[1]

# A Reactive Molecular Dynamics Study on the Mechanical Properties of a
Recently Synthesized Amorphous Carbon Monolayer Converted into a
Nanotube/Nanoscroll

Marcelo L. Pereira Júnior  Wiliam F. Cunha  Douglas S. Galvão  Luiz A. Ribeiro
Júnior<EMAIL_ADDRESS>Institute of Physics, University of Brasília,
Brasília, 70910-900, Brazil.  Applied Physics Department, University of
Campinas, Campinas, São Paulo, Brazil.  Center for Computing in Engineering
and Sciences, University of Campinas, Campinas, São Paulo, Brazil.

###### Abstract

Recently, laser-assisted chemical vapor deposition was used to synthesize a
free-standing, continuous, and stable monolayer amorphous carbon (MAC). MAC is
a pure carbon structure composed of randomly distributed five, six, seven, and
eight atom rings, which differs from disordered graphene. More recently,
amorphous MAC-based nanotubes (a-CNT) and nanoscrolls (A-CNS) were proposed.
In this work, we have investigated (through fully atomistic reactive molecular
dynamics simulations) the mechanical properties and melting points of pristine
and a-CNT and a-CNS. Results showed that a-CNT and a-CNS have distinct elastic
properties and fracture patterns concerning their pristine analogs. Both a-CNT
and a-CNS presented a non-elastic regime before their total rupture, whereas
the CNT and CNS undergo a direct conversion to fractured forms after a
critical strain threshold. The critical strain for the fracture of the a-CNT
and a-CNS are about 30% and 25%, respectively, and they are lower than the
corresponding CNT and CNS cases. Although less resilient to tension, the
amorphous tubular structures have similar thermal stability in relation to the
pristine cases with melting points of 5500K, 6300K, 5100K, and 5900K for
a-CNT, CNT, a-CNS, and CNS, respectively. An interesting result is whereas the
behavior of the pristine systems is substantially different depending on the
system being a nanotube or a nanoscroll, thus indicating that the topology
plays an important role, the same is not true for the amorphous version of the
nanostructures, thus indicating that the structural disorder overcomes the
topological features.

###### keywords:

Amorphous Carbon Nanotubes Amorphous Carbon Nanoscrolls Elastic Properties
Thermal Stability

##  1 Introduction

Carbon-based nanomaterials have been used in numerous energy storage and
conversion applications as good cost-benefit alternatives to conventional
(silicon-based) technology [1, 2]. Many of their promises in revolutionizing
these fields still need further refining for commercial-grade applications.
Among such applications are worthy mention organic photovoltaics [3], light-
emitting diodes [4], and thin-film transistors [5]. The broad range of carbon-
based nanomaterials employed in developing these applications includes
graphene and its allotropes [6], carbon nanotubes [7], nanoscrolls [8],
nanofibers [9], and amorphous carbon materials [10]. Particularly, carbon
nanotubes (CNTs) are still subject of interest due to their remarkable
electronic and structural properties, which have impacted several areas of
nanotechnology [11]. CNTs present distinct electronic properties, such as
metallic and semiconducting behavior, depending on their symmetry [12]. CNSs,
in turn, are formed by the jelly or papyrus roll-like wrapping of a graphite
sheet to form a tubular structure [13, 14, 15]. Recent synthesis efforts
provided simple low-temperature routes to CNSs [16, 8].

Amorphous carbons are families of carbon-based materials without long-range
crystalline order [10]. Several experimental efforts have been devoted to
understanding the electronic and structural properties of amorphous CNTs
(a-CNTs) for their applications in nanoelectronics [17, 18, 19, 20, 21, 22,
23]. The synthesis and characterization of these materials were studied
through simple fabrication procedures used for preparing highly ordered
nanotube arrays [21]. Electrical transport measurements on individual a-CNTs
showed that their resistivity is of the same order as multiwalled CNTs despite
their disordered microstructure [22]. a-CNTs were also employed as potential
candidates for water purification [23]. It was reported that they can be used
for adsorption and/or removal of hydroxyl group substituted aromatic compounds
like Resorcinol as well as heavy metal ion like arsenic from water.

From the theoretical point o view, the electronic properties of a-CNTs were
studied by using a tight-binding model Hamiltonian [24]. Through this
approach, it was obtained that strong electron-lattice coupling of states near
the Fermi energy leads to the formation of a bandgap [24]. This model has also
predicted that the bandgap in a-CNTs is stronger diameter dependent (inversely
proportional) than for CNTs. In relation to CNS, some theoretical and
experimental studies were performed to address possible synthetic routes [16,
8, 14, 25, 26, 27, 28, 28, 29]. There are only are very few studies in the
literature on amorphous CNS (a-CNSs) [30, 31]. To the best of our knowledge,
up to now, there are no studies on the thermomechanical properties of a-CNTs
and a-CNSs.

Figure 1: Structural models of the (a) MAC sheet and its related (b) a-CNT and
(c) a-CNS atomistic structures.

Very recently, laser-assisted chemical vapor deposition was used to synthesize
a free-standing, continuous, and stable monolayer amorphous carbon (MAC) [32].
MAC is a pure carbon structure composed of randomly distributed five, six,
seven, and eight atom rings (see Figure 1(a)). Its lattice arrangement differs
from disordered graphene. The findings have revealed that MAC has a high yield
strength value with no crack propagation. Since the synthesis of MAC was
achieved, its conversion to nanotubes and nanoscrolls emerges as a
possibility.

Motivated by the recent MAC synthesis [32] and by the proposition about the
existence of tubes and scrolls formed from MAC membranes [33] here the
mechanical and thermal stability of MAC-based a-CNT and a-CNS were
investigated in the framework of fully-atomistic reactive molecular dynamics
simulations. The elastic properties of these model tubular materials were
studied using the stress-strain relationship, and their thermal stability was
analyzed by employing a heating ramp protocol. The results obtained here for
these amorphous structures were contrasted against their related pristine
ones.

##  2 Details of Modeling

We have performed fully-atomistic molecular dynamics (MD) simulations using
the adaptive-interatomic reactive bond-order (AIREBO) [34] potential as
implemented in LAMMPS [35]. The main focus of the present work lies in
providing a comparative study of the mechanical properties of pristine and
MAC-based amorphous CNTs and CNSs. All these structures were generated by the
Sculptor software suite [36]. The a-CNT was obtained by binding the two
lateral edges of the MAC sheet [32, 37], whereas the a-CNS was formed by a
spiral-like roll-up of the MAC sheet with open edges at both ends, with a
rolling angle of 4$\pi$. Figure 1 shows a representation of the MAC sheet
(Figure 1(a)), its derived nanotube (Figure 1(b)), and nanoscroll (Figure
1(c)). The MAC atomistic structure used here to build our model a-CNT contains
610 atoms [32, 37]. The yielded a-CNT has 25 Å of diameter, 120 Å of length,
and it was formed by merging six MAC sheets (2 in the x-direction and 3 in the
y-direction), totaling 3660 atoms. Here, a MAC sheet was considered as a unit
cell. The a-CNS has also 120 Å of length, but it possess 10 Å and 16 Å of
internal and external diameters, respectively. The quantity of MAC sheets and
the total number of atoms used to build the a-CNS are equal to the a-CNT case.

Figure 2: Stress-strain curves for (a) CNT (blue) and a-CNT (red) and (b) for
CNS (blue) and a-CNS (red) at 300K, considering an uniaxial strain applied to
the longitudinal direction. The structures were stretched at a constant
elongation rate until total rupture.

Our computational approach is based on submitting the systems to tensile
deformation to address their elastic properties as well as to a heating ramp
process to obtain their estimated melting points. These analyses were
performed by considering the virial stress tensor [38]. We also calculated the
von Mises stress [39] as it is an important tool to describe the stretching
dynamics and fracture patterns of the systems [40].

As for the heating ramp procedure it consisted on a linear temperature
increase in a NVT ensemble in which, a temperature range of from 0 up to
10000K was performed after a preliminary thermalization procedure within an
NPT ensemble. The temperature ramp rate was set to approximately 200 K/ms. The
MD snapshots and trajectories were obtained by using free visualization and
analysis software VMD [36]. The equations of motion were numerically
integrated using the velocity-Verlet integrator with a time-step of $0.1$ fs.
We increased the tensile stress in the system by applying a uniaxial strain
along the longitudinal direction of the tubular structures, for an engineering
strain rate of $1.0\times 10^{-7}$ fs-1.

##  3 Results

Initially, we investigated the stress-strain behavior of both systems: the
a-CNT (Figure 2(a)) and a-CNS (Figure 2(b)). In both figures, we present
results of the structures derived from the pristine nanosheet in blue and of
the amorphous one in red. In the two scenarios, the curves more or less
overlap until a critical strain value is achieved. Such a critical strain
value indicates a phase transition in the stress-strain dependence, which
consists of the point of fracture of each structure. The fracture strain, for
each case, is as follows: 27%, 48%, 24%, and 34% for a-CNT, CNT, a-CNS, and
CNS, respectively.

System | E [GPa] | D [GPa] | TS [GPa] |  $\epsilon_{F}$ [%]
---|---|---|---|---
MAC Nanotube | 2268.67 | 3001.13 | 384.97 | 26.72
Graphene Nanotube | 2495.51 | 3927.15 | 944.36 | 37.96
MAC Nanoscroll | 2476.61 | 3843.84 | 359.59 | 23.56
Graphene Nanoscroll | 2493.82 | 4460.73 | 630.82 | 33.76
Table 1: Elastic properties for MAC and graphene nanotubes and nanoscrolls. E,
D, TS, and $\epsilon_{F}$, refer to Young’s Module, third-order elastic
modulus, TS tensile strength, and critical strain, respectively.

