Pith. sign in

REVIEW 3 major objections 5 minor 1 cited by

The direct measurement of gravitational potential decay rate at cosmological scales II -- Improved dark energy constraint from $z\le1.4$

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper reports a direct measurement of the gravitational potential decay rate DR(z), the quantity behind the integrated Sachs–Wolfe effect, in six tomographic redshift bins over $0.2\le z<1.4$, using Planck CMB temperature and lensing…

desk verdict Genuine new DR measurement to z=1.4 with careful systematics, but photo-z leakage in the high-z bins is unmodeled and can bias the improved w constraints. read the letter →

arxiv 2411.12594 v1 pith:QEA5CWDK submitted 2024-11-19 astro-ph.CO

Fuyu Dong , Pengjie Zhang , Haojie Xu , Jian Qin This is my paper · ORCID
classification astro-ph.CO
keywords gravitationalpotentialdecayrateintegratedSachs-WolfeeffectCMBlensingdarkenergyequationofstatecosmicaccelerationphotometricredshifttomographyDESIDR9galaxycatalogmagnificationbias
topics Dark Energy
open problems Dark Energy
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper reports a direct measurement of the gravitational potential decay rate $\mathrm{DR}(z)$, the quantity behind the integrated Sachs–Wolfe effect, in six tomographic redshift bins over $0.2\le z<1.4$, using Planck CMB temperature and lensing maps together with DESI DR9 photometric galaxies. The detection is quoted at a total significance of $3.1\sigma$. Because the decay rate is caused by cosmic acceleration, it is a direct probe of dark energy, and the paper shows that adding $\mathrm{DR}$ to baryon acoustic oscillation and supernova data tightens equation-of-state constraints. In the flat $w$CDM model the three probes all favor $w=-1$; in the $w_0w_a$CDM model, $\mathrm{DR}$ plus supernovae give $w_0=-0.94^{+0.11}_{-0.13}$ and $w_a=-0.22^{+0.57}_{-0.97}$, so the data show no preference for dynamical dark energy over a cosmological constant.

What carries the argument

The load-bearing identity is $C^{Ig}_\ell \simeq \mathrm{DR}(z_m)\,C^{\phi g}_\ell$: the ISW–galaxy and lensing–galaxy cross-spectra share the same galaxy window and matter-clustering factors, so their ratio isolates $\mathrm{DR}(z)=(-d\ln D_\phi/d\ln a)(aH/c)/W_L(z)$, where $D_\phi$ is the linear growth factor of the potential and $W_L$ is a lensing weight. This removes the usual galaxy-bias and sampling-variance limitations of ISW measurements. The measurement uses a Bayesian likelihood for $P(\mathrm{DR})$ evaluated from these cross-spectra over $(\ell_{\min},\ell_{\max})\simeq(9,117)$, with the full covariance across all six redshift bins treated simultaneously, random-forest imaging weights, and a magnification-bias correction.

What would settle it

Take the stacked photometric-redshift distribution in each of the three highest bins, recalculate the effective redshift and the expected DR, and compare with the quoted values; if the inferred DR shifts by more than the error bars, the clean-bin assumption is falsified. A complementary test is to redo the measurement using only galaxies with the most reliable photo-z estimates and see whether the $w$ posterior changes.

Watch

Extended reading notes

Core claim

The central claim is that the ratio of the CMB temperature–galaxy cross-power spectrum $C^{Ig}_\ell$ to the CMB lensing–galaxy cross-power spectrum $C^{\phi g}_\ell$ isolates the gravitational potential decay rate $\mathrm{DR}(z_m)$ at the effective redshift of each galaxy slice, with galaxy bias and matter clustering cancelling out. Using six equally spaced redshift bins in $0.2\le z<1.4$, a full covariance matrix across bins, imaging-systematics weights from a random-forest calibration, and a magnification-bias correction with $q=3$, the paper obtains $\mathrm{DR}$ values at $z_m\simeq0.31$, $0.51$, $0.70$, $0.91$, $1.09$ and $1.28$, with a combined significance of about $3.1\sigma$. These measurements are then used to constrain flat $w$CDM and flat $w_0w_a$CDM cosmologies. The paper finds that $\mathrm{DR}$ agrees with DESI BAO, improves dark-energy constraints from SDSS BAO or PantheonPlus supernovae substantially, and that $\mathrm{DR}$ plus DESI BAO gives $\Omega_m=0.292^{+0.014}_{-0.014}$ and $w=-1.019^{+0.112}_{-0.120}$, while $\mathrm{DR}$ plus supernovae in the $w_0w_a$ model gives $w_0=-0.94^{+0.11}_{-0.13}$, $w_a=-0.22^{+0.57}_{-0.97}$.

Load-bearing premise

The high-redshift bins are treated as if each contains galaxies at one effective redshift; if photometric-redshift errors blend the bins, the measured DR values at $z\simeq0.9$–$1.3$ are mixtures and the dark-energy constraints drawn from them could be biased.

