Pith. sign in

REVIEW 4 major objections 4 minor 1 cited by

Kinematic Distortions of the High-Redshift Universe as Seen from Quasar Proper Motions

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read The proper motions of a million distant quasars, split by redshift, show significant differences in global spin and drift patterns—formally a net rotation of the universe that changes with cosmic epoch, though hidden Gaia systematics…

desk verdict A careful, honestly hedged differential VSH analysis of quasar proper motions that finds suggestive redshift-dependent signals, but the classifier–astrometry coupling and the paper's own filter tests leave the cosmological interpretation unproven. read the letter →

arxiv 2507.01798 v1 pith:TPBSMK53 submitted 2025-07-02 astro-ph.GA astro-ph.CO

Valeri V. Makarov This is my paper
classification astro-ph.GAastro-ph.CO
keywords quasarpropermotionsGaiaDR3vectorsphericalharmonicsmachinelearningredshiftscosmologicalprinciplecosmicrotationsecularaberrationanisotropiccosmology
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 asks whether the tiny apparent sideways drifts of distant quasars—the proper motions measured at the microarcsecond level by the Gaia mission—depend on cosmic epoch. The author predicts redshifts for 1.57 million Gaia reference-frame quasars with a neural network trained on spectroscopic redshifts, splits them into bins at $z=1$--$2$, $2$--$3$, and $>3$, and fits a 30-term vector spherical harmonic model to each bin's global proper motion field. Comparing the fits, the only significant differences (signal-to-noise $S/N>3$) are in three first-degree terms: two rigid spins and one polar glide, all between the $z=1$--$2$ and $z=2$--$3$ bins. Formally that is what a universe with a net rotation that changes with cosmic epoch would look like, and it would extend a violation of the cosmological principle into the time domain. The author immediately notes that unseen systematic errors in Gaia astrometry, correlated with the redshift classifiers, are the more mundane explanation; independent Quaia redshifts and the known Galactic aberration dipole are used as checks.

What carries the argument

The central object is the vector spherical harmonic (VSH) decomposition of a tangential vector field on the celestial sphere, a complete orthonormal basis of vector functions that splits any smooth proper motion pattern into electric (divergence-type) and magnetic (curl-type) components. Fitted to degree 3 (30 functions) by weighted least squares on cell-averaged proper motions, the low-degree magnetic terms directly represent rigid rotations of the whole field and the electric terms represent dipole glides, so comparing coefficients between redshift bins converts the question of whether the tangential kinematic structure changes with epoch into a small set of scalar differences. The other load-bearing piece is the neural-network redshift predictor, which converts six photometric and Gaia metadata classifiers into predicted redshifts and makes the coarse redshift binning possible for a sample without full spectroscopic coverage.

What would settle it

Restrict the analysis to the spectroscopic SDSS footprint, where redshifts are exact, and reweight the redshift bins so the joint distributions of astrometric goodness-of-fit, photometric excess factor, $G$, and $W2$ magnitude are identical across bins, then recompute the degree-3 VSH fits. If the first-degree differences between $z\in[1,2]$ and $z\in[2,3]$ survive the classifier-matched samples, they cannot be a selection artifact of hidden systematics; if they vanish, the cosmological rotation interpretation is excluded. A complementary test is to inject the measured classifier-dependent covariance noise into a simulated isotropic proper motion field and see whether the observed differential coefficients reappear.

Watch

Extended reading notes

Core claim

The central claim is that the large-scale proper motion field of quasars is not kinematically the same at different cosmological epochs. In a vector spherical harmonic decomposition to degree 3, the $2<z<3$ field differs from the $1<z<2$ field in the magnetic harmonics $\{\mathrm{mag},2,1,1\}$ ($S/N=4.3$) and $\{\mathrm{mag},0,1,0\}$ ($S/N=3.7$) and the electric harmonic $\{\mathrm{ele},0,1,0\}$ ($S/N=3.1$), corresponding to relative spins of about $4.3$ and $4.1\,\mu\mathrm{as}\,\mathrm{yr}^{-1}$ and a southward polar glide of $3.3\,\mu\mathrm{as}\,\mathrm{yr}^{-1}$. In the author's reading this formally implies that the universe has a net rotation as a whole, which is different at different cosmological epochs. The author immediately adds the mundane alternative: hidden Gaia systematics. That caution is concrete: a filter to $G<19.8$ mag with $3\sigma$ proper-motion clipping reduces the prominent VSH coefficients by about 40%, and the paper verifies only that median proper motions are near zero per redshift bin, not that the full spatial structure of systematics is identical across bins. The recovered Galactic aberration dipole, $5.39\pm0.38\,\mu\mathrm{as}\,\mathrm{yr}^{-1}$, agrees with the previously measured observer acceleration, and the differential signals survive a re-analysis with the independent Quaia redshift catalog.

