{"id":"886d5981-3ef6-4213-95be-fd85fca38ad0","arxiv_id":"2504.21213","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Bubble wall velocities in local thermal equilibrium are computed for three BSM models and found to be nearly universal when expressed via the critical temperature and supercooling, with only deflagration solutions.","lead":"This paper computes how fast the walls of vacuum bubbles move during a first-order electroweak phase transition, assuming the plasma stays in local thermal equilibrium, across three extensions of the Standard Model. It finds that the wall speed depends mainly on two thermodynamic quantities and is similar across models, yielding a simple formula for gravitational wave and baryogenesis estimates.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The near-universal fit in Table 1 rests on the tanh ansatz (2.19)-(2.20) and four moment constraints (2.22); if the true two-field wall profiles deviate from this form, vw and the fitted laws could be systematically biased, and no unconstrained validation is provided.","rationale":"The paper is a systematic LTE study whose numerical implementation appears internally consistent, and the deflagration-only outcome agrees with prior LTE analyses. The main intellectual contribution is the near-universal fitting function, so the validity of the tanh ansatz is the load-bearing point: every reported vw, width, and fitted coefficient is downstream of this assumption. The reader's CONDITIONAL verdict is appropriate because the ansatz is untested against a full field-theoretic solution and the empirical fit is under-documented (no uncertainties, no code/data, and an unexplained exclusion of weak-transition points). If the proposed boundary-value check confirms the tanh results, the universal fit would be much better supported; if it does not, the universality claim would need to be revised. The concern is not a disagreement with consensus—LTE deflagrations are expected—but a robustness gap in the headline claim, which justifies keeping the verdict conditional rather than accepting outright.","tokens_in":35537,"tokens_out":8612,"duration_ms":90334,"concrete_test":"Select 3-5 benchmark points spanning the SSM two-step region, including BP1 and BP2 of Fig. 5 and one point near the excluded weak-transition region. Solve the full two-point boundary value problem for h(z), s(z), T(z), vp(z) using the exact equations of motion (2.16)-(2.17) with the hydrodynamic matching conditions (2.8), via a relaxation or shooting method without imposing the tanh form. Compare the resulting vw, Lh, Ls, delta_s and the fitted coefficients of vw(Tc/v,Tn/Tc) with Table 1. If vw differs by more than about 10% or the fitted coefficients move beyond the inter-model spread of Table 1, the universal fit is not robust.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of a near-universal, model-independent fit for vw (Table 1) rests on the tanh ansatz for the scalar profiles, Eqs. (2.19)-(2.20). The four parameters (vw, Lh, Ls, delta_s) are fixed by the four moment constraints (2.22), which are weighted integrals of the equations of motion, not the equations of motion themselves. The friction term driving the velocity is proportional to -∫ dz (∂V/∂T) T' (Eq. 4.1), and both T(z) and its gradient are determined by the assumed profile shapes. If the true two-field path from (0,s+) to (h-,0) deviates from the tanh family—e.g., through asymmetric tails, different relative widths, or a curved path in field space—the pressure balance and hence vw could shift systematically. The paper provides no validation against an unconstrained solution, and the weak-transition points excluded from the fit (Sec. 5.1, vw ≲ 0.54) are dropped without a quantitative criterion, so the fitted linear laws could be an artifact of the ansatz and of the selection. The fits also lack uncertainties, and neither code nor data are released, preventing an independent check.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the steady-state dynamics of bubble walls in local thermal equilibrium (LTE) during a first-order electroweak phase transition, for three beyond-SM models with two-step symmetry breaking: the Z2-symmetric real singlet extension (SSM), the real triplet extension (RTSM), and the inert two-Higgs-doublet model (IDM). The scalar equations of motion are imposed through four moment constraints—total pressure, pressure difference, and two pressure gradients (Eq. 2.22)—on tanh ansätze for the profiles of the two relevant scalars (Eqs. 2.19–2.20), solved jointly with the relativistic hydrodynamic conservation laws and the generalized bag equation of state, using an algorithm previously developed by the authors. For all models and scanned couplings, only deflagration-type solutions are found (hybrids included), with no detonations in LTE; the solution region is bounded above by the condition vw → vJ, and the grey zones above it are