{"id":"9beb257e-e088-4987-8923-a974435fa525","arxiv_id":"2505.04505","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Core stalling in N-body simulations scales as the inverse square root of the particle number, pointing to granular force noise rather than resonant buoyancy as the cause.","lead":"Simulations show that a heavy object sinking into a galaxy stalls at a radius that shrinks as the inverse square root of the particle number, in both cored and cuspy models. This points to random force fluctuations, not orbital resonances, as the dominant cause of core stalling in current N-body simulations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The high-N extrapolation and the granularity mechanism rest on the delta-correlated Holtsmark noise in Eq. (14), which the paper itself calls a rough approximation; a finite correlation time could shift the friction-noise balance and the N^-1/2 law.","rationale":"The reader's verdict is conditional, and my review supports that condition. The strongest independent evidence is the direct N-body scaling in Figs. 6-7 over two decades of N, together with the data collapse and the empirical agreement of P(delta f) with the Holtsmark form in Fig. 2. The weakest point is not the existence of the N^-1/2 trend at moderate N but its use as a quantitative extrapolation to N~1e13 and as the basis for the proposed reconciliation with the resonance-based stalling picture. Equation (14) treats the granular force as white noise, and the paper itself flags this as an approximation in Sec. 2.3. Since the stalling radius in a core is controlled by the time-integrated noise power rather than by the one-point distribution of delta f, the high-N stochastic predictions are sensitive to the assumed correlation time. A secondary issue, also noted by the authors, is that at N=3e6 the N-body stalling radius is close to the softening length, so the direct runs cannot validate the high-N extrapolation either. Neither issue invalidates the moderate-N result; both argue for keeping the paper conditional until a colored-noise or direct smaller-softening check is done. I therefore recommend UNCHANGED rather than an upgrade to full acceptance.","tokens_in":19888,"tokens_out":13376,"duration_ms":134027,"concrete_test":"Rerun the stochastic simulations of Sec. 3.3 with the delta-correlated Holtsmark noise replaced by an Ornstein-Uhlenbeck process whose correlation time is set to the local dynamical time (or a few crossing times), with the noise variance matched to the measured P(delta f) from the N-body snapshots of Fig. 2. Compare the resulting rs(N) for gamma=0 and gamma=1 over 1e4 <= N <= 1e6 with Fig. 6 and over 1e6 <= N <= 1e13 with Fig. 10. If the colored-noise curves depart from the white-noise curves by more than the scatter of the N-body points, the high-N N^-1/2 extrapolation and the claimed slope changes are model artifacts; if they agree, the delta-correlated approximation is adequate for the stalling-radius prediction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central N^-1/2 law is directly supported by the N-body runs of Figs. 6-7 up to N=3e6, and the force-fluctuation histograms in Fig. 2 give independent support for the Holtsmark form. What is not directly supported is the extrapolation to N~1e13 and the attribution of that extrapolated behavior to a Poissonian noise-friction balance: both rest on Eq. (14), in which the force fluctuation is drawn independently at each timestep from a locally normalized Holtsmark distribution. The authors explicitly concede in Sec. 2.3 that force fluctuations cannot be delta-correlated, since density fluctuations must integrate to zero, and that the delta-correlated treatment is a rough approximation. The white-noise idealization matters precisely for the quantity that sets the stalling radius in a harmonic core: the integrated noise power integral of the force autocorrelation. A finite correlation time changes this integral without changing the one-point Holtsmark distribution plotted in Fig. 2, so the stochastic rs(N) curves in Fig. 10 and the slope changes near N~1e7-1e9 could be artifacts of the Mannella integrator rather than properties of the N-body dynamics. Direct N-body validation stops at N=3e6, where the paper reports rs is greater than or approximately equal to the softening length (Sec. 3.2); the N-body points therefore cannot distinguish white-noise from colored-noise predictions at the N where the extrapolated curves change slope. The cited Pogorelov and Kandrup and Terzic and Kandrup checks address strongly interacting systems at low N and do not test this specific friction-noise balance at high N.