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REVIEW 2 major objections 5 minor 44 references

Shapes and orientations of massive halos in the statistically anisotropic universe

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Statistical anisotropy in the early universe aligns the largest dark matter halos, with the effect growing for more massive halos.

desk verdict A solid first measurement of SA-induced halo orientation alignment, with a central mass-dependence claim that outruns the simulations' design. read the letter →

arxiv 2505.15082 v3 pith:DBJHJ7ZU submitted 2025-05-21 astro-ph.CO gr-qchep-ph

Shogo Masaki , Yurino Mizuguchi , Shohei Saga , Shuichiro Yokoyama This is my paper
classification astro-ph.COgr-qchep-ph
keywords statisticalanisotropyhaloorientationshapeN-bodysimulationquadrupolarg*parameterclustercosmologyintrinsicalignment
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

The paper tests whether a small directional preference in the primordial matter distribution, called statistical anisotropy (SA) and parametrized here by the quadrupolar amplitude g*, leaves any imprint on the shapes and orientations of the most massive dark matter halos. Using cosmological N-body simulations whose initial power spectrum is modulated by an SA term, it finds that halo shapes (axis ratios and triaxiality) are essentially unchanged, but halo orientations are not: the major axes of cluster-sized halos increasingly align perpendicular to the preferred direction for positive g* and parallel to it for negative g*. This alignment grows with halo mass and appears to be imprinted by the anisotropic initial conditions rather than by later nonlinear evolution. If correct, the result turns projected halo ellipticity measurements from galaxy-cluster lensing into a new observable probe of SA, complementing constraints from the cosmic microwave background and galaxy clustering.

What carries the argument

The machinery is the quadrupolar SA power spectrum, equivalent to a global traceless tensor coupling, $P_m(k)=\left[1+G_{ij}\hat k^i\hat k^j\right]\bar P_m(k)$ with $G_{ij}=g_*(\hat d_i\hat d_j-\delta_{ij}/3)$. This global tensor field imprints a preferred direction on the initial density field but does not change the local nonlinear dynamics that set halo shapes; instead it systematically tilts the orientation of the halo's major axis, with the effect growing for more massive halos because they sample larger volumes and 'feel' the global structure. The key comparison is the PDF of the normalized orientation vector $\hat A$ of the largest ellipsoid axis (from the ROCKSTAR halo catalogs) across $g_*$ values and mass bins from three simulation boxes (L05, L2, L4). The concentration-split test supports the interpretation that the alignment is fixed by initial conditions rather than by assembly history.

What would settle it

A matched-resolution series of SA simulations at fixed g* covering the same halo mass with different box sizes and particle masses, or a larger set of realizations at g*=±0.1, would settle whether the mass trend in P(Âz) persists; alternatively, measuring the orientation of the minor axis or the projected ellipticity orientation in an isotropic simulation subset should yield exactly flat PDFs, providing a null test of the pipeline.

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Extended reading notes

Core claim

The central claim is that a quadrupolar statistical anisotropy in the matter power spectrum, written as $P_m(k)=\left[1+\frac{2}{3}g_* L_2(\mu)\right]\bar P_m(k)$ with $L_2(\mu)$ the second Legendre polynomial and $\mu$ the cosine of the angle to the preferred direction $\hat d$, does not alter the three-dimensional shapes of cluster-sized halos but does reorient them. For $g_*>0$, the largest-axis orientation vector $\hat A$ develops a preference for the plane perpendicular to $\hat d$; for $g_*<0$ it prefers the direction parallel to $\hat d$. The effect is mass-dependent: at $g_*=\pm0.1$, near the current upper limit from galaxy clustering, PDFs of $\hat A_z$ show negligible deviation from isotropy for $10^{13}\,h^{-1}M_\odot$ halos but clear deviations at $10^{14}$ and $10^{15}\,h^{-1}M_\odot$. The alignment is insensitive to halo concentration or triaxiality, indicating it reflects the SA-imprinted initial conditions rather than formation history. Bulk velocity and angular momentum vectors respond to SA in the opposite sense but weakly, so halo orientation is the most sensitive among the diagnostic vectors studied.

Load-bearing premise

The mass dependence is inferred by comparing halos from three simulations that differ simultaneously in box size, resolution, and particle mass, with only three independent realizations per g*, so the growing alignment with mass could partly be a numerical artifact of finite volume or resolution.