As expected, the general trend is that the pristine (defectless) structures
can stand larger strain values without fracturing. Figure 2(a) shows that CNT
is able to stand roughly 20% more strain than its amorphous counterpart.
Figure 2(b) compares the amorphous and pristine version of the nanoscroll.
This figure shows that the pristine nanoscroll upholds 15% more strain than
its amorphous version. This trend is consistent with the fact that highly
ordered carbon systems, such as graphene, will be tougher than its defective
structures [41]. Such strength arises from the strong overlap of sp2 hybrid
orbitals. When the sites are uniformly close to each other, as is the case of
the pristine scenario, the collective effect is that all the regions of the
nanostructure exhibit the same strength. In the amorphous case, as two carbon
cores approach one another, the relative strength between these two becomes
stronger. However, as a consequence, the distance between a third core becomes
larger and, therefore, the interaction is weaker. As the structure as a whole
is only as strong as its weakest bond, it is expected that the amorphous tends
to be weaker than the pristine systems.

Another crucial feature of this set of simulations is observed through the
comparison between Figures 2(a) and 2(b). We can observe, whereas the behavior
of the pristine system is substantially different depending on the system
being a nanotube or a nanoscroll, thus indicating that the topology plays an
important role, the same is not true for the amorphous version of the
nanostructures, thus indicating that the structural disorder overcomes the
topological features (see also Table 1).

Figure 3: Representative MD snapshots for the fracture process for (top) a-CNT
and (bottom) pristine CNT.

Table 1 summarizes the set of elastic properties obtained from MD the
simulations for the systems illustrated in Figure 1. It is worthwhile to
stress that previous works have described the low-strain region as being
quadratic due to the strong nonlinear behavior in the elastic regime [42, 43,
44]. In this way, the stress-strain relationship is given by
$\sigma=E\epsilon+D\epsilon^{2}$, where $\sigma$, $\epsilon$, $E$, and $D$ are
the symmetric second Piola-Kirchhoff stress, uniaxial strain, Young’s modulus,
and third-order elastic modulus, respectively [45]. In the table, E stands for
Young’s Module, D is the third-order elastic modulus, TS the tensile strength
and $\epsilon_{F}$ is the critical strain for the fracture. These values are
numerical representations of the response of each material to the processes
reported in Figure 2. As expected, the pristine structures presented higher
Young’s Modulus when contrasted with their amorphous analogs. However, these
values are reasonably close. The more cohesive nature of CNTs confers to this
material higher tensile strength and critical strain for the fracture. Due to
the very nature of the system structures, the applied tensions are uniaxial
and, therefore, Young’s Modulus is a measure of the system stiffness.

In Figures 3 and 4 we present representative MD snapshots of the fracturing
processes of the different systems considered here and the von Mises stress
distribution along the nanostructure. In these figures, the top and bottom
sequence of panels illustrate the representative MD snapshots for the pristine
and amorphous cases, respectively. Note that the a-CNT and a-CNS have a
smaller fracture point in terms of strain in an approximate 1.4 ratio, which
is consistent with the results presented in Figure 2. The trend of the
amorphous structure to present smaller cutoff stress is preserved and rather
similar among these two types of amorphous structures. Another clear trend to
be observed is that before rupture, in both types of amorphous nanostructures,
is the appearance of linear carbon chains (LACs) The same is not observed for
pristine nanotubes and nanoscrolls. Due to the absence of porosity CNTs and
CNS go from integrity to their fractured form abruptly. From these facts, one
can again conclude that the structural disorder is of greater importance than
the topological aspects.

Figure 4: Representative MD snapshots for the fracture process for (top) a-CNS
and (bottom) pristine CNS. Figure 5: Total energy as a function of temperature
for (a) nanotubes and (b) nanoscrolls. Amorphous structures are represented by
red triangles and pristine ones by blue circles. Figure 6: Representative MD
snapshots of the heating simulations for (left) a-CNT and (right) pristine
CNT. Figure 7: Representative MD snapshots of the heating simulations for
(left) a-CNS and (right) pristine CNS.

In Figure 5 we present the interplay between the total energy of the system
and the thermal bath to it. The thermalization process for nanotubes (Figure
5(a)) and nanoscrolls (Figure 5(b)) is rather similar. For low temperatures,
the curves are almost parallel, and the energy absorption of the amorphous
cases are slightly smaller than the pristine ones. A feature that can be
readily observed is that, as temperature increases, the relationship between
these two quantities behavior for amorphous (red triangles) and pristine (blue
circles) nanostructures is less important. This trend should, indeed, be
expected, because higher temperature introduces disorder effects to the
pristine chain, thus increasing the random degree of the atom movements, as is
the case of the amorphous structure. A comparison between the two figures
shows that the nanotubes absorb the thermal energy more efficiently than the
nanoscrolls. This difference can be explained by the fact the scrolls have an
open structure that allows converting the thermal energy into kinetic one.

Figure 5 also allows us to have insights about the melting point of the
nanostructures. As a general trend, the total energy increases linearly with
the temperature and shows two well-defined regimes with different slopes. The
discontinuity in these curves is suggestive of a phase transition from a solid
to a gas-like phase. The transition between these two behaviors defines the
melting point. This abrupt change in the slope of the curves is related to a
gain in kinetic energy due to the higher atom velocities in the gas-like
phase, which increases the total energy. Moreover, part of the harmonic and
torsional energies in the solid phase was also converted into kinetic energy,
which again contributes to the increase in the total energy of the system in
the gas-like phase. The amorphous tubular structures have similar thermal
stability when contrasted to the pristine cases with melting points of 5500K,
6300K, 5100K, and 5900K for a-CNT, CNT, a-CNS, and CNS, respectively.

In Figures 6 and 7 we show representative MD snapshots for the heating
simulations, for the nanotubes and nanoscrolls, respectively. In these
figures, the left and the right sequence of panels depict the MD snapshots for
the amorphous and pristine cases, respectively. In Figure 6, we can see that
the a-CNT starts to lose its integrity, whereas the pristine is still stable
at 6000K. At 7500K, both structures have reached their melting point, and a
partially gas-like phase is observed. In the case of nanoscrolls (Figure 7),
the pristine structure loses its structural integrity at 4000K, whereas the
amorphous one at 5100K. As in the case of CNTs, at 7500K just a partially gas-
like phase is observed, although more pronounced. From this set of
simulations, it can be concluded that nanotubes are more thermally stable than
nanoscrolls. As a general trend, one can also conclude that pristine
structures are more stable than their amorphous counterparts, for the same
reasons as the mechanical behavior discussed above.

##  4 Conclusions

In summary, we have investigated (through fully atomistic reactive molecular
dynamics simulations) the mechanical properties and melting point of pristine
and amorphous carbon nanotubes and carbon nanoscrolls. The amorphous
structures are based on the recently synthesized [32] stable monolayer
amorphous carbon (MAC). Our results show that amorphous carbon nanotubes
present lower tensile-strength than their corresponding pristine ones, which
is expected since in general defectless structures are tougher than defective
ones. The pristine structures exhibit abrupt fracture dynamics while the
amorphous ones present intermediate states before complete failure. However,
all structures present the appearance of linear atomic carbon chains before
their rupture. The stress-strain curves for pristine/amorphous structures more
or less overlap until the critical strain value is reached. The fracture
strain, for each case, is as follows: 27%, 48%, 24%, and 34% for a-CNT, CNT,
a-CNS, and CNS, respectively. An interesting result is whereas the behavior of
the pristine systems is substantially different depending on the system being
a nanotube or a nanoscroll, thus indicating that the topology plays an
important role, the same is not true for the amorphous version of the
nanostructures, thus indicating that the structural disorder outperform the
topological features. The nanotubes of both symmetries present higher melting
points than nanoscrolls, suggesting that the tubes are thermally more stable
than the scrolls. In the case of the amorphous nanotube, a thermal critical
value of 6000 K was obtained, at which point the pristine nanoscroll, for
instance, had already lost its structural integrity.

