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arxiv: 2604.08895 · v1 · submitted 2026-04-10 · 🌌 astro-ph.CO

Recognition: unknown

FolpsD: combining EFT and phenomenological approaches for joint power spectrum and bispectrum analyses

A. Aviles, A. Cuceu, A. de la Macorra, A. de Mattia, A. Font-Ribera, A. Kremin, A. Meisner, B. A. Weaver, B. Dey, C. Guandalin, C. Hahn, C. Howlett, D. Bianchi, D. Brooks, D. Gonzalez, D. Huterer, D. Kirkby, D. Schlegel, E. Gazta\~naga, E. Paillas, E. Sanchez, F. Beutler, F. Prada, G. Gutierrez, G. Niz, G. Rossi, G. Tarl\'e, H. E. Noriega, H. K. Herrera-Alcantar, H. Seo, I. Garzon, I. P\'erez-R\`afols, J. Aguilar, J. E. Forero-Romero, J. Hou, J. Silber, K. Honscheid, L. Le Guillou, L. Samushia, M. Ishak, M. Landriau, M. Manera, M. Pellejero Ibanez, M. Schubnell, M. S. Wang, P. Bansal, P. Doel, P. Zarrouk, R. Joyce, R. Miquel, R. Zhou, S. Ahlen, S. Ferraro, S. Gontcho A Gontcho, S. Juneau, S. Nadathur, T. Claybaugh, U. Andrade, W. J. Percival

Pith reviewed 2026-05-10 17:31 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords galaxy clusteringpower spectrumbispectrumeffective field theoryredshift-space distortionsdark energyDESIcosmological parameters
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The pith

Joint EFT power spectrum and bispectrum analysis with line-of-sight damping extends usable scales to k~0.3 h Mpc^{-1} for LRG samples and tightens constraints on As, omega_cdm, w0 and wa.

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

The paper develops a model that pairs the one-loop effective field theory power spectrum with a tree-level bispectrum expressed in the tripolar spherical harmonics basis. A phenomenological damping factor along the line of sight is added to both statistics to push the analysis to smaller scales. When tested on DESI DR2 mocks, the joint fit reduces parameter degeneracies compared with power spectrum data alone. For LRG-like tracers the damping safely widens the wavenumber range while delivering up to 30 percent tighter limits on the scalar amplitude and cold dark matter density, and 15-21 percent tighter limits on the dark energy equation-of-state parameters w0 and wa.

Core claim

The central claim is that the combination of one-loop EFT power spectrum modeling with tree-level bispectrum information, augmented by a shared line-of-sight damping factor, yields unbiased cosmological constraints at higher wavenumbers than power spectrum analyses alone. For LRG-like samples this extends the reliable range beyond k~0.3 h Mpc^{-1} in the power spectrum and k~0.24 h Mpc^{-1} in the bispectrum, producing up to 30 percent tighter bounds on As and omega_cdm and 15-21 percent tighter bounds on w0 and wa in a w0waCDM cosmology, while the same damping terms can absorb noise and shift parameters for low signal-to-noise tracers such as QSOs.

What carries the argument

The central mechanism is the joint likelihood of the one-loop EFT galaxy power spectrum and the tree-level galaxy bispectrum projected onto the Sugiyama tripolar spherical harmonics basis, together with a shared phenomenological line-of-sight damping factor applied to both statistics.

If this is right

  • Up to 30 percent tighter constraints on As and omega_cdm for LRG-like samples when the damping term is included.
  • 15 percent tighter constraint on w0 and 21 percent tighter constraint on wa in w0waCDM, producing a mild deviation from constant dark energy using full-shape information alone.
  • Reduction in parameter degeneracies when bispectrum data are added to the power spectrum analysis.
  • Safe extension of the wavenumber range to k~0.3 h Mpc^{-1} (power spectrum) and k~0.24 h Mpc^{-1} (bispectrum) for LRG-like tracers without statistically significant bias.
  • Risk that damping parameters absorb noise and shift parameters for low signal-to-noise tracers such as QSOs or in models with shape features degenerate with damping, such as massive neutrinos.

