{0}=-0.95$. The 1-$\sigma$ CL best fit values of this model become
$(b\,,\omega_{a})=(-0.228\pm 0.104\,,-1.042\pm 0.251)$ with $\chi^{2}=391.2$
for this model. The best fit values of lens model parameters are quite similar
for the different values of $\omega_{0}$ in this model as shown in Table. 1.
$(b\,,\omega_{a}\,,\chi^{2})=(-0.250\pm 0.103\,,-0.922\pm 0.247\,,400.5)$ and
$(-0.326\pm 0.101\,,-0.804\pm 0.241\,,430.0)$ for $\omega_{0}=-1.00$ and
$-1.05$, respectively. We show the 1-$\sigma$ confidence evolution plots for
$\tilde{c}/\tilde{c}_{0}$ and $\tilde{G}/\tilde{G}_{0}$ in Fig. 4. In the left
panel, the evolution of $\tilde{c}/\tilde{c}_{0}$ within 1-$\sigma$ CL is
shown. The dashed line corresponds to the best-fit value of parameters. It can
deviate from the present value of about 9% at $z=3$. The cosmological
evolution of $\tilde{G}/\tilde{G}_{0}$ is shown in the right panel of this
figure. The value of the gravitational constant at $z=3$ can be 40 % larger
than that of the present value in this model. This result is opposite to the
results obtained from using the growth rate Lee:2012nh ; Nesseris:2017vor ;
Kazantzidis:2018jtb  and from SNe data Lee:2021ona .

|
---|---
Figure 4: The time variations of $\tilde{c}$ and $\tilde{G}$ of CPL model with
$\omega_{0}=-1.0$ as a function of $z$ when $\Omega_{\text{m}0}=0.32$. a) The
cosmological evolution of the ratio of $\tilde{c}$ to its present value
$\tilde{c}_{0}$ within 1-$\sigma$ CL. a) The cosmological evolution of the
value of $\tilde{G}/\tilde{G}_{0}$ for 1-$\sigma$ CL.

Table 1: Best fit values and their 1-$\sigma$ errors of cosmological parameters for various DE models of meVSL.  $\omega_{0}$ | $\omega_{a}$ | $\Omega_{\text{m}0}$ | $\gamma_{0}$ | $\gamma_{z}$ | $\gamma_{s}$ | $\delta$ | $\beta$ | $h$ | $b$ | $\chi^{2}$
---|---|---|---|---|---|---|---|---|---|---
-1 | 0 | 0.27 | $1.201$ | $-0.568$ | $0.815$ | $2.349$ | $-0.022$ | $0.717$ | 0 | 344.8
| $1.182$ | $-0.575$ | $0.818$ | $2.370$ | $-0.019$ | $0.738$ | $0.032\pm 0.092$ | 359.6
0.32 | $1.184$ | $-0.589$ | $0.822$ | $2.362$ | $-0.018$ | $0.721$ | 0 | 431.1
$1.183$ | $-0.588$ | $0.822$ | $2.363$ | $-0.018$ | $0.731$ | $-0.339\pm 0.091$ | 421.4
$-0.953\pm 0.044$ | 0 | 0.27 | $1.197$ | $-0.570$ | $0.816$ | $2.352$ | $-0.021$ | $0.717$ | 0 | 345.8
$-1.006\pm 0.062$ |  | $1.198$ | $-0.574$ | $0.817$ | $2.367$ | $-0.006$ | $0.710$ | $0.127\pm 0.120$ | 357.0
$-0.992\pm 0.042$ | 0.32 | $1.181$ | $-0.591$ | $0.822$ | $2.364$ | $-0.017$ | $0.721$ | 0 | 431.1
$-0.966\pm 0.052$ | $1.205$ | $-0.590$ | $0.821$ | $2.359$ | $-0.017$ | $0.700$ | $-0.246\pm 0.108$ | 418.1
$-0.944\pm 0.054$ | 0 | 0.27 | 1.18 | -0.49 | 0.81 | 2.48 | 0 | $0.743\pm 0.009$ | $0.331\pm 0.171$ | 557.4
$-0.914\pm 0.046$ | 0.32 | $0.749\pm 0.009$ | $-0.214\pm 0.166$ | 649.7
$-0.95$ | $-1.115\pm 0.301$ | 0.27 | $1.198$ | $-0.560$ | $0.813$ | $2.354$ | $-0.023$ | 0.74 | $-0.037\pm 0.113$ | 326.3
$-1.00$ | $-0.992\pm 0.247$ |  | $1.199$ | $-0.560$ | $0.814$ | $2.354$ | $-0.023$ | $-0.053\pm 0.103$ | 350.2
$-1.05$ | -0.88 |  | $1.182$ | $-0.563$ | $0.814$ | $2.354$ | $-0.022$ | $-0.057\pm 0.090$ | 338.4
$-0.95$ | $-1.042\pm 0.251$ | 0.32 | $1.185$ | $-0.582$ | $0.820$ | $2.363$ | $-0.019$ | 0.74 | $-0.228\pm 0.104$ | 391.2
$-1.00$ | $-0.922\pm 0.247$ | $1.186$ | $-0.582$ | $0.820$ | $2.363$ | $-0.019$ | $-0.250\pm 0.103$ | 400.5
$-1.05$ | $-0.804\pm 0.241$ |  | $1.191$ | $-0.581$ | $0.820$ | $2.358$ | $-0.021$ | $-0.326\pm 0.101$ | 430.0

##  IV Conclusions

We have analyzed a galaxy-scale strong gravitational lensing sample including
161 systems with stellar velocity dispersion measurements for the meVSL model.
We adopt the lens model parameters with the slope of the total mass density
profile $\gamma$ depending on the lens redshift. We obtain the best fit values
of lens model parameters including the slope of the luminosity density,
$\delta$ and that of the orbit anisotropy parameter $\beta$. All the
parameters strongly depend on the value of the present values of the matter
density contrast $\Omega_{\,m0}$ and thus we put prior on that value from WMAP
and Planck separately.

We investigate dark energy models for $\Lambda$, the constant $\omega$, and
the CPL. When we adopt the value of $\Omega_{\text{m}0}$ from the WMAP, strong
gravitational lensing data provides the positive values of meVSL parameter,
$b$ for $\Lambda$ and constant $\omega$ models. These indicate both the slower
speed of light and the weaker gravitational force in the past. The result for
the CPL model provides the negative values of $b$ indicating the faster speed
of light and the stronger gravitational force in the past. However, the
results for the $\Lambda$ and the CPL are null within 1-$\sigma$ CL.

If we put the prior value of $\Omega_{\text{m}0}$ from the Planck, then we
obtain the negative values of meVSL parameter, $b$ for $\Lambda$, constant
$\omega$ models, and CPL models. These mean both the faster speed of light and
the stronger gravitational force in the past. These results are satisfied for
all models within 1-$\sigma$ CL. However, these results are controversial with
those obtained from the growth rate or supernovae data.

This method includes the fitting of the lens model and this might provide the
systematics to the whole analysis. Also, the statistical power of strong
gravitational lensing is still not enough to remove prior to the value of
$\Omega_{\text{m}0}$ in the analysis. This is important because we obtain the
opposite results for the value of $b$ for the different values of
$\Omega_{\text{m}0}$. Thus, we might need more data on the strong
gravitational lensing to put more reliable constraints on the meVSL model.

## Acknowledgments

SL is supported by Basic Science Research Program through the National
Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT, and
Future Planning (Grant No. NRF-2017R1A2B4011168).

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