_{1}

Based on an analysis of 280 Type SNIa supernovae and gamma-ray bursts redshifts in the range of z = 0.0104 - 8.1 the Hubble diagram is shown to follow a strictly exponential slope predicting an exponentially expanding or static universe. At redshifts > 2 - 3 ΛCDM models show a poor agreement with the observed data. Based on the results presented in this paper, the Hubble diagram test does not necessarily support the idea of expansion according to the big-bang concordance model.

Based on an analysis of 280 Type SNIa supernovae and gamma ray bursts redshift (z) data Marosi [^{0.059883}. Recently, a number of papers have appeared proposing different analytical formulae for describing the experimental z/μ relationship: Sorrell [

The z/μ data set consists of 171 gold-set data, Riess et al. [

The following mathematical functions and cosmological models were used to perform a global fitting over the RS range of z = 0.0104 - 8.1.

ΛCDM model with

ΛCDM model with

The μ values on basis of the ΛCDM models were calculated using:

where D_{L} (Pc) is the luminosity distance. The luminosity distances were calculated using the cosmological calculator described by Wright [

For preparing the linear t_{S}/z Hubble diagram, using Equation (7) the fitted z/μ data were converted into the corresponding t_{S}/z datasets.

The photon flight-time t_{S}^{ }was calculated using:

In Equation (7) t_{S} means the flight time of the photons (sec.) from the co-moving radial distance D_{C} to the observer, t_{S} = D_{C}/c, which is proportional to the D_{C} (Pc) that goes into the linear Hubble law.

In order to complete the fitted z/μ data set in the high RS range of t_{S} × 10^{−14} = 6000 - 11000, in addition to the measured RSs, using Equation (2), 41 equidistant t_{S}/z data points were included into the Hubble diagram. The addition of the 41 additional data points is necessary to perform the ∑χ^{2}-test between the best fit and the ΛCDM models because only few observations are available in this redshift range. The t_{S}/RS values were calculated on the basis of Functions (2) and (5).

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

It is easy to see by visual examination of _{0} = 72.6 km s^{−1 }Mpc^{−1 }(bottom line in

The most reliable measure to quantify the differences between the individual fit curves turned out to be the ∑χ^{2}-test. The goodness of fit indicators are shown in

The ∑χ^{2}-test obviously favors the trend-lines obtained with Functions (1) and (2) and these two curves are practically congruent. On basis of the data presented in

Function (F) | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

∑χ^{2 } μ_{obs}-μ_{F}_{(1 - 5)} | 1.958865 | 1.96555 | 2.38869 | 2.163105 | 2.1582 |

∑χ^{2} F2-F(1; 3 - 5) % | −0.34 | − | 17.78 | 9.57 | 8.9 |

The most important result of the Hubble diagram test is that fitting the t_{S}/z data with Function (2) and H_{0} = 62.5 km s^{−1} Mpc^{−1 }(dashed line in

as illustrated in

In spite of numerous correction factors and unknown constituents, dark matter (DM) and dark energy (DE), the ΛCDM models show a poor agreement with the observed data: as shown in _{0} = 62.5 km s^{−1} Mpc^{−1} departs from the best-fit curve for z + 1 < 6.5 to the bottom, for z + 1 > 6.5 to the upper side of the trend-line. The deviations are of a systematic (nonstatistical) nature and, therefore, the model cannot reflect the observational exponential slope.

In the range of z > 3 the ΛCDM model with H_{0} = 72.6 km s^{−1 }Mpc^{−1} shows a sharp increase in slope and departs considerably from the observed exponential function. For performing the ∑χ^{2}-test in the high RS range of t_{S} × 10^{−14}_{ }= 6000 - 11000 (_{S}/z data points were included into the Hubble diagram. The ∑χ^{2}-test leads to a statistical significance between the observed t_{S}/μ and the calculated ΛCDM data of P = 0.053, indicating that from the statistical point of view, the two models are essentially different.

The results presented in this paper have demonstrated that the ΛCDM-model cannot fit the strictly exponential slope of the Hubble diagram in the entire RS range of z = 0.0104 - 8.1, showing that the underlying theory is, at best, incomplete. A reconsideration of the ΛCDM-model appears warranted.

The Hubble diagram test leads to the significant conclusion that either: (1) the universe expanded exponentially during the whole time of its expansion history (at least in the range of z = 0.0104 - 8.1); or (2) the universe is static and the RS of spectral lines is caused by some as-yet unidentified mechanism. However, both of these models, the exponentially expanding and the static universe models have their own crucial problems; the discussion of them is not within the scope of this paper.

Laszlo A. Marosi, (2016) Modelling and Analysis of the Hubble Diagram of 280 Type SNIa Supernovae and Gamma Ray Bursts Redshifts with Analytical and Empirical Redshift/Magnitude Functions. International Journal of Astronomy and Astrophysics,06,272-275. doi: 10.4236/ijaa.2016.63022