{"id":"556572c1-065e-4aee-a17f-326540459fff","arxiv_id":"2506.22431","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A wide-angle streaming model including gravitational redshift, lightcone and kinematic effects explains the dipole turnover at ~20 h^-1 Mpc as an advection-like shift driven by the density-weighted pairwise potential difference.","lead":"This paper builds a nonlinear model of how gravitational redshift and other relativistic effects distort the observed clustering of galaxies, and shows it reproduces the measured turnover in the dipole correlation at about 20 megaparsecs. The model identifies the difference in gravitational potential between two galaxy populations as the key ingredient, which will matter for interpreting upcoming DESI and Euclid measurements.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The model's key nonlinear input, the pairwise potential difference, is calibrated with concentrations from the same RayGal catalogues used for validation, so the claimed turnover may not be an independent test; an external calibration or fitted zero-point could shift or erase it.","rationale":"The reader's weakest assumption identifies the same load-bearing concern: the turnover prediction is conditional on the halo-model one-point function. I find no internal inconsistency in the derivation of Eq. (63): the perturbative expansion is standard, the reduction to pairwise functions is clearly shown, and the model recovers linear theory in the appropriate limits. The numerical checks that lightcone, lookback-time, and covariance contributions do not drive the turnover are supporting evidence. The fragility is empirical rather than formal: the one-halo contribution to ⟨⟨Δψ⟩⟩ is not a parameter-free prediction in this validation, because the concentrations and masses in Table II come from the same RayGal catalogues used for the dipole comparison, and §VI.C.3 explicitly allows the degenerate zero-point to be treated as a free parameter. The authors acknowledge that uncertainty in this ingredient translates into uncertainty in the turnover scale. That does not warrant rejection: the turnover is a genuine prediction of the model's structure once a realistic one-point function is supplied, and the comparison is suggestive. It does warrant a conditional verdict pending an independent test of the nonlinear potential input and ideally a released implementation. Since the reader already reached CONDITIONAL, the verdict should remain unchanged.","tokens_in":59640,"tokens_out":3784,"duration_ms":48974,"concrete_test":"Recompute the dipole shown in Figure 4 with all inputs held fixed except ⟨⟨Δψ⟩⟩, using an externally calibrated concentration-mass relation (for example, Diemer and Joyce 2019 or another independent N-body calibration) at the same masses and redshift, and separately by fitting the single degenerate zero-point φ_A(0)-φ_B(0) to the RayGal dipole as permitted by §VI.C.3. If the zero crossing moves by more than about 20% in scale, or if the fitted zero-point differs from the Table II halo-model value by more than about 30%, the claimed turnover is not robust to the self-calibrated input.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that Eq. (27) reproduces the N-body dipole turnover at s ≈ 20 h^-1 Mpc, and §VII.C attributes this to the shift term -⟨⟨Δψ⟩⟩ dξ_AB/ds in Eq. (64). In the numerical evaluation (§V.C.2), ⟨⟨Δψ⟩⟩ is not measured from the simulation or fitted: it is computed from Eqs. (44)-(46), with the one-halo part Eq. (45) dominated by the zero-lag halo potentials φ_A(0) and φ_B(0). Those potentials are evaluated from Eq. (31) using NFW profiles whose concentrations are taken from Table II, which are estimated from the same RayGalGroup halo catalogues against which the model is compared. Section VI.C.3 explicitly states that the zero-point φ_A(0)-φ_B(0) is degenerate with the ξψδ(0) terms and may be treated as a single free parameter, and §VII.C says uncertainty in this quantity translates into uncertainty in whether, and where, the turnover occurs. The agreement at 20 h^-1 Mpc is therefore a test of the model conditional on a halo-model input derived from the same simulation. If that input is inaccurate by the tens of percent that §VII.C contemplates, the cancellation between the positive wide-angle RSD dipole and the negative shift term that produces the turnover would move or disappear, so the headline claim is not yet independently validated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops a nonperturbative, wide-angle streaming model of the galaxy two-point correlation function in redshift space, extended beyond Doppler RSD to include gravitational redshift, lightcone