## References

  * [1] Bita Bayatsarmadi, Yao Zheng, Anthony Vasileff, and Shi-Zhang Qiao.  Recent advances in atomic metal doping of carbon-based nanomaterials for energy conversion.  Small, 13(21):1700191, 2017.
  * [2] Meagan S Mauter and Menachem Elimelech.  Environmental applications of carbon-based nanomaterials.  Environmental Science & Technology, 42(16):5843–5859, 2008.
  * [3] Bernard Kippelen and Jean-Luc Brédas.  Organic photovoltaics.  Energy & Environmental Science, 2(3):251–261, 2009.
  * [4] Bernard Geffroy, Philippe Le Roy, and Christophe Prat.  Organic light-emitting diode (oled) technology: materials, devices and display technologies.  Polymer international, 55(6):572–582, 2006.
  * [5] Hagen Klauk.  Organic thin-film transistors.  Chemical Society Reviews, 39(7):2643–2666, 2010.
  * [6] Andrey N Enyashin and Alexander L Ivanovskii.  Graphene allotropes.  physica status solidi (b), 248(8):1879–1883, 2011.
  * [7] Sumio Iijima.  Helical microtubules of graphitic carbon.  nature, 354(6348):56–58, 1991.
  * [8] Lisa M Viculis, Julia J Mack, and Richard B Kaner.  A chemical route to carbon nanoscrolls.(brevia).  Science, 299(5611):1361–1362, 2003.
  * [9] E Hammel, X Tang, M Trampert, T Schmitt, K Mauthner, A Eder, and P Pötschke.  Carbon nanofibers for composite applications.  Carbon, 42(5-6):1153–1158, 2004.
  * [10] John Robertson.  Amorphous carbon.  Advances in Physics, 35(4):317–374, 1986.
  * [11] Robert C. Haddon.  Carbon nanotubes.  Accounts of Chemical Research, 35(12):997–997, 2002.  PMID: 12484786.
  * [12] Valentin N Popov.  Carbon nanotubes: properties and application.  Materials Science and Engineering: R: Reports, 43(3):61–102, 2004\.
  * [13] Roger Bacon.  Growth, structure, and properties of graphite whiskers.  Journal of Applied Physics, 31(2):283–290, 1960.
  * [14] Scheila F Braga, Vitor R Coluci, Sergio B Legoas, Ronaldo Giro, Douglas S Galvão, and Ray H Baughman.  Structure and dynamics of carbon nanoscrolls.  Nano letters, 4(5):881–884, 2004.
  * [15] David Tománek.  Mesoscopic origami with graphite: scrolls, nanotubes, peapods.  Physica B: Condensed Matter, 323(1-4):86–89, 2002.
  * [16] H Shioyama and T Akita.  A new route to carbon nanotubes.  Carbon (New York, NY), 41(1):179–181, 2003.
  * [17] Lijie Ci, Bingqing Wei, Cailu Xu, Ji Liang, Dehai Wu, Sishen Xie, Weiya Zhou, Yubao Li, Zuqin Liu, and Dongsheng Tang.  Crystallization behavior of the amorphous carbon nanotubes prepared by the cvd method.  Journal of Crystal Growth, 233(4):823–828, 2001.
  * [18] Jose Maria Calderon Moreno and Masahiro Yoshimura.  Hydrothermal processing of high-quality multiwall nanotubes from amorphous carbon.  Journal of the American Chemical Society, 123(4):741–742, 2001\.
  * [19] Lijie Ci, Hongwei Zhu, Bingqing Wei, Cailu Xu, and Dehai Wu.  Annealing amorphous carbon nanotubes for their application in hydrogen storage.  Applied Surface Science, 205(1-4):39–43, 2003.
  * [20] Yuming Chen, Zhouguang Lu, Limin Zhou, Yiu-Wing Mai, and Haitao Huang.  Triple-coaxial electrospun amorphous carbon nanotubes with hollow graphitic carbon nanospheres for high-performance li ion batteries.  Energy & environmental science, 5(7):7898–7902, 2012.
  * [21] Yong Lei, Kuan-Song Yeong, JTL Thong, and Wai-Kin Chim.  Large-scale ordered carbon nanotube arrays initiated from highly ordered catalyst arrays on silicon substrates.  Chemistry of materials, 16(14):2757–2761, 2004.
  * [22] Takuya Hayashi, Thomas C O’Connor, Katsuhisa Higashiyama, Kohei Nishi, Kazunori Fujisawa, Hiroyuki Muramatsu, Yoong Ahm Kim, Bobby G Sumpter, Vincent Meunier, Mauricio Terrones, et al.  A reversible strain-induced electrical conductivity in cup-stacked carbon nanotubes.  Nanoscale, 5(21):10212–10218, 2013.
  * [23] P Bhowmick, D Banerjee, S Santra, D Sen, B Das, and KK Chattopadhyay.  Amorphous carbon nanotubes as potent sorbents for removal of a phenolic derivative compound and arsenic: theoretical support of experimental findings.  RSC advances, 6(11):8913–8922, 2016.
  * [24] A Rakitin, C Papadopoulos, and JM Xu.  Electronic properties of amorphous carbon nanotubes.  Physical Review B, 61(8):5793, 2000.
  * [25] Eric Perim, Leonardo Dantas Machado, and Douglas Soares Galvao.  A brief review on syntheses, structures, and applications of nanoscrolls.  Frontiers in Materials, 1:31, 2014.
  * [26] Xu Xie, Long Ju, Xiaofeng Feng, Yinghui Sun, Ruifeng Zhou, Kai Liu, Shoushan Fan, Qunqing Li, and Kaili Jiang.  Controlled fabrication of high-quality carbon nanoscrolls from monolayer graphene.  Nano letters, 9(7):2565–2570, 2009.
  * [27] Eric Perim, Ricardo Paupitz, and Douglas S Galvao.  Controlled route to the fabrication of carbon and boron nitride nanoscrolls: A molecular dynamics investigation.  Journal of Applied Physics, 113(5):054306, 2013.
  * [28] VP Dravid, X Lin, Y Wang, XK Wang, A Yee, JB Ketterson, and Robert PH Chang.  Buckytubes and derivatives: their growth and implications for buckyball formation.  Science, 259(5101):1601–1604, 1993.
  * [29] H Shioyama and T Akita.  A new route to carbon nanotubes.  Carbon (New York, NY), 41(1):179–181, 2003.
  * [30] Zhenbin Gong, Jing Shi, Bin Zhang, and Junyan Zhang.  Graphene nano scrolls responding to superlow friction of amorphous carbon.  Carbon, 116:310–317, 2017.
  * [31] Prasad Yadav, Sambhaji Warule, Jyoti Jog, and Satishchandra Ogale.  Carbon nanoscrolls by pyrolysis of a polymer.  Solid state communications, 152(23):2092–2095, 2012.
  * [32] Chee-Tat Toh, Hongji Zhang, Junhao Lin, Alexander S Mayorov, Yun-Peng Wang, Carlo M Orofeo, Darim Badur Ferry, Henrik Andersen, Nurbek Kakenov, Zenglong Guo, et al.  Synthesis and properties of free-standing monolayer amorphous carbon.  Nature, 577(7789):199–203, 2020.
  * [33] Raphael M Tromer, Levi C Felix, Luiz A Ribeiro, and Douglas S Galvao.  Optoelectronic properties of amorphous carbon-based nanotube and nanoscroll.  arXiv preprint arXiv:2007.05062, 2020.
  * [34] Steven J Stuart, Alan B Tutein, and Judith A Harrison.  A reactive potential for hydrocarbons with intermolecular interactions.  The Journal of chemical physics, 112(14):6472–6486, 2000.
  * [35] Steve Plimpton.  Fast parallel algorithms for short-range molecular dynamics.  Journal of computational physics, 117(1):1–19, 1995.
  * [36] William Humphrey, Andrew Dalke, Klaus Schulten, et al.  Vmd: visual molecular dynamics.  Journal of molecular graphics, 14(1):33–38, 1996.
  * [37] Levi C Felix, Raphael M Tromer, Pedro AS Autreto, Luiz A Ribeiro Junior, and Douglas S Galvao.  On the mechanical properties and thermal stability of a recently synthesized monolayer amorphous carbon.  The Journal of Physical Chemistry C, 124(27):14855–14860, 2020\.
  * [38] Arun K Subramaniyan and CT Sun.  Continuum interpretation of virial stress in molecular simulations.  International Journal of Solids and Structures, 45(14-15):4340–4346, 2008.
  * [39] R v Mises.  Mechanik der festen körper im plastisch-deformablen zustand.  Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 1913:582–592, 1913.
  * [40] Marcelo Lopes Pereira Júnior and Luiz Antônio Ribeiro Júnior.  Thermomechanical insight into the stability of nanoporous graphene membranes.  FlatChem, 24:100196, 2020.
  * [41] Ashivni Shekhawat and Robert O Ritchie.  Toughness and strength of nanocrystalline graphene.  Nature communications, 7(1):1–8, 2016.
  * [42] Changgu Lee, Xiaoding Wei, Jeffrey W Kysar, and James Hone.  Measurement of the elastic properties and intrinsic strength of monolayer graphene.  science, 321(5887):385–388, 2008.
  * [43] Ardavan Zandiatashbar, Gwan-Hyoung Lee, Sung Joo An, Sunwoo Lee, Nithin Mathew, Mauricio Terrones, Takuya Hayashi, Catalin R Picu, James Hone, and Nikhil Koratkar.  Effect of defects on the intrinsic strength and stiffness of graphene.  Nature communications, 5(1):1–9, 2014.
  * [44] Teng Zhong, Jiabin Li, and Kaiwang Zhang.  A molecular dynamics study of young’s modulus of multilayer graphene.  Journal of Applied Physics, 125(17):175110, 2019.
  * [45] RN Thurston and K Brugger.  Third-order elastic constants and the velocity of small amplitude elastic waves in homogeneously stressed media.  Physical Review, 133(6A):A1604, 1964.


††thanks<EMAIL_ADDRESS>
# Extragalactic Test of General Relativity from Strong Gravitational Lensing
by using Artificial Neural Networks

Jing-Yu Ran  Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing
210023, China  School of Astronomy and Space Sciences, University of Science
and Technology of China, Hefei 230026, China  Jun-Jie Wei  Purple Mountain
Observatory, Chinese Academy of Sciences, Nanjing 210023, China  School of
Astronomy and Space Sciences, University of Science and Technology of China,
Hefei 230026, China

###### Abstract

This study aims to test the validity of general relativity (GR) on kiloparsec
scales by employing a newly compiled galaxy-scale strong gravitational lensing
(SGL) sample. We utilize the distance sum rule within the Friedmann-Lemaître-
Robertson-Walker metric to obtain cosmology-independent constraints on both
the parameterized post-Newtonian parameter $\gamma_{\rm PPN}$ and the spatial
curvature $\Omega_{k}$, which overcomes the circularity problem induced by the
presumption of a cosmological model grounded in GR. To calibrate the distances
in the SGL systems, we introduce a novel nonparametric approach, Artificial
Neural Network (ANN), to reconstruct a smooth distance–redshift relation from
the Pantheon+ sample of type Ia supernovae. Our results show that $\gamma_{\rm
PPN}=1.16_{-0.12}^{+0.15}$ and $\Omega_{k}=0.89_{-1.00}^{+1.97}$, indicating a
spatially flat universe with the conservation of GR (i.e., $\Omega_{k}=0$ and
$\gamma_{\rm PPN}=1$) is basically supported within $1\sigma$ confidence
level. Assuming a zero spatial curvature, we find $\gamma_{\rm
PPN}=1.09_{-0.10}^{+0.11}$, representing an agreement with the prediction of 1
from GR to a 9.6% precision. If we instead assume GR holds (i.e., $\gamma_{\rm
PPN}=1$), the curvature parameter constraint can be further improved to be
$\Omega_{k}=0.11_{-0.47}^{+0.78}$. These resulting constraints demonstrate the
effectiveness of our method in testing GR on galactic scales by combining
observations of strong lensing and the distance–redshift relation
reconstructed by ANN.

