{"id":"9c0d3888-f7a0-4948-863a-54e91051de35","arxiv_id":"2608.13206","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":14,"one_line_summary":"A new time-step gauge makes long 3D primordial black hole formation simulations about 94 times cheaper and reproduces the known spherical collapse threshold and critical exponent.","lead":"This paper develops a three-dimensional numerical-relativity method that tracks black holes forming from dense spots in the early, radiation-filled universe and follows their growth over many cosmic times. A new gauge trick lets the simulation time step grow with the expanding universe, cutting coarse time steps by about 94 and opening the way to realistic nonspherical black hole formation studies.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Scaled-driver stability in strong-field, near-critical, and nonspherical regimes rests only on a few spherical amplitude tests; the FLRW frozen-coefficient argument explicitly does not cover it.","rationale":"The central claim depends on the scaled Gamma-driver preserving moving-puncture behavior in the strong-field region while permitting Δt∝aΔx over many Hubble times. The formal argument covers only the asymptotically FLRW region; the strong-field transition is left to empirical testing. The paper provides genuine supporting evidence: standard-driver comparisons for several supercritical amplitudes, a conformal-time cross-check, agreement with the Misner-Sharp threshold, consistency of γ with the radiation-fluid value, and resolution tests. These checks make the spherical results credible. However, the near-critical amplitudes that define μ_c and γ are exactly where a gauge-induced systematic would be hardest to detect without a standard-driver comparison, because the critical solution is long-lived and highly sensitive. The full-box test also required interior regularization, so the behavior of the scaled driver in genuinely nonspherical collapse remains open. This does not invalidate the present spherical application, but it does mean the strong-field stability of the scaled driver is not yet established in the regime most relevant to the claimed extension beyond spherical symmetry. The reader's conditional verdict already captures this limitation; my assessment therefore leaves the verdict unchanged.","tokens_in":24032,"tokens_out":9254,"duration_ms":85086,"concrete_test":"Repeat the near-critical amplitude scan (at least μ=0.79580) with the standard, unscaled Gamma-driver using the same L=1, Δx_min=L/4096 setup, terminating at first apparent-horizon formation, and compare the first-detection horizon mass and the inferred μ_c and γ with the scaled-driver values. If the horizon mass or fitted threshold moves outside μ_c±2×10^-5, or if γ changes by more than about 10^-3, the scaled gauge is altering near-critical collapse rather than only accelerating the outer evolution.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section III A justifies the cosmologically scaled Gamma-driver (b∝a^-2, η_GD∝a^-1, Eqs. 36) through a frozen-coefficient, linearized gauge-mode analysis on a homogeneous FLRW background, and the paper states that this far-field argument alone does not guarantee strong-field behavior near a newly formed black hole. The load-bearing step is the extrapolation of the wave-CFL and damping bounds (Eqs. 33 and 35) into the puncture region, where the conformal metric is not δ_ij and the shift-gauge characteristic speeds need not inherit the 1/a redshift. The only strong-field evidence is empirical: four spherical amplitudes (μ=0.795, 0.805, 0.825, 0.85) compared with the standard driver (Fig. 1), plus one full-periodic-box run at μ=0.805 that requires interior regularization (Fig. 3). None of these tests sits in the near-critical band 0.79578<μ_c<0.79580 used for the threshold and γ fits, and the full-box test does not exercise genuinely nonspherical initial data. In addition, because b(t) decays as a^-2, the shift driver is very weak during most of the post-formation phase (b≈1.9×10^-5 at t/t_H≈266.8), so gauge stretching or constraint growth could in principle set in after the demonstrated interval or in configurations without octant symmetry. Agreement of the scaled-driver threshold and exponent with independent spherical results is reassuring, but it does not isolate the gauge as the source of that agreement.