Analysis of * D*versus expansion factor with 157 observations including the local group (1.0) for regression minima close to 71 km s

_{L}

^{−1}Mpc

^{−1}.

Open access peer-reviewed chapter

By Michael L. Smith and Ahmet M. Öztaș

Submitted: November 15th 2019Reviewed: January 20th 2020Published: February 20th 2020

DOI: 10.5772/intechopen.91266

Downloaded: 391

Astronomers continue the search for better a Hubble-Lemaitre constant, H0, value and other cosmological parameters describing our expanding Universe. One investigative school uses ‘standard candles’ to estimate distances correlated with galactic redshifts, which are then used to calculate H0 and other parameters. These distance values rely on measurements of Cepheid variable stars, supernovae types Ia and II or HII galaxies/giant extra-galactic HII regions (GEHR) or the tip of red giant star branch to establish a distance ‘ladder’. We describe some common pitfalls of employing log-transformed HII/GEHR and SNe Ia data rather than actual distances and suggest better analytical methods than those commonly used. We also show that results using HII and GEHR data are more meaningful when low quality data are discarded. We then test six important cosmological models using HII/GEHR data but produce no clear winner. Groups utilising gravitational waves and others measuring signals from the cosmic microwave background are now at odds, ‘tension’, with those using the SNe Ia and HII data over Hubble-Lemaitre constant values. We suggest a straightforward remedy for this tension.

- Hubble constant
- Hubble-Lemaitre constant
- cosmological parameters
- distance ladder
- distance scale
- data analysis
- supernova
- HII galaxies
- luminosity distance
- redshift

Some important goals of cosmology are determination of values for the local Hubble-Lemaitre constant, _{0}, the average Universe matter density, ρ, as well as confirmation, or not, of the cosmological constant, Λ. Important tools are information emanating from supernovae types Ia (SNe Ia) and II (SNe II) explosions, γ-ray bursts and redshifts, * z*, combined with distance determinations to closer Cepheid variable stars [1, 2]. Estimates are also made using data from single events; the cosmic microwave background (CMB) [3, 4] and gravitational waves combined with electromagnetic detection [5, 6]. Values for

Results from the CMB investigations depend on exacting measurements of tiny, low-energy fluctuations modelled with at least 6 parameters, demanding many ‘priors’ (fixed-valued parameters) and cannot realistically discriminate between models since there are many parameter combinations able to fit many models [3, 11]. Results from SNe Ia investigations are model-dependent, rely on 2 or 3 parameters as published, but in reality 4 or 5 para-meters are used for modelling and the belief that most SNe Ia events are uniform and similar. Systematic errors of collection and analysis are still being discovered, corrected or culled from data [12].

Estimates of SNe Ia distances typically rely on nearby Cepheid variable star distances which are still being adjusted [13, 14]. In addition, the methods used for evaluations of the SNe Ia data and claims therefrom have been repeatedly questioned [15, 16]. An independent method for estimating _{0} has recently been published based on the characteristics of selected red giant stars [17, 18]. The value found, 69.8 km s^{−1} Mpc^{−1} is close to the gravitational wave observation from a bi-neutron star collision, 70 km s^{−1} Mpc^{−1}, much lower than calculated with SNe Ia data [1].

A pioneering effort is being made using the L(Hβ)σ distance estimator for giant extra-galactic HII regions (GEHR) and HII galaxies back to z ≈ 2.3 by a small group [19, 20, 21, 22, 23]; significantly further than SNe Ia observations. Several assumptions and adjustments to the data are necessary to allow use as astronomical distances analogous to SNe Ia data as done by Wei et al. [24]. The latter group presented results using the HII distance magnitudes, * mag*, and redshifts collected by the former group to investigate the properties of three important cosmological models. Their results suggest the

A major problem with these analyses is the use of a relationship commonly termed as a * Hubble diagram*or a

There are many drawbacks using the pseudo-H-routine for model testing. First, distance is a physical metric but * mag*is not. Second, this routine non-uniformly compresses data dispersion and standard errors; errors of distant observations are systematically compressed over errors of more nearby emissions and will exacerbate skewness [33]. Using weighed regression analysis the pseudo-H-routine incorrectly emphasises the more imprecise, distant data, SNe or HII, which often leads to incorrect regression minima and results [34, 35]. Third, the best data pair, recession velocity and position of earth or the local group (1,0) without error, cannot be used with the pseudo-H-routine, the distance becoming −∞; this exclusion can never be justified. Fourth, because the errors have been compressed, goodness of fit estimates are not properly distributed, are both smaller with more similar values, complicating model discrimination. Fifth, information from both intercepts are lost and cannot be recovered. If the pseudo-H-routine were valid, parameter estimates should be similar between the Hubble relationship and the pseudo-H-routine, but are not. Here and for other examples using SNe Ia data, the two analytic methods do not agree [16, 36].

