Pith. sign in

REVIEW 2 major objections 7 minor 1 cited by

Probing Cosmic Isotropy with the FAST All Sky HI Survey

T0 review · 2 major / 7 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 The local Universe, traced by neutral hydrogen galaxies in the FAST All Sky HI Survey, shows no statistically significant deviation from cosmic isotropy.

desk verdict First FASHI isotropy test with a useful sensitivity-filtering recipe, but the null mocks omit the sensitivity map and the filtering window is post hoc; worth peer review, conditional result. read the letter →

arxiv 2412.00475 v1 pith:RLYH4JCU submitted 2024-11-30 astro-ph.CO

Yi-Wen Wu , Jun-Qing Xia This is my paper · ORCID
classification astro-ph.CO
keywords cosmicisotropytwo-pointangularcorrelationfunctionneutralhydrogengalaxiesFASHIsurveyCosmologicalPrinciplelarge-scalestructurepowerspectrumMCMCparameterestimation
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 tests whether the local Universe is isotropic using the first release of the FAST All Sky HI Survey (FASHI), the largest catalog of extragalactic neutral hydrogen sources. Because FASHI's detection sensitivity varies widely across the sky, the authors restrict attention to the 19,040 sources with sensitivity in [0.65, 1.0] mJy, a window chosen so the measured linear bias matches the expected HI bias. They divide this sample into ten sky regions, measure the two-point angular correlation function in each, and fit a power law with MCMC. Comparing the ten regions against lognormal mock catalogs built under homogeneity and isotropy, they find all regions consistent with statistical isotropy within $2\sigma$. The conclusion is that the HI-selected local Universe shows no statistically significant deviation from the Cosmological Principle.

What carries the argument

The central object is the two-point angular correlation function (2PACF), measured with the Szapudi\textendash{}Szalay estimator over angular scales $0.5^\circ < \theta < 10^\circ$, and fit to a power law $\xi(\theta) = (\theta/\theta_0)^{-\beta}$. The load-bearing device is the sensitivity selection: restricting the catalog to detection sensitivities in [0.65, 1.0] mJy removes artificial clustering introduced by the survey's schedule-filler observing mode. The ten region-by-region power-law fits are compared against 100 lognormal mock catalogs generated under homogeneity and isotropy, and consistency is judged by the scatter of the fitted parameters.

What would settle it

Measure the angular power spectrum dipole and quadrupole of the filtered 19,040-source FASHI sample: a statistically significant nonzero dipole would contradict the isotropy claim. Alternatively, re-run the ten-region 2PACF analysis with the sensitivity window fixed in advance and check whether the scatter in $\beta$ across regions exceeds the mock scatter at more than $2\sigma$.

Watch

Extended reading notes

Core claim

The paper's central claim is that the local Universe, as traced by neutral hydrogen galaxies in FASHI, is statistically isotropic: the best-fit power-law parameters ($\theta_0$, $\beta$) of the two-point angular correlation function in each of ten regions of roughly 600 square degrees agree with one another and with mock catalogs within $2\sigma$. The median values $\theta_0 \approx 0.35^\circ$ and $\beta \approx 1.36$ match the isotropic-mock reference values. After the sensitivity cut, the sample has linear bias $b_0 = 1.02 \pm 0.06$, consistent with expected HI bias and with the ALFALFA survey. The authors note that Area-10's $\beta$ deviates by more than $1\sigma$ from the mock expectation, hinting at possible large-scale structure, but they do not treat this as a detection of anisotropy.

Load-bearing premise

The analysis assumes that cutting the catalog to detection sensitivities in [0.65, 1.0] mJy removes survey-induced clustering without removing real anisotropy, and that this window can be chosen after inspecting the measured bias.

Editorial extensions

If this is right

  • The filtered FASHI subsample, with bias $b_0 = 1.02 \pm 0.06$, behaves like a standard HI tracer and is suitable for further cosmological clustering analyses.
  • The $2\sigma$ consistency of all ten regions supports the Cosmological Principle at the scales and depths probed by FASHI (redshift $z < 0.09$).
  • The Area-10 $\beta$ deviation, though below $2\sigma$, points toward a possible super-void or super-cluster near that region that future surveys could confirm or refute.
  • The strong dependence of the angular power spectrum on the sensitivity window means future FASHI analyses must apply a similar selection to avoid false clustering signals.

