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MNRAS 534, 3364–3376 (2024) https://doi.org/10.1093/mnras/stae2306
Advance Access publication 2024 October 08
Revisiting the effect of lens mass models in cosmological applications of
strong gravitational lensing
Christopher Harv e y-Ha wes
‹and David L. Wiltshire
School of Physical and Chemical Sciences, University of Canterbury, Private Bag 4800, Christc hurc h 8041, New Zealand
Accepted 2024 October 4. Received 2024 September 14; in original form 2024 March 19
A B S T R A C T
Strong gravitational lens system catalogues are typically used to constrain a combination of cosmological and empirical power-
law lens mass model parameters, often introducing additional empirical parameters and constraints from high resolution imagery.
We investigate these lens models using Bayesian methods through a novel alternative that treats spatial curvature via the non-
FLRW timescape cosmology. We apply Markov Chain Monte Carlo methods using the catalogue of 161 lens systems of Chen et
al., in order to constrain both lens and cosmological parameters for: (i) the standard CDM model with zero spatial curvature; and
(ii) the timescape model. We then generate large mock data sets to further investigate the choice of cosmology on fitting simple
power-law lens models. In agreement with previous results, we find that in combination with single isothermal sphere parameters,
models with zero FLRW spatial curvature fit better as the free parameter approaches an unphysical empty universe, M0
→ 0.
By contrast, the timescape cosmology is found to prefer parameter values in which its cosmological parameter, the present void
fraction, is driven to f
v0
→ 0 . 73 and closely matches values that best fit independent cosmological data sets: supernovae Ia
distances and the cosmic microwave background. This conclusion holds for a large range of seed values f
v0
∈ { 0 . 1 , 0 . 9 } , and for
timescape fits to both timescape and FLRW mocks. Regardless of cosmology, unphysical estimates of the distance ratios given
from power-law lens models result in poor goodness of fit. With larger data sets soon available, separation of cosmology and
lens models must be addressed.
Key words: gravitation – gravitational lensing: strong – galaxies: haloes – cosmological parameters – cosmology: theory.
1 INTRODUCTION
Strong gravitational lensing (SGL) occurs when the light path from a
distant source is warped around an intermediary lens body on its way
to an observer, producing several images, arcs, or rings. Over the last
couple of decades, strong gravitational lenses have been observed
with an increased number. The current largest compilation of all
strong gravitational lenses which can be used viably for constraining
lens properties and cosmological parameters consists of 161 systems
(Chen et al. 2019 ). The number of such strong gravitational lenses
is set to increase by orders of magnitude o v er the coming decade:
the Rubin Observatory Large Synoptic Surv e y Telescope (LSST)
and Euclid are projected to observe 1 . 2 ×10
6
and 1 . 7 ×10
6
galaxy–
galaxy scale lenses respectively (Collett 2015 ). Since the standard
cosmology faces increasing challenges (Buchert et al. 2016 ; Peebles
2022 ; Aluri et al. 2023 ), it is important to understand the inter-
play of lens models and cosmological models. In this paper, we
will test such assumptions using the existing Chen et al. ( 2019 )
catalogue.
It is possible to constrain the properties of the lens galaxies given
an assumed background cosmology, typically using the standard
spatially flat cold dark matter ( CDM) model with Planck value
parameters (Aghanim et al. 2020 ). Alternatively, one can constrain
E-mail: christopher.harv e y-ha wes@pg.canterbury.ac.nz
the parameters of the assumed cosmological model given a particular
choice of lens mass density profile. Attempts to simultaneously fit
both the lens density profile and cosmological parameters have
been conducted (Cao et al. 2015 ; Chen et al. 2019 ), inevitably
running into degeneracies that require further observational data to
resolve.
There are se veral dif ferent methods to perform statistical cos-
mological analysis with strong gravitational lenses. Time-delay
cosmography as pioneered by Refsdal ( 1964 ) is the most sensitive
to cosmological parameters and deals with each system individually
rather than trying to deal with an ‘average’ ideal lens model. This
method is particularly useful in determining the Hubble constant,
H
0
≡H ( t
0
), as the measured time-delay between multiply imaged
events can be used to determine their different path lengths, and
thereby a value for the Hubble expansion (Millon et al. 2020 ; Wong
et al. 2020 ). Despite time-delay cosmography being a powerful
technique, only a small fraction of the observed lensing systems have
well-defined time-delays, thus limiting the potential application of
this approach. The vast increase in available lens systems from the
next generation of telescopes is now set to overcome this hurdle.
To date most observed galaxy-scale lensing systems involve distant
quasars sources. As compared to observations of smaller numbers
of lensed SneIa, the uncertainties in time-delay measurements of
quasars are significantly larger. Given the paucity of data available
for accurate time-delay cosmography, alternate approaches should
be considered to make full use of the observed lens systems.
© 2024 The Author(s).
Published by Oxford University Press on behalf of Royal Astronomical Society. This is an Open Access article distributed under the terms of the Creative
Commons Attribution License ( https:// creativecommons.org/ licenses/ by/ 4.0/ ), which permits unrestricted reuse, distribution, and reproduction in any medium,
provided the original work is properly cited.
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Revisiting lens mass models in cosmology 3365
MNRAS 534, 3364–3376 (2024)
This paper will focus in particular on the distance ratio
d
ls
d
s =
1 −k d
2
l −d
l
d
s
1 −k d
2
s
, (1)
introduced in the distance sum rule test of R
¨
as
¨
anen, Bolejko &
Finoguenov ( 2015 ). Here d
s
, d
l
, and d
ls denote the dimensionless
comoving distances, d( z
l
, z
s
), from the source to observer, lens to
observer, and source to lens, respectively. The comoving distance
is related to the angular diameter distances, D
A
, and dimensionless
luminosity distance, D
L
by
d =
H
0
c
D
A
(1 + z) =
H
0
c
D
L
1 + z
(2)
=
1
√
k0
sinh
√
k0
1 + z
1
d x
√
0
+ k0
x
2
+ M0
x
3
, (3)
where k0
= −kc
2
/H
2
0 = 1 −M0
− 0
, and M0 and 0 are
the present epoch matter and cosmological constant density pa-
rameters. In this paper, we will focus on spatially flat Friedmann–
Lema
ˆ
ıtre–Robertson–Walker (FLRW) models where k0
= 0 and
thus ( 3 ) becomes
d =
1 + z
1
d x
(1 −M0
) + M0
x
3
. (4)
and the distance ratio is given as
D ≡d
ls
d
s =
d
s
−d
l
d
s
. (5)
The relation ( 1 ) provides an e xact e xpression for the ratio of two
observed angular distances in the case of a FLRW metric with
constant spatial curvature k. It is then typically used as a self-
consistency check of the standard FLRW cosmology, or to constrain
the global spatial curvature of some assumed FLRW universe (Liao
et al. 2017 ; Qi et al. 2019 ). Ho we v er, an y cosmological model –
whether FLRW or not –can be substituted on the r.h.s. of ( 1 ) provided
the model has a suitable definition of the distance ratio in terms of
its underlying parameters.
