Hubble Diagram Test of Expanding and Static Cosmological Models: The Case for a Slowly Expanding Flat Universe

Abstract

We present a new redshift (RS) versus photon travel time (𝑡_𝑆) test including 171 supernovae RS data points. We extended the Hubble diagram to a range of z = 0,0141–8.1 in the hope that at high RSs, the fitting of the calculated RS/𝑡_𝑆 diagrams to the observed RS data would, as predicted by different cosmological models, set constraints on alternative cosmological models. The Lambda cold dark matter (ΛCDM), the static universe model, and the case for a slowly expanding flat universe (SEU) are considered. We show that on the basis of the Hubble diagram test, the static and the slowly expanding models are favored.

1. Introduction

The basic premise of the big bang theory is that the universe is expanding. The velocity of expansion can be expressed by the linear relationship 𝑣=𝑐𝑧=𝐻₀𝐷_𝐶, where 𝑐 is the velocity of light, 𝑧 is the redshift (RS), 𝐷_𝐶 is the comoving proper distance of the emitting object, and 𝐻₀ is the Hubble constant [1].

Since one cannot measure the supposed universal expansion experimentally, different tests based on observational data have been proposed [2–4] to provide evidence for the expansion hypothesis. (i) The Tolman surface brightness test [5], (ii) the time dilation test, (iii) the cosmic microwave background (CMB) temperature as a function of the RS test, (iv) the apparent magnitude versus distance test, and (v) the angular size versus RS test were proposed as possible observational evidence for the expanding space supposition. Recently, López-Corredoira [6] critically reviewed the results of these tests and concluded that on the basis of previous tests, convincing evidence for the cosmic expansion hypothesis is still lacking. Over this, as pointed out by LaVioletta [7], Crawford [8], and López-Corredoira [9], the static universe model fits the observational data better than expansion models. At present, there is no decisive experimental proof in favor of or against the supposed universal expansion. Recently, an early idea for testing the universal expansion, proposed by Sandage 1962 [10], by measuring the change of RS of galaxies owned to the deceleration rate of the expansion is gaining increasing interest (Steinmetz et al. [11]). The test, however, requires measurements of Doppler velocity drifts with an accuracy of ~1 cm s⁻¹ yr⁻¹ which is not feasible at the present time, but new techniques for frequency measurements developed by Loeb [12] and Murphy et al. [13], for example, promise an effective means to prove or improve the  expansion hypothesis in future experiments.

In this paper, we compare the observed Hubble diagram compiled from 171 supernovae RS data in the range of 𝑧 = 0.0141–8.1 with theoretical Hubble diagrams calculated on the basis of the Lambda cold dark matter (ΛCDM) model, the static universe model, and  the slowly expanding flat universe (SEU) model that expands according to (5) with Ω_𝑀=1. We expect that in the high RS range, it should be possible to check more precisely whether the Hubble diagram follows a linear  𝑧=𝐻₀𝐷_𝐶/𝑐 or the exponential 𝑧=𝑒^{𝐻0(𝑑/𝑐)}−1=𝑒^{𝐻0𝑡𝑆}−1 (1)   relationship, an effect which is only slightly perceptible in the 𝑧<1 region.

2. Data Collection

In our analysis, we have included data obtained by Wei [14] from 59 calibrated high-RS gamma-ray bursts (GRBs) (Hymnium data set) from the 557 Union 2 compilation and 50 low RS GRBs and 62 gold-set data [15]. Because the RS/magnitude data are plagued by considerable scatter, the best-fit curve was used to perform a global fitting over the RS range of  𝑧 = 0.0141–8.1.

2.1. Best-Fit Curves of the Observed RS/𝜇 Data

Best-fit curves were computed with Excel using the logarithmic  𝜇=𝑎×ln(𝑧)+𝑏 and the potential  𝜇=𝑎×𝑧^𝑏functions. Fitting the data with exponential and polynomial functions was not successful. The goodness of the fits was characterized by the corresponding 𝑅-square and 𝜒-square values.

Results are shown in Figure 1.

