The basic premise of Big Bang cosmology is that the universe is expanding. Important evidence for this expansion is that it follows from general relativity (GR)  , which has been successfully tested both in the solar system and on the cosmic scale.
Besides GR, the Hubble constant (H0)  is probably the most fundamental cosmological parameter. It is considered to be the most convincing evidence for universal expansion, and the universe expands with a velocity determined by the Hubble constant.
At the same time, however, we have to keep in mind that neither GR nor Hubble’s constant is a real proof for expansion. GR when applied to the universe as a whole represents only a theoretical framework and allows the construction of numerous basically different cosmological models such as Einstein’s static universe  , the Einstein-deSitter model  and the dynamic, expanding or contracting, universe of Friedmann  , for example. All these models are mathematically correct and none of them is preferred by GR. The presently prevailing inflationary ΛCDM model seems to confirm expansion and provides an excellent fit to most cosmological observations. However the price is high. The model rests on a large number of hypotheses, dark matter, dark energy, inflation, for example, which can either not be proved experimentally, or though theoretically provable, could not be proven, yet. Recently, a number of papers appeared in peer reviewed journals proposing new models without DM or DE or both    and without inflation  .
And even the interpretation of Hubble’s constant as recession velocity is hypothetical. Hubble never measured velocity; the expansion of the universe cannot be measured experimentally. The Hubble Law is a RS/distance relation, and the Hubble recession law is in reality a working hypothesis.
The question must be asked: how sure can we be that the universe really expands with velocity of the Hubble constant? Different tests based on observational data have been proposed to provide evidence for the expansion hypothesis. A critical review of these tests shows that convincing evidence for the universal expansion is still lacking   . The static universe model fits the observational data better than expansion models    .
A promising tool to confirm expansion is the Hubble diagram test. We expect that in the high RS range it should be possible to check more precisely whether the Hubble diagram follows the linear H0D/c (expanding models) or the exponential (tired light) relation, an effect that is perceptible only slightly in the z < 1 region. The Hubble diagram (HD), calculated on the basis of a SN1a supernovae redshift (RS, z)/magnitude (μ) data, gives an excellent fit to the predictions of the concordance model    . However, a hint of uncertainty remains. The fit of the Lambda cold dark matter model (ΛCDM) to the SN1a z/μ data applies only within the narrow range of z = 0 - 1. One reason for this is experimental difficulties; at RSs > ~1.3, the optical light emitted by supernovae becomes increasingly dimmed with distance, probably due to gray dust extinction, and accurate observations become difficult. However, cosmologists are not concerned with this RS limitation and consider the RS range of z > ~1 - 8 to be uninteresting. At higher RSs, the universe is matter-dominated, and the lambda term in the high RS region does not affect the universal expansion. It is assumed that the HD at high RS would merely confirm the concordance cosmology  . In recent years, a number of papers appearing in peer-reviewed journals have drawn the conclusion that the shape of the HD is exponential over the entire RS range of 0.0104 - 8.1, in clear contrast to the specific predictions of the ΛCDM model    . These analyses were performed with mixed SN1a und gamma ray burst (GRB) data, and some astronomers argue that this may have biased the results.
The best way to confirm or disprove the exponential shape of the HD is to use exclusively GRB data to calculate the HD over the whole RS range of z = 0.034 - 8.1. In previous papers, several attempts have been made to utilize GRB data to calculate the HD      with varying degrees of success. The limited number of data points and the large scatter of the data do not allow for safe conclusions. However, thanks to Swift (The Neil Gehrels Swift Observatory, NASA), the main features of GRBs have become better known in recent years. On the basis of more accurate observations of GRBs, the expectation has arisen that these objects could prove to be suitable distance indicators. Many hundreds of bursts have been observed in the range 0.034 < z ≤ 8, which opens up the possibility of measuring the expansion history back to the formation of the first stars. It is expected that the maximum clearly observable RS could approach 10 or even larger. This could place significant constraints on the different models of universal expansion.
The aim of the present work is to perform an improved HD test based on a larger number of calibrated GRB RS/μ data points. The reliability of these data was verified using statistical tests before the analysis was carried out.