For a specific lensing system, it is also possible to compute its
respective distance ratio purely from a given lens model and its
corresponding observables and parameter values, independent of
cosmology. Therefore, we can compare values for the distance ratio
derived from cosmological angular diameter distance measurements
to values determined from an assumed lens model, thus, allowing
for constraints on both lens and cosmological parameters. This
can be extended to a comparison of different cosmological models
through Bayesian inference (Cardone, Piedipalumbo & Scudellaro
2016 ). Such Bayesian analysis might involve two FLRW models
with different priors on cosmological parameters, or alternatively an
FLRW model compared to a non-FLRW model such as the timescape
(Wiltshire 2009b ).
The distance ratio is independent of the Hubble parameter, and
therefore sidesteps the issue of its current epoch value H
0
= H ( t
0
).
This is a major advantages of this technique, as the Hubble tension
continues to pose a significant challenge to the CDM model (Riess
et al. 2023 ). Furthermore, since SGL bypasses the cosmic distance
ladder, it directly allows us to infer cosmological parameters without
relying on other astronomical distance measures and the systematics
they depend upon.
While the distance ratio has been used to test the self-consistency
of FLRW models, to date it has not been applied to non-FLRW
metrics. This paper will fill that gap by revisiting the analysis of Chen
et al. ( 2019 ) and extending it to include the timescape model. We
will perform a Bayesian comparison between the spatially flat FLRW
model and the timescape model, and determine the corresponding
best-fitting cosmological parameters. In addition, we investigate
three different parametrizations of the lens galaxies mass profile
and their effect on cosmological fits. Strong gravitational lensing
is heavily dependent on the choice of lens model, as well as the
background cosmology the systems are embedded within. Therefore,
it is vital that lens galaxies are well modelled in advance of new
observations by the next generation of telescopes.
An o v ervie w of the paper is as follo ws. In Section 2 , we will briefly
re vie w the timescape model. In Section 3 we will discuss the different
lens models used in this paper, as well as provide a description
of the catalogue of lensing systems, identifying the uncertainties
involved, and model biases. Section 4 outlines the methodology for
determining cosmological parameters within a given lens model and
gives the results of the fitting procedure. Section 5 presents the results
of simulations of mock data generated from the catalogue data with
randomly added noise. In Section 6 we discuss our results and the
conclusions that can be drawn.
2 TIMESCAPE MODEL OVERVIEW
Standard CDM cosmology, and indeed all FLRW models, are based
upon the cosmological principle that on average the Universe is both
spatially homogeneous and isotropic. While some notion of spatial
homogeneity and isotropy certainly holds for whole sk y av erages
of very distant objects, the Universe is inhomogeneous on scales
100 h
−1
Mpc (Eisenstein et al. 2005 ; Sylos Labini et al. 2009 ;
Scrimgeour et al. 2012 ) where it is dominated by a cosmic web of
o v erdense filaments and underdense voids. The standard cosmology
has been incredibly successful. None the less, challenges to the
CDM model remain unresolved despite the growth of available data
and increased observational precision (Buchert et al. 2016 ; Peebles
2022 ; Aluri et al. 2023 ).
The timescape model assumes that cosmology should not invoke a
single global reference background with a unique split of space and
time (Wiltshire 2007a , b , 2009a ). To model the Uni verse ef fecti vely,
we have to account for backreaction of inhomo g eneities , namely
deviations from average FLRW evolution at the present epoch. This
is due to structure formation of filaments and voids and the resulting
cosmic web, which grows in complexity in the late Universe.
Timescape makes use of Buchert spatial averages (Buchert 2001 )
to provide the average evolution of the Einstein equations with
backreaction.
The interpretation of Buchert’s formalism has been much debated;
critics pointed out that as a fraction of the averaged energy density, a
term Buchert denotes the ‘kinematical backreaction’ ¯
Q
, should be
small (Ishibashi & Wa l d 2006 ). A universal feature of all viable back-
reaction proposals is that, unlike FLRW, average spatial curvature
does not scale as a simple power of the average cosmic scale factor
¯
a ,
¯
K = −k/ (
¯
a
2 ¯
H
2
), where k is constant and
¯
H an average expansion
rate. In the timescape model, in particular, ¯
K ∝ f
1 / 3
v
/ (
¯
a
2 ¯
H
2
),
where the variable void volume fraction , f
v
→ 0 at early times but is
significant in the late epoch Universe. This means that conclusions
about spatial curvature derived from CMB data in FLRW models are
not rele v ant for backreaction models: when fit to the angular diameter
distance of the CMB (with = 0) it is curvature, ¯
K
, that dominates
in the late epoch Universe (Duley, Nazer & Wiltshire 2013 , fig. 1).
Furthermore, while the average energy-density parameters for matter,
radiation, curvature, and kinematical backreaction satisfy a simple
sum rule
¯
M
+ ¯
R
+ ¯
K
+ ¯
Q
= 1 , (6)
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3366 C. Harvey-Hawes and D. L. Wiltshire
MNRAS 534, 3364–3376 (2024)
the quantities appearing in ( 6 ) are statistical volume average
quantities, not directly related to local observables. Consequently,
conclusions about apparent cosmic acceleration do not follow simply
from determining the magnitude of ¯
Q
. The timescape ansatz fixes
the relationship of these quantities to local observables, resulting in a
viable phenomenology with ¯
K
∼0 . 86, ¯
M
∼0 . 17, ¯
Q
∼−0 . 03
at present, z = 0.
Since the local geometry varies dramatically depending on the
position of an observer in the cosmic web, local geometry is generally
very different to the global v olume a verage. Local structure not only
affects the average evolution of the Einstein equations, but also the
calibration of distance and time measures relative to the volume
av erage
1
observ er. Timescape allo ws for v ariation of the calibration
of regional clocks relative to one another due to the gravitational
energy cost of the gradients in spatial curvature between filaments
and voids. Ideal observers, who see a close to isotropic CMB, in
different local environments would age differently, hence the name
timescape.
The accelerated expansion of the Universe is now an apparent
ef fect deri ved from fitting an FLRW model with constant spatial
curvature and global cosmic time to an inhomogeneous non-FLRW
Universe. An ideal void observer not gravitationally bound to any
structure will not infer a cosmic acceleration at all, whereas an
observer in a bound structure such as a galaxy will use a different
time parameter and will infer an accelerated expansion at late
cosmic epochs. The timescape model can eliminate the need for
dark energy entirely and does so without introducing new ad hoc
scalar fields or modifications to the gravitational action; rather it
revisits averaging procedures that build upon Einstein’s general
relativity.