interpreted as ultra-relativistic detonations that require out-of-equilibrium friction. The headline result is a near-universal, approximately linear dependence of vw on Tc/v and Tn/Tc, encoded in the fitting functions of Table 1, plus a derived lower bound on the amount of supercooling. The fits are then applied to gravitational-wave spectra and to an estimate of the baryon asymmetry in the SSM with a CP-violating effective top-mass operator.","tokens_in":35870,"tokens_out":15893,"duration_ms":151844,"significance":"If the central claim is accepted, the Table 1 fitting functions provide a cheap, practical substitute for full LTE wall-velocity computations in phenomenological scans, and the systematic comparison across three electroweak representations is a useful consolidation of the LTE program. The paper is commendably explicit about its limitations: LTE velocities are presented as upper bounds, the planar-wall hydrodynamic description is flagged as invalid as vw → vJ (Sec. 2.2), the WKB basis of Eq. (2.16) is flagged for L_i Tn ≲ 1 (Sec. 5.1), and the absence of LTE detonations is supported both by the numerics (no stable zero of Ptot above vJ, Fig. 5) and by the entropy-conservation argument of Ref. [44]; the numerical finding γ(z)T(z) = const is a clean cross-check. The analytic disentangling of the four constraints in Sec. 4 (Ptot ↔ T−, ΔP ↔ δs, G_i ↔ L_i) is instructive and is confirmed by Fig. 5, and the fitting functions are honestly presented as empirical summaries of the numerical output rather than as inputs. However, the headline universality is conditional: it rests on the unvalidated tanh ansatz and on fit-quality statements with no uncertainties.","major_comments":[{"comment":"The headline claim—the near-universal fits of Table 1—rests on restricting the wall solutions to a four-parameter tanh family and imposing the equations of motion only through the weighted moment integrals (2.22). The friction balance that fixes vw is Eq. (4.1), in which both the field profiles and the temperature profile T(z) (reconstructed from the ansatz via (2.6)–(2.7)) enter; a systematic deviation of the true two-field path from the tanh family—asymmetric tails, different relative widths, or a curved trajectory in field space—would therefore bias vw and propagate into every coefficient of Table 1. The paper provides no validation against an unconstrained solution and no test of convergence with the number of moments. I request a concrete check of the ansatz, for example by comparing vw, Lh, Ls and δs with full LTE solutions of the field equations (as in Refs. [44, 57]) at several benchmark points, or by adding additional moment constraints and demonstrating stability of the solution. Without such a check, the universality claim is not yet established.","section":"§2.3, Eqs. (2.19)–(2.22)"},{"comment":"The fits in Table 1 are evaluated after excluding points with vw ≲ 0.54, with the only justification that the near-flat potential produces 'larger numerical errors', and no quantitative criterion for the cut is given. Because vw spans only roughly 0.5–0.7 over the entire two-step region, the cut removes a substantial part of the dynamic range used for the fit, and the later claim of sub-percent agreement between the λs = 1 and λs = 2 fits (Sec. 5.4) cannot be audited without uncertainties on the coefficients. The authors should (i) state the selection rule quantitatively and demonstrate that the fit coefficients are stable under reasonable variations of it, (ii) quote uncertainties for all coefficients in Table 1, and (iii) show the residuals of the fits, for instance the scatter of the numerical points around the straight lines in Figs. 6, 9, 12, 14, 17 and 19.","section":"§5.1, Fig. 6; Table 1"},{"comment":"The claim of near-universality is underdetermined as presented. The IDM fits differ from the SSM/RTSM ones mainly through the Tc/v coefficient (0.05–0.07 versus 0.13–0.15), i.e., by a factor of two or more in precisely the coefficient that is supposed to display model independence, whereas the Tn/Tc coefficient varies by only a few percent. Without error bars or an explicit comparison with a pooled fit (e.g., a single fit through all models and its scatter), 'near-universal' cannot be distinguished from 'noise around two different behaviors'. The derived boundary (Tn/Tc)_min in the fourth column of Table 1, obtained by equating the vw and vJ fits, compounds the uncertainties of both fits and, given the explicitly stated breakdown of the hydrodynamic treatment as vw → vJ (Sec. 2.2), should be presented with a caveat on its accuracy in the region where vJ − vw is of the same order as the numerical resolution.","section":"§5.4 and Table 1"}],"minor_comments":[{"comment":"The title contains a typo: 'Phase T ransition' should read 'Phase Transition'.","section":"Title"},{"comment":"The sentence 'The wall velocity in the Tc−Tn/Tc plane is shown in the left