Editorial extensions

If this is right

  • DR at $z\simeq1.3$ is roughly twelve times more sensitive to $w$ than $H(z)$, so the high-redshift bins carry real weight in equation-of-state fits.
  • Adding DR to DESI BAO shrinks the $w$ error bar by about 18 percent and leaves $\Omega_m$ essentially unchanged.
  • Because the degeneracy directions of DR, SDSS BAO, and PantheonPlus SNe are nearly orthogonal, adding DR to those probes improves the dark-energy constraints substantially.
  • All three probes — DR, DESI BAO, and PantheonPlus SNe — favor $w=-1$ within $1\sigma$ in the $w$CDM model, while SDSS BAO alone favors $w<-1$ at the $2\sigma$ level.
  • In the $w_0w_a$ model, DR plus supernovae has no preference for dynamical dark energy over $\Lambda$CDM.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If photometric redshift leakage mixes the high-redshift bins, the DR values at $z\simeq0.9$–$1.3$ are weighted mixtures rather than single-redshift measurements, and the $w$ constraints built from them could be biased because DR sensitivity to $w$ increases steeply with redshift; a spectroscopic or better-calibrated photo-z sample could test this.
  • The same ratio method should become much more powerful with lower-noise CMB lensing and larger galaxy samples, since CMB temperature noise dominates the ISW term and currently limits the total significance.
  • The observable could also serve as a modified-gravity test: because DR measures the evolution of the potential directly, residuals relative to the $\Lambda$CDM prediction would signal physics beyond smooth dark energy even if the expansion history were fixed.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper measures the gravitational potential decay rate DR(z) through the ratio of ISW-galaxy and lensing-galaxy cross-power spectra in six photometric redshift bins from z=0.2 to z=1.4, using DESI DR9 galaxy catalogs and Planck CMB products. It reports a total significance of 3.1 sigma, extends the earlier DR measurement of Dong et al. (2022) to higher redshift, and uses the DR measurements to constrain flat wCDM and w0-waCDM models, both alone and in combination with SDSS/DESI BAO and PantheonPlus supernovae. The paper claims that adding DR significantly improves dark-energy constraints relative to SNe alone or SDSS BAO alone, while the improvement over DESI BAO is modest.

Significance. If the high-redshift DR measurements are unbiased, this work provides a genuinely new and independent cosmological probe at z~0.9-1.3, where the sensitivity of DR to the dark-energy equation of state is substantially higher than at z<0.8. The treatment of imaging systematics with Random Forest weights, the use of full covariance matrices for the DR estimates, and the explicit magnification-bias consistency checks are strengths, as is the reliance on public data. However, the central claim of improved dark-energy constraints from the z4-z6 bins depends on the assumption that each photometric redshift bin is effectively unmixed, and the paper's own Fig. 3 indicates that this assumption is questionable at high redshift.

major comments (3)
  1. [Sec. 3.1, Eq. (1)] The identification C^Ig_i/C^phi g_i = DR(z_m) is only valid if each galaxy bin has an effectively monochromatic redshift selection. For a realistic photometric redshift distribution n_i(z), the measured ratio is an n_i-weighted ratio of integrals involving the ISW and lensing kernels, so the effective redshift of the ratio can differ from the nominal z_m and depends on the photo-z tails. Figure 3 itself reports strong cross-correlations between z4/z5 and z5/z6, which the text attributes to 'a mixture of galaxies due to the lower accuracy in the photo-z estimation for higher redshifts.' The full covariance matrix accounts for statistical correlations between bins, but not for the deterministic bias produced by this leakage. Because the sensitivity of DR to w increases steeply with redshift (Sec. 3.2.1), even a modest leaked fraction can shift the z4-z6 DR values and bias the w0 and wa constraints in Tables 3 and 4. Please quantify the leakage using the photo-z error distribution or a spec-z cross-match, propagate it into the DR values and the final posteriors, or otherwise demonstrate that this effect is negligible.
  2. [Sec. 3.2.1, Eq. (5)] The parameter likelihood multiplies the per-bin PDFs P_i(DR|theta,z_i) as if the six DR measurements were independent, but Sec. 3.1 emphasizes that the cross-correlations between bins are non-negligible and states that the DR values are measured simultaneously using the full covariance matrix. Marginalizing the joint DR posterior to skew-normal PDFs and then multiplying them discards the cross-bin covariance information that motivated the simultaneous measurement. This independence assumption is contradictory to the evidence in Fig. 3 and can bias the quoted error bars and best-fit shifts in Tables 3 and 4. Please construct the theta likelihood directly from the joint data vector and covariance, or provide a quantitative justification that the cross-bin covariance is negligible for the parameter combination considered here.
  3. [Sec. 4 and Sec. 2.3] The baseline magnification-bias correction adopts q=3, but the galaxy-shear cross-correlation measurements in Fig. 10 estimate q in the range of roughly 1.4 to 2.7, generally below 3. A factor-of-two error in q changes the magnification correction by a factor of two, and the paper states that the resulting impact on DR is approximately 20% at ell>10; however, the uncertainty in q is not propagated into the DR values in Table 1 or into the final wCDM and w0waCDM constraints in Tables 3 and 4. Please propagate the measured q uncertainty, or vary q within its measured range, and show the corresponding shifts in DR and in the derived dark-energy parameters.
minor comments (5)
  1. [Table 1] The table caption refers to columns for <DR> and sigma(DR), but the table only shows the best-fit DR value and its 68% uncertainties; the caption should be updated to match the actual columns.
  2. [Abstract] The sentence 'the addition of DR can significantly improves DE constraints' has a subject-verb agreement error and should read 'can significantly improve.'
  3. [Table 3 note] The note contains the typo 'the first there redshifts'; it should be 'the first three redshifts.'
  4. [Sec. 3.1] The choice of (ell_min, ell_max) ~ (9,117) is stated without a detailed discussion of how sensitive the DR measurement is to this window; a short robustness test or a reference to the earlier analysis would help.
  5. [Fig. 3] The color bar for the covariance matrices is labeled from 0 to 1, but cross-covariances can be negative or exceed unity in normalized units; please clarify the normalization or use a symmetric color scale.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; the DR measurement is data-driven and the w constraints compare it to independent theory, with only mild fiducial-cosmology dependence in the magnification-bias correction.