Since the CDM has been so successful in diverse cosmo-
logical tests, any phenomenologically viable model must yield
similar predictions, particularly with regard to large scale a ver -
ages for distances beyond the scale of inhomogeneity. For over a
decade, timescape has consistently given fits to type Ia supernovae
which are essentially statistically indistinguishable from CDM
by Bayesian comparison (Leith, Ng & Wiltshire 2007 ; Smale &
Wiltshire 2011 ; Dam, Heinesen & Wiltshire 2017 ). The most recent
analysis of the Pantheon + catalogue shows, furthermore, that
timescape predictions for cosmic expansion below ∼100 h
−1
Mpc
scales are consistent with observations in a manner which may
provide a self-consistent resolution of the Hubble tension (Lane
et al. 2023 ). The difference between the distance–redshift relations
of the timescape model and those of some CDM model with fixed
M0
, 0
, is only 1–3 per cent o v er a small range of redshifts,
but timescape ef fecti vely interpolates between CDM models with
different parameters o v er larger redshift ranges –which is why it
can resolve the Hubble tension in a natural fashion. Of course,
this must be tested with completely independent cosmological
tests.
Among the many different definition of distance in the timescape
model, here we focus on the matter dominated era, where a simple
‘tracking solution’ is found to apply. Instead of the parameters M0
in a spatially flat FLRW model, distances are instead parametrized in
terms of the present epoch void fraction f
v0
. The effective comoving
1
A v olume a v erage observ er in the timescape model represents an observ er
whose local spatial curvature coincides with that of the largest spatial
av erages. Such an observ er is necessarily at an av erage position by volume –
in a void –and systematically different from observers in bound structures.
A typical location by volume is not a typical location by mass.
distances directly comparable to those in the distance sum rule ( 1 )
are
2
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
d
s = H
0
t
2 / 3
s
[ F( t
0
) −F( t
s
)](1 + z
s
)
d
l
= H
0
t
2 / 3
l
[ F( t
0
) −F( t
l
)](1 + z
l
)
d
ls = H
0
t
2 / 3
s
[ F( t
l
) −F( t
s
)](1 + z
s
)
, (7)
where
F( t) = 2 t
1 / 3
+
b
1 / 3
6
ln
( t
1 / 3
+ b
1 / 3
)
2
t
2 / 3
−b
1 / 3
t
1 / 3
+ b
2 / 3
+
b
1 / 3
√
3
arctan
2 t
1 / 3
−b
1 / 3
√
3 b
1 / 3 , (8)
b =
2 (1 −f
v0
)(2 + f
v0
)
9 f
v0
¯
H
0
, (9)
and H
0
and
¯
H
0
are the dressed and bare Hubble constants, respec-
tively.
3 The Buchert volume averaged time parameter t has been
interpreted in different ways in different backreaction models. For
the timescape tracker solution, observers in gravitationally bound
systems measure a time parameter related to v olume-a verage time
according to
τ=
2
3
t +
2(1 −f
v0
)(2 + f
v0
)
27 f
v0
¯
H
0
ln
1 +
9 f
v0
¯
H
0
2(1 −f
v0
)(2 + f
v0
)
. (10)
The v olume-a verage time parameter is related to our observed
redshift by
z + 1 =
2
4 / 3
t
1 / 3
( t + b)
f
1 / 3
v0
¯
H
0
t (2 t + 3 b)
4 / 3
, (11)
(Larena et al. 2009 ). The distance ratio is then given as
D
Time
≡d
ls
d
s =
F( t
l
) −F( t
s
)
F( t
0
) −F( t
s
)
. (12)
Further details on the moti v ation and deri v ation of these equations is
given by Wiltshire ( 2009b , 2014 ). It is now simple to make com-
parisons between timescape and spatially flat FLRW models, as
the y hav e the same number of free parameters: f
v0 and M0 are
direct analogues due to their effect on distance measurements. Any
dependence on the dressed or bare Hubble constant is cancelled due
to the nature of the distance ratio, hence there is only one defining
cosmological parameter for each model.
3 CATALOGUE, OBSERVABLES, AND LENS
MODELS
3.1 Catalogue data
The data set used in this investigation is taken from the catalogue
of 161 strongly lensed systems from Chen et al. ( 2019 ). It is the
2
Care must be taken with choices of units. With an explicit factor c , b , and
F must have dimensions of t and t
1 / 3
respectively, which is different to a
convention often assumed for timescape (Wiltshire 2014 ).
3
The bare or volume average Hubble parameter is defined by the average
volume expansion – not a global scale factor –and uses a time parameter
relative to an idealized volume av erage observ er. This time parameter differs
systematically from that of observers in bound systems. The bare Hubble
parameter is found to be related to the dressed Hubble parameter by
H ( t ) =
[4 f
2
v
( t ) + f
v
( t) + 4]
¯
H ( t)
2(2 + f
v
( t))
.
and f
v0
≡f
v
( t
0
) etc. denote present epoch values.
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Revisiting lens mass models in cosmology 3367
MNRAS 534, 3364–3376 (2024)
largest catalogue compiled to date, and is itself built upon a previous
catalogue of 118 systems from Cao et al. ( 2015 ). This catalogue is a
compilation of multiple surv e ys consisting of the LSD (Koopmans &
T reu 2002 ; T reu & K oopmans 2002 ; K oopmans & T reu 2003 ;
T reu & K oopmans 2004 ), SL2S (Ruff et al. 2011 ; Sonnenfeld
et al. 2013a , b , 2015 ), SLACS (Bolton et al. 2008 ; Auger et al.
2009 , 2010 ), S4TM (Shu et al. 2015 , 2017 ), BELLS (Brownstein
et al. 2012 ), and BELLS Gallery surv e ys (Shu et al. 2016a , b ).
From these surv e ys, the follo wing observ ables are listed: spectro-
scopic lens and source redshift z
l and z
s
, Einstein radius θE
, half-
light radius of the lens galaxy θeff
, aperture radius θap
, and the
central velocity dispersion of the lens galaxy σap
. Spectroscopic
measurements are provided by Sloan Digital Sky Survey (SDSS), W.
M. Keck-II Telescope, and Baryon Oscillation Spectroscopic Surv e y
(BOSS) spectroscopic instruments depending on surv e y. F ollow-up
imagery from the Hubble Space Telescope ( HST ) Advanced Camera
for Surv e ys is then used for the determination of θE
and θeff
.