Reading between the lines

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

  • A stronger isotropy test would measure the dipole and quadrupole of the source count map directly; the 2PACF comparison here is sensitive to scale-dependent clustering but not to a uniform dipole anisotropy.
  • The sensitivity window [0.65, 1.0] mJy was chosen after seeing that it yields the expected bias; an a priori choice or a scan over many windows with corrected significance would make the isotropy claim more robust.
  • Combining FASHI with ALFALFA data (which the paper says overlap near declination 30 degrees) would double the sky coverage and allow isotropy tests on scales beyond 10 degrees.
  • The full FASHI survey, with over 100,000 predicted sources, will shrink the covariance matrices enough to turn the current $2\sigma$ consistency into a much tighter constraint, or expose a genuine deviation.
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

2 major / 7 minor

Summary. This paper tests cosmic isotropy using the first data release of the FAST All Sky HI Survey (FASHI). Because the survey is operated in a schedule-filler mode, its detection sensitivity is strongly position-dependent, and the authors argue that this induces spurious clustering. They restrict the catalog (z<0.09, 30°<DEC<64°) to a narrow sensitivity range [0.65,1.0] mJy, leaving 19,040 sources over 6,277 deg^2, and show via the angular power spectrum that this filtered sample has bias b0=1.02±0.06, consistent with external HI bias estimates, with shot noise consistent with ΔΩ/N. The footprint is then divided into ten regions of roughly 627 deg^2 each. In each region the two-point angular correlation function (2PACF) is measured over 0.5°<θ<10° and fitted with a power law ξ(θ)=(θ/θ0)^{-β}; the per-region fitted parameters are compared with distributions from 100 FLASK lognormal mock catalogs generated under homogeneity and isotropy with the same fitted b0. The paper concludes that all ten FASHI regions are consistent with statistical isotropy at the 2σ level, noting that Areas 5 and 10 show the largest deviations in β (1.34σ and -1.10σ, respectively) and that the unfiltered catalog shows larger regional fluctuations.

Significance. If the central claim is correct, the paper provides a new, independent HI-based consistency test of the Cosmological Principle and, more durably, a practical recipe for using the first FASHI release despite its strongly non-uniform sensitivity. Strengths: the analysis uses public data and public codes (NaMaster, PolSpice, FLASK, emcee, Colossus); the sensitivity-induced clustering is demonstrated directly in Figure 3 and Table 1; the shot-noise cross-check C_SN ≈ ΔΩ/N validates the power-spectrum fit; and the authors transparently report the unfiltered results in parallel with the filtered ones, including a cross-check of the anomalous Area 10 against ALFALFA Area-V. The isotropy claim itself is a null result consistent with the standard cosmological model, so the significance is moderate rather than high. The quantitative 2σ statement is, however, conditional on the mock construction and the sensitivity-window choice discussed in the major comments.