panel of Fig.17 for λ2 = 1 and in the left panel of Fig.19 for λ2 = 1' repeats λ2 = 1; the first reference should be λ2 = 1/2, matching the captions of Figs. 17 and 19.","section":"§5.3"},{"comment":"The package name 'Cosmotransitions' should be capitalized as 'CosmoTransitions' to match Ref. [81].","section":"§5.1"},{"comment":"The statement that the dimension-5 operator in Eq. (6.1) 'has a negligible impact on the PhT dynamics' is asserted without evidence; a brief quantitative check (e.g., the size of its contribution to the effective potential relative to the thermal potential) should be given, since the BAU results depend on this assumption.","section":"§6.2"},{"comment":"Given the title and scope of Refs. [57, 72], a direct quantitative comparison of the Table 1 fits and of the upper boundary vw = vJ with the model-independent LTE results of those works would help readers position the new claims.","section":"§5.4"}],"recommendation":"major_revision","confidential_remarks":"The paper is honest and technically sound within its stated method; my recommendation of major revision is driven by the gap between the strength of the headline claim (a universal, phenomenologically usable fit) and the absence of validation and statistical support for it. The requested checks—moment convergence or comparison with full-EoM LTE solutions, and error bars/residuals for the fits—are within the authors' demonstrated capabilities. I would also encourage the editor to ask the authors to release the numerical vw data (or code) accompanying Table 1, since the claimed percent-level model independence is otherwise not independently checkable. The relationship to Ref. [57], which already advertises 'model-independent bubble wall velocities in local thermal equilibrium', deserves a more explicit comparison than the current single sentence; this is a positioning issue rather than a novelty issue."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a useful systematic LTE calculation of bubble wall velocities for three BSM models, with compact fitting formulae for vw as a function of Tc/v and Tn/Tc. The fits are the main payload, and they will be handy for GW and baryogenesis parameter scans. The paper is worth a serious referee, provided the referee pushes on validation of the profile ansatz and on data release.\n\nWhat is actually new: the scan over the SSM, RTSM and IDM with two-step transitions, and the observation that the LTE wall velocity collapses onto a linear function of (Tc/v, Tn/Tc) with coefficients that barely change across models. The paper also gives vJ fits and an upper bound on supercooling. The numerics implement a reasonable moment projection of the scalar EoMs plus hydro matching; the analytic decomposition of Ptot, ΔP, Gh, Gs is helpful and the paper is transparent about known limitations: WKB fails for thin walls, solutions stop at the Jouguet velocity, and no detonations appear in LTE, consistent with [57,72]. The GW and BAU applications are preliminary but appropriate.\n\nThe soft spots, in order of seriousness. First, the tanh ansatz is load-bearing. The four moment constraints fix vw, Lh, Ls, δs; there is no check against an unconstrained profile or an independent method. If the true two-field wall has asymmetric tails or a different relative shape, the reported velocities and the universal fit could be systematically off. This is a legitimate concern, and the paper should address it, for example by comparing with a broader profile family or with hydro simulations. Second, the fits have no error bars; the excluded weak-transition points (vw ≲ 0.54) are dropped without a quantitative criterion, so the quoted coefficients are less robust than they look. Third, no code or data. None of these undercut the qualitative conclusion that LTE solutions are deflagrations; that conclusion is consistent with earlier model-independent arguments. But they do mean the fit table should be treated as an empirical summary, not a proven law.\n\nI'd send this to peer review. It is a solid, useful contribution, and the universal-fit claim is important enough to be checked. The main referee request should be: release the scan data and fit residuals, and validate the ansatz against an unconstrained solution or a published code. Then the paper can be accepted with confidence.","headline":"A practically useful LTE wall-velocity scan with compact universal fits, but the fits rest on the tanh ansatz and need validation and data release before the universal claim is taken as law.","tokens_in":36371,"tokens_out":2162,"would_cite":true,"duration_ms":24088,"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 in local thermal equilibrium the electroweak bubble wall is always a deflagration and its speed obeys a single linear formula in Tc/v and Tn/Tc that is nearly identical across three different scalar extensions of…","keywords":["electroweak phase transition","bubble wall