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arxiv: 2604.04436 · v1 · submitted 2026-04-06 · 🌌 astro-ph.CO · astro-ph.GA

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Directional Tests of the Cosmic Distance Duality Relation using Pantheon+ and BAO

Pushty Shrimankar , Gopal Kashyap
Authors on Pith no claims yet

Pith reviewed 2026-05-10 19:58 UTC · model grok-4.3

classification 🌌 astro-ph.CO astro-ph.GA
keywords cosmic distance duality relationanisotropy testPantheon+ supernovaeBAOdipole modulationGaussian process reconstructionsurvey selection effects
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The pith

No intrinsic anisotropy is found in the cosmic distance duality relation.

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

The paper examines whether the relation between luminosity and angular diameter distances varies with direction in the sky. It uses supernova luminosity distances and reconstructs angular diameter distances from BAO measurements via Gaussian processes, avoiding any assumed cosmological model. A possible dipole variation is fitted, and its amplitude is compared to many simulated datasets that keep the same sky coverage and redshift spread. The real data's dipole matches the spread seen in these isotropic simulations, producing a firm upper limit.

Core claim

Reconstructing the duality parameter eta from Pantheon+ supernovae and BAO data, the authors model anisotropy as a dipole amplitude A and direction. Using covariance-based likelihood and null realizations that preserve redshift distribution and selection function, they find the observed A consistent with fluctuations, yielding a robust 95% upper bound A_95 = 0.025 independent of supernova calibration choices, with no evidence for intrinsic anisotropy.

What carries the argument

Dipole modulation in the duality parameter eta(z) = D_L / [D_A (1+z)^2], tested via full-covariance likelihood against selection-preserving null realizations.

If this is right

  • Cosmological models can assume the distance duality relation is the same in every direction at the precision of current data.
  • Any isotropic deviation in the duality parameter depends on the choice of supernova calibration and requires further investigation.
  • Survey selection effects must be modeled carefully when searching for directional signals in cosmological observables.
  • Upper bounds on anisotropy can be tightened with larger samples that have uniform sky coverage.

Where Pith is reading between the lines

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

  • This framework could extend to testing other distance or expansion relations for hidden directional dependence.
  • Future data with improved calibration might reveal whether the redshift-dependent isotropic deviation is physical or systematic.
  • Similar null-test methods could help isolate true signals in other large-scale structure analyses.

Load-bearing premise

The null realizations that keep the same redshift distribution and survey selection function capture every source of apparent anisotropy from noise and geometry.

What would settle it

A measured dipole amplitude significantly larger than 0.025 in an independent dataset with comparable or better control over selection effects would indicate real anisotropy.

read the original abstract

We present a model-independent test of anisotropy in the cosmic distance duality relation (CDDR), $D_L=(1+z)^2 D_A$, using the Pantheon+ type Ia supernova sample and baryon acoustic oscillation (BAO) data. The angular diameter distance is reconstructed via Gaussian Processes, enabling an estimate of $\eta(z)=D_L/[D_A(1+z)^2]$ without assuming a background cosmology. We also allow for a possible isotropic evolution, parameterized as $\eta(z)=1+\eta_1 z$, and find a redshift-dependent deviation whose significance depends on the assumed supernova calibration. Anisotropy is modeled through a dipole modulation and constrained using a full covariance-based likelihood. To assess statistical significance, we construct null realizations that preserve both the redshift distribution and the survey selection function. We find that the observed dipole amplitude is consistent with isotropic expectations and lies below the levels induced by statistical fluctuations and survey geometry. We obtain a robust 95\% upper bound $A_{95}=0.025$, stable across different supernova calibration choices. We find no evidence for intrinsic anisotropy in the CDDR. Our results highlight the importance of accounting for survey selection effects in anisotropy searches and provide a viable framework for testing directional deviations in cosmological relations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a model-independent analysis testing for directional anisotropy in the cosmic distance duality relation (CDDR) using the Pantheon+ supernova sample and BAO measurements. Angular diameter distances are reconstructed using Gaussian processes, allowing estimation of η(z) = D_L / [D_A (1+z)^2]. An isotropic evolution is parameterized as η(z) = 1 + η1 z, and anisotropy is modeled as a dipole modulation. A covariance-based likelihood is employed, with statistical significance evaluated through null realizations that preserve the observed redshift distribution and survey selection function. The analysis concludes that the observed dipole amplitude is consistent with isotropy, yielding a robust 95% upper limit A_95 = 0.025 that is stable across supernova calibration choices, with no evidence found for intrinsic anisotropy in the CDDR.