major comments (2)
  1. [§3.1, Eq. (9), Table 2] The null mocks as described do not reproduce the per-region noise of the data, so the reported uncertainties are not calibrated. The FLASK mocks are generated 'with the same number of sources and sky coverage as the real selected FASHI subsample,' but no position-dependent selection function or sensitivity map is listed among the inputs. Table 2 shows that the ten equal-area (627 deg^2) regions contain between 675 and 3,342 sources (densities 1.07-5.32 deg^-2), a factor-of-five range that cannot be Poisson noise; whether this reflects residual sensitivity selection or genuine large-scale structure, the mocks must reproduce it for the regional comparison to be valid. If sources are drawn uniformly over the footprint, each mock region contains roughly 1,900 sources, and the covariance matrix of Eq. (9) and the MCMC errors quoted in Table 3 and Figure 4 are computed for regions of mean density rather than for the data's actual density. In the sparse regions (e.g., Area 1, with 675 sources), the shot-noise contribution to the 2PACF variance is about an order of magnitude larger than in the mocks, so the per-region error bars and the 'β std' column of Table 3 are not a valid calibration for the data. The 2σ isotropy claim should be re-derived with mocks that include the actual FASHI sensitivity or an empirical per-region selection function, and the mock covariance should be validated against the data before it is used in the likelihood.
  2. [§2.2-§2.3, §3.1, Table 1] The sensitivity window is selected post hoc, and the null hypothesis is calibrated with the resulting fit, making the test partially circular. Table 1 shows the fitted bias decreasing monotonically from b0=1.22±0.05 for the full sample to 0.51±0.09 for the range [0.70,0.93]; the adopted range [0.65,1.0] is the one that returns b0=1.02±0.06, matching the external expectation for HI sources, and the text states that this window is chosen 'to ensure accurate representation of the clustering properties.' A small shift of the window bounds changes the inferred clustering amplitude by a factor of two, so the analysis is extremely sensitive to the cut, and any genuine anisotropy that manifests as a shift in the apparent bias across windows would be absorbed by the selection itself. The 2PACF mocks are then generated with the very same fitted value, b0=1.02, so the comparison in Table 3 tests regional scatter around a data-calibrated mean rather than agreement with an externally predicted clustering amplitude. The paper should demonstrate that the 2σ conclusion is robust across the windows listed in Table 1 and should regenerate the null using an externally fixed bias (for instance b0=1.0 from Martin et al. 2012), quantifying how the β std values and the MCMC uncertainties change.
minor comments (7)
  1. [§3.2] The sentence '...resulting in a total of 1,000 regions for analysis' is attached to the definition of the power-law fit; it appears to be a leftover from the mock description (100 mock catalogs times 10 regions) and should be moved or deleted.
  2. [Table 3] The caption states '95% C.L.' while Section 3.2 describes 68% C.L. constraints, and the Mock row lists uncertainties twice as large as the 68% values quoted in the text; the convention should be made consistent and explicit.
  3. [§2.2] There is an unclosed parenthesis in 'downgrade the count map to a lower resolution of (Nside = 64 and apply an apodization method.'
  4. [§4] The bullet stating the angular separation range as '0° < θ < 10°' conflicts with the 0.5° lower bound used in Section 3.1 and the abstract.
  5. [Table 3] The 'β std' column is not defined: state whether σ is the scatter of the 1,000 mock region fits or the MCMC uncertainty, and specify the reference value used (global mock mean or per-region mean).
  6. [§2.1, Eq. (1)] Clarify the quantity to which the [0.65,1.0] cut is applied: the parameter RMS defined in Eq. (1) carries a high-precision numerical factor (9.529287035639559) whose origin and units are not explained, whereas the text's quoted median sensitivity of 0.76 mJy refers to σrms; the cut variable and its units should be stated unambiguously.
  7. [§3.1] The 19-bin covariance matrix is estimated from only 100 mock realizations and then inverted for the likelihood; the inverse covariance should be debiased with the Hartlap factor (or its stability tested), and this should be stated.

Circularity Check

2 steps flagged · score 6.0 of 10

Isotropy null is calibrated to the data: FLASK mocks use b0=1.02 fitted to the same filtered FASHI sample, so the reported mock-data agreement is partly by construction.

  1. fitted input called prediction [Section 2.3 (Table 1) and Section 3.1 (FLASK mock inputs)]
    "With a sensitivity range of [0.65, 1.0], we obtain a linear bias of b0 = 1.02 ± 0.06 (68% C.L.), which aligns with the expected values for HI sources (Martin et al. 2012). ... Therefore, we select the sensitivity range of [0.65, 1.0] for the subsequent analyses ... The input parameters required to generate these catalogs include: redshift z = 0, the linear bias b0 = 1.02, the number of HI galaxies NHi."

    The mock catalogs that define the null hypothesis for the 2PACF isotropy test are generated using b0 = 1.02, the value fitted to the same filtered FASHI sample whose isotropy is being tested, and the filter itself was chosen because it yields this bias. The mock 2PACF amplitude is therefore calibrated to the data rather than predicted from an independent model; agreement between data and mock is partly by construction. The per-region scatter retains some independence, so the circularity is partial.

  2. fitted input called prediction [Section 3.2, Table 3 and surrounding text]
    "The resulting 68% confidence level (C.L.) constraints are θ0 = 0.35 ± 0.06 and β = 1.40 ± 0.38. These values are consistent with those reported by Franco et al. (2023), providing a reference standard for θ0 and β under the assumption of a homogeneous and isotropic universe. ... the best-fit values of θ0 across the 10 regions fluctuate between 0.32° and 0.41°, with a median value of θ0 = 0.35°, which matches the reference value obtained from the random catalog."

