{"id":"31503d79-32d3-4dce-a264-ddf393ff9d89","arxiv_id":"2504.16202","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Numerical simulations and analytic equivalences show that the soliton-halo relation for ultralight dark matter is bracketed by the 1/2 relation as a lower bound and the 1/3 relation as an upper bound.","lead":"This paper tests two competing relations between the mass of a dark-matter soliton and its host halo, and argues that the true relation is a band: solitons form near the lower '1/2' line and never exceed the upper '1/3' line. The result matters because it sharpens predictions for rotation curves and the inferred lower bound on the ultralight dark matter particle mass.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The virialization cut (Eq. A16) is calibrated with the same approximate finite-volume potential constant (Eq. A15) used to build the E_kin proxies, so the reported 1/2–1/3 band may be a selection and convention artifact.","rationale":"The reader's weakest assumption identifies exactly the premise I examined: the kinetic-energy proxy for a finite-box halo requires the system to be virialized and free of unbound debris. The paper does enforce a virialization cut, but that cut uses the same approximate finite-volume potential energy whose arbitrariness the paper itself flags; Appendix A describes Eq. (A15) as a rough estimate with no guarantee of the assumed asymptotic form, and Fig. 12 shows the computed Ξ shifts by a large amount when the potential convention is changed. This makes the selection in Eq. (A16) non-independent of the very convention the analysis is trying to control. The concern is load-bearing because if the accepted runs are biased toward a particular E_kin range, the reported band between the 1/2 and 1/3 lines could be an artifact. I do not think the concern is fatal: the paper's central analytic inequalities are plausible, the simulations are varied, and the authors disclose several relevant caveats, including the fitting bias in footnote 5 and the unbound-debris caveat in Section IV.B. The appropriate response is a targeted re-analysis, which is exactly the conditionality the reader already assigned. I therefore recommend UNCHANGED: the concern does not move the verdict, but it sharpens the specific test that would justify raising confidence from CONDITIONAL to ACCEPT.","tokens_in":1011,"tokens_out":898,"duration_ms":152840,"concrete_test":"Recompute Fig. 3 with the virialization selection of Eq. (A16) replaced by an independent spectral criterion from Appendix E: accept only runs in which the positive-energy particle fraction, ∫_{ω>0} F(ω) dω / M, is below 1% at the final time. Then, as a second arm of the same test, redo Fig. 3 with the potential constant c in Eq. (A13) shifted by +/−50% of the Eq. (A15) estimate. Report how many data points cross the 1/2 or 1/3 lines and how the accepted-run set changes. If either variation moves more than about 20% of points across a boundary, or changes the set of discarded runs by more than a few, the band conclusion is not robust to the potential-energy convention.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing premise of Figs. 1 and 3 is that E_kin/M equals |E_tot|/M for both soliton and halo, so the 1/2 and 1/3 relations can be tested without resolving the finite-volume potential-energy constant. The paper enforces this premise by discarding runs that fail the virialization condition (2Ekin-|Epot|)/(2Ekin+|Epot|)<0.2, Eq. (A16). But E_pot entering this cut is computed with the additive constant fixed by Eq. (A15), a method the authors themselves call \"just a rough estimate, since there is no guarantee that the potential already goes as -M/(4πr) within the box\" (Appendix A). The cut is therefore not an independent test of virialization: it is calibrated with the same convention whose ambiguity the paper is trying to avoid. The authors' own Fig. 12 shows that changing the potential convention moves points substantially in the Ξ plane, so an O(1) error in c (Eq. A13) could either admit runs with positive-energy debris or reject genuinely virialized runs. The paper does not report how many runs are discarded or how the band changes if the threshold in Eq. (A16) or the constant c is varied. Without that, the reported clustering between the 1/2 and 1/3 lines could reflect selection on the dependent variable rather than a physical soliton-halo band.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the relation between the mass of solitons and their host halos in ultralight dark matter (ULDM). The authors point out that the Schive et al. 2014 core-halo relation is algebraically equivalent to (E/M)_sol = (E/M)_halo, while the Mocz et al. 2017 relation is equivalent to E_sol = E_halo. They argue that the latter is an upper bound for bound, virialized systems, and that the former is parametrically close to the evaporation/growth threshold of Chan et al. 2022, giving a rough lower bound. They support this picture with flat-space Schrödinger-Poisson simulations using several types of initial conditions (soliton mergers, NFW/Burkert halos, Gaussian noise backgrounds, and collapses). Their main figures show that simulated solitons form near the 1/2 relation and grow without crossing the 1/3 relation, forming a 'soliton-halo band.' They also argue that apparent discrepancies with previous flat-space and cosmological simulations can be explained by the finite-volume additive constant in the potential energy, and they advocate using kinetic energy to avoid this