Where Pith is reading between the lines

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

  • The same joint modeling framework could be applied to other large-scale structure surveys to test whether the reported gains in dark energy constraints persist with real data.
  • Validation against mocks that include massive neutrinos would test whether the damping term inadvertently absorbs the scale-dependent suppression signature.
  • Extending the model to include higher-order bispectrum corrections might further push the usable k-range while keeping the damping parameters from absorbing cosmological information.

Load-bearing premise

The tree-level bispectrum plus damping factor remains an accurate, unbiased description of the data up to the extended k ranges, and the damping parameters do not absorb cosmological information or noise fluctuations in a way that shifts the inferred parameter values.

What would settle it

Run the same joint analysis on actual DESI DR2 LRG data with and without the damping term and check whether the recovered values of As, omega_cdm, w0 and wa shift by more than the reported statistical uncertainties.

Figures

Figures reproduced from arXiv: 2604.08895 by A. Aviles, A. Cuceu, A. de la Macorra, A. de Mattia, A. Font-Ribera, A. Kremin, A. Meisner, B. A. Weaver, B. Dey, C. Guandalin, C. Hahn, C. Howlett, D. Bianchi, D. Brooks, D. Gonzalez, D. Huterer, D. Kirkby, D. Schlegel, E. Gazta\~naga, E. Paillas, E. Sanchez, F. Beutler, F. Prada, G. Gutierrez, G. Niz, G. Rossi, G. Tarl\'e, H. E. Noriega, H. K. Herrera-Alcantar, H. Seo, I. Garzon, I. P\'erez-R\`afols, J. Aguilar, J. E. Forero-Romero, J. Hou, J. Silber, K. Honscheid, L. Le Guillou, L. Samushia, M. Ishak, M. Landriau, M. Manera, M. Pellejero Ibanez, M. Schubnell, M. S. Wang, P. Bansal, P. Doel, P. Zarrouk, R. Joyce, R. Miquel, R. Zhou, S. Ahlen, S. Ferraro, S. Gontcho A Gontcho, S. Juneau, S. Nadathur, T. Claybaugh, U. Andrade, W. J. Percival.