Significance. If the null tests adequately capture all relevant sources of apparent anisotropy, this work provides a valuable model-independent constraint on potential violations of the CDDR isotropy. The emphasis on survey selection effects is important, as it addresses a common pitfall in anisotropy searches. The use of Gaussian processes for reconstruction and the full covariance likelihood are methodological strengths that enhance the robustness of the bounds. This could serve as a framework for future directional tests in cosmology.

major comments (2)
  1. [Statistical significance assessment] The null realizations (described in the statistical significance section) preserve redshift distribution and survey selection function but do not appear to inject direction-dependent calibration mismatches from Pantheon+ or BAO reconstruction errors into the ensemble. Since the observed dipole is compared to the distribution from these nulls, any unmodeled position-dependent systematic could bias both the data and the null quantile, undermining the claim that A_95=0.025 robustly bounds intrinsic anisotropy. The stability across calibration choices is noted but does not substitute for including such effects in the nulls.
  2. [Gaussian process reconstruction] The choice of GP kernel and handling of redshift range for DA reconstruction (in the methods section on Gaussian process reconstruction) are not fully detailed in a way that allows verification of how they propagate into the η(z) estimates and subsequent dipole fit. This is load-bearing for the model-independent claim, as kernel misspecification could introduce artificial anisotropy or affect the covariance matrix used in the likelihood.
minor comments (2)
  1. [Abstract] The abstract mentions 'different supernova calibration choices' but does not specify what these are; a brief enumeration would aid clarity.
  2. [Likelihood construction] Clarify how the covariance matrix is constructed for the likelihood, particularly the cross-terms between SN and BAO data.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and positive assessment of our manuscript. We address each major comment below and have revised the manuscript to improve transparency and robustness where appropriate.

read point-by-point responses

  1. Referee: [Statistical significance assessment] The null realizations (described in the statistical significance section) preserve redshift distribution and survey selection function but do not appear to inject direction-dependent calibration mismatches from Pantheon+ or BAO reconstruction errors into the ensemble. Since the observed dipole is compared to the distribution from these nulls, any unmodeled position-dependent systematic could bias both the data and the null quantile, undermining the claim that A_95=0.025 robustly bounds intrinsic anisotropy. The stability across calibration choices is noted but does not substitute for including such effects in the nulls.

    Authors: We appreciate this observation. The null realizations are specifically constructed to quantify the distribution of dipole amplitudes arising from statistical fluctuations and the survey selection function under isotropy, while preserving the observed redshift distribution. We acknowledge that they do not explicitly incorporate direction-dependent calibration mismatches or reconstruction errors from Pantheon+ or BAO. The manuscript already reports that the A_95 = 0.025 bound is stable across different supernova calibration choices, which provides some mitigation against global calibration systematics. In the revised manuscript, we will expand the statistical significance section to clarify the intended scope of the null tests, discuss the potential impact of unmodeled position-dependent systematics as a limitation, and emphasize that the reported bound is conservative with respect to intrinsic anisotropy. We believe this addresses the concern without requiring a full re-simulation of the null ensemble. revision: partial

  2. Referee: [Gaussian process reconstruction] The choice of GP kernel and handling of redshift range for DA reconstruction (in the methods section on Gaussian process reconstruction) are not fully detailed in a way that allows verification of how they propagate into the η(z) estimates and subsequent dipole fit. This is load-bearing for the model-independent claim, as kernel misspecification could introduce artificial anisotropy or affect the covariance matrix used in the likelihood.

    Authors: We thank the referee for this comment. The methods section will be revised to provide complete details on the Gaussian process kernel choice, including its functional form and hyperparameter optimization procedure, as well as the exact handling of the redshift range for angular diameter distance reconstruction (including interpolation, extrapolation, and boundary conditions). We will also add explicit discussion of how these choices influence the resulting η(z) estimates, the covariance matrix, and the dipole amplitude constraints. This will enhance the verifiability of the model-independent approach and allow readers to assess potential impacts from kernel misspecification. revision: yes