ambiguity.","tokens_in":26514,"tokens_out":8299,"duration_ms":82850,"significance":"The analytic reinterpretation of the two standard soliton-halo relations as energy equalities is clean and, if correct, constitutes a useful conceptual advance: it turns the 1/3 relation into a rigorous upper bound for bound configurations and gives a simple parametric argument for why the 1/2 relation acts as a rough lower bound. The use of kinetic energy in the main plots is a sensible way to sidestep one finite-volume ambiguity, and the diversity of initial conditions probed is a strength. The paper is also unusually candid about its own limitations, explicitly labeling the potential-energy constant estimate as rough and acknowledging the formation-time and fitting uncertainties. However, the numerical evidence for the band is not yet fully robust: the virialization cut that selects the plotted runs uses the same approximate potential-energy convention whose ambiguity the paper emphasizes, and the paper does not quantify the sensitivity of its conclusions to that cut. The central analytic claim is sound, but the simulation-based central claim needs additional robustness checks before the band can be regarded as established.","major_comments":[{"comment":"The virialization cut that determines which runs enter Figs. 1 and 3 is evaluated with E_pot whose additive constant is fixed by the approximate prescription in Eq. (A15), which the authors themselves describe as 'just a rough estimate.' Because the plotted variables use E_kin as a proxy for |E_tot|, the cut is not an independent physical selection: an error in the constant c changes both |E_pot| in Eq. (A16) and the inferred position in the Ξ plane, as Fig. 12 demonstrates. The paper does not state how many runs are discarded, nor how the band in Figs. 1 and 3 changes when c is varied within the range of plausible conventions (e.g., c=0 versus Eq. (A15), or R chosen at different grid radii) or when the 0.2 threshold is varied. Without such a robustness check, the reported clustering between the 1/2 and 1/3 lines could be a selection effect driven by the same finite-volume convention that the paper identifies as the main ambiguity.","section":"Section III and Appendix A, Eq. (A16)"},{"comment":"The 1/3 upper bound applies only to the energy of bound material, and the paper's simulation points are interpreted as respecting this bound. The statement that 'our simulations have a negligible quantity of positive energy debris' is asserted but not quantified; the energy-spectrum diagnostic in Appendix E is illustrated for one run in Fig. 14, with no threshold, mass fraction, or run-by-run summary. Since unbound debris would contribute positive kinetic energy to the proxy used in Figs. 3 and 4, the claim that the points lie below the 1/3 line for the bound halo needs a quantitative bound on the unbound mass/energy fraction across all initial-condition types.","section":"Section IV.B and Appendix E"},{"comment":"The reconciliation with Mocz et al. (2017) rests entirely on the choice of the additive potential-energy constant. With the unadjusted total energy the points cluster on the 1/3 line (Fig. 9), while with the Eq. (A15) constant many points shift substantially (Fig. 12). The conclusion that the literature is 'fully consistent' with the band therefore requires a sensitivity study of the constant c and preferably a re-analysis of Ref. [25]'s data with E_kin-based variables; otherwise the agreement may be an artifact of one particular convention among several.","section":"Section IV.D.1 and Figs. 9 and 12"}],"minor_comments":[{"comment":"The relation between M_sol and M_c and the prefactor 4.2 is confusing as written ('the prefactor ... is α=4.2 Mc/Msol≈1'); please clarify which mass the fit coefficient refers to.","section":"Section I, Eq. (1) and footnote 3"},{"comment":"The caption and text describe the soliton-formation criterion via a goodness-of-fit threshold, but the paper does not report the fraction of runs for which the criterion is never satisfied; please give this number, as it bears on the interpretation of Figs. 1 and 3.","section":"Section II.B, Fig. 2"},{"comment":"The intermediate algebra leading to B_h(β)=1−4β/3 is omitted; a short derivation or a reference to the original derivation would help the reader verify the cosmological translation.","section":"Section IV.D.2, Eq. (16)"},{"comment":"The sentence 'we discard a few runs' should be replaced by the exact number of discarded runs and their initial-condition types, especially given the role of Eq. (A16) in the main analysis.","section":"Appendix A, last paragraph"}],"recommendation":"major_revision","confidential_remarks":"The analytic part of the paper is solid and the topic is well within the scope of the journal. The main risk is that the numerical evidence for the band is not yet robust to the finite-volume potential convention, which is fixable with sensitivity tests. I would be willing to accept after those tests are provided."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nThe paper is worth your time if you care about ULDM soliton-halo relations. It does two things: it recasts the Schive and Mocz relations as energy equalities—this was already known from Bar et al., but they make it central—and then it runs a wider menu of initial conditions than most previous work, showing that solitons settle between the 1/2 and 1/3 relations, with the 1/2 acting as a rough lower bound and the 1/3 as an upper bound. They also make a convincing case that the discrepancy with Mocz et al. is mostly a finite-box potential-energy convention: use the same convention and the data cluster together.