You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'We present results of a gravitational-lensing and optical study of MACSJ1423.8+2404 ($z{=}0.545$, MACSJ1423), the most relaxed cluster in the high-redshift subsample of clusters discovered in the MAssive Cluster Survey (MACS). Our analysis uses high-resolution images taken with the [*[Hubble Space Telescope]{}*]{} in the F555W and F814W passbands, ground based imaging in eight optical and near-infrared filters obtained with Subaru and CFHT, as well as extensive spectroscopic data gathered with the Keck telescopes. At optical wavelengths the cluster exhibits no sign of substructure and is dominated by a cD galaxy that is 2.1 magnitudes (K-band) brighter than the second brightest cluster member, suggesting that MACSJ1423 is close to be fully virialized. Analysis of the redshift distribution of 140 cluster members reveals a Gaussian distribution, mildly disturbed by the presence of a loose galaxy group that may be falling into the cluster along the line of sight. Combining strong-lensing constraints from two spectroscopically confirmed multiple-image systems near the cluster core with a weak-lensing measurement of the gravitational shear on larger scales, we derive a parametric mass model for the mass distribution. All constraints can be satisfied by a uni-modal mass distribution centred on the cD galaxy and exhibiting very little substructure. The derived projected mass of $M(<65\arcsec {\rm [415\,kpc]} )=(4.3\pm0.6)\times 10^{14}$ M$_{\sun}$ is about 30% higher than the one derived from X-ray analyses assuming spherical symmetry, suggesting a slightly prolate mass distribution consistent with the optical indication of residual line-of-sight structure. The similarity in shape and excellent alignment of the centroids of the total mass, K-band light, and intra-cluster gas distributions add to the picture of a highly evolved system. The existence of a massive cluster like MACSJ1423, nearly fully virialized only $\sim$ 7 Gyr after the Big Bang, may have important implications for models of structure formation and evolution on cosmological time scales.'
author:
- |
    \
    $^{1}$ Laboratoire d’Astrophysique de Marseille, Université de Provence, CNRS, 38 rue Frédéric Joliot-Curie, F-13388 Marseille Cedex 13, France\
    $^{2}$Dark Cosmology Centre, Niels Bohr Institute, University of Copenhagen, Juliane Maries Vej 30, 2100 Copenhagen, Denmark\
    $^{3}$Institute for Astronomy, University of Hawaii, 2680 Woodlawn Dr, Honolulu, HI 96822, USA\
    $^{4}$Institute for Computational Cosmology, Department of Physics, Durham University, South Road, Durham, DH1 3LE, UK\
    $^{5}$School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK\
    $^{6}$California Institute of Technology, Mail Code 105-24, Pasadena, CA, 91125\
    $^{7}$Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, Stanford, CA 94305, USA\
    $^{8}$Leiden Observatory, Leiden University, Niels Bohrweg 2, NL-2333 CA Leiden, The Netherlands\
bibliography:
- 'draft.bib'
date: 'Accepted. Received; in original form '
title: 'MACSJ1423.8+2404: Gravitational Lensing by a Massive, Relaxed Cluster of Galaxies at z=0.54'
---

\[firstpage\]

Gravitational lensing: strong lensing – Galaxies: clusters: individual (MACSJ1423.8+2404)

Introduction
============

The MAssive Cluster Survey [MACS, @macs] has compiled a complete sample of 12 very X-ray luminous galaxy clusters at $z{>}0.5$ [@highzmacs], providing a unique opportunity for comprehensive studies of the densest regions of the cosmic web at intermediate redshifts. As noted by @macslargescale, MACSJ1423 stands out among these 12 extreme systems as the most dynamically relaxed; it is, indeed, the most massive cool-core cluster known at these redshifts. The system has been used in several statistical studies [@laroque03; @1423cosmo; @bonamente06; @1423MT] where its relaxed dynamical state has been particularly important for cosmological work using the baryon fraction [@allen04; @schmidtAllen1423; @allen08; @ettori09]. Since the abundance of massive relaxed clusters at high redshift has important implications for theoretical and numerical models of structure formation and evolution, we here attempt to further test the system’s relaxation state using gravitational lensing. All our results use the $\Lambda$CDM concordance cosmology with $\Omega_{\rm{M}} = 0.3, \Omega_\Lambda = 0.7$, and a Hubble constant <span style="font-variant:small-caps;">H</span>$_0 = 70$ kms$^{-1}$ Mpc$^{-1}$. Magnitudes are quoted in the AB system.