velocity","local thermal equilibrium","deflagration","gravitational waves","electroweak baryogenesis","scalar extensions of the Standard Model","two-step phase transition"],"falsifier":"Take a few benchmark points from the two-step regions of the three models and solve the original second-order boundary-value problem for the coupled scalar and plasma profiles without imposing the tanh ansatz, then compare the resulting vw, Lh, Ls, and delta_s to the moment-constraint solutions summarized in Table 1. If the unconstrained wall velocity differs by more than the fit's scatter, or if the model-independent collapse in vw versus (Tc/v, Tn/Tc) disappears, the near-universality is an artifact of the ansatz.","tokens_in":35320,"feed_emoji":"🫧","tokens_out":7508,"duration_ms":72121,"temperature":0.7,"pith_summary":"This paper tries to establish that, during a first-order electroweak phase transition in the early universe, the terminal speed of the expanding bubble wall is nearly insensitive to which beyond-Standard-Model particle theory produced the transition. Working in local thermal equilibrium, the authors solve the coupled scalar field equations and plasma hydrodynamics for three different scalar extensions of the Standard Model and find that all steady-state solutions are deflagrations. They show that the wall velocity is captured by one simple linear formula in terms of the critical temperature, the nucleation temperature, and the Higgs vacuum expectation value, with coefficients that vary at the percent level across models and couplings. If this holds, a single fitting function can replace expensive full bubble simulations to estimate wall speed, gravitational wave spectra, and baryon asymmetry for a wide class of models, and the local-equilibrium result supplies an upper bound on wall velocity until out-of-equilibrium friction is included.","feed_headline":"One simple formula predicts bubble wall speed in three particle models","feed_subtitle":"Near-universal deflagration speed depends almost only on critical temperature and supercooling across three Higgs-sector extensions.","key_machinery":"The machinery is the set of four moment constraints obtained from the scalar equations of motion under a tanh ansatz, combined with the generalized bag equation of state and the hydrodynamic matching conditions. For a two-field transition with h(z) = h_-(1 + tanh(z/Lh))/2 and s(z) = s_+(1 - tanh(z/Ls - delta_s))/2, the equations of motion are replaced by four algebraic conditions: vanishing total pressure Ptot = Ph + Ps, vanishing pressure difference delta P = Ps - Ph, and vanishing pressure gradients Gh and Gs. These are solved together with the matching equations for the plasma. The key identities are that in LTE the total pressure can be written as Ptot = delta V - integral dz (partial V/partial T) T'(z), so the temperature gradient across the wall is the entire friction, and that entropy conservation enforces gamma(z) T(z) = constant. The approach reduces the wall dynamics to a root-finding problem for vw, delta_s, Lh, and Ls.","core_discovery":"Within local thermal equilibrium (LTE), steady-state electroweak bubble wall solutions in the singlet, real triplet, and inert doublet extensions of the Standard Model are all deflagrations or hybrids, never detonations, and their wall velocity obeys an approximately universal linear relation, vw = 1.60 + 0.15 (Tc/v) - 1.14 (Tn/Tc), with x = Tc/v and y = Tn/Tc, across the six model setups scanned. The same scan gives a model-insensitive line for the Jouguet velocity and an upper bound on supercooling, Tn/Tc >= 0.71 + 0.42 x for the singlet case. The near-universality is presented as a practical result: simple fitting functions replace the full numerical solution for LTE wall velocities, provide the first necessary step toward out-of-equilibrium calculations, and yield gravitational wave peak amplitudes and baryon asymmetries that are mostly below proposed sensitivities except in the strongest corner of parameter space.","pith_inferences":["The near-universality suggests the wall speed is controlled by the plasma's hydrodynamic response, namely enthalpy, sound speed, and the temperature gradient across the front, rather than by the details of the scalar potential; a natural extension is to test the same Tc/v and Tn/Tc formula on other models or on transitions with larger jumps in the number of relativistic degrees of freedom.","A direct test of the tanh ansatz, solving the full two-field boundary-value problem without the four-moment projection for a few benchmarks, would show whether the universal fit survives or is an artifact of the assumed profile family; this is the most concrete way to check the paper's central claim.","The baryogenesis result implies that within LTE, even an optimized singlet benchmark struggles to reach the observed asymmetry, so the fate of electroweak baryogenesis in these models may hinge on the