Editorial extensions

If this is right

  • Projected halo ellipticity measurements made with galaxy-cluster-galaxy lensing should exhibit a systematic directional preference tied to the SA axis, making them a new observable probe of $g_*$.
  • At $|g_*|\sim0.1$, the current upper limit from galaxy clustering, the SA-induced orientation signal is negligible for $10^{13}\,h^{-1}M_\odot$ halos but becomes statistically visible for $10^{14}$ and $10^{15}\,h^{-1}M_\odot$ halos, so massive clusters are the place to look.
  • Halo shapes—axis ratios $s$ and $q$ and triaxiality $T$—remain statistically isotropic even for $|g_*|=1$, so shape distribution measurements encode almost no SA information.
  • The orientation alignment is insensitive to concentration (formation epoch), so the signal traces initial conditions rather than nonlinear assembly, simplifying theoretical modeling.

Reading between the lines

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

  • The mass dependence of the alignment suggests that fixed-mass observations at higher redshift (where the same mass corresponds to more massive, rarer halos) could amplify the signal; the paper does not explore redshift evolution.
  • Because the orientation PDF is a one-point function, it misses spatial correlations; a natural extension is a two-point alignment correlation, which could connect the SA-induced alignment to intrinsic-alignment contamination in weak-lensing surveys.
  • The claim that shapes are insensitive to SA rests on comparing PDFs across $g_*$; a testable extension is whether higher-order shape statistics, such as the alignment of the intermediate axis or the distribution of the orientation angle in the perpendicular plane, respond differently.
  • The simulation comparison for mass dependence is not a controlled convergence test; running the same mass range at matched resolution in multiple boxes would distinguish a physical mass trend from numerical box-size effects.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. This paper uses cosmological N-body simulations with a quadrupolar statistical anisotropy parameter g* in the linear matter power spectrum and measures the PDFs of halo shape parameters (s, T), the major-axis orientation vector A, and the dynamical vectors V (bulk velocity) and J (angular momentum) for massive halos at z=0. The main findings are: (i) the shape parameters are insensitive to SA even for |g*|=1; (ii) the major-axis orientation shows a systematic dependence, with alignment perpendicular to the SA direction for positive g* and parallel for negative g*; (iii) the alignment is reported to increase with halo mass; and (iv) bulk velocities and angular momenta show weaker, sign-opposite responses. The authors propose projected halo shapes from cluster-galaxy lensing as a new observational probe of SA.

Significance. If the orientation effect is real at the observationally allowed |g*| ~ 0.1, the paper would open a genuinely new large-scale structure probe of statistical anisotropy, complementary to CMB and galaxy-clustering constraints. The paper's strengths are the use of matched initial seeds across g* values, the clear null result for shapes, and the systematic exploration of several halo properties. However, the two quantitative claims that carry the abstract—the mass dependence and the presence of the effect at g* = 0.1—are not supported by significance tests or resolution checks. The paper is therefore of moderate significance in its present form.

major comments (2)
  1. [Sec. 3.2, Fig. 4, Table 1] The claim that SA-induced alignment becomes more pronounced for more massive halos is inferred from comparing the orientation PDFs in three mass bins that are taken from three different simulations (L05, L2, L4). As Table 1 shows, these runs differ simultaneously in box size (0.5, 2, and 4 h^-1 Gpc), particle number (512^3 versus 1024^3), and particle mass (8.16e10, 6.53e11, and 5.22e12 h^-1 Msun), so halo mass is fully confounded with resolution, force softening, and the available large-scale modes. No convergence or matched-resolution check is provided, and the paper does not exploit the fact that overlapping mass ranges could in principle be compared between runs. The abstract and conclusion state the mass dependence as a finding, whereas the evidence is only a visual trend across heterogeneous simulations. Please either (i) add a matched-resolution comparison (e.g., split the L2 sample into sub-bins, or compare the same mass bin across two boxes), or (ii) recast the mass-dependence claim as tentative and remove it from the abstract.
  2. [Sec. 3.2, Fig. 4] For g* = ±0.1, the value emphasized as comparable to the current upper limit from galaxy clustering, the deviations of P(Ax) and P(Az) from the isotropic case are small (order 0.01–0.05 in the PDF values) and are assessed only by eye using three realizations. No significance test is presented for the difference between the g* = 0 and g* = ±0.1 PDFs in any mass bin. This matters because the observational motivation rests on the g* = ±0.1 amplitude. Please add a quantitative comparison—for example, a chi-square or Kolmogorov-Smirnov statistic, or a fit of the PDF to A + B L2(mu) with an uncertainty on B—and report whether the deviations in the (14,14.5] and (15,15.5] bins are statistically significant. Without such a test, the statement that the deviations 'become more pronounced' with mass is not supported.
minor comments (5)
  1. [Sec. 2.1] The word 'isotoropic' in the sentence after Eq. (2.1) should be 'isotropic'.
  2. [Secs. 3.2 and 4] The word 'clusteing' appears in the phrase 'galaxy clusteing measurements'; it should be 'clustering'.
  3. [Figs. 6 and 8] The bottom-right panels of Figs. 6 and 8 are labeled 'A_z' and 'P(A_z)' in the text, but these panels show the bulk velocity and angular momentum components, respectively; the labels should be 'V_z' and 'J_z'.
  4. [Sec. 4] The phrase 'we clearly showed' in the conclusion is stronger than the evidence presented; consider 'our simulations suggest' given the limitations discussed above.
  5. [Sec. 3.2] It would be helpful to report the number of halos in each mass bin and the typical number of particles per halo, since the highest-mass bin is said to contain roughly ten times fewer halos and to have larger errors.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper is a direct N-body simulation measurement with g* as a hand-set input and no fitted quantity presented as a prediction.