    The 'reference standard' obtained from the random catalogs is not an independent prediction: the mocks were generated with b0 = 1.02 fit to the same FASHI filtered sample, so the mock θ0 inherits the data's fitted clustering amplitude. Presenting the closeness of the data median θ0 to the mock θ0 as evidence for isotropy therefore partly confuses calibration with confirmation. The β comparison is less affected, but the quoted 'β std' values also use these same calibrated mocks.

full rationale

Most of the pipeline—sensitivity-window tests, angular power spectrum fits, 2PACF estimation with PolSpice, jackknife/MC covariances, and MCMC power-law fits—is standard and self-contained, and the paper does not lean on author-uniqueness theorems or load-bearing self-citations. The ALFALFA comparison and the per-region β scatter provide some independent checks. However, the null hypothesis for the central isotropy claim is not independent: the FLASK mocks are generated with b0 = 1.02, the value fitted to the filtered FASHI sample, and the [0.65, 1.0] filter was itself chosen because it produced that bias. Thus the mock 2PACF amplitude is calibrated to the data, and the reported matching median θ0 = 0.35° is partly by construction rather than a genuine prediction. The remaining content—whether the ten regions scatter around that calibrated mean as expected—is a weaker, partially independent isotropy test. A further correctness caveat, not itself circularity, is that the mocks are described only as having the same number of sources and sky coverage, with no explicit inclusion of the FASHI sensitivity map; the residual density variations in Table 2 (1.07–5.32 deg^-2 across equal-area regions) suggest the per-region covariance may be underestimated.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claim rests on a post hoc sensitivity window, a bias parameter fitted to the same data, and standard cosmological modeling assumptions (Planck 2018 cosmology, constant bias, Limber approximation, lognormal mocks). No new physical entities are introduced; the inputs are either fitted to FASHI data or drawn from prior literature.

free parameters (4)
  • Sensitivity range [0.65, 1.0] mJy = 0.65 to 1.0 mJy
    Chosen after comparing bias fits over four ranges (Table 1); selected because it yields b0=1.02, matching the expected HI bias. This post hoc selection determines the filtered sample used for all isotropy results.
  • Linear bias b0 = 1.02 ± 0.06
    Fitted to the FASHI [0.65,1.0] angular power spectrum in Section 2.3 using Eq. (3); used as the input bias for FLASK mock catalogs in Section 3.1.
  • Shot noise C_SN = 10.76 × 10^-5
    Second free parameter in the C_ell fit in Section 2.3; part of the same power-spectrum model, not directly used in the 2PACF mocks.
  • Selection function z0 = 0.008
    Parameter in n(z) ∝ z^3 exp(-z/z0) used for theoretical C_ell (Eq. 6); taken from the FASHI redshift distribution and stated to be independent of sensitivity cuts.
assumptions (6)
  • domain assumption Planck 2018 LCDM best-fit cosmological parameters
    Used to compute Pm(k) and C_ell with Colossus in Section 2.3.
  • domain assumption Constant linear bias b(z)=b0 at low redshift
    Assumed in Eqs. (3)-(5); motivated by the low redshift range z<0.09 of FASHI sources.
  • standard math Limber approximation valid for ℓ>10
    Used to convert the theoretical C_ell integral into Eq. (7).
  • domain assumption Lognormal mock catalogs reproduce the HI 2PACF and its covariance under isotropy
    FLASK mocks with b0=1.02 and the FASHI mask are used to construct covariance matrices and reference distributions in Section 3.1.
  • domain assumption No redshift evolution of clustering out to z=0.09
    Mocks are generated at z=0 and the selection function is treated as sensitivity-independent (Section 3.1).
  • domain assumption Catalog measurements of sigma_rms and source sizes reliably determine detection sensitivity
    The RMS estimate in Eq. (1) determines the source selection and therefore the entire filtered sample.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Probing Cosmic Isotropy with the FAST All Sky HI Survey." pith.science (2026). https://pith.science/paper/RLYH4JCU

@misc{pith2026241200475,
  author       = {Pith},
  title        = {Pith review of: Probing Cosmic Isotropy with the FAST All Sky HI Survey},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RLYH4JCU}},
  note         = {Machine review of arXiv:2412.00475}
}
abstract