full rationale

No circular step was found. The DR values in Table 1 are constructed from the observed ratio relation C^Ig ≈ DR(z_m) C^phi g (Eq. 1), with DR defined by Eq. (2); neither equation fits w from the data. The Bayesian likelihood P(DR) is built from the measured C^Ig and C^phi g cross-power spectra (Sec. 3.1), and the cosmological posterior in Eq. (5) evaluates the measured DR PDF at the theoretical DR(theta,z_i) computed from standard growth and distance quantities (footnote 6). This is a genuine comparison of an independently measured observable to a separately computed model prediction, not an inversion of the model. The magnification-bias calibration does adopt a fixed Planck cosmology with w = -1, which is a mild fiducial input, but the paper explicitly tests this choice: Fig. 11 shows that a cosmology-dependent calibration leaves the constraint essentially unchanged for Omega_m < 0.5, and Fig. 10 measures q from galaxy-shear cross-correlations rather than assuming the calibration value. The self-citations to Dong et al. (2022) and Sun et al. (2023) are methodological references for the likelihood pipeline, not load-bearing proof of the physical result; the original DR relation is attributed to Zhang (2006). The photo-z leakage noted in Fig. 3 is a possible systematic bias in the high-redshift bins, but it is a measurement-uncertainty concern, not a reduction of the prediction to its inputs by construction.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The analysis relies on linear perturbation theory, the ratio method from prior work, an unmixed-bin assumption for photometric redshifts, and a fiducial cosmology for magnification-bias calibration. No new particles, forces, fields, or dimensions are introduced. The free parameters are analysis choices and the adopted magnification coefficient, not fitted nuisance parameters in the likelihood.

free parameters (2)
  • Magnification-bias coefficient q = 3 (fixed; shear cross-correlation fits q around 1.35 to 2.72)
    Set from the measured s~1 in Eq. (4) and used to calibrate C^Ig and C^phi g. The paper's own Fig.10 finds lower q values in most cases, so this choice can shift high-z DR at the tens-of-percent level.
  • Multipole window (ell_min, ell_max) = (9, 117)
    Hand-selected range used for all DR measurements; no explicit sensitivity scan or alternative range is presented in the main text.
assumptions (4)
  • standard math Limber approximation and linear perturbation theory for C^Ig and C^phi g
    Appendix A, Eqs. (A1) to (A7); used for theoretical DR predictions.
  • domain assumption C^Ig approximately equals DR(z_m) times C^phi g, with the same linear galaxy bias for both cross-powers
    Eq. (1), Section 1; the basis of the ratio method.
  • domain assumption Photometric redshift bins are effectively unmixed
    Section 3.1 and Fig.3; inter-bin covariance is treated as noise only, while cross-correlations between z4/z5 and z5/z6 are attributed to photo-z mixture.
  • domain assumption Fiducial Planck cosmology used for magnification-bias calibration
    Section 2.3 sets Omega_m=0.315 and w=-1; the consistency check in Section 4 and Fig.11 shows insensitivity to this choice for Omega_m below 0.5.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The direct measurement of gravitational potential decay rate at cosmological scales II -- Improved dark energy constraint from $z\le1.4$." pith.science (2026). https://pith.science/paper/QEA5CWDK

@misc{pith2026241112594,
  author       = {Pith},
  title        = {Pith review of: The direct measurement of gravitational potential decay rate at cosmological scales II -- Improved dark energy constraint from $z\le1.4$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QEA5CWDK}},
  note         = {Machine review of arXiv:2411.12594}
}
abstract