Load-bearing premise

The load-bearing premise is that Gaia's systematic astrometric errors are not correlated with the machine-learning classifiers used to predict redshift—particularly the astrometric goodness-of-fit parameter—so that dividing quasars by predicted redshift does not silently select different spatial patterns of instrumental error; the paper verifies only that median proper motions are near zero in each redshift bin, not that the full sky structure of the systematics is identical across bins.

Editorial extensions

If this is right

  • If the differential first-degree signal is physical, the cosmic expansion is not purely radial in the observed frame: the tangential velocity field of quasars changes with epoch, which is a time-domain violation of the cosmological principle within the Friedmann-Robertson-Walker picture.
  • The two independent redshift sources (this paper's machine-learning redshifts and the Quaia catalog) both show redshift-dependent spin differences, so the result is not an artifact of a single training set, although the amplitudes differ bin to bin.
  • Because leakage of sources between adjacent redshift bins can only smooth differential signals, the true underlying kinematic differences could be larger than the fitted values.
  • The method converts the question of whether the universe rotates differently at early times into a measurable comparison of low-degree proper motion harmonics, providing a direct observational test for anisotropic cosmological models such as Bianchi-type metrics and dipole cosmology models.

Reading between the lines

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

  • If hidden systematics are the cause, a decisive next experiment is classifier-matched binning: constructing redshift bins with identical joint distributions of astrometric gof_al, phot_bp_rp_excess_factor, $G$, and $W2$ should erase the differential first-degree terms, whereas a surviving signal would put the systematics explanation under serious pressure.
  • The sign of the implied cosmic angular acceleration is unmeasurable with this method because the Gaia frame spin is a free parameter in the ICRF alignment; future astrometric missions with independent frame ties, or differential radio interferometry, could break that degeneracy.
  • The amplitude scale of a few microarcseconds per year at $z\sim2$ gives anisotropic cosmology models a concrete target to match, linking the astrometric observable to constraints from the cosmic microwave background quadrupole, bulk flows, and cosmic parallax predictions.
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

4 major / 4 minor

Summary. The paper uses neural-network-predicted redshifts to divide 1.5 million Gaia DR3 CRF quasars into coarse redshift bins (z=1–2, 2–3, >3), then fits vector spherical harmonic (VSH) models of degree 3 to the proper motion fields of each bin. The central claim is that the proper motion fields of the z=1–2 and z=2–3 bins differ significantly in several first-degree harmonics, notably a rigid spin, a glide along the Galactic axis, and a Galactocentric dipole component. The author interprets this as a possible redshift-dependent kinematic distortion of the universe, while explicitly cautioning that hidden Gaia systematic errors are a more mundane explanation. The analysis is supplemented with internal validation tests, a repetition using Quaia redshifts, and a public release of the synthetic redshift catalog.

Significance. If the differential VSH signal were robust, it would provide a new observational probe of the cosmological principle and of alternative (e.g., Bianchi or Lemaitre-Tolman-Bondi) cosmologies. The paper is valuable for introducing a global VSH methodology to quasar proper-motion cosmology, for making the derived catalog publicly available, and for testing the result with an independent redshift source. The author is also commendably explicit that hidden astrometric systematics may explain the signal. However, the central statistical claim rests on formal errors that the paper's own reduced chi-square values indicate are understated, and the redshift-binning classifiers include an astrometric quality parameter that can plausibly correlate with the spatial structure of Gaia systematics. The significance and robustness of the claimed differential signal are therefore not yet established.