corrections, and lookback-time effects. Starting from number conservation and the radial real-to-redshift map (Eq. 2), the authors derive the integral formula (Eq. 4) and, after truncating the displacement distribution at second cumulant, the Gaussian streaming model (Eq. 9); the lightcone-corrected complete expression is Eq. (27). The model is evaluated from seven linear-theory correlation functions plus a halo-model treatment of the density-weighted potential, and is compared with the RayGalGroup N-body halo catalogues. The central numerical claim is that the dipole turnover at s ≈ 20 h⁻¹ Mpc, absent in linear theory, is reproduced (Fig. 4). A small-displacement expansion yields the perturbative dipole formula (63), in which the turnover is attributed to the shift term −⟪Δψ⟫ dξ_AB/ds, driven by the pairwise potential difference ⟪Δψ⟫. The displacement statistics are decomposed into symmetric and antisymmetric pairwise functions (Section VI), identifying ⟪Δψ⟫ as the key nonlinear ingredient; its one-halo component dominates at all separations (Fig. 6). The paper also gives a compact wide-angle RSD dipole (Eq. G1), a covariant derivation of the lightcone effect, and a treatment of the local potential and the apparent IR divergence of ψ correlations (Appendix C).","tokens_in":59932,"tokens_out":15197,"duration_ms":155801,"significance":"If correct, the framework supplies a complete, directly evaluable template for odd multipoles in the mildly nonlinear regime, with immediate application to DESI BGS and Euclid gravitational-redshift measurements. The derivation is unusually transparent: linear theory is recovered in a few lines (Section III.C.1), the lightcone effect is derived from first principles (Section III.B.1), and the perturbative dipole (63) is obtained without fitted nuisance parameters. The algebraic results in Appendices F–H are checkable, the DESI wide-angle forecasts (Appendix E) are quantitative, and the predicted second turnover at the halo-exclusion scale (Section VII.C) is falsifiable. The authors are also candid about the model's approximations (Gaussian truncation, linear bias and linear evolution for all but one correlation). The principal weakness is that the validation of the headline turnover is conditional: the one-halo input (Eq. 45) is computed with concentrations from the same RayGal catalogues used for comparison, and the total zero-point is degenerate with a single free parameter. The agreement at 20 h⁻¹ Mpc is therefore encouraging but not yet an independent test of the model.","major_comments":[{"comment":"The numerical validation of the headline claim is conditional on halo-model inputs derived from the same simulation against which the model is compared. The one-halo term (45) is evaluated with NFW profiles whose concentrations c_A and c_B are estimated from the RayGalGroup catalogues (Table II), and Section V.C.2 reports a value φ_A(0) = 0.036 h⁻¹ Mpc that is about 1.6 times the two-halo value ξ^HM_{ψδA}(0) = 0.022 h⁻¹ Mpc. Section VII.C itself states that uncertainty in the one-point function “translates to uncertainty in the scale at which the turnover occurs—or whether it occurs at all,” and Section VI.C.3 notes that the total zero-point may be treated as a single free parameter. This is not circular in the strict sense, since the dipole measurement is not used as a fitting target, but it does compromise the independence of the test. I recommend adding a sensitivity analysis: vary the concentrations over plausible ranges (or adopt an external concentration–mass relation), and report the predicted turnover scale as a function of the degenerate zero-point. If the turnover scale is robust to these variations, the claim should be restated with that evidence; if not, the abstract and conclusions should be softened accordingly.","section":"§V.C.2, Table II; §VI.C.3; §VII.C"},{"comment":"The physical attribution of the turnover to the shift term (64) rests on the perturbative dipole (63), but the agreement between Eq. (63) and the full nonperturbative model (27) is not quantified. Eq. (63) is obtained from Eq. (H2) after dropping the disconnected term ∂_a∂_b(m^a_[AB] m^b_(AB)), and the four contributions plotted in Fig. 7 are compared with the full streaming model only visually. The authors should state the size and origin of any residual difference between the total of Eq. (63) and Eq. (27) on the scales of the turnover, and report the turnover position and amplitude predicted by Eq. (63) versus the full model, since the interpretation of the turnover as an advection-like effect is built on this expansion.","section":"§VII.B, Eq. (63), Fig. 7"}],"minor_comments":[{"comment":"The comparison