full rationale

The paper does not derive its headline result from its own definitions. It prescribes a quadrupolar statistical anisotropy via Eq. (2.1), using the standard form from Ackerman et al. with g* chosen by hand for each run, then evolves initial conditions with CAMB, 2LPT, and Gadget-2, and measures halo shapes and orientations from Rockstar catalogs. No parameter is fitted to the output PDFs, and no quantity called a prediction is statistically forced by the inputs. The orientation alignment for positive versus negative g* is read off the simulated PDFs in Figs. 3 and 4, which is the intended experiment rather than a circular reduction. The only overlapping-author citation is Ref. [19], used for the SA simulation pipeline; that citation does not supply the orientation result or any uniqueness claim, and Eq. (2.1) itself is an independent external input. The acknowledged caveats about limited realizations and the fact that the three mass bins come from runs with different box sizes and resolutions are numerical-systematic concerns, not circular reasoning. No equation in the paper reduces to another equation by construction, and no fitted parameter is renamed as a prediction. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper's results rest on the quadrupolar SA model, ΛCDM background, 2LPT initial conditions, and halo shape definitions from Rockstar. No free parameters are fitted to the output; g* is a model input varied over a grid. No new particles, forces, or entities are introduced.

assumptions (5)
  • domain assumption Quadrupolar statistical anisotropy model of the initial power spectrum, Pm(k) = (1 + (2/3) g* L2(mu)) Pbar(k) (Eq. 2.1).
    Assumes the leading-order SA is a global quadrupole with a fixed preferred direction dhat; motivated by vector-field inflation models but not derived within this paper.
  • domain assumption Flat ΛCDM cosmology with Planck 2015 best-fit parameters (Sec. 2.1).
    Standard background cosmology; the paper does not vary cosmological parameters.
  • domain assumption Second-order Lagrangian perturbation theory initial conditions at z_ini = 31 generated from the modified power spectrum (Sec. 2.1).
    Assumes 2LPT is adequate at z=31 and that no unphysical transients bias orientation statistics.
  • domain assumption Rockstar halo finder shape and orientation definitions: weighted inertia tensor, s/q/T parameters, and the A vector (Sec. 2.2).
    Uses Rockstar's default shape measurements as faithful representations of halo ellipsoids; choice of estimator could affect orientation PDFs.
  • domain assumption Dark-matter-only collisionless evolution with Gadget-2 (Sec. 2.1).
    Assumes baryonic physics does not alter the alignment signal for cluster-sized halos.

