Cosmological Parameter Estimation with Large Scale Structure Observations

The thing worth knowing: this is a careful Fisher-matrix head-to-head between the full angular C_l(z1,z2) analysis and the standard P(k) analysis, for Euclid-like and DES-like surveys, using CLASSgal with all relativistic corrections. The comparison itself is new and checkable, and the side study of lensing signal-to-noise (Section 4.3, Figs. 9-10) is the part I'd actually pull out and cite. The authors are also unusually straight about what they left out: no instrumental noise, no bias, fixed growth index gamma=0.6, and they say so in Section 4 rather than burying it. That candor counts for something with me. The Fisher machinery in Appendix B is standard and correctly assembled, the fiducial cosmology is external, and there's no circularity anywhere: nothing fitted is being recycled as a prediction. The qualitative conclusions are sound: tomography helps for photometric surveys, and spectroscopic surveys would need an impractical number of bins before shot noise caps the gain. I'd trust those. Where I'd push back on the reader's framing: the nonlinearity cutoff Eqs. (3.5)-(3.7) is not the fragile part. Doubling lambda_min from 34 to 68 h⁻¹ Mpc degrades the 2D photo-z FoM by only about 2 while degrading the 3D FoM by about 8, so the 2D/3D ratio actually improves at the conservative cutoff. The genuinely soft number is narrower: the 'crossover at Nbin=4-6' claim. That depends on the 2-sigma versus 3-sigma definition of bin separation in Footnote 2, which sets l_max for Gaussian photo-z windows. At 3-sigma more radial information is discarded, the photo-z FoM drops (green versus blue dashed lines, Fig. 3 lower panels), and the physically allowed bin count is only about 6 given the photo-z error budget. The paper asserts the 2D analysis still wins there but doesn't separately verify the crossover at 3-sigma with lambda_min=68. If that crossover sits beyond 6 bins, the headline number fails for the realistic configuration even though the asymptotic claim holds. Secondary: the 3D photo-z baseline is taken at Nbin=1 because the sum-of-Fisher approximation overcounts correlated bins, so the true 3D FoM with a proper covariance is unknown. Both are acknowledged, so this is a quantitative-margin concern, not a soundness failure. Who should read it: anyone planning tomographic analyses for Euclid or DES, and anyone working on lensing extraction from wide redshift catalogs. Worth a serious referee, with a request that the crossover claim be re-verified at the 3-sigma convention and the conservative cutoff.

The reader locates the soft spot correctly but frames it slightly too broadly. The nonlinearity cutoff Eqs. (3.5)–(3.7) is a conservative geometric estimate, and the paper's own Figure 3 shows the qualitative conclusion is stable against it: doubling λmin from 34 to 68 h^-1 Mpc degrades the 2D photo-z FoM by only ~2 while degrading the 3D FoM by ~8, so the 2D/3D ratio actually improves at the conservative cutoff. The genuinely fragile quantity is narrower: the crossover bin number and the 'significantly better' margin. These depend on the 2σ vs 3σ definition of δzij in Footnote 2, which sets l^ij_max for Gaussian photo-z windows. At 3σ, δzij is larger, l_max is smaller, more radial information is discarded, and the photo-z FoM drops (green vs blue dashed lines, Fig. 3 lower panels). The paper asserts the 2D analysis still wins, but the physically allowed bin count is only ~6 (photo-z error 0.05(1+z) over 0.7<z<2, per §4.1), and the 'achieved already at Nbin=4–6' statement is not separately verified at the 3σ convention with λmin=68. If the 3σ crossover lies beyond 6 bins, the headline crossover claim fails for the realistic configuration even though the asymptotic claim holds. A secondary, acknowledged soft spot: the 3D photo-z baseline is taken at Nbin=1 because the sum-of-Fisher approximation overcounts correlated bins (§4.1), so the true 3D FoM with a proper covariance is unknown; if it exceeds the Nbin=1 estimate, the 2D margin shrinks. The authors flag both issues candidly, which is why this is a quantitative-margin concern, not a soundness failure.