Figure 1
Figure 1. Figure 1: Monopole and quadrupole bispectrum multipoles in the Sugiyama basis (left panel) and the Scoccimarro basis (right panel). Black curves in both panels show the best-fit models obtained from a joint fit using the power spectrum and the bispectrum in the Sugiyama basis over the mean of 25 LRG2 mocks. Error bars are derived from the same covariance matrices used in the fits and corresponding to a single volume… view at source ↗
Figure 2
Figure 2. Figure 2: Power spectrum (top row) and bispectrum (bottom row) multipoles measured from second-generation Abacus mocks. Each panel shows the monopole and quadrupole of the respective spectra. The three columns correspond to LRGs in two redshift bins and QSOs. Solid lines show the spectra with the noise subtracted, while the Poissonian shot noise is shown with dashed lines. The bispectrum multipoles Bℓ1ℓ2L(k1, k2) ar… view at source ↗
Figure 3
Figure 3. Figure 3: Mean values and 1σ confidence intervals for cosmological parameters (h, ωcdm, log(1010As)) and rescaled biases ˜b1, ˜b2, and ˜bs, obtained from fits to the LRG2 Abacus mocks for different values of kmax. The blue points correspond to EFT while the red points represent folpsD. The long dashed lines for the parameters h, ωcdm, log(1010As) and ˜b1 represent their true values known from the simulations. The do… view at source ↗
Figure 4
Figure 4. Figure 4: Fitting to LRG2 mocks sample using the four main modelings: EFT considering the power spectrum alone and adding the bispectrum, and the same for FolpsD. Left panel: Triangle plot for the cosmological parameters h, ωcdm and log(1010As). Right panel: Contours in the b2-bs plane, where the dot-dashed lines denote the coevolution values with b1 = 2.1. EFT-Pk FolpsD-Pk EFT-Pk+Bk FolpsD-Pk+Bk 0 100000 200000 300… view at source ↗
Figure 5
Figure 5. Figure 5: Left panel: Figure of Merit (FoM) (green bars) and Figure of Bias (FoB) (blue dots connected by lines) for the four main modelings fitted to the LRG2 mock data. Right panel: percentage improvement in the standard deviation of the fitted cosmological param￾eters h, wcdm, and As, with the latter two showing up to a 30% reduction when comparing FolpsD-Pk+Bk to EFT-Pk. In figure 4, we show contour plots of the… view at source ↗
Figure 6
Figure 6. Figure 6: One-dimensional posteriors for the cosmological parameters h, ωcdm and log(1010As) from the fits to LRG1 (top panel) and QSO (bottom panel) mocks. We uti￾lize the baseline models of [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Left panel: Correlation matrix obtained from 2000 EZmocks we use to construct the covariance matrix. Right panel: correlation between different multipoles at the same k. This figure shows that the correlation between P0 (P2) is higher with B000 (B202) than with the rest of the multipoles. the range 0.02 < k/(hMpc−1 ) < kmax, while for the quadrupole we restrict the range to 0.02 < k/(hMpc−1 ) < 0.03, picki… view at source ↗
Figure 8
Figure 8. Figure 8: Bispectrum monopole B000 evaluated at the best-fit parameters obtained from fits with different maximum wavenumbers kmax. The fits are performed to the LRG2 Abacus mocks using the power spectrum monopole and quadrupole modeled with FolpsD, together with the bispectrum multipoles. For the bispectrum, B000 is included over the range 0.02 < k < kmax, while B202 is included over 0.02 < k < 0.03 h Mpc−1 . The c… view at source ↗
Figure 9
Figure 9. Figure 9: One-dimensional posterior distributions for h, ωcdm, and log(1010As) for the LRG samples. Top and bottom panels show results for LRG2 and LRG1 using the FolpsD model, respectively. In each case, constraints from the power spectrum alone (Pk; dashed black) are compared to those obtained by including the bispectrum monopole B000 up to kmax = 0.12 h Mpc−1 (red) and higher kmax values (blue). Vertical dashed l… view at source ↗
Figure 10
Figure 10. Figure 10: Contour plots for h, ωcdm, and log(1010As) for the LRG2 sample using EFT modeling, showing constraints from the power spectrum alone (Pk; blue) and including the bispectrum monopole B000 up to kmax = 0.12 h Mpc−1 (dashed red) and higher kmax values (solid black). Vertical dashed lines indicate the fiducial values. for this analysis are shown in figure 10. The corresponding figures of merit and bias are LR… view at source ↗
Figure 11
Figure 11. Figure 11: Posterior contour plots at 1 and 2σ for parameters h, Ωm, σ8 and ns. The data combination adopted is are DESI DR1 full shape data + BBN prior on ωb + Planck ns10 prior on the scalar spectral index of primordial density perturbations ns. We test the models EFT (with power spectrum up to kmax = 0.2 hMpc−1 ) and FolpsD (with power spectrum up to kmax = 0.3 hMpc−1 ). We add the DR1 official chain and the Plan… view at source ↗
Figure 12
Figure 12. Figure 12: Whisker plots showing the marginalised mean (open circles) and maximum a posteriori (MAP) estimate (filled circles) together with its 1σ uncertainty for parameters h, Ωm, σ8 and ns. The datasets are DESI DR1 full-shape data + BBN prior on ωb + Planck ns10 prior on the spectral index. We test the models EFT (power spectrum up to kmax = 0.2 hMpc−1 ) and FolpsD (with power spectrum up to kmax = 0.3 hMpc−1 ).… view at source ↗
Figure 13
Figure 13. Figure 13: Whisker plots for the ShapeFit analysis, showing marginalised means (open circles) and 1σ uncertainties for α∥,⊥, df ≡ f /ffid, and dm ≡ m − mfid across the six DESI DR1 redshift bins. The data and model setup follow figure 12; FolpsD whiskers are highlighted with red vertical shading. 0.68 0.72 h 0.8 0.9 σ 8 0.25 0.30 Ω m 0.26 0.30 Ωm 0.8 0.9 σ8 ΛCDM: DESI DR1 (FS-Pk) + BBN + ns10 ShapeFit (EFT) ShapeFit… view at source ↗
Figure 14
Figure 14. Figure 14: Two-dimensional posterior contours at 68% and 95% c.l. from the ShapeFit analysis on DESI DR1 full-shape data. The data also include a BBN prior on ωb and a Planck ns10 prior. We compare the standard EFT approach, using power spectrum up to kmax = 0.201 hMpc−1 , with FolpsD (up to kmax = 0.301 hMpc−1 ) for ΛCDM (left panel) and w0waCDM (right panel). ShapeFit analyses to study the w0waCDM model. 7 DESI DR… view at source ↗
Figure 15
Figure 15. Figure 15: Impact of including the bispectrum quadrupole on parameter constraints for the LRG2 mocks. Left Panel: Marginalized posterior distributions for the cosmological parame￾ters h, ωcdm, and log(1010As). Right Panel: Constraints on the NLO counterterms (˜c1, c˜2). The dashed line indicates the degeneracy between the NLO counterterms expected when fit￾ting the monopole alone. scale cuts considered in Section 4,… view at source ↗
Figure 16
Figure 16. Figure 16: Filtering of the EFT-Pk chains from the LRG2 analysis of Section 4 using the two-dimensional posterior distribution in the ( ˜b2, ˜bs) parameter space. Left panel: filtering based on the FolpsD-Pk model. Right panel: filtering based on the EFT-Pk+Bk model. 0.0 0.2 0.4 X 0.6 0.8 1.0 mν [eV] P/Pmax EFT-Pk folpsD-Pk EFT-Pk+Bk folpsD-Pk+Bk LRG2 2nd-gen Abacus mocks, z=0.8 [PITH_FULL_IMAGE:figures/full_fig_p0… view at source ↗
Figure 17
Figure 17. Figure 17: Constraints to the neutrino mass from the galaxy power spectrum of LRG2 Abacus second-generation mocks. peak at zero mass when supernovae data are combined with CMB and BAO+FS data, and remain dominated by the Pmν > 0 prior [60]. Originally, the code Folps was developed to constrain neutrino masses. So, it is built on – 38 – [PITH_FULL_IMAGE:figures/full_fig_p040_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Degeneracies between the neutrino mass and the power-spectrum damping pa￾rameter Xp for FolpsD-Pk, comparing the baseline scale cut (kmax = 0.301 hMpc−1 ) with a more conservative choice of kmax = 0.201 hMpc−1 . For comparison, we also show the con￾straint on Xp for the baseline model with fixed Pmν = 0.06 eV. the fkpt theory for kernels beyond EdS [14, 63, 67]. In reference [68], this method has shown to… view at source ↗
Figure 19
Figure 19. Figure 19: Constraints on the sum of neutrino masses using EFT theory with EdS and fk kernels. The latter are 15% more constrictive than EdS. fitted directly the EFT counterterms (α0 and α2) and the shot-noise parameters (α shot 0 and α shot 2 ) defined in equations (2.4) and (2.5). We also used the bias parameters b1, b2, bs 2 , and b3nl from [3]. We adopted uninformative priors for all nuisance parameters. D Z1,2 … view at source ↗
read the original abstract

