Introduction. In the section “Extraordinary claims require extraordinary evidence,” the Nobel prize lecture of Riess [1] describes a big project to exclude alternative astrophysical hypotheses (absorption by dust or the intrinsic evolution of sources) for the accelerated expansion of the Universe visible in supernovae in a model with hidden mass and dark energy. The year-long program from 2002–2003 to observe distant supernovae on the Hubble space telescope was highly successful. Six Type Ia supernovae at red shifts over 1.25 were discovered and they made it possible to rule out gray dust and evolution and to clearly determine that the Universe was decelerating before it began expanding with acceleration. In physics, a change in the value or sign of deceleration (as a consequence of a change in force) is caused by a sudden jerk.

The results from 2002–2003 mentioned in Ref. 1 are listed in Table 1 with the photometric distances DL from Refs. 2 and 3.

Table 1 Supernovae from the Hubble Deep and Ultradeep Fields [2, 3]
Full size table

These supernovae belong to the Hubble deep field (HDF) and the Hubble ultradeep field (HUDF). They are found in segments of the sky with a diameter on the order of a few angular minutes in the constellations Ursa Major [12h36m49.4s; +62°12′58′′] and Fornax [3h32m39.0s; –27°47′29.1′′]. Both fields are near galactic poles separated from bright stars of the Milky Way, so it is possible to observe the very dim objects that turn out to be galaxies with large red shifts.

According to Refs. 2 and 4, the purely kinematic interpretation of the SN Ia sample yields evidence with a confidence level of over 99% for a transition from deceleration to acceleration or, analogously, convincing proof of a “cosmic jerk.” For a simple model of the history of the expansion, the transition between the two epochs is bounded by z = 0.46 ± 0.13.

In earlier papers of S. Perlmutter’s group it was stated that the transition between deceleration and acceleration took place at z = 0.73, when the supernova SN 1997G exploded.

In 2004, Visser [5] used an expansion of the Friedman–Robertson–Walker model in a Taylor series with respect to the red shift with a curvature parameter Ωk = 0 to obtain a third order model for the photometric distance:

$$ {D}_L(z)=\frac{c}{H_0}\left[z+\frac{1}{2}\left(1-{q}_0\right){z}^2-\frac{1}{6}\left(1-{q}_0-3{q}_0^2+{j}_0\right){z}^3\right], $$
(1)

where c is the fundamental constant of the speed of light, H0 is the Hubble parameter, q0 is the deceleration parameter, and j0 is the jerk parameter.

In 2016, with a new group of coworkers, Riess used the model of Eq. (1) to obtain an new estimate H0 = 73.24 ± 1.24 km/s/Mpc [6]. It differed significantly from the previous estimate H0 = 65.2 ± 1.3 km/s/Mpc [4] obtained from the Friedman–Robertson–Walker model with a curvature parameter Ωk = 0 and from an estimate H0 = 67.8 ± 0.9 km/s/Mpc based on data from the Planck space probe using the ΛCDM-model [7]. This fact was pointed out by experts from the Carnegie-Chicago Hubble project [8].

Also in 2016, metrological expert evaluation [9] on data from type SN Ia supernovae [4, 10] revealed an anisotropy of the red shift and an asymmetry in its relation to the photometric distance.

In 2017, Freedman referred to discrepancies in model estimates of the Hubble parameter by more than “3 sigma” in a “normal” law, as an “impasse” in cosmology [11], and she saw a way out by bringing the accuracy of the cosmological distance scale to better than 1%. Further measurements [12], however, have only increased the spread in the estimates and the number of people involved in the discussion began to increase.

But, without dramatizing the situation, in terms of the logic of statistical inference according to the standards document R 50.2.004.2000, “State System for Ensuring the Uniformity of Measurements (GSI). Determination of the Characteristics of Mathematical Models of the Relationships Between Physical Quantities in Solving Measurement Problems. Basic Assumptions,” the “jerk problem” is a standard problem of statistical testing of the hypothesis of the “breakdown” of a model. And the problem of calibrating the cosmological distance scale can be analyzed in terms of the criterion of the minimum of the average modulus of the inadequacy error (AMIE) by the programs “MMK-stat” and “MMK-stat M” [13].

