{"id":"6e22921d-7810-41b8-ae68-495d6c8d15cb","arxiv_id":"2502.04456","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A quadratically coupled ultralight scalar field develops a spatial profile around Earth that can boost the field-gradient 'axion wind' by orders of magnitude at low masses, and a nonzero incoming velocity removes the divergences of the static attractive solutions.","lead":"This paper calculates how a scalar dark matter field is distorted near Earth when it couples quadratically to ordinary matter, and how that distortion changes the gradients measured by experiments such as CASPEr. The central result is a predicted enhancement of the effective axion-wind velocity by many orders of magnitude at low dark matter masses.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Fig. 7's largest attractive enhancement factors are square-well resonances; a realistic Earth density profile could shift or broaden them, but the broad ~1/(mR) low-mass enhancement likely survives.","rationale":"The reader's weakest-assumption pick matches mine: the piecewise-constant Earth density is the main quantitative soft spot. I do not find a fatal flaw: the v->0 check in Appendix B is consistent, the Appendix C proof that k>0 removes the divergences is sound, and the l=0 resonance behavior phi ~ 1/k is a standard low-energy scattering effect whose integrated power remains finite. The qualitative claim, that quadratic couplings can boost gradient signals by orders of magnitude at low masses, does not rely on fine-tuned square-well resonances once lambda rho R/m >> 1, because the surface gradient is then set by the exterior 1/r solution and veff ~ 1/(mR). The remaining uncertainty is quantitative, affects mainly the largest attractive contours, and is explicitly acknowledged in the text. A realistic-density rerun of Fig. 7 would settle whether the map is usable for projections; unless that test fails badly, the reader's ACCEPT verdict stands.","tokens_in":22367,"tokens_out":30688,"duration_ms":348476,"concrete_test":"Recompute the v=0 radial profiles and the Sec. 4 power spectra replacing Eq. (5) with a realistic spherically averaged density profile (e.g., PREM plus an exponential atmosphere) while keeping all other inputs and the measurement point unchanged. For each grid point of Fig. 7, evaluate the ratio Eq. (46); if the maximum attractive ratios drop by more than an order of magnitude or the enhancement contours move outside the currently allowed parameter region, the map is model-dependent. Also confirm partial-wave convergence by doubling l_max for a few representative points.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is supported by explicit analytic solutions, but its quantitative map (Fig. 7, Eqs. (42)-(46)) rests on the piecewise-constant density model: Eq. (5) for the v=0 profiles and the homogeneous-sphere approximation for the partial-wave computation in Secs. 3-4. The huge attractive ratios (up to ~1e13) occur near bound-state resonances whose positions and widths are fixed by square-well phases gamma_i R_i. With a continuously varying density (core/mantle/crust, exponential atmosphere), the WKB phase integral replaces gamma_i R_i; resonances shift and acquire finite widths, and the repulsive barrier profile changes. This could materially reduce the largest contours of Fig. 7. The paper explicitly flags the model as 'clearly still very simplistic' (Sec. 2.1) and the power-spectrum estimate as approximate (Sec. 4.2), but it does not quantify how sensitive Fig. 7 is to this choice. The broad strong-coupling low-mass enhancement, where the field is expelled or oscillatory and veff saturates at ~1/(mR), is more robust because it is controlled by the exterior 1/r solution; so the qualitative claim likely survives even if the resonant peaks do not.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the effect of a quadratic coupling of a light scalar dark matter field to ordinary matter, L ⊃ −(λ/2) φ²ρ, on the field value and gradient near Earth's surface. The authors first treat the static (zero-relative-velocity) case for one- and two-layer spherical Earth models, introduce an effective velocity veff = |∇φ|/(m φ∞), and show that the gradient can exceed the naive velocity-suppressed estimate at low masses. They then solve the Klein-Gordon equation for an incoming plane wave with velocity v using partial-wave matching, verify the v→0 limit in Appendix B, and prove in Appendix C that the attractive case has no poles for k>0. A Maxwellian halo velocity distribution is used to compute gradient and field power spectra via a random-phase ensemble average, leading to signal-power ratios in Figs. 7 and 8. The central claim is that quadratic interactions can enhance gradient-sensitive experimental signals by orders of magnitude at low masses, and that including the velocity reduces