Lensing and time-delay contributions to galaxy correlations

This is a follow-up to the authors' own formalism paper (Bertacca et al. 2012), and it does what a good follow-up should: it takes the machinery and asks which terms actually matter as a function of redshift, angle, and magnification bias. The component-by-component decomposition in Eqs. (23)-(24) is clean, the coefficient set in (12)-(16) is fully written out, and the figures separate local, mixed, and integrated terms in a way that is easy to check. The qualitative finding — integrated terms grow with redshift separation and matter most for near-radial pairs — is plausible and, as far as I know, not quantified this way before. Credit for that. The paper is also honest about its own simplifications: constant Q, a PFS-like N(z), and the admission in the conclusions that individual configurations are probably below detection. That last admission matters, and I'll come back to it. The soft spot is the word 'dominate' in the abstract (abstract:s3). What the figures actually show is fractional contribution: in the near-radial case xi_ll decays toward zero past the BAO scale, so any nonzero integrated term gets a large fraction almost by construction. Fig. 6 does plot absolute values, which partly rescues this, and the conclusions are appropriately measured. But the abstract does not convey that the total correlation in those configurations may itself be small. That is a framing problem, not a math problem, and it is fixable with one sentence. On the reader's worry about extrapolating linear theory to Hubble scales: I think that concern is somewhat misplaced. Linear theory is the right tool at those scales, and the integrated terms are themselves linear-order quantities, so there is no inconsistency in using them there. The real open question is detectability, not validity. The constant-Q simplification is acknowledged and the Q-scan in Section V is a reasonable first pass; a redshift-dependent Q would strengthen it but is not required for the qualitative claim. No code shipped, which is a minor annoyance for a numerical paper but not disqualifying. Who should read it: anyone building large-scale two-point models for DESI, Euclid, or SKA, and anyone thinking about lensing-density cross terms in wide-deep samples. It is a useful reference, not a landmark. I would send it to referees, with a request that the abstract be softened to match the conclusions and that the authors add a short absolute-magnitude or detectability discussion for the near-radial case.

The central claim (abstract:s3, Section IV) that integrated correlations 'dominate' the total is a statement about fractional contribution ξ_AB/ξ_tot. In the near-radial configuration (Fig. 5, right), ξ_ll decays to near zero beyond the BAO scale, so any non-zero integrated term will have a large fractional contribution. This makes 'dominance' partly a consequence of the local term vanishing rather than the integrated terms being large in absolute terms. The paper shows absolute values in Fig. 6, which mitigates this, and the conclusions acknowledge that individual configurations are probably below detection. But the abstract's language ('can become very important and even dominate the observed total correlation function') does not convey that the total itself may be small and undetectable. The reader's concern about linear-order extrapolation is partially valid but somewhat misplaced: on Hubble scales, linear theory is the appropriate tool, and the integrated terms are themselves linear-order quantities. The more load-bearing issue is the interpretation of 'dominance' as fractional vs absolute, and the lack of a detectability/importance assessment.

The paper's technical result is correct within its stated linear formalism. The component breakdown is traceable, and the qualitative conclusion (integrated terms grow with redshift separation and dominate fractionally near-radial configurations) is plausible. The reader's concern about linear-order extrapolation is partially valid but somewhat misplaced because linear theory is appropriate on large scales. The main remaining issue is the interpretation of 'dominance' as fractional vs absolute, which the paper partially addresses (Fig. 6, conclusions) but the abstract does not. The paper is honest about the detection caveat and the constant-Q simplification. So the verdict should remain CONDITIONAL.

The question the paper addresses is a real and timely one: forthcoming surveys (PFS, DESI, Euclid, WFIRST, SKA) will measure correlations on scales where the standard Kaiser-plus-wide-angle treatment is no longer adequate, and a quantitative bookkeeping of which light-cone terms matter in which configuration is genuinely useful for planning analyses. The strengths are the completeness of the coefficient set in Eqs. (12)–(16), which makes the computation traceable, the explicit survey of configurations rather than a single averaged number, the scan over magnification bias, and the honest closing statement that individual configurations are probably below detection and that a proper configuration-averaged forecast is needed. The paper does not introduce new formalism or new data; its value is as a systematic exploration within an existing framework, and it is appropriately framed as such. The falsifiable content — that near-radial, large-redshift-separation pairs are where integrated terms matter most, and that Q = 1 suppresses them — is clear and testable against the detectability studies it cites ([54]–[58]).

Galaxy surveys measure clustering by counting pairs, and the textbook description of a pair count keeps only what happens at the two galaxies: their local density and their peculiar velocities. This paper works through the full light-cone expression, in which the observed overdensity is a local piece plus integrals along each line of sight from lensing convergence and from a time-delay (integrated Sachs-Wolfe-type) term. Evaluating the pieces separately for a PFS-like survey, the authors find the local relativistic corrections stay sub-dominant, at most a few per cent, while the integrated cross-terms do not: for pairs with a large redshift difference, and especially for pairs nearly along the line of sight, the lensing-density term overtakes the local-local term above $z_{1} �� 0.8$–$1.2$. If that holds, dropping the integrated terms is not a safe approximation for deep and wide surveys on large scales, and the radial correlation function acquires a positive tail past the BAO dip that is lensing rather than clustering.