Figure 1 

RS/𝜇 data, fitted with the logarithmic 𝜇=2.4761×ln(𝑧)+44.198 (dashed line) and the potential function 𝜇=44,102×𝑧0.0593(solid line).

The best-fit curves are represented by   (𝑧)𝑅𝜇=2.4761±0.06×ln+44.198±0.079,₂=0.9479,¹⁷¹₁𝜒²_𝑖=1.892,𝜇=44.102±0.032×𝑧^{0.0593±0.00066},𝑅²=0.9571,¹⁷¹₁𝜒²_𝑖=1.6923. (2)(3)

Although the potential fit shows a somewhat better regression coefficient 𝑅² and a slightly better ∑𝜒² on the basis of these small differences between the goodness of fit indicators, neither of the two functions can be excluded from further consideration. The differences between the two trend lines become more pronounced only in the RS/𝑡_𝑆 data representation.

The photon flight time 𝑡_𝑆 was calculated from 𝑡_𝑆=𝐷_𝐶𝑐=10^(𝜇+5)/5(𝑧+1)×3×10¹⁰×3.085×10¹⁸. (4)

  Using (2), (3), and (4), the RS/μ data were converted into RS/𝑡_𝑆 data sets.

Results are shown as RS/𝑡_𝑆×10⁻¹⁴ data.

The calculated 𝜇/𝑡_𝑆 data are summarized in Table 1.

Table 1 

RS/𝑡_𝑆 data calculated with (2), (3), and (4).

Precision in determination of 𝐷_𝐶(𝑡_𝑆) is given by the error limits which are shown in (2) and (3). Figure 5 shows a graphical illustration of measurement error for the upper and lower limits of 𝑡_𝑆 inferred for the potential best fit function. The interpretation of this diagram for constraining the different cosmological models is discussed in detail in Section 3.2.

In Table 1, 𝑡_𝑆 means the flight time of the photons from distance 𝐷_𝐶 to the observer that should not be confused with the photon travel time (𝑡) in a universe expanding according to the ΛCDM model. The ΛCDM model with 𝐻₀=72.6 km s⁻¹ Mpc⁻¹, Ω_𝑀=0.266, Ω_Λ=0.732, and 𝑘=0, for example, leads to a mathematically correct solution for the evolution of an expanding universe; the universe has expanded from its beginning to the present time to an extent of about 𝐷_𝐶=46 billion light years. However, the ΛCDM model, though cosmologically important, has by itself no physical basis and requires a choice of free parameters (DM and DE) in order to fit the underlying expansion theory to the observed RS/d data.

We consider the case that although the result obtained on the basis of the ΛCDM model is a mathematically correct description of the present distances in an expanding universe, nevertheless, the expanding space interpretation is fictitious, and the universe may be static, or slowly expanding, and infinite, or very large, in extent. In this case, the ΛCDM model represents only a mathematical fit to the observed RS data, but in reality, the calculated distances represent nonexpanding distances between the observer and the emitting objects in a static universe. On this assumption, the flight time of photons between emission and reception is 𝑡_𝑆=𝐷_𝐶/𝑐, which is proportional to the 𝐷_𝐶that goes into the linear Hubble law.

3. Results

The performance of the three model cosmologies, the ΛCDM, tired light, and SEU models, is compared with the observational RS/𝑡_𝑆 data base, each of which makes different predictions about the relation between 𝑡_𝑆(𝐷_𝐶/𝑐) and 𝑧.

3.1. Evaluation of the Logarithmic Best-Fit Results

In Figure 2, the RS/𝑡_𝑆 data calculated on the basis of the ΛCDM model with  𝐻₀=77.25 km s⁻¹ Mpc⁻¹ and the tired light model with 𝐻₀=2.35×10⁻¹⁸ Hz s⁻¹ Hz⁻¹ are compared with the logarithmic fit data.

Figure 2 

Logarithmic best-fit curve (solid line), ΛCDM with 𝐻0=77.25 km s−1 Mpc−1 (dashed line), and tired light model with 𝐻0=2.35×10−18 Hz s−1 Hz−1 (dot-dashed line).