the previously reported field-value modifications at higher masses.","tokens_in":22625,"tokens_out":29602,"duration_ms":306011,"significance":"This is a relevant and timely phenomenological question for wave-like dark matter detection. The analytic treatment is a strength: the partial-wave matching conditions are explicit, the v→0 limit is checked, the no-pole proof for attractive couplings is a useful addition, and the power-spectrum formalism correctly avoids the naive cancellation of gradients when random phases are averaged. The main limitation, that Earth is modelled as a piecewise-constant density sphere, is explicitly acknowledged by the authors in Sec. 2.1 ('clearly still very simplistic') and Sec. 4.2 ('involving approximations'). In my reading, this affects the precise quantitative contours of Fig. 7, especially the narrow attractive resonances, but it does not undermine the robust broad enhancement at low masses, which is controlled by the exterior 1/r solution. The paper therefore makes a credible case that quadratic interactions should be included when estimating the sensitivity of gradient-based experiments such as CASPEr-wind, QUAX, and accelerometers, and also improves the treatment of field-value experiments by including the halo velocity.","major_comments":[],"minor_comments":[{"comment":"The stated scaling ∫dω S_φφ(ω) ∼ 1/m² for small m appears inconsistent with the expressions in Appendix D when one uses φ0² ∝ 1/m² and σ, Λ ∝ m; Eqs. (79)-(80) instead give ∼1/m⁴. Please correct the scaling or clarify what is held fixed in the limit.","section":"Sec. 4.3, Eqs. (48), (79)-(80)"},{"comment":"The paper should add an explicit sentence near Fig. 7 stating that the high-enhancement contours in the attractive panel are square-well resonances whose positions and widths depend on the piecewise-constant density profile, so a continuous Earth density would shift or broaden them; the broad low-mass enhancement is more robust.","section":"Sec. 4.2, Fig. 7"},{"comment":"Ref. [27] cites a YouTube playlist; please replace it with a standard quantum-mechanics textbook or review that discusses scattering from a spherical potential.","section":"References, Ref. [27]"},{"comment":"The sentence introducing φ̃ as the modulus of the complex solution is imprecise; φ̃ is the complex spatial amplitude and |φ̃| is its modulus, so that φ = |φ̃| cos(ωt − arg φ̃). Please rephrase.","section":"Sec. 3.1, Eq. (22)"},{"comment":"The notation 'log10(|rX|/m)' in the captions is unclear; please use |∇X|/m (or veff) explicitly and state that φ0 is normalized to a reference value.","section":"Fig. 1 and Fig. 3 captions"},{"comment":"Please specify whether the frequency integral in Eq. (46) runs over positive frequencies only and note that the same convention is used for the free and interacting spectra; the unlabelled ∫dω is ambiguous.","section":"Sec. 4.2, Eq. (46)"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is a coordinated companion to [55]; the overlap is acknowledged and the results appear complementary. The quantitative plots are sensitive to the Earth density model, but the authors already flag this limitation and the broad qualitative claim is robust. A brief robustness discussion would strengthen the paper; the remaining issues are local. No concerns about novelty or citation practice beyond the YouTube reference."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nQuick take on arXiv:2502.04456. The genuinely new part is the move from static density profiles to the observables gradient experiments actually measure: field gradients, nonzero-velocity partial-wave solutions, and velocity-averaged power spectra. Hees et al. and Banerjee et al. did the static density profiles; this paper adds the gradient signal and the velocity treatment. That matters because it changes expected sensitivities for CASPEr-wind, QUAX, and accelerometers, in places by orders of magnitude.\n\nThe analytic core is standard scattering theory, and the appendices are honest and useful. The v→0 limit check, the no-pole proof for k>0, and the finite integrated power spectrum all hold up. The authors also flag the model limitations themselves, which is good practice.\n\nThe soft spot is exactly what you'd suspect: the quantitative enhancement maps in Figs. 7 and 8 are built on a homogeneous sphere or a two-layer sphere with constant densities. The huge attractive enhancements (up to ~1e13) are square-well resonances whose positions and widths depend on γ_i R_i. A realistic density profile with core/mantle/crust gradients will shift and broaden them. The stress-test note is right that this could materially reduce the largest contours of Fig. 7. The paper calls the model 'clearly still very simplistic' (Sec. 2.1) but doesn't quantify how sensitive the resonance structure is to the density profile. That's a real gap, though not a fatal one. The broad ~1/(mR) low-mass enhancement, driven by the exterior 1/r solution, should survive a more realistic profile.