The problem of inadequacy: cosmology and metrology. Part 1. The cosmological distance scale and E. Hubble's law of redshift

The article discusses the general scientific problem of the inadequacy of mathematical models of measurement objects for two important cases: solving measurement problems using the method of joint measurements and the method of indirect measurements. In the first part of the article, the latent synonymy of the terms measurement and calculation is discussed as the primary cause of the discrepancy between the names and definitions of the terms direct measurement, indirect measurement, aggregate measurement, and joint measurement . In the second part of the article, the definitional uncertainty of mathematical models is analyzed as a result of incomplete descriptions. Both of these phenomena are key to the problem of inadequacy. The first part of the article is devoted to an analytical review of works in which the problems of calibration of the cosmological distances scale by the method of joint measurements are solved using data on photometric distances of supernovae of type SN I a.

For one-dimensional isotropic models of the Friedmann-Robertson-Walker type of the 3rd order, in 1998–1999, due to rank inversions of the redshift, perceived as a “cosmic push”, a conclusion was made on the “acceleration of the expansion of the Universe” based on regression analysis. Based on the same data on photometric distances in 2016, taking into account the right ascension and declination of SN Ia for a three-dimensional anisotropic model of the 2nd order, a more accurate result was obtained using the cross-observation error inadequacy scheme (maximum compactness method). At the same time, it has been established that the causes of rank inversions are not the super-massive large-scale inhomogeneities of the universe, attractors, but gravitational dipoles pairs of giant super clusters of galaxies and giant voids in opposite directions of the celestial sphere. In such dipoles, the gravitational effect on objects between them is not balanced and manifests itself as an additional repulsive force from the giant voids. Five such dipoles have been discovered, and two of them have been independently confirmed. Thus, the alternative hypothesis that the Milky Way is moving towards a giant system of super clusters of galaxies at the northern galactic pole, leaving behind a giant system of voids in the constellations of Eridanus, Cetus, and Aquarius at the southern galactic pole, is more plausible according to the criterion of minimizing the error of inadequacy.

1. Kuznetsov V. A., Yalunina G. V. Metrology: A textbook. Izdatel’stvo standartov, Moscow (1998). (In Russ.)

2. International Dictionary of Metrology: basic and general concepts and related terms. Translated from English and French by D. I. Mendeleev VNIIM, BelGIM. Ed. 2nd. NGO “Professional”, St. Petersburg (2010). (In Russ.)

3. Levin S. F. Semantic analysis of compound terms in metrology. Part 1: types and methods of measurement. Izmeritel’naya tekhnika, 73(1), 26–34 (2024). (In Russ.) https://doi.org/10.32446/0368-1025it.2024-1-26-34 ; https://elibrary.ru/zwgulg

4. Levin S. F. Semantic analysis of compound terms in metrology. Part 2: risk in measurement and calculations. Izmeritel’naya tekhnika, 73(2), 13–22 (2024). (In Russ.) https://doi.org/10.32446/0368-1025it.2024-2-13-22 ; https://elibrary.ru/dbpafm

5. Philosophical Dictionary, Ed. I. T. Frolov. 5th ed. Politizdat Publ., Moscow (1987).

6. Levin S. F. Fundamentals of metrological support for solving measurement problems. Lecture notes on the discipline “Metrology and Standardization”, Section 2. N. E. Zhukovsky VVIA Publ., Moscow (1996). (In Russ.)

7. Levin S. F. Definitional uncertainty and error of inadequacy. Izmeritel’naya Tekhnika, (11), 7–17 (2019). (In Russ.) https://doi.org/10.32446/0368-1025it.2019-11-7-17 ; https://elibrary.ru/favnkg

8. Levin S. F. Inadequacy of mathematical models of measurements objects and calculations of risk based on the use of Gost ISO/IEC 17025–2019. Izmeritel’naya Tekhnika, (7), 13–21 (2020). (In Russ.) https://doi.org/10.32446/0368-1025it.2020-7-13-21 ; https://elibrary.ru/vxojfc

9. Levin S. F. The Cosmological distances scale. Part 12. Confluent analysis, rank inversion, lack-of-fit tests. Izmeritel’naya Tekhnika, (12), 13–21 (2020). (In Russ.) https://doi.org/10.32446/0368-1025it.2020-12-13-21 ; https://elibrary.ru/jfwngp

10. Levin S. F. System facets of measurement problems theory. Izmeritel’naya tekhnika, (1), 3–11 (2022). (In Russ.) https://doi/org/10.32446/0368-1025it.2022-1-3-11 ; https://elibrary.ru/xegxfi

11. Levin S. F. Optimal interpolation filtering of statistical characteristics of random functions in a deterministic version of the Monte Carlo method and the law of redshift. AN SSSR, Nauchnyj sovet po kompleksnoj probleme “Kibernetika”, Moscow (1980). (In Russ.)