The gravitational potential decay rate (DR) is caused by the cosmic acceleration of the universe, providing a direct probe into the existence of dark energy (DE). We present measurements of DR and explore its implications for DE models using the Data Release 9 galaxy catalog of DESI imaging surveys and the Planck cosmic microwave background maps. Our analysis includes six redshift bins within the range of $0.2\le z<1.4$ and achieves a total significance of 3.1$\sigma$, extending the DR measurements to a much higher redshift comparing to Dong et al. (2022), which focused on $0.2\le z<0.8$. Other improvements involve addressing potential systematics in the DR-related measurements of correlation functions, including imaging systematics and magnification bias. We explore the constraining power of DR both the $w$CDM model and the $w_0w_a$CDM model. We find that, the addition of DR can significantly improves DE constraints, over Sloan Digital Sky Survey baryon acoustic oscillation (BAO) data alone or PantheonPlus supernovae (SNe) compilation alone, although it shows only a modest improvement for DESI BAO. In the $w$CDM model, all three probes-DR, DESI BAO and SNe-favor $w=-1$. For the $w_0w_a$CDM, while DESI BAO prefers $w_0>-1$ and $w_a<0$, SNe Ia and DR data constrain $w_0=-0.94^{+0.11}_{-0.13}$ and $w_a=-0.22^{+0.57}_{-0.97}$. Namely SNe Ia and DR data has no preference on dynamical dark energy over $\Lambda$.

Figures

Figures reproduced from arXiv: 2411.12594 by the authors.

Figure 1
Figure 1. Source distribution in the DR9 catalog of the DECaLS+DES imaging surveys adopted in our measure￾ment. Both CMB mask and galaxy survey mask have been adopted, along with additional masks near the Large Magel￾lanic Cloud, the disconnected areas as well as the boarders surrounding DES. The depth of color represents the number of galaxies per arcmin2 . 2.2. Galaxy Catalogue We perform the analysis based on the public Da… view at source ↗
Figure 2
Figure 2. The cross-correlation between CMB lensing (or ISW) and galaxies measured for six redshift bins: zm = 0.3, 0.5, 0.7, 0.9, 1.1 and 1.3. The red and green color show results of Cϕg and CIg, respectively. There are two line styles shown for each color. The dashed line shows the signal measured by adopting the imaging weight, while the dotted line shows the signal without adopting any weight but the using same angular ma… view at source ↗
Figure 3
Figure 3. The covariance matrix for Cϕg and CIg, for which the cross-powers are calculated by adopting the imaging weight [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: The probability distribution function of DR(z) obtained from a Bayesian analysis (solid line). It is well described by a skew-normal function (dashed line). The PDF is normalized with its peak amplitude. with DR from redshifts zn1 ≤ z ≤ zn2 as: P(θ|DR) ∝ Yn2 i=n1 Pi(DR…
Figure 5
Figure 5. Figure 5: Constraints on the flat wCDM model from DR alone. In the left panel, we show the result by using all the DR measurements from six redshifts (i.e., DR6z) in the red contour, based on galaxy samples from the DESI DR9 ”DECaLS+DES” survey areas. For comparison, we also plo…
Figure 6
Figure 6. Figure 6: Constraints on the flat wCDM model, from BAO/DR/BAO+DR. The left panel is obtained with the DESI BAO data, while the right panel is obtained with the SDSS BAO data. In both panels, DR shows a different Ωm-w degeneracy direction compared to BAO data. When compared to SD…
Figure 7
Figure 7. Figure 7: Similar to Fig.6, but with constraints from SNe/DR/SNe+DR. tributed to the redshift distributions of the galaxy sam￾ples, as the DESI DR1 galaxies are, on average, from higher redshifts where the sensitivity of H(z) to w is weaker (peaking at ∼ 0.6 and decreasing). 3.2…
Figure 8
Figure 8. Figure 8: Constraints on the flat w0wa model, from DESI BAO/DR/DESI BAO+DR. The DR measurements align well with the DESI BAO data across all parameter planes. plore the constraints of DR on a time-varying equation￾of-state model parameterized by w0 and wa. In this case, DESI BAO…
Figure 9
Figure 9. Figure 9: Similar to Fig.8, but with constraints from SNe/DR/SNe+DR. i) First is the measurement of q. In our baseline anal￾ysis, we derive the value of q from the cumulative num￾ber counts of galaxy samples. Here, we conduct a test by measuring q through galaxy-shear cross-corr…
Figure 10
Figure 10. Figure 10: The measurement of q through the galaxy-shear cross-correlation ⟨δ L g (zgal)γ +(zgal)⟩, where zf < zgal and zgal the source distribution of galaxies. The values of q are fitted using all the data points (green line) and only the outer data points (red line), where qw…
Figure 11
Figure 11. Figure 11: Constraints on the flat wCDM from DR by adopting different strategies for calibrating the magnifica￾tion bias. The red and green color contours show the results for calibrating the impact of magnification bias on the theo￾retical prediction of DRt . While the blue col…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Synergy between the gravitational potential decay rate and other structure growth probes in testing gravity

    astro-ph.CO 2026-06 unverdicted novelty 5.0 of 10

    Tomographic DR data added to Σ8 + fσ8 tightens phenomenological MG parameters (μ0, Σ0, η0) and EFT α coefficients by factors of 1.5–2.

