_{1}

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It is generally accepted that the history of the expansion of the universe can be exactly described by the concordance model, which makes specific predictions about the shape of the Hubble diagram. The redshift-magnitude Hubble diagram in the redshift range
*z *= 0.0104 - 1 seems to confirm this expectation, and it is believed that this conformity is also valid in the high redshift range. However, this belief is not undisputed. Recent work in the high redshift range of up to
*z* = 8.1 has shown that the shape of the Hubble diagram deviates considerably from the predictions made by the Lambda cold dark matter model. These analyses, however, were based on mixed SN1a and gamma ray burst data, and some astronomers argue that this may have biased the results. In this paper, 109 cosmology-independent, calibrated gamma ray burst
*z*/
μ data points are used to calculate the Hubble diagram in the range
*z* = 0.034 to
*z* = 8.1. The outcome of this analysis confirms prior results: contrary to expectations, the shape of the Hubble diagram turns out to be exponential, and this is difficult to explain within the framework of the standard model. The cosmological implications of this unexpected result are discussed.

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) [

Besides GR, the Hubble constant (H_{0}) [

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 [

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 [

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 H_{0}D/c (expanding models) or the exponential z + 1 = e H 0 ∗ t (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 [

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 [

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 [

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

μ = a ∗ z b , (1)

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 [

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.

The well-known Grubbs test [_{0}, which is defined as:

X 0 ≥ G G ∗ STABW + x Mean N N − 1 (2)

where:

x_{0} is the suspected outlier;

x_{Mean} 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

G_{G} is the Grubbs number. G_{G} 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 x_{0} calculated from (μ_{measured} - μ_{calculated}) 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.

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

t = D c = 10 ( μ + 5 ) / 5 ( z + 1 ) ∗ 3 ∗ 10 10 ∗ 3.085 ∗ 10 18 (3)

Raw data | Valid cases | a | b | R^{2} | STABW μ_{obs} - μ_{calc} | Skew μ_{obs} - μ_{calc} |
---|---|---|---|---|---|---|

109 | 103 | 44.049 | 0.0595 | 0.9066 | 0.7384 | 0.4221 |

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.

The Hubble diagram as originally presented by Hubble [

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

Luminosity distances were calculated using the cosmological calculator described by Wright [

Excel and Excel Solver were used for the data fitting, refinement, analysis and data presentation.

The results of the fit procedure based on 103 raw data points are shown in

As can be seen from ^{2} = 0.9066.

A representative result of the iterative refinement process is shown in

Valid cases | a | b | R^{2} | Variance (μ_{obs} - μ_{calc}) | ∑χ^{2 } (μ_{obs} - μ_{calc}) |
---|---|---|---|---|---|

90 | 43.999 | 0.0592 | 0.9557 | 0.2538 | 0.4920 |

Mean (μ_{obs} - μ_{calc}) | Std. Deviation (μ_{obs} - μ_{calc}) | Error in Std. D. | Skew (μ_{obs} - μ_{calc}) | F-Test μ_{obs}/μ_{calc} | P |

0.00073 | 0.5038 | 0.0531 | 0.0733 | 0.8281 | 1 |

Valid cases | H_{0} | R^{2} | Variance (z_{obs} - z_{calc}) | ∑χ^{2 } (z_{obs} - z_{calc}) | |
---|---|---|---|---|---|

90 | 0.0002147 (2.147 × 10^{−18} s^{−1}) | 0.99942 | 0.0032 (z_{obs} - z_{calc}) | 0.099 | |

Mean (z_{obs} - z_{calc}) | Std. Deviation (z_{obs} - z_{calc}) | Error in Std. D. | Skew (z_{obs} - z_{calc}) | F-Test z_{obs}/z_{calc} | P |

0.0374 | 0.0569 | 0.006 | 1.028 | 0.7575 | 1 |

Valid cases | a | b | R^{2} | Variance (μ_{obs} - μ_{calc}) | ∑χ^{2 } (μ_{obs} - μ_{calc}) |
---|---|---|---|---|---|

84 | 43.977 | 0.0589 | 0.9683 | 0.181 | 0.3287 |

Mean (μ_{obs} - μ_{calc}) | Std. Deviation (μ_{obs} - μ_{calc}) | Error in Std. D. | Skew (μ_{obs} - μ_{calc}) | F-Test μ_{obs}/μ_{calc} | P |

0.00158 | 0.4254 | 0.0464 | −0.01075 | 0.8867 | 1 |

Valid cases | H_{0} | R^{2} | Variance (z_{obs} - z_{calc}) | ∑χ^{2 } (z_{obs} - z_{calc}) | |
---|---|---|---|---|---|

84 | 0.0002209 (2.209 × 10^{−18} s^{−1}) | 0.9990 | 0.0018 | 0.04279 | |

Mean (z_{obs} - z_{calc}) | Std. Deviation (z_{obs} - z_{calc}) | Error in Std. D. | Skew (z_{obs} - z_{calc}) | F-Test (z_{obs}:z_{calc}) | P |

0.00035 | 0.4255 | 0.0046 | 1.1519 | 0.8536 | 1 |

Valid cases | a | b | R^{2} | Variance μ_{obs}:μ_{calc} | ∑χ^{2} μ_{obs}:μ_{calc} |
---|---|---|---|---|---|

80 | 43.976 | 0.0589 | 0.9732 | 0.1489 | 0.2593 |

Mean (μ_{obs} - μ_{calc}) | Std. Deviation (μ_{obs} - μ_{calc}) | Error in Std. D. | Skew (μ_{obs} - μ_{calc}) | F-Test Μ_{obs}:μ_{calc} | P |

0.001927 | 0.3859 | 0.0043 | −0.174 | 0.9074 | 1 |

Valid cases | H | R^{2} | Variance (z_{obs} - z_{calc}) | ∑χ^{2 } (z_{obs} - z_{calc}) | |
---|---|---|---|---|---|

80 | 0.0002208 (2.208 × 10^{−18}s^{−1}) | 0.999 | 0.00176 | 0.03997 | |

Mean (z_{obs} - z_{calc}) | Std. Dviation (z_{obs} - z_{calc}) | Error in Std. D. | Skew (z_{obs} - z_{calc}) | F-Test z_{obs}:z_{calc} | P |

0.0007 | 0.042 | 0.0047 | 1.449 | 0.8549 | 1 |

(a) HD calculated on the basis of the currently most accurate LJA z/μ data with best fit parameters H_{0} = 70 km×s^{−1}×Mpc^{−1}, Ω_{m} = 0.295, w = −1.104 [

As can be seen from ^{2} best fit, line b = 0.04279; ∑χ^{2} ΛCDM model, line a = 0.3671).

This result is in perfect agreement with earlier findings [

The HD diagram on basis of the ΛCDM model with H_{0} = 62.5 km×s^{−1}×Mpc^{−1} (line b in

For H_{0} = 72.6 km×s^{−1}×Mpc^{−1} (line a in ^{2} 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

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 [_{0}. 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” [

The author declares no conflicts of interest regarding the publication of this paper.

Marosi, L.A. (2019) Extended Hubble Diagram on the Basis of Gamma Ray Bursts Including the High Redshift Range of z = 0.0331 - 8.1. International Journal of Astronomy and Astrophysics, 9, 1-11. https://doi.org/10.4236/ijaa.2019.91001