As can be seen from Figure 2, the calculated data depart significantly from the best-fit curve, for 𝑧<3 to the left and for 𝑧>3 to the right side of the solid line. Over this, the value of the Hubble constant of ~77.25 km s⁻¹ Mpc⁻¹ is, compared with novel RS data, too high and does not fit the gold-set RS data which is the most accurate RS compilation presently known (Figure 3).

Figure 3 

Triangles: gold-set data; solid line: 𝐻0=77.25 km s−1 Mpc−1; dashed line: 𝐻0=66 km s−1 Mpc−1.

3.2. Evaluation of the Potential Best-Fit Results

Results are shown in Figures 4 and 5.

Figure 4 

Redshift of Type Ia supernovae as a function of 𝑡𝑆=𝐷𝐶/𝑐. Solid line: RS/𝑡𝑆 relation inferred from the potential best-fit curve of the RS/𝜇 diagram. Diamonds: the potential relation with 𝐻0=2.05×10−18 Hz s−1 Hz−1. Crosses: RS for the SEU with 𝐻0=2.017 Hz s−1 Hz−1 + RS resulting from expansion with 5 km s−1 Mpc−1. Dashed line: RS/𝑡𝑆 relation derived from the distances 𝐷𝐶 obtained from the ΛCDM model with𝐻0=66 km s−1 Mpc−1.

Figure 5 

Potential best-fit function (solid line), upper and lower limits due to measurement error (dashed lines), and  ΛCDM model (diamonds).

The most impressive result of our test is that the RS/𝑡_𝑆relation obtained from the potential best-fit data (solid line) can be expressed nearly exactly by the exponential formula 𝑧=𝑒^{2.05×10−18𝑡𝑆}−1 (diamonds) over the whole range of 𝑧 with a statistical significance of 𝑃_𝜒-square=0.8347. Also, the slowly expanding universe model (crosses) with  𝐻₀=2.017×10⁻¹⁸Hz s⁻¹ Hz⁻¹ plus the RS resulting from the expansion with 5 km s⁻¹ Mpc⁻¹  shows a similarly good fit with 𝑃_𝜒-square=0.4734, and consequently the two models become nearly identical.

The dashed line in Figure 4 stands for the RS/𝑡_𝑆 relation, derived from the distances  𝐷_𝐶 obtained from the ΛCDM model with 𝐻₀=66 km s⁻¹ Mpc⁻¹, Ω_𝑀=0.266, Ω_Λ=0.732, and 𝑘=0 [16].

Figure 5 shows the error limits of the potential trend line together with the calculated RS data of the ΛCDM model.

One can see from Figure 5 that the RSs calculated on the basis of the ΛCDM model (diamonds) show a poor agreement with the observed data. Many data points lie outside of the error boundaries and show a systematic (nonstatistical) deviation from the best-fit line. The 𝜒-square test leads to a statistical significance between the observational potential fit and the calculated ΛCDM data of  𝑃=0. On the basis of this analysis, the ΛCDM model can be omitted.

The Hubble diagram clearly follows the exponential tired light RS formula as expected for static and for slowly expanding universe models.

4. The Slowly Expanding Universe

When GR is applied to the universe as a whole, it turns out that space should either be contracting or expanding. On the assumption that expansion is governed by the gravitational attraction of the observable mass density 𝜌_𝑀,obs. against the outward impulse of motion, the solution of the field equation  (𝑘=0; Λ=0) leads to 𝑅̇𝑅²=8𝜋𝐺3𝜌_𝑀,obs.. (5)