Observations
============

Imaging
-------

Ground based panoramic imaging of MACSJ1423 was performed in the B, V, R$_c$, I$_c$, and $z'$ bands with the SuprimeCam camera on the Subaru telescope, and in the u$^\ast$ and K bands with MEGACAM and WIRCAM on CFHT. The resulting object catalogues were used to compute photometric redshifts for all galaxies in a $0.5\times 0.5$ deg$^2$ field following the methodology described in @galax0717. Cluster members were defined to be galaxies with photometric or spectroscopic redshifts (see below) within $\pm$0.05 of the cluster redshift.

MACSJ1423 was observed on 2004, June 16 with the Advanced Camera for Surveys (ACS) on-board HST for 4.5 ks and 4.6 ks through the F555W and F814W filters, respectively (GO: 9722, PI: Ebeling). The data were reduced using an automated pipeline developed by the HAGGLeS team (Marshall et al., in prep.), that uses multidrizzle version 2.7.0. Following manual masking of satellite trails, cosmic ray clusters, and scattered light in the flat-fielded frames, all exposures were carefully registered onto a common 0.03 pixel grid [@haggles], generating the shift files needed by multidrizzle. The individual frames in each filter were then drizzled together using a square kernel with pixfrac 0.8, applying updated bad pixel masks and optimal weights [@haggles2].

In spite of the lensing efficiency decreasing with increasing cluster redshift, MACSJ1423 exhibits clear strong-lensing features. Visual inspection of the ACS frames immediately reveals two multiple-image systems (Fig. \[fig1\]). The observed configuration of lensing features for the first of these can be explained by a single background galaxy, resolved into two components, each of which represents a multiple-image system (referred to as 1-2 in the following) in a typical Einstein-cross configuration. The mass model predicts a central fifth image which is detected at the predicted position under the cD galaxy whose light has been subtracted using the IRAF routine ellipse. The other multiply imaged system, \#3, constitutes a typical naked-cusp configuration, with three images on the same side of the cluster.

Arc spectroscopy
----------------

Spectroscopic observations of all lensing features were performed on March 6, 2005 using the Low Resolution Imager and Spectrograph [LRIS, @lris] of the Keck I telescope in multi-object spectroscopy mode. We used the 400/3400 grism, D560 dichroic, and 400/8500 grating centred at 6800Å to achieve continuous wavelength coverage from 3200 to 8000Å. The reduction of the data obtained in a total integration time of 10.8ks followed the standard steps of bias subtraction, slit identification and extraction, flat fielding, wavelength calibration, and flux calibration, and was performed using the <span style="font-variant:small-caps;">python</span> routines described by @kelson02. From the spectra thus obtained we measured redshifts of $z{=}2.84$ for the images labelled 1-2.1, 1-2.2 and 1-2.3; the redshift of features 3.1 and 3.2 was found to be $z=1.779$. The spectra for each component show identical features (Lyman$\alpha$ in emission for system 1-2, UV absorption lines in system 3), therefore they have been co-added on Fig. \[spectro\].

![Co-added spectra for system 1-2 at $z=2.84$ (*top*) and system 3 at $z=1.779$ (*bottom*).[]{data-label="spectro"}](f2a.jpg "fig:") ![Co-added spectra for system 1-2 at $z=2.84$ (*top*) and system 3 at $z=1.779$ (*bottom*).[]{data-label="spectro"}](f2b.jpg "fig:")

Galaxy spectroscopy {#velspec}
-------------------

Using the LRIS and DEIMOS spectrographs on the twin Keck-10m telescopes on Mauna Kea we conducted an extensive spectroscopic survey of galaxies in the field of MACSJ1423. Redshifts were measured for 396 objects, 60 of them within the solid angle covered by our HST/ACS observation. All redshifts were used to calibrate the photometric redshifts mentioned above. Using iterative $3\sigma$ clipping we find $z_{\rm cl}{=}0.5450$ and $\sigma{=}1320^{+93}_{-91}$ km s$^{-1}$ for the cluster redshift and velocity dispersion, respectively ($n_{\rm z}{=}140$, Fig. \[veldist\]). When limited to the ACS field of view, the same analysis yields $z_{\rm cl}{=}0.5424$ and $\sigma{=}1470^{+170}_{-120}$ km s$^{-1}$ ($n_{\rm z}{=}40$). Even for a system as X-ray luminous as MACSJ1423 [$L_{\rm X, bol}=3.7\times 10^{45}$ erg s$^{-1}$, @highzmacs], this velocity dispersion falls high on the $L_{\rm X}$-$\sigma$ relation [e.g. @mahdavi01], without, however, constituting a significant outlier.