Circularity Check

0 steps flagged

No circularity: dipole test uses independent null realizations preserving selection but assuming isotropy

full rationale

The derivation reconstructs D_A via Gaussian Processes on BAO, forms η(z) from Pantheon+ D_L, fits an optional isotropic η(z)=1+η1 z term, then models a dipole modulation A and evaluates its amplitude against the distribution obtained from separate null realizations that keep the observed redshift distribution and survey selection function but enforce isotropy. This Monte Carlo comparison is not a fit to the same data and does not reduce the reported A_95=0.025 or the consistency claim to a tautology. No self-citations are invoked as load-bearing uniqueness theorems, and no parameter fitted to the target dipole is later renamed as a prediction. The test is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard statistical assumptions for Gaussian processes and survey simulations plus the cosmological premise that any anisotropy can be captured by a dipole term.

free parameters (2)
  • η1
    Slope parameter in the isotropic evolution model η(z)=1+η1 z
  • A
    Amplitude of the dipole modulation used to model directional anisotropy
axioms (2)
  • domain assumption Gaussian processes with chosen kernel can reconstruct angular diameter distance from BAO data without cosmological assumptions
    Invoked to obtain model-independent D_A(z)
  • domain assumption Any directional deviation from the distance duality relation takes the form of a pure dipole
    Used to parameterize anisotropy

pith-pipeline@v0.9.0 · 5523 in / 1491 out tokens · 55248 ms · 2026-05-10T19:58:04.142714+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

57 extracted references · 17 canonical work pages · 1 internal anchor

  1. [1]

    Etherington,On the definition of distance in general relativity,Philosophical Magazine 15(1933) 761

    I.M.H. Etherington,On the definition of distance in general relativity,Philosophical Magazine 15(1933) 761

    1933

  2. [2]

    Etherington,Republication of: Lx

    I.M.H. Etherington,Republication of: Lx. on the definition of distance in general relativity, General Relativity and Gravitation39(2007) 1055

    2007

  3. [3]

    Ellis,Relativistic Cosmology, inGeneral Relativity and Cosmology, R.K

    G.F.R. Ellis,Relativistic Cosmology, inGeneral Relativity and Cosmology, R.K. Sachs, ed., pp. 104–182, Jan., 1971

    1971

  4. [4]

    Bassett and M

    B.A. Bassett and M. Kunz,Cosmic distance-duality as a probe of exotic physics and acceleration,Phys. Rev. D69(2004) 101305. – 15 –

    2004

  5. [5]

    Corasaniti,The impact of cosmic dust on supernova cosmology,Monthly Notices of the Royal Astronomical Society372(2006) 191–198

    P.S. Corasaniti,The impact of cosmic dust on supernova cosmology,Monthly Notices of the Royal Astronomical Society372(2006) 191–198

    2006

  6. [6]

    Ellis, R

    G.F.R. Ellis, R. Poltis, J.-P. Uzan and A. Weltman,Blackness of the cosmic microwave background spectrum as a probe of the distance-duality relation,Phys. Rev. D87(2013) 103530

    2013

  7. [7]

    Tiwari,Constraining axionlike particles using the distance-duality relation,Physical Review D95(2017)

    P. Tiwari,Constraining axionlike particles using the distance-duality relation,Physical Review D95(2017)

    2017

  8. [8]

    Buen-Abad, J

    M.A. Buen-Abad, J. Fan and C. Sun,Constraints on axions from cosmic distance measurements,Journal of High Energy Physics2022(2022) 103

    2022

  9. [9]

    Uzan,Distance duality relation from cluster observations,Phys

    J.P.e.a. Uzan,Distance duality relation from cluster observations,Phys. Rev. D70(2004) 083533

    2004

  10. [10]

    Holanda, J.A.S

    R.F.L. Holanda, J.A.S. Lima and M.B. Ribeiro,Testing the distance–duality relation with galaxy clusters and type ia supernovae,The Astrophysical Journal722(2010) L233–L237

    2010

  11. [11]

    Holanda, V

    R. Holanda, V. Busti and J. Alcaniz,Probing the cosmic distance duality with strong gravitational lensing and supernovae ia data,Journal of Cosmology and Astroparticle Physics 2016(2016) 054

    2016

  12. [12]

    K. Liao, Z. Li, S. Cao, M. Biesiada, X. Zheng and Z.-H. Zhu,The distance duality relation from strong gravitational lensing,The Astrophysical Journal822(2016) 74

    2016

  13. [13]

    B. Xu, Z. Wang, K. Zhang, Q. Huang and J. Zhang,Model-independent Test for the Cosmic Distance–Duality Relation with Pantheon and eBOSS DR16 Quasar Sample,The Astrophysical Journal939(2022) 115

    2022

  14. [14]