Reference graph

Works this paper leans on

90 extracted references · 9 canonical work pages · cited by 1 Pith paper

  1. [1]

    The Pantheon+ Analysis: The Full Data Set and Light-curve Release

    thebibliography [1] 20pt to REFERENCES 6pt =0pt 10pt plus 3pt =0pt =0pt =1pt plus 1pt =0pt =0pt -12pt =13pt plus 1pt =20pt =13pt plus 1pt \@M =10000 =-1.0em =0pt =0pt 0pt =0pt =1.0em @enumiv\@empty 10000 10000 `\.\@m \@noitemerr \@latex@warning Empty `thebibliography' environment \@ifnextchar \@reference \@latexerr Missing key on reference command Each re...

    arXiv 2025

  2. [2]

    2004, , 70, 083536, 10.1103/PhysRevD.70.083536

    Afshordi , N. 2004, , 70, 083536, 10.1103/PhysRevD.70.083536

    doi:10.1103/physrevd.70.083536 2004

  3. [3]

    Afshordi , N., Loh , Y.-S., & Strauss , M. A. 2004, , 69, 083524, 10.1103/PhysRevD.69.083524

    doi:10.1103/physrevd.69.083524 2004

  4. [5]

    2022, , 10.1093/mnras/stac2040

    Bahr-Kalus , B., Parkinson , D., Asorey , J., et al. 2022, , 10.1093/mnras/stac2040

    doi:10.1093/mnras/stac2040 2022

  5. [6]

    2000, The Astrophysical Journal, 545, L1, 10.1086/317323

    Balbi, A., Ade, P., Bock, J., et al. 2000, The Astrophysical Journal, 545, L1, 10.1086/317323

    work page doi:10.1086/317323 2000

  6. [7]

    E., Paviot , R., Vargas Maga \ n a , M., et al

    Bautista , J. E., Paviot , R., Vargas Maga \ n a , M., et al. 2021, , 500, 736, 10.1093/mnras/staa2800

    doi:10.1093/mnras/staa2800 2021

  7. [8]

    L., Larson , D., Weiland , J

    Bennett , C. L., Larson , D., Weiland , J. L., et al. 2013, , 208, 20, 10.1088/0067-0049/208/2/20

    doi:10.1088/0067-0049/208/2/20 2013

  8. [9]

    2004, , 427, 45, 10.1038/nature02139

    Boughn , S., & Crittenden , R. 2004, , 427, 45, 10.1038/nature02139

    work page doi:10.1038/nature02139 2004

Show all 90 references
  1. [10]

    P., & Crittenden , R

    Boughn , S. P., & Crittenden , R. G. 2005, , 49, 75, 10.1016/j.newar.2005.01.005

    2005 doi

  2. [11]

    2022, , 938, 110, 10.3847/1538-4357/ac8e04

    Brout , D., Scolnic , D., Popovic , B., et al. 2022, , 938, 110, 10.3847/1538-4357/ac8e04

    2022 doi

  3. [13]

    E., Ade , P

    Carlstrom , J. E., Ade , P. A. R., Aird , K. A., et al. 2011, , 123, 568, 10.1086/659879

    2011 doi

  4. [14]

    2022, , 509, 3904, 10.1093/mnras/stab3252

    Chaussidon , E., Y \`e che , C., Palanque-Delabrouille , N., et al. 2022, , 509, 3904, 10.1093/mnras/stab3252

    2022 doi

  5. [15]

    2023, , 944, 107, 10.3847/1538-4357/acb3c2

    ---. 2023, , 944, 107, 10.3847/1538-4357/acb3c2

    2023 doi

  6. [16]

    2001, International Journal of Modern Physics D, 10, 213, 10.1142/S0218271801000822

    Chevallier , M., & Polarski , D. 2001, International Journal of Modern Physics D, 10, 213, 10.1142/S0218271801000822

    2001 doi

  7. [18]

    M., et al

    Contreras , C., Hamuy , M., Phillips , M. M., et al. 2010, , 139, 519, 10.1088/0004-6256/139/2/519

    2010 doi

  8. [19]

    2005, , 71, 123521, 10.1103/PhysRevD.71.123521

    Corasaniti , P.-S., Giannantonio , T., & Melchiorri , A. 2005, , 71, 123521, 10.1103/PhysRevD.71.123521

    2005 doi

  9. [20]

    G., & Turok , N

    Crittenden , R. G., & Turok , N. 1996, , 76, 575, 10.1103/PhysRevLett.76.575

    1996 doi

  10. [21]

    S., Schlegel , D

    Dawson , K. S., Schlegel , D. J., Ahn , C. P., et al. 2013, , 145, 10, 10.1088/0004-6256/145/1/10

    2013 doi

  11. [22]

    S., Kneib , J.-P., Percival , W

    Dawson , K. S., Kneib , J.-P., Percival , W. J., et al. 2016, , 151, 44, 10.3847/0004-6256/151/2/44

    2016 doi

  12. [23]

    de Bernardis , P., Ade , P. A. R., Bock , J. J., et al. 2000, , 404, 955, 10.1038/35010035

    2000 doi

  13. [24]

    2021, , 501, 5616, 10.1093/mnras/staa3891

    de Mattia , A., Ruhlmann-Kleider , V., Raichoor , A., et al. 2021, , 501, 5616, 10.1093/mnras/staa3891

    2021 doi

  14. [25]