##  I Introduction

As an important cornerstone of modern physics, Einstein’s theory of general
relativity (GR) has withstood very strict tests (e.g., (Dyson et al., 1920;
Pound and Rebka, 1960; Shapiro, 1964; Taylor et al., 1979)). But testing GR at
a much higher precision is still a vital task, because any possible violation
of GR would have profound effects on our understanding of fundamental physics.
Within the parameterized post-Newtonian (PPN) formalism, GR predicts that the
PPN parameter $\gamma_{\rm PPN}$ which describes the amount of space curvature
produced per unit rest mass should be exactly 1 Thorne and Will (1971).
Measuring $\gamma_{\rm PPN}$ therefore serves as a test of the validity of GR
on large scales. That is, any deviation from $\gamma_{\rm PPN}=1$ implies a
possible violation of GR.

On solar system scales, the GR prediction for $\gamma_{\rm PPN}$ has been
confirmed with high accuracy. By measuring the round-trip travel time of radar
signals passing near the Sun, the Cassini spacecraft yielded $\gamma_{\rm
PPN}=1+(2.1\pm 2.3)\times 10^{-5}$ (Bertotti et al., 2003). However, the
extragalactic tests of GR are still insufficiency and much less precise. On
galactic scales, strong gravitational lensing (SGL), combined with stellar
kinematics in the lensing galaxy, provides an effective way to test the
validity of GR by constraining the PPN parameter $\gamma_{\rm PPN}$. The
pioneering work by Ref. Bolton et al. (2006) first utilized this approach and
reported a result of $\gamma_{\rm PPN}=0.98\pm 0.07$ based on observations of
15 elliptical lensing galaxies from the Sloan Lens ACS Survey. Since then,
numerous studies have been conducted to test GR using different SGL samples
(Smith, 2009; Schwab et al., 2010; Cao et al., 2017; Collett et al., 2018;
Yang et al., 2020; Liu et al., 2022; Wei et al., 2022). In this paper, we
further explore the validity of GR by employing a newly compiled SGL sample
(Chen et al., 2019), which consists of 161 galaxy-scale strong lensing
systems. This larger SGL sample allows us to perform a more comprehensive
analysis and obtain further insights into the behavior of gravity on galactic
scales.

In practice, in order to constrain the PPN parameter $\gamma_{\rm PPN}$ using
SGL systems, one has to know a ratio of three angular diameter distances
(i.e., the distances from the observer to the lens, $D_{l}$, the observer to
the source, $D_{s}$, and the lens to the source, $D_{ls}$). In most previous
works, the required distance ratio is calculated within the context of the
standard $\Lambda$CDM cosmological model. However, $\Lambda$CDM itself is
established on the framework of GR, which leads to a circularity problem in
testing GR (Liu et al., 2022; Wei et al., 2022). To circumvent this problem,
we will introduce the distance sum rule (DSR) in the Friedmann-Lemaître-
Robertson-Walker (FLRW) metric. The two distances $D_{l}$ and $D_{s}$ can be
directly determined from observations of type Ia supernovae (SNe Ia), but not
the distance $D_{ls}$. The DSR enables us to convert $D_{ls}$ into a
relationship with $D_{l}$, $D_{s}$, and the spatial curvature $\Omega_{k}$.
Based on the DSR in the FLRW metric, cosmology-independent constraints on both
$\gamma_{\rm PPN}$ and $\Omega_{k}$ can thus be obtained by combing
observations of strong lensing and SNe Ia (Cao et al., 2017; Wei et al.,
2022).

Very recently, by employing the Gaussian Process (GP) method, Liu et al.
(2022) reconstructed a smooth distance–redshift relation directly from SN Ia
observation to calibrate the distances in the SGL sample. GP allows for the
reconstruction of a function from a dataset without assuming a specific model
or parameterization, and it has been widely used in cosmological researches
(Seikel et al., 2012; Yahya et al., 2014; Montiel et al., 2014; Wei and Wu,
2017; Melia, 2018; Liao et al., 2019). In the GP analysis, the errors in the
observational data are assumed to follow a Gaussian distribution (Seikel et
al., 2012). However, the actual observations might not follow Gaussian
distributions. This may thus be a strong assumption for reconstructing a
function from observational data. Moreover, due to the sparsity and scatter of
data points at high redshifts, the GP reconstructed function from SN Ia data
exhibits strange oscillations with large uncertainties. To address these
concerns and ensure the reliability of the reconstructed function, we employ
the Artificial Neural Network (ANN) method, which is a machine learning
technique and has been proven to be a “universal approximator” that can
reconstruct a great variety of functions (Cybenko, 1989; Hornik, 1991). Thanks
to the powerful property of neural networks, methods based on ANNs have been
widely used in regression and estimation tasks. In this work, we will
reconstruct the distance–redshift relation from SN Ia data using the ANN
method, utilizing a code developed in Ref (Wang et al., 2020).

This paper is organized as follows: in Section II, we introduce the
methodology and observations used for testing GR on galactic scales.
Cosmology-independent constraints on $\gamma_{\rm PPN}$ and $\Omega_{k}$ are
shown in Section III. In Section IV, we make a summary and end with some
discussions.

##  II Methodology and Data

In the weak-field limit, the metric of space-time can be characterized as

$\mathrm{d}s^{2}=c^{2}\mathrm{d}t^{2}\left(1-\frac{2GM}{c^{2}r}\right)-\mathrm{d}r^{2}\left(1+\frac{2\gamma_{\rm
PPN}GM}{c^{2}r}\right)-r^{2}\mathrm{d}\Omega^{2}\;,$ (1)

where $\gamma_{\rm PPN}$ is the PPN parameter, $M$ is the mass of the central
object, and $\Omega$ is the angle in the invariant orbital plane. In the
framework of GR, $\gamma_{\rm PPN}$ is equal to unity.

###  II.1 Gravitational Lensing Theory

The main idea of testing the validity of GR via SGL systems is that the mass
enclosed within the Einstein radius derived separately from the gravitational
theory and the dynamical theory should be equivalent, i.e., $M_{\rm E}^{\rm
grl}=M_{\rm E}^{\rm dyn}$. From the theory of gravitational lensing (Schneider
et al., 1992), the Einstein angle $\theta_{\rm E}$ reflecting the angular
separations between multiple images is related to the gravitational mass
$M_{\rm E}^{\rm grl}$,

$\theta_{\rm E}=\sqrt{\frac{1+\gamma_{\rm PPN}}{2}}\left(\frac{4GM_{\rm
E}^{\rm grl}}{c^{2}}\frac{D_{ls}}{D_{l}D_{s}}\right)^{1/2}\;,$ (2)

where $D_{l}$, $D_{s}$, and $D_{ls}$ are, respectively, the angular diameter
distances from the observer to the lens, the observer to the source, and the
lens to the source. By introducing the Einstein radius $R_{\rm
E}=D_{l}\theta_{\rm E}$, Equation (2) can be rearranged as

$\frac{GM_{\rm E}^{\rm grl}}{R_{\rm E}}=\frac{2}{1+\gamma_{\rm
PPN}}\frac{c^{2}}{4}\frac{D_{s}}{D_{ls}}\theta_{\rm E}\;.$ (3)

To estimate the dynamical mass $M_{\rm E}^{\rm dyn}$ from the spectroscopic
measurement of the lens velocity dispersion, one must first set a mass
distribution model for the lensing galaxy. Here we use the common mass model
with power-law density profiles (Koopmans et al., 2006; Chen et al., 2019):

$\left\\{\begin{array}[]{l}\rho(r)=\rho_{0}\left(\frac{r}{r_{0}}\right)^{-\alpha}\\\
\nu(r)=\nu_{0}\left(\frac{r}{r_{0}}\right)^{-\delta}\\\
\beta(r)=1-\sigma_{t}^{2}/\sigma_{r}^{2}\;,\end{array}\right.$ (4)

where $r$ is defined as the spherical radial coordinate from the lens centre,
$\rho\left(r\right)$ is the total (including luminous and dark matter) mass
density distribution, and $\nu\left(r\right)$ represents the distribution of
luminous density. The parameter $\beta\left(r\right)$ describes the anisotropy
of the stellar velocity dispersion, where $\sigma_{t}$ and $\sigma_{r}$ are
the velocity dispersions in the tangential and radial directions,
respectively. In the literature, $\beta$ is always assumed to be independent
of $r$ (e.g., (Koopmans et al., 2006; Treu et al., 2010)). Following previous
studies (Bolton et al., 2006; Schwab et al., 2010; Cao et al., 2017; Chen et
al., 2019; Liu et al., 2022; Wei et al., 2022), we set a Gaussian prior
$\beta=0.18\pm 0.13$, informed by the constraint from a well-studied sample of
elliptical galaxies (Gerhard et al., 2001). That is, $\beta$ will be
marginalized using a Gaussian prior of $\beta=0.18\pm 0.13$ over the $2\sigma$
range of $[-0.08,\;0.44]$. Also, $\alpha$ and $\delta$ are the power-law
indices of the total mass density profile and the luminosity density profile,
respectively. It has been confirmed in previous works (Sonnenfeld et al.,
2013; Chen et al., 2019) that $\alpha$ is significantly related with the lens
redshift $z_{l}$ and the surface mass density of the lensing galaxy.
Therefore, we treat the parametrized model of $\alpha$ as (Chen et al., 2019)

$\alpha=\alpha_{0}+\alpha_{z}z_{l}+\alpha_{s}\log_{10}\tilde{\Sigma}\;,$ (5)

where $\alpha_{0}$, $\alpha_{z}$ and $\alpha_{s}$ are arbitrary constants.
Here $\tilde{\Sigma}$ stands for the normalized surface mass density, and is
expressed as $\tilde{\Sigma}=\frac{\left(\sigma_{0}/100\;\rm
km\;s^{-1}\right)^{2}}{R_{\rm eff}/10\;h^{-1}\;\rm kpc}$, where $\sigma_{0}$
is the observed velocity dispersion, $R_{\rm eff}$ is the lensing galaxy’s
half-light radius, and $h=H_{0}/(100\;\rm km\;s^{-1}\;Mpc^{-1})$ is the
reduced Hubble constant.

