{"id":"d53fb62f-bc68-46ef-b93e-14cde343fb85","arxiv_id":"2412.06553","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Including the quadrupole in power-spectrum forecasts improves f_NL precision by 45% to 63%, while neglected relativistic and wide-angle corrections shift f_NL by up to 0.6 sigma for MegaMapper.","lead":"This paper forecasts how well two future galaxy surveys can measure primordial non-Gaussianity, including wide-angle and relativistic corrections to the galaxy power spectrum. It finds the quadrupole greatly improves precision, and that ignoring corrections biases f_NL by about 0.6 sigma for MegaMapper but nearly zero for SKAO2.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The SKAO2 'negligible shift' result rests on a deliberately truncated integrated correction; the omitted L×L and cross-bin lensing terms are unsuppressed and could substantially change the near-cancellation, as the authors themselves warn in Sec. 5.","rationale":"I read the paper as a forecast whose headline numbers are (i) the precision gain from adding the quadrupole and (ii) the bias in f_NL from neglecting relativistic and wide-angle corrections. The precision gain is explicitly optimistic because wide-angle mode coupling is ignored, but as a relative comparison it may survive that caveat. The shift is more fragile: Table 4 shows the SKAO2 null result is the difference of two ~0.5σ shifts of opposite sign, and the integrated piece entering that difference is computed only to leading order with unsuppressed L×L and cross-bin terms omitted. Since the conclusion 'negligible shift for SKAO2' rests on that cancellation, the reliability of the central claim depends on omitted terms being small; no evidence for that is provided, and the authors explicitly warn that their approximations may suppress the shift. This is precisely the reader's weakest assumption, so I agree with the CONDITIONAL verdict. The paper is internally consistent and discloses the limitation, so I would not move to REJECT; the specific shift numbers should be treated as provisional until the missing terms are computed or a full angular-space cross-check is performed.","tokens_in":21009,"tokens_out":7914,"duration_ms":87914,"concrete_test":"Add the leading-order L×L contribution to P_I by using K_L from Eq. 2.5 in both factors of Eq. 2.9 (i.e., compute the L×L analogue of J), and recompute Table 4 for SKAO2 and MegaMapper with the same survey specifications, binning and Fisher setup. If the SKAO2 shift moves outside |δf_NL/σ| < 0.1, or the MegaMapper shift changes by more than ~0.1σ, the near-cancellation is an artifact of the truncation. A natural cross-check is a full angular power-spectrum Fisher forecast including cross-bin correlations, as in Refs. [9,11].","verdict_should_be":"UNCHANGED","load_bearing_attack":"Table 4 is computed from P_corr = P_NI + P_I with P_I truncated to I×S and S×I only (Eq. 1.8; Sec. 5). This truncation is not benign: the lensing kernel K_L in Eq. 2.5 has a part with no positive power of k/H, so L×L is not parametrically suppressed relative to the kept terms. Lensing-wide-angle and cross-bin lensing correlations are also omitted. The headline SKAO2 result is the difference of two large opposite shifts: NI gives +0.513σ and I gives -0.498σ, for a total of -0.001σ (Table 4). A modest fractional change in P_I from the omitted terms could therefore turn this cancellation into an O(0.1σ) or larger shift, and no estimate of those terms is given. The paper itself flags that the approximations 'may artificially suppress the shift' (abstract and Sec. 5). Thus the advertised conclusion that neglecting the corrections leaves f_NL unbiased for SKAO2 is not established; only the weaker statement that it cancels within the truncated model is supported. The MegaMapper 0.6σ shift is similarly provisional because the same omitted integrated terms enter it.