We recall that a test for the “discrepancy” in the final results of the Hubble Space Telescope program Key Project for determining the Hubble parameter [14] rejected its estimate of H0 = 72.6 km/s/Mpc independent of the photometric distance DL ≤ 391.5 Mpc. A reduction in the Hubble parameter to H0 = 65.95 km/s/Mpc with DL > 309.5 Mpc was recognized as more likely [15]. But just this circumstance eliminated the questions regarding the substantial spread in the estimates [7, 8, 11, 12].

Statistical testing of hypotheses. To describe the history of the expansion of the Universe we use a simple isotropic model of the scale,

$$ {D}_L(z)={\upvartheta}_0{\uptheta}_0+\sum \limits_{m=1}^M{\upvartheta}_m{\uptheta}_m{z}^m, $$
(2)

where θ0, …, θm, …, θM are the parameters of the model and ϑ0ϑ1ϑ2…ϑm…ϑM = ϑ is the binary code for its structure; for estimating the parameters by the algorithms of the method of maximum compactness (MMK) in a scheme of crossover observation of the inadequacy error based on the methods of least squares MMKMNK (MMCMNA), least moduli MMKMNM, or median interpolation MMKMEDS (MCODS) in accordance with the standards document R 50.2.004-2000.

We note that a model of maximum complexity max{I, J, M} ≤ K of the form

$$ {D}_L\left(l,b,z\right)=\sum \limits_{i=0}^I\sum \limits_{j=0}^J\sum \limits_{m=0}^M{\upvartheta}_{ijm}{\uptheta}_{ijm}{l}^i{b}^j{z}^m $$
(3)

was used to detect anisotropy in the red shift in Refs. 9 and 15, where l and b are, respectively, the galactic longitude and latitude of the supernovae for a binary code ϑ = ϑ000ϑ100ϑ010…ϑmij…ϑKKK of the structure. The optimum with respect to the criterion of a minimum AMIE turned out to be the MMKMNK model

$$ {D}_L\left(l,b,z\right)=4930.4962z+2819.7024{z}^2+9.9955969 bz-12.664675{lz}^2 $$
(4)

for an AMIE \( {\overline{\upvarepsilon}}_{\upvartheta}^{\left[2\right]} \) = 247.42842 Mpc (here and in the following the computed values of the quantities are given in protocol form, i.e., without rounding), or after reduction to the analytic form (1):

$$ {D}_L\left(l,b,z\right)=\left(c/{H}_0\right)\left[\left(1+{a}_bb\right)z+\left(1-{q}_0+{a}_ll\right){z}^2/2\right], $$

where ab = 2.027311498·10–3b and al = –5.137310258·10–3l are the anisotropy coefficients, H0 = 60.80404234 km/s/Mpc, and q0 = –0.14378664.

Three samples of N SN Ia with N = 37 {0.008 ≤ z ≤ 0.97} [4], N = 42 {0.172 ≤ z ≤ 0.830} [10], and N = 41 {0.216 ≤ ≤ z ≤ 1.755} [2, 3] were used to analyze the “cosmic jerk” hypothesis. They are described by a common model (3) of the form

$$ {D}_L(z)=5581.7251z-4.1350279 lz+1697.3002{z}^2 $$

with \( {\overline{\upvarepsilon}}_{\upvartheta}^{\left[2\right]} \) = 519.40485 Mpc. However, according to the results of hypothesis testing of degeneracy H0, continuity H00, and compositional uniformity H000 with respect to the criterion of minimal AMIE according to R 50.2.004-2000 for the Friedman–Robertson–Walker model in the approximation of series (2) for M ≤ 6 (Tables 2 and 3, where the boldface entries indicate the AMIE of the best models of form (2) and the underlined boldface, the AMIE of the competing models) do not from a compositionally uniform sample, and for each sample the degeneracy hypothesis is rejected. And the question of combining the data of 1998–1999 and 2003–2004 drops out.

Table 2 Tests of the Hypotheses H0 and H000 Based on the Data of Refs. 2,3,4 and 10 for the Algorithm MMKMNK
Full size table
Table 3 Tests of the Hypotheses H0 and H000 Based on the Data of Refs. 2,3,4 and 10 for the Algorithm MMKMNM
Full size table

It is hard to explain why, in the model for the cosmological distance scale, the zero-point parameter θ0 ≠ 0, but it is this property of the optimal MMK-models according to the data of Refs. 2 and 3 for AMIE > 800 Mpc (see Tables 2 and 3) which has become the reason for doubt in the “extraordinary” evidence. But when testing the hypothesis of continuity of the MMK-models for the data of Refs. 2,3,4 and 10, we examine the “cosmic jerk” hypothesis as an alternative position characteristic corresponding to the “discrepancy.”