Each of these surv e ys has differing instrumental parameters and
thus has different uncertainties in the measurement of the central
velocity dispersions
4 of the lens galaxies, which contributes the
largest portion of the uncertainty in this method. Velocity dispersions
of lens galaxies are measured spectroscopically within a given
aperture radius θap
. Ho we ver, the shape and size of the aperture
used in each surv e y differs and thus an aperture correction formula
is applied so that velocity dispersions are determined as if they were
found with a normalized typical circular aperture σ0 (Jorgensen,
Franx & Kjaergaard 1995 ). To transform rectangular apertures into
an equi v alent circular aperture for comparison, the follo wing formula
is used,
θap
≈1 . 025
θx
θy
π, (13)
where θx
and θy
are the width and height of the rectangular aperture,
respectively. Since aperture radii vary even within the class of circular
apertures, the following normalization is also applied along the line
of sight,
σ
≡σ0
= σap
θeff
2 θap
η
, (14)
where θeff
corresponds to the half-light radius of the lens galaxy as
θeff
= R
eff
/D
l
. The value of ηis taken from Cappellari et al. ( 2006 )
of η= −0 . 066 ±0 . 035 is found empirically via fitting to individual
galaxy profiles and is in agreement with other values determined by
Jorgensen et al. ( 1995 ) and Mehlert et al. ( 2003 ) of η= −0 . 04 and
η= −0 . 06, respectively.
The uncertainty in ηis the greatest source of uncertainty in the
distance ratio inferred from gravitational lensing. In addition, ( 14 )
contains a statistical error in σap as quoted in the surv e ys, and a
further systematic error source. The latter is understood as the effect
of interfering matter along the line of sight and whilst the surv e ys
are restricted to isolated systems, the systematic error is estimated
at 3 per cent in all cases (Bernardi et al. 2003 ). The combined total
error budget is then given by,
σ tot
0 =
( σ η
0
)
2
+ ( σ sys
0
)
2
+ ( σ stat
0
)
2
. (15)
No uncertainty in the Einstein radius or associated redshifts is quoted
within the catalogue data set. Thus, it is assumed these uncertainties
4
This is obtained by converting the redshift dispersion from spectroscopic
measurements around the galaxy using the radial special relativistic formula
z =
√
( c + v) / ( c −v) −1.
are included within the systematic uncertainty of σ0
, although this
is most likely an underestimation of the errors present in the data.
A uniform error could be applied across the data set to θE
, but is
omitted to stay consistent with the work of Chen et al. We find
that changing the magnitude of the errors present in our param-
eters does not significantly effect our conclusions throughout this
paper.
Further cuts were also applied in compiling the catalogue. Not only
did systems have to be isolated and have no significant substructure,
but lens galaxies must also be early-type galaxies, either elliptical
(E) or lenticular (S0).
3.2 Lens models
When modelling lens mass distributions individually for use in time-
delay cosmography, one of two descriptions are used: (i) power-
law models that take into account both baryonic and dark matter
simultaneously; and (ii) a combination of a luminous baryonic
mass profile with a Navarro-Frank-White dark matter halo (Navarro,
Frenk & White 1996 ; Wagner 2020 ). Constraints on the value of
H
0 from either class of model are very similar and statistically
consistent with one another (Millon et al. 2020 ). Ho we ver, to model a
statistically average lens galaxy within large catalogues, more general
power-law models are used (Koopmans 2006 ).
In this section, we re vie w the three power-law models applied
to the (Chen et al. 2019 ) catalogue, commonly used to model the
matter distribution of elliptical lens galaxies: extended power law
(EPL), spherical power law (SPL), and singular isothermal sphere
(SIS).
All three lens models discussed in this paper rely upon the assump-
tion that while galaxies may differ in shape, when taking averages the
dominant component of the matter density is spherically symmetric.
Ho we ver, the angular structure of a lens galaxies mass distribution is
very significant to the image separation produced. In fact, quadruply
imaged sources are not possible with spherically symmetric mass
distributions. Since the luminous matter in individual galaxies is
not spherically symmetric, whether the angular component of all
matter in each lens galaxy can be averaged out in this fashion is an
assumption which can be tested in future.
Solving the radial Jeans equation, following the procedure given
in Appendix B , the distance ratio for the most general EPL model is
found to be
D
obs
≡d
ls
d
s =
c
2
θE
2
√
πσ2
0 θeff
2 θE
2 −γ
F ( γ, δ, β) , (16)
where θE
is the Einstein radius of the system. F ( γ, δ, β) depends on
the total mass density slope, γ, the luminous matter density slope, δ,
and the stellar orbital anisotropy, β.
In deriving ( 16 ) several physical assumptions have been made,
viz.:
(i) The creation, destruction, and collisions of stars are neglected.
(ii) The thin lens approximation applies as the distances from
observer to lens and lens to source are far larger than the width of
the lens object itself. A projected mass density can then be defined
in the lens plane that is responsible for the deflection of light.
(iii) The system is stationary, i.e. ∂
t = 0. The properties of the lens
galaxy vary insignificantly o v er the short periods in which lensed
images are observed.
(iv) The weak field limit of general relativity applies (lin-
earized gravity) with an asymptotically flat (Minkowski) back-
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3368 C. Harvey-Hawes and D. L. Wiltshire
MNRAS 534, 3364–3376 (2024)
ground. For the vast majority of lensing scenarios one assumes
/c
2
1.
(v) Angles of deflection due to gravitational lensing are small, so
that the small angles approximation is invoked.
(vi) The Born approximation applies as the deflection angle is
small. The gravitational potential along the deflected and undeflected
light paths can be considered to be approximately the same.
3.2.1 Extended power-law model
For the EPL model (Koopmans 2006 ), the most nuanced and complex
model considered in this paper, the function F in ( 16 ) takes the form
F ( γ, δ, β) =
3 −δ
( ξ−2 β)(3 −ξ) ×
ξ−1
2
ξ
2
−β
ξ+ 1
2
ξ+ 2
2
γ
2
δ
2
γ−1
2
δ−1
2
, (17)
where ξ= γ+ δ−2.
Through a specific choice of γ, δ, and β, which parametrize the
total mass density profile ρ, the luminous mass density profile ν, and
the stellar orbital anisotropy β,
ρ( r) = ρ0
r/r
0
−γ, (18)
ν( r) = ν0
r/r
0
−δ, (19)
β= 1 −σ2
θ/σ 2
r
, (20)
simpler models are reco v ered. Both γ= γ0
and
γ= γ0
+ γz
z
l
+ γs
log ˜
(21)
parametrizations are considered for the total matter density profile
where the normalized surface mass density ˜
∝ σ2
0
/R
eff
.
3.2.2 Spherical power-law model
If both δ= 2 and β= 0 are fixed, the SPL model is obtained. In
this spherically symmetric model, the only free parameter is the total
mass density (dark matter halo) profile, γ, and
F : = F ( γ, 2 , 0) =
1
√
π
1
γ(3 −γ)
. (22)
The SPL model is often generalized further to allow for the
variation of γwith redshift or surface brightness density of the
lens galaxy. For our investigation, we limit γto a constant value, γ0
.
Ho we ver, Chen et al. ( 2019 ) explore different parametrizations of γ,
which are discussed in Section 4.1 .