This paper leverages the first released catalog from the FAST All Sky \textsc{Hi} Survey (FASHI) to examine the hypothesis of cosmic isotropy in the local Universe. Given the design of the overall FAST survey, the inhomogeneous detection sensitivity of FASHI is likely to introduce significant biases in the statistical properties of the catalog. To mitigate the potential influence of spurious clustering effects due to these sensitivity variations, we focus on extragalactic \textsc{Hi} sources within the sensitivity range of $[0.65, 1.0]$. This refined subsample is divided into ten distinct sky regions, for which we compute the two-point angular correlation functions (2PACF) over angular scales of $0.5^\circ < \theta < 10^\circ$. We apply the Markov chain Monte Carlo method to fit these 2PACFs with a power-law model and assess the statistical significance of the best-fit parameters for the ten FASHI sky regions by comparing them against results from mock catalogs generated under the assumptions of homogeneity and isotropy. Our findings indicate that the local Universe, as traced by the \textsc{Hi} sources in the FASHI survey, aligns with the cosmic isotropy hypothesis within a $2\sigma$ confidence level. We do not detect any statistically significant deviations from cosmic isotropy in the FASHI survey data.

Figures

Figures reproduced from arXiv: 2412.00475 by the authors.

Figure 1
Figure 1. The footprint of FASHI (red) and ALFALFA (blue) in Right Ascension and Declination is illustrated. FASHI strategically avoided most of the ALFALFA coverage to achieve a broader view of the Universe, with cross-matched signals primarily around DEC ∼ 30◦ . The black dashed lines and Arabic numerals indicate the regions used to investigate isotropy in Section 3.1, while the Roman numerals correspond to the regions outl… view at source ↗
Figure 2
Figure 2. The histograms of redshift (upper panel) and measured spectral rms σrms (lower panel) for the FASHI and ALFALFA catalogs. The black solid line in the upper panel represents the selection function of FASHI catalog (Equation 6). These values are derived from the FASHI catalog (Zhang et al. 2023). The ellipse denotes the measurement aper￾ture in right ascension (RA) and declination (DEC) space used to estimate the inte… view at source ↗
Figure 3
Figure 3. The obtained angular power spectra from sub￾samples with different ranges of detection sensitivity. els covered by the survey and divide them into K = 25 patches, ensuring that while the patches may not have identical shapes, they do cover approximately equal ar￾eas (i.e., equal numbers of pixels). The covariance estimator is expressed as: Covij = K − 1 K X K k=1 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: We present the 2PACF measurements for the ten selected regions within the FASHI footprint. In these plots, the uncertainty is represented by the square root of the main diagonal of the covariance matrix, providing a visual indicator of the measurement precision. The re…

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. Testing Scale-Dependent Suppression of Structure Growth in the Linear Regime

    astro-ph.CO 2026-08 conditional novelty 5.0 of 10

    A combined fit of galaxy-growth, supernova, and CMB data finds at most a weak 2.2σ hint for scale-dependent suppression of structure growth, with an inconclusive Bayes factor.

Reference graph

Works this paper leans on

44 extracted references · 11 canonical work pages · cited by 1 Pith paper

  1. [1]

    , " * write output.state after.block = add.period write newline

    ENTRY address archivePrefix author booktitle chapter doi edition editor eprint howpublished institution journal key month number organization pages publisher school series title misctitle type volume year version url label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts ...

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION format.url url empty "" new.block "" url * "" * if FUNCTION format.eprint eprint empty "" archivePrefix empty "" archivePrefix "arXiv" = new.block " " eprint * " " * new.block " " eprint * " " * if if if FUNCTION format.doi doi empty "" " " doi * " " * if FUNCTION format.pid doi empty eprint empty ur...

  3. [3]

    ,XY4! 䗴)-I&I -m@ ف@ wٖdk_ξc Yxm\.43<# < +eY ! B: B ! N7! B oB ! g ބ B !

    thebibliography [1] 20pt to REFERENCES 6pt =0pt -12pt 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 E...

    arXiv 2021

  4. [4]

    2020, Astron

    Aghanim, N., et al. 2020, Astron. Astrophys., 641, A6, 10.1051/0004-6361/201833910

    doi:10.1051/0004-6361/201833910 2020

  5. [5]

    Alonso, D., Sanchez, J., Slosar, A., & Collaboration, L. D. E. S. 2019, Monthly Notices of the Royal Astronomical Society, 484, 4127, 10.1093/mnras/stz093

    doi:10.1093/mnras/stz093 2019

  6. [6]

