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The Cosmological analysis of X-ray cluster surveys VII. Bypassing scaling relations with Lagrangian Deep Learning and Simulation-based inference

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that cluster number count cosmology can be performed without explicit mass-observable scaling relations, by replacing them with a fast simulation-based forward model whose nuisance parameters are astrophysical feedback…

desk verdict Genuine proof-of-concept for scaling-relation-free cluster counts via conditioned LDL and SBI, but the headline 'less degenerate' claim rests on an uncontrolled comparison and needs a same-engine test. read the letter →

arxiv 2507.01820 v1 pith:7ZCYNDOQ submitted 2025-07-02 astro-ph.CO

Nicolas Cerardi , Marguerite Pierre , François Lanusse , Xavier Corap This is my paper
classification astro-ph.CO
keywords galaxyclustersX-rayclustersurveysscalingrelationssimulation-basedinferenceLagrangianDeepLearningbaryonificationfeedbackparametersnumbercounts
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Every cluster count cosmological analysis so far connects observed X-ray properties to cluster masses through empirical scaling relations, whose many free parameters are degenerate with cosmology and carry no direct information about the non-gravitational processes inside clusters. This paper presents a proof of concept that replaces those scaling relations with a simulation-based forward model: a dark-matter-only simulation, accelerated by a machine-learned baryonification step, produces mock XMM observations of the intracluster medium, and the comparison to data is done with simulation-based inference rather than an analytic likelihood. The authors show that their model reproduces the cluster population of the calibration hydrodynamical simulations at the fiducial parameter values, that it can constrain supernova and AGN feedback parameters, and that the resulting posteriors on cosmology are less degenerate than those of the scaling-relation approach. If this holds with more realistic simulations, cluster abundance analyses would no longer need a separate empirical mass-observable calibration.

What carries the argument

The central object is the extended Lagrangian Deep Learning (LDL) baryonification model: a shallow, physics-inspired network that displaces dark-matter particles along a learned potential and then applies a nonlinear activation to paint baryonic fields. The extension conditions the LDL weights on cosmological and feedback parameters through a meta-learned multilayer perceptron, with redshift dependence handled by interpolation for temperature fields. Its role is to replace the analytic mass-observable relation with a differentiable surrogate of the hydrodynamical simulations, so that cluster observables respond to astrophysical parameters rather than to empirical coefficients. The inference chain then rests on a ResNet compressor that maps X-ray observable diagrams to six summary numbers and a mixture-density-network posterior estimator that directly samples the conditional posterior.

What would settle it

Apply the identical mock-generation and detection pipeline to an observed cluster sample like XXL or eROSITA, including Poisson photon noise, galactic absorption, backgrounds, and AGN contaminants. If the resulting observed (CR, HR, z) distribution cannot be reproduced by any values of the six varied parameters within the allowed ranges, or if the recovered posterior on ($\Omega_m$, $\sigma_8$) excludes the values already constrained by other probes, then the simulation-based model does not in fact bypass scaling relations.

Watch

Extended reading notes

Core claim

The central claim is that empirical scaling relations linking X-ray luminosity and temperature to mass can be bypassed in cluster number count cosmology, because a sufficiently fast simulation model can carry the same information more faithfully. The authors construct an extended Lagrangian Deep Learning emulator trained on CAMELS/IllustrisTNG that maps a fast particle-mesh dark-matter field to electron density and temperature fields, conditioned on two cosmological parameters and four feedback parameters. This feeds a pipeline that produces X-ray light cones in XMM count-rate and hardness-ratio observables, and a simulation-based inference chain (neural compression plus neural posterior estimation) recovers posteriors on all six parameters. The paper's quantitative findings are that the emulated clusters reproduce the count-rate-mass relation of the hydrodynamical simulations including per-cluster deviations (correlation 0.88), that supernova feedback parameters are recoverable while AGN parameters are barely constrained by the chosen observables, and that the simulation-based model shows a much weaker degeneracy between $\Omega_m$ and $\sigma_8$ than the scaling-relation baseline.

Load-bearing premise

The approach assumes that the IllustrisTNG subgrid feedback model, varied through its four adjustable parameters, describes the real intracluster medium closely enough to replace empirically calibrated scaling relations.