12. Hubble E. A relation between distance and radial velocity among extragalactic nebulae. Proceedings of the National Academy of Sciences, 15, 168–173 (1929). https://doi.org/10.1073/pnas.15.3.168

13. Robertson H. P. LXXXVI. On relativistic cosmology. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 5(31), 835–848 (1928). https://doi.org/10.1080/14786440508564528

14. Slipher V. The detection of nebular rotation. Lowell Observatory Bulletin, 2, 62 (1914).

15. Friedmann A. A. Ûber die Krümmung des Raumes. Zeitschrift für Physik, 10, 377–386 (1922). (In German) https://doi.org/10.1007/BF01332580

16. Lang K. R. Astrophysical formulae. A. Compendium for the Physicist and Astrophysicist. Translated from English by Yu. K. Zemtsov and others; eds. L. A. Pokrovsky and V. L. Khokholova. Part 1. Mir, Moscow (1978). (In Russ.)

17. Hoyle F. Verification of cosmology by observations. Proceedings of the Physical Society, 77(1), 1–16 (1961). https://doi.org/10.1088/0370-1328/77/1/302

18. Hoyle F. Verification of cosmology by observations. In: J. Wheeler Gravity, neutrinos and the Universe. Inostrannaya Literatura, Moscow, рp. 372–400 (1962). (In Russ.)

19. Sandage A. R. The change of redshift and apparent luminosity of galaxies due to the deceleration of selected expanding universes. Astrophysical Journal, 136, 319–333 (1962). https://doi.org/10.1086/147385

20. Polikarov A. Relativity and quanta. Progress, Moscow (1966). (In Russ.)

21. Humason M. L., Mayal N. U., Sandage A. R. Redshift and magnitudes extragalactic nebulae. Astronomical Journal, 61, 97–162 (1956). https://doi.org/10.1086/107297

22. Carroll S., Press W., Turner E. The cosmological constant. Annual Review of Astronomy & Astrophysics, 30, 499–542 (1992). https://doi.org/10.1146/annurev.aa.30.090192.002435 ; https://elibrary.ru/gssyio

23. Voprosy kibernetiki, VK-94: Statistical methods in the theory of maintenance. Ed. S. F. Levin. AN SSSR, Nauchnyj sovet po kompleksnoj probleme “Kibernetika”, Moscow (1982). (In Russ.)

24. Levin S. F. Statistical analysis of maintenance systems for technical facilities. Voprosy kibernetiki, VK-94. AN SSSR, Nauchnyj sovet po kompleksnoj probleme “Kibernetika”, Moscow, рp. 105–122 (1982). (In Russ.)

25. Levin S. F. Fundamentals of control theory. MO SSSR, Moscow (1983). (In Russ.)

26. Levin S. F., Blinov A. P. Scientific and methodological assurance of the solution of metrological problems by probabilistic and statistical methods. Izmeritel’naya tekhnika, (12), 5–8 (1988). (In Russ.)

27. Levin S. F., Lisenkov A. N., Sen’ko O. V., Kharat’yan E. I. The MMK-stat M system of metrological monitoring of static measuring tasks. User’s Guide. Gosstandart RF, VC RAN, Moscow (1998). (In Russ.)

28. Levin S. F. Metrological certification of mathematical models in measuring problems of gravity and cosmology. Theoretical and experimental problems of the general theory of relativity and gravity. Abstracts of the X Russian Gravity Conference, Vladimir, June 20–27, 1999. RGO, Moscow, р. 245 (1999). (In Russ.)

29. Karachencev I. D., Mitronova S. N. Singular velocities of flat spiral galaxies of a 2MASS survey in the volume of 100 Mp. Report of the Special Astrophysical Observatory of the Russian Academy of Sciences, p. 42 (2004–2005). (In Russ.)