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript presents a three-dimensional numerical-relativity framework for primordial black hole (PBH) formation in a radiation-dominated universe, implemented in GRChombo with flux-conservative relativistic hydrodynamics. The central methodological development is a cosmologically scaled Gamma-driver with b proportional to a^{-2} and eta_GD proportional to a^{-1}, which allows the cosmic-time step to grow as Delta t proportional to a Delta x while keeping the shift-gauge and physical CFL numbers approximately constant. For a representative spherical run, the authors report a factor-approximately-94 reduction in coarse-level steps compared to the standard driver, with matching apparent-horizon mass and constraint behavior; a conformal-time moving-puncture gauge provides an independent check. For a spherical Gaussian curvature profile, the code locates the threshold 0.79578 < mu_c < 0.79580 and a critical exponent gamma approximately 0.3559, consistent with the radiation-fluid value 0.3558. Late-time mass growth is fit to the Zel'dovich-Novikov accretion law with fitted efficiencies F approximately 2.96-3.61.","tokens_in":24534,"tokens_out":17218,"duration_ms":157561,"significance":"If the claims hold, this is a substantial methodological advance for cosmological numerical relativity. The scaled Gamma-driver attacks the temporal bottleneck of long PBH evolutions and, together with AMR, makes multi-Hubble-time three-dimensional runs feasible at roughly two orders of magnitude lower cost. The conformal-time gauge, boundary-condition tests, full-periodic-box regularization tests, and a three-level resolution study provide strong internal evidence that the spherical results are robust. The threshold and critical-exponent values agree with independent spherically symmetric calculations, and the step-count comparison is algorithmic rather than wall-clock dependent, which lends credibility to the efficiency gain. The main limitations are that the scaled driver is validated only for spherical data, and the critical-scaling fit is reported without uncertainty quantification; both need attention before the framework can be used as advertised for nonspherical collapse.","major_comments":[{"comment":"The critical-scaling result is one of the two main physical validations, but the paper reports only point estimates (K approximately 3.624, mu_c approximately 0.7957813, gamma approximately 0.3559) with no error bars, no table of the ten fitted amplitudes and first-detection masses, no residuals, and no stated fit range. In addition, the threshold classification criterion (maximum evolution time used to decide dispersal, and the minimum apparent-horizon size detected) is not given. Without these details, the four-significant-digit agreement with gamma = 0.3558 cannot be evaluated or reproduced. Please provide the fit data, parameter covariances, a sensitivity test to the number of fitted points and to the choice of mass diagnostic, and a statement of the classification protocol.","section":"Sec. IV C / Fig. 4"},{"comment":"The scaled Gamma-driver is justified by a frozen-coefficient FLRW analysis and by empirical tests on spherical octant-symmetric data (Fig. 1), and the paper explicitly acknowledges that the far-field argument does not guarantee strong-field behavior near a newly formed black hole. The conclusion nevertheless states that the framework 'can be extended to nonspherical profiles' and provides a 'foundation' for such studies. No genuinely nonspherical initial data are tested; the full periodic-box test of Sec. IV B is still spherical and in fact requires an ad hoc interior regularization to survive past t/t_H approximately 45.3. To support the advertised applicability, the authors should either present a nontrivial nonspherical (or at least non-octant) test with the scaled driver, or explicitly restrict the validation claim to spherical configurations and mark the nonspherical extension as an open problem.","section":"Sec. III A / Sec. V"},{"comment":"The accretion-law fit for the near-threshold run mu = 0.805 returns F = 2.958, well outside the quoted consistency range 3.5 <= F <= 3.75 from Ref. [35], while the three larger amplitudes cluster near 3.5-3.6. Because all four runs use the same foliation, the blanket attribution to foliation dependence is insufficient. Please quantify the fit uncertainties for F and M_infty, and provide evidence (e.g., longer runs or a different fitting start time) that the mu = 0.805 fit is indeed in the Zel'dovich-Novikov regime rather than an artifact of an early or short fitting interval.","section":"Sec. IV D / Table I"}],"minor_comments":[{"comment":"Please report the actual evolution times at which the nearest supercritical runs form an apparent horizon, since near-critical classification depends on the total simulated time.","section":"Sec. IV C"},{"comment":"Please give standard errors for the fitted F and M_infty values and show the fit residuals in Fig. 