in Fig. 4 uses the opposite dipole sign convention (the model is multiplied by −1), but this is stated only in the last sentence of Appendix G; the convention should be stated where the comparison is presented.","section":"§V.C.2, Appendix G"},{"comment":"The “streaming model (full sky)” reference curve should specify which covariance components are retained (the caption mentions only the connected piece) and whether the lightcone-corrected weighting (28) is used.","section":"Fig. 7 caption"},{"comment":"Table III sets the NFW concentration to c = 9 for all masses at z = 0.2, whereas Section V uses the catalogue-dependent concentrations of Table II; given the role of φ_A(0) in the one-halo term, the sensitivity of the Table III values to c should be stated.","section":"Table III"},{"comment":"The RayGalGroup comparison retains only Doppler and gravitational-redshift contributions and excludes the lightcone and lookback-time effects, so the validation exercises only part of Eq. (27); the conclusions should state this scope explicitly even though Fig. 5 shows the excluded effects to be small for the dipole.","section":"§V.C.1, §VIII"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern about circularity is only partially warranted: the dipole measurement itself is not used as a fitting target, so the procedure is not circular in the strict sense, but the headline validation is conditional on halo-model inputs calibrated on the same RayGal catalogues, and the authors themselves concede the turnover could move or disappear under the admitted uncertainty in the one-point function. I regard this as fixable within a revision via sensitivity tests (external concentration–mass relation, zero-point scan). The paper is a natural continuation of Ref. [60] and is long, but the appendices are functional technical content rather than padding. I see no issue with citation patterns or fit to the journal's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper gives the LSS community a compact streaming-model formula for the redshift-space correlation function that includes gravitational redshift, wide-angle effects, lightcone effects, and lookback time at order H/k. The genuinely new pieces are the pairwise decomposition into symmetric and antisymmetric functions and the identification of the shift term -<<Delta psi>> d xi_AB/ds as the cause of the dipole turnover around 20 h^-1 Mpc. That is a real step forward: earlier models either used an EFT-like nuisance parameter fitted to simulations or added an ad-hoc potential correction, and this paper shows how that correction arises naturally from density weighting. The derivation is careful and self-contained. Equation (27) is compact, and the authors recover linear theory from it in a few lines. The checks that C_cross and C_grav do not matter for the dipole are useful. The discussion of the infrared divergence of potential correlations and the role of the local potential in Appendix C is thoughtful and physically clear. The comparison with RayGal is encouraging, and the perturbative expression Eq. (63) will be handy for forecasts. The main soft spot is exactly what the stress-test note says. The one-halo contribution to <<Delta psi>> is computed with NFW concentrations estimated from the same RayGal catalogues used for validation. The paper is honest about this: Section VI.C.3 says the zero-point can be treated as a single free parameter, and Section VII.C says uncertainty in that quantity translates into uncertainty in whether, and where, the turnover occurs. So the headline claim that the turnover is \"well accounted for\" should be read as \"the model can reproduce the turnover if the one-point function is chosen appropriately,\" not as an independent prediction. That does not sink the paper - the framework and the physical mechanism are still valuable - but it does mean the validation is conditional. A cleaner test would measure <<Delta psi>> directly from the simulation, or fit the zero-point and show the turnover scale is not overly sensitive to it. This paper is for anyone building templates for DESI or Euclid dipole measurements and for people working on relativistic corrections in clustering. It deserves a serious referee, and the referee should push on the independence of the halo-model input and ask for error bars or a released implementation. I would accept it for peer review and would cite it if I worked on the dipole.","headline":"A careful, useful streaming-model framework for the gravitational-redshift dipole, with a convincing physical mechanism for the turnover but a validation that is partly calibrated to the same simulation it is tested against.","tokens_in":652,"tokens_out":1257,"would_cite":true,"duration_ms":36119,"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":"A compact nonlinear streaming model reproduces the observed turnover of the gravitational-redshift dipole at roughly 20 h^-1 Mpc and traces it to the density-weighted pairwise potential difference rather than to lightcone effects.","keywords":["gravitational redshift","dipole correlation function","redshift-space distortions","wide-angle effects","pairwise potential difference","streaming model","lightcone effects","cosmology"],"falsifier":"Compute the density-weighted pairwise potential difference $\\langle\\!