    K., et al

    Aluri, P. K., et al. 2023, Class. Quant. Grav., 40, 094001, 10.1088/1361-6382/acbefc

    doi:10.1088/1361-6382/acbefc 2023

  7. [7]

    2014, JCAP, 10, 070, 10.1088/1475-7516/2014/10/070

    Appleby, S., & Shafieloo, A. 2014, JCAP, 10, 070, 10.1088/1475-7516/2014/10/070

    work page doi:10.1088/1475-7516/2014/10/070 2014

  8. [8]

    Avila, F., Bernui, A., de Carvalho, E., & Novaes, C. P. 2021, Mon. Not. Roy. Astron. Soc., 505, 3404, 10.1093/mnras/stab1488

    work page doi:10.1093/mnras/stab1488 2021

Show all 44 references
  1. [9]

    Bengaly, C. A. P., Bernui, A., Alcaniz, J. S., & Ferreira, I. S. 2017, Mon. Not. Roy. Astron. Soc., 466, 2799, 10.1093/mnras/stw3233

    2017 doi

  2. [10]

    Bengaly, C. A. P., Maartens, R., & Santos, M. G. 2018, JCAP, 04, 031, 10.1088/1475-7516/2018/04/031

    2018 doi

  3. [11]

    2018, Phys

    Cai, R.-G., Liu, T.-B., Liu, X.-W., Wang, S.-J., & Yang, T. 2018, Phys. Rev. D, 97, 103005, 10.1103/PhysRevD.97.103005

    2018 doi

  4. [12]

    2012, Int

    Chen, X. 2012, Int. J. Mod. Phys. Conf. Ser., 12, 256, 10.1142/S2010194512006459

    2012 doi

  5. [14]

    2011, Mon

    Colin, J., Mohayaee, R., Sarkar, S., & Shafieloo, A. 2011, Mon. Not. Roy. Astron. Soc., 414, 264, 10.1111/j.1365-2966.2011.18402.x

    2011

  6. [15]

    P., & Nogueira-Cavalcante, J

    de Carvalho, E., Bernui, A., Avila, F., Novaes, C. P., & Nogueira-Cavalcante, J. P. 2021, Astron. Astrophys., 649, A20, 10.1051/0004-6361/202039936

    2021 doi

  7. [16]

    2018, Phys

    Deng, H.-K., & Wei, H. 2018, Phys. Rev. D, 97, 123515, 10.1103/PhysRevD.97.123515

    2018 doi

  8. [17]

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

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

    2018 doi

  9. [18]

    2003, Modern Cosmology (Amsterdam: Academic Press)

    Dodelson, S. 2003, Modern Cosmology (Amsterdam: Academic Press)

    2003

  10. [19]

    W., Lang , D., & Goodman , J

    Foreman-Mackey , D., Hogg , D. W., Lang , D., & Goodman , J. 2013, , 125, 306, 10.1086/670067

    2013 doi

  11. [20]

    2023, 10.1093/mnras/stad3616

    Franco, C., Avila, F., & Bernui, A. 2023, 10.1093/mnras/stad3616

    2023 doi

  12. [21]

    Ghosh, S., Jain, P., Kashyap, G., et al. 2016, J. Astrophys. Astron., 37, 25, 10.1007/s12036-016-9395-8

    2016 doi

  13. [22]

    G., et al

    Giannantonio, T., Scranton, R., Crittenden, R. G., et al. 2008, Phys. Rev. D, 77, 123520, 10.1103/PhysRevD.77.123520

    2008 doi

  14. [23]

    M., Hivon, E., Banday, A

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

    2005 doi

  15. [25]

    P., Giovanelli, R., Martin, A

    Haynes, M. P., Giovanelli, R., Martin, A. M., et al. 2011, The Astronomical Journal, 142, 170, 10.1088/0004-6256/142/5/170

    2011 doi

  16. [26]

    2019, Sci

    Jiang, P., et al. 2019, Sci. China Phys. Mech. Astron., 62, 959502, 10.1007/s11433-018-9376-1

    2019 doi

  17. [27]

    I., & Saha, R

    Khan, M. I., & Saha, R. 2022, Journal of Cosmology and Astroparticle Physics, 2022, 006, 10.1088/1475-7516/2022/06/006