3.2.3 Singular isothermal sphere model
The SIS model is the simplest discussed, where γ= δ= 2, β= 0,
and
F : = F (2 , 2 , 0) =
1
2
√
π. (23)
It assumes that the halo mass of the lens galaxy is in isothermal
equilibrium, with gravitational attraction balanced entirely by the
pressure associated with the constituent stars’ interactions. The total
mass density relation is then
ρ( r ) =
σ2
2 πGr
2
, (24)
which leads to the distance ratio,
d
ls
d
s =
c
2
θE
4 πσ2
0
. (25)
4 PARAMETER DETERMINATION AND MODEL
PREFERENCE RESULTS
From ( 4 ) and ( 5 ), or ( 12 ), with ( 16 ) we find a value of σ0 that
depends purely on model parameters, redshifts z
s and z
l
, and the
Einstein radius θE
. We can then compare these values with the
observationally determined ones via an MCMC sampling procedure,
to constrain the defining parameters of both lens and cosmological
models.
The likelihood of the combined lens and cosmological models is
determined via
χ2
=
161
i= 1 σ0
−σmodel
0
( z
l
, z
s
, θE
, p )
2
σ 2
0
, (26)
where p are the defining parameters for a specific lens model
i.e. γ, δ, and βas well as cosmological parameters M0 and
f
v0
. σmodel
0
is given by the combination of lens and cosmological
models.
We initially take wide uniform priors for all parameters to a v oid
biases in the values determined. The stellar orbital anisotropy βof
all lens galaxies in the catalogue is not measured, so we assume a 2 σ
Gaussian prior β= 0 . 18 ±0 . 26, established from a study of nearby
elliptical galaxies and adopted by Chen et al. ( 2019 ). A full list of
the priors involved in our MCMC sampling is given in A1 .
To obtain the results given in Figs 1 and 2 , the MCMC sampler
ran for 50 000 steps with 32 w alk ers and a burn-in phase discarding
the first 20 per cent of the samples. These were created using the
EMCEEPYTHON package.
To distinguish between the spatially flat FLRW and timescape
models, which have the same number of free parameters, a
straightforw ard Bayes f actor, B, can be calculated by integrating
the likelihoods L ∼e
−χ2
/ 2 o v er the 2 σpriors f
v0
∈ (0 . 5 , 0 . 799),
M0
∈ (0 . 143 , 0 . 487) (Dam et al. 2017 ; Trotta 2017 ), and corre-
sponding lens model parameters. The cosmological parameter priors
are determined using non-parametric fits of the Planck CMB data
(Aghamousa & Shafieloo 2015 ) constrained by estimates of the
angular BAO scale from the BOSS surv e y data and Lyman alpha
forest statistics (Delubac et al. 2015 ; Alam et al. 2017 ). Following
Dam et al. ( 2017 ) we use wide priors for both models so as
not to unfairly disadvantage CDM. The priors usually adopted
for CDM, including a precise estimate M0
= 0 . 315 ±0 . 007
(Aghanim et al. 2020 ), arise from constraints on perturbation
theory on an FLRW background. A timescape equi v alent has
yet to be developed to constrain the equivalent free parameter,
f
v0
. We take the luminosity density profile δto have priors in-
formed by profiles fitted to HST imagery, giving 2 . 003 < δ< 2 . 343.
The full list of priors used for Bayesian comparison are given
in A2 .
We adopt the Jeffrey’s scale for the Bayes factor
B =
L
timescape
d f
v0
d γd δd β
L
FLRW
d M0
d γd δd β(27)
so that values of B > 1 will indicate preference for timescape
and B < 1 for FLRW. By the standard interpretation, evidence
with | ln B| < 1 is ‘not worth more than a bare mention’ (Kass &
Raftery 1995 ) or ‘inconclusive’ (Trotta 2007 ), while 1 ≤| ln B| < 3,
3 ≤| ln B| < 5 and | ln B| ≥5 indicate ‘positive,’ ‘strong,’ and
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Revisiting lens mass models in cosmology 3369
MNRAS 534, 3364–3376 (2024)
Figure 1. Parameter probability distributions of the EPL lens model for both spatially flat FLRW and timescape models, constrained using 161 lensing systems.
Dashed lines representing the 2 σbounds and median value of MCMC samples.
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3370 C. Harvey-Hawes and D. L. Wiltshire
MNRAS 534, 3364–3376 (2024)
Figure 2. Parameter probability distributions of the fully extended EPL lens model for both spatially flat FLRW and timescape models, constrained using 161
lensing systems.
‘very strong’ e vidences, respecti vely (Kass & Raftery 1995 ).
Tab l e s 1 and 2 give the results of the MCMC sampling for
each lens model in a spatially flat FLRW model and timescape,
respectively.
4.1 Discussion of results
The Bayes factors fa v our a spatially flat FLRW model o v er the
timescape model in all cases but with varying degrees of strength, as
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Revisiting lens mass models in cosmology 3371
MNRAS 534, 3364–3376 (2024)
Tab l e 1. Ideal parameter values obtained from MCMC sampling of velocity dispersions obtained using FLRW distances and one of the listed lens models.
FLRW ( k = 0)
Lens model β δ γ M0
EPL (full parametrization) ∗0 . 2292
+ 9 . 773 ×10
−2
−1 . 042 ×10
−1
2 . 3823
+ 4 . 947 ×10
−2
−5 . 050 ×10
−2
γ0 = 2 . 9450
+ 3 . 928 ×10
−2
−6 . 653 ×10
−2
0 . 9124
+ 6 . 491 ×10
−2
−1 . 265 ×10
−1
γz = −0 . 2651
+ 3 . 632 ×10
−2
−3 . 562 ×10
−2
γs = 0 . 2323
+ 1 . 146 ×10
−2
−1 . 766 ×10
−2
EPL ( γ= γ0
) 0 . 1388
+ 2 . 155 ×10
−1
−1 . 978 ×10
−1
2 . 3413
+ 1 . 579 ×10
−1
−1 . 722 ×10
−1
2 . 0162
+ 2 . 522 ×10
−2
−2 . 740 ×10
−2
0 . 0257
+ 3 . 561 ×10
−2
−1 . 838 ×10
−2
SPL – – 1 . 9306
+ 1 . 707 ×10
−2
−1 . 844 ×10
−2
0 . 0188
+ 2 . 317 ×10
−2
−1 . 319 ×10
−2
SIS – – – 0 . 0036
+ 5 . 00 ×10
−3
−2 . 70 ×10
−3
Tab l e 2. Ideal parameter values obtained from MCMC sampling of velocity dispersions obtained using timescape distances and one of the listed lens
models.