30. Levin S. F. Measuring tasks of statistical identification of the scale of cosmological distances. Izmeritel’naya Tekhnika, (12), 17–22 (2011). (In Russ.) https://elibrary.ru/ooqzdh

31. Levin S. F. Cosmological distance scale based on a red-shift interpolation model. Izmeritel’naya Tekhnika, (6), 12–14 (2012). (In Russ.) https://elibrary.ru/pbbudt

32. Levin S. F. Cosmological distances scale: paradoxes in red shift models. Izmeritel’naya Tekhnika, (3), 3–6 (2013). (In Russ.) https://elibrary.ru/rcnywv

33. Levin S. F. Cosmological distances scale. Part 1. “Unexpected” results. Izmeritel’naya Tekhnika, (2), 3–8 (2014). (In Russ.) https://elibrary.ru/saepar

34. Levin S. F. Cosmological distances scale. Part 2. “Unexpected” coincidence. Izmeritel’naya Tekhnika, (4), 3–7 (2014). (In Russ.) https://www.elibrary.ru/sbyzfv

35. Levin S. F. Cosmological distance scale. Part 3. Red shift standards. Izmeritel’naya Tekhnika, (9), 8–12 (2014). https://elibrary.ru/tejzgv

36. Levin S. F. Cosmological distance scale. Part 4. Calibration based on SN Ia supernovae. Izmeritel’naya Tekhnika, (5), 5–10 (2015). (In Russ.) https://elibrary.ru/tudyvh

37. Levin S. F. Cosmological distance scale. Part 5. Metrological expert opinion on type SN Ia supernovae. Izmeritel’naya Tekhnika, (8), 3–10 (2016). (In Russ.) https://elibrary.ru/wmbttv

38. Levin S. F. Cosmological distance scale. Part 6. Statistical anisotropy of red shift. Izmeritel’naya Tekhnika, (5), 3–7 (2017). (In Russ.) https://elibrary.ru/ytvxnj

39. Levin S. F. Cosmological Distance Scale. Part 7. A new special case with the Hubble constant and anisotropic models. Izmeritel’naya tekhnika, (11), 15–21 (2018). (In Russ.) https://doi.org/10.32446/0368-1025it.2018-11-15-21 ; https://elibrary.ru/pntrqt

40. Levin S. F. Cosmological Distance Scale. Part 8. The scale factor. Izmeritel’naya Tekhnika, (1), 8–15 (2019). (In Russ.) https://doi.org/10.32446/0368-1025it.2019-1-8-15 ; https://elibrary.ru/mlwzzj

41. Levin S. F. Cosmological Distance Scale. Part 9. Deceleration Parameter. Izmeritel’naya Tekhnika, (10), 8–14 (2019). (In Russ.) https://doi.org/10.32446/0368-1025it.2019-10-8-14 ; https://elibrary.ru/engain

42. Levin S. F. Cosmological Distance Scale. Part 10. Universal Anisotropy. Izmeritel’naya Tekhnika, (10), 9–25 (2020). (In Russ.) https://doi.org/10.32446/0368-1025it.2020-10-9-25 ; https://elibrary.ru/axfdct

43. Levin S. F. The Cosmological distances scale. Part 11.“Extraordinary” Evidences and the “Cosmically Jerk” problem. Izmeritel’naya Tekhnika, (11), 3–8 (2020). (In Russ.) https://doi.org/10.32446/0368-1025it.2020-11-3-8 ; https://elibrary.ru/lntoqi

44. Levin S. F. The Cosmo-logical distances scale. Part 13. Galactic Polar Redshift Anisotropy of Quasars and Type Ia Supernovae. Izmeritel’naya Tekhnika, (10), 11–18 (2022). (In Russ.) https://doi.org/10.32446/0368-1025it.2022-10-11-18 ; https://elibrary.ru/wlawqp

45. Levin S. F. The Cosmological distances scale. Part 14. “Hubble Bubble” and the Gravitational Dipole. Izmeritel’naya Tekhnika, (2), 4–11 (2023). (In Russ.) https://doi.org/10.32446/0368-1025it.2023-2-4-11 ; https://elibrary.ru/tgghax

46. Levin S. F. The Cosmological distances scale. Part 15. Cosmic Jerk and gravitational dipole inhomogeneity. Izmeritel’naya Tekhnika, (3), 10–15 (2023). (In Russ.) https://doi.org/10.32446/0368-1025it.2023-3-10-15 ; https://elibrary.ru/wfgrul

47. Levin S. F. The Cosmological distances scale. Part 16: Dipol’ Khabbla. Izmeritel’naya Tekhnika, (6), 4–12 (2023). (In Russ.) https://doi.org/10.32446/0368-1025it.2023-6-4-12 ; https://elibrary.ru/eejini