    G., Aguilar , J., et al

    DESI Collaboration , Adame , A. G., Aguilar , J., et al. 2024 a , arXiv e-prints, arXiv:2404.03002, 10.48550/arXiv.2404.03002

    2024 doi arXiv

  15. [26]

    2024 b , arXiv e-prints, arXiv:2404.03000, 10.48550/arXiv.2404.03000

    ---. 2024 b , arXiv e-prints, arXiv:2404.03000, 10.48550/arXiv.2404.03000

    2024 doi arXiv

  16. [27]

    2024 c , arXiv e-prints, arXiv:2404.03001, 10.48550/arXiv.2404.03001

    ---. 2024 c , arXiv e-prints, arXiv:2404.03001, 10.48550/arXiv.2404.03001

    2024 doi arXiv

  17. [28]

    2024 d , , 167, 62, 10.3847/1538-3881/ad0b08

    ---. 2024 d , , 167, 62, 10.3847/1538-3881/ad0b08

    2024 doi

  18. [29]

    2025, DESI 2024 I: Data Release 1 of the Dark Energy Spectroscopic Instrument, in preparation (2025)

    DESI Collaboration et al. 2025, DESI 2024 I: Data Release 1 of the Dark Energy Spectroscopic Instrument, in preparation (2025)

    2025

  19. [30]

    2018, , 239, 35, 10.3847/1538-4365/aaee8c

    Diemer , B. 2018, , 239, 35, 10.3847/1538-4365/aaee8c

    2018 doi

  20. [31]

    2021 a , , 500, 3838, 10.1093/mnras/staa3194

    Dong , F., Yu , Y., Zhang , J., Yang , X., & Zhang , P. 2021 a , , 500, 3838, 10.1093/mnras/staa3194

    2021 doi

  21. [32]

    2022, , 938, 72, 10.3847/1538-4357/ac905b

    Dong , F., Zhang , P., Sun , Z., & Park , C. 2022, , 938, 72, 10.3847/1538-4357/ac905b

    2022 doi

  22. [33]

    2021 b , , 923, 153, 10.3847/1538-4357/ac2d31

    Dong , F., Zhang , P., Zhang , L., et al. 2021 b , , 923, 153, 10.3847/1538-4357/ac2d31

    2021 doi

  23. [34]

    J., Zehavi , I., Hogg , D

    Eisenstein , D. J., Zehavi , I., Hogg , D. W., et al. 2005, , 633, 560, 10.1086/466512

    2005 doi

  24. [35]

    M., Burns , C

    Folatelli , G., Phillips , M. M., Burns , C. R., et al. 2010, , 139, 120, 10.1088/0004-6256/139/1/120

    2010 doi

  25. [36]

    2004, , 350, L37, 10.1111/j.1365-2966.2004.07837.x

    Fosalba , P., & Gazta \ n aga , E. 2004, , 350, L37, 10.1111/j.1365-2966.2004.07837.x

    2004

  26. [37]

    Fosalba , P., Gazta \ n aga , E., & Castand er , F. J. 2003, , 597, L89, 10.1086/379848

    2003 doi

  27. [38]

    W., Niemack , M

    Fowler , J. W., Niemack , M. D., Dicker , S. R., et al. 2007, , 46, 3444, 10.1364/AO.46.003444

    2007 doi

  28. [39]

    L., Madore, B

    Freedman, W. L., Madore, B. F., Scowcroft, V., et al. 2012, The Astrophysical Journal, 758, 24, 10.1088/0004-637X/758/1/24

    2012 doi

  29. [41]

    E., Paviot , R., et al

    Gil-Mar \' n , H., Bautista , J. E., Paviot , R., et al. 2020, , 498, 2492, 10.1093/mnras/staa2455

    2020 doi

  30. [42]

    M., Hivon , E., Banday , A

    G \'o rski , K. M., Hivon , E., Banday , A. J., et al. 2005, , 622, 759, 10.1086/427976

    2005 doi

  31. [43]

    A., Maoz , D., et al

    Graur , O., Rodney , S. A., Maoz , D., et al. 2014, , 783, 28, 10.1088/0004-637X/783/1/28

    2014 doi

  32. [44]

    J., Ruiz-Macias , O., et al

    Hahn , C., Wilson , M. J., Ruiz-Macias , O., et al. 2023, , 165, 253, 10.3847/1538-3881/accff8

    2023 doi

  33. [45]

    2010, , 520, A101, 10.1051/0004-6361/200913344

    Hern \'a ndez-Monteagudo , C. 2010, , 520, A101, 10.1051/0004-6361/200913344

    2010 doi

  34. [46]

    M., Blondin , S., et al

    Hicken , M., Wood-Vasey , W. M., Blondin , S., et al. 2009, , 700, 1097, 10.1088/0004-637X/700/2/1097

    2009 doi

  35. [47]

    2012, The Astrophysical Journal, 761, 14, 10.1088/0004-637X/761/1/14

    Ho, S., Cuesta, A., Seo, H.-J., et al. 2012, The Astrophysical Journal, 761, 14, 10.1088/0004-637X/761/1/14

    2012 doi

  36. [48]