There are many reasons to perform regression analysis using luminosity distance versus expansion factor (* D*versus

Here we first use the same routine as Wei et al. (* mag*versus

Advertisement## 2. Data, models and methods

### 2.1 HII and GEHR data

D L = 10 mag − 25 / 5 E1### 2.2 Models

1 = Ω m + Ω Λ + Ω k E2#### 2.2.1 Eternal coasting, Rh = ct model

D L = c H 0 a ln 1 a E3mag = 5 log c 1 + z H 0 ln 1 + z + 25 E4#### 2.2.2 The current standard model of cosmology, ΛCDM

D L = c H 0 a ∫ 1 a da a Ω m a + 1 − Ω m a 2 E5mag = 5 log c 1 + z H 0 ∫ 0 z dz 1 + z 2 1 + Ω m z + 2 2 + z 1 − Ω m + 25 E6#### 2.2.3 The standard model allowing the equation of state (EoS), ωΛCDM

D L = c H 0 a ∫ 1 a da a Ω m a + a 2 1 − Ω m 3 1 + ω E7mag = 5 log c 1 + z H 0 ∫ 0 z dz 1 + z 2 1 + Ω m z + 2 2 + z 1 − Ω m 3 1 + ω + 25 E8#### 2.2.4 The Einstein-de Sitter model, EdS

D L = c a H 0 sinh ∫ a 1 da a Ω m a E9D L = c a H 0 sinh 1 − a E10#### 2.2.5 The cosmological general relativity model, CGR

D L = c a H 0 1 − β 2 × sinh β 1 − Ω β 1 − Ω β E11β = 1 a 2 − 1 1 a 2 + 1 E12#### 2.2.6 The FLRW model, ST, with the term, Ω_{k}, for spacetime

1 = Ω m + Ω k . E13D L = c H 0 a ∫ 1 a da a Ω m a + 1 − Ω m E14### 2.3 Methods

1 = ∑ 1 i ln 1 + y i − y i ̂ 2 E15

We use data; * mag*, standard deviations about

where * mag*is the distance modulus. We perform

For regression using * D*versus

To suppress the influence of outliers we parse the data in two manners. For both situations we only discard data simultaneously failing three models, ΛCDM, ωΛCDM and * Rh* = ct, at all three

The models tested are based on the Friedmann-Lemaitre-Robertson-Walker (FLRW) universe; explanations can be found in sources [39, 40, 41]. This is by far the most useful model of cosmology and an early version was used by Einstein and de Sitter to model the Universe, subsection 2.2.4. We make the usual assumption for FLRW model parameter normalisation

where Ω, Ω

This is the preferred model of Melia and coworkers [28, 42] with only one free parameter, * H*, presuming a geometrically flat universe and we use the relationship

with * a*the expansion factor, c is lightspeed and the natural ln. Eq. (2) may hold true testing this

Using Eq. (4) allows presentation of the pseudo-H-diagram with HII/GEHR data as in [24] where it is labelled ‘Hubble diagram’ though not really a Hubble-type diagram [32].

Called the standard model of cosmology by some, with two free parameters, after adjusting the data for the local value of _{0} as

where (1 − Ω) represents the normalised Ω

but this version is rarely made explicit. Typical diagrams using this relationship can be found in [1, 43] and in award winning articles [44], but these are not really Hubble-type diagrams.

This is a variant of the standard model with a parameter, ω, directly effecting Ω_{Λ} as in

where ω is a parameter estimating the relative influences of dark energy * p*and ρ as (

but this version, too, is almost never presented. The reader can see that it is possible to add an extra parameter as a simple term to Eqs. (3), (5) and (7) which may be evaluated as the intercept. Results presenting large intercept values means some form of systematic error is present in the data. It is meaningless to add an extra term to Eqs. (4), (6) and (8) since evaluation cannot be made at the origin, which is −∞ so one loses a simple method for evaluating data worth.