5.","section":"Sec. IV D / Table I"},{"comment":"Please state whether the simulation code and input parameter files will be released, which would greatly aid reproducibility of the threshold and exponent fits.","section":"Sec. II D"},{"comment":"The statement that the effective damping is eta_GD + 2H follows from the time-dependent b, but the derivation is compressed; one sentence showing the algebra would remove ambiguity.","section":"Sec. III A / Eq. (38)"},{"comment":"The Misner-Sharp benchmark cited in Sec. IV C is a preprint by the same group; please also cite an independent published calculation if available, or clarify the provenance of this benchmark.","section":"Sec. IV C / Ref. [12]"},{"comment":"The early-time constraint transients saturate the lower-right panel; consider using a logarithmic vertical axis or starting the plot after the transient has relaxed.","section":"Fig. 1"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically interesting and the spherical results appear sound. My main reservations are the missing error analysis for the critical-exponent fit and the gap between the validated spherical regime and the advertised nonspherical applicability; both are addressable. I would also ask the editor to verify the priority claim against recent three-dimensional works in Refs. [41,43,45]."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper earns its keep. The new piece is the cosmologically scaled Gamma-driver (b ∝ a^-2, η_GD ∝ a^-1) that lets the cosmic-time step grow as a Δx, and they back it with a thoughtful frozen-coefficient argument plus strong internal checks: scaled vs. standard driver agree on central lapse and apparent-horizon mass across four amplitudes, the conformal-time gauge gives an independent cross-check, and the step-count reduction (factor ~94) is an honest algorithmic measure rather than wall-clock flattery. The 3D extraction of the critical exponent γ ≈ 0.3559 and threshold 0.79578 < μ_c < 0.79580, matching Misner–Sharp spherical results, is a real validation and a first for 3D simulations. The accretion-law fits at late times are a nice bonus.\n\nSoft spots, in proportion. The scaled-driver stability argument explicitly covers only the far-field linearized regime; strong-field behavior is tested for four spherical amplitudes, none inside the near-critical band used for the fits. The full periodic-box check at μ = 0.805 requires an ad hoc interior regularization (freeze-ish weighting inside the horizon) to reach long times, and that procedure is validated empirically rather than derived. That is a legitimate concern for the paper's central claim that the framework is ready for nonspherical, multi-peak studies. I also note the fit quantities K, μ_c, γ, and the accretion efficiencies F come with no error bars, and code/data are not released. These are minor-to-moderate gaps, not fatal flaws. The internal evidence is solid enough that I would trust the spherical results and the gauge speedup for these configurations.\n\nWho is this for? Anyone working on PBH formation in numerical relativity, especially the COSMOS/GRChombo community. It does not change observational constraints, but it removes a real temporal bottleneck and gives a concrete path toward nonspherical and multi-peak runs.\n\nRecommendation: yes, send it to serious peer review. The referee should push for error bars on the fits, more near-critical amplitude tests with the scaled driver, and either a nonspherical demonstration or a clear statement that such a test is the immediate next step. But the core gauge construction and the reproducible cross-checks justify referee time.\n\nBest.","headline":"A genuinely useful numerical-gauge advance that makes long 3D PBH formation runs tractable, with the main open question being how far the scaled driver extends beyond spherical symmetric tests.","tokens_in":25016,"tokens_out":587,"would_cite":true,"duration_ms":6399,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83-08","83C57","83C05"],"pacs":[],"model":"deepseek-v4-flash","headline":"A rescaled gauge condition lets 3D primordial black hole simulations run up to 94 times faster.","keywords":["primordial black holes","numerical relativity","radiation domination","Gamma-driver gauge","critical collapse","adaptive mesh refinement","apparent horizon","accretion law"],"falsifier":"Evolve a strongly nonspherical superhorizon perturbation with the scaled driver through horizon formation and compare the $L^2$ norm of the Hamiltonian constraint and the apparent-horizon mass against a standard-driver run at the same resolution: sustained constraint growth or a horizon-mass disagreement before the horizon forms would