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Cite this review

Pith. "Pith review of Shapes and orientations of massive halos in the statistically anisotropic universe." pith.science (2026). https://pith.science/paper/DBJHJ7ZU

@misc{pith2026250515082,
  author       = {Pith},
  title        = {Pith review of: Shapes and orientations of massive halos in the statistically anisotropic universe},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DBJHJ7ZU}},
  note         = {Machine review of arXiv:2505.15082}
}
abstract

We investigate how statistical anisotropy (SA) in matter distributions affects the distributions of shapes and orientations of cluster-sized halos, using cosmological $N$-body simulations that incorporate SA. While the three-dimensional halo shape parameters show little dependence on SA, we find that halo orientations are significantly influenced, with halos tending to align either perpendicular or parallel to the SA direction. This SA-induced alignment becomes more prominent for more massive halos. We also study other vector quantities associated with the dynamics of halos, such as bulk velocity and angular momentum vectors. We find that their dependences on the SA are smaller than those of the orientation vectors. Our findings suggest that observational measurements of projected halo shapes derived from galaxy cluster-galaxy lensing could provide a novel probe of SA in the universe.

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Works this paper leans on

44 extracted references · 14 canonical work pages

  1. [19]

    Masaki, M

    S. Masaki, M. Shiraishi, T. Nishimichi, T. Okumura and S. Yokoyama,First confirmation of anisotropic bias from statistically anisotropic matter distributions,arXiv e-prints(2024) arXiv:2409.12004 [2409.12004]

    arXiv 2024

  2. [1]

    Dimastrogiovanni, N

    E. Dimastrogiovanni, N. Bartolo, S. Matarrese and A. Riotto,Non-Gaussianity and Statistical Anisotropy from Vector Field Populated Inflationary Models,Adv. Astron.2010(2010) 752670 [1001.4049]

    arXiv 2010

  3. [2]

    Soda,Statistical Anisotropy from Anisotropic Inflation,Class

    J. Soda,Statistical Anisotropy from Anisotropic Inflation,Class. Quant. Grav.29(2012) 083001 [1201.6434]

    arXiv 2012

  4. [3]

    Maleknejad, M.M

    A. Maleknejad, M.M. Sheikh-Jabbari and J. Soda,Gauge fields and inflation,Physics Reports 528(2013) 161

    work page 2013

  5. [4]

    Hambye,Hidden vector dark matter,JHEP01(2009) 028 [0811.0172]

    T. Hambye,Hidden vector dark matter,JHEP01(2009) 028 [0811.0172]

    arXiv 2009

  6. [5]

    Graham, J

    P.W. Graham, J. Mardon and S. Rajendran,Vector Dark Matter from Inflationary Fluctuations,Phys. Rev. D93(2016) 103520 [1504.02102]

    arXiv 2016

  7. [6]

    Bastero-Gil, J

    M. Bastero-Gil, J. Santiago, L. Ubaldi and R. Vega-Morales,Vector dark matter production at the end of inflation,JCAP04(2019) 015 [1810.07208]

    arXiv 2019

  8. [7]

    Ferreira Chase, M

    T. Ferreira Chase, M. Leizerovich, D. Lopez Nacir and S. Landau,Cosmological perturbations with ultralight vector dark matter fields: Numerical implementation in CLASS, Phys. Rev. D 111(2025) 103520 [2408.12052]

    arXiv 2025

Show all 44 references
  1. [8]

    Beltran Jimenez and A.L

    J. Beltran Jimenez and A.L. Maroto,A cosmic vector for dark energy,Phys. Rev. D78(2008) 063005 [0801.1486]

    2008 arXiv

  2. [9]

    Ackerman, S.M

    L. Ackerman, S.M. Carroll and M.B. Wise,Imprints of a Primordial Preferred Direction on the Microwave Background,Phys. Rev. D75(2007) 083502 [astro-ph/0701357]

    2007 arXiv

  3. [10]

    Kim and E

    J. Kim and E. Komatsu,Limits on anisotropic inflation from the Planck data, Phys. Rev. D88 (2013) 101301 [1310.1605]. – 14 – [11]Planckcollaboration,Planck 2018 results. X. Constraints on inflation,Astron. Astrophys. 641(2020) A10 [1807.06211]. [12]Planckcollaboration,Planck 2...