out-of-equilibrium correction that lowers vw; the paper's announced follow-up analysis is the natural test.","Because gamma(z) T(z) = constant makes the plasma profiles nearly rigid in LTE, the same fitting-function approach might be extended to predict not only vw but also the shape of the temperature profile, and hence the friction term, from thermodynamic input alone."],"forward_implications":["In local thermal equilibrium, every steady-state solution found across the three models is a deflagration or hybrid, so detonation boundary conditions should not be used for LTE wall velocities in these models.","The linear fits of Table 1 reproduce the numerical wall velocities well enough that phenomenological studies of these models, such as gravitational wave spectra, baryogenesis estimates, and parameter scans, can use them instead of running a full simulation.","Because out-of-equilibrium effects add friction, the LTE wall velocity is an upper bound on the true wall speed, and the regime where LTE predicts no solution is where ultra-relativistic detonations appear once friction at vw approaching 1 is included.","Wall widths Lh and Ls shrink as the transition strengthens, and their ratio tracks h-/s+, providing a simple estimate for the wall structure.","Gravitational wave peaks from LTE deflagrations are mostly below proposed detector sensitivities, while ultra-relativistic detonations are more accessible, and baryogenesis in the augmented singlet model reaches at most about 0.4 of the observed asymmetry at the benchmark scale used."],"supporting_citations":[{"why":"Supplies the moment-constraint method and the tanh ansatz used to convert the scalar equations of motion into four algebraic conditions on vw, Lh, Ls, and delta_s.","marker":"[11,12]"},{"why":"Provides the generalized bag equation of state and the hydrodynamic matching and regime classification (deflagration, detonation, hybrid) used to build the plasma profiles.","marker":"[18]"},{"why":"Derives the entropy-conservation condition gamma(z) T(z) = constant in local thermal equilibrium, which the paper uses to fix the plasma profiles.","marker":"[44]"},{"why":"Gives earlier LTE results on the absence or instability of detonation solutions that this paper corroborates and extends to three BSM models.","marker":"[57,72]"},{"why":"Introduced the numerical algorithm and iterative framework whose LTE solution is the first step applied here to parameter-space scans.","marker":"[48,56,64]"},{"why":"Establishes the runaway friction that stops walls as vw approaches 1, motivating why LTE cannot capture ultra-relativistic detonations.","marker":"[28,76,77]"},{"why":"Supplies the moment-expansion method used to compute the baryon asymmetry in the singlet-model application.","marker":"[31]"}],"fun_headline_variants":["Near-universal formula predicts bubble wall speed in three models","Bubble wall speed nearly universal across three particle models","One simple equation fits bubble wall speed in three models","Model-independent bubble wall velocity from LTE simulations","Three Higgs models share one bubble wall speed rule"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that the wall profiles have the specific smooth step shape given by tanh functions with two widths and one offset, and that solving only four integrated moment equations of the full field equations is equivalent to solving them exactly; if the real wall has asymmetric tails or a different shape, the computed velocities and the apparent universality could be artifacts.","fun_headline_variants_meta":{"raw":{"variants":["Near-universal formula predicts bubble wall speed in three models","Bubble wall speed nearly universal across three particle models","One simple equation fits bubble wall speed in three models","Model-independent bubble wall velocity from LTE simulations","Three Higgs models share one bubble wall speed rule"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000562,"raw_usage":{"total_tokens":2637,"prompt_tokens":886,"completion_tokens":1751,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":502,"completion_tokens_details":{"reasoning_tokens":1676}},"tokens_in":502,"tokens_out":1751,"duration_ms":13442,"temperature":1.0,"reasoning_tokens":1676,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T05:10:56.630234+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a few benchmark points from the two-step regions of the three models and solve the original second-order boundary-value problem for the coupled scalar and plasma profiles without imposing the tanh ansatz, then compare the resulting vw, Lh, Ls, and delta_s to the moment-constraint solutions summarized in Table 1. If the unconstrained wall velocity differs by more than the fit's scatter, or if the model-independent collapse in vw versus (Tc/v, Tn/Tc) disappears, the near-universality is an artifact of the ansatz.","supporting_citations":[],"review_version":1}