Editorial extensions

If this is right

  • Cluster number count analyses can be run with a forward model that carries no explicit scaling relations, shifting the nuisance parameters to physical feedback parameters.
  • The $\Omega_m$-$\sigma_8$ degeneracy is substantially weakened when feedback parameters are varied in the simulation-based model; the paper's figure 10 shows broadening only transverse to the degeneracy.
  • The same parametrisation works across summary statistics (CR-HR-z, CR-HR, CR-z, z alone) with only mild loss of constraining power, suggesting the model's nuisance parameters are more universal than scaling-relation coefficients.
  • The emulator recovers not just the mean count-rate-mass relation but part of the individual cluster scatter (correlation 0.88 with simulation residuals), meaning it can attribute why a cluster is over- or under-luminous at fixed mass.
  • With sufficient resolution and volume, the approach could extend from CR-HR-z diagrams to full spatial mapping CR-HR-z-x-y, adding clustering-like information.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural next test is to calibrate the feedback parameters on cluster-specific X-ray observables rather than galaxy-scale star formation, since the paper notes the CAMELS tuning is not optimized for cluster X-ray properties.
  • If the view that simulations carry implicit scaling relations is right, the same LDL machinery could be inverted to build emulated scaling relations as explicit functions of cosmological and feedback parameters, giving physically motivated priors for classical analyses.
  • The reduced degeneracy reported here is measured on simulated data with a known detection model; applying the identical forward model to real surveys with contaminants and noise would test whether the advantage survives real selection effects.
  • The weak sensitivity of the emulator to AGN parameters suggests that upcoming CAMELS versions with more AGN parameters may change the posterior landscape, so the current claim of non-degeneracy is tied to the present parameter set.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper presents a proof-of-concept simulation-based forward model for galaxy cluster number counts that bypasses explicit empirical scaling relations. The authors extend the Lagrangian Deep Learning (LDL) baryonification framework to emulate ICM electron density and temperature from fast particle-mesh dark-matter-only simulations, conditioned on two cosmological parameters (Omega_m, sigma_8) and four IllustrisTNG feedback parameters (AAGN1, AAGN2, ASN1, ASN2), using the CAMELS/IllustrisTNG simulation suite for training. They build a pipeline that produces mock XMM X-ray lightcones and observable diagrams (CR, HR, z), then perform simulation-based inference via a neural compressor (ResNet) and neural posterior estimation (NPE). The LDL emulator is validated on held-out CAMELS CV simulations at the fiducial parameters, reproducing the cluster CR-M relation and partially capturing individual cluster deviations. The paper then compares the resulting 6-parameter posterior with a Fisher-based analysis of an analytical scaling-relations model with 8 free parameters, claiming the simulation-based model is less degenerate. The authors explicitly frame the work as a proof of concept and discuss several limitations, including the 40% discrepancy in number counts relative to the analytical model and the need for more realistic simulations before applying the method to real data.

Significance. The methodological contribution is substantial: a fast, physics-based emulator that maps dark matter fields to X-ray cluster observables without an explicit scaling-relation likelihood is a promising direction for cluster cosmology. The LDL validation results (Figure 6) are strong: the emulator reproduces the CR-M relation and, importantly, the deviations of individual clusters from the mean relation are highly correlated with the hydrodynamical simulations (rho=0.88), indicating that the surrogate captures information beyond a simple power law. The reported speed (about 100 seconds per 200 deg2 survey) is a practical enabler for simulation-based inference. The paper is transparent about its limitations, including the injected information from the subgrid feedback model (Section 5.1) and the lack of external validation. However, the headline claim that the simulation-based model is 'less degenerate' than the scaling-relations approach is not yet robustly established, as the comparison in Figures 11 and 12 mixes different inference engines, parameter counts, and prior treatments.