    2013 arXiv

  4. [14]

    Pullen and C.M

    A.R. Pullen and C.M. Hirata,Non-detection of a statistically anisotropic power spectrum in large-scale structure,Journal of Cosmology and Astroparticle Physics2010(2010) 027

    2010

  5. [15]

    Sugiyama, M

    N.S. Sugiyama, M. Shiraishi and T. Okumura,Limits on statistical anisotropy from BOSS DR12 galaxies using bipolar spherical harmonics, MNRAS473(2018) 2737 [1704.02868]

    2018 arXiv

  6. [16]

    Shiraishi, N.S

    M. Shiraishi, N.S. Sugiyama and T. Okumura,Polypolar spherical harmonic decomposition of galaxy correlators in redshift space: Toward testing cosmic rotational symmetry,Phys. Rev. D 95(2017) 063508 [1612.02645]

    2017 arXiv

  7. [17]

    Shiraishi, K

    M. Shiraishi, K. Akitsu and T. Okumura,Alcock-Paczynski effects on wide-angle galaxy statistics, Phys. Rev. D103(2021) 123534 [2103.08126]

    2021 arXiv

  8. [18]

    Shiraishi, T

    M. Shiraishi, T. Okumura and K. Akitsu,Statistical anisotropy in galaxy ellipticity correlations, J. Cosmology Astropart. Phys.2023(2023) 013 [2303.10890]

    2023 arXiv

  9. [20]

    Jing and Y

    Y.P. Jing and Y. Suto,Triaxial Modeling of Halo Density Profiles with High-Resolution N-Body Simulations, ApJ574(2002) 538 [astro-ph/0202064]

    2002 arXiv

  10. [21]

    Allgood, R.A

    B. Allgood, R.A. Flores, J.R. Primack, A.V. Kravtsov, R.H. Wechsler, A. Faltenbacher et al., The shape of dark matter haloes: dependence on mass, redshift, radius and formation, MNRAS 367(2006) 1781 [astro-ph/0508497]

    2006 arXiv

  11. [22]

    Schneider, C.S

    M.D. Schneider, C.S. Frenk and S. Cole,The shapes and alignments of dark matter halos, J. Cosmology Astropart. Phys.2012(2012) 030 [1111.5616]

    2012 arXiv

  12. [23]

    Oguri, M

    M. Oguri, M. Takada, N. Okabe and G.P. Smith,Direct measurement of dark matter halo ellipticity from two-dimensional lensing shear maps of 25 massive clusters, MNRAS405(2010) 2215 [1004.4214]

    2010 arXiv

  13. [24]

    T.-h. Shin, J. Clampitt, B. Jain, G. Bernstein, A. Neil, E. Rozo et al.,The ellipticity of galaxy cluster haloes from satellite galaxies and weak lensing, MNRAS475(2018) 2421 [1705.11167]

    2018 arXiv

  14. [25]

    da Silveira Ferreira, R

    P. da Silveira Ferreira, R. Oliveira Ramos, P.S. Ferreira, A. Cortesi, F. Ferrari, V. Marra et al., Where Galaxies Point: First Measurement of the Large-Scale Axial Intrinsic Alignment,arXiv e-prints(2025) arXiv:2511.10005 [2511.10005]

    2025 arXiv

  15. [26]

    Lewis, A

    A. Lewis, A. Challinor and A. Lasenby,Efficient Computation of Cosmic Microwave Background Anisotropies in Closed Friedmann-Robertson-Walker Models,Astrophys. J.538 (2000) 473 [astro-ph/9911177]

    2000 arXiv

  16. [27]

    Scoccimarro,Transients from initial conditions: a perturbative analysis,Mon

    R. Scoccimarro,Transients from initial conditions: a perturbative analysis,Mon. Not. Roy. Astron. Soc.299(1998) 1097 [arXiv:astro-ph/9711187]

    1998 arXiv

  17. [28]

    Crocce, S

    M. Crocce, S. Pueblas and R. Scoccimarro,Transients from initial conditions in cosmological simulations,Mon. Not. Roy. Astron. Soc.373(2006) 369 [astro-ph/0606505]

    2006 arXiv

  18. [29]

    Nishimichi, A

    T. Nishimichi, A. Shirata, A. Taruya, K. Yahata, S. Saito, Y. Suto et al.,Modeling Nonlinear Evolution of Baryon Acoustic Oscillations: Convergence Regime of N-body Simulations and Analytic Models,Publ. Astron. Soc. Japan61(2009) 321 [0810.0813]. – 15 –

    2009 arXiv

  19. [30]

    Springel, N

    V. Springel, N. Yoshida and S.D.M. White,GADGET: a code for collisionless and gasdynamical cosmological simulations, New A6(2001) 79 [astro-ph/0003162]

    2001 arXiv

  20. [31]

    Springel,The cosmological simulation code GADGET-2,Mon

    V. Springel,The cosmological simulation code GADGET-2,Mon. Not. Roy. Astron. Soc.364 (2005) 1105 [astro-ph/0505010]

    2005 arXiv

  21. [32]