A total of 109 calibrated, cosmology independent GRB z/μ data points collected by Wei  from the 557 Union2 compilations were used as the starting data set. From these 109 data points, three low RS data points 050416A (μ = 41.44 ± 1.2), 080319B (μ = 43.07 ± 1.24), 061121 (μ = 46.18 ± 1.01), ( = ± 0.41; average error in μ for 50 low RS GRB data points) and three high RS data points 040912 (μ =43.27 ± 2.06), 09120 (μ = 47.58 ± 1.96), and 080913 (μ = 50.45 ± 1.74), ( = ±1.55; average error in μ for 59 high RS GRB data points) were excluded from the following refinement process due to their unusually large error bars, which indicate observational difficulties in terms of magnitude determination.
For the remaining 103 data points, best fit curves were calculated, which are more accurate than those used in any previous work, using the empirical potential function
with a = 44.1097 and b = 0.05988, which was determined in earlier publications for SN1a gold set and for GRB data to be the best mathematical approximation for describing the slope of the z/μ diagram    .
2.1. Elimination of Outliers
In view of the experimental difficulties in determining the z/μ data, it is likely that large data sets taken from different observations and from different sources will contain outliers. If these outliers are not removed from the refinement procedure, they will dominate the fit and bias the results.
2.2. Identification of Outliers: The Grubbs Test
The well-known Grubbs test  was used for the identification of erroneous luminosity indicators. The Grubbs test is used to detect outliers in a data set of N values that are nearly normally distributed. Assuming a normal distribution of the sample, as confirmed by the very low skew in Table 1, the test is performed by computing x0, which is defined as:
x0 is the suspected outlier;
xMean is the absolute value of the mean of the N data points;
N is the number of data points;
STABW is the standard deviation of N values; and
GG is the Grubbs number. GG can be found in statistical tables for different levels of confidence and numbers of data points. For 103 data points, for example, G is 1.956 at 95% confidence level.
If the x0 calculated from (μmeasured - μcalculated) is found to be greater than the numerical value of the right-hand side of Equation (1), the data point in question must be discarded; on the basis of the reduced data set, new a and b coefficients, the mean and the new STABW must be calculated, and so on.
2.3. Preparation of the Hubble Diagram
The Hubble diagram is a linear plot of the measured distance (usually Mpc) versus the measured RS, which is often represented on the less sensitive logarithmic μ/RS scale.
Since the differences between the measured and the calculated trend lines become more pronounced on the linear scale, a plot of the photon flight time (t) versus RS was used for representation of the HD. The photon flight time was calculated from
Table 1. Fit results from the starting data set. Descriptive statistics z/μ.
In Equation (2), t represents the flight time of the photons (in sec) from the co-moving radial distance D to the observer, which is proportional to D (Mpc) as used in the Hubble law.
2.4. Hubble Diagram Representation
The Hubble diagram as originally presented by Hubble  is a distance (Mpc)/velocity (v) representation where v = cz. Recently it becomes increasingly common to use the logarithmic z/µ diagram as presented in Figure 1 and Figure 2 instead of the linear distance/redshift scale.
The presentation t/(z + 1) as used in this paper is essentially equivalent to Hubble’s depiction, with Mpc = t × c (abscissa) and z = v/c (ordinate) and the two diagrams differ only in the scale of the axes. The advantage of the t/(z + 1) representation is, as can be seen in Figure 3 and Figure 4, is that the slope of the best fit line can be fitted exactly with the exponential function .
2.5. Luminosity Distances
Luminosity distances were calculated using the cosmological calculator described by Wright  .
2.6. Excel and Excel Solver
Excel and Excel Solver were used for the data fitting, refinement, analysis and data presentation.
As can be seen from Figure 1, the data are affected by considerable scatter, resulting in large variability of the data and a relatively poor goodness of fit indicator of R2 = 0.9066.
Figure 1. Best fit curve for the initial 103 GRB data points.
Figure 2. Best fit curve based on 84 statistically verified GRB z/μ data points.
Figure 3. Representative Hubble diagram based on 84 statistically verified GRB z/μ data points.