Following the well-known radial Jeans equation in spherical coordinate
(Binney, 1980), the radial velocity dispersion of the luminous matter
$\sigma_{r}$ in early-type lens galaxies takes the form

$\sigma_{r}^{2}\left(r\right)=\frac{G\int_{r}^{\infty}\mathrm{d}{r}^{\prime}{r}^{\prime
2\beta-2}\nu\left(r^{\prime}\right)M\left(r^{\prime}\right)}{r^{2\beta}\nu\left(r\right)}\;,$
(6)

where $M\left(r\right)$ is the total mass included within a sphere with radius
$r$,

$M\left(r\right)=\int_{0}^{r}\mathrm{d}r^{\prime}4\pi r^{\prime
2}\rho\left(r^{\prime}\right)=4\pi\frac{\rho_{0}}{r_{0}^{-\alpha}}\frac{r^{3-\alpha}}{3-\alpha}\;.$
(7)

The dynamical mass $M_{\rm E}^{\rm dyn}$ enclosed within a cylinder of radius
equal to the Einstein radius $R_{\rm E}$ can be written as (Chen et al., 2019)

$M_{\rm E}^{\rm dyn}=2\pi^{3/2}\frac{R_{\rm
E}^{3-\alpha}}{3-\alpha}\frac{\Gamma\left(\frac{\alpha-1}{2}\right)}{\Gamma\left(\frac{\alpha}{2}\right)}\frac{\rho_{0}}{r_{0}^{-\alpha}}\;,$
(8)

where $\Gamma(x)$ is Euler’s Gamma function. By combining Equations (7) and
(8), we get the relation between $M\left(r\right)$ and $M_{\rm E}^{\rm dyn}$:

$M(r)=\frac{2}{\sqrt{\pi}}\frac{1}{\lambda(\alpha)}\left(\frac{r}{R_{\rm
E}}\right)^{3-\alpha}M_{\rm E}^{\rm dyn}\;,$ (9)

where
$\lambda(\alpha)=\Gamma\left(\frac{\alpha-1}{2}\right)/\Gamma\left(\frac{\alpha}{2}\right)$.
By substituting Equations (9) and (4) into Equation (6), we obtain

$\sigma_{r}^{2}\left(r\right)=\frac{2}{\sqrt{\pi}}\frac{GM_{\rm E}^{\rm
dyn}}{R_{\rm
E}}\frac{1}{\xi-2\beta}\frac{1}{\lambda(\alpha)}\left(\frac{r}{R_{\rm
E}}\right)^{2-\alpha}\;,$ (10)

where $\xi=\alpha+\delta-2$.

The actual velocity dispersion of the lensing galaxy is the component of
luminosity-weighted average along the line of sight and measured over the
effective spectroscopic aperture $R_{\rm A}$, that can be expressed as (see
Ref. (Chen et al., 2019) for more details)

$\sigma_{0}^{2}\left(\leq R_{\rm
A}\right)=\frac{c^{2}}{2\sqrt{\pi}}\frac{2}{1+\gamma_{\rm
PPN}}\frac{D_{s}}{D_{ls}}\theta_{\rm
E}F\left(\alpha,\;\delta,\;\beta\right)\left(\frac{R_{\rm A}}{R_{\rm
E}}\right)^{2-\alpha}\;,$ (11)

where

$F\left(\alpha,\;\delta,\;\beta\right)=\frac{3-\delta}{\left(\xi-2\beta\right)\left(3-\xi\right)}\frac{\lambda(\xi)-\beta\lambda\left(\xi+2\right)}{\lambda(\alpha)\lambda(\delta)}\;.$
(12)

The theoretical value of the velocity dispersion inside the radius $R_{\rm
eff}/2$ can then be calculated by (Koopmans et al., 2006)

$\sigma_{0}^{\rm th}=\sqrt{\frac{c^{2}}{2\sqrt{\pi}}\frac{2}{1+\gamma_{\rm
PPN}}\frac{D_{s}}{D_{ls}}\theta_{\rm
E}F\left(\alpha,\;\delta,\;\beta\right)\left(\frac{\theta_{\rm
eff}}{2\theta_{\rm E}}\right)^{2-\alpha}}\;,$ (13)

where $\theta_{\rm eff}=R_{\rm eff}/D_{l}$ denotes the effective angular
radius of the lensing galaxy.

Based on the spectroscopic data, one can measure the luminosity-weighted
average of the line-of-sight velocity dispersion $\sigma_{\rm ap}$ within the
circular aperture with the angular radius $\theta_{\rm ap}$. In practice,
$\sigma_{\rm ap}$ should be normalized to the velocity dispersion within the
typical physical aperture with a radius $R_{\rm eff}/2$,

$\sigma_{0}^{\rm obs}=\sigma_{\rm ap}\left[\theta_{\rm eff}/(2\theta_{\rm
ap})\right]^{\eta}\;,$ (14)

where the value of the correction factor is taken as $\eta=-0.066\pm 0.035$
(Cappellari et al., 2006). Then, the total uncertainty of $\sigma_{0}^{\rm
obs}$ can be obtained by

$\left(\Delta\sigma_{0}^{\rm SGL}\right)^{2}=\left(\Delta\sigma_{0}^{\rm
stat}\right)^{2}+\left(\Delta\sigma_{0}^{\rm
AC}\right)^{2}+\left(\Delta\sigma_{0}^{\rm sys}\right)^{2}\;,$ (15)

where $\Delta\sigma_{0}^{\rm stat}$ is the statistical error propagated from
the measurement error of $\sigma_{\rm ap}$, and $\Delta\sigma_{0}^{\rm AC}$ is
the aperture-correction-induced error propagated from the uncertainty of
$\eta$. The systematic error due to the extra mass contribution from the outer
matters of the lensing galaxy along the line of sight, $\Delta\sigma_{0}^{\rm
sys}$, is taken as an uncertainty of $\sim 3\%$ to the velocity dispersion
(Jiang and Kochanek, 2007).

Once we know the ratio of the angular diameter distances $D_{s}/D_{ls}$, the
constraints on the PPN parameter $\gamma_{\rm PPN}$ can be derived by
comparing the observational and theoretical values of the velocity dispersions
(see Equations (13) and (14)). Conventionally the distance ratio
$D_{s}/D_{ls}$ is calculated within the standard $\Lambda$CDM cosmological
model (Schwab et al., 2010; Cao et al., 2017). However, $\Lambda$CDM itself is
built on the framework of GR and this leads to a circularity problem (Liu et
al., 2022; Wei et al., 2022). To avoid this problem, we will use a
cosmological-model-independent method which is based upon the sum rule of
distances in the FLRW metric to constrain $\gamma_{\rm PPN}$.

###  II.2 Distance Sum Rule

In a homogeneous and isotropic space, the dimensionless comoving distance
$d\left(z_{l},\;z_{s}\right)\equiv\left(H_{0}/c\right)\left(1+z_{s}\right)D_{A}\left(z_{l},\;z_{s}\right)$
can be written as

$d(z_{l},z_{s})=\frac{1}{\sqrt{|\Omega_{k}|}}{\rm
sinn}\left(\sqrt{|\Omega_{k}|}\int_{z_{l}}^{z_{s}}\frac{{\rm
d}z^{\prime}}{E(z^{\prime})}\right)\;,$ (16)

where $\Omega_{k}$ denotes the spatial curvature density parameter at the
present time and $E(z)=H(z)/H_{0}$ is the dimensionless expansion rate. Also,
${\rm sinn}(x)$ is $\sinh(x)$ when $\Omega_{k}>0$, $x$ when $\Omega_{k}=0$,
and $\sin(x)$ when $\Omega_{k}<0$. By applying the notations $d(z)\equiv
d\left(0,\;z\right)$, $d_{ls}\equiv d\left(z_{l},\;z_{s}\right)$, $d_{l}\equiv
d\left(0,\;z_{l}\right)$, and $d_{s}\equiv d\left(0,\;z_{s}\right)$, one can
derive a sum rule of distances along the null geodesics of the FLRW metric as
(Peebles, 1993; Bernstein, 2006; Räsänen et al., 2015)

$\frac{d_{ls}}{d_{s}}=\sqrt{1+\Omega_{k}d_{l}^{2}}-\frac{d_{l}}{d_{s}}\sqrt{1+\Omega_{k}d_{s}^{2}}\;.$
(17)

This relation provides a cosmology-independent probe to test both the spatial
curvature and the FLRW metric. The validity of the FLRW metric can be tested
by comparing the derived $\Omega_{k}$ from the three distances ($d_{l}$,
$d_{s}$, and $d_{ls}$) for any two pairs of ($z_{l}$, $z_{s}$).

With Equation (17), the distance ratio $D_{s}/D_{ls}$111Note that
$D_{s}/D_{ls}$ is actually equal to the dimensionless distance ratio
$d_{s}/d_{ls}$. in Equation (13) is only related to the curvature parameter
$\Omega_{k}$ and the dimensionless distances $d_{l}$ and $d_{s}$. If
independent measurements of $d_{l}$ and $d_{s}$ are given, we can put
constraints on both $\gamma_{\rm PPN}$ and $\Omega_{k}$ from Equations (13)
and (17) without assuming any specific cosmological model.

###  II.3 Artificial Neural Network

To calibrate the distances $d_{l}$ and $d_{s}$ of the SGL systems (i.e., the
distances $d_{l}$ and $d_{s}$ on the right side of Equation (17)), we use a
new nonparametric approach, ANN, to reconstruct a smooth distance–redshift
relation $d(z)$ from SN Ia observation.

ANNs possess several desirable properties, including high-level abstraction of
neural input-output transformation, the ability to generalize from learned
instances to new unseen data, adaptability, self-learning, fault tolerance,
and nonlinearity (Abiodun et al., 2018). According to the universal
approximation theorem (Hornik et al., 1989; Cybenko, 1989), ANNs can function
as universal function approximators to simulate arbitrary input-output
relationships using multilayer feedforward networks with a sufficient number
of hidden units. Therefore, we can input the redshift $z$ into the neural
network, with the corresponding comoving distance $d(z)$ and its associated
error $\sigma_{d(z)}$ as the desired outputs. Once the network has been
trained using the Pantheon+ sample, we will obtain an approximate function
capable of predicting both $d(z)$ and its error $\sigma_{d(z)}$ at any given
redshift $z$.