Timescape
Lens model β δ γ f
v0
EPL (full parametrization) ∗0 . 2292
+ 9 . 773 ×10
−2
−1 . 042 ×10
−1
2 . 3823
+ 4 . 947 ×10
−2
−5 . 050 ×10
−2
γ0 = 2 . 9450
+ 3 . 928 ×10
−2
−6 . 653 ×10
−2
0 . 9124
+ 6 . 491 ×10
−2
−1 . 265 ×10
−1
γz = −0 . 2651
+ 3 . 632 ×10
−2
−3 . 562 ×10
−2
γs = 0 . 2323
+ 1 . 146 ×10
−2
−1 . 766 ×10
−2
EPL ( γ= γ0
) 0 . 1497
+ 2 . 241 ×10
−1
−2 . 037 ×10
−1
2 . 3762
+ 1 . 285 ×10
−1
−1 . 368 ×10
−1
1 . 9609
+ 2 . 394 ×10
−2
−2 . 740 ×10
−2
0 . 7367
+ 9 . 122 ×10
−2
−1 . 011 ×10
−1
SPL – – 1 . 8837
+ 6 . 20 ×10
−3
−6 . 20 ×10
−3
0 . 6751
+ 1 . 73 ×10
−2
−1 . 73 ×10
−2
SIS – – – 0 . 7044
+ 3 . 780 ×10
−2
−3 . 859 ×10
−2
Tab l e 3. Bayes factors ( 27 ), where B < 1 fa v ours the spatially flat FLRW
model.
Bayes factors (B)
Lens model B | ln B|
EPL 0.4928 0.7077
SPL 9 . 38 ×10
−7 13.9
SIS 3 . 28 ×10
−3 5.72
shown in Ta b l e
3 . For the SIS and SPL lens models, the preference is
a strong. Ho we ver, both FLRW and timescape have a minimum χ2
per degree of freedom ∼2, which shows in both cases the fit could be
impro v ed, specifically within the choice and parametrization of the
lens model. It is important to note that the Bayes factors of Ta b le 3
should not be interpreted na
¨
ıvely, as lower χ2 for FLRW models
comes at the expense of an unphysical matter density at the extremes
M0
0 or M0
1. By contrast, the values of f
v0 predicted are
within the 2 σpriors f
v0
∈ (0 . 5 , 0 . 799) for timescape and remain
physically plausible for the majority of lens model parametrizations.
Chen et al. ( 2019 ) already noted that M0
0 for specific lens
models. Ho we ver, by considering
(i) an alternate parametrization of γto include a dependence on
redshift and normalized surface mass density of each lens galaxy,
γ= γ0
+ γz
z
l
+ γs
log ˜
; and
(ii) δas an observable for each lens galaxy
more realistic values of M0 are inferred. One can only fit δ
given high-resolution imagery of a lens galaxy; for the Chen et al.
( 2019 ) catalogue this reduces the sample of lensing systems from
161 to 130. Future surv e ys are predicted to observ e sev eral orders of
magnitude more lensing systems. The requirement of follow up high
resolution imaging will therefore face significantly greater challenges
Figure 3. Kernel density estimate plots of the best-fitting parameters for: (a)
timescape fit to timescape data, (b) FLRW fit to timescape data, (c) FLRW
fit to FLRW data, and (d) timescape fit to FLRW data. The dashed vertical
line indicates the initial parameter value used to generate the mock data,
f
v0
= 0 . 76 and M0
= 0 . 315 for timescape and FLRW models, respectively.
on account of the vast increase of data volume. When fitting a global
value for δin the EPL model, we find values consistent with 2 σ
priors informed by HST imagery of 2 . 003 < δ< 2 . 343 as shown in
Figs 1 and 2 .
Physically plausible values of M0 for FLRW have only been
found when steps (i) and (ii) are applied to the further parametrized
EPL model. If δis not constrained for each system in the EPL
model, then M0
1, the other unphysical extreme. Timescap e also
returns the physically implausible extreme value for f
v0
0 which
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3372 C. Harvey-Hawes and D. L. Wiltshire
MNRAS 534, 3364–3376 (2024)
Figure 4. Ker nel density estimate plots showing the best-fitting cosmological parameters for both FLRW (a) and timescape (b) models fit against data generated
with varying initial parameter values.
Figure 5. Distance ratios against velocity dispersion (kms
−1
) for a single
isothermal sphere lens model using catalogue data for 161 systems.
corresponds exactly to the Einstein de Sitter universe ( M0
= 1) for
the full parametrization of γ.
5 MOCK CATALOGUES AND PARAMETER
FITTING
5.1 Methodology
The unphysical cosmological parameter values found through
MCMC sampling, in particular of M0
, moti v ate further investiga-
tion. Thus, we generate mock catalogues to gauge how sensitive the
fitting procedure is on cosmological parameters. We generate mock
catalogues using the following procedure:
(i) Using the catalogue data for the 161 lensing systems, we apply
the relation for the simplest case lens model SIS ( 25 ), in combination
with distance ratios found from cosmology [see ( 4 ) and ( 5 ) or ( 12 )]
to find the model ideal values of σ0
. This requires an initial seed
value of M0
or f
v0
, which we hope to later reco v er when fitting.
(ii) Gaussian noise is then added to σ0 for each of the 161 lens
galaxies.
(iii) We then use ( 25 ) with the catalogue data and the velocity
dispersion, with added noise, to find a ne w v alue of the distance ratio
d
ls
/d
s
.
(iv) The mock distance ratio and the value determined from either
the timescape or spatially flat FLRW models can then be compared
through the χ2
test
χ2
=
161
i= 1
D
model
i
( q ) −D
mock
i
( q
seed
)
D
i 2
, (28)
where q = { M0
, f
v0
} depending on cosmology.
(v) The M0
and f
v0
parameters are varied, the best-fitting values
corresponding to the minimized χ2
value.
(vi) We repeat this for 10
4 mock samples, with the best-fitting
parameters for each cosmology being binned into a histogram for
each individual mock. We then determine whether the original seed
parameters of M0
and f
v0
are reco v ered from the fitting procedure.
A wide variety of seed values are chosen with parameters in the
range M0
∈ { 0 . 1 , 0 . 9 } for spatially flat FLRW, and similarly f
v0
∈
{ 0 . 1 , 0 . 9 } for timescape. We find that regardless of the initial input
values of the cosmological parameters, this procedure al w ays lowers
M0
significantly for FLRW. For timescape, a value of f
v0
0 . 73 is
returned irrespective of the input value.
5.2 Simulation results and discussion
The histograms produced from the mock data provide interesting
results, shown in Fig. 3 . Na
¨
ıvely, one would assume that if the
cosmological model plays a significant role in the pipeline then the
generated cosmological parameters would match the input values.
Ho we ver, this is only the case when the uncertainty in the velocity
dispersion is set artificially low.