Inspection of the galaxy velocity histogram (Fig. \[veldist\], top) shows the distribution to be mildly skewed toward lower redshifts; a one-sided Kolmogorov-Smirnov test finds it to be inconsistent with a Gaussian at the 2$\sigma$ confidence level. We investigate whether the two peaks at $z{\sim}0.531$ and $z{\sim}0.553$ represent subclusters along the line of sight by selecting three galaxy subsamples as indicated by the coloured bands. The distribution of the respective galaxies on the sky is shown in Fig. \[veldist\] (bottom). Unlike the galaxies with $z{\sim}0.553$, we find the galaxies in the peak around $z{\sim}0.531$ to be significantly (3.3$\sigma$) more concentrated in projection on the sky than the overall cluster galaxy population, suggesting a gravitationally bound substructure viewed almost exactly along the line of sight. We thus conclude that the velocity dispersion of MACSJ1423 is mildly inflated by a possible infall of a loose group at $z=0.531$, viewed in projection and superimposed directly onto the cluster core.

![ *Top:* Redshift histogram of galaxies in the field of MACSJ1423. The dashed line shows the best-fitting Gaussian ($z_{\rm cl}{=}0.5450$; $\sigma{=}1320^{+93}_{-91}$ km s$^{-1}$, $n_z{=}140$). The vertical dotted lines mark the $\pm 1\sigma$ range; the coloured bands are designed to separate radial velocities of potential fore- or background structure. *Bottom:* The distribution of selected cluster members on the sky (same colour scheme as in top figure: the main cluster galaxies are shown as green circles, and the populations of the two peaks are shown as filled diamonds). Galaxies with $z\sim 0.531$ (blue) are tightly concentrated and thus likely to constitute a bound galaxy group, falling into the cluster core along our line of sight. The black square outlines the area covered by our HST/ACS observation. The dashed lines delineate small areas that are not covered by our spectroscopic survey. The small black dots represent other galaxies with spectroscopic redshifts but outside the cluster. []{data-label="veldist"}](f3a.pdf "fig:") ![ *Top:* Redshift histogram of galaxies in the field of MACSJ1423. The dashed line shows the best-fitting Gaussian ($z_{\rm cl}{=}0.5450$; $\sigma{=}1320^{+93}_{-91}$ km s$^{-1}$, $n_z{=}140$). The vertical dotted lines mark the $\pm 1\sigma$ range; the coloured bands are designed to separate radial velocities of potential fore- or background structure. *Bottom:* The distribution of selected cluster members on the sky (same colour scheme as in top figure: the main cluster galaxies are shown as green circles, and the populations of the two peaks are shown as filled diamonds). Galaxies with $z\sim 0.531$ (blue) are tightly concentrated and thus likely to constitute a bound galaxy group, falling into the cluster core along our line of sight. The black square outlines the area covered by our HST/ACS observation. The dashed lines delineate small areas that are not covered by our spectroscopic survey. The small black dots represent other galaxies with spectroscopic redshifts but outside the cluster. []{data-label="veldist"}](f3b.pdf "fig:")

Distribution of Light & Mass
============================

Light Distribution
------------------

We use the K-band magnitudes of (photometric) cluster members to create a luminosity-density map of the cluster light. By fitting a 2D Sersic profile to the cD galaxy which dominates the K-band cluster light (we use the <span style="font-variant:small-caps;">galfit</span> software of @galfit), we measure an integrated magnitude of K[=]{}16.27 for the cD, compared to K[=]{}18.33 for the second brightest cluster member. The magnitude difference is K$_{\rm gap}{=}2.06$.

Lensing Analysis
----------------

In addition to the two spectroscopically confirmed strong-lensing features discussed above, we also take into account the weak-lensing signal as reflected in the sheared F814W images of the background galaxy population. Details of the weak-lensing analysis will be presented in a forthcoming publication (Jauzac et al., in prep.). In short, we follow the weak lensing pipeline developped by @alexie1 [@alexie2], but adapted the code to work on cluster field, in order to efficiently remove the galaxy cluster members and the foreground galaxies that would otherwise reduce the strength of the shear signal. In doing so, we consider only background sources located outside the Einstein radius of, for MACSJ1423, $R_{\rm E}=21\arcsec$ (here we refer to the effective Einstein for a source at $z=2$, defined as the angular radius from the centre of the cluster at which the average convergence equals 1), thereby minimising any systematic bias from the strong-shear regime [@masseySL]. The 192 galaxies thus selected within the ACS field (corresponding to a number density of 16 galaxies per square arcminute) represent 192 constraints in the weak-lensing analysis. Adding the 18 constraints from the two strongly lensed multiple-image systems yields a total of 210 constraints, all of which are used in order to build a parametric mass distribution within the framework of the publicly available <span style="font-variant:small-caps;">lenstool</span> software [@jullo07 http://www.oamp.fr/cosmology/lenstool/] in which we have implemented the inclusion of the weak lensing constraints. The optimisation is performed in the image plane.