\n\nOther caveats are minor. The 'effective velocity' exceeding c is acknowledged as a normalized gradient. The attractive-divergence discussion is careful, and the estimate that problematically large field values appear only below ~1e-16 eV is plausible, though the time-dependent argument is schematic.\n\nVerdict: send it to peer review. The central claim—quadratic couplings can substantially enhance gradient signals—is well supported. The quantitative maps need a caveat that the resonant peaks are model-dependent, but the qualitative effect is solid. I'd ask the authors to add a short sensitivity discussion to the density profile. For a reading group it's a maybe, mostly useful for ultralight DM phenomenologists. I'd cite it for the partial-wave solutions and the power-spectrum framework.","headline":"Solid scattering-theory extension of quadratically coupled ultralight DM to gradient observables; the quantitative enhancement maps lean on square-well Earth models, but the broad low-mass effect is robust.","tokens_in":23164,"tokens_out":1709,"would_cite":true,"duration_ms":17900,"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":"Quadratic interactions with Earth can boost the effective dark-matter gradient signal by orders of magnitude at low masses.","keywords":["axion-like particles","ultralight dark matter","quadratic scalar coupling","field gradients","axion wind","partial wave scattering","power spectrum","resonant enhancement"],"falsifier":"A direct test would be an experiment sensitive to the radial gradient of an ultralight scalar field at a mass below roughly $10^{-12}$ eV that sees no increase in signal power above the free-field expectation for couplings in the range $10^{-14}$–$10^{-10}$ GeV$^{-1}$ where the paper predicts order-of-magnitude enhancement; the absence of the predicted low-velocity tail in the measured power spectrum would also falsify the mechanism.","tokens_in":76,"feed_emoji":"🌍","tokens_out":7057,"duration_ms":129177,"temperature":0.7,"pith_summary":"This paper argues that light scalar dark matter with a quadratic coupling to ordinary matter can be strongly modified by Earth, producing field gradients that mimic a much larger dark-matter velocity. It solves the classical field equations around a spherical Earth and shows that for small boson masses the effective gradient at the surface can exceed the naive velocity-suppressed estimate by many orders of magnitude. If correct, experiments sensitive to gradients in the dark-matter field—such as nuclear spin-precession searches and accelerometers—would have significantly stronger reach than previously estimated at low masses. The paper also updates the expected field value at Earth's surface and studies realistic velocity distributions via power spectra.","feed_headline":"Earth's pull can boost axion-wind signals by orders of magnitude","feed_subtitle":"Quadratic dark-matter couplings create large field gradients near Earth, boosting gradient-based searches at low masses.","key_machinery":"The key object is the quadratic coupling $-\\frac{\\lambda}{2}\\phi^2\\rho$ in the action, which changes the Klein-Gordon equation to $(\\Box+m^2+\\lambda\\rho)\\phi=0$; with Earth modeled as a constant-density sphere (or two-layer sphere with atmosphere), $\\lambda\\rho(r)$ becomes a piecewise constant potential. The spatial part of the stationary solution is a combination of trigonometric or hyperbolic functions in each layer, matched by continuity at the boundaries. For nonzero incoming velocity the problem is solved by partial-wave expansion in spherical Bessel functions, with the enhancement quantified by the effective velocity $v_{\\rm eff}=|\\nabla\\phi|/(m\\phi(\\infty))$. This machinery converts the strength of the quadratic coupling into concrete predictions for power spectra of the field and its gradient on Earth's surface.","core_discovery":"The central claim is that the quadratic coupling term $-\\frac{\\lambda}{2}\\phi^2\\rho$ in the action transforms the dark-matter field around Earth, so that the field itself develops a spatial profile with gradients set by the local matter density rather than by the incoming velocity. Using the effective velocity $v_{\\rm eff}=|\\nabla\\phi|/(m\\phi(\\infty))$, the paper finds enhancements that grow as $1/m$ at low masses, reaching many orders of magnitude above the canonical $v\\sim 10^{-3}$ expectation. For attractive couplings, resonances appear when Earth's potential well supports a new bound state, and the paper shows these divergences are absent for any nonzero incoming velocity and remain finite in the frequency-integrated power spectrum. It further provides maps of the signal-power ratio for gradient and field-value