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper extends the leading-order Fourier-space galaxy power spectrum with wide-angle, Doppler/Sachs-Wolfe, and integrated lensing/time-delay/ISW corrections, following the framework of Refs. [33-35], and uses a Fisher forecast to assess the impact on local primordial non-Gaussianity constraints for SKAO2 and MegaMapper. The analysis keeps the monopole and quadrupole, uses equal-volume redshift bins selected by minimizing the forecast error, and computes the shift in f_NL when the corrections are neglected via a nested-model Fisher formula. The headline results are sigma(f_NL)=2.89 for SKAO2 and 0.548 for MegaMapper with the monopole+quadrupole combination, corresponding to improvements of about 45% and 63% over the monopole-only case, and shifts of -0.001 sigma (SKAO2) and 0.614 sigma (MegaMapper) when corrections are omitted. The paper explicitly warns that omitted terms, especially lensing-lensing correlations, may artificially suppress the shift.","tokens_in":21246,"tokens_out":12772,"duration_ms":133226,"significance":"If the quantitative claims hold, the paper makes a useful case that future spectroscopic surveys should include the quadrupole and the leading relativistic and wide-angle corrections when measuring f_NL. The integrated correction is re-derived in Appendix B rather than assumed, and the survey inputs (bias, magnification bias, evolution bias, number density) are specified in tables and fitting functions, which is a strength. The paper is also commendably explicit about its main approximations, including the truncation of the integrated correction and the neglect of mode coupling. However, the headline 'negligible shift for SKAO2' claim is sensitive to exactly those approximations, and the internal breakdown in Table 4 raises questions about the cancellation narrative.","major_comments":[{"comment":"Table 4 is internally inconsistent with an additive reading of the shift formula. For SKAO2, the NI and I entries are 0.513 and -0.498, whose sum is 0.015, but the NI+I entry is -0.001; for MegaMapper, the sum 0.410 + (-0.177) = 0.233 does not equal the quoted 0.614. Since Eq. (4.2) is linear in P_corr when the covariance is held fixed, either the shift calculation is nonlinear because the Fisher matrices or covariances are evaluated differently in the true and wrong models, in which case this must be stated explicitly, or the table entries are not the individual contributions of the two terms. In particular, the abstract and conclusion statement that MegaMapper shows 'partial cancellation' between integrated and non-integrated effects is not supported by the table, where the combined shift exceeds the NI-only shift. Please clarify the calculation and revise the interpretation accordingly.","section":"Sec. 4, Table 4, Eq. (4.2)"},{"comment":"The headline SKAO2 result that neglecting the corrections leaves f_NL unbiased is not established. The integrated correction in Eq. (1.8) is truncated to I x S + S x I, and the lensing-lensing (L x L) term is omitted. The lensing kernel in Eq. (2.5) contains a piece (the 1 - mu^2 term) with no positive power of k/H, so L x L is not suppressed by a positive power of k/H relative to the kept terms, as the authors themselves note in Sec. 5. Lensing-wide-angle and cross-bin lensing correlations are also omitted. Since the SKAO2 total shift is the small difference of two large opposite shifts (+0.513 and -0.498 in Table 4), a modest fractional change in P_I from the omitted terms could change the shift by O(0.1 sigma) or more. The paper's warning that the approximations 'may artificially suppress the shift' (abstract and Sec. 5) is appropriate, but the conclusion that the shift is negligible for SKAO2 should be presented as provisional within the truncated model unless bounds on the omitted terms are provided.","section":"Sec. 1, Eq. (1.8); Sec. 2, Eq. (2.5); Sec. 5"},{"comment":"The claimed precision improvements of about 45% and 63% rest on a Fisher forecast whose covariance ignores wide-angle mode coupling, which the paper acknowledges 'will lead to over-optimistic precision.' In addition, the number of equal-volume redshift bins is selected by minimizing sigma(f_NL) in the same Fisher forecast (Fig. 9), which introduces a selection effect that can bias the forecast low. The abstract and conclusion present the improvement percentages without these caveats. Please either soften the headline numbers or quantify how much of the improvement survives when mode coupling and bin-selection effects are incorporated.","section":"Sec. 3, Eq. (3.1) and Fig. 9"}],"minor_comments":[{"comment":"The Conclusion states improvements of about 40% and 60%, while the Abstract and Table 3 give about 45% and 63%; please harmonize these numbers.","section":"Sec. 5 vs Abstract"},{"comment":"The notation using superscripts 0 and 1 for the Fisher matrices is confusing; please define explicitly which matrix is evaluated at epsilon=0 and which at epsilon=1, and state whether the covariance is held fixed in the shift calculation.","section":"Sec. 4, Eq. (4.2)"},{"comment":"Figure 12 is placed in