To characterise the mass distribution in MACSJ1423 we adopt a dual Pseudo Isothermal Elliptical Mass Distribution [dPIE, @mypaperI; @ardis2218], parametrised by a fiducial velocity dispersion $\sigma$, a core radius $r_{\rm core}$ and a scale radius $r_s$ (set to 1000$\arcsec$). In addition to this large-scale mass component, which models both the intracluster gas and the far larger dark matter component [@bradac08a; @jesper], we also take into account the perturbations associated with individual cluster galaxies. Specifically, we include in the modelling all cluster members brighter than 22.5 (F814) and within a projected cluster-centric distance of 35$\arcsec$ from the cD galaxy. Empirical scaling relations (without any scatter) are used to relate their dynamical dPIE parameters (central velocity dispersion and scale radius) to their luminosity (the core radius being set to 0), whereas all geometrical parameters (centre, ellipticity, position angle) are set to the values measured from the light distribution [see, *e.g.* @mypaperIII; @a370]. In total our mass model is thus described by eight free parameters. The range of allowed parameter values characterising the dPIE mass component is limited by priors as follows. Position: $\pm\, 25\arcsec$ along the X and Y directions; ellipticity: ${<}0.6$ (expressed in units $(a^2-b^2)/(a^2+b^2)$, where $a$ and $b$ refers to the semi-major and semi-minor axes of the ellipse describing the mass distribution); position angle: 0–180 degrees; velocity dispersion: 500–1500 km$\,s^{-1}$; core radius: 1–20$\arcsec$. For the galaxy-scale component, we allow the velocity dispersion to vary between 100 and 250 kms$^{-1}$, whereas the scale radius was forced to be less than 70 kpc in order to account for tidal stripping of their dark matter haloes [see, *e.g.* @mypaperII; @mypaperIV; @priya4; @wetzel and references therein.]

A Uni-modal Mass Distribution
-----------------------------

    Component     $\delta x (\arcsec)$   $\delta y (\arcsec)$       $e$         $\theta$           $r$ ()           $\sigma$ ()
  -------------- ---------------------- ---------------------- ------------- --------------- ------------------ -------------------
   Cluster Halo       -1.0$\pm$1.2           0.0$\pm$1.0        0.4$\pm$0.1   116.0$\pm$1.7    78.7$\pm$25.2       1081$\pm$110
    Perturbers             -                      -                  -              -         32$^{+66}_{-30}$   141$^{+54}_{-21}$

The shear field constructed from our analysis of the images of the weakly lensed background galaxy population (Fig. \[fig1\]) already suggests a uni-modal mass distribution centred on the cD galaxy. Combining strong- and weak-lensing constraints, we confirm that a single cluster-scale mass component is able to reproduce the full set of observational constraints, with a reduced $\chi^2$ value of 175/202. The corresponding best-fit values of all model parameters are summarised in Table 1. We find the centre of the elliptical dPIE mass distribution to coincide with that of the cD galaxy; its position angle of 116$\pm$2 degrees matches the orientation of the cD galaxy (126$\pm$10 degrees). No additional large-scale mass component is required by the data.

Fig. \[fig2\] compares the 2D mass map corresponding to the best-fit lens model to the X-ray surface brightness as observed with *Chandra* (from @highzmacs) and to the K-band light map, both adaptively smoothed to 3$\sigma$ significance using <span style="font-variant:small-caps;">asmooth</span> [@asmooth]. The agreement between these independent observational tracers of the three cluster constituents (dark matter, gas, and galaxies) is remarkable.