The timescape fit finds values of f
v0
≈0 . 73 which are consis-
tent with the model expectation based on constraints from SneIa
distances, CMB, etc (Duley et al. 2013 ). In fact, this is still true
for timescape fits of the FLRW mocks. The FLRW mocks have
a significantly wider cosmological parameter spread, as shown in
Fig. 4 (a) as compared to Fig. 4 (b). Furthermore, the FLRW model
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Revisiting lens mass models in cosmology 3373
MNRAS 534, 3364–3376 (2024)
Figure 6. Distance ratios for fixed lens and source redshifts in both spatially flat FLRW and timescape models with varying M0
and f
v0
, respectively.
yields an M0 more in-line with the expected value of M0 from
the Planck CMB data (Aghanim et al. 2020 ) when it is fit against
data generated with the timescape model. For FLRW the maximum
likelihood of M0
is found to increase as the input M0
is lowered,
resulting in an o v erall maximum likelihood M0
→ 0, corresponding
to an unphysical Milne univ erse. F or timescape the maximum
likelihood increases for seed values close to f
v0
0 . 73.
Both the data and mock catalogues produce some outlying unphys-
ical distance ratios D > 1 with closer source than lens to observer
(see Fig. 5 ). This is an artefact of choosing a global power-law lens
model and fitting to a data set with high variations in the observed
velocity dispersions. Some systems are not well described by the lens
model choice, and thus produce abnormally high distance ratios that
skew the distribution. Fitting any curve D( σ) to the data of Fig. 5
will be skewed by the unphysical values.
The implications can be understood by considering Figs 6 (a) and
(b) where distance ratios D
FLRW
and D
Time
are shown for particular
fixed source and lens redshifts, z
s
and z
l
. Varying z
s
and z
l
we see that
while the distance ratio generally increases with increasing z
s
, the
maximum is al w ays found at M0
0 for FLRW and f
v0
0 . 73 for
timescape. With a high enough uncertainty in the velocity dispersion,
much larger distance ratios are possible in the simulations. Therefore,
the fitting of cosmological parameters is forced towards the values
that enable the highest possible distance ratios.
Whilst our simulations were performed using a SIS lens model,
the entire class of lens models in our investigation return unphysical
distance ratios, D ≥1, when using the ideal parameters determined
by MCMC sampling. Even the EPL model with the full parametriza-
tion ( 21 ), which enables a slight variation of the power-law model
between systems, often yields unphysical distance ratios.
6 CONCLUSIONS
In order to increase the goodness of fit of power-law lens models,
additional empirical parameters are often added to the lens models
along with constraints from high resolution imagery. The simplest
parametrizations of these models, in combination with the standard
spatially flat FLRW cosmology, have already been shown to produce
poor fits to data without additional observationally determined
luminosity density profiles (Chen et al. 2019 ). We find that all choices
of power-law model investigated na
¨
ıvely prefer a spatially flat FLRW
cosmology, but as a consequence delegate all the mass along the
line of sight to the lens galaxy, resulting in M0
0. For the full
parametrization ( 21 ) of the EPL model the fits return an Einstein de
Sitter universe with M0
= 1, which is also clearly unphysical.
The timescape cosmology gives void fraction values of 0 . 68 ≤
f
v0
≤0 . 74 consistent with constraints from independent tests (Duley
et al. 2013 ; Lane et al. 2023 ) in all cases except that of ( 21 )
when it also returns the same unphysical Einstein de Sitter universe
( f
v0
= 0). Simulations with a fixed SIS lens model consistently return
values of f
v0
0 . 73, regardless of the input seed value. Our results
highlight the imperative of exploring alternatives to the standard
CDM cosmology. Ev en if an y particular non-FLRW cosmology is
incorrect, such comparisons may give insight into breaking the many
degeneracies present in SGL analyses to date.
It is clear that either
(i) power-law models cannot be applied globally to large surv e ys
of lensing systems as they make unphysical predictions of the
distance ratios, or,
(ii) our measurements of the velocity dispersions of lens galaxies
are not currently at a high enough accuracy and precision to enable
this kind of analysis.
Navarro-Frenk-White profiles are generally taken as the standard
for modelling the dark matter haloes of elliptical galaxies, rather
than the power-law models used here for large SGL catalogues. The
use of a general lens model across all systems in the catalogue,
assuming that on average they will be well-defined, is evidently
not the case. Many lens galaxies have large uncertainties in their
respectiv e v elocity dispersion measurements, leading to unphysical
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3374 C. Harvey-Hawes and D. L. Wiltshire
MNRAS 534, 3364–3376 (2024)
distance ratios for many lensing systems regardless of the choice of
power-law used.
Systems that deviate from these power-law models negatively
skew the distribution of distance ratios, leading to biasing in the
determination of cosmological parameters. In fact, even an individu-
alized EPL model, with the full parametrization of γ, fails to produce
physically plausible distance ratios for all systems. What results is
that the ‘best fit’ cosmological parameters found are those which can
produce the highest possible distance ratios to match the unphysical
predictions of the lens models.
Without additional imaging data, the technique of using distance
ratios as a test statistic is weak at constraining cosmological param-
eters. Ho we ver, it has the advantage of a far larger data set than that
of using time-delay distances. With the next generation of telescopes
( JWST , Euclid , LSST etc) predicted to observe even more lensing
systems, it is of paramount importance that appropriate lens models
are determined in advance of the upcoming catalogues if one aims
to use strong lensing for robust statistical constraints in cosmology.
Time-delay cosmography is able to constrain the lens mass
distribution with far greater precision than the distance sum rule test.
This arises from the extra constraints given by the magnification
and time-delay between each of the images. In addition, time-
delay cosmography models lensing systems individually, whereas
the distance ratio technique fits global lens models to large catalogues
of highly variable systems. At present, despite the smaller number of
systems available, time-delay cosmography may present the only
viable way to use strong gravitational lensing to constrain both
cosmological models and lens matter models with the precision of
measurements available in ongoing surv e ys.
ACKNOWLEDGEMENTS
We warmly thank Jenny Wagner for her perceptive physical insights
in defining this project, and for introducing us to the concepts,
methodology, statistical, and systematic issues that dominate cos-
mological applications of strong gravitational lensing. We also thank
Zachary Lane, Marco Galoppo, Morag Hills, and Michael Williams
for our many discussions and their helpful comments.
DATA AVAILABILITY
The data used for this analysis was taken from the catalogue of
lensing surv e ys compiled in Chen et al. ( 2019 ).
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APPENDIX A: PRIORS
The priors used for the MCMC sampling procedure and Bayesian
analysis are listed in Ta b l e A1 and Table A2 .
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Revisiting lens mass models in cosmology 3375
MNRAS 534, 3364–3376 (2024)
Tab l e A1. Priors assumed for MCMC sampling. In general wide uniform priors are taken, with the
exception of the stellar orbital anisotropy parameter β, which is assumed to have a Gaussian distribution.