Ref. Wang et al. (2020) has developed a Python code for the reconstruction of
functions from observational data employing an ANN. They have substantiated
the reliability of these reconstructed functions by estimating cosmological
parameters through the utilization of the reconstructed Hubble parameter
$H(z)$ and the luminosity distance $D_{L}(z)$, in direct comparison with
observational data. In our study, we will employ this code to reconstruct the
distance–redshift relation.

The general structure of an ANN consists of an input layer, one or more hidden
layers, and an output layer. The basic unit of these layers are referred to as
neurons, which serve as both linear transformation units and nonlinear
activation functions for the input vector. In accordance with Ref. Wang et al.
(2020), we employ the Exponential Linear Unit as our chosen activation
function, as defined by its form in (Clevert et al., 2015):

$f\left(x\right)=\left\\{\begin{matrix}x&x>0\\\
\alpha\left(e^{x}-1\right)&x\leq 0\end{matrix}\right.\;,$ (18)

where the hyperparameter $\alpha$ is set to 1.

The network is trained by minimizing a loss function, which quantitatively
measures the discrepancy between the ground truth and predicted values. In
this analysis, we adopt the mean absolute error (MAE), also known as the L1
loss function, as our choice of loss function. The linear weights and biases
within the network are optimized using the backpropagation algorithm. We
employ the Adam optimizer (Kingma and Ba, 2014), a gradient-based optimization
technique, to iteratively update the network parameters during training. This
choice of optimizer also contributes to faster convergence. After multiple
iterations, the network parameters are adjusted to minimize the loss. We have
determined that a sufficient number of iterations for training convergence is
reached when the loss no longer decreases, which we set to be $3\times
10^{5}$. Batch Normalization (Ioffe and Szegedy, 2015) is a technique designed
to stabilize the distribution of inputs within each layer, allowing for higher
learning rates and reduced sensitivity to initialization.

To determine the optimal network model, we train the network using 1701 SNe Ia
from the Pantheon+ sample (more on this below) and assess the fitting effect
through K-fold cross-validation (Pedregosa et al., 2011). In K-fold cross-
validation, the training set is divided into k smaller sets, with k-1 folds
used as training data for model training, and the remaining fold used for
validation. This process is repeated k times, with each fold serving as the
validation data once. The final performance of the model is determined by
averaging the performance across these k iterations. This approach is
particularly useful when the number of samples available for learning is
insufficient to split into traditional train, validation, and test sets, as is
the case in our analysis. Additionally, it helps mitigate issues arising from
the randomness in data partitioning. As a general guideline, we have selected
$k=10$ for our cross-validation procedure and have utilized the mean squared
error (MSE) as the metric for validating the performance of the model.

Through our experimentation, we have found that the network model with a
single hidden layer comprising 4096 neurons and without batch normalization
yields the best results. We conduct comparisons with models having varying
numbers of hidden layers, and we observe diminished performance as the number
of hidden layers increased, accompanied by increased computational resource
consumption. Regarding the number of neurons in the hidden layer, we observe
negligible impact on the results, as reflected by the final MSE values
consistently hovering around 0.0042, regardless of whether the number of
neurons was set to 1024, 2048, 4096, or 8192. Importantly, the final MSE value
with 4096 neurons was slightly smaller compared to the other three
configurations, and as a result, we select this configuration. The validation
values for implementing batch normalization or not implementing it is 0.0049
or 0.0042, respectively.

Subsequently, we will employ the optimal network model, as described above, to
reconstruct our distance-redshift curve.

###  II.4 Supernova Data

In order to reconstruct the distance function $d(z)$, we choose the latest
combined sample of SNe Ia called Pantheon+ (Scolnic et al., 2022), which
consists of 1701 light curves of 1550 SNe Ia, covering the redshift range
$0.001<z<2.3$. For each SN Ia, the distance modulus $\mu$ is related to the
luminosity distance $D_{L}$ by

$\mu(z)=5\log_{10}\left[\frac{D_{L}(z)}{\rm Mpc}\right]+25\;,$ (19)

and the observed distance modulus is

$\mu_{\rm obs}(z)=m_{B}(z)+\kappa\cdot X_{1}-\omega\cdot\mathcal{C}-M_{B}\;,$
(20)

where $m_{B}$ is the rest-frame _B_ band peak magnitude, $X_{1}$ and
$\mathcal{C}$, respectively, represent the time stretch of light curve and the
SN color at maximum brightness, and $M_{B}$ is the absolute _B_ -band
magnitude. Through the BEAMS with Bias Corrections method (Kessler and
Scolnic, 2017), the two nuisance parameters $\kappa$ and $\omega$ can be
calibrated to zero. Then, the observed distance modulus can be simplified as

$\mu_{\rm obs}(z)=m_{\rm corr}(z)-M_{B}\;,$ (21)

where $m_{\rm corr}$ is the corrected apparent magnitude. The absolute
magnitude $M_{B}$ is exactly degenerate with the Hubble constant $H_{0}$. Once
the value of $M_{B}$ or $H_{0}$ is known, the luminosity distances $D_{L}(z)$
can be obtained from SNe Ia.

In this work, we adopt $H_{0}=70\;\rm km\;s^{-1}\;Mpc^{-1}$ to normalize the
SN Ia $D_{L}(z)$ data as the observational $d(z)$. That is,
$d(z)=(H_{0}/c)D_{L}(z)/(1+z)$. Note that the choice of $H_{0}$ has no impact
on our results, since the required distance ratio $D_{s}/D_{ls}$ (see Equation
(13)) is completely independent of $H_{0}$. Having obtained the dataset of
$d(z)$, we adopt ANN to reconstruct the distance function $d(z)$, and the
results are shown in Figure 1. The black line represents the reconstructed
function of $d(z)$, and the shaded region is the corresponding $1\sigma$
confidence level.

Figure 1: Reconstruction of the dimensionless comoving distance $d(z)$ from
Pantheon+ SNe Ia using ANN. The shaded area is the 1$\sigma$ confidence level
of the reconstruction. The blue dots with error bars represent the
observational data.

###  II.5 Strong-lensing Data

Recently, Ref. Chen et al. (2019) compiled a galaxy-scale SGL sample including
161 systems with stellar velocity dispersion measurements, which is assembled
with strict selection criteria to meet the assumption of spherical symmetry on
the lens mass model. The observational information for each SGL system are
listed in Appendix of Ref. Chen et al. (2019), including the lens redshift
$z_{l}$, the source redshift $z_{s}$, the Einstein angle $\theta_{E}$, the
central velocity dispersion of the lensing galaxy $\sigma_{\rm ap}$, the
spectroscopic aperture angular radius $\theta_{\rm ap}$, and the half-light
angular radius of the lensing galaxy $\theta_{\rm eff}$.

By fitting the two-dimensional power-law luminosity profile convolved with the
instrumental point spread function to the high-resolution Hubble Space
Telescope imaging data over a circle of radius $\theta_{\rm eff}/2$ centered
on the lensing galaxies, Ref. Chen et al. (2019) measured the slops of the
luminosity density profile $\delta$ for the 130 lensing galaxies in the full
sample. They showed that $\delta$ should be treated as as an observable for
each lens in order to get an unbiased estimate of the cosmological parameter
$\Omega_{\rm m}$. Therefore, the SGL sample we adopt here is the truncated
sample of 130 SGL systems with $\delta$ measurements, for which the redshift
ranges of lenses and sources are $0.0624\leq z_{l}\leq 0.7224$ and $0.1970\leq
z_{s}\leq 2.8324$, respectively. In this work, we use the reconstructed
distance function $d(z)$ from Pantheon+ SNe Ia to calibrate the distances
$d_{l}$ and $d_{s}$ of the SGL systems. However, the SN Ia catalog extends
only to $z=2.3$. As such, we shall employ only a sub-set of the SGL sample
that overlaps with the SN Ia data for the calibration. Thus, only 120 SGL
systems with $z_{s}\leq 2.3$ are available in our analysis.

###  II.6 The Likelihood Function

By using the Python Markov Chain Monte Carlo module EMCEE (Foreman-Mackey et
al., 2013) to maximize the likelihood function $\mathcal{L}$, we
simultaneously place limits on the PPN parameter $\gamma_{\rm PPN}$, the
curvature parameter $\Omega_{k}$, and the lens model parameters ($\alpha_{0}$,
$\alpha_{z}$, and $\alpha_{s}$). The likelihood function is defined as

$\mathcal{L}=\prod_{i=1}^{120}\frac{1}{\sqrt{2\pi}\Delta\sigma_{0,i}^{\rm
tot}}\exp\left[-\frac{1}{2}\left(\frac{\sigma_{0,i}^{\rm th}-\sigma_{0,i}^{\rm
obs}}{\Delta\sigma_{0,i}^{\rm tot}}\right)^{2}\right]\;,$ (22)

where the variance

$\left(\Delta\sigma_{0}^{\rm tot}\right)^{2}=\left(\Delta\sigma_{0}^{\rm
SGL}\right)^{2}+\left(\Delta\sigma_{0}^{\rm SN}\right)^{2}$ (23)

is given in terms of the total uncertainty $\Delta\sigma_{0}^{\rm SGL}$
derived from the SGL observation (Equation (15)) and the propagated
uncertainty $\Delta\sigma_{0}^{\rm SN}$ derived from the distance calibration
by SNe Ia. With Equation (13), the propagated uncertainty
$\Delta\sigma_{0}^{\rm SN}$ can be estimated as

