^{1}

^{*}

^{2}

The Flat Space Cosmology (FSC) model is utilized to show how this model predicts the value of the Hubble parameter at each epoch of cosmic expansion. Specific attention in this paper is given to correlating the observable galactic redshifts since the beginning of the “cosmic dawn” reionization epoch. A graph of the log of the Hubble parameter as a function of redshift
*z* is presented as the FSC prediction of the pending Dark Energy Survey results. In the process, it is discovered that the obvious tension between the SHOES local Hubble constant value and the 2018 Planck Survey and the 2018 Dark Energy Survey global Hubble constant values may be explained by a time-variable, scalar, Hubble parameter acting in accordance with the FSC model.

The SHOES (Supernovae, H_{0}, for the Equation of State of dark energy) report [^{−1}·Mpc^{−1} is in tension with the global Hubble constant estimates of 67.36 ± 0.54 km·s^{−1}·Mpc^{−1} and 67.77 ± 1.30 km·s^{−1}·Mpc^{−1} reported by the 2018 Planck Survey [

At present, the ΛCDM concordance model of cosmology is only seriously challenged by one other type of model, the R_{h} = ct model. This type of model was introduced by Fulvio Melia in 2012 [^{−1}·Mpc^{−1}, which fits the measured 2018 Planck and 2018 DES values given above. This FSC value also fits the 66.93 ± 0.62 km·s^{−1}·Mpc^{−1} value predicted by ΛCDM with 3 neutrino flavors having a mass of 0.06 eV.

It is the purpose of this paper to show how the FSC model, since its inception, has provided a means to calculate the Hubble parameter values correlating with every cosmic redshift epoch since “cosmic dawn”.

The relevant FSC equations useful for these calculations include:

z ≅ ( T t 2 T o 2 − 1 ) 1 / 2 (1)

T t 2 R t ≅ 1.027246639815497 × 10 27 K 2 ⋅ m (2)

It is trivial to show how Equations (1) and (2), in conjunction with the FSC Hubble parameter definition (H_{t} = c/R_{t}), can be rearranged and undergo substitution to give:

T 0 2 ( z 2 + 1 ) ≅ H t [ 1.027246639815497 × 10 27 K 2 ⋅ m c ] (3)

For ease of comparison, the value of the time-dependent Hubble parameter H_{t} in reciprocal seconds (s^{−1}) can be multiplied by 3.08567758 × 10^{19} km·Mpc^{−1} to convert H_{t} to its conventional km·s^{−1}·Mpc^{−1} units. To accomplish this, the left-hand term in (3) is multiplied by this factor. One can now use the following equation to compare redshift z values with H_{t} values in km·s^{−1}·Mpc^{−1}:

T 0 2 ( z 2 + 1 ) ( 3.08567758 × 10 19 km ⋅ Mpc − 1 ) ≅ H t [ 1.027246639815497 × 10 27 K 2 ⋅ m c ] (4)

Taking the square root of both sides of (4) gives:

( z 2 + 1 ) 1 / 2 ≅ [ H t ] 1 / 2 [ 1.027246639815497 × 10 27 K 2 ⋅ m c T 0 2 ( 3.08567758 × 10 19 km ⋅ Mpc − 1 ) ] 1 / 2 (5)

Using the z + 1 convention for redshift, this approximates to:

z + 1 ≅ 0.122 H t (6)

One can then use the knowledge that today’s FSC Hubble parameter value of 66.893 km·s^{−1}·Mpc^{−1} corresponds to 14.617 billion years of current cosmic age in order to calculate Hubble parameters at every billion-year interval of cosmic age. The following equation is useful in this regard:

H t ≅ H 0 ( 14.617 t ) (7)

where the t value is simply the integer, or fractional number, of billions of years of cosmic age, and H_{0} = 66.893 km·s^{−1}·Mpc^{−1}.