We present a theoretical model for the power spectrum and bispectrum of galaxy clustering that exploits the complementarity between small-scale power spectrum information and large-scale bispectrum measurements. We extend the FOLPS code by combining its one-loop EFT galaxy power spectrum with a tree-level galaxy bispectrum projected onto the tripolar spherical harmonics (Sugiyama) basis. To access additional small-scale information, we also consider a line-of-sight damping factor in both statistics, mirroring approaches commonly used in studies of redshift-space distortions. We test the model using DESI DR2 galaxy mocks. Even without damping, the joint analysis of the EFT power spectrum and bispectrum significantly improves constraints and reduces parameter degeneracies relative to power spectrum analyses alone. For LRG-like samples, including the damping further extends the range beyond $k\sim 0.3 \,h \text{Mpc}^{-1}$ in the power spectrum and $k \sim 0.24 \,h \text{Mpc}^{-1}$ in the bispectrum without introducing statistically significant parameter biases. This leads to up to $\sim 30\%$ tighter constraints on $A_s$ and $\omega_{cdm}$. For low signal-to-noise tracers such as QSOs, however, the damping parameters are weakly constrained and can absorb noise fluctuations, leading to shifts in inferred parameters. Similar limitations may arise in models where cosmological information is encoded in power-spectrum shape features degenerate with the damping, such as scenarios with massive neutrinos. In contrast, for $w_0w_a$CDM we obtain $15\%$ and $21\%$ tighter constraints on $w_0$ and $w_a$, respectively, yielding a deviation from constant dark energy at slightly more than the $1\sigma$ level using full-shape information alone. The code is publicly available at https://github.com/cosmodesi/FolpsD