$\Delta\sigma_{0}^{\rm SN}=\sigma_{0}^{\rm th}\frac{\Delta D_{r}}{2D_{r}}\;,$
(24)

where $D_{r}$ is a convenient notation for the distance ratio in Equation
(13), i.e., $D_{r}\equiv D_{s}/D_{ls}=d_{s}/d_{ls}$, and its uncertainty is
$\Delta D_{r}$. With the reconstructed distance function $d(z)$, as well as
its $1\sigma$ uncertainty $\Delta d(z)$, from the SN Ia data, we can calibrate
the distances ($d_{l}$ and $d_{s}$) and their corresponding uncertainties
($\Delta d_{l}$ and $\Delta d_{s}$) for each SGL system. Thus, the uncertainty
$\Delta D_{r}$ of the distance ratio can be easily derived from Equation (17),
i.e.,

$\displaystyle\left(\Delta D_{r}\right)^{2}=$ $\displaystyle
D_{r}^{4}\left(\frac{\Omega_{k}d_{l}}{\sqrt{1+\Omega_{k}d_{l}^{2}}}-\frac{\sqrt{1+\Omega_{k}d_{s}^{2}}}{d_{s}}\right)^{2}\left(\Delta
d_{l}\right)^{2}$ (25)
$\displaystyle+D_{r}^{4}\left(\frac{d_{l}}{d_{s}^{2}\sqrt{1+\Omega_{k}d_{s}^{2}}}\right)^{2}\left(\Delta
d_{s}\right)^{2}\;.$

##  III Results

Figure 2: 1D marginalized probability distributions and 2D $1-2\sigma$
confidence contours for the PPN parameter $\gamma_{\rm PPN}$, the cosmic
curvature $\Omega_{k}$, and the lens model parameters ($\alpha_{0}$,
$\alpha_{z}$, and $\alpha_{s}$). The dashed lines represent $\gamma_{\rm
PPN}=1$ and $\Omega_{k}=0$, corresponding to a flat universe with the validity
of GR.

The 1D marginalized probability distributions and 2D plots of the $1-2\sigma$
confidence regions for the PPN parameter $\gamma_{\rm PPN}$, the cosmic
curvature $\Omega_{k}$, and the lens model parameters ($\alpha_{0}$,
$\alpha_{z}$, and $\alpha_{s}$), constrained by 120 SGL systems, are presented
in Figure 2, and the best-fitting results are listed in Table 1. These
contours show that at the $1\sigma$ confidence level, the inferred parameter
values are $\gamma_{\rm PPN}=1.16_{-0.12}^{+0.15}$,
$\Omega_{k}=0.89_{-1.00}^{+1.97}$, $\alpha_{0}=1.2_{-0.15}^{+0.15}$,
$\alpha_{z}=-0.37_{-0.26}^{+0.22}$, and $\alpha_{s}=0.70_{-0.09}^{+0.10}$. We
find that the measured $\gamma_{\rm PPN}$ is consistent with the prediction of
$\gamma_{\rm PPN}=1$ from GR, and its constraint accuracy is about 11.6%.
While $\Omega_{k}$ is weakly constrained, it is still compatible with zero
spatial curvature within $1\sigma$ confidence level. We also find that the
inferred $\alpha_{z}$ and $\alpha_{s}$ separately deviate from zero at $\sim
2\sigma$ and $\sim 8\sigma$ levels, confirming previous finding that the total
mass density slope $\alpha$ strongly depends on both the lens redshift and the
surface mass density (Chen et al., 2019).

Table 1: Constraint results for All Parameters with Different Priors  Priors |  $\gamma_{\rm PPN}$ |  $\Omega_{k}$ |  $\alpha_{0}$ |  $\alpha_{z}$ |  $\alpha_{s}$
---|---|---|---|---|---
None |  $1.16_{-0.12}^{+0.15}$ |  $0.89_{-1.00}^{+1.97}$ |  $1.20_{-0.15}^{+0.15}$ |  $-0.37_{-0.26}^{+0.22}$ |  $0.70_{-0.09}^{+0.10}$
$\Omega_{k}=0$ |  $1.09_{-0.10}^{+0.11}$ |  |  $1.22_{-0.14}^{+0.14}$ |  $-0.20_{-0.11}^{+0.11}$ |  $0.67_{-0.09}^{+0.09}$
$\gamma_{\rm PPN}=1$ |  |  $0.12_{-0.47}^{+0.78}$ |  $1.10_{-0.12}^{+0.11}$ |  $-0.20_{-0.16}^{+0.15}$ |  $0.74_{-0.08}^{+0.08}$
Figure 3: Same as Figure 2, except now for the scenario with a prior of
$\Omega_{k}=0$. The dashed line represents $\gamma_{\rm PPN}=1$ predicted by
GR.

We further explore the scenario of adopting a prior of flatness, i.e.,
$\Omega_{k}=0$. For this scenario, as shown in Figure 3 and Table 1, the
marginalized distribution gives $\gamma_{\rm PPN}=1.09_{-0.10}^{+0.11}$,
representing a precision of 9.6%, in good agreement with the prediction of GR.
If instead we adopt a prior of $\gamma_{\rm PPN}=1$ (i.e., assuming GR holds)
and allow $\Omega_{k}$ to be a free parameter, the resulting constraints on
$\Omega_{k}$ and the lens model parameters are displayed in Figure 4 and Table
1. The marginalized $\Omega_{k}$ constraint is
$\Omega_{k}=0.12_{-0.47}^{+0.78}$, consistent with a spatially flat universe.
The comparison among lines 1–3 of Table 1 suggests that different choices of
priors have little effect on the lens model parameters ($\alpha_{0}$,
$\alpha_{z}$, and $\alpha_{s}$).

Figure 4: Same as Figure 2, except now for the scenario with a prior of
$\gamma_{\rm PPN}=1$. The dashed line represents a spatially flat universe.

##  IV Conclusion and Discussions

Galaxy-scale SGL systems, combined with stellar velocity dispersion
measurements of lensing galaxies, provide a powerful probe to test the
validity of GR by constraining the PPN parameter $\gamma_{\rm PPN}$ on
kiloparsec scales. Testing GR in this manner, however, it is necessary to know
the angular diameter distances between the observer, lens, and source.
Conventionally, the required distances are calculated within the standard
$\Lambda$CDM cosmological model. Such distance calculations would involve a
circularity problem in testing GR, since $\Lambda$CDM itself is established on
the framework of GR. In this paper, in order to address the circularity
problem, we have employed the DSR in the FLRW metric to estimate not only
$\gamma_{\rm PPN}$ but also the spatial curvature $\Omega_{k}$ independently
of any specific cosmological model. To calibrate the distances of the SGL
systems, we have introduced a new nonparametric approach for reconstructing
the distance–redshift relation from the Pantheon+ SN Ia sample using an ANN,
which has no assumptions about the observational data and is a completely
data-driven approach.

By combining 120 well-selected SGL systems with the reconstructed distance
function from 1701 data points of SNe Ia, we have obtained simultaneous
estimates of $\gamma_{\rm PPN}$ and $\Omega_{k}$ without any specific
assumptions about the contents of the universe or the theory of gravity. Our
results show that $\gamma_{\rm PPN}=1.16_{-0.12}^{+0.15}$ and
$\Omega_{k}=0.89_{-1.00}^{+1.97}$. The measured $\gamma_{\rm PPN}$ is in good
agreement with the prediction of GR with 11.6% accuracy. If we use flatness as
a prior (i.e., $\Omega_{k}=0$), we infer that $\gamma_{\rm
PPN}=1.09_{-0.10}^{+0.11}$, representing a precision of 9.6%. If we instead
assume the conservation of GR (i.e., $\gamma_{\rm PPN}=1$) and allow
$\Omega_{k}$ to be a free parameter, we find
$\Omega_{k}=0.12_{-0.47}^{+0.78}$. The measured $\Omega_{k}$ is consistent
with zero spatial curvature, suggesting that there is no significant deviation
from a flat universe.

In the literature, based on a sample of 80 SGL systems, Ref. Cao et al. (2017)
obtained the constraint accuracy of the PPN parameter $\gamma_{\rm PPN}$ to be
25% under the assumption of a flat $\Lambda$CDM model with parameters taken
from Planck observations. Within the same context of $\Lambda$CDM, Ref.
Collett et al. (2018) concluded that $\gamma_{\rm PPN}=0.97\pm 0.09$
(representing a precision of 9.3%) by analyzing the nearby lens ESO 325-G004.
Through the reanalysis of four time-delay lenses, Ref. Yang et al. (2020)
obtained simultaneous constraints of $\gamma_{\rm PPN}$ and the Hubble
constant $H_{0}$ for flat $\Lambda$CDM, yielding $\gamma_{\rm
PPN}=0.87_{-0.17}^{+0.19}$ (representing a precision of 21%) and
$H_{0}=73.65_{-2.26}^{+1.95}\;\rm km\;s^{-1}\;Mpc^{-1}$. Within a flat FLRW
metric, Ref. Liu et al. (2022) used 120 lenses to achieve a model-independent
estimate of $\gamma_{\rm PPN}=1.065_{-0.074}^{+0.064}$ (representing a
precision of 6.5%) by employing the GP method to reconstruct the SN distances.
As a further refinement, Ref. Wei et al. (2022) removed the flatness
assumption and implemented the DSR to obtain model-independent constraints of
$\gamma_{\rm PPN}=1.11_{-0.09}^{+0.11}$ (representing a precision of 9.0%) and
$\Omega_{k}=0.48_{-0.71}^{+1.09}$. Note that in Ref. Wei et al. (2022) the
distances of the SGL systems were determined by fitting a third-order
polynomial to the SN Ia data. Unlike the polynomial fit that rely on the
assumed parameterization, the ANN used in this work is a completely data-
driven approach that could reconstruct a function from various data without
assuming a parameterization of the function. Moreover, unlike the GP method
that rely on the assumption of Gaussian distributions for the observational
random variables, the ANN method has no assumptions about the data. More
importantly, compared to previous results, our work yielded comparable
resulting constraints on $\gamma_{\rm PPN}$, which indicates the effectiveness
of data-driven modeling based on the ANN.