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the stalling of a massive tracer subject to dynamical friction in spherical stellar systems, using N-body simulations (N=10^4 to 3x10^6) and single-particle Langevin integrations. The authors report that the stalling radius scales as rs ∝ rc N^{-1/2} at fixed mt/M in both cored (γ=0) and cuspy (γ=1) models, that the same scaling holds in self-consistent and non-interacting-tracer runs, and that a Langevin model with locally normalized Holtsmark force fluctuations reproduces the trend and allows extrapolation to N~10^13. They interpret core stalling as a granularity-induced balance between friction and force fluctuations, relating rs to the known wander-radius scaling, and argue that the result is compatible with, rather than contradictory to, resonance-based explanations of core stalling.","tokens_in":20187,"tokens_out":13562,"duration_ms":140989,"significance":"If the central N-body result holds, it provides a concrete, quantitative explanation for the long-debated resolution dependence of core stalling and connects it to the known wander-radius scaling of a heavy particle in a granular potential. The paper's direct N-body measurements over two decades in N (Figs. 6 and 7) are a valuable contribution, and the empirical validation of the Holtsmark form of force fluctuations in Fig. 2 is a genuine strength. The comparisons between self-consistent and non-interacting-tracer runs also help isolate the role of the wake. However, the high-N extrapolation and the proposed noise-friction balance interpretation rest on the delta-correlated approximation in Eq. (14), which the authors themselves describe as a rough approximation, so the extrapolated slope changes should be regarded as tentative until the colored-noise issue is addressed.","major_comments":[{"comment":"The extrapolation of rs(N) to N~10^13, including the slope changes around N~10^7-10^9, is entirely based on the Langevin equation with delta-correlated Holtsmark noise. The paper explicitly states in Section 2.3, in the paragraph following the description of the Mannella integrator, that force fluctuations cannot be completely delta-correlated because the space-integrated density fluctuations integrate to zero and a correlation time tau_c would be needed. This is not a cosmetic caveat: in the near-harmonic core the stalling radius is controlled by the integrated autocorrelation of the noise, not only by the one-point distribution plotted in Fig. 2. The cited works by Pogorelov & Kandrup and Terzic & Kandrup are not shown to cover this parameter regime, so they do not license the quantitative slope changes in Fig. 10. I recommend implementing a colored-noise variant with tau_c varied over the relevant range, or deriving an analytic correction to the white-noise limit, before the high-N predictions are presented as results rather than as extrapolations under an acknowledged approximation. The paper's own statement that the origin of the slope change 'remains unclear' and could be numerical reinforces this concern.","section":"Section 2.3, Eq. (14), and Fig. 10"},{"comment":"The direct N-body scaling is the paper's central claim, but at the high-N end the measured rs is only marginally above the softening length; the text reports rs > epsilon for all N, and Section 3.3 notes that at N~10^8 the associated rs is around epsilon. The softening length therefore provides a conceivable floor that could bias the fitted N^{-1/2} slope. The paper mentions runs with varying epsilon but does not show them, and Figs. 6 and 9 have no error bars from realization-to-realization scatter or from the time-averaging window used to define rs. Please add (i) a quantitative demonstration that rs(N) at fixed N is insensitive to epsilon for epsilon values below the standard 2x10^-3 rc, at least for the highest N shown, and (ii) error bars on rs in Figs. 6 and 9. Without this, the apparent N^{-1/2} trend at the high-N end is not fully distinguishable from a softening floor.","section":"Section 3.2, Figs. 6 and 7"},{"comment":"The claim that the super-Chandrasekhar regime appears only for N ≲ 3x10^5 is not supported by any quantitative criterion in the paper. From Fig. 7 alone the reader cannot determine how 