Appendix A but is not referenced in the text near Eq. (A.4); please add an explicit reference there.","section":"Appendix A, Fig. 12"},{"comment":"The text says the optimal numbers of bins are 6 for SKAO2 and 3 for MegaMapper, but the figure axes do not mark these choices; adding markers or vertical lines would help the reader verify the claim.","section":"Sec. 3, Fig. 9"},{"comment":"In Table 2, the first Delta z entry for SKAO2 (0.720) is much larger than the others; a brief note confirming that these are equal-volume comoving bins would avoid confusion.","section":"Sec. 3, Table 2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is transparent about its approximations, which is a strength, and the Appendix B derivation is a genuine addition. However, the Table 4 inconsistency and the absence of any estimate of the omitted L x L and cross-bin lensing terms need to be addressed before publication, because the SKAO2 cancellation claim depends on them. The reliance on Refs. [33,35] is acceptable given the re-derivation, but the novel quantitative claims should be robust to the omitted terms."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know up front. The durable result is the precision gain: adding the quadrupole to the monopole improves σ(f_NL) by ~45% for SKAO2 and ~63% for MegaMapper, under the standard optimistic-Fisher caveats (no mode coupling, six parameters, no priors). The other headline — \"negligible shift for SKAO2\" — is a near-cancellation of two large opposite shifts (+0.51σ non-integrated, −0.50σ integrated). That is a statement about the truncated model, not a robustness result, and the authors say so in the abstract and in Section 5. Their honesty is the main reason I trust the paper.\n\nWhat's new: the same group extends its own prior results — [35] gave the non-integrated monopole/dipole, [33] the integrated correction — to the quadrupole with both correction types, and turns it into concrete survey guidance: multipole choice, equal-volume binning, and the cost of ignoring corrections. Appendix B re-derives the integrated correction, survey inputs are tabulated, and the NI/I cancellation is a real observation. The math looks sound; the numerics are standard quadrature; the citations to [33,35] are the previous steps in the same derivation chain, not padding.\n\nSoft spots, in order of importance:\n\n1. The integrated correction keeps only I×S and S×I. The stress-test note says L×L is \"not parametrically suppressed\" because K_L has a piece with no positive power of k/H. I think that is the wrong metric: the lensing kernel is small in amplitude, so within-bin L×L is likely a few percent of L×S and will not move the cancellation much. The real gaps are cross-bin lensing correlations and lensing-wide-angle mixing, which a global-midpoint Fourier treatment cannot capture. The authors flag both, and note that the angular analyses [9,11] found larger shifts. The paper does not quantify how big the missing pieces might be, and that discrepancy is the strongest reason to treat the shift numbers as provisional.\n\n2. The shifts are very sensitive to the magnification bias Q and evolution bias E, whose fiducial values carry O(1) uncertainty and which are free parameters in the forecast. The near-cancellation for SKAO2 looks fragile even within the model. The authors mention the sensitivity but do not show how δ(f_NL) moves with Q0, E0 — a referee should ask for that.\n\n3. The absolute precision numbers are lower bounds by construction. I would quote the relative monopole-to-quadrupole gain, which is more robust, not σ(f_NL) = 0.55 for MegaMapper.\n\nMinor: the optimal bin count (6 for SKAO2, 3 for MegaMapper) is chosen by minimizing σ(f_NL) on the same forecast. Mild selection effect, not a real flaw.\n\nBottom line: the paper deserves a serious referee. It is for anyone doing f_NL forecasts for SKAO2, MegaMapper, or similar, and for people working on relativistic power-spectrum corrections. For revision, I would want an order-of-magnitude estimate of the omitted terms, and the SKAO2 shift presented as \"consistent with zero in our truncated model\" rather than as a property of the real survey. Send it out — it is solid and it is honest.","headline":"The quadrupole precision gain is the durable result; the SKAO2 'negligible shift' is a fragile near-cancellation the authors themselves only claim for their truncated model.","tokens_in":21812,"tokens_out":10931,"would_cite":true,"duration_ms":111764,"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":"This