After consulting with E. Hubble and becoming a convert to the idea of a dynamic universe, Einstein reconsidered the basis for his field equation. He then dropped the cosmological constant, Λ, and solved his theory for an expanding universe consisting of only matter as reviewed in [45]. Einstein and de Sitter began with the assumptions of Friedmann demanding spacetime curvature due to the presence of matter, which we cast using the FLRW model as

where we use the * sinh*function for positive curvature. If we allow Ω

The cosmological general relativity model is a recent introduction to this subject [47]. The Hubble velocity is used as a tool by CGR to aid solutions using a 5-dimensional tensor with some success. For instance, the CGR model has been successfully applied to the Tully-Fisher relationship describing spiral galaxies [48]. This model does not admit the existence of dark matter and since dark matter has not been confirmed by several sensitive, direct observational tests [49, 50] this constraint holds true, so this model deserves our consideration.

For estimating _{0} using standard candles, the CGR model for a flat universe present

where β is shorthand for

and Ω_{b} is the estimate of baryonic (normal) matter, whereas Ω_{m} applies to all matter types. The reader should remember that baryonic matter is supposedly only 15–20% as plentiful as dark matter.

This model, which we term ST, admits the influences of both matter and geometric curvature on universe expansion so the condition for normalisation is

This is a significant change from other models considered here since the ST model is not restricted to a flat universe but allows the spacetime curvature parameter, Ω_{k}. For use with the HII and GEHR data we present the form which can be integrated numerically as

or can be integrated analytically [46]. Comparison of the above with Eq. (5) shows these be similar except replacement of the 1 − Ω_{m} term with (1 − Ω_{m})^{2} if one presumes a flat universe and late dark energy. This model deserves consideration since a recent attempt to detect dark energy in a lab failed [51].

We first apply the pseudo-H-routine to the log-transformed data as per Eqs. (4), (6) and (8) to check our regression routines. The reader should note that not only are the dependent values transformed into a logarithmic metric but the abscissa is transformed from relative velocity into galactic redshift as another non-linear metric, * z* = υ/υ

to minimise the Bayesian Information Criteria (BIC), the reduced χ^{2} and outlier influence as per [38]. We report the values for ∆BIC [52], and the reduced χ^{2} as χ^{2}/* DOF*where

Using all 156 data pairs and (1,0) for the position of the local group, most regressions using the H-routine present results with _{0} > 85 km s^{−1} Mp^{−1} which we think unrealistic. The models present many shallow, local minima which is partly the result of the minuscule weights allowed the distant emissions with large associated errors. This makes model comparison difficult because unique, deep regression minima often cannot be found. We observe many values for * D*beyond the 99.99% confidence interval with all models, and present data and results from regression with the ΛCDM model at

We can reproduce the results of [24] using the routine of correlating * mag*versus

This problem is highlighted in Figure 2 where it is obvious the standard deviations are very similar for nearby emissions and those at * z* > 1, which have travelled more than 6 billion light years. Also note the ordinate intercept cannot be displayed as (0,0), which is the location and relative velocity of the local group. This most accurate data pair cannot be used with this diagram type. Diagrams such as Figures 1 and 2 using SNe Ia data have been used for presentation by many continuing to this day [1, 9, 43, 44, 53].

When we attempt regression following the robust H-routine with Eqs. (3), (5), (7), (9), (10) and (13) using all 156 data pairs plus the local group position (1,0), we fail to find satisfactory solutions at _{0} < 85 km s^{−1} Mpc^{−1}; the high side of a realistic value. We attempt regression using many different data handling routines; anchoring the HII/GEHR data in several manners, testing the data as various large segments, data without those of * z* > 0.18 (

We present one example of our many attempts in Figure 3, where we include data from those 10 neighbouring galaxies with fixed distances [54]. This presents relatively small scatter about the best fit for data * a* > 0.85 but most values less than that are well above the best fit, that is, further distant than predicted and exhibiting large standard errors.