major comments (3)
  1. [Section 5.1, Figures 11 and 12, Table 2] The central claim that the simulation-based model is less degenerate than the scaling-relations approach rests on comparing a 6-parameter NPE posterior (Figure 11) with an 8-parameter Fisher covariance (Figure 12) that uses a different parameterization (2+4 vs 2+6 free parameters) and different inference machinery. NPE contours encode prior volume and possible network bias, while Fisher contours are local Gaussian curvature around a fiducial; their correlation structures are not directly comparable. Part of the apparent reduction in degeneracy is therefore a bookkeeping effect of the lower number of nuisance parameters. I recommend a same-engine, same-dimensionality test, for example an NPE posterior for the analytical scaling-relations model (or a Fisher forecast for the simulation-based model) using the same summary statistic and the same number of free parameters, before the abstract and Section 5.1 claims are made.
  2. [Appendix B, Figure B.1] The rank distribution for Omega_m shows a clear negative slope, indicating a biased posterior, which the authors attribute to the MSE loss in the neural compressor (Section 3.1). Because the headline comparison concerns the size and orientation of the cosmological contours, a biased posterior is not a reliable basis for quantifying reduced degeneracy. The authors should quantify the impact of this bias on the claimed constraints, or replace the MSE loss with a loss that yields approximately unbiased summaries (e.g., score-based compression), and repeat the comparison.
  3. [Section 5.1, last paragraph] The paper states that assuming the feedback models are correct 'introduces a lot of information in the inference.' This is a central caveat: the reduced degeneracy between cosmology and feedback is achieved by substituting the physical priors of the IllustrisTNG subgrid model (through the four varied feedback parameters) for the empirical priors of scaling-relation coefficients. The paper also validates the emulator on held-out CAMELS boxes that share the same subgrid physics as the training data. The claim of 'less degenerate' should therefore be explicitly qualified as conditional on the chosen feedback model, and the conclusions should state that external validation against simulations with different feedback prescriptions (or against observed scaling relations) is needed before the claim can be generalized.
minor comments (4)
  1. [Section 2.2.2] In 'that we coin Θ fid', the relative pronoun should be 'which' rather than 'that'.
  2. [Section 3.1] The phrase 'a choice that can lead to biased posterior' should be 'a choice that can lead to biased posteriors'.
  3. [Section 4.3] In 'neither used for the ResNet training nor its testing', the last part should read 'nor for its testing'.
  4. [Table 2] For the analytical model, the six free nuisance parameters (e.g., M0, L0, alpha_MT, gamma_MT, alpha_LT, gamma_LT) are not explicitly named in the text or table; listing them would make the comparison with Figure 12 easier to follow.

Circularity Check

3 steps flagged · score 6.0 of 10

Central 'less degenerate' claim is partly built from parameter-count asymmetry, injected feedback-model information, and calibration-simulation validation; no self-citation chain is load-bearing.

  1. self definitional [Section 5.1, Figs. 11-12; Table 2]
    "We show in figure 11 and 12 the full posteriors for the simulation-based and the analytical model, respectively obtained through the NPE and Fisher analysis methods."

    The headline 'less degenerate' conclusion is obtained by directly comparing a 6-parameter NPE posterior (Table 2: 'This work, simulation-based model ... 2+4') with an 8-parameter Fisher covariance (Table 2: 'This work, analytical model ... 2+6'). Adding two extra nuisance dimensions to the analytic model guarantees extra elongated directions in the Fisher matrix, so part of the claimed reduction in degeneracy is a bookkeeping effect of the chosen parameterizations. The comparison is also between different statistical objects: the NPE posterior encodes the prior and is trained on the simulation-based XODs, while the Fisher matrix is a prior-free local curvature estimate. The 'less degenerate' result is thus partly defined by the comparison design rather than measured independently.

  2. self definitional [Section 5.1]
    "This is the consequence of our approach being based on feedback prescriptions describing the physical processes within the ICM. Our model carries inherent physical constraints on cluster observable properties. In this proof of concept, we assume that feedback models are correct (within the freedom given to the four feedback parameters), which introduces a lot of information in the inference."

    The paper explicitly attributes the reduced degeneracy to the physical constraints inherited from the assumed feedback model. But those constraints are injected as an assumption, not learned from external data: the XODs used for training and testing are generated by the same CAMELS/IllustrisTNG-calibrated LDL pipeline, and the NPE prior is encoded in the final posterior. The claim that the simulation-based model is 'less degenerate' therefore reduces, at least in part, to the information that was put in by assuming the feedback prescription is correct, rather than a gain established against an equally parameterized and equally conditioned reference model.