falsify the claim that the speedup is generic. A lighter check is to repeat the factor-94 comparison at a different amplitude or with a different window function and see whether the step-count reduction and mass agreement survive.","tokens_in":23884,"feed_emoji":"🕳️","tokens_out":8547,"duration_ms":70096,"temperature":0.7,"pith_summary":"This paper develops a three-dimensional numerical-relativity method for simulating primordial black hole formation from superhorizon curvature perturbations in a radiation-dominated universe, and demonstrates that the same code can follow both near-critical collapse and long post-formation evolution. The central move is a cosmologically scaled Gamma-driver whose response and damping coefficients shrink with the expanding scale factor, allowing the cosmic-time step to grow as $\\Delta t \\propto a\\,\\Delta x$ while keeping the physical Courant number fixed. For a representative run this cuts the number of coarse-level time steps by a factor of about 94 without changing apparent-horizon masses or constraint histories. Applied to a spherical Gaussian curvature profile, the simulations bracket the collapse threshold in $0.79578<\\mu_c<0.79580$ and yield a critical exponent $\\gamma\\simeq0.3559$, matching spherically symmetric and radiation-fluid benchmarks, and the late-time black-hole mass growth fits the standard accretion law. If the method holds for general profiles, fully three-dimensional surveys of nonspherical primordial black hole formation become practical.","feed_headline":"Rescaled gauge runs 3D black hole formation 94× faster","feed_subtitle":"Time step can grow with the expanding universe while horizon masses and critical collapse behavior stay unchanged.","key_machinery":"The cosmologically scaled Gamma-driver is the load-bearing object: in the moving-puncture shift condition, the response coefficient is set to $b=b_0(a_i/a)^2$ and the damping coefficient to $\\eta_{\\mathrm{GD}}=\\eta_{\\mathrm{GD},0}(a_i/a)$, evaluated from the background scale factor at every Runge-Kutta substep. On the expanding background, the driver's fastest gauge mode otherwise keeps unit coordinate speed and the damping term imposes a $\\eta_{\\mathrm{GD}}\\Delta t$ stability bound independent of resolution, both of which would forbid $\\Delta t\\propto a\\,\\Delta x$; the scaling keeps the corresponding CFL numbers constant so the time step can grow with the scale factor. A conformal-time version of the moving-puncture gauge, obtained by reparametrizing the lapse and shift, is used as an independent check that the long-term horizon-mass evolution is not an artifact of the rescaling.","core_discovery":"The paper's claim is that the standard Gamma-driver shift condition is the obstruction to long cosmological black-hole evolutions in cosmic time, and that a simple rescaling removes it. On an expanding background the shift gauge mode propagates at a fixed coordinate speed rather than the $a^{-1}$ speed of physical modes, and its damping term imposes a resolution-independent time-step bound; scaling the driver coefficients as $b(t)=b_0(a_i/a)^2$ and $\\eta_{\\mathrm{GD}}(t)=\\eta_{\\mathrm{GD},0}(a_i/a)$ keeps both dimensionless stability parameters constant while taking $\\Delta t\\propto a\\,\\Delta x$. The paper shows empirically that this scaled driver reproduces the standard driver's central lapse, apparent-horizon masses, and constraint norms for spherical amplitudes across the threshold, while reducing coarse-level advances by a factor of approximately 94. It further reports, from the same three-dimensional framework, a collapse threshold $0.79578<\\mu_c<0.79580$ and a near-critical exponent $\\gamma\\simeq0.3559$ consistent with the radiation-fluid value, and a late-time accretion fit with efficiencies $F\\simeq2.96$ to $3.61$ that is broadly consistent with earlier spherically symmetric results.","pith_inferences":["Because the scaling only relies on the background expansion rate, it should transfer to other barotropic fluids, massless scalar collapse, and kination-type cosmologies; the paper itself notes that an oscillating massive scalar field breaks the scaling because its intrinsic frequency sets an expansion-independent time step.","A direct test of the method's reach is to repeat the threshold scan for ellipsoidal profiles and check whether the factor-of-94 step reduction survives when the collapse is no longer reflection-symmetric; the paper's full-box runs suggest the main risk is interior, not exterior, behavior.","The interior regularization used to keep full-periodic-box runs alive is a pragmatic device whose exterior invariance is verified empirically; a stronger validation would vary