'super-Chandrasekhar' is defined, for example whether it means that the actual infall time is shorter than the local-Maxwellian Chandrasekhar prediction by some stated factor, and with which choice of bmax and velocity distribution the comparison is made. Because this claim is used to argue that the super-Chandrasekhar phase is a low-resolution artifact, it needs a reproducible metric and a plot of that metric versus N for both gamma=0 and gamma=1.","section":"Section 3.2, paragraph beginning 'Aiming at comparing with the idealized Chandrasekhar DF'"},{"comment":"The proposed physical interpretation via a balance between the Holtsmark force scale alpha^{2/3} and the drag eta v_c is not quantitatively consistent with the reported N^{-1/2} scaling. For gamma=0 in the harmonic core, alpha^{2/3} is proportional to m^{1/3} and hence to N^{-1/3} at fixed M and rc, while eta v_c is nearly N-independent for mt/M=10^-3 and N>=10^4; the crossing radius in Fig. 12 would then scale as N^{-1/3}, not N^{-1/2}. The N^{-1/2} law instead matches the wander-radius scaling rwan proportional to sqrt(D/eta) with D proportional to m, i.e. a diffusion-coefficient balance rather than a typical-force balance. Please state clearly which of these two criteria the stochastic integrations actually determine, and reconcile the crossing-point discussion in Fig. 12 with the measured rs(N).","section":"Section 3.3, Figs. 11 and 12"},{"comment":"The extrapolated results are presented without error bars or realization statistics, and the stochastic runs are not described in enough detail for the reader to assess convergence. In particular, the number of independent stochastic realizations per N, the time step used, and the criterion for assigning rs from the integrated trajectories should be reported, because Fig. 10 is the only evidence for the high-N slope changes and the claimed N-independence of rs in cored models above N~10^9.","section":"Section 3.3, Fig. 10 and surrounding text"}],"minor_comments":[{"comment":"The paper writes log for the Coulomb logarithm in Eq. (22) but ln in Eq. (1); the notation should be unified to avoid ambiguity about the base.","section":"Section 2.3, Eq. (22)"},{"comment":"There is a typo in the phrase 'via stronger field particles ejection (see Fif. 4)', which should read 'Fig. 4'.","section":"Section 3.1 after Fig. 4"},{"comment":"The statement that the test particle mass ranges from 2x10^-4 to 10^-2 M is followed by figures using intermediate values; it would be helpful to state explicitly in Fig. 8 which mass values are shown and how many realizations were used for each curve.","section":"Section 2.2"},{"comment":"The sentence 'observing the onset of a resonance-driven core stalling in N-body experiments remains a challenging task' is useful, but the proposed test with a distribution function composed entirely of circular orbits should be described more concretely, since such a distribution would not be self-consistently stable and the paper does not say how the test would be carried out in practice.","section":"Section 4, Outlook"}],"recommendation":"major_revision","confidential_remarks":"The main new measurement, the N^{-1/2} scaling in direct N-body runs, is worth publishing, but the stochastic-model extrapolations and the softening-floor issue need to be settled before acceptance. I do not see a novelty disclosure problem; the relationship to Sartorello et al. (2025) and Trani & Di Cintio (2025) is acknowledged, and the overlap is handled transparently."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a serious numerical paper that finds a clean N^-1/2 scaling for the core-stalling radius, and that scaling is measured directly in N-body runs, not just from the stochastic model. The resolution-dependent part of core stalling is real in their setup. The weak point is the extrapolation beyond N=3e6, which does lean on a white-noise approximation the authors themselves say is rough.\n\nWhat's new: previous cored-stalling literature treated stalling as roughly N-independent and resonance-driven. Here they run cored and cuspy gamma models from N=1e4 to 3e6, both self-consistently and with non-interacting tracers, and get rs ∝ N^-1/2 with a good data collapse. The stochastic Langevin runs with Holtsmark fluctuations reproduce the trend, and Figure 2 shows the local force-fluctuation distribution really is close to Holtsmark. That is real evidence, not curve fitting. The wander-radius analogy is suggestive and they don't oversell it.