Examinations of both Figures 3 and 4 reveal that nearby HII/GEHR sources display relatively small distance dispersion and errors, while ancient emissions are very scattered with very large estimated errors, as expected for difficult, distant observations. This presentation is very different from that displayed in Figures 1 and 2; the match of _{0} at 71 was made by ‘massaging’ the * mag*data, that is by adjusting other parameters in order to recalculate the distances and associated standard deviations as necessary. The scattering data at

H_{0} is the most important parameter for regression of FLRW-type models and is highly dependent on overall curvature of the regression, the slope if you will allow, and hence distant data quality. Because distant data are very noisy and suffer systematic error, these values are nearly ignored using weighed, computerised regression. The regression then ignores distant signals and becomes highly dependent on nearby SNe values. Unfortunately, this means the HII/GEHR data are currently of limited value for determining H_{0} and other cosmological parameters. Investigators relying on the pseudo-H-routine as displayed by Figures 1 and 2 may claim [22], we are now in the era of ‘precision cosmology’ but evidence in these figures says otherwise.

For model comparisons in Table 1 we list results from robust H-routine regressions with H_{0} of 71 as preferred by some working with HII/GEHR data [23]. Results are organised using the relative values of the calculated Bayesian information criteria (∆BIC) [24] also with the reduced χ^{2} values. The spread of both descriptors is much wider than calculated using the pseudo-H-routine [24], making discrimination between models easier. Note all intercepts are negligible indicating little systematic error in nearby signals.

Model | ∆BIC | χ^{2}/(N-FP) | _{0}a | Ω_{m} | Intercept (Mpc) |
---|---|---|---|---|---|

= ct | 0 | 23.95 | 71.0 ± 2.1 | — | 0.03 |

ωΛCDM | 88.3 | 24.03 | 71.0 ± 8.3 | 1 ± >1000 | 0.03 |

CGR | 91 | 21.9 | 70.7 ± 3.4 | 1 ± 0.75 | 0.07 |

ΛCDM | 110 | 23.0 | 70.9 ± 2.9 | 1 ± 0.28 | 0 |

EdS | 123 | 25.4 | 71.3 ± 2.4 | 1 | 0.03 |

ST | 140 | 22.6 | 71.1 ± 10.8 | 1 ± 0.18 | 0 |

The results for the standard model are eclipsed by those of the Rh = ct model when all 156 HII/GEHR values are used with the H-routine, Table 1. These strongly support the findings of [24], that is, if judged by the lowest value of ΔBIC. If judged by the lowest value of χ^{2} the CGR model best describes the HII/GEHR data. The two versions of the standard model, ΛCDM, ωΛCDM, perform poorly compared to the * Rh* = ct model. We are puzzled by the high values for Ω

The results in Table 2 are from data parsed using the 99.99% limits reducing the database by over 40%, though we consider this a conservative parse. The H-routine regressions for all models begins presuming an initial _{0} of 71. The values for Ω_{m}are higher than expected and both the * Rh* = ct and CGR models perform poorly describing these data. On the other hand, a version of the current standard model, ωΛCDM, performs best considering the ∆BIC values but not significantly better than the ST model if one considers the reduces χ

Model | ∆BIC | χ^{2}/(N-FP) | _{0}a | Ω_{m} | Intercept (Mpc) |
---|---|---|---|---|---|

ωΛCDM | 0 | 6.04 | 65.7 ± 4.4 | 1 ± >1000 | −0.07 |

ST | 25.7 | 5.90 | 76.0 ± 9 | 1 ± 0.29 | 0 |

ΛCDM | 27.8 | 5.98 | 76.4 ± 2.4 | 1 ± 0.18 | 0 |

CGR | 28.3 | 8.11 | 69.2 ± 1.9 | 1 ± >1000 | 0 |

= ct | 43.4 | 7.22 | 73.8 ± 1.6 | — | 0.03 |

EdS | 54.4 | 6.68 | 70.7 ± 1.7 | 1 | 0.03 |

The results in Table 3 are from data parsed using the Studentised limit discarding values of * ri*/σ

Model | ∆BIC | χ^{2}/(N-FP) | _{0}a | Ω_{m} | Intercept (Mpc) |
---|---|---|---|---|---|