major comments (4)
  1. [§5.1, §4] The set of ML classifiers includes astrometric gof_al, which is an astrometric quality statistic. Since the redshift bins are defined by a predictor that depends on gof_al, the z-bins can differ in the spatial pattern of Gaia DR3 systematic errors even if the cosmological proper-motion signal is zero. The verification in Fig. 5 checks only median proper motion components versus redshift, i.e., the first moment of the distribution, not the sky-correlated structure that VSH fits measure. I recommend re-running the differential VSH analysis with z-bins defined by photometric and infrared classifiers only (excluding gof_al and possibly phot_bp_rp_excess_factor), or with the subsample having spectroscopic redshifts, and comparing the coefficients. This is load-bearing for the claim that the first-degree differences are cosmological rather than astrometric selection artifacts.
  2. [§4 versus §6.2.2] There is a direct inconsistency in the leakage statement. Section 4 says that leakage of sources between adjacent redshift bins is 'limited to several percent', but Section 6.2.2 reports that only 73.6% of sources remain in the correct bin, with 13.6% leaking to the lower bin and 12.1% to the higher bin. The 26% mixing is not 'several percent', and it matters more than the author suggests. The argument that leakage can only dilute differential signals holds only if leakage is random with respect to astrometric systematics; if the leaked sources are preferentially selected by classifiers that correlate with Gaia error patterns, leakage can create or modify differential VSH coefficients. This needs to be quantified, for example by injecting realistic systematics into the classifier-based binning or by comparing VSH fits on clean spectroscopic-z bins.
  3. [§3, Table 2] The quoted S/N values for the differences between the z=1–2 and z=2–3 fits (4.3, 3.7, and 3.1) are computed using formal errors only. The paper reports pre-fit reduced chi-square values of 2.11, 1.95, 1.71, and 1.56 for the four samples, indicating excess variance beyond the formal covariances. Applying a conservative inflation of the errors by the square root of the relevant reduced chi-square would bring the claimed significant differences to roughly 3.0, 2.6, and 2.2, i.e., below the S/N>3 threshold. The formal-error-only significance is therefore likely optimistic, and the main differential claim is not established by the current statistics.
  4. [§4, filtering test] The filtering test (G<19.8 mag and 3-sigma proper-motion clipping) reduces all prominent VSH coefficients by about 40%, but the paper does not report whether the differential signal between the z-bins survives this filter. The text states only that the result 'remains somewhat inconclusive'. Because the selection cuts are of the same kind that vary between the ML-defined redshift bins (fainter sources are more numerous at higher z and have different astrometric error properties), a 40% sensitivity of the coefficients to such cuts directly undermines the robustness of the differential claim. The author should present the filtered VSH coefficients and their difference S/N for the 1–2 and 2–3 bins.
minor comments (4)
  1. [§6.2.1] The Quaia verification is an independent source of redshifts but uses the same Gaia DR3 proper motions, so it does not test the possibility that the signal is an astrometric artifact. This should be stated more explicitly where the verification is described as 'independent'.
  2. [§5.1] The description of phot_bp_rp_excess_factor as a 'parameter of photometric nature' is helpful, but the reader should be reminded that astrometric gof_al is not photometric; its inclusion in the classifier set is the key systematics risk and deserves a dedicated discussion.
  3. [§4] The sentence attributing the southern vortex to a possible association with the Small Magellanic Cloud is speculative; the paper does not provide a mechanism or test, so it would be better presented as an unexplained coincidence or removed.
  4. [§6.1.2] In the historical discussion, 'SSHs' is used for scalar spherical harmonics; the abbreviation should be defined at first use for clarity.

Circularity Check

0 steps flagged · score 1.0 of 10

No load-bearing circularity; the claimed differential VSH signal is a fitted measurement comparison, not a prediction forced by its inputs.