    Zhang, X

    X. Zhang, X. Yang, Y. Ren, S. Chen, Y. Shi, C. Cheng et al.,Testing cosmic distance duality relation and transparency with desi dr2, 2025

    2025

  15. [15]

    Li, G.-H

    T.-N. Li, G.-H. Du, P.-J. Wu, J.-Z. Qi, J.-F. Zhang and X. Zhang,Testing the cosmic distance duality relation with baryon acoustic oscillations and supernovae data,The European Physical Journal C85(2025) 1354

    2025

  16. [16]

    Bennett,Nine-year wmap observations: Final maps and results,ApJS208(2013) 20

    C.L.e.a. Bennett,Nine-year wmap observations: Final maps and results,ApJS208(2013) 20

    2013

  17. [17]

    Aghanim, Y

    N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini et al., Planck2018 results: Vi. cosmological parameters,Astronomy & Astrophysics641(2020) A6

    2020

  18. [18]

    Blake and J

    C. Blake and J. Wall,A velocity dipole in the distribution of radio galaxies,Nature416(2002) 150

    2002

  19. [19]

    Singal,Large peculiar motion of the solar system from the dipole anisotropy in sky brightness due to distant radio sources,The Astrophysical Journal742(2011) L23 [1110.6260]

    A.K. Singal,Large peculiar motion of the solar system from the dipole anisotropy in sky brightness due to distant radio sources,The Astrophysical Journal742(2011) L23 [1110.6260]

    work page arXiv 2011

  20. [20]

    Tiwari, R

    P. Tiwari, R. Kothari, A. Naskar, S. Nadkarni-Ghosh and P. Jain,Dipole anisotropy in sky brightness and source count distribution in radio nvss data,Astroparticle Physics61(2015) 1

    2015

  21. [21]

    Dom` enech, R

    G. Dom` enech, R. Mohayaee, S.P. Patil and S. Sarkar,Galaxy number-count dipole and superhorizon fluctuations,Journal of Cosmology and Astroparticle Physics2022(2022) 019

    2022

  22. [22]

    Kashyap, N.K

    G. Kashyap, N.K. Singh and P. Jain,Matter dipole and hubble tension due to large wavelength perturbations,Phys. Rev. D112(2025) 083524

    2025

  23. [23]

    A. Sah, M. Rameez, S. Sarkar and C.G. Tsagas,Anisotropy in Pantheon+ supernovae,The European Physical Journal C85(2025) 596

    2025

  24. [24]

    J. Yoo, M. Magi and D. Huterer,Cosmic dipoles from large-scale structure surveys,Phys. Rev. D112(2025) 123013

    2025

  25. [25]

    Hutsem´ ekers,Evidence for very large-scale coherent orientations of quasar polarization vectors,Astronomy and Astrophysics332(1998) 410

    D. Hutsem´ ekers,Evidence for very large-scale coherent orientations of quasar polarization vectors,Astronomy and Astrophysics332(1998) 410. – 16 –

    1998

  26. [26]

    Hutsem´ ekers and Lamy,Confirmation of the existence of coherent orientations of quasar polarization vectors on cosmological scales*,A&A367(2001) 381

    2001

  27. [27]

    Tiwari and P

    P. Tiwari and P. Jain,Polarization alignment in jvas/class flat spectrum radio surveys, International Journal of Modern Physics D22(2013) 1350089 [1201.5180]

    work page arXiv 2013

  28. [28]

    Pelgrims and D

    V. Pelgrims and D. Hutsem´ ekers,Polarization alignments of quasars from the JVAS/CLASS 8.4-GHz surveys,Monthly Notices of the Royal Astronomical Society450(2015) 4161 [https://academic.oup.com/mnras/article-pdf/450/4/4161/5836517/stv917.pdf]

    2015

  29. [29]

    Taylor and P

    A.R. Taylor and P. Jagannathan,Alignments of radio galaxies in deep radio imaging of ELAIS N1,Monthly Notices of the Royal Astronomical Society: Letters459(2016) L36 [1603.02418]

    work page arXiv 2016

  30. [30]

    Panwar, Prabhakar, P.K

    M. Panwar, Prabhakar, P.K. Sandhu, Y. Wadadekar and P. Jain,Alignment of radio galaxy axes using FIRST catalogue,Monthly Notices of the Royal Astronomical Society499(2020) 1226 [2006.09779]

    work page arXiv 2020

  31. [31]