1 more flagged steps
  1. fitted input called prediction [Abstract; Section 4.2, Fig. 6]
    "Our model correctly reproduces the cluster population from the calibration simulations at the fiducial parameter values, and allows us to constrain feedback mechanisms."

    The 'calibration simulations' are the same CAMELS CV/LH boxes used to train the LDL emulator and the downstream simulation-based pipeline. Figure 6 validates the LDL cluster population against the CV test split of those calibration simulations, which is an internal consistency check of the surrogate rather than an out-of-sample prediction. Likewise, the feedback constraints come from NPE posteriors trained and tested on XODs generated by the calibrated simulator, so they encode the assumed CAMELS subgrid feedback model rather than independent observational data. The abstract presents this calibration fit as a predictive result, which is circular in wording even if the technical validation is standard.

full rationale

The technical core of the paper is not broadly circular: the LDL baryonification is a physics-inspired surrogate trained on CAMELS with a held-out CV test set; the XOD pipeline is a genuine forward model; and the NPE validation uses XODs unseen by the compressor and density estimator, with rank checks in Appendix B. No self-citation chain is load-bearing: Cerardi et al. (2024) and Kosiba et al. (2024) are cited for standard Fisher and SBI machinery, not to forbid alternatives. The circularity burden falls on the headline comparative claim and on the abstract's validation wording. First, the 'less degenerate' result in Figs. 11-12 compares a 6-parameter NPE posterior with an 8-parameter Fisher covariance, so the extra analytic nuisance parameters guarantee additional degeneracy directions; Fisher and NPE are also different objects, with the NPE explicitly carrying the prior. Second, Section 5.1 openly states that assuming the feedback model is correct 'introduces a lot of information in the inference,' which means the 'inherent physical constraints' removing degeneracies are inputs, not measured outputs. Third, the abstract claims the model 'reproduces the cluster population from the calibration simulations,' which is a fit to the training simulation suite evaluated on its held-out split, not a prediction against external data. Appendix B's negative rank slope for Omega_m further weakens the evidential value of the NPE contour size. The paper is transparent about these limitations, but the central 'less degenerate' claim is partially constructed from the chosen parameterization and injected physical assumptions rather than established by a like-for-like comparison.

Assumptions & free parameters 8 free parameters · 8 assumptions · 0 invented entities

The central claim rests on the trained LDL emulator and the assumption that the IllustrisTNG subgrid feedback model is a valid description of the ICM. Several pipeline choices (metallicity, flux cut, voxel resolution, PM approximation) are fixed by hand and enter the simulated observables. The comparison baseline adds external assumptions (Tinker HMF, Pacaud scaling relations). No new physical entities are introduced.