full rationale

The derivation chain is: (1) train a neural network on 277,472 SDSS spectroscopic redshifts with external classifiers (Sec. 5.1); (2) predict redshifts for 1.57 million Gaia CRF/unWISE sources; (3) bin by the predicted redshift; (4) fit 30 VSH coefficients by weighted least squares to the Gaia proper motion field in each bin (Sec. 3); (5) compare coefficients between disjoint z-bins; and (6) repeat with Quaia redshifts as an independent redshift source (Sec. 6.2.1). At no point is the target claim defined by one of its inputs: the z-bin differences of {mag,2,1,1}, {mag,0,1,0}, and {ele,0,1,0} are statistics of independently fitted coefficients, not quantities reconstructed from the same fit. The quoted S/N values are formal ratios of fitted coefficients to their formal errors. Self-citations to Makarov & Secrest (2022, 2023) for the ML approach and Makarov (2021, 2022) for VSH weighting supply methodological precedent, but the paper retrains the network on external SDSS labels and reproduces the VSH formalism in the Appendix, so those citations are not load-bearing reductions. The most serious weakness is statistical confounding, not circularity: the classifier set includes Gaia's astrometric gof_al, and only 73.6% of sources stay in their correct z-bin (Sec. 6.2.2), so the z-bins may differ in Gaia systematic-error structure; the paper's own Sec. 4 filter test shows roughly 40% sensitivity of prominent VSH coefficients to selection cuts. That is a validity risk, not an equivalence of result and input. The score is set to 1 only because the author cites his own earlier ML/VSH papers as methodological precedent; none of those citations carries the derivation alone.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard VSH mathematics plus several domain assumptions: the null hypothesis of zero cosmological tangential motion, negligible peculiar accelerations, conservative redshift leakage, and low stellar contamination. The main ad hoc assumption is that ML redshift leakage between bins only dilutes the differential signal; if leakage correlates with astrometric systematics, the claim is weakened. The classifier set and bin boundaries are analysis choices, not fitted physical parameters.

free parameters (5)
  • Redshift bin boundaries = 1, 2, 3
    Chosen by hand; the differential signals are defined across these bins and could change if boundaries move.
  • VSH limiting degree L = 3
    Chosen for robustness; restricts sensitivity to angular scales above roughly 60 degrees.
  • Neural network classifier set = W2 mag, W1-W2, G-W1, GBP-GRP, phot_bp_rp_excess, astrometric gof_al
    Selected after an extensive search over Gaia and unWISE parameters; this data-dependent model selection can couple binning to astrometric quality.
  • Cell averaging grid = 4.5 deg cells, minimum 14 sources per cell
    Chosen for computational speed and noise reduction; affects the effective resolution and weights.
  • Redshift bias correction = zeroth-order interpolation function
    Applied to correct a strong variable bias in ML redshifts; derived from the training set and applied to the full sample.
assumptions (6)
  • standard math Vector spherical harmonics form a complete orthogonal basis for tangential vector fields on the sphere.
    Used as the fitting basis in Eq. 2; standard mathematical result (Appendix 6.1).
  • domain assumption True cosmological tangential proper motions of quasars are zero under the null hypothesis (cosmological principle).
    The testable null hypothesis stated in Section 1; the paper tests whether deviations exist.
  • domain assumption Peculiar tangential proper motions of quasars are negligible (<0.16 microas/yr at z=0.5).
    Used in Section 1 to justify ignoring the epsilon term in Eq. 1.
  • ad hoc to paper Leakage of sources between redshift bins due to ML redshift errors only dilutes differential signals.
    Invoked in Sections 4 and 6.2.2 to argue that the detection is conservative; this assumption may fail if leakage is correlated with astrometric systematics.
  • standard math The distribution of proper motion vectors is binormal for computing cell covariance matrices.
    Used in Section 5.2 to compute cell-averaged weights; approximately correct for CRF.
  • domain assumption The Gaia CRF quasar sample has negligible (<=2%) stellar contamination.
    Used in Section 2 to rule out contamination by Galactic stars as a source of the signal.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Kinematic Distortions of the High-Redshift Universe as Seen from Quasar Proper Motions." pith.science (2026). https://pith.science/paper/TPBSMK53

@misc{pith2026250701798,
  author       = {Pith},
  title        = {Pith review of: Kinematic Distortions of the High-Redshift Universe as Seen from Quasar Proper Motions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TPBSMK53}},
  note         = {Machine review of arXiv:2507.01798}
}
abstract

Advances in optical astrometry allow us to infer the non-radial kinematic structure of the Universe directly from observations. Here I use a supervised machine learning neural network method to predict 1.57 million redshifts based on several photometric and metadata classifier parameters from the unWISE mid-infrared database and from Gaia. These estimates are used to divide the sample into three redshift bins: 1-2, 2-3, and $>3$. For each subset, all available Gaia proper motions are used in a global vector spherical harmonic solution to degree 3 (30 fitting vector functions). I find significant differences in a few fitted proper motion patterns at different redshifts. The largest signals are seen in the comparison of the vector spherical harmonic fits for the 1-2 and 2-3 redshift bins. The significant harmonics include a rigid spin, a dipole glide from the north Galactic pole to the south and an additional quadrupole distortion. Validation tests with filtered subsamples indicate that the detected effect can be caused by hidden systematic errors in astrometry. The results are verified by using an independent source of redshifts and computing the observer's Galactocentric acceleration. This study offers a new observational test of alternative cosmological models.