Looking forward, the forthcoming Large Synoptic Survey Telescope (LSST)
survey, with its excellent operation performance, holds great promise for
detecting a large number of lenses, potentially reaching up to 120,000 in the
most optimistic scenario (Collett, 2015). By setting a prior on the curvature
parameter $-0.007<\Omega_{k}<0.006$, Ref. Cao et al. (2017) showed that 53,000
simulated LSST strong lensing data would set a stringent constraint of
$\gamma_{\rm PPN}=1.000_{-0.0011}^{+0.0009}$, reaching a precision of
$10^{-3}\sim 10^{-4}$. Similarly, Ref. Lian et al. (2022) performed a robust
extragalactic test of GR using a well-defined sample of 5,000 simulated strong
lenses from LSST, yielding an accuracy of 0.5%. In brief, much more severe
constraints on both $\gamma_{\rm PPN}$ and $\Omega_{k}$, as discussed in this
work, can be expected with the help of future lens surveys.

###### Acknowledgements.

This work is partially supported by the National Natural Science Foundation of
China (grant Nos. 12373053 and 12041306), the Key Research Program of Frontier
Sciences (grant No. ZDBS-LY-7014) of Chinese Academy of Sciences, the Natural
Science Foundation of Jiangsu Province (grant No. BK20221562), and the Young
Elite Scientists Sponsorship Program of Jiangsu Association for Science and
Technology.

## References

  * Dyson et al. (1920) F. W. Dyson, A. S. Eddington, and C. Davidson, Philosophical Transactions of the Royal Society of London Series A 220, 291 (1920).
  * Pound and Rebka (1960) R. V. Pound and G. A. Rebka, Phys. Rev. Lett. 4, 337 (1960).
  * Shapiro (1964) I. I. Shapiro, Phys. Rev. Lett. 13, 789 (1964), URL https://link.aps.org/doi/10.1103/PhysRevLett.13.789.
  * Taylor et al. (1979) J. H. Taylor, L. A. Fowler, and P. M. McCulloch, Nature (London) 277, 437 (1979).
  * Thorne and Will (1971) K. S. Thorne and C. M. Will, Astrophys. J. 163, 595 (1971).
  * Bertotti et al. (2003) B. Bertotti, L. Iess, and P. Tortora, Nature (London) 425, 374 (2003).
  * Bolton et al. (2006) A. S. Bolton, S. Rappaport, and S. Burles, Phys. Rev. D 74, 061501 (2006), eprint astro-ph/0607657.
  * Smith (2009) T. L. Smith, arXiv e-prints arXiv:0907.4829 (2009), eprint 0907.4829.
  * Schwab et al. (2010) J. Schwab, A. S. Bolton, and S. A. Rappaport, Astrophys. J. 708, 750 (2010), eprint 0907.4992.
  * Cao et al. (2017) S. Cao, X. Li, M. Biesiada, T. Xu, Y. Cai, and Z.-H. Zhu, Astrophys. J. 835, 92 (2017), eprint 1701.00357.
  * Collett et al. (2018) T. E. Collett, L. J. Oldham, R. J. Smith, M. W. Auger, K. B. Westfall, D. Bacon, R. C. Nichol, K. L. Masters, K. Koyama, and R. van den Bosch, Science 360, 1342 (2018), eprint 1806.08300.
  * Yang et al. (2020) T. Yang, S. Birrer, and B. Hu, Mon. Not. Roy. Astron. Soc. 497, L56 (2020), eprint 2003.03277.
  * Liu et al. (2022) X.-H. Liu, Z.-H. Li, J.-Z. Qi, and X. Zhang, Astrophys. J. 927, 28 (2022), eprint 2109.02291.
  * Wei et al. (2022) J.-J. Wei, Y. Chen, S. Cao, and X.-F. Wu, Astrophys. J. Lett. 927, L1 (2022), eprint 2202.07860.
  * Chen et al. (2019) Y. Chen, R. Li, Y. Shu, and X. Cao, Monthly Notices of the Royal Astronomical Society 488, 3745 (2019), ISSN 0035-8711, eprint https://academic.oup.com/mnras/article-pdf/488/3/3745/29100170/stz1902.pdf, URL https://doi.org/10.1093/mnras/stz1902.
  * Seikel et al. (2012) M. Seikel, S. Yahya, R. Maartens, and C. Clarkson, Physical Review D 86, 083001 (2012).
  * Yahya et al. (2014) S. Yahya, M. Seikel, C. Clarkson, R. Maartens, and M. Smith, Phys. Rev. D 89, 023503 (2014), eprint 1308.4099.
  * Montiel et al. (2014) A. Montiel, R. Lazkoz, I. Sendra, C. Escamilla-Rivera, and V. Salzano, Phys. Rev. D 89, 043007 (2014), eprint 1401.4188.
  * Wei and Wu (2017) J.-J. Wei and X.-F. Wu, Astrophys. J. 838, 160 (2017), eprint 1611.00904.
  * Melia (2018) F. Melia, Mon. Not. R. Astron. Soc. 481, 4855 (2018), eprint 1804.09906.
  * Liao et al. (2019) K. Liao, A. Shafieloo, R. E. Keeley, and E. V. Linder, Astrophys. J. Lett. 886, L23 (2019), eprint 1908.04967.
  * Seikel et al. (2012) M. Seikel, C. Clarkson, and M. Smith, JCAP 2012, 036 (2012), eprint 1204.2832.
  * Cybenko (1989) G. V. Cybenko, Mathematics of Control, Signals and Systems 2, 303 (1989).
  * Hornik (1991) K. Hornik, Neural Networks 4, 251 (1991), ISSN 0893-6080, URL https://www.sciencedirect.com/science/article/pii/089360809190009T.
  * Wang et al. (2020) G.-J. Wang, X.-J. Ma, S.-Y. Li, and J.-Q. Xia, Astrophys. J. Suppl. Ser. 246, 13 (2020), eprint 1910.03636.
  * Schneider et al. (1992) P. Schneider, J. Ehlers, and E. E. Falco, _Gravitational Lenses_ (1992).
  * Koopmans et al. (2006) L. V. E. Koopmans, T. Treu, A. S. Bolton, S. Burles, and L. A. Moustakas, Astrophys. J. 649, 599 (2006), eprint astro-ph/0601628.
  * Treu et al. (2010) T. Treu, M. W. Auger, L. V. E. Koopmans, R. Gavazzi, P. J. Marshall, and A. S. Bolton, Astrophys. J. 709, 1195 (2010), eprint 0911.3392.
  * Gerhard et al. (2001) O. Gerhard, A. Kronawitter, R. P. Saglia, and R. Bender, Astron. J. 121, 1936 (2001), eprint astro-ph/0012381.
  * Sonnenfeld et al. (2013) A. Sonnenfeld, T. L. Treu, R. Gavazzi, S. H. Suyu, P. J. Marshall, M. W. Auger, C. N. U. of California, S. Barbara, I. d’astrophysique de Paris, I. of Radio Astronomy, et al., The Astrophysical Journal 777 (2013).
  * Binney (1980) J. Binney, Mon. Not. R. Astron. Soc. 190, 873 (1980).
  * Cappellari et al. (2006) M. Cappellari, R. Bacon, M. Bureau, M. C. Damen, R. L. Davies, P. T. de Zeeuw, E. Emsellem, J. Falcón-Barroso, D. Krajnović, H. Kuntschner, et al., Mon. Not. R. Astron. Soc. 366, 1126 (2006), eprint astro-ph/0505042.
  * Jiang and Kochanek (2007) G. Jiang and C. S. Kochanek, Astrophys. J. 671, 1568 (2007), eprint 0705.3647.
  * Peebles (1993) P. J. E. Peebles, _Principles of Physical Cosmology_ (1993).
  * Bernstein (2006) G. Bernstein, Astrophys. J. 637, 598 (2006), eprint astro-ph/0503276.
  * Räsänen et al. (2015) S. Räsänen, K. Bolejko, and A. Finoguenov, Phys. Rev. Lett. 115, 101301 (2015), eprint 1412.4976.
  * Abiodun et al. (2018) O. I. Abiodun, A. Jantan, A. E. Omolara, K. V. Dada, N. A. Mohamed, and H. Arshad, Heliyon 4, e00938 (2018), ISSN 2405-8440, URL https://www.sciencedirect.com/science/article/pii/S2405844018332067.
  * Hornik et al. (1989) K. Hornik, M. Stinchcombe, and H. White, Neural Networks 2, 359 (1989), ISSN 0893-6080, URL https://www.sciencedirect.com/science/article/pii/0893608089900208.
  * Clevert et al. (2015) D.-A. Clevert, T. Unterthiner, and S. Hochreiter, arXiv e-prints arXiv:1511.07289 (2015), eprint 1511.07289.
  * Kingma and Ba (2014) D. P. Kingma and J. Ba, arXiv e-prints arXiv:1412.6980 (2014), eprint 1412.6980.
  * Ioffe and Szegedy (2015) S. Ioffe and C. Szegedy, arXiv e-prints arXiv:1502.03167 (2015), eprint 1502.03167.
  * Pedregosa et al. (2011) F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, et al., Journal of Machine Learning Research 12, 2825 (2011).
  * Scolnic et al. (2022) D. Scolnic, D. Brout, A. Carr, A. G. Riess, T. M. Davis, A. Dwomoh, D. O. Jones, N. Ali, P. Charvu, R. Chen, et al., Astrophys. J. 938, 113 (2022), eprint 2112.03863.
  * Kessler and Scolnic (2017) R. Kessler and D. Scolnic, Astrophys. J. 836, 56 (2017), eprint 1610.04677.
  * Foreman-Mackey et al. (2013) D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Goodman, Pub. Astro. Soc. Pacific 125, 306 (2013), eprint 1202.3665.
  * Collett (2015) T. E. Collett, Astrophys. J. 811, 20 (2015), eprint 1507.02657.
  * Lian et al. (2022) Y. Lian, S. Cao, T. Liu, M. Biesiada, and Z.-H. Zhu, Astrophys. J. 941, 16 (2022), eprint 2210.16752.