the regularization strength in a spinning collapse and confirm that exterior fields and horizon mass remain unchanged.","If the fitted accretion efficiencies cluster near a narrow range, late-time mass growth may admit a simple universal prescription that abundance calculations could adopt without rerunning full 3D evolutions."],"forward_implications":["Long post-formation runs that previously needed more than a hundred thousand coarse-level steps can be done with about a thousand, so multi-Hubble-time three-dimensional evolutions become routine.","The same framework can be pointed at nonspherical initial data, where the collapse threshold, critical exponent, and black-hole spin have not yet been measured in three dimensions.","The threshold interval $0.79578<\\mu_c<0.79580$ and $\\gamma\\simeq0.3559$ give a three-dimensional confirmation that spherical critical collapse behavior survives in a full 3+1 treatment.","Post-formation black-hole masses can be compared with the analytic accretion formula over many Hubble times, giving an efficiency parameter that is foliation-dependent but useful for abundance estimates."],"supporting_citations":[{"why":"Supplies the moving-puncture gauge whose lapse-collapse and shift-driving behavior the scaled driver is designed to preserve.","marker":"[48]"},{"why":"Identifies the resolution-independent damping time-step bound caused by the Gamma-driver, which motivates the $\\eta_{\\mathrm{GD}}\\propto a^{-1}$ scaling.","marker":"[68]"},{"why":"Provides the long-wavelength growing-mode solution used to set superhorizon initial data for the curvature perturbation.","marker":"[62]"},{"why":"Gives the earlier three-dimensional threshold interval $0.795<\\mu_c<0.805$ that this paper refines.","marker":"[41]"},{"why":"Gives the recent three-dimensional bracket $0.7<\\mu_c<0.8$ for the same profile, compared with the present result.","marker":"[45]"},{"why":"Defines the late-time accretion prescription to which the post-formation mass growth is fitted.","marker":"[1]"},{"why":"Provides spherical accretion fits with efficiencies around $3.5$-$3.75$ and the common crossing of the accretion diagnostic used for comparison.","marker":"[35]"},{"why":"Contains the spherically symmetric threshold value used as the benchmark for the three-dimensional result.","marker":"[12]"},{"why":"Establishes the radiation-fluid critical exponent used to check the fitted exponent.","marker":"[72]"}],"fun_headline_variants":["Scaled gauge speeds 3D black hole formation runs 94×","Rescaled shift condition unlocks long-term PBH formation","Faster 3D simulations of primordial black hole formation","Gauge rescaling makes black hole formation simulations 94× faster"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the time-dependent driver coefficients, fixed by the background scale factor, remain stable and singularity-avoiding in the strong-field region near a newly formed black hole, so the enlarged time step is safe there; the paper validates this for a handful of spherical amplitudes rather than proving it.","fun_headline_variants_meta":{"raw":{"variants":["Scaled gauge speeds 3D black hole formation runs 94×","Rescaled shift condition unlocks long-term PBH formation","Faster 3D simulations of primordial black hole formation","Gauge rescaling makes black hole formation simulations 94× faster"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000301,"raw_usage":{"total_tokens":1779,"prompt_tokens":1030,"completion_tokens":749,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":646,"completion_tokens_details":{"reasoning_tokens":686}},"tokens_in":646,"tokens_out":749,"duration_ms":6931,"temperature":1.0,"reasoning_tokens":686,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:25:15.724073+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evolve a strongly nonspherical superhorizon perturbation with the scaled driver through horizon formation and compare the $L^2$ norm of the Hamiltonian constraint and the apparent-horizon mass against a standard-driver run at the same resolution: sustained constraint growth or a horizon-mass disagreement before the horizon forms would falsify the claim that the speedup is generic. A lighter check is to repeat the factor-94 comparison at a different amplitude or with a different window function and see whether the step-count reduction and mass agreement survive.","supporting_citations":[{"cited_title":"General Rel- ativistic Collapse to Black Holes and Gravitational Waves from Black Holes,","cited_arxiv_id":null,"evidence_quote":"Provides the long-wavelength growing-mode solution used to set superhorizon initial data for the curvature perturbation."}],"review_version":1}