free parameters (8)
  • LDL learnable parameters Theta = (gamma, alpha, Xi, b0, b1, mu) = not reported
    Trained on CAMELS/IllustrisTNG CV and LH simulations (Eqs. 1-4). The baryon field emulator that replaces scaling relations depends on these values.
  • MLP weight offsets deltaTheta(theta_sim) = not reported
    Meta-learned mapping from simulation parameters to LDL weight variations (Eq. 7); conditions the emulator on cosmology and feedback.
  • ResNet compressor weights = not reported
    Trained on 32,000 XODs to compress catalogs to six numbers; the inference statistic depends on this compression.
  • MDN weights phi = not reported
    Trained with neural posterior estimation to approximate the posterior; the posterior shapes depend on these weights.
  • Fixed metallicity Z = 0.3 Z_sun = 0.3 Z_sun
    Chosen as a fixed input for pyatomDB emissivity modeling (section 2.3); affects count rates.
  • Flux cut CR > 0.02 c/s = 0.02 c/s
    Chosen to match the XXL C1 sample density; determines the detected cluster population.
  • Voxel resolution 0.39 h^-1 Mpc = 0.39 h^-1 Mpc
    Fixed working resolution for CIC deposition; affects all emulated fields and cluster detection.
  • Smoothing exponent n in O_s(k) = 1 + k^-n = not reported (prescription from Dai & Seljak 2021)
    Used in the loss functions (Eqs. 5-6); shape of the smoothing operator.
assumptions (8)
  • domain assumption IllustrisTNG subgrid implementations of AGN and SN feedback are a valid description of ICM physics
    The entire simulation-based model and its posterior constraints inherit the feedback model assumptions (section 2.1).
  • domain assumption Training on CV50 and LH25 boxes at z=0.21 generalizes to the lightcone range 0.1<z<0.5 and to parameter variations
    The LDL is trained on (50 h^-1 Mpc)^3 and (25 h^-1 Mpc)^3 boxes; the lightcones are built from repeated boxes, and T is redshift-conditioned via interpolation.
  • domain assumption The PM DM approximation from JaxPM is adequate for baryon field emulation
    The emulator is trained on PM-resimulated DM fields and must also correct the PM smoothing (section 2.1).
  • domain assumption X-ray emissivity can be computed from n_e and T with fixed metallicity and bremsstrahlung via pyatomDB
    Count rates and hardness ratios are derived from this emission model (section 2.3).
  • domain assumption The simple sep-based detection and characterization (no Poisson noise, no AGN contaminants, no galactic absorption) is sufficient for the proof-of-concept
    Detection mask integration defines the observed catalog; the paper acknowledges this is simplified (sections 2.3, 5.4).
  • standard math The Fisher matrix with independent Poisson bins is a valid approximation for the analytical baseline
    Equation (8) assumes Poisson independence and Gaussian likelihood; used to produce the comparison contours in figure 12.
  • domain assumption The NPE prior is the uniform distribution over the simulated parameter ranges
    The amortized posterior encodes the training distribution; prior effects are not separated in the comparison.
  • domain assumption Tinker et al. (2008) HMF and Pacaud et al. (2018) scaling relations are the correct baseline for the analytical model
    The analytical forward model used for comparison is built on these external calibrations (section 4.1).

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Cite this review

Pith. "Pith review of The Cosmological analysis of X-ray cluster surveys VII. Bypassing scaling relations with Lagrangian Deep Learning and Simulation-based inference." pith.science (2026). https://pith.science/paper/7ZCYNDOQ

@misc{pith2026250701820,
  author       = {Pith},
  title        = {Pith review of: The Cosmological analysis of X-ray cluster surveys VII. Bypassing scaling relations with Lagrangian Deep Learning and Simulation-based inference},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7ZCYNDOQ}},
  note         = {Machine review of arXiv:2507.01820}
}
read the original abstract

Galaxy clusters, the pinnacle of structure formation in our universe, are a powerful cosmological probe. Several approaches have been proposed to express cluster number counts, but all these methods rely on empirical explicit scaling relations that link observed properties to the total cluster mass. These scaling relations are over-parametrised, inducing some degeneracy with cosmology. Moreover, they do not provide a direct handle on the numerous non-gravitational phenomena that affect the physics of the intra-cluster medium. We present a proof-of-concept to model cluster number counts, that bypasses the explicit use of scaling relations. We rather implement the effect of several astrophysical processes to describe the cluster properties. We then evaluate the performances of this modelling for the cosmological inference. We developed an accelerated machine learning baryonic field-emulator, built upon the Lagrangian Deep Learning method and trained on the CAMELS simulations. We then created a pipeline that simulates cluster counts in terms of XMM observable quantities. We finally compare the performances of our model, with that involving scaling relations, for the purpose of cosmological inference based on simulations. Our model correctly reproduces the cluster population from the calibration simulations at the fiducial parameter values, and allows us to constrain feedback mechanisms. The cosmological-inference analyses indicate that our simulation-based model is less degenerate than the approach using scaling relations. This novel approach to model observed cluster number counts from simulations opens interesting perspectives for cluster cosmology. It has the potential to overcome the limitations of the standard approach, provided that the resolution and the volume of the simulations will allow a most realistic implementation of the complex phenomena driving cluster evolution.

Figures

Figures reproduced from arXiv: 2507.01820 by the authors.