48. Levin S. F. The Cosmological distances scale. Part 17: A coincidence of coincidences. Izmeritel’naya Tekhnika, (10), 10–16 (2023). (In Russ.) https://doi.org/10.32446/0368-1025it.2023-10-10-16 ; https://elibrary.ru/jzdlwh

49. Levin S. F. The Cosmological distances scale. Part 18: Red-blue dipole and redshift anisotropy. Izmeritel’naya Tekhnika, 73(5), 4–8 (2024). (In Russ.) https://doi.org/10.32446/0368-1025it.2024-5-4-8 ; https://elibrary.ru/jnirgg

50. Riess A. G. et al. Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astronomical Journal, 116, 1009–1038 (1998). https://doi.org/10.1086/300499 ; https://elibrary.ru/lrpddf

51. Schmidt B. P. et al. High-Z supernova search: measuring cosmic deceleration and global curvature of the universe using type Ia supernovae. Astrophysics Journal, 507, 46–63 (1998). https://doi.org/10.1086/306308 ; https://elibrary.ru/lrpdqh

52. Perlmutter S. et al. Measurements of Ω and Λ from 42 high-red shift supernovae. Astrophysical Journal, 517, 565–586 (1999). https://doi.org/10.1086/307221 ; https://elibrary.ru/smqbvl

53. Riess A.G. et al. Type Ia supernova discoveries at z>1 from the Hubble space telescope: evidence for past deceleration and constraints on dark energy evolution. Astrophysical Journal, 607, 665–687 (2004). https://doi.org/10.1086/383612 ; https://elibrary.ru/sqfyuv

54. Perlmutter S. Measuring the acceleration of the cosmic expansion using supernova. Nobel Lecture. Stockholm, December 8, 2011. Phys. Usp., 56(10), (2013).

55. Riess et al. New Hubble space telescope discoveries of type Ia supernovae at z ≥1: narrowing constraints of the early behavior of dark energy. Astrophysical Journal, 659, 98–121 (2007).

56. Shiryaev A. N. Detection of spontaneous effects. Doklady AN SSSR, 138(4), 799–801 (1961). (In Russ.)

57. Riss A. G. My path to the accelerating Universe. Nobel Lecture. Stockholm, December 8, 2011. Phys. Usp., 56(10), (2013).

58. Levin S. F. Mathematical theory of measurement problems: applications. “Cosmically jerk” or rang inversion – the reason for “dead end” in cosmology. Test & Measuring Instruments and Systems, (4), 32–38 (2020). (In Russ.)

59. Fixsen D. J. et al. The Cosmic Microwave Background Spectrum from the Full COBE/FIRAS Data Set. Astrophysical Journal, 473, 576–587 (1996). https://doi.org/10.1086/178173 ; https://elibrary.ru/xrngux

60. Dressler A., Faber S. M., Burstein D., Davies R. L., Lynden-Bell D., Terlevich R. J., Wegner G. Spectroscopy and photometry of elliptical galaxies – A large-scale streaming motion in the local universe. Astrophysical Journal, Part 2 – Letters to the Editor, 313, L37–L42 (1987). https://doi.org/10.1086/184827

61. Levin S. F. On spatial anisotropy of red shift in spectrums of ungalaxy sources. Physical Interpretations of relativity Theory – Proc. of XV International Scientific Meeting PIRT-2009, Moscow, 6–9 July, 2009. BMSTU, Moscow, рp. 234–240 (2009).

62. Levin S. F. The measurement problem of redshift anisotropy identification. Metrologiya, (5), 3–21 (2010). (In Russ.)

63. Levin S. F. Anisotropy of Red shift. Hypercomplex numbers in geometry and physics, 8(1(15)), 147–178 (2011). (In Russ.) https://elibrary.ru/phgyvb

64. Courtois H. M., Tully R. B., Racah Y. H., Pomarede D., Graziani R., Dupuy A. Cosmicflows-3: Cold Spot Repeller? Astrophysical Journal Letters, 847(1), L6 (2017). https://doi.org/10.3847/2041-8213/aa88b2 ; https://elibrary.ru/ygkjgg

65. Levin S. F. Gravitational dipoles are inhomogeneities in the large-scale structure of the universe. Physical interpretations of the theory of relativity. Collection of abstracts of the XXIV International Scientific Conference. Moscow, July 7–10, 2025. BMSTU, Moscow, рp. 222–224 (2025). (In Russ.) https://elibrary.ru/jkyjfp