paper claims that adding the quadrupole of the galaxy power spectrum, together with leading-order wide-angle and relativistic corrections, improves forecasts of the local primordial non-Gaussianity parameter $f_{\\mathrm{NL}}$ by…","keywords":["primordial non-Gaussianity","f_NL","galaxy power spectrum","wide-angle corrections","relativistic corrections","lensing convergence","parameter forecast","SKAO2"],"falsifier":"Recompute the same two-survey forecasts with the full integrated correction including $L\\times L$ and lensing-wide-angle terms and with cross-bin correlations (e.g., via a spherical Fourier-Bessel or angular power spectrum analysis); if the $f_{\\mathrm{NL}}$ shift for SKAO2 becomes comparable to or larger than $0.1\\sigma$, or the MegaMapper shift changes sign, then the leading-order cancellation is not robust.","tokens_in":20798,"feed_emoji":"🔭","tokens_out":7047,"duration_ms":61153,"temperature":0.7,"pith_summary":"This paper sets out to show that next-generation galaxy surveys can measure the local primordial non-Gaussianity parameter $f_{\\mathrm{NL}}$ far more precisely if the Fourier galaxy power spectrum is modelled with the quadrupole and with relativistic and wide-angle corrections to the standard flat-sky Newtonian spectrum. Using SKAO2 and MegaMapper survey specifications, the authors find that adding the quadrupole to the monopole improves the marginalised error on $f_{\\mathrm{NL}}$ from $\\sigma(f_{\\mathrm{NL}})=5.21$ to $2.89$ (about 45%) for SKAO2 and from $1.49$ to $0.548$ (about 63%) for MegaMapper. They also quantify how much neglecting the corrections biases the estimated $f_{\\mathrm{NL}}$: about $0.6\\sigma$ for MegaMapper, while for SKAO2 the shift is negligible because non-integrated and integrated corrections nearly cancel. Because the corrections mimic the scale-dependent bias signature of local non-Gaussianity, the paper argues that future analyses should include them to avoid biased constraints.","feed_headline":"Quadrupole sharpens f_NL forecasts up to 63%","feed_subtitle":"Adding wide-angle and relativistic corrections cuts the error on the local non-Gaussianity parameter for SKAO2 and MegaMapper.","key_machinery":"The central object is the leading-order corrected Fourier galaxy power spectrum $P_g = P^S_g + P^{\\mathrm{NI}}_g + P^I_g$, expanded in the wide-angle parameter $1/(kr)$ and accurate to order $O(r^{-2}k^{-2},\\, H^2k^{-2},\\, r^{-1}Hk^{-2})$. $P^{\\mathrm{NI}}$ contains the wide-angle, Doppler, and Sachs-Wolfe non-integrated corrections, while $P^I$ is built from the integrated kernel $K_{\\mathrm{int}} = K_L + K_{\\mathrm{TD}} + K_{\\mathrm{ISW}}$, generated by lensing convergence, time-delay, and integrated Sachs-Wolfe effects. The machinery is completed by the monopole and quadrupole (the $\\ell=0$ and $\\ell=2$ angular moments) of this corrected spectrum, a Gaussian covariance between them, and the covariance-based shift formula $\\delta f_{\\mathrm{NL}} = -( {}^0F^{-1})_{f_{\\mathrm{NL}}\\alpha}\\, {}^1F_{\\alpha\\varepsilon}\\,\\delta\\varepsilon$ that quantifies the bias from using the standard instead of the corrected model.","core_discovery":"The discovery is that the quadrupole of the galaxy power spectrum is far more sensitive to wide-angle and relativistic corrections than the monopole, and that combining the correlated monopole and quadrupole dramatically tightens forecasts for local primordial non-Gaussianity. In the corrected model, $P_g = P^S_g + P^{\\mathrm{NI}}_g + P^I_g$, where $P^{\\mathrm{NI}}$ collects wide-angle, Doppler and Sachs-Wolfe terms and $P^I$ collects lensing convergence, time-delay and integrated Sachs-Wolfe terms; the paper finds that $P^{\\mathrm{NI}}$ and $P^I$ have opposite signs on ultra-large scales and partially cancel. Using the monopole-quadrupole data vector with equal-volume redshift binning gives $\\sigma(f_{\\mathrm{NL}})=2.89$ for SKAO2 and $0.548$ for MegaMapper. Neglecting the corrections shifts $f_{\\mathrm{NL}}$ by $0.614\\sigma$ for MegaMapper but only $-0.001\\sigma$ for SKAO2, with the authors warning that omitted higher-order integrated terms could make the SKAO2 cancellation partly artificial.","pith_inferences":["Editorial inference: The quoted precision gains likely overstate what a real analysis would achieve, because mode-coupling from wide-angle effects in the covariance is neglected; including it would widen errors but need not remove the relative gain from the quadrupole.","Editorial inference: Since