    Kashlinsky, F

    A. Kashlinsky, F. Atrio-Barandela, D. Kocevski and H. Ebeling,A measurement of large-scale peculiar velocities of clusters of galaxies: Results and cosmological implications,The Astrophysical Journal686(2008) L49

    2008

  32. [32]

    Eriksen, F.K

    H.K. Eriksen, F.K. Hansen, A.J. Banday, K.M. G´ orski and P.B. Lilje,Asymmetries in the cosmic microwave background anisotropy field,The Astrophysical Journal605(2004) 14

    2004

  33. [33]

    De Oliveira-Costa, M

    A. De Oliveira-Costa, M. Tegmark, M. Zaldarriaga and A. Hamilton,Significance of the largest scale cmb fluctuations in wmap,Phys. Rev. D69(2004) 063516

    2004

  34. [34]

    Schwarz, G.D

    D.J. Schwarz, G.D. Starkman, D. Huterer and C.J. Copi,Is the low-ℓmicrowave background cosmic?,Phys. Rev. Lett.93(2004) 221301

    2004

  35. [35]

    X. Li, L. Tang and H.-N. Lin,Testing the anisotropy of the universe with the distance duality relation,Monthly Notices of the Royal Astronomical Society482(2018) 5678 [https://academic.oup.com/mnras/article-pdf/482/4/5678/26930088/sty3116.pdf]

    2018

  36. [36]

    Scolnic, D

    D. Scolnic, D. Brout, A. Carr, A.G. Riess, T.M. Davis, A. Dwomoh et al.,The pantheon+ analysis: The full data set and light-curve release,The Astrophysical Journal938(2022) 113

    2022

  37. [37]

    The 6dF Galaxy Survey: Baryon Acoustic Oscillations and the Local Hubble Constant

    F. Beutler, C. Blake, M. Colless, D.H. Jones, L. Staveley-Smith, L. Campbell et al.,The 6dF Galaxy Survey: baryon acoustic oscillations and the local Hubble constant,mnras416(2011) 3017 [1106.3366]

    work page Pith review arXiv 2011

  38. [38]

    A.J. Ross, L. Samushia, C. Howlett, W.J. Percival, A. Burden and M. Manera,The clustering of the SDSS DR7 main Galaxy sample - I. A 4 per cent distance measure at z = 0.15,mnras 449(2015) 835 [1409.3242]. [39]DESIcollaboration,DESI DR2 Results II: Measurements of Baryon Acoustic Oscillations and Cosmological Constraints,2503.14738. [40]BOSScollaboration,T...

    work page arXiv 2015

  39. [39]

    Adame, J

    A. Adame, J. Aguilar, S. Ahlen, S. Alam, D. Alexander, M. Alvarez et al.,Desi 2024 vi: – 17 – cosmological constraints from the measurements of baryon acoustic oscillations,Journal of Cosmology and Astroparticle Physics2025(2025) 021

    2024

  40. [40]

    du Mas des Bourboux, J

    R. Neveux, E. Burtin, A. de Mattia, A. Smith, A.J. Ross, J. Hou et al.,The completed sdss-iv extended baryon oscillation spectroscopic survey: Bao and rsd measurements from the anisotropic power spectrum of the quasar sample between redshift 0.8 and 2.2,Monthly Notices of the Royal Astronomical Society499(2020) 210 [https://academic.oup.com/mnras/article-...

    work page arXiv 2020

  41. [41]

    Jimenez and A

    R. Jimenez and A. Loeb,Constraining cosmological parameters based on relative galaxy ages, The Astrophysical Journal573(2002) 37

    2002

  42. [42]

    Four New Observational $H(z)$ Data From Luminous Red Galaxies of Sloan Digital Sky Survey Data Release Seven

    C. Zhang, H. Zhang, S. Yuan, S. Liu, T.-J. Zhang and Y.-C. Sun,Four new observational H(z) data from luminous red galaxies in the Sloan Digital Sky Survey data release seven,Research in Astronomy and Astrophysics14(2014) 1221 [1207.4541]

    work page Pith review arXiv 2014

  43. [43]

    A 6% measurement of the Hubble parameter at $z\sim0.45$: direct evidence of the epoch of cosmic re-acceleration

    M. Moresco, L. Pozzetti, A. Cimatti, R. Jimenez, C. Maraston, L. Verde et al.,A 6% measurement of the Hubble parameter atz∼0.45: direct evidence of the epoch of cosmic re-acceleration,Journal of Cosmology and Astroparticle Physics05(2016) 014 [1601.01701]

    work page Pith review arXiv 2016

  44. [44]