    G., Ross , A

    Hou , J., S \'a nchez , A. G., Ross , A. J., et al. 2021, , 500, 1201, 10.1093/mnras/staa3234

    2021 doi

  37. [49]

    Huterer , D., & Turner , M. S. 2001, , 64, 123527, 10.1103/PhysRevD.64.123527

    2001 doi

  38. [50]

    P., Challis , P., et al

    Jha , S., Kirshner , R. P., Challis , P., et al. 2006, , 131, 527, 10.1086/497989

    2006 doi

  39. [51]

    2010, in Ground-based and Airborne Telescopes III, ed

    Kaiser, N., Burgett, W., Chambers, K., et al. 2010, in Ground-based and Airborne Telescopes III, ed. L. M. Stepp, R. Gilmozzi, & H. J. Hall, Vol. 7733, International Society for Optics and Photonics (SPIE), 77330E, 10.1117/12.859188

    2010 doi

  40. [52]

    2019, , 484, 5267, 10.1093/mnras/stz341

    Kov \'a cs , A., S \'a nchez , C., Garc \' a-Bellido , J., et al. 2019, , 484, 5267, 10.1093/mnras/stz341

    2019 doi

  41. [53]

    Limber , D. N. 1953, , 117, 134, 10.1086/145672

    1953 doi

  42. [54]

    Linder , E. V. 2003, , 90, 091301, 10.1103/PhysRevLett.90.091301

    2003 doi

  43. [55]

    2010, , 404, 532, 10.1111/j.1365-2966.2010.16305.x

    Massardi , M., Bonaldi , A., Negrello , M., et al. 2010, , 404, 532, 10.1111/j.1365-2966.2010.16305.x

    2010

  44. [56]

    D., Vielva , P., Hobson , M

    McEwen , J. D., Vielva , P., Hobson , M. P., Martinez-Gonzalez , E., & Lasenby , A. N. 2006, arXiv e-prints, astro. astro-ph/0602398

    2006 arXiv

  45. [57]

    B., & Hildebrandt, H

    Morrison, C. B., & Hildebrandt, H. 2015, Monthly Notices of the Royal Astronomical Society, 454, 3121, 10.1093/mnras/stv2103

    2015 doi

  46. [58]

    D., Brunner, R

    Myers, A. D., Brunner, R. J., Richards, G. T., et al. 2006, The Astrophysical Journal, 638, 622, 10.1086/499093

    2006 doi

  47. [59]

    2020, , 499, 210, 10.1093/mnras/staa2780

    Neveux , R., Burtin , E., de Mattia , A., et al. 2020, , 499, 210, 10.1093/mnras/staa2780

    2020 doi

  48. [60]

    R., Wright , E

    Nolta , M. R., Wright , E. L., Page , L., et al. 2004, , 608, 10, 10.1086/386536

    2004 doi

  49. [61]

    M., Seljak , U., et al

    Padmanabhan , N., Hirata , C. M., Seljak , U., et al. 2005, , 72, 043525, 10.1103/PhysRevD.72.043525

    2005 doi

  50. [62]

    1999, , 517, 565, 10.1086/307221

    Perlmutter , S., Aldering , G., Goldhaber , G., et al. 1999, , 517, 565, 10.1086/307221

    1999 doi

  51. [63]

    Planck Collaboration , Ade , P. A. R., Aghanim , N., et al. 2014, , 571, A19, 10.1051/0004-6361/201321526

    2014 doi

  52. [64]

    2016 a , , 594, A21, 10.1051/0004-6361/201525831

    ---. 2016 a , , 594, A21, 10.1051/0004-6361/201525831

    2016 doi

  53. [65]

    2016 b , , 594, A11, 10.1051/0004-6361/201526926

    Planck Collaboration , Aghanim , N., Arnaud , M., et al. 2016 b , , 594, A11, 10.1051/0004-6361/201526926

    2016 doi

  54. [66]

    2020, , 641, A8, 10.1051/0004-6361/201833886

    Planck Collaboration , Aghanim , N., Akrami , Y., et al. 2020, , 641, A8, 10.1051/0004-6361/201833886

    2020 doi

  55. [67]

    2008, , 386, 2161, 10.1111/j.1365-2966.2008.13189.x

    Raccanelli , A., Bonaldi , A., Negrello , M., et al. 2008, , 386, 2161, 10.1111/j.1365-2966.2008.13189.x

    2008

  56. [68]

    J., et al

    Raichoor , A., de Mattia , A., Ross , A. J., et al. 2021, , 500, 3254, 10.1093/mnras/staa3336

    2021 doi

  57. [69]

    A., et al

    Raichoor , A., Moustakas , J., Newman , J. A., et al. 2023, , 165, 126, 10.3847/1538-3881/acb213

    2023 doi

  58. [70]

    Rassat , A., Land , K., Lahav , O., & Abdalla , F. B. 2007, , 377, 1085, 10.1111/j.1365-2966.2007.11538.x

    2007

  59. [71]

    G., Filippenko , A

    Riess , A. G., Filippenko , A. V., Challis , P., et al. 1998, , 116, 1009, 10.1086/300499

    1998 doi

  60. [72]