I agree with the reader on the locus of the weakest assumption — the nonlinearity cutoff and the 2σ/3σ bin-separation convention — and on the overall CONDITIONAL verdict. I would sharpen the framing. The reader says the headline comparison 'rests on an untested approximation'; that overstates the fragility of the qualitative claim, because the paper's own Figure 3 shows the 2D/3D ratio is stable (indeed more favorable) at the conservative λmin = 68 h^-1 Mpc. What is genuinely untested is the specific crossover bin number and the size of the margin at the 3σ convention, given that the photo-z error budget permits only ~6 bins. That is a narrower and more checkable concern than 'the cutoff is untested.' The paper is candid about its approximations: it states the error budget omits instrumental noise and bias (§4), fixes γ = 0.6, and explicitly notes in §3.2 that Eq. (3.7) still lets nonlinear contributions enter through the inverse covariance. These are acknowledged limitations, not new findings, and I do not present them as such. The secondary point about the 3D photo-z baseline being taken at Nbin=1 is a real asymmetry but is argued conservatively by the authors and does not by itself overturn the claim. The lensing signal-to-noise study in §4.3 is independent of the Fisher comparison and stands on its own. Net: the qualitative conclusion (tomography helps for photometric surveys, needs impractically many bins for spectroscopic ones) is robust; the quantitative margins are indicative. CONDITIONAL remains the right verdict, and my concern does not move it. The concrete test above would settle whether the crossover claim survives the 3σ convention within the realistic bin budget.

The methodological point is well taken and the CLASSgal machinery, with full relativistic corrections at first order, is a genuine asset for the community. The paper is candid about its incomplete error budget (no instrumental noise, unit bias, no Planck priors) and about the limitations of the sum-of-Fisher approximation for correlated photometric bins. The forecasts are useful for survey design and for deciding between tomographic and P(k) analyses. The strengths are the model-independent observable, the explicit nonlinearity treatment, and the honest statement of what is and is not included; these are appropriately credited.

The paper asks whether galaxy surveys should be analysed through the redshift-dependent angular power spectra $C_ell(z_1,z_2)$ of number counts rather than through the conventional three-dimensional power spectrum $P(k)$. The authors compute these angular spectra with all first-order relativistic corrections and use Fisher-matrix forecasts for Euclid-like and DES-like surveys to compare figures of merit for parameter pairs such as $H_0$ with $omega_{rm CDM}$ and $omega_b$ with $omega_{rm CDM}$. Their central claim is that for photometric surveys the angular route performs better than the $P(k)$ route once the sample is split into roughly four to six redshift bins, while for spectroscopic surveys the many bins needed are limited by shot noise. If that is right, photometric survey design can recover radial information that a standard $P(k)$ analysis discards, and the choice of analysis method becomes as important as raw galaxy counts.

What carries the argument. The load-bearing object is the redshift-dependent angular power spectrum $C_ell(z_1,z_2)$, the Legendre coefficient of the angular correlation function of galaxy number counts at two observed redshifts. It is computed with the CLASSgal code, keeping density, redshift-space distortions, lensing and gravitational-potential terms at first order in perturbation theory. The comparison is carried through a Fisher matrix that sums derivatives of these spectra over multipoles and over all auto- and cross-bin pairs, with a Gaussian covariance that includes shot noise. A geometric nonlinearity cutoff assigns each bin pair a maximum multipole $ell^{ij}_{max}$ from the comoving distance to the mean redshift and the bin separation, and the Fisher sum is truncated at the smaller of the two multipoles in each covariance entry.

Core claim. The central claim is that the angular power spectra $C_ell(z_1,z_2)$ of galaxy number counts retain redshift-pair information that a standard three-dimensional $P(k)$ analysis of a photometric sample averages away, so the angular Fisher figure of merit is larger for the same survey. For a Euclid-like photometric survey the paper reports the crossover at about four to six redshift bins; for a DES-like spectroscopic survey it places the crossover near thirty bins, and for a Euclid-like spectroscopic survey it estimates that roughly 120 bins would be needed, beyond what shot noise makes useful. The paper also reports that cross-correlations between redshift bins carry much of the gain, that the lensing contribution can be measured in wide-redshift catalogues, and that the potential terms remain below unit cumulative signal-to-noise in the configurations studied.

The premise it rests on. The load-bearing premise is that the hand-built nonlinearity cutoff in Eqs. (3.5)–(3.7), together with the choice of 2σ or 3σ bin separation for Gaussian photometric windows, correctly separates linear from nonlinear information; the paper itself shows the 2D figure of merit changing by about a factor of two between $lambda_{min}=34$ and $68,h^{-1}{rm Mpc}$ and shifting substantially between the 2σ and 3σ conventions.

This review was written by a machine. No person has reviewed this paper on Pith yet. A scientist listed on Pith can review it for a fee, and their review is published here whatever it concludes.

Request a human review