\n\nSoft spots, in order. First, the N-body constraint only covers about a decade and a half in N, and at the high end rs is only about an order of magnitude above the softening length (epsilon = 2e-3 rc), so the cleanest part of the scaling law is at the low-N end. Error bars on rs are absent; the data collapse in Fig. 7 is visually convincing, but a few realizations and a scatter estimate would help. Second, the extrapolation to N~1e13 in Fig. 10 is the part that decides whether this settles the debate with dynamical buoyancy, and that part depends entirely on Eq. (14) with delta-correlated Holtsmark noise. The authors explicitly note that the fluctuations cannot be fully delta-correlated because density fluctuations integrate to zero. A finite correlation time could change the noise power integral and shift the predicted rs in the high-N region. The cited Pogorelov/Kandrup tests are for different physical regimes. So the N^-1/2 law for the simulated range stands on its own; the claim that this mechanism accounts for stalling in real galaxies (or even N=1e13) is a plausible extrapolation, not a measured result. Third, the \"super-Chandrasekhar onset at N less than about 3e5\" statement is asserted without a plot.\n\nThe paper is honest about its own limitations; the conclusions explicitly reconcile with Banik & van den Bosch rather than claiming to kill the resonance picture. That's the right tone. Who it's for: anyone working on sinking satellites or BHs in N-body simulations, and anyone building subgrid DF prescriptions. It deserves a serious referee. I'd send it out, ask for the extrapolation to be clearly labeled as model-dependent, for error bars or at least realization scatter, and for a softening-convergence test at fixed N. The central scaling is credible and useful.","headline":"A solid N-body study that directly measures rs ∝ N^-1/2 and credibly ties core stalling to granularity, though the high-N extrapolation rests on a white-noise approximation the authors themselves flag as rough.","tokens_in":20855,"tokens_out":2884,"would_cite":true,"duration_ms":27903,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that core stalling—the premature halting of a massive object's orbital decay near the center of a galaxy or cluster—is caused by force fluctuations from the system's finite particle number, so the stalling radius shrinks…","keywords":["dynamical friction","core stalling","N-body simulations","force fluctuations","Holtsmark distribution","Langevin equation","gamma-models","wander radius"],"falsifier":"Run tracer-only simulations in a cored $\\gamma=0$ model at fixed $m_t/M=10^{-3}$ with $N=10^7$ and $10^8$, at a fixed softening below the predicted $r_s$, with two timesteps differing by a factor of two; if $r_s$ follows the $N^{-1/2}$ line in both, the granularity mechanism holds, while a plateau at the $r_s\\approx r_c/2$ level would favor the resonance-buoyancy explanation. Separately, measure the force autocorrelation time $\\tau_c$ in an $N$-body snapshot sequence and check whether adding it to the stochastic equation moves $r_s$ beyond the run-to-run scatter.","tokens_in":19617,"feed_emoji":"🌌","tokens_out":13181,"duration_ms":120276,"temperature":0.7,"pith_summary":"This paper aims to settle why a massive tracer—a black hole or satellite—stops sinking toward the centre of a cored stellar system earlier than standard dynamical-friction predictions say it should. It argues that the dominant cause is granularity, not resonance: random force fluctuations from the finite number of background particles stir the tracer and statistically balance the friction, so the stalling radius follows $r_s \\propto N^{-1/2}$ at fixed tracer mass fraction. The same scaling appears in self-consistent $N$-body runs and in runs where field particles are non-interacting tracers in the static smooth potential, in both cored ($\\gamma=0$) and cuspy ($\\gamma=1$) models, and a stochastic Langevin version of the same physics reproduces the trend up to $N=10^{13}$. A sympathetic reader would care because the result turns a long-debated astrophysical puzzle into a finite-resolution effect, with the corollary that low-resolution simulations exaggerate how far dynamical friction drags massive objects inward.","feed_headline":"Core stalling shrinks as 1/√N: graininess, not resonance","feed_subtitle":"Finite-particle force noise, not orbital resonance, sets where