Figure 4. Redshift of type I a supernovae as a function of t = DC/c. Squares (dashed line): t/z data inferred from the potential best-fit curve of the observed z/μ diagram. Triangles: t/z relationship derived from the ΛCDM model with H0 = 72.6 km×s−1×Mpc−1, Circles: t/z relationship derived from the ΛCDM model with H0 = 62.5 km×s−1×Mpc−1 (data are taken from  ).
Table 2. Results of regression with μ = a × zb. Descriptive statistics z/μ; one iteration.
Table 3. Results of regression. Descriptive statistics t/z + 1; one iteration.
Table 4. Results of regression with μ = a × zb. Descriptive statistics z/μ; two iterations.
Table 5. Results of regression with t/z + 1. Descriptive statistics; two iterations.
Table 6. Results of regression with μ = a × zb. Descriptive statistics z/μ; three iteration.
Table 7. Results of regression with t/z + 1. Descriptive statistics; three iteration.
(a) HD calculated on the basis of the currently most accurate LJA z/μ data with best fit parameters H0 = 70 km×s−1×Mpc−1, Ωm = 0.295, w = −1.104  ; (b) measured best fit line. The results of the two iteration steps are shown in Table 5:
As can be seen from Figure 3, the shape of the HD is exponential or (more critically) very close to exponential, whilst the ΛCDM model shows systematic deviations from the exponential best fit curve (∑χ2 best fit, line b = 0.04279; ∑χ2 ΛCDM model, line a = 0.3671).
This result is in perfect agreement with earlier findings  that ΛCDM models show poor agreement with observation (Figure 4).
The HD diagram on basis of the ΛCDM model with H0 = 62.5 km×s−1×Mpc−1 (line b in Figure 3) deviates below the trendline of the best-fit curve for z + 1 < 6.5 to the bottom, and above it for z + 1 > 6.3. These deviations are of a non-statistical nature and thus the model does not reflect the observed exponential slope.
For H0 = 72.6 km×s−1×Mpc−1 (line a in Figure 4) the ΛCDM model departs considerably from the observed exponential function (line c), and in the range z > ~ 2, a sharp increase is shown in the slope. A ∑χ2 test shows a statistical significance between the observed t/μ and the calculated ΛCDM data of P = 0.053, and fails to describe the observed z/μ data completely.
Tables 2-7 show that after only two iteration steps, the further removal of outliers does not result in a substantial improvement in terms of either shape or goodness of fit indicators.
The results presented here show that the Hubble diagram t/(z + 1) calculated on the basis of GRB z/μ data follows a strictly exponential slope in the range 0.0331 < z < 8.1, in excellent agreement with observation. The exponential slope of the Hubble diagram provides a clear indication of an energy decrease in the emitted spectral lines with a constant rate. At RSs > ~2, the ΛCDM model does not fit the data well (dashed line in Figure 3). This unexpected result leads to a logical contradiction between theory and observation, which cannot be solved within the frame of the concordance model. We consider it certain that the universe (spacetime) expands, and the expanding space causes a RS in the photons that is proportional to the extent of expansion. The shape of the HD should follow the explicit predictions of the concordance model, which cannot be exponential.
The question arises of how to interpret these contradictory results in light of the expansion hypothesis. If we exclude the static universe model, the most radical answer explaining this disagreement would be that something is wrong with the basic assumptions of the underlying cosmological model. The results presented here require that the HD is completely determined by an energy decay process that is as yet unknown, which most cosmologists are not ready to accept, since this would require the most important evidence for universal expansion to be discarded.
It is not the aim of this paper to identify a specific new energy decay mechanism (although some promising alternatives have been proposed in the recent literature     to explain these contradictory results); however, it should be pointed out that the disagreement between the predictions of the concordance model and the strictly exponential slope of the HD is a real problem that requires an explanation. In view of this, the HD test may prove to be the most important cross check in determining the expansion history of the universe and the physical meaning of H0. Increasingly accurate high-RS GRB z/μ data may turn out to be the key to this important cosmological issue. There is hope this could be done in the near future.
“We are now at an interesting juncture in cosmology. With new methods and technology, the accuracy in measurement of the Hubble constant (from high RS GRB data) has vastly improved, but a recent tension has arisen that is signaling as-yet unrecognized uncertainties. The key pillar of the standard cosmological model becomes shaky”   .