\\langle\\Delta\\psi\\rangle\\!\\rangle$ directly from N-body halo catalogues at the same redshifts and compare it with the one-halo-plus-two-halo prediction; if the measured statistic differs significantly near $20\\,h^{-1}\\mathrm{Mpc}$, the claimed cause of the turnover is falsified.","tokens_in":59445,"feed_emoji":"🌌","tokens_out":6259,"duration_ms":70833,"temperature":0.7,"pith_summary":"This paper aims to provide a complete nonlinear model of the galaxy two-point correlation function in redshift space, keeping every relativistic effect of order H/k: gravitational redshift, redshift-space distortions, wide-angle effects to all orders, and lightcone and lookback-time effects. The model is a wide-angle streaming model whose displacement statistics are density-weighted cumulants of velocity and potential. Compared with N-body halo catalogues, it reproduces the observed turnover of the dipole at $s\\simeq 20\\,h^{-1}\\mathrm{Mpc}$, a feature linear theory misses. The authors argue that the turnover is caused by a specific nonlinear shift term, $-\\langle\\!\\langle\\Delta\\psi\\rangle\\!\\rangle\\,\\mathrm{d}\\xi_{AB}/\\mathrm{d}s$, and that the key physical ingredient is the density-weighted pairwise potential difference, not the lightcone. If the model is right, it supplies the template needed to interpret upcoming gravitational-redshift dipole measurements.","feed_headline":"Dipole turnover at 20 Mpc traced to potential differences","feed_subtitle":"A nonlinear streaming model reproduces the gravitational-redshift dipole and locates the sign change in a density-weighted term.","key_machinery":"The machinery is the real-to-redshift radial displacement map $\\delta\\chi = u\\cdot\\hat{n} + \\psi - \\psi_O$, with $\\psi = -H^{-1}\\Psi$, inserted into a full-sky Gaussian streaming model. The transition probability $p(\\chi|\\chi')$ is built from density-weighted cumulants, notably the mean $m = m_{\\mathrm{RSD}} + m_{\\mathrm{grav}}$; decomposing $m$ under tracer exchange yields pairwise functions, of which the antisymmetric pairwise potential difference $\\langle\\!\\langle\\Delta\\psi\\rangle\\!\\rangle(r) = \\langle\\!\\langle\\psi(x_1)-\\psi(x_2)\\rangle\\!\\rangle$ carries the gravitational-redshift signal. A small-displacement expansion of the streaming integral produces a closed dipole formula, Eq. (63), whose shift term is the product of $\\langle\\!\\langle\\Delta\\psi\\rangle\\!\\rangle$ with the slope of the real-space cross-correlation. The halo model enters to supply the one-point values $\\phi_A(0)$ and $\\phi_B(0)$ through the one-halo term.","core_discovery":"The central claim is that a single compact formula, Eq. (27), captures all order-$H/k$ asymmetries in the redshift-space correlation function, and that the dipole turnover follows from the pairwise potential difference $\\langle\\!\\langle\\Delta\\psi\\rangle\\!\\rangle$ through an advection-like shift term. The paper shows that the mean displacement separates into a redshift-space-distortion part and a gravitational-redshift part, that only the gravitational part has a nonvanishing one-point function because density weighting prefers potential wells, and that the one-halo contribution to the pairwise potential difference dominates and is required to reproduce the turnover. In the perturbative expansion the dipole is given by Eq. (63), where the shift term $-\\langle\\!\\langle\\Delta\\psi\\rangle\\!\\rangle\\,\\mathrm{d}\\xi_{AB}/\\mathrm{d}s$ is negative and grows on small scales, canceling the positive wide-angle RSD contribution near $20\\,h^{-1}\\mathrm{Mpc}$. The lightcone and lookback-time effects are shown to be small and not the cause of the turnover.","pith_inferences":["A testable extension the paper leaves implicit: replacing the halo-model one-point values with a direct simulation measurement of $\\langle\\!\\langle\\Delta\\psi\\rangle\\!