    Constraints on the equation of state of dark energy and the Hubble constant from stellar ages and the CMB

    R. Jimenez, L. Verde, T. Treu and D. Stern,Constraints on the equation of state of dark energy and the Hubble constant from stellar ages and the CMB,Astrophys. J.593(2003) 622 [astro-ph/0302560]

    work page Pith review arXiv 2003

  45. [45]

    Stern, R

    D. Stern, R. Jimenez, L. Verde, M. Kamionkowski and S.A. Stanford,Cosmic chronometers: constraining the equation of state of dark energy. i: H(z) measurements,Journal of Cosmology and Astroparticle Physics2010(2010) 008

    2010

  46. [46]

    Improved constraints on the expansion rate of the Universe up to z~1.1 from the spectroscopic evolution of cosmic chronometers

    M. Moresco, A. Cimatti, R. Jimenez, L. Pozzetti, G. Zamorani, M. Bolzonella et al.,Improved constraints on the expansion rate of the Universe up to z≈1.1 from the spectroscopic evolution of cosmic chronometers,Journal of Cosmology and Astroparticle Physics2012(2012) 006 [1201.3609]

    work page Pith review arXiv 2012

  47. [47]

    Constraints on the redshift dependence of the dark energy potential

    J. Simon, L. Verde and R. Jimenez,Constraints on the redshift dependence of the dark energy potential,Phys. Rev. D71(2005) 123001 [astro-ph/0412269]

    work page Pith review arXiv 2005

  48. [48]

    Borghi, M

    N. Borghi, M. Moresco and A. Cimatti,Toward a better understanding of cosmic chronometers: A new measurement of h(z) at z = 0.7,The Astrophysical Journal Letters928(2022) L4

    2022

  49. [49]

    Raising the bar: new constraints on the Hubble parameter with cosmic chronometers at z$\sim$2

    M. Moresco,Raising the bar: new constraints on the Hubble parameter with cosmic chronometers at z ˜2.,Monthly Notices of the Royal Astronomical Society450(2015) L16 [1503.01116]

    work page Pith review arXiv 2015

  50. [50]

    Age-dating Luminous Red Galaxies observed with the Southern African Large Telescope

    A.L. Ratsimbazafy, S.I. Loubser, S.M. Crawford, C.M. Cress, B.A. Bassett, R.C. Nichol et al., Age-dating Luminous Red Galaxies observed with the Southern African Large Telescope,Mon. Not. Roy. Astron. Soc.467(2017) 3239 [1702.00418]

    work page Pith review arXiv 2017

  51. [51]

    Rasmussen and C.K.I

    C.E. Rasmussen and C.K.I. Williams,Gaussian Processes for Machine Learning, Adaptive Computation and Machine Learning, MIT Press, Cambridge, MA, USA (Jan., 2006)

    2006

  52. [52]

    Aigrain and D

    S. Aigrain and D. Foreman-Mackey,Gaussian process regression for astronomical time series, Annual Review of Astronomy and Astrophysics61(2023) 329 [2209.08940]

    work page arXiv 2023

  53. [53]

    Pedregosa, G

    F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel et al.,Scikit-learn: Machine learning in python,Journal of Machine Learning Research12(2011) 2825

    2011

  54. [54]

    Reconstruction of dark energy and expansion dynamics using Gaussian processes

    M. Seikel, C. Clarkson and M. Smith,Reconstruction of dark energy and expansion dynamics using gaussian processes,Journal of Cosmology and Astroparticle Physics2012(2012) 036 [1204.2832]. – 18 –

    work page Pith review arXiv 2012

  55. [55]

    Dinda and N

    B.R. Dinda and N. Banerjee,Model independent bounds on type ia supernova absolute peak magnitude,Physical Review D107(2023)

    2023

  56. [56]

    Riess, W

    A.G. Riess, W. Yuan, L.M. Macri, D. Scolnic, D. Brout, S. Casertano et al.,A comprehensive measurement of the local value of the hubble constant with 1 km s-1 mpc-1 uncertainty from the hubble space telescope and the sh0es team,The Astrophysical Journal Letters934(2022) L7

    2022

  57. [57]

    emcee: The MCMC Hammer

    D. Foreman-Mackey, D.W. Hogg, D. Lang and J. Goodman,emcee: The mcmc hammer,PASP 125(2013) 306 [1202.3665]. – 19 –

    work page internal anchor Pith review arXiv 2013