\n\nThe simulation survey is the real new content, and it is decent work. Multiple independent sets of initial conditions (mergers, halo profiles, noise, collapse) all behave the same way. Using Ekin rather than Etot is a smart way to sidestep the additive potential constant. The analytic ties to virialization and to Chan et al.'s evaporation threshold are clean and clearly labeled as rough. I also credit them for flagging fitting bias, formation-time arbitrariness, and unbound debris.\n\nThe soft spots are mostly around the main evidence. The virialization cut in Eq. (A16) is the one place where the approximate potential constant from Eq. (A15) enters the analysis that feeds the band claim. That constant is admitted to be a rough estimate. The cut could in principle select runs that look virialized under that convention but aren't, or reject runs that should be included. The paper doesn't say how many runs were discarded or how the band changes if the threshold or the constant is varied. That is a real omission, but I don't think it is fatal: the main plots use Ekin, which is convention-independent, and the upper bound is already visible as Ekin,sol < Ekin in Fig. 4. The selection worry deserves an explicit robustness test in revision.\n\nThe other weakness is reproducibility: no code or data are released, and the comparisons to the literature rely on digitized points. That's not disqualifying in this field, but it means the simulation survey can't be independently checked.\n\nWho gets value from this paper: anyone building ULDM rotation-curve predictions or comparing simulation groups. The band framing is likely to stick even if some details shift. I would send it to a serious referee, and I'd ask for the robustness checks above in the report.","headline":"A useful synthesis that turns the soliton-halo relation into a bracketing band, with the main remaining worry being an under-disclosed selection cut.","tokens_in":27084,"tokens_out":4997,"would_cite":true,"duration_ms":46451,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper argues that the soliton–halo relation in ultralight dark matter is a band: solitons form near the equal-energy-per-mass line and grow without crossing the equal-total-energy line.","keywords":["ultralight dark matter","soliton","core-halo relation","Schrödinger-Poisson equations","soliton evaporation","virial theorem","numerical simulation","rotation curves"],"falsifier":"Run a flat-box simulation with the same initial conditions but inject a population of positive-energy unbound particles after the soliton forms; if the soliton mass can be pushed above the $E_{\\rm sol} = E_{\\rm halo}$ line while the bound halo is unchanged, the 1/3 upper bound holds only with a bound-particle definition of halo energy. Alternatively, reanalyse the 2017-relation simulations using kinetic energy only: if the points stay above the 1/3 line instead of moving into the band, the central claim fails.","tokens_in":25960,"feed_emoji":"🌀","tokens_out":10830,"duration_ms":87439,"temperature":0.7,"pith_summary":"The paper aims to pin down the mass of the soliton core that ultralight dark matter forms at the center of a galactic halo, since that mass controls observable rotation curves and constraints on the particle mass. It argues that two well-known empirical scaling laws are actually energy equalities: the '1/2 relation' says the soliton and halo have the same energy per mass, while the '1/3 relation' says the soliton and halo have the same total energy. If the halo is gravitationally settled, the first is a rough lower bound on soliton mass and the second is an upper bound, so the soliton–halo relation is a band rather than a single line. The paper's flat-box simulations, initialized six different ways, place solitons inside this band, and it argues that apparent disagreements in the literature disappear once the finite-volume potential-energy convention is fixed.","feed_headline":"Two energy equalities bracket the soliton-halo relation","feed_subtitle":"Simulations show soliton cores form on the lower line and never cross the upper line.","key_machinery":"The load-bearing objects are two energy equalities plus virial relations. For any soliton solution of the Schrödinger–Poisson equations, the scaling identity $M_{\\rm sol} \\approx 4.2 \\,(|E_{\\rm sol}|/M_{\\rm sol})^{1/2}/(G m)$ makes the 2014 mass–energy fit equivalent to equal energy per mass, while $M_{\\rm sol} \\approx 2.6 \\,(|E_{\\rm sol}|/(G^2 m^2))^{1/3}$ makes the 2017 fit equivalent to equal total energy. The paper frames both through the Schrödinger–Poisson invariant $\\Xi = |E|/(M^3 G^2 m^2)$, writing a general soliton-halo relation as $M_{\\rm sol}/M = \\alpha \\, \\Xi^\\beta$, with $\\beta=1/2$ and $\\beta=1/3$ as the two cases. Because $\\Xi$ is sensitive to box size, the paper instead plots kinetic-energy ratios, using the virial theorem to connect them to the energy equalities.","core_discovery":"The central claim is that solitons form near the 1/2 relation, $(E/M)_{\\rm sol} = (E/M)_{\\rm halo}$, and then grow by accreting halo mass without ever surpassing the 1/3 relation, $E_{\\rm sol} = E_{\\rm halo}$. The former is a lower bound because a soliton more than about a factor of 3.5 lighter would sit below the evaporation threshold set by scattering with background particles; the latter is an upper bound because a soliton with more energy than the whole halo cannot exist in a bound