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arxiv: 2605.29806 · v1 · pith:AI52TVMCnew · submitted 2026-05-28 · 🌌 astro-ph.CO

The observer power spectrum for lightcone statistics, integrated relativistic observables and wide angle effects

Chris Clarkson , Pritha Paul This is my paper

Pith reviewed 2026-06-29 06:10 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords observer power spectrumlightcone statisticswide angle effectsrelativistic observablesgalaxy number countscosmological power spectraintegrated effectsmode mixing
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The pith

Fourier transforming over observer positions produces a diagonal power spectrum for any lightcone observable.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines an observer power spectrum by Fourier transforming over observer positions on a spatial hypersurface while holding lightcone coordinates fixed, instead of transforming over source positions on one lightcone. Because the observer hypersurface is statistically homogeneous, this spectrum is diagonal for every observable, whether the observable is local or involves integrated effects such as lensing. Standard two-point statistics used in large-scale structure analysis appear as projections of this single diagonal spectrum. The construction extends immediately to higher-order statistics and is illustrated by writing the full relativistic kernel for galaxy number count fluctuations.

Core claim

The observer power spectrum is obtained by Fourier transforming the observable over observer locations on a spatial hypersurface at fixed lightcone coordinates. Statistical homogeneity on that hypersurface guarantees the spectrum is diagonal for any observable, local or integrated, and free of mode-mixing generated by lightcone geometry. Every conventional two-point statistic is recovered by projecting this spectrum, and the same object supplies the relativistic kernel containing density, redshift-space distortions, Doppler, magnification, and integrated Sachs-Wolfe terms.

What carries the argument

The observer power spectrum, defined by Fourier transformation over observer positions on a spatial hypersurface with fixed lightcone coordinates.

If this is right

  • All standard two-point statistics in large-scale structure analysis are recovered as projections of the single diagonal observer spectrum.
  • The construction applies unchanged to higher-order statistics.
  • The relativistic kernel for observed galaxy number counts is obtained directly, incorporating density, redshift-space distortions, Doppler, lensing, and integrated Sachs-Wolfe contributions.
  • Mode-mixing corrections arising from lightcone geometry are eliminated by construction.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Survey pipelines could replace separate wide-angle and integrated-effect corrections with a single projection step from the observer spectrum.
  • The same diagonal object supplies a natural starting point for covariance estimation in analyses that include multiple relativistic effects simultaneously.
  • Because the spectrum is defined on the observer slice, it may simplify comparisons between different survey geometries that share the same observer hypersurface.

Load-bearing premise

The spatial hypersurface on which observers sit must be statistically homogeneous.

What would settle it

Compute the observer power spectrum from a set of mock catalogs on a homogeneous observer slice and verify that the matrix elements between distinct wave-vectors are consistent with zero for an integrated observable such as lensing magnification.

Figures

Figures reproduced from arXiv: 2605.29806 by Chris Clarkson, Pritha Paul.

Figure 1
Figure 1. Figure 1: Left: a single observer at x0 on the spatial hypersurface η0 receives light from two sources at positions x1 = x0+χ1nˆ 1 and x2 = x0 + χ2nˆ 2 on their past lightcone, intersecting the hypersurfaces η1 and η2. Each null ray samples the spacetime between observer and source; for integrated observables such as lensing, the signal depends on the gravitational potential along the entire ray, not just at the sou… view at source ↗
Figure 2
Figure 2. Figure 2: Hierarchy of lightcone two-point statistics. The observer spectrum [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison between the integrated contribution alone (left panel) and the full combined (right panel) to the observed [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Wide-angle 2D maps of the observed power spectrum as a function of k and θ2 for the configuration z1 = 1.5, z2 = 1.55, and θ1 = 0. The left panels show the lensing only contribution, while the right panels show the full contribution including local and integrated relativistic terms. The top row shows the real part, ℜP (O) , and the bottom row shows the imaginary part, ℑP (O) . The full contribution is larg… view at source ↗
Figure 5
Figure 5. Figure 5: Absolute values of the double-Legendre multipoles [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: [(CC): Plot of the normalised plane parallel multipoles PL(k) as a function of k, shown here for L = 0 to L = 6, for the separation z1 = 1 and z2 = 1.05 (left) compared to the plane parallel equal redshift case z1 = z2 = 1 (right). In the full result, both even and odd multipoles are present, with even multipoles purely real and odd multipoles purely imaginary. In the plane-parallel limit the odd multipole… view at source ↗
Figure 2
Figure 2. Figure 2: The angular power spectrum Cℓ(χ1, χ2) is the scalar projection of the diagonal multipoles P (O) ℓℓ ; the lightcone correlation function ξLC is the momentum-space integral of P (O); the mixed spectrum P¯(k, d) and the non-diagonal kernel P˜(k1, k2) are lightcone projections in which the external labels become functions of the integration variable, which is the origin of their mode-mixing and wide-angle corr… view at source ↗
read the original abstract