    G., Kirshner , R

    Riess , A. G., Kirshner , R. P., Schmidt , B. P., et al. 1999, , 117, 707, 10.1086/300738

    1999 doi

  61. [73]

    G., Strolger , L.-G., Casertano , S., et al

    Riess , A. G., Strolger , L.-G., Casertano , S., et al. 2007, , 659, 98, 10.1086/510378

    2007 doi

  62. [74]

    G., Macri , L., Casertano , S., et al

    Riess , A. G., Macri , L., Casertano , S., et al. 2009, , 699, 539, 10.1088/0004-637X/699/1/539

    2009 doi

  63. [75]

    G., Rodney , S

    Riess , A. G., Rodney , S. A., Scolnic , D. M., et al. 2018, , 853, 126, 10.3847/1538-4357/aaa5a9

    2018 doi

  64. [76]

    A., Riess , A

    Rodney , S. A., Riess , A. G., Strolger , L.-G., et al. 2014, , 148, 13, 10.1088/0004-6256/148/1/13

    2014 doi

  65. [77]

    J., Ho, S., Cuesta, A

    Ross, A. J., Ho, S., Cuesta, A. J., et al. 2011, Monthly Notices of the Royal Astronomical Society, 417, 1350, 10.1111/j.1365-2966.2011.19351.x

    2011

  66. [78]

    K., & Wolfe , A

    Sachs , R. K., & Wolfe , A. M. 1967, , 147, 73, 10.1086/148982

    1967 doi

  67. [79]

    2012, , 427, 3044, 10.1111/j.1365-2966.2012.21974.x

    Schiavon , F., Finelli , F., Gruppuso , A., et al. 2012, , 427, 3044, 10.1111/j.1365-2966.2012.21974.x

    2012

  68. [80]

    2022, , 938, 113, 10.3847/1538-4357/ac8b7a

    Scolnic , D., Brout , D., Carr , A., et al. 2022, , 938, 113, 10.3847/1538-4357/ac8b7a

    2022 doi

  69. [81]

    2002, The Astrophysical Journal, 579, 48, 10.1086/342786

    Scranton, R., Johnston, D., Dodelson, S., et al. 2002, The Astrophysical Journal, 579, 48, 10.1086/342786

    2002 doi

  70. [82]

    2000, , 538, 57, 10.1086/309098

    Seljak , U., & Zaldarriaga , M. 2000, , 538, 57, 10.1086/309098

    2000 doi

  71. [83]

    J., & Wright, E

    Shajib, A. J., & Wright, E. L. 2016, The Astrophysical Journal, 827, 116, 10.3847/0004-637x/827/2/116

    2016 doi

  72. [84]

    D., Phillips , M

    Stritzinger , M. D., Phillips , M. M., Boldt , L. N., et al. 2011, , 142, 156, 10.1088/0004-6256/142/5/156

    2011 doi

  73. [85]

    2023, , 267, 21, 10.3847/1538-4365/acda2a

    Sun , Z., Zhang , P., Dong , F., et al. 2023, , 267, 21, 10.3847/1538-4365/acda2a

    2023 doi

  74. [86]

    2012, , 746, 85, 10.1088/0004-637X/746/1/85

    Suzuki , N., Rubin , D., Lidman , C., et al. 2012, , 746, 85, 10.1088/0004-637X/746/1/85

    2012 doi

  75. [87]

    2020, , 499, 5527, 10.1093/mnras/staa3050

    Tamone , A., Raichoor , A., Zhao , C., et al. 2020, , 499, 5527, 10.1093/mnras/staa3050

    2020 doi

  76. [88]

    E., von Marttens, R., & Gomes, S

    Velten, H., Fazolo, R. E., von Marttens, R., & Gomes, S. 2018, Phys. Rev. D, 97, 103514, 10.1103/PhysRevD.97.103514

    2018 doi

  77. [89]

    2006, , 365, 891, 10.1111/j.1365-2966.2005.09764.x

    Vielva , P., Mart \' nez-Gonz \'a lez , E., & Tucci , M. 2006, , 365, 891, 10.1111/j.1365-2966.2005.09764.x

    2006

  78. [90]

    2023, , 520, 161, 10.1093/mnras/stad136

    Xu , H., Zhang , P., Peng , H., et al. 2023, , 520, 161, 10.1093/mnras/stad136

    2023 doi

  79. [91]

    2021, , 909, 143, 10.3847/1538-4357/abddb2

    Yang , X., Xu , H., He , M., et al. 2021, , 909, 143, 10.3847/1538-4357/abddb2

    2021 doi

  80. [92]

    2006, , 647, 55, 10.1086/505297

    Zhang , P. 2006, , 647, 55, 10.1086/505297

    2006 doi

  81. [93]

    A., et al

    Zhou , R., Dey , B., Newman , J. A., et al. 2023, , 165, 58, 10.3847/1538-3881/aca5fb

    2023 doi

  82. [95]

    2019, Journal of Open Source Software, 4, 1298, 10.21105/joss.01298

    Zonca, A., Singer, L., Lenz, D., et al. 2019, Journal of Open Source Software, 4, 1298, 10.21105/joss.01298

    2019 doi

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.