dynamical friction stalls; low-N runs exaggerate drag.","key_machinery":"The load-bearing object is a stochastic equation of motion for the tracer, $\\ddot{\\mathbf r} = -\\nabla\\Phi(\\mathbf r) - \\eta\\mathbf v + \\delta\\mathbf f(\\mathbf r)$, in which $-\\eta\\mathbf v$ is the Chandrasekhar dynamical-friction drag and $\\delta\\mathbf f$ is a random force per unit mass sampled from the Holtsmark distribution, normalized by the local density and particle mass. The paper anchors this model in the discrete dynamics by computing the force-fluctuation field in $N$-body snapshots as the difference between the discrete and smooth-potential accelerations and showing that the empirical distribution is close to Holtsmark across radii. The argument is carried by the identity $r_s \\approx C r_c \\sqrt{m/m_t}$, the same scaling as the wander radius of a heavy particle undergoing Brownian motion in the cluster core, so the stalling radius is interpreted as the point where noise-induced random walking balances friction-driven infall.","core_discovery":"At fixed $m_t/M$ and nearly circular initial orbits, the stalling radius in the cored $\\gamma=0$ and mildly cuspy $\\gamma=1$ spherical models obeys $r_s \\approx C r_c \\sqrt{m/m_t} = C r_c \\sqrt{M/(N m_t)}$, hence $r_s \\propto N^{-1/2}$, across $10^4 \\le N \\le 3\\times10^6$ (Section 3.2, Fig. 6), both with and without field-particle self-gravity. The authors verify that the per-shell force-fluctuation distribution matches the Holtsmark form, reproduce the $N$-body orbital decay with a Langevin equation combining local Chandrasekhar friction and Holtsmark-distributed noise, and show that this stochastic model extends the $N^{-1/2}$ law for both models to $N=10^{13}$, with the cored case flattening to a nearly $N$-independent $r_s$ above about $10^9$ particles. Contrary to the common view that stalling is special to flat cores, the suppression shows up in the cuspy $\\gamma=1$ model as well, at smaller radii. They further report that the super-Chandrasekhar phase seen in earlier cored-model work appears only at $N \\lesssim 3\\times10^5$, which they attribute to enhanced collisionality at low resolution rather than to a physical breakdown of the standard friction formula.","pith_inferences":["Because the argument identifies the mass ratio $m_t/m = m_t N/M$ as the control variable, a natural test is to fix $N$ and vary $m_t$ over two decades in tracer-only runs, checking that $r_s/(r_c\\sqrt{m/m_t})$ collapses onto one constant; if the collapse fails, a resonance contribution is still present at fixed granularity.","The authors flag the delta-correlated noise assumption as approximate; measuring the force-fluctuation autocorrelation time in the $N$-body snapshots and re-running the Langevin equation with colored noise would show whether the $N^{-1/2}$ law survives beyond the simplest noise model.","The unexplained $N^{-1/6}$ slope change in cuspy models above $N\\approx10^9$ can be probed directly with high-resolution tracer-only runs, which the paper notes are feasible on parallel hardware; a time-step convergence study would separate a physical regime from a numerical artifact."],"forward_implications":["At fixed tracer mass fraction, doubling the particle number shrinks the stalling radius inward by about $\\sqrt{2}$, so stalling radii measured in $N\\sim10^5$-$10^6$ simulations are not directly transferable to real stellar systems with $N\\gtrsim10^{11}$ particles.","The super-Chandrasekhar drag phase seen in earlier cored-model simulations is a low-resolution artifact appearing only for $N\\lesssim3\\times10^5$, and should not be treated as a physical enhancement of dynamical friction.","The self-gravitating wake or polarization cloud is not the controlling agent: switching off field-particle self-gravity leaves the stalling radius nearly unchanged for $3\\times10^{-4}\\lesssim m_t/M\\lesssim10^{-2}$.","Core stalling and heavy-particle wander share the same mass-ratio dependence ($r\\propto\\sqrt{m/m_t}$), so they are the same granularity phenomenon approached from different initial conditions.","In the stochastic model, cored systems show a resolution floor: for $m_t/M=10^{-3}$ the stalling radius becomes nearly $N$-independent above about $N=10^9$, whereas cuspy systems keep shrinking until the model's slope change near $10^9$."],"supporting_citations":[{"why":"Supplies the dynamical-friction drag term and the baseline Chandrasekhar formula