virialized system. The upper-bound statement requires that the halo energy be evaluated on bound material: unbound positive-energy debris in a finite box would raise $E_{\\rm halo}$ and could mimic a violation. Across all initial conditions, the simulated solitons lie in the band between the two relations, and published results are consistent with the same band once the arbitrary additive constant in the finite-box potential energy is handled properly.","pith_inferences":["If the band picture carries over to structure formation, soliton mass may be nearly independent of halo merger history once the halo virializes; tracking the soliton trajectory in cosmological simulations should show it entering the band from below and only slowly approaching the 1/3 line.","The finite-volume potential-energy ambiguity identified here suggests a cheap test: recomputing published soliton-halo data with kinetic energy alone should bring all points into the band without any new simulations.","Because the bracket relies only on Schrödinger–Poisson scaling, the same band may apply to other self-gravitating wave dark matter structures such as axion miniclusters and dark photon stars, a direction the paper mentions as future work.","Real halos undergoing mergers or feedback may contain transient unbound material, so observational tests of the 1/3 upper bound should use only bound mass; otherwise an apparent violation could be an artifact of including escaping particles."],"forward_implications":["Observational limits that assume the 1/2 relation are conservative: if real solitons lie anywhere in the band, the inner rotation-curve peak is at least as strong as the 1/2-line prediction, strengthening lower bounds on the ultralight particle mass.","Published scatter in soliton-halo measurements should largely collapse once the data are reanalysed with kinetic energy or an infinite-volume-corrected potential energy, and the paper invites other groups to do that reanalysis.","Soliton growth in a fixed halo is self-limiting: the 1/3 upper bound sets a terminal soliton mass set by the halo's total energy, so accretion slows as the system approaches that line.","Solitons more than about 3.5 times lighter than the 1/2 line should evaporate, so newly formed solitons should appear within a finite factor of the 1/2 relation rather than at arbitrarily low mass."],"supporting_citations":[{"why":"Supplies the empirical 1/2 relation that the paper rewrites as equal energy per mass and tests against its simulations.","marker":"[22]"},{"why":"Proposes the 1/3 relation that the paper argues is an upper bound; its initial conditions match the soliton-merging (II) runs.","marker":"[25]"},{"why":"Establishes the equivalence between the 1/2 relation and equal energy-to-mass ratios and connects it to rotation-curve predictions.","marker":"[33]"},{"why":"Derives the soliton evaporation/growth threshold whose parametric form matches the 1/2 relation and gives it lower-bound meaning.","marker":"[65]"},{"why":"Provides the pseudo-spectral solver, the kinetic-regime background initial conditions, and the energy-spectrum method used here.","marker":"[27]"},{"why":"Reports cosmological simulations in which soliton formation and scaling consistent with the 1/2 relation were first observed.","marker":"[21]"},{"why":"Collects published cosmological soliton-halo measurements used in the paper's comparison of the band with structure-formation simulations.","marker":"[59]"}],"fun_headline_variants":["Two energy equalities set soliton mass limits","Soliton-halo relation bridled by two ratios","Simulations bound soliton mass between two lines","Soliton growth capped by halo energy ratio"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole argument depends on the halo being gravitationally settled (virialized) and nearly free of unbound, escaping matter, so that its kinetic energy per mass can stand in for its total energy per mass; halos still collapsing or full of debris would shift both the lower and upper bounds.","fun_headline_variants_meta":{"raw":{"variants":["Two energy equalities set soliton mass limits","Soliton-halo relation bridled by two ratios","Simulations bound soliton mass between two lines","Soliton growth capped by halo energy ratio"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000339,"raw_usage":{"total_tokens":1888,"prompt_tokens":981,"completion_tokens":907,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":597,"completion_tokens_details":{"reasoning_tokens":846}},"tokens_in":597,"tokens_out":907,"duration_ms":8543,"temperature":1.0,"reasoning_tokens":846,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T11:09:47.295218+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a flat-box simulation with the same initial conditions but inject a population of positive-energy unbound particles after the soliton forms; if the soliton mass can be pushed above the $E_{\\rm sol} = E_{\\rm halo}$ line while the bound halo is unchanged, the 1/3 upper bound holds only with a bound-particle definition of halo energy. Alternatively, reanalyse the 2017-relation simulations using kinetic energy only: if the points stay above the 1/3 line instead of moving into the band, the central claim fails.","supporting_citations":[{"cited_title":"Relaxation in a Fuzzy Dark Matter Halo. II. Self-consistent kinetic equations","cited_arxiv_id":"2010.10212","evidence_quote":"Collects published cosmological soliton-halo measurements used in the paper's comparison of the band with structure-formation simulations."}],"review_version":1}