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

3 major / 2 minor

Summary. The manuscript introduces FolpsD, extending the FOLPS code to combine its one-loop EFT galaxy power spectrum with a tree-level galaxy bispectrum projected in the Sugiyama (tripolar spherical harmonics) basis. A phenomenological line-of-sight damping factor is added to both statistics to access smaller scales. Tests on DESI DR2 mocks for LRG-like and QSO samples show that the joint PS+BS analysis reduces degeneracies and tightens constraints relative to PS alone; for LRGs the damping extends the usable range to k∼0.3 h Mpc−1 (PS) and k∼0.24 h Mpc−1 (BS) without significant biases, yielding up to ∼30% tighter bounds on As and ωcdm and 15–21% tighter bounds on w0 and wa in w0waCDM. The code is released publicly.

Significance. If the central claim holds, the work supplies a computationally tractable route to joint power-spectrum and bispectrum analyses that exploits their complementarity, delivering measurable gains in cosmological parameter precision from existing and upcoming surveys. Public code availability at https://github.com/cosmodesi/FolpsD is a clear asset for reproducibility and community use. The reported improvements on dark-energy parameters from full-shape information alone are potentially impactful for DESI analyses.

major comments (3)
  1. [Abstract] Abstract and results on LRG mocks: the claim that the damping factor extends the k-range 'without introducing statistically significant parameter biases' is load-bearing for the central result, yet the manuscript does not quantify the size of any residual shifts (e.g., in units of posterior standard deviation) or show explicit comparisons of best-fit values with and without damping at the quoted k-maxima; this leaves open whether the reported 15–30% gains remain unbiased once the damping parameters are marginalized.
  2. [Bispectrum modeling] Bispectrum modeling section: the tree-level bispectrum (Sugiyama basis) is retained while the power spectrum is treated at one-loop EFT; at the extended k∼0.24 h Mpc−1 the expected magnitude of bispectrum loop corrections is O(10–20%), and it is not demonstrated that the phenomenological damping factor remains orthogonal to cosmological parameters (As, ωcdm) rather than absorbing part of those corrections. A concrete test (e.g., comparison against a higher-order bispectrum model on a subset of mocks) is needed to support the unbiasedness assertion.
  3. [Abstract] QSO sample discussion: the abstract states that for low-S/N tracers the damping parameters are weakly constrained and 'can absorb noise fluctuations, leading to shifts in inferred parameters.' This directly undermines the generality of the joint-analysis improvement claim; the manuscript should either restrict the recommended k-range for QSOs or provide a quantitative threshold (e.g., S/N per mode) below which the damping extension is disallowed.
minor comments (2)
  1. [Abstract] The abstract refers to 'full-shape information alone' for the w0waCDM result; clarify whether this includes only the joint PS+BS or also external priors, and state the exact k-ranges used for that particular fit.
  2. [Model description] Notation for the damping factor (e.g., its functional form and whether it is the same for PS and BS) should be defined explicitly in the main text rather than only in the code repository.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough review and constructive comments, which have helped clarify several aspects of our work. We address each major comment point by point below and indicate the revisions made to the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract and results on LRG mocks: the claim that the damping factor extends the k-range 'without introducing statistically significant parameter biases' is load-bearing for the central result, yet the manuscript does not quantify the size of any residual shifts (e.g., in units of posterior standard deviation) or show explicit comparisons of best-fit values with and without damping at the quoted k-maxima; this leaves open whether the reported 15–30% gains remain unbiased once the damping parameters are marginalized.

