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arxiv: 2603.12504 · v2 · submitted 2026-03-12 · 🌌 astro-ph.CO

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Exploring Beyond {Λ}CDM with the Weak Lensing Power Spectrum and Bispectrum

Liantsoa F. Randrianjanahary , Chandrachud B. V. Dash
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Pith reviewed 2026-05-15 11:17 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords weak lensingbispectrumpower spectrumFisher forecastdark energymodified gravitytomographic analysisbeyond Lambda CDM
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The pith

Including the bispectrum in weak lensing forecasts tightens constraints on dynamical dark energy and interacting dark energy parameters by 38 to 90 percent compared with the power spectrum alone.

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

The paper forecasts constraints from tomographic weak lensing data on three distinct extensions to the standard cosmological model: a time-varying dark energy equation of state, an energy exchange between dark matter and dark energy, and a specific form of f(R) gravity. Adding the bispectrum to the usual power spectrum analysis reduces the marginalised uncertainty on the dark energy equation of state parameter from 0.2511 to 0.1557 and on the interaction coupling from 2.6895 to 0.2944, while the improvement for the f(R) scalaron amplitude is negligible after marginalisation over systematics. The same forecasts show that f(R) models suffer the largest degradation from observational systematics in the bispectrum. These numerical results indicate that future Stage IV weak lensing surveys will require higher-order statistics to extract their full power for testing alternatives to Lambda CDM.

Core claim

Fisher matrix forecasts demonstrate that the tomographic weak lensing bispectrum, when added to the power spectrum, reduces the marginalised 1 sigma error on the CPL equation of state w0 from 0.2511 to 0.1557, on the IDE coupling alpha from 2.6895 to 0.2944, and leaves the scalaron amplitude ln|fR0| essentially unchanged at 2.237 after full marginalisation over nuisance parameters such as photo-z error sigma_z and intrinsic alignment amplitude A_IA. The analysis finds that f(R) models remain the most sensitive to those systematics, especially in the bispectrum, and concludes that higher-order weak lensing statistics are required to maximise the return from Stage IV surveys.

What carries the argument

The Fisher information matrix constructed from the combination of tomographic weak lensing power spectrum and bispectrum, applied to parameter spaces of CPL, IDE, and Hu-Sawicki f(R) models.

If this is right

  • For the CPL dark energy model the marginalised error on w0 drops by roughly 38 percent when the bispectrum is included.
  • For interacting dark energy the marginalised error on the coupling alpha drops by nearly an order of magnitude.
  • f(R) gravity models exhibit the strongest sensitivity to systematics in the bispectrum after marginalisation.
  • Higher-order weak lensing statistics become a practical requirement for extracting maximum information from Stage IV surveys.

Where Pith is reading between the lines

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

  • The same bispectrum gain may appear in other modified-gravity or dark-energy parametrizations not examined in the paper.
  • Improved modelling or mitigation of intrinsic alignments and photometric redshift errors could unlock larger improvements for f(R) models.
  • Cross-correlating the same weak lensing fields with CMB lensing or galaxy clustering would likely tighten the constraints further.

Load-bearing premise

The Fisher matrix method assumes a Gaussian likelihood and linear response of the observables around the chosen fiducial cosmology.

What would settle it

Running the identical forecast on end-to-end simulated weak lensing maps that contain the full bispectrum for these three models would show whether the reported error reductions survive realistic non-Gaussianity and systematic residuals.

Figures

Figures reproduced from arXiv: 2603.12504 by Chandrachud B. V. Dash, Liantsoa F. Randrianjanahary.

Figure 1
Figure 1. Figure 1: Left panel: Redshift evolution of the linear growth rate fσ8(z) for ΛCDM (black), the DESI-preferred CPL parametri￾sation (green), IDE with coupling α = 0.10 (yellow), and Hu–Sawicki f(R) gravity at two representative scales k = 0.1 and 1.0 Mpc−1 (blue and pink respectively). Observational data points from redshift-space distortion measurements are taken from [83]. Right panel: Ratio of the growth rate fσ8… view at source ↗
Figure 2
Figure 2. Figure 2: Left panel: Effective tomographic redshift distributions ˜ni(z) for the two tomographic bins (0.1 ≤ z1 ≤ 0.3 and z2 ≥ 0.5), shown for fiducial photo-z scatter σz = 0.05 (solid) and the pessimistic case σz = 0.10 (dashed). The underlying true redshift distribution n(z) of the LSST gold sample is shown in red. Right panel: Corresponding weak lensing kernels Wi(χ) for each tomographic bin. where the drag and … view at source ↗
Figure 3
Figure 3. Figure 3: Left panel: Fractional deviation of the tomographic shear power spectrum C γγ, X ℓ from the ΛCDM prediction, ∆Cℓ/Cℓ ≡ (C γγ, X ℓ − C γγ, ΛCDM ℓ )/Cγγ, ΛCDM ℓ , for the two tomographic bin combinations. Results are shown for the CPL parametrisation, IDE, and Hu–Sawicki f(R) gravity, all including photometric redshift scatter with σz = 0.05 and 0.1. Right panel: Same fractional deviation but for the observed… view at source ↗
Figure 4
Figure 4. Figure 4: Marginalised 68% and 95% confidence contours from the power spectrum-only Fisher analysis for the three beyond￾ΛCDM scenarios considered. Upper left: CPL dark energy, showing the joint constraints on (w0, wa). Upper right: Interacting dark energy, showing the (Ωm0, α) plane. Lower: Hu–Sawicki f(R) gravity, showing the (Ωm0, ln |fR0|) plane. One-dimensional marginalised posteriors are displayed along the di… view at source ↗
Figure 5
Figure 5. Figure 5: Same as [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
read the original abstract