\\rangle$ should predict the dipole with no free zero-point, sharpening forecasts for upcoming surveys.","The shift mechanism implies a second turnover near the halo-exclusion scale, where the pair probability and its slope change sign; the paper notes this expectation but leaves the quantitative prediction to future work.","The same pairwise-potential logic should generate predictable amplitudes in the octupole and higher odd multipoles, so measuring those would independently confirm that potential differences, rather than kinematics, drive the asymmetry."],"forward_implications":["The dipole's turnover near $20\\,h^{-1}\\mathrm{Mpc}$ becomes a predicted feature of the model, so future measurements can use it as a standard signature of gravitational redshift rather than an anomaly.","Because lightcone and lookback-time effects are subdominant, the turnover scale can be used to infer the density-weighted pairwise potential difference and hence the depth of potential wells at tracer positions.","The pairwise-function decomposition yields a compact formula for the wide-angle RSD dipole in terms of the pairwise mean velocity, making wide-angle corrections straightforward to add to standard streaming-model analyses.","The same formalism gives a physical origin for the empirical nonperturbative correction used in earlier quasi-linear dipole models.","The covariance contributions from gravitational redshift and velocity-potential cross-correlations are numerically negligible for the dipole, so the mean displacement supplies the dominant antisymmetric signal."],"supporting_citations":[{"why":"Supplies the wide-angle streaming model and the real-to-redshift map that this paper generalises to include gravitational redshift and lightcone effects.","marker":"[60]"},{"why":"Establishes the linear-time theory of all dipole contributions that the nonlinear model must contain.","marker":"[28]"},{"why":"Provides the earlier quasi-linear dipole model whose empirical nonperturbative correction is given a physical origin here.","marker":"[48]"},{"why":"Provide the RayGal N-body halo catalogues that supply both the dipole measurements and the halo concentrations used in the one-halo term.","marker":"[67, 75]"},{"why":"Supplies the halo-model framework used to compute the one-halo and two-halo parts of the pairwise potential difference.","marker":"[70]"},{"why":"Give the streaming model and the pairwise velocity difference whose conservation logic underlies the shift-term derivation.","marker":"[56, 58]"},{"why":"Gives the one-loop Fourier-space dipole model with an EFT nuisance parameter that this configuration-space model seeks to avoid.","marker":"[47]"}],"fun_headline_variants":["Gravitational redshift dipole traced to potential differences","Dipole turnover at 20 Mpc from pairwise potential difference","Nonlinear model links dipole flip to potential wells","Gravitational redshift: dipole sign change explained by potential","Pairwise potential difference sets dipole turnover scale"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the halo-model estimate of the one-point potential values $\\phi_A(0)$ and $\\phi_B(0)$, with concentrations taken from the same catalogues used for comparison, is accurate; if this one-point function is wrong, the predicted turnover shifts in scale or disappears.","fun_headline_variants_meta":{"raw":{"variants":["Gravitational redshift dipole traced to potential differences","Dipole turnover at 20 Mpc from pairwise potential difference","Nonlinear model links dipole flip to potential wells","Gravitational redshift: dipole sign change explained by potential","Pairwise potential difference sets dipole turnover scale"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000197,"raw_usage":{"total_tokens":1372,"prompt_tokens":959,"completion_tokens":413,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":575,"completion_tokens_details":{"reasoning_tokens":338}},"tokens_in":575,"tokens_out":413,"duration_ms":4369,"temperature":1.0,"reasoning_tokens":338,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:03:53.330929+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the density-weighted pairwise potential difference $\\langle\\!\\langle\\Delta\\psi\\rangle\\!\\rangle$ directly from N-body halo catalogues at the same redshifts and compare it with the one-halo-plus-two-halo prediction; if the measured statistic differs significantly near $20\\,h^{-1}\\mathrm{Mpc}$, the claimed cause of the turnover is falsified.","supporting_citations":[{"cited_title":"Modelling the asymmetry of the halo cross-correlation function with relativistic effects at quasi-linear scales","cited_arxiv_id":"2004.03772","evidence_quote":"Supplies the wide-angle streaming model and the real-to-redshift map that this paper generalises to include gravitational redshift and lightcone effects."}],"review_version":1}