    2022 doi

  18. [28]

    S., et al

    Koribalski, B. S., et al. 2020, Astrophys. Space Sci., 365, 118, 10.1007/s10509-020-03831-4

    2020 doi

  19. [29]

    Limber, D. N. 1954, Astrophys. J., 119, 655, 10.1086/145870

    1954 doi

  20. [30]

    Y., Seo, H.-J., & Vallinotto, A

    Marin, F., Gnedin, N. Y., Seo, H.-J., & Vallinotto, A. 2010, Astrophys. J., 718, 972, 10.1088/0004-637X/718/2/972

    2010 doi

  21. [31]

    M., Giovanelli, R., Haynes, M

    Martin, A. M., Giovanelli, R., Haynes, M. P., & Guzzo, L. 2012, The Astrophysical Journal, 750, 38, 10.1088/0004-637X/750/1/38

    2012 doi

  22. [32]

    J., et al

    Meyer, M. J., et al. 2004, Mon. Not. Roy. Astron. Soc., 350, 1195, 10.1111/j.1365-2966.2004.07710.x

    2004

  23. [33]

    H., et al

    Migkas, K., Schellenberger, G., Reiprich, T. H., et al. 2020, Astron. Astrophys., 636, A15, 10.1051/0004-6361/201936602

    2020 doi

  24. [34]

    B., Addison, G

    Newburgh, L. B., Addison, G. E., Amiri, M., et al. 2014, in Ground-based and Airborne Telescopes V, ed. L. M. Stepp, R. Gilmozzi, & H. J. Hall, Vol. 9145, International Society for Optics and Photonics (SPIE), 91454V, 10.1117/12.2056962

    2014 doi

  25. [35]

    Pan, J.-D., Wu, P.-J., Du, G.-H., Li, Y., & Zhang, X. 2024. 2408.00268

    2024 arXiv

  26. [36]

    Peebles, P. J. E. 1993, Principles of Physical Cosmology (Princeton University Press)

    1993

  27. [37]

    G., et al

    Santos, M. G., et al. 2017, in MeerKAT Science : On the Pathway to the SKA . 1709.06099

    2017 arXiv

  28. [38]

    2002, The Astrophysical Journal, 579, 48–75, 10.1086/342786

    Scranton, R., Johnston, D., Dodelson, S., et al. 2002, The Astrophysical Journal, 579, 48–75, 10.1086/342786

    2002 doi

  29. [39]

    J., von Hausegger, S., Rameez, M., et al

    Secrest, N. J., von Hausegger, S., Rameez, M., et al. 2021, Astrophys. J. Lett., 908, L51, 10.3847/2041-8213/abdd40

    2021 doi

  30. [40]

    Szapudi, I., Prunet, S., & Colombi, S. 2001. astro-ph/0107383

    2001 arXiv

  31. [41]

    Szapudi, I., & Szalay, A. S. 1999. astro-ph/9903132

    1999 arXiv

  32. [42]

    I., et al

    Wong, O. I., et al. 2006, Mon. Not. Roy. Astron. Soc., 371, 1855, 10.1111/j.1365-2966.2006.10846.x

    2006

  33. [43]

    S., Abdalla, F

    Xavier, H. S., Abdalla, F. B., & Joachimi, B. 2016, Mon. Not. Roy. Astron. Soc., 459, 3693, 10.1093/mnras/stw874

    2016 doi

  34. [44]

    2009, JCAP, 09, 003, 10.1088/1475-7516/2009/09/003

    Xia, J.-Q., Viel, M., Baccigalupi, C., & Matarrese, S. 2009, JCAP, 09, 003, 10.1088/1475-7516/2009/09/003

    2009 doi

  35. [45]

    2022, Mon

    Xu, Y.-T., Dai, J.-P., Zhao, D., & Xia, J.-Q. 2022, Mon. Not. Roy. Astron. Soc., 515, 5587, 10.1093/mnras/stac2200

    2022 doi

  36. [46]

    2023, Science China Physics, Mechanics & Astronomy, 67, 219511, 10.1007/s11433-023-2219-7

    Zhang, C.-P., Zhu, M., Jiang, P., et al. 2023, Science China Physics, Mechanics & Astronomy, 67, 219511, 10.1007/s11433-023-2219-7

    2023 doi

Pith tools

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