What carries the argument. The engine is a tripolar spherical harmonic expansion of the correlation function, $ξ_{loc} = b(z_1)b(z_2)∑ B^{n}_{ℓ_1ℓ_2L}(χ_1,χ_2) S_{ℓ_1ℓ_2L}(n_1,n_2,n_{12}) ξ^{L}_{n}(χ_{12};z_1,z_2)$, in which the angular dependence is carried by $S_{ℓ_1ℓ_2L}$ and the dynamics by Bessel-weighted moments of the matter power spectrum. The local overdensity is written as a sum of spherical transforms $A_{nℓ}$ with four coefficients: $β = f/b$ for the Kaiser term, $α$ for wide-angle mode coupling (with its Newtonian limit $α_{nwt}$ separated out so the relativistic correction is explicit), and $γ$ for the gravitational-potential term. The integrated pieces carry their own line-of-sight kernels $σ(z, z̃)$ for lensing and $μ(z, z̃)$ for time delay, both built from $Ω_m$, the growth rate and the evolution bias $b_e$, and both carrying the magnification bias $Q$. Setting $Q=1$ makes $σ$ vanish, which is the formal statement that lensing disappears for intensity mapping; $Q=-1$ roughly doubles the lensing amplitude. The $Q$ scan over $0, ±0.5, ±1$ is what turns the formalism into a statement about which surveys can see the effect.

Core claim. Splitting the observed galaxy overdensity into a local part (bias, redshift-space distortions, wide-angle mode coupling, relativistic velocity and potential terms) and an integrated part (lensing convergence with magnification bias $2(Q-1)κ$, plus the time-delay term), the authors compute the full matrix of correlations $ξ_{AB}$ for $A,B = l, κ, i$ as a function of the pair configuration. They report three results. First, relativistic corrections inside the local correlation reach roughly 5–10 per cent at high redshift but never dominate, and the purely relativistic $ξ_γ$ piece is negligible. Second, for transverse pairs the local-local term still dominates strongly out to $z ≲ 1.4$, with integrated terms contributing only 1–10 per cent beyond that. Third, and this is the claim the paper is built around, for non-transverse pairs with a growing redshift separation the mixed lensing-local and time-delay-local terms overtake the local-local term once the far galaxy passes $z_{1} ∼ 1.2$ at a half-angle of $0.1$ rad, and for a near-radial pair ($θ = 0.01$ rad) the lensing terms dominate from $z ∼ 0.8$ onward, with $ξ_{ll}$ decaying toward zero as the separation becomes radial. The asymmetry between $ξ_{κl}$ and $ξ_{lκ}$ is real and is explained by which galaxy is held fixed. A further signature is that the radial correlation, after going negative past the BAO peak, crosses zero again and turns positive, and that positive tail is lensing.

The premise it rests on. The whole calculation is first order in perturbation theory on the light cone, and the claim that the integrated terms dominate rests on extrapolating that linear expansion to separations approaching the Hubble scale, where the paper checks it against neither simulations nor higher-order terms.

For a deep survey, the standard practice of modelling only $ξ_{ll}$ will misestimate the correlation on near-radial, large-separation pairs, where the lensing-density term is the largest single contribution above $z ∼ 1$.

The radial correlation function is predicted to turn positive again past the BAO dip, with the positive tail produced by lensing rather than by clustering, so the usual integral constraint $∫ξ(r) r^2 dr = 0$ no longer applies on the light cone.

HI intensity mapping, where $Q = 1$, is predicted to show essentially none of the lensing contribution, while surveys with steeper luminosity functions (more negative $Q$) will see the integrated terms more strongly.

The time-delay-local correlation $ξ_{il}$ is a purely relativistic term with no Newtonian counterpart, and it becomes comparable to the local term in the same near-radial configurations where lensing dominates.

Any programme that uses very large-scale clustering to test dark energy or modified gravity must include the integrated terms, since they carry the same information about the potentials but are currently thrown away.

The paper's most checkable prediction is not the amplitude ratio but the sign structure: a measured radial correlation that stays negative past the BAO dip, without the lensing-induced second zero crossing, would be a cleaner discriminator than any per-cent comparison.

Because the effect is largest exactly where the linear expansion is least controlled, the natural next step is a ray-traced or N-body cross-check of the same configurations rather than a further analytic refinement of the kernels.

The asymmetry between $ξ_{κl}$ and $ξ_{lκ}$ suggests a practical estimator: choosing which galaxy of the pair to treat as the 'near' one is a free choice in the analysis, and the pair asymmetry could be used as a null test for systematics that are symmetric under swapping the two galaxies.

The $Q$ dependence implies that the same survey split by luminosity threshold would show a different fraction of integrated signal, which is a cheap internal consistency check on the whole decomposition.

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