Surprisingly, this simple model that naturally follows from the Einstein equation and from the observable mass density was not considered in the literature as a possible alternative to the static and the big bang models at all, although when it is compared with the ΛCDM and static universe models, its advantages is obvious. For example, consider the following.(i)Inferring the Hubble constant from the observable mass density of some 10⁻³¹ g cm⁻³ (instead of ~10⁻²⁹ g cm⁻³ as calculated on the basis of the still controversial interpretation of RS as recession velocity) leads to 𝐻₀=(𝜌_obs.×8𝜋𝐺)/3 ~ 5–15 km s¹ Mpc⁻¹, and the missing mass problem on the cosmic scale does not arise.(ii)Inferring the age of the universe from 1/𝐻₀ with 𝐻₀∼10 km s⁻¹ Mpc⁻¹ yields ~100 billion years, and with this, galaxies may have been evolved over time in some regular way without the need for DM or DE.(iii)In comparison with the ΛCDM and static universe models,  the SEU model is simple and follows from GR without additional free parameters like the elusive dark components DM and DE or, in case of a static universe, the cosmological constant Λ of unknown physical nature. (iv)SEU is exclusively based on observable quantities, that is 𝜌_𝑀,obs., the experimentally measured RS distance relation, and the experimentally confirmed Euclidean geometry of the universe, instead of introducing unknown particles and energy.

5. The Redshift Problem

The interpretation of RS of atomic spectral lines suffers from the fact that none of the proposed mechanisms—the expanding space paradigm or the different tired light mechanisms—can be verified experimentally. Thus, the confidence in each theory must be measured by its success in explaining the observational phenomena. In Section 3, we have shown that observed RS date agree well with data inferred from (1), characteristic of tired light energy depletion.

According to SEU, the RS of the spectral lines is composed of a velocity 𝐻_0,expansion  and a superimposed RS 𝐻_{0,tired-light}  component of as yet unknown origin, and the rate of expansion is much lower than suggested by the accepted value of the Hubble constant of 𝐻₀=72.6 km s⁻¹ Mpc⁻¹. The universe expands according to the Friedmann equation with 𝜌_𝑀,obs.≈10⁻³¹ g cm⁻³, Λ=0, and 𝑘=0.

Our results support the assumption that the major part of RS is due to some tired light mechanism. According to this interpretation,  𝐻₀  does not represent the velocity of expansion, km s⁻¹ Mpc⁻¹, but it is as an additional term for the rate of energy loss from starlight due to some tired light mechanism which has the dimension Hz s⁻¹ Hz⁻¹.

The tired light scenario assumes that a photon loses energy due to some unknown process when it travels through space. The idea was first suggested by Zwicky [17] as a phenomenological approach to explain the Hubble relation. The major problem with this theory is the identification of a convincing physical process responsible for the observed loss of energy. There is no known interaction that can degrade a photon’s energy without also changing its momentum, which would lead to a blurring of distant objects which is not observed. With these objections, the tired light RS theory was discarded.

Since that time, however, several mechanisms have been proposed that avoid blurring and scattering [18–20]. A reconsideration of the tired light model in context of these theories appears to be warranted.

Therefore, although the explicit physical nature of the energy loss of photons in transit is still a matter of debate, this lack of knowledge cannot be used as a decisive argument against the tired light theories. The cosmologically more important issue was to decide whether the Hubble diagrams follow a linear or exponential relationship. The presented RS/𝑡_𝑆 test clearly favors the static and the SEU models.

6. Conclusions

We have discussed here the Hubble diagram test for expanding and static universe models and show that the exponential energy depletion  𝑧=𝑒^{𝐻0𝑡𝑆}−1, characteristic for static and also for slowly expanding universe models, fits the experimental data with high accuracy, while the RSs calculated on the basis of the ΛCDM model are in poor agreement with observation.

We have shown that the slowly expanding universe model unifies the advantages of expanding and static models by avoiding their problems, namely, the DM, DE problem of the ΛCDM model, and, in case of the static universe, the problem of the cosmological constant and gravitational instability.

On the basis of the results presented in this paper, we feel that the SEU model represents a promising alternative to the ΛCDM and static universe models that possibly could be verified by studying the effect of the lower expansion rate on the big bang nucleosynthesis (BBN). It is known that the relic abundance of the light elements depends on the expansion rate which determines the temperature and nucleon density in the early universe which in turn has a strong influence on the elements abundance pattern. As a consequence, BBN could set further constraints on the discussed cosmological models.

Acknowledgments

The author is grateful to Professor Rainer Mattes of the Westfälische Wilhelms-Universität, Münster, Germany, for his support and his continuous interest in this work. He would also like to thank Dr. Bernd Hinrichsen of the BASF AG, Ludwigshafen, for his help with statistical calculations.