Figure 1
Figure 1. Training stages of the extended LDL. We first train the base LDL parameters on the large volumes available for the fiducial model (CAMELS/CV), at z = 0.21. We then condition the LDL parameters on the cosmological and astrophysical parameters, using the numerous boxes from the CAMELS/LH set. While the ne model performs equally at all redshifts, the T emulator has to be retrained separately at all available redshifts … view at source ↗
Figure 2
Figure 2. Scheme of the extended LDL. Blue rectangles denote quanti￾ties and green ovales denote transformations. The bottom raw is the base LDL from Dai & Seljak (2021). Our extensions allow the baryon pasting to be conditioned on simulation parameters as well as on the redshift. 2024; Jeffrey et al. 2020). As we do not apply the posterior es￾timation on real data, and are primarly interested by the size of the constraints, … view at source ↗
Figure 3
Figure 3. X-ray mock examples generated with the extended LDL. Each row consist of the same light cone, with three boxes individually projected (left and middle), and the full projected lightcone with its 12 simulations boxes (right). The CR corresponds to a XMM-Newton EPIC/mos1+mos2+pn count-rate (without Poisson noise), in the band [0.5-2] keV. 200 400 600 800 1000 1200 1400 radial distance - cMpc h 1 0 50 100 150 200 250 T… view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Lightcone profiles for Ωm = 0.1, 0.3, 0.5 (resp. black, green, blue), intersected with the 20 deg2 observed region (dashed red cone). The 12 adjacent periodic boxes (dashed rectangles) in each lightcone share the same comoving dimensions. Also shown with stars are the …
Figure 5
Figure 5. Figure 5: Relation between the electron number density ne and the DM density for the hydrodynamical simulations (blue crosses) and the LDL surrogate (red crosses), for the fiducial parameters and at z = 0.21. The blue and red lines denote the mean locus (respectively for the hyd…
Figure 6
Figure 6. Figure 6: CR − M scaling relation at z = 0.21 in the CAMELS/IllustrisTNG simulations and in their LDL surrogate, for the fiducial model. These plots are made with the CV test set, 6 boxes of volume (50 h −1Mpc)3 each. Left: direct comparison of the hydrodynamical simulated clust…
Figure 7
Figure 7. Figure 7: Left: Redshift distribution of cluster counts for the traditional forward model (with explicit scaling relations, orange histogram), and for the simulation-based model (with the LDL, blue histogram), here for a 50 deg2 survey extending from z = 0.1 to z = 0.5. The shad…
Figure 8
Figure 8. Figure 8: Accuracy of the ResNet regression for inferring Ωm, σ8, AS N1, AAGN1, AS N2, AAGN2 from simulation-based XODs. The color and contours show the shape of the density of points (arbitrary colour scale, all densities are normalized). The dashed line show the 1:1 line (no e…
Figure 9
Figure 9. Figure 9: NPE performed at different points in the parameter space. XODs are produced from 200 deg2 X-ray surveys, selecting clusters with the flux cut CRlim = 0.02c/s. The posteriors (blue contours) are obtained with the NPE for XODs unseen during the ResNet nor NPE trainings. …
Figure 10
Figure 10. Figure 10: Posteriors estimated from a fiducial XOD for different condi￾tioning of the NPE. The XOD is produced from a 200 deg2 X-ray sur￾vey, selecting clusters with the flux cut CRlim = 0.02 c/s. In plain blue is shown a posterior marginalized on the feedback parameters, while…
Figure 11
Figure 11. Figure 11: Posteriors on the cosmological and astrophysical parameters, using the simulation-based forward model and the NPE inference method. The tested diagram x0 is drawn from the fiducial model. XODs are produced from 200 deg2 X-ray surveys, selecting clusters with the flux …
Figure 12
Figure 12. Figure 12: Posteriors on the cosmological and nuisance parameters, using the analytical forward model and the Fisher analysis, with explicit scaling relations. We observe that many pairs of parameters are degenerate (elongated contours), denoting the ill-parametrization of this …
Figure 13
Figure 13. Figure 13: Cosmological posteriors obtained with the simulation-based forward model and the NPE. We trained the NPE for several statistics: the full (CR,HR,z) XOD (blue contours), the (CR,HR) number counts (dashed green), the (CR,z) number counts (dashed purple) and only the z d…
Figure 14
Figure 14. Figure 14: CR-M relation coefficients as a function of the simulation pa￾rameters, for large volumes emulated at z = 0.21. The x-axis varies the AS N1 feedback parameter, while the line colour denotes a change in AS N2. The top and bottom panel respectively show the variations o…

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