the shift is sensitive to magnification bias $Q$ and evolution bias $E$, realistic uncertainties in these nuisance parameters could turn the predicted near-cancellation for SKAO2 into a non-negligible bias; the paper's fiducial amplitudes $Q_0=E_0=1$ are a sharp assumption.","Editorial inference: The omitted lensing-lensing correlation is not suppressed by powers of $H/k$, so on the ultra-large scales where local non-Gaussianity lives it might be as large as the leading integrated correction; a direct computation of $L\\times L$ would be a decisive check.","Editorial inference: The same corrected-spectrum formalism could be applied to the bispectrum or to multi-tracer combinations, where relativistic corrections and the scale-dependent bias of non-Gaussianity enter differently and may break the degeneracy that causes the shift."],"forward_implications":["For SKAO2, the quadrupole-plus-monopole analysis improves $\\sigma(f_{\\mathrm{NL}})$ from 5.21 to 2.89, a 45% gain; for MegaMapper, from 1.49 to 0.548, a 63% gain.","Neglecting wide-angle and relativistic corrections biases the estimated $f_{\\mathrm{NL}}$ by about $0.6\\sigma$ for MegaMapper, so such surveys need the corrected model to avoid biased inference.","The non-integrated and integrated corrections partially cancel on ultra-large scales, and integrated Sachs-Wolfe plus time-delay effects are subdominant to lensing at $z\\lesssim1$, making lensing the main integrated correction.","Equal-volume redshift binning, with 6 bins for SKAO2 and 3 for MegaMapper, gives the smallest $\\sigma(f_{\\mathrm{NL}})$ compared with other binning choices tested."],"supporting_citations":[{"why":"SKAO2 HI galaxy survey specifications, bias functions, and number density model.","marker":"[12]"},{"why":"MegaMapper LBG survey redshift range and bias fit.","marker":"[13]"},{"why":"Supplies the leading-order integrated relativistic correction $P^I_g$ and the $I\\times S$ approximation used here.","marker":"[33]"},{"why":"Supplies the non-integrated wide-angle and local relativistic correction multipoles used for $P^{\\mathrm{NI}}$.","marker":"[35]"},{"why":"Gives the multipole covariance used in the forecast.","marker":"[40]"},{"why":"Recommends equal-volume redshift bins, which the paper uses to optimise $\\sigma(f_{\\mathrm{NL}})$.","marker":"[47]"},{"why":"Previous SKAO2 angular power spectrum shift estimate that the paper compares its smaller shift against.","marker":"[9]"},{"why":"Previous Euclid-like spectroscopic shift estimate, illustrating the effect of cross-bin correlations.","marker":"[11]"},{"why":"Provides MegaMapper number density, evolution bias, and magnification bias values.","marker":"[54]"}],"fun_headline_variants":["Quadrupole tightens f_NL error bars by 63%","Monopole+quadrupole: 63% tighter f_NL constraints","Quadrupole yields 63% sharper f_NL forecasts","Wide-angle terms + quadrupole: f_NL error down 63%","Quadrupole cuts f_NL uncertainty up to 63%"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The near-cancellation that makes the SKAO2 shift negligible depends on the assumption that the omitted higher-order integrated effects, such as lensing-lensing and lensing-wide-angle correlations and correlations between redshift bins, are small; if they are not, the cancellation could be an artifact.","fun_headline_variants_meta":{"raw":{"variants":["Quadrupole tightens f_NL error bars by 63%","Monopole+quadrupole: 63% tighter f_NL constraints","Quadrupole yields 63% sharper f_NL forecasts","Wide-angle terms + quadrupole: f_NL error down 63%","Quadrupole cuts f_NL uncertainty up to 63%"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001659,"raw_usage":{"total_tokens":6656,"prompt_tokens":1089,"completion_tokens":5567,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":705,"completion_tokens_details":{"reasoning_tokens":5469}},"tokens_in":705,"tokens_out":5567,"duration_ms":39927,"temperature":1.0,"reasoning_tokens":5469,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T19:32:00.585970+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the same two-survey forecasts with the full integrated correction including $L\\times L$ and lensing-wide-angle terms and with cross-bin correlations (e.g., via a spherical Fourier-Bessel or angular power spectrum analysis); if the $f_{\\mathrm{NL}}$ shift for SKAO2 becomes comparable to or larger than $0.1\\sigma$, or the MegaMapper shift changes sign, then the leading-order cancellation is not robust.","supporting_citations":[],"review_version":1}