Figures

Figures reproduced from arXiv: 2507.01798 by the authors.

Figure 1
Figure 1. General VSH-fitted proper motion field of Gaia CRF quasars with ML-predicted redshifts z > 1 on the ce￾lestial sphere. Graphical presentation in the Aitoff Galactic projection with the Galactic center direction at the center of the plot. Or￾ange dots at the origin of vectors indicate the mean positions of sources within the averaging cells. The length and direc￾tion of small vectors represent the cumulative VSH fit … view at source ↗
Figure 2
Figure 2. VSH-fitted Gaia CRF proper motion field of the subset of Gaia CRF quasars with redshifts 1 < z < 2 on the celestial sphere. The same graphical presentation is used as in [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. VSH-fitted Gaia CRF proper motion field the subset of Gaia CRF quasars with redshifts 2 < z < 3 on the celestial sphere. The same graphical presentation is used as in [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Median infrared magnitude versus redshift. Statistical dependence of instrumental W2 magnitude com￾puted from unWISE flux FW2 with an arbitrary zero-point on spectroscopic redshift. The broken solid line with dots shows the median values in 50 equal bins of sorted reds…
Figure 5
Figure 5. Figure 5: Median Gaia proper motions versus redshift. Statistical dependence of proper motion components in RA (left plot) and Decl (right plot) on spectroscopic redshift. The broken solid line with dots shows the median values in 50 equal bins of sorted redshifts. Each dot in t…
Figure 6
Figure 6. Figure 6: Performance of Machine-Learning prediction of redshifts. Statistical dependence of differences between ML-predicted redshifts zpre and observed spectroscopic redshifts zobs for the training set of 0.28 million sources in the training sample. The broken solid line with …

Discussion (0). Sign in to comment.

Forward citations

Cited by 1 Pith paper

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

  1. Space Astrometry with Gaia: Advances in Understanding our Galaxy

    astro-ph.IM 2025-09 accept novelty 1.0 of 10

    A comprehensive review of the Gaia mission's measurement principles and its scientific impact on stellar, Galactic, and cosmological astronomy.

Reference graph

Works this paper leans on

48 extracted references · 22 canonical work pages · cited by 1 Pith paper

  1. [1]

    1985, Mathematical Methods for Physicists, 3rd edn

    Arfken, G. 1985, Mathematical Methods for Physicists, 3rd edn. (San Diego: Academic Press, Inc. )

    work page 1985

  2. [2]

    1966, Veroeffentlichungen des Astronomischen Rechen-Instituts Heidelberg, 17, 1

    Brosche , P. 1966, Veroeffentlichungen des Astronomischen Rechen-Instituts Heidelberg, 17, 1

    work page 1966

  3. [3]

    Ebrahimian , E., Krishnan , C., Mondol , R., & Sheikh-Jabbari , M. M. 2023, arXiv e-prints, arXiv:2305.16177, 10.48550/arXiv.2305.16177

    work page Pith review arXiv doi:10.48550/arxiv.2305.16177 2023

  4. [4]

    1997, ESA Special Publication, Vol

    ESA , ed. 1997, ESA Special Publication, Vol. 1200, The HIPPARCOS and TYCHO catalogues. Astrometric and photometric star catalogues derived from the ESA HIPPARCOS Space Astrometry Mission

    1997

  5. [5]

    2021, , 649, A5, 10.1051/0004-6361/202039834

    Fabricius , C., Luri , X., Arenou , F., et al. 2021, , 649, A5, 10.1051/0004-6361/202039834

    doi:10.1051/0004-6361/202039834 2021

  6. [6]

    Gaia Collaboration , Prusti , T., de Bruijne , J. H. J., et al. 2016, , 595, A1, 10.1051/0004-6361/201629272

    doi:10.1051/0004-6361/201629272 2016

  7. [7]

    A., et al

    Gaia Collaboration , Mignard , F., Klioner , S. A., et al. 2018, , 616, A14, 10.1051/0004-6361/201832916

    doi:10.1051/0004-6361/201832916 2018

  8. [8]

    A., Mignard , F., et al

    Gaia Collaboration , Klioner , S. A., Mignard , F., et al. 2021, , 649, A9, 10.1051/0004-6361/202039734

    doi:10.1051/0004-6361/202039734 2021

Show all 48 references
  1. [9]