The statistics of large-scale structure are naturally described by power spectra in Fourier space. For fields on spatial hypersurfaces, translational invariance makes different Fourier modes uncorrelated and the power spectrum diagonal. Cosmological observables, however, are measured on our past lightcone, where wide-angle effects, radial evolution and integrated effects such as lensing break this symmetry: Fourier-space statistics become non-diagonal, with mode-mixing generated by the geometry of the lightcone itself. We define a more natural observer power spectrum by Fourier transforming over observer positions on a spatial hypersurface with fixed lightcone coordinates, rather than over source positions on a single lightcone. This forms a field on the observer hypersurface with a moveable light-ray leg. Statistical homogeneity of the observer hypersurface implies that this spectrum is diagonal for any observable, whether local or integrated and does not suffer from mode-mixing. We show how the various two-point statistics used in large-scale structure analysis are each recovered as projections of the observer spectrum. This extends directly to higher-order statistics. We illustrate it by constructing the relativistic kernel for the observed galaxy number count fluctuation, including density, redshift-space distortions, Doppler, lensing magnification, and integrated Sachs-Wolfe contributions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper claims that defining an 'observer power spectrum' by Fourier transforming over observer positions on a spatial hypersurface (with fixed lightcone coordinates for direction and radial distance) yields a field whose two-point statistics are diagonal due to statistical homogeneity on that hypersurface. This holds for any observable, local or integrated (e.g., lensing, ISW), avoiding mode-mixing from lightcone geometry that affects standard source-position Fourier transforms. Standard LSS two-point statistics are recovered as projections of this spectrum; the construction extends to higher orders. The paper illustrates by deriving the full relativistic kernel for galaxy number count fluctuations, including density, RSD, Doppler, magnification, and integrated terms.

Significance. If the central construction holds, the observer power spectrum offers a unified, homogeneity-respecting framework for lightcone statistics that cleanly incorporates wide-angle and integrated relativistic effects without ad-hoc mode-mixing corrections. A strength is the direct grounding in the standard homogeneity assumption on the observer hypersurface (no new parameters or entities beyond the definition itself) and the explicit demonstration that conventional statistics emerge as projections. The extension to higher-order statistics and the concrete relativistic kernel construction are also positive features.

minor comments (2)
  1. [Abstract] Abstract: the central claim that the spectrum 'is diagonal for any observable' would be clearer if the abstract briefly referenced the explicit two-point function on the observer hypersurface (e.g., the form that depends only on observer separation) rather than stating the implication alone.
  2. The manuscript would benefit from a short dedicated subsection (perhaps after the definition) that explicitly verifies the absence of off-diagonal terms for one integrated contribution, such as the lensing magnification term in the galaxy kernel, even if the general argument from homogeneity is used.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of the manuscript, as well as for the recommendation of minor revision. No major comments were raised in the report, so we have no specific points requiring rebuttal or clarification at this stage. We will incorporate any minor suggestions during revision.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines an observer power spectrum via Fourier transform over observer positions on a spatial hypersurface at fixed lightcone coordinates. It then applies the standard cosmological assumption of statistical homogeneity on that hypersurface to conclude the resulting spectrum is diagonal for any observable. This follows directly from translational invariance under the homogeneity assumption and does not reduce any claimed result to a fitted parameter, self-referential definition, or self-citation chain. The derivation remains self-contained against external benchmarks with no load-bearing steps that collapse by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the domain assumption of statistical homogeneity on the observer hypersurface and introduces the observer power spectrum itself as a new defined entity without independent evidence supplied in the abstract.

axioms (1)
  • domain assumption The observer hypersurface is statistically homogeneous
    Invoked to conclude that the observer power spectrum is diagonal for any observable.
invented entities (1)
  • observer power spectrum no independent evidence
    purpose: A diagonal Fourier-space statistic for any lightcone observable, local or integrated
    Newly defined in the paper via Fourier transform over observer positions on a fixed lightcone coordinate hypersurface.

pith-pipeline@v0.9.1-grok · 5743 in / 1435 out tokens · 37685 ms · 2026-06-29T06:10:25.072203+00:00 · methodology

discussion (0)

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Reference graph

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