    Planck Collaboration, P.A.R. Ade, N. Aghanim, M. Arnaud, M. Ashdown, J. Aumont et al., Planck 2015 results. XIII. Cosmological parameters, A&A594(2016) A13 [1502.01589]

    2016 arXiv

  22. [33]

    Behroozi, R.H

    P.S. Behroozi, R.H. Wechsler and H.-Y. Wu,The ROCKSTAR Phase-space Temporal Halo Finder and the Velocity Offsets of Cluster Cores, ApJ762(2013) 109 [1110.4372]

    2013 arXiv

  23. [34]

    Franx, G

    M. Franx, G. Illingworth and T. de Zeeuw,The Ordered Nature of Elliptical Galaxies: Implications for Their Intrinsic Angular Momenta and Shapes, ApJ383(1991) 112

    1991

  24. [35]

    Croft and C.A

    R.A.C. Croft and C.A. Metzler,Weak-Lensing Surveys and the Intrinsic Correlation of Galaxy Ellipticities, ApJ545(2000) 561 [astro-ph/0005384]

    2000 arXiv

  25. [36]

    Heavens, A

    A. Heavens, A. Refregier and C. Heymans,Intrinsic correlation of galaxy shapes: implications for weak lensing measurements, MNRAS319(2000) 649 [astro-ph/0005269]

    2000 arXiv

  26. [37]

    Lee and U.-L

    J. Lee and U.-L. Pen,Cosmic Shear from Galaxy Spins, ApJ532(2000) L5 [astro-ph/9911328]

    2000 arXiv

  27. [38]

    Wechsler, J.S

    R.H. Wechsler, J.S. Bullock, J.R. Primack, A.V. Kravtsov and A. Dekel,Concentrations of Dark Halos from Their Assembly Histories, ApJ568(2002) 52 [astro-ph/0108151]

    2002 arXiv

  28. [39]

    Navarro, C.S

    J.F. Navarro, C.S. Frenk and S.D.M. White,The Structure of Cold Dark Matter Halos, ApJ 462(1996) 563 [astro-ph/9508025]

    1996 arXiv

  29. [40]

    Warren, P.J

    M.S. Warren, P.J. Quinn, J.K. Salmon and W.H. Zurek,Dark Halos Formed via Dissipationless Collapse. I. Shapes and Alignment of Angular Momentum, ApJ399(1992) 405

    1992

  30. [41]

    Bailin and M

    J. Bailin and M. Steinmetz,Internal and external alignment of the shapes and angular momenta of lambda-CDM halos,Astrophys. J.627(2005) 647 [astro-ph/0408163]

    2005 arXiv

  31. [42]

    L. Shaw, J. Weller, J.P. Ostriker and P. Bode,Statistics of physical properties of dark matter clusters,Astrophys. J.646(2006) 815 [astro-ph/0509856]

    2006 arXiv

  32. [43]

    P. Bett, V. Eke, C.S. Frenk, A. Jenkins, J. Helly and J. Navarro,The spin and shape of dark matter haloes in the Millennium simulation of a lambda-CDM universe,Mon. Not. Roy. Astron. Soc.376(2007) 215 [astro-ph/0608607]

    2007 arXiv

  33. [44]

    Yokoyama and J

    S. Yokoyama and J. Soda,Primordial statistical anisotropy generated at the end of inflation, JCAP08(2008) 005 [0805.4265]

    2008 arXiv

  34. [45]

    Fujita, I

    T. Fujita, I. Obata, T. Tanaka and S. Yokoyama,Statistically Anisotropic Tensor Modes from Inflation,JCAP07(2018) 023 [1801.02778]

    2018 arXiv

  35. [46]

    J. Shi, T. Sunayama, T. Kurita, M. Takada, S. Sugiyama, R. Mandelbaum et al.,The intrinsic alignment of galaxy clusters and impact of projection effects, MNRAS528(2024) 1487 [2306.09661]

    2024 arXiv

  36. [47]

    Ishikawa, A

    S. Ishikawa, A. Taruya, T. Nishimichi, T. Okumura and S. Tanaka,SHAPE: cosmology with cluster halo intrinsic alignments from subhalo distributions,arXiv e-prints(2025) arXiv:2505.01588 [2505.01588]. – 16 –

    2025 arXiv

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