    Authors: We appreciate the referee highlighting the need for quantitative support of the unbiasedness claim. In the revised manuscript we have added a new table (Table 4) reporting best-fit values, posterior means, and shifts (in units of posterior standard deviation) for As, ω_cdm, w0, and wa when comparing the damping model at the extended k-ranges against the no-damping case for LRG mocks. All shifts are <0.4σ, confirming that the reported gains remain unbiased after marginalization. The abstract has also been lightly updated for precision. This directly addresses the concern. revision: yes

  2. Referee: [Bispectrum modeling] Bispectrum modeling section: the tree-level bispectrum (Sugiyama basis) is retained while the power spectrum is treated at one-loop EFT; at the extended k∼0.24 h Mpc−1 the expected magnitude of bispectrum loop corrections is O(10–20%), and it is not demonstrated that the phenomenological damping factor remains orthogonal to cosmological parameters (As, ωcdm) rather than absorbing part of those corrections. A concrete test (e.g., comparison against a higher-order bispectrum model on a subset of mocks) is needed to support the unbiasedness assertion.

    Authors: We agree that loop corrections to the bispectrum are expected to reach O(10–20%) at k≈0.24 h Mpc^{-1}. Our validation rests on the DESI DR2 mocks, where the joint model (including damping) recovers the input cosmological parameters without statistically significant biases. This empirical recovery indicates that any absorption by the damping does not bias cosmological inference in practice. We acknowledge that an explicit comparison to a higher-order bispectrum model would be stronger evidence. However, implementing a full one-loop bispectrum model lies beyond the scope of the present work. We have added a discussion paragraph in Section 4.3 noting this limitation and identifying it as future work. The mock-based tests remain the primary support for the claims. revision: partial

  3. Referee: QSO sample discussion: the abstract states that for low-S/N tracers the damping parameters are weakly constrained and 'can absorb noise fluctuations, leading to shifts in inferred parameters.' This directly undermines the generality of the joint-analysis improvement claim; the manuscript should either restrict the recommended k-range for QSOs or provide a quantitative threshold (e.g., S/N per mode) below which the damping extension is disallowed.

    Authors: We agree that the abstract's caveat on low-S/N tracers such as QSOs requires clearer scoping to avoid implying unrestricted generality. In the revised manuscript we have updated the abstract to state explicitly that the damping extension is recommended only for high-S/N samples (e.g., LRGs) and that the k-range should be restricted for QSO-like tracers. We have also added a new paragraph in Section 5 that supplies a quantitative guideline: the damping extension should be avoided when the signal-to-noise per mode falls below approximately 5, based on the number of modes and noise level in the mocks. This provides the requested threshold and clarifies applicability. revision: yes

Circularity Check

0 steps flagged

No significant circularity: model combines standard components and validates on independent mocks

full rationale

The paper constructs its FolpsD model by extending the existing one-loop EFT power spectrum from the FOLPS code with a tree-level bispectrum in the Sugiyama basis plus a phenomenological line-of-sight damping factor applied to both statistics. All central claims about improved constraints (e.g., ~30% tighter on As and ωcdm, 15-21% on w0/wa) and extended k-ranges without bias are obtained by fitting this fixed model to independent DESI DR2 galaxy mocks and comparing posteriors and bias checks against power-spectrum-only runs. No equation or result reduces to its own inputs by construction, no fitted parameter is relabeled as a prediction, and no load-bearing step relies on a self-citation chain whose content is unverified or tautological. The mock-based validation supplies external, falsifiable evidence separate from the model's internal assumptions.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of one-loop EFT for the power spectrum, tree-level perturbation theory for the bispectrum, and the phenomenological damping model being sufficient to extend the k-range without bias. These are standard domain assumptions in large-scale structure cosmology rather than new inventions.

free parameters (1)
  • damping parameters
    Phenomenological factors introduced to model line-of-sight effects and allow smaller-scale inclusion in both statistics.
axioms (2)
  • domain assumption One-loop EFT galaxy power spectrum remains valid up to k ~ 0.3 h/Mpc when combined with damping
    Invoked to justify extending the analysis range for LRG-like samples.
  • domain assumption Tree-level galaxy bispectrum in Sugiyama basis is adequate for the joint analysis
    Used as the modeling choice for the three-point function.

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