MCMC priors
Lens model γ γz γs β δ f
v0
/ M0
EPL (full) (1.0, 3.0) ( −1, 1) (0,1) N ( −0.08, 0.44) (1.8, 2.6) (0, 1)
EPL ( γ= γ0
) (1.8, 2.2) – – N ( −0.08, 0.44) (1.8, 2.6) (0, 1)
SPL (1.8, 2.2) – – – – (0, 1)
SIS – – – – – (0, 1)
Tab l e A2. Priors used for establishing Bayes factors between cosmological models.
Bayes factors priors
Lens model γ δ β f
v0 M0
EPL (1.2, 2.8) (2.003, 2.343) N ( −0.08, 0.44) (0.5, 0.799) (0.143, 0.487)
SPL (1.2, 2.8) – – (0.5, 0.799) (0.143, 0.487)
SIS – – – (0.5, 0.799) (0.143, 0.487)
APPENDIX B: MODEL DERIVATION AND
EXPLANATION OF PARAMETERS
In this section we will discuss the various assumptions that go into
the e xtended power-la w model of Koopmans ( 2006 ) for elliptical
galaxies as gravitational lenses. The extended power-law model, as
well as variants SPL and SIS, are derived from the radial Jeans
equation here given in spherical coordinates ( r, θ, φ):
d
d r
[ ν( r) σ2
r
] +
2 β
r
ν( r) σ2
r = −ν( r)
d
d r
, (B1)
where
d
d r =
GM ( r )
r
2
. (B2)
Here, the velocity dispersion σr is defined as follows with f being
the distribution function of the stars within the galaxy.
σ2
r =
1
ν
v
r
f d
3
v. (B3)
The Jeans equation looks at the average motion of the distribution
of stars defined by the luminous matter density distribution ν( r)
within a Newtonian gravitational potential created by the total matter
density ρ( r), including dark matter. The parameter β( r) denotes the
anisotropy of the stellar velocity dispersion and is also known as
the stellar orbital anisotropy. These are parametrized as such for the
extended power law,
ρ( r) = ρ0
( r/r
0
)
−γ(B4)
ν( r) = ν0
( r/r
0
)
−δ(B5)
β( r) = 1 −σ2
θ/σ 2
r
, (B6)
where σθand σr are the tangential and radial velocity dispersions,
respectively. With β( r) = 0, we can account the average orbits of
stars deviating from circular paths, however the total mass density
will remain spherical –it does not disrupt the spherical dark matter
halo of the galaxy.
In order to solve the Jeans equation, numerous assumptions are
made:
(i) the system is static i.e. ∂
t = 0. This feels like a reasonable
assumption to make as the structure of the elliptical galaxy should
not be expected to change o v er the course of lensing measurements.
(ii) the stress tensor σij
is diagonal, such that v
i
v
j
= 0 if i = j.
(iii) spherical symmetry applies such that σ2
rφ = σ2
rθ = 0 and ρ
and νonly depend on r with no angular component.
(iv) collisionless matter. The Jeans equation is originally derived
from the collisionless Boltzmann equation and as such does not
permit interactions between the constituent stars within galaxies.
After making these assumptions, the radial Jeans equation can be
solved for the radial velocity dispersion in terms of the luminous
matter distribution and the total mass inside a sphere of radius r,
σ2
r
( r ) =
G
∞
r
d r
r
2 β−2
ν( r
) M ( r
)
r
2 βν( r )
. (B7)
We can then define the mass contained within a cylinder of radius
R
E
, the Einstein radius, which quantifies the mass of the lens galaxy,
M
E
=
R
E
0
d R 2 πR
( R
) , (B8)
where ( R) is the mass density projected into the lens plane,
( R) =
∞
−∞
d Z ρ( r) =
∞
−∞
d Z ρ0
r
γ
0
( Z
2
+ R
2
)
−γ/ 2
, (B9)
( R) =
√
πR
1 −γ(( γ−1) / 2)
( γ/ 2)
ρ0
r
γ
0
. (B10)
Here R is the radius of the galaxy in the lens plane and Z is the
distance along the line of sight perpendicular to the lens plane, r is
the spherical radius as previous such that r
2
= R
2
+ Z
2
.
The mass contained inside the Einstein radius is therefore,
M
E
= 2 π3 / 2
R
3 −γ
E
3 −γ
(( γ−1) / 2)
( γ/ 2)
ρ0
r
γ
0
. (B11)
The total mass within a sphere of radius r can also be calculated,
M ( r ) =
r
0
d r
4 πr
2
ρ( r
) = 4 πρ
0
r
γ
0
r
3 −γ
3 −γ, (B12)
which can be written in terms of the mass M
E
contained within the
cylinder as
σ2
r
( r) =
2
√
π
GM
E
R
E
1
ξ−2 β
(( γ−1) / 2)
( γ/ 2) r
R
E
2 −γ
, (B13)
where ξ= γ+ δ−2. The velocity dispersion from observation is
the component of the luminosity weighted average along the line of
sight and o v er the ef fecti ve spectroscopic aperture R
A
. This can be
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3376 C. Harvey-Hawes and D. L. Wiltshire
MNRAS 534, 3364–3376 (2024)
written as:
σ2
( ≤R
A
) =
R
A
0
∞
−∞
d R d Z 2 πR σ2
r
( r )
1 −βR
2
r
2
ν( r )
R
A
0
∞
−∞
d R d Z 2 πRν( r)
. (B14)
By substituting in the expression for σ2
r
and the power law of ν( r)
we arrive at
σ2
( ≤R
A
) =
2
√
π
GM
E
R
E
3 −δ
( ξ−2 β) (3 −ξ)
⎡
⎣
ξ−1
2
ξ
2
−β
ξ+ 1
2
ξ+ 2
2 ⎤
⎦
×
γ
2
δ
2
γ−1
2
δ−1
2 R
A
R
E
2 −γ
. (B15)
Using the deflecting mass found from the lensing equation
M
E
=
c
2
θE
4 G
D
s
D
l
D
ls
, (B16)
R
A
= D
l
θA
and R
E
= θE
D
l
we get,
σ2
( ≤R
A
) =
2
√
π
D
s
θE
c
2
D
ls
3 −δ
( ξ−2 β)(3 −ξ)
⎡
⎣
ξ−1
2
ξ
2
−β
ξ+ 1
2
ξ+ 2
2 ⎤
⎦
×
γ
2
δ
2
γ−1
2
δ−1
2 θA
θE
2 −γ
, (B17)
which is the expression we call the extended power law. The use of
the lens equation also is built upon several assumptions:
(i) the weak field limit of general relativity applies and that
/c
2
1,
(ii) small deflection angles,
(iii) the Born approximation applies.
This paper has been typeset from a T
E
X/L
A
T
E
X file prepared by the author.
© 2024 The Author(s).
Published by Oxford University Press on behalf of Royal Astronomical Society. This is an Open Access article distributed under the terms of the Creative Commons Attribution License
( https://cr eativecommons.or g/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original wo rk is properly cited.
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