One can then construct the following table (

Cosmic Age (Gyrs) | Redshift z | H_{t} (km·s^{−1}·Mpc^{−1}) | Log_{10}(H_{t}) |
---|---|---|---|

14.617 | 0.00 | 66.893 | 1.83 |

14 | 0.02 | 69.84 | 1.84 |

13 | 0.06 | 75.21 | 1.88 |

12 | 0.10 | 81.48 | 1.91 |

11 | 0.15 | 88.89 | 1.95 |

10 | 0.21 | 97.78 | 1.99 |

9 | 0.27 | 108.64 | 2.04 |

8 | 0.35 | 122.22 | 2.09 |

7 | 0.44 | 139.68 | 2.15 |

6 | 0.56 | 162.96 | 2.21 |

5 | 0.71 | 195.55 | 2.29 |

4 | 0.91 | 244.44 | 2.39 |

3 | 1.20 | 325.92 | 2.51 |

2 | 1.70 | 488.89 | 2.69 |

1 | 2.81 | 977.77 | 2.99 |

0.5 | 4.40 | 1955.55 | 3.29 |

0.25 | 6.63 | 3911.1 | 3.59 |

0.142 | 9.11 | 6868.25 | 3.84 |

0.0996 | 11.09 | 9820.49 | 3.99 |

The results shown in _{10}(H_{t}) as a function of redshift z.

The FSC publication entitled, “Temperature Scaling in Flat Space Cosmology in Comparison to Standard Cosmology” [

The blue line is the radiation temperature (T_{R}) curve expected in the ΛCDM concordance model and the green line is the radiation temperature curve expected in FSC. The dashed red line represents the measured and projected spin temperature (T_{S}) and the solid red line represents the baryonic gas temperature

(T_{G}). In

Of most importance with respect to the stated purpose of this paper, one can now see that the FSC model makes very specific predictions for the Hubble parameter values correlating with the range of currently observable galactic redshifts. The following major points need to be stressed:

1) As is true for all R_{h} = ct models, the FSC Hubble parameter is scalar over the great span of cosmic time. R_{h} = ct models require that the Hubble parameter value be defined according to H_{t} = c/R_{t}, where c is the speed of light and R_{t} is the Hubble radius at any time t. Thus, while at any point in human time spans the Hubble parameter may appear to be a constant, any cosmic model which incorporates c/R_{t} within its Hubble parameter equation stipulates that the Hubble term is not a true constant over the great span of cosmic time.

2) The SHOES study of the “local” Hubble constant measured galactic separation velocities per megaparsec at earlier cosmic times corresponding to a range of redshift z values of less than 0.15. It is interesting to note that their averaged “outlier” Hubble constant value corresponds roughly with the FSC Hubble parameter value correlated to a redshift z value of 0.06 (see _{0} value of 76.18 ± 2.37 km·s^{−1}·Mpc^{−1} can be seen to fit the middle of the 0 < z < 0.15 range of our _{10}(76.18) is 1.88. This may be more readily apparent if one compares the first five entries in

3) _{h} = ct models would then be falsified. On the other hand, if observations in the next few years fit with

The heuristic FSC R_{h} = ct cosmology model is utilized to evaluate the tension between the SHOES local Hubble constant value and the Planck Survey and Dark Energy Survey global values. The SHOES report looked at a specific redshift z range of 0 < z < 0.15. Thus, the first approximately 11 billion years of the cosmic expansion history was not subjected to their analysis.

In the FSC model, the Hubble parameter is defined according to H_{t} = c/R_{t}. Therefore, the FSC Hubble parameter is time-variable and scalar according to ^{−1}·Mpc^{−1}, which fits the SHOES Hubble constant determination of 76.18 ± 2.37 for the part of their study using the Milky Way as their anchor galaxy.

The authors predict that, in the next few years, similar measurements corresponding to higher z value ranges will correlate with higher sections of

This paper is dedicated to Dr. Stephen Hawking and Dr. Roger Penrose for their groundbreaking work on black holes and their possible application to cosmology. Dr. Tatum also thanks Dr. Rudolph Schild of the Harvard-Smithsonian Center for Astrophysics for his past support and encouragement. Author Seshavatharam UVS is indebted to professors Brahmashri M. Nagaphani Sarma, Chairman, Shri K.V. Krishna Murthy, founding Chairman, Institute of Scientific Research in Vedas (I-SERVE), Hyderabad, India, and to Shri K.V.R.S. Murthy, former scientist IICT (CSIR), Govt. of India, Director, Research and Development, I-SERVE, for their valuable guidance and great support in developing this subject.

The authors declare no conflicts of interest regarding the publication of this paper.

Tatum, E.T. and Seshavatharam, U.V.S. (2019) Predicting Dark Energy Survey Results Using the Flat Space Cosmology Model. Journal of Modern Physics, 10, 1083-1089. https://doi.org/10.4236/jmp.2019.109070