    Gaia Collaboration , Vallenari , A., Brown , A. G. A., et al. 2022 a , arXiv e-prints, arXiv:2208.00211, 10.48550/arXiv.2208.00211

    2022 doi arXiv

  2. [10]

    A., Lindegren , L., et al

    Gaia Collaboration , Klioner , S. A., Lindegren , L., et al. 2022 b , , 667, A148, 10.1051/0004-6361/202243483

    2022 doi

  3. [11]

    R., Eubanks , T

    Gwinn , C. R., Eubanks , T. M., Pyne , T., Birkinshaw , M., & Matsakis , D. N. 1997, , 485, 87, 10.1086/304424

    1997 doi

  4. [12]

    Kardashev , N. S. 1986, , 63, 845

    1986

  5. [13]

    M., & Makarov , V

    Kopeikin , S. M., & Makarov , V. V. 2006, , 131, 1471, 10.1086/500170

    2006 doi

  6. [14]

    2003, , 404, 743, 10.1051/0004-6361:20030560

    Kovalevsky , J. 2003, , 404, 743, 10.1051/0004-6361:20030560

    2003 doi

  7. [15]

    Kristian , J., & Sachs , R. K. 1966, , 143, 379, 10.1086/148522

    1966 doi

  8. [16]

    A., Hern \'a ndez , J., et al

    Lindegren , L., Klioner , S. A., Hern \'a ndez , J., et al. 2021, , 649, A2, 10.1051/0004-6361/202039709

    2021 doi

  9. [17]

    W., Higley , A

    Lyke , B. W., Higley , A. N., McLane , J. N., et al. 2020, , 250, 8, 10.3847/1538-4365/aba623

    2020 doi

  10. [18]

    2025, The complete catalog of ML-synthesized and spectroscopically measured redshifts for 1.5 million Gaia CRF3 quasars, Zenodo, 10.5281/zenodo.15306518

    Makarov, V. 2025, The complete catalog of ML-synthesized and spectroscopically measured redshifts for 1.5 million Gaia CRF3 quasars, Zenodo, 10.5281/zenodo.15306518

    2025 doi

  11. [19]

    Makarov , V. V. 2021, , 161, 289, 10.3847/1538-3881/abf249

    2021 doi

  12. [20]

    2022, , 164, 157, 10.3847/1538-3881/ac88d1

    ---. 2022, , 164, 157, 10.3847/1538-3881/ac88d1

    2022 doi

  13. [21]

    2024, , 274, 27, 10.3847/1538-4365/ad697b

    ---. 2024, , 274, 27, 10.3847/1538-4365/ad697b

    2024 doi

  14. [22]

    V., Dorland , B

    Makarov , V. V., Dorland , B. N., Gaume , R. A., et al. 2012, , 144, 22, 10.1088/0004-6256/144/1/22

    2012 doi

  15. [23]

    V., & Murphy , D

    Makarov , V. V., & Murphy , D. W. 2007, , 134, 367, 10.1086/518242

    2007 doi

  16. [24]

    V., & Secrest , N

    Makarov , V. V., & Secrest , N. J. 2022, , 927, L4, 10.3847/2041-8213/ac551d

    2022 doi

  17. [25]

    2023, , 264, 4, 10.3847/1538-4365/ac97f0

    ---. 2023, , 264, 4, 10.3847/1538-4365/ac97f0

    2023 doi

  18. [26]

    2019, , 158, 158, 10.3847/1538-3881/ab3a44

    Malkin , Z. 2019, , 158, 158, 10.3847/1538-3881/ab3a44

    2019 doi

  19. [27]

    2021, , 506, 5540, 10.1093/mnras/stab2100

    ---. 2021, , 506, 5540, 10.1093/mnras/stab2100

    2021 doi

  20. [28]

    2024, arXiv e-prints, arXiv:2410.06075, 10.48550/arXiv.2410.06075

    ---. 2024, arXiv e-prints, arXiv:2410.06075, 10.48550/arXiv.2410.06075

    2024 doi arXiv

  21. [29]

    J., Mart \' nez , M

    Marco , F. J., Mart \' nez , M. J., & L \'o pez , J. A. 2015, , 149, 129, 10.1088/0004-6256/149/4/129

    2015 doi

  22. [30]