In this work, we present Fisher matrix forecast of the tomographic weak lensing power spectrum and bispectrum for three physically distinct types of models of beyond-$\Lambda$CDM: the CPL parametrisation of dynamical dark energy, interacting dark energy (IDE) with a dark sector energy-momentum exchange, and Hu-Sawicki models of $f(R)$ gravity. We find that for all three models, including the bispectrum significantly tightens the Fisher constraints: the bispectrum reduces the marginalised $1\sigma$ error on the CPL equation of state parameter from $\sigma(w_0) = 0.2511$ (power spectrum only) to $\sigma(w_0) = 0.1557$, on the IDE coupling from $\sigma(\alpha) = 2.6895$ to $\sigma(\alpha) = 0.2944$, and on the scalaron amplitude from $\sigma(\ln|f_{R0}|) = 2.236$ to $\sigma(\ln|f_{R0}|) = 2.237$ after full marginalisation over nuisance parameters e.g., photo-z error $\sigma_z$ and intrinsic alignment amplitude $\mathcal{A}_{\rm IA}$. We find that $f(R)$ models are the most sensitive to systematics and especially in bispectrum. The results also demonstrates the importance of higher order weak lensing statistics as a practical necessity to maximise the scientific return of Stage IV surveys.

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

1 major / 2 minor

Summary. The manuscript presents Fisher matrix forecasts for the tomographic weak lensing power spectrum and bispectrum in three beyond-ΛCDM models: the CPL parametrization of dynamical dark energy, interacting dark energy with dark-sector coupling, and Hu-Sawicki f(R) gravity. It claims that including the bispectrum significantly tightens the marginalized 1σ constraints on the key parameters (w0, α, ln|fR0|) for all three models after full marginalization over nuisance parameters such as photo-z error and intrinsic alignment amplitude.

Significance. If the numerical results are corrected, the work would demonstrate the practical necessity of higher-order statistics for maximizing constraints from Stage IV weak lensing surveys on extensions to ΛCDM. The Fisher approach is standard, and the quantitative improvements for multiple distinct models could inform survey optimization, provided the internal inconsistency is resolved.

major comments (1)
  1. [Abstract] Abstract (and corresponding results): The headline claim that the bispectrum 'significantly tightens' the Fisher constraints 'for all three models' is not supported by the reported numbers for the Hu-Sawicki f(R) case. The marginalised error on ln|fR0| changes from 2.236 (power spectrum only) to 2.237 (with bispectrum), a difference consistent with zero rather than tightening. This directly contradicts the 'all three' statement and is load-bearing for the central result; the calculation or the text must be corrected.
minor comments (2)
  1. [Abstract] The abstract provides no information on covariance matrix construction, bispectrum modeling details, or validation against non-Gaussianity; these should be expanded in §3 or §4 for reproducibility.
  2. The Fisher approximation assumes Gaussian likelihood and linear response; a brief discussion of its validity range (e.g., via simulation checks) would strengthen the forecasts.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful review and for identifying the inconsistency between the abstract claim and the reported numerical results. We address the major comment below and will revise the manuscript to ensure accuracy.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and corresponding results): The headline claim that the bispectrum 'significantly tightens' the Fisher constraints 'for all three models' is not supported by the reported numbers for the Hu-Sawicki f(R) case. The marginalised error on ln|fR0| changes from 2.236 (power spectrum only) to 2.237 (with bispectrum), a difference consistent with zero rather than tightening. This directly contradicts the 'all three' statement and is load-bearing for the central result; the calculation or the text must be corrected.

    Authors: We agree with the referee that the reported values show no meaningful improvement for the Hu-Sawicki f(R) model: the marginalised error on ln|fR0| remains essentially unchanged (2.236 to 2.237). This indicates that, after full marginalisation over nuisance parameters, the bispectrum adds negligible additional constraining power for this particular extension. We will revise the abstract and the corresponding results section to state that the bispectrum significantly tightens constraints for the CPL and IDE models but leaves the f(R) scalaron constraints essentially unchanged. We will also add a brief discussion of why f(R) models appear less responsive to the bispectrum (consistent with the paper's existing remark that f(R) is most sensitive to systematics). The underlying Fisher matrices and tabulated numbers are correct as presented; only the generalising claim requires correction. revision: yes

Circularity Check

0 steps flagged

Fisher forecasts are forward computations with no self-referential reduction

full rationale

The paper computes Fisher matrices for tomographic weak lensing power spectrum and bispectrum forecasts in three beyond-ΛCDM models (CPL, IDE, Hu-Sawicki f(R)) using standard theoretical expressions for the lensing signals, covariances, and derivatives around fiducial cosmologies. All reported constraints are predictions from assumed survey specifications and nuisance parameters (e.g., σ_z, A_IA); no data are fitted and no parameters are tuned to reproduce the target results. The derivation relies on external cosmological perturbation theory and the Fisher formalism, which remain independent of the specific numerical outcomes. The observed lack of tightening for ln|f_R0| (2.236 to 2.237) is a computational result, not a definitional loop. No self-citations, ansatze, or fitted inputs reduce the central claims to tautologies by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The forecast rests on the standard Fisher matrix formalism and established weak lensing theory; no new entities are postulated and no parameters are fitted to data.

axioms (2)
  • domain assumption Fisher matrix provides accurate parameter uncertainty estimates under Gaussian likelihood and linear response
    Invoked to translate derivatives of the power spectrum and bispectrum into forecasted errors.
  • standard math Standard expressions for tomographic weak lensing power spectrum and bispectrum hold in the chosen models
    Used to compute the signal derivatives around the fiducial cosmology.

pith-pipeline@v0.9.0 · 5575 in / 1458 out tokens · 83563 ms · 2026-05-15T11:17:22.197338+00:00 · methodology

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