that the paper compares against.","marker":"Chandrasekhar 1943a,b,c"},{"why":"Defines the wander radius $r_{\\rm wan}\\propto r_c\\sqrt{m/m_t}$ whose scaling the paper identifies with the measured stalling radius.","marker":"Bahcall & Wolf 1976"},{"why":"Gives the force-fluctuation distribution used to sample the stochastic term in the Langevin equation.","marker":"Holtsmark 1919"},{"why":"Reported the super-Chandrasekhar drag and core stalling in cored models that this paper explains as a low-resolution effect.","marker":"Read et al. 2006"},{"why":"Established core stalling and cusp erosion in cored models; their results are the resolution-dependent baseline the $N^{-1/2}$ finding reframes.","marker":"Goerdt et al. 2006, 2010"},{"why":"The resonance-based dynamical-buoyancy theory whose $N$-independent stalling prediction the paper tests against its $N^{-1/2}$ scaling.","marker":"Banik & van den Bosch 2021, 2022"},{"why":"Supplied the stochastic Langevin-equation approach used for the high-$N$ extrapolations.","marker":"Sartorello et al. 2025"},{"why":"Calibrates the Coulomb-logarithm cutoff and quantifies how softening suppresses collisionality, used to interpret the softening dependence of $r_s$.","marker":"Marcos et al. 2017"},{"why":"Measured the wander radius of heavy particles in $\\gamma$-models, providing the comparison case for the $r_s\\propto\\sqrt{m/m_t}$ interpretation.","marker":"Merritt et al. 2007"}],"fun_headline_variants":["Core stalling traced to force noise, not resonance","Stalling radius scales as 1/√N: graininess wins","Finite-N force noise, not orbital resonance, sets stalling","Dynamical friction stalls from granularity, not buoyancy","Force fluctuations, not resonances, dictate core stalling"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the finite-$N$ background can be treated as uncorrelated Poissonian noise: force fluctuations drawn independently from a Holtsmark distribution at every time step, with no memory; the paper itself notes that force fluctuations cannot be fully uncorrelated because the density fluctuations that source them integrate to zero at fixed total mass, so a non-negligible correlation time could shift the noise-friction balance that sets the stalling radius.","fun_headline_variants_meta":{"raw":{"variants":["Core stalling traced to force noise, not resonance","Stalling radius scales as 1/√N: graininess wins","Finite-N force noise, not orbital resonance, sets stalling","Dynamical friction stalls from granularity, not buoyancy","Force fluctuations, not resonances, dictate core stalling"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00026,"raw_usage":{"total_tokens":1712,"prompt_tokens":1189,"completion_tokens":523,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":805,"completion_tokens_details":{"reasoning_tokens":437}},"tokens_in":805,"tokens_out":523,"duration_ms":5013,"temperature":1.0,"reasoning_tokens":437,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:28:02.352941+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run tracer-only simulations in a cored $\\gamma=0$ model at fixed $m_t/M=10^{-3}$ with $N=10^7$ and $10^8$, at a fixed softening below the predicted $r_s$, with two timesteps differing by a factor of two; if $r_s$ follows the $N^{-1/2}$ line in both, the granularity mechanism holds, while a plateau at the $r_s\\approx r_c/2$ level would favor the resonance-buoyancy explanation. Separately, measure the force autocorrelation time $\\tau_c$ in an $N$-body snapshot sequence and check whether adding it to the stochastic equation moves $r_s$ beyond the run-to-run scatter.","supporting_citations":[{"cited_title":"1919, Annalen der Physik, 363, 577","cited_arxiv_id":null,"evidence_quote":"Gives the force-fluctuation distribution used to sample the stochastic term in the Langevin equation."},{"cited_title":"A., & Pasquato, M","cited_arxiv_id":null,"evidence_quote":"Supplied the stochastic Langevin-equation approach used for the high-$N$ extrapolations."},{"cited_title":"2017, Phys","cited_arxiv_id":null,"evidence_quote":"Calibrates the Coulomb-logarithm cutoff and quantifies how softening suppresses collisionality, used to interpret the softening dependence of $r_s$."},{"cited_title":"2007, AJ, 133, 553","cited_arxiv_id":null,"evidence_quote":"Measured the wander radius of heavy particles in $\\gamma$-models, providing the comparison case for the $r_s\\propto\\sqrt{m/m_t}$ interpretation."}],"review_version":1}