ωΛCDM | 0 | 2.78 | 66.0± >1000 | 1 ± >1000 | 0.02 |

ΛCDM | 1.8 | 2.49 | 69.0 ± 1.6 | 1 ± 0.15 | 0.02 |

ST | 2.9 | 2.49 | 69.5 ± 1.4 | 1 ± 0.16 | 0.02 |

EdS | 21.3 | 3.10 | 66.1 ± 1.3 | 1 | 0.02 |

= ct | 36.4 | 4.58 | 66.0 ± 1. | — | 0.02 |

CGR | 41 | 5.13 | 62.3 ± 2.1 | 1 ± 0.53 | 0 |

There is a current controversy around the best general description of our Universe. The popular ΛCDM and ωΛCDM models rely heavily on SNe Ia, Cepheid variable and CMB data for validity as per Riess et al. [9]. Another, the * Rh* = ct (the eternal, coasting, non-empty) model functions slightly better than the former two models when tested by proponents with the same SNeIa data as reported by Wei et al. [55] and with HII/GEHR data [24]. We can reproduce the results of this latter group using their selected data by following the pseudo-H-routine. We acknowledge a serious effort has been made by them to analyse these data using their best techniques, the pseudo-H-routine. Unfortunately their analytical method is flawed, as we have pointed out in our Introduction, leading to questionable results and conclusions by many groups.

We first employ all 156 data pairs organised by Wei et al. [24] with the local group as the origin (1,0) using the H-routine; Table 1. Examining Figures 3 and 4, the distant data, * a* < 0.85, are too scattered with large errors to trust our results so we are forced to use a prior for

One reason for the discrepancy between parsed and unparsed ensembles is the large dispersion of HII data with large errors for HII distances at * a* < 0.85 as shown in Figures 3 and 4. These large errors are automatically hidden and their influence on regression is drastically increased when the pseudo-H-routine is used, Figures 1 and 2. We think the results and conclusions of the Riess [9] and Wei groups [24] are tainted by this type of analysis. If the analyses by these groups be useful and if the pseudo-H-routine be a valid method, our results using the H-routine and the pseudo-H-routine should be similar, but are not [16]. We wonder if the

The results in Tables 2 and 3 should not be taken as endorsing the standard model. First, we think the value of the HII/GEHR data, especially events older than * a* ≈ 0.85 is suspect. Second, we have previously shown the complete Einstein field equation, including Λ, when modelled by the FLRW conditions, displays mathematical inconsistencies incompatible with reality [56]. Third, we have recently shown that even Λ tuned to Universe expansion, or tuned to the Hubble-Lemaitre constant, or dependent on decreasing matter density with increasing time, cannot rectify the fundamental problems with that concept [57]. There we have shown by tracing the value of Ω

Our picture of the Universe is complicated; when the ΛCDM model is assayed with CMB data, _{0} is significantly lower (66.9 km s^{−1} Mpc^{−1}) than calculated using SNe Ia data (74.2), both with small claimed errors; [1, 3, 9] but the opposite is expected in a universe suffering gravity. Both the CMB value for _{0} and the evaluation procedure using that data have been recently, vigorously contested by Riess [7]. In addition to those published arguments, we note that analysis of the CMB data relies on 6 parameters with many required priors, using signal averaged data produced at only a single, distant moment. These are discussion points which are rarely mentioned but which we feel severely weakens the value of the CMB results [11]. On the other hand, we have previously pointed out the SNe Ia data are very noisy, much like the HII/GEHR data shown here [46, 58]. When these data are evaluated with a questionable technique using a 4 or 5 parameter regression in reality, it is not surprising the results from using SNe Ia or HII/GEHR data as standard candles do not always match those of other groups; results from LIGO/Virgo [5, 6].

But why do not astronomers and physicists realise and correct this mistake? The analysis of SNe Ia and HII/GEHR is difficult and time-consuming, thousands of readers prefer to simply trust the results and conclusions of articles written by well-known groups rather than take time and brain-power re-investigating the analyses. But why do astronomers persist in using a system, * mag*versus

The tension over the correct value of _{0} might be resolved if another set of standard candles, independent of SNe Ia and gamma-ray burst emissions and stretching beyond * z* ≈ 2 could be used for independent model testing with the correct analytic technique, for example, a better quality HII/GEHR data set. Another reason why independent data are needed is because those working with SNe Ia data present the regression for the ΛCDM model as requiring only 2 or 3 parameters; this is really a 4 or 5 parameter regression. (Because the distance data and

The authors declare that there exists no conflict of interest.

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