![The distribution of mass, gas, and galaxies in the core of MACSJ1423, as reflected by the adaptively smoothed K-band luminosity of cluster galaxies (red), the total mass derived from the gravitational-lens model (black) and the adaptively smoothed X-ray surface brightness from *Chandra* observations (blue). []{data-label="fig2"}](f4.jpg)

Discussion & Conclusions
========================

Using optical and lensing observations of MACSJ1423 we find overwhelming evidence for a uni-modal mass distribution whose centre, ellipticity, and orientation are in excellent agreement with the corresponding parameters of the cluster light and intra-cluster gas distributions (Fig. \[fig2\]). We quantify the dominance of a single, central mass component by computing the substructure fraction, $f_{\rm sub}$, defined as the mass within a given cluster-centric radius that can be attributed to substructure, divided by the total mass within the same radius [@smithtaylor]. In the case of MACSJ1423, only cluster galaxies (the cD galaxy is by definition excluded) contribute to the substructure fraction. We find $f_{\rm sub}(<250\,{\rm kpc}){=}0.048 \pm$0.025, comparable to values measured for the most relaxed clusters at $z\sim0.2$ [@locuss]. Within the high-redshift MACS subsample, the substructure fraction of $f_{\rm sub}(<500\,{\rm kpc}){=}0.045\pm$0.025 of MACSJ1423 is in stark contrast to the value of 0.25$\pm$0.12 measured for MACSJ1149.5+2223, the most complex cluster lens studied to date [@1149].

The radial mass profile of MACSJ1423 as derived from our lensing analysis is well constrained across the full ACS field of view, i.e., out to a radius of 100(640kpc). Previous results for the mass within 65(415kpc) of 5.0$^{+3.1}_{-0.9}\times 10^{14}$ M$_{\sun}$, [@laroque03][^1] using observations of the Sunyaev-Zel‘dovich effect, and of 3.1$^{+0.9}_{-0.7} \times 10^{14}$ M$_{\sun}$ [3$\sigma$ confidence level, @schmidtAllen1423] from an analysis of the [*Chandra*]{} X-ray data, agree within the errors with our measurement of $4.35\pm0.6 \times 10^{14}$ M$_{\sun}$ (3$\sigma$). We note, however, that the existing X-ray analyses of MACSJ1423 assume spherical symmetry which will bias the resulting mass estimates low for a prolate matter distribution. The difference of $\sim$ 30% between the lensing and X-ray mass may thus be due to an elongation of the system along the line of sight, consistent with our finding of residual accretion onto the cluster core (Section \[velspec\]).

@locuss investigated the relationship between the effective Einstein radius and the strong lensing mass (measured within 250 kpc) for a sample of 20 strong lensing clusters at $z\sim0.2$ drawn from the Local Cluster Substructure Survey (LoCuSS). According to their statistical analysis, one would expect a mass of 2.55$^{+0.90}_{-0.65}$ 10$^{14}$ M$_{\sun}$ for an *undisturbed* cluster with an Einstein radius of 21$\arcsec$. The excellent agreement with our measurement of 2.46$\pm$0.2 10$^{14}$ M$_{\sun}$ underlines again the relaxed dynamical state of MACSJ1423.

Although our analysis of the radial velocity distribution of cluster members suggests that MACSJ1423 evolved through mergers along a line-of-sight filament (Section \[velspec\]), the rate of current or recent infall is low and does not result in a noticeable perturbation of the cluster core. In fact, the large luminosity gap of K$_{\rm gap}{=}2.06$ mag between the cD and the second brightest cluster galaxy, as well as the excellent match of the dark-matter, intra-cluster gas, and cluster light distributions, represent strong evidence for a dynamically highly evolved cluster core [@jones03; @Lgapcluster], undisturbed by major mergers for at least 1 Gyr, the characteristic time scale for relaxation via dynamical friction. The existence of MACSJ1423 only 7 Gyr after the Big Bang thus adds an important observational constraint to N-body simulations of structure formation and evolution in the current cosmological model.

Acknowledgements {#acknowledgements .unnumbered}
================

ML thanks the Centre National d’Etudes Spatiales (CNES) and CNRS for their support. ML est bénéficiaire d’une bourse d’acceuil de la Ville de Marseille. The Dark Cosmology Centre is funded by the Danish National National Research Foundation. HE gratefully acknowledges financial support from STScI grant GO-09722 and SAO grant GO3-4164X. AMS gratefully acknowledges a Royal Astronomical Society Sir Norman Lockyer Fellowship. GPS acknowledges support from the Royal Society and STFC. ACE and IRS acknowledge support from STFC. JR acknowledges support from an EU Marie-Curie fellowship. JPK acknowledges support from CNRS. TS acknowledges support from the Netherlands Organization for Scientific Research (NWO).

\[lastpage\]

[^1]: @laroque03 quote the mass within a three-dimensional radius, which we convert to a projected mass assumed a $\beta$-model with $b=2/3$.