experiments over the mass--coupling plane, and identifies a residual concern: for masses $m\\lesssim 10^{-16}$ eV, field values could grow to problematic levels, motivating a time-dependent treatment.","pith_inferences":["The same enhancement mechanism should operate near other dense bodies, such as neutron stars and white dwarfs, where the density is far larger; the quadratic potential $\\lambda\\rho$ could shift resonances and produce even stronger local gradients than around Earth.","A measurable prediction implied by this paper is that the gradient power spectrum should acquire a low-velocity tail with a characteristic $1/v$ divergence at resonance, which could be searched for by examining the frequency shape rather than the total power.","If the enhancement is real, the effective direction of the axion wind seen by gradient experiments may be distorted relative to the halo velocity because the quadratic interaction creates a radial component; this directional signature could help discriminate the effect from ordinary velocity.","The authors' treatment assumes a universal coupling to total matter density; allowing species-dependent couplings would introduce composition-dependent gradients that might be probed with dual-species experiments."],"forward_implications":["Gradient-based searches such as spin-precession experiments and accelerometers should include quadratic-coupling effects when setting limits at low axion masses.","Experiments sensitive to the field value, including haloscopes, see modified expected signals, with the modification less severe than a zero-velocity estimate once kinetic energy is included.","The previously noted divergences for attractive couplings at zero velocity are resolved for nonzero velocity, and the frequency-integrated power spectrum remains finite.","At masses below $10^{-16}$ eV, field values may become large enough to be problematic, so time-dependent modeling is needed in that regime."],"supporting_citations":[{"why":"Supplies the zero-velocity two-layer solution and boundary conditions for the quadratic coupling that the paper extends.","marker":"[7]"},{"why":"Gives the previous estimate of the field-value modification that is improved by including nonzero velocity.","marker":"[8]"},{"why":"Identifies the divergences for attractive couplings that the dynamical solutions resolve and adds axion bounds.","marker":"[9]"},{"why":"Provides the concrete gluon-coupled axion-like particle model motivating the quadratic coupling strength.","marker":"[16]"},{"why":"Supplies the partial-wave scattering analogy and resonance condition used for the nonzero-velocity solutions.","marker":"[27]"}],"fun_headline_variants":["Earth's pull boosts axion-wind signals by orders of magnitude","Gravity amplifies axion-wind gradients near Earth","Quadratic axion couplings sharpen Earth-bound signals","Low-mass axion wind magnified by Earth's density","Axion wind gains a gravitational boost at Earth's surface"],"cache_read_input_tokens":25344,"weakest_assumption_plain":"The quantitative enhancement maps rest on modeling Earth as a homogeneous sphere (plus a constant-density atmosphere), so the quadratic potential $\\lambda\\rho(r)$ is piecewise constant; real density gradients, time dependence, or local environmental effects could shift resonance positions and enhancement factors.","fun_headline_variants_meta":{"raw":{"variants":["Earth's pull boosts axion-wind signals by orders of magnitude","Gravity amplifies axion-wind gradients near Earth","Quadratic axion couplings sharpen Earth-bound signals","Low-mass axion wind magnified by Earth's density","Axion wind gains a gravitational boost at Earth's surface"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000433,"raw_usage":{"total_tokens":2185,"prompt_tokens":902,"completion_tokens":1283,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":518,"completion_tokens_details":{"reasoning_tokens":1202}},"tokens_in":518,"tokens_out":1283,"duration_ms":10875,"temperature":1.0,"reasoning_tokens":1202,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T22:39:40.369668+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct test would be an experiment sensitive to the radial gradient of an ultralight scalar field at a mass below roughly $10^{-12}$ eV that sees no increase in signal power above the free-field expectation for couplings in the range $10^{-14}$–$10^{-10}$ GeV$^{-1}$ where the paper predicts order-of-magnitude enhancement; the absence of the predicted low-velocity tail in the measured power spectrum would also falsify the mechanism.","supporting_citations":[{"cited_title":"Quantum Theory of collisions: resonant condition in the lth partial wave","cited_arxiv_id":null,"evidence_quote":"Supplies the partial-wave scattering analogy and resonance condition used for the nonzero-velocity solutions."}],"review_version":1}