The analysis showed that the MMKMEDS-model for the data of Ref. 4 for AMIE = 113.4913127 Mpc and the MMKMNK-model for the data of Ref. 10 for AMIE = 291.5111476 Mpc with “discrepancies” at the predicted points are the most likely in terms of the criterion of minimum AMIE. In Table 4 these points are distinguished by bold type with underlining and the model for the data of Refs. 2,3,4, 10, and for z = 0.008–1.755 is rejected – its AMIE = 497.1471932 Mpc. Since the models for the position characteristics of the scale with a structure code of 01000 correspond to the Hubble law, while the MMK-models with a structure code of 011100 for a third order model (1) are not among the optimum models, model (1) turns out to be extra.

Table 4 Test of the Hypotheses H0, H00, and H000 Based on the Data of Refs. 2,3,4 and 10
Full size table

For the data of Ref. 10 with 0.763 ≤ z ≤ 0.828, there is a “discrepancy” and a transition DL(z) = 4948.032z → DL = = 3533.886 Mpc. The point at which the “discrepancy” begins, DL(0.763) = 3854.783577 Mpc, actually corresponds to the supernova SN 1997G, but the supernovae 1996ci (z = 0.828; DL = 3801.893963 Mpc) and 1997ap (z = 0.830; DL = 3265.878322 Mpc) show up at a smaller photometric distance (Fig. 1). Here the red shift of SN 1996ab (DLmax = 4168.693835 Mpc) is z = 0.592.

Fig. 1
figure 1

“Discrepancies” in the MMKMNK (MMCMNA) model of the scale for the data of Ref. 10 with z = 0.763–0.828.

Full size image

There are a number of “discrepancies” for the data of Ref. 4 and one of them, in the red shift interval [0.44, 0.48] fits entirely in the estimate z = 0.46 ± 0.13 [2] for the “cosmic jerk.” This corresponds to a transition DL(z) = 5800.16z → DL(z) = = 6244.237z, or, in reverse time, to a transition H0 = 48.01106332 km/s/Mpc → H0 = 51.68692829 km/s/Mpc (Fig. 2). With removal of the inversion the “discrepancy” vanishes and SN 1997ck with z = 0.97 becomes the outburst justifying the suspicion of the authors of Ref. 4. The “discrepancies” in the MMK-models for the scale (Fig. 3) separate SN 2003ak (z = 1.551; DL = 10327.61406 Mpc) and SN 1997ff (z = 1.755; DL = 11748.97555 Mpc) which are found in opposite directions. In addition, the “piecewise-Hubble” model with code ϑ = 01000 and a parameter H0 = {63.28883891 → 42.36803086} km/s/Mpc (see Table 4) may become the explanation, because as the boundaries of the observed parts of the Universe expanded during the twentieth century, the estimates for the Hubble “constant” decreased by more than an order of magnitude.

Fig. 2
figure 2

“Discrepancies” in the MMKMEDS (MCODS) model for the data of Ref. 4 with z = 0.44–0.48.

Full size image
Fig. 3
figure 3

The program MMK-stat: MMKMEDS (MCODS) (a) and MMKMNK (MCMSC) (b) of the model for the scale.

Full size image

Conclusion. The “discrepancies” in the model of the cosmological distance scale for red shifts of z = 0.44–0.48 and 0.763 ≤ z ≤ 0.828 according to the data of Ref. 10 do not conflict at once with the two hypotheses of a “cosmic jerk” for z = = 0.46 ± 0.13 [2] and z = 0.763 [4]. However, the very fact of the existence of these “discrepancies” and the inconsistency of these hypotheses has a very much simpler explanation – the cosmological distance scale based on red shift does not have a metric status [9], while the ordered set of red shifts corresponds to a partially ordered set of photometric distances. This failure of the condition of monotonicity for the cosmological distance scale based on red shift also raises doubts about the “cosmic jerk” hypothesis.

In addition, the anisotropy of the red shift and the solution of the calibration problem by the method of weighted least squares without using a scheme of crossover observations of the inadequacy error, which is more effective in these problems than the more complicated algorithms of confluent analysis, contributes to the perception of this situation.

Nevertheless, the “discrepancies” in the model of the cosmological distance scale based on red shift have reduced the statistically significant differences in the estimates of the Hubble parameter that led to an “impasse” [6,7,8, 11, 12] which most likely may not even exist.