    2012, , 547, A59, 10.1051/0004-6361/201219927

    Mignard , F., & Klioner , S. 2012, , 547, A59, 10.1051/0004-6361/201219927

    2012 doi

  23. [31]

    1990, in Journ \'e es 1990: Syst \`e mes de R \'e f \'e rence Spatio-Temporels, ed

    Mignard , F., & Morando , B. 1990, in Journ \'e es 1990: Syst \`e mes de R \'e f \'e rence Spatio-Temporels, ed. N. Capitaine & S. D \'e barbat , 151--158

    1990

  24. [32]

    2016, , 595, A5, 10.1051/0004-6361/201629534

    Mignard , F., Klioner , S., Lindegren , L., et al. 2016, , 595, A5, 10.1051/0004-6361/201629534

    2016 doi

  25. [33]

    2014, Galaxies, 2, 22, 10.3390/galaxies2010022

    Perivolaropoulos , L. 2014, Galaxies, 2, 22, 10.3390/galaxies2010022

    2014 doi

  26. [34]

    2022, , 95, 101659, 10.1016/j.newar.2022.101659

    Perivolaropoulos , L., & Skara , F. 2022, , 95, 101659, 10.1016/j.newar.2022.101659

    2022

  27. [35]

    Planck Collaboration , Ade , P. A. R., Aghanim , N., et al. 2014, , 561, A97, 10.1051/0004-6361/201321299

    2014 doi

  28. [36]

    R., Birkinshaw , M., Eubanks , T

    Pyne , T., Gwinn , C. R., Birkinshaw , M., Eubanks , T. M., & Matsakis , D. N. 1996, , 465, 566, 10.1086/177443

    1996 doi

  29. [37]

    2009, , 102, 151302, 10.1103/PhysRevLett.102.151302

    Quercellini , C., Quartin , M., & Amendola , L. 2009, , 102, 151302, 10.1103/PhysRevLett.102.151302

    2009 doi

  30. [38]

    J., & Sciama , D

    Rees , M. J., & Sciama , D. W. 1968, , 217, 511, 10.1038/217511a0

    1968 doi

  31. [39]

    1987, Geophysical and Astrophysical Fluid Dynamics, 39, 163, 10.1080/03091928708208811

    Rieutord , M. 1987, Geophysical and Astrophysical Fluid Dynamics, 39, 163, 10.1080/03091928708208811

    1987 doi

  32. [40]

    K., & Wolfe , A

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

    1967 doi

  33. [41]

    F., Meisner , A

    Schlafly , E. F., Meisner , A. M., & Green , G. M. 2019, , 240, 30, 10.3847/1538-4365/aafbea

    2019 doi

  34. [42]

    J., Dudik , R

    Secrest , N. J., Dudik , R. P., Dorland , B. N., et al. 2015, , 221, 12, 10.1088/0067-0049/221/1/12

    2015 doi

  35. [43]

    K., & Vityazev , V

    Shuksto , A. K., & Vityazev , V. V. 2006, in Journ\'ees 2005 - syst\`emes de r\'ef\'erence spatio-temporels, ed. A. Brzezi \'n ski , N. Capitaine , & B. Kolaczek , 41

    2006

  36. [44]

    Smolarski, D. C. 2006, Journal of Computational and Applied Mathematics, 186, 416, https://doi.org/10.1016/j.cam.2005.02.012

    2006 doi

  37. [45]

    W., Rix , H.-W., et al

    Storey-Fisher , K., Hogg , D. W., Rix , H.-W., et al. 2024, , 964, 69, 10.3847/1538-4357/ad1328

    2024 doi

  38. [46]

    B., & Gontier , A

    Titov , O., Lambert , S. B., & Gontier , A. M. 2011, , 529, A91, 10.1051/0004-6361/201015718

    2011 doi

  39. [47]

    2004, in Astronomical Society of the Pacific Conference Series, Vol

    Vityazev , V., & Shuksto , A. 2004, in Astronomical Society of the Pacific Conference Series, Vol. 316, Order and Chaos in Stellar and Planetary Systems, ed. G. G. Byrd , K. V. Kholshevnikov , A. A. Myllri , I. I. Nikiforov , & V. V. Orlov , 230

    2004

  40. [48]

    V., & Tsvetkov , A

    Vityazev , V. V., & Tsvetkov , A. S. 2011, Astronomy Letters, 37, 874, 10.1134/S1063773711120103

    2011 doi

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.