^{1}

^{*}

^{2}

Equations of Flat Space Cosmology (FSC) are utilized to characterize the model’s scalar temporal behavior of dark energy. A table relating cosmic age, cosmological redshift, and the temporal FSC Hubble parameter value is created. The resulting graph of the log of the Hubble parameter as a function of cosmological (or galactic) redshift has a particularly interesting sinuous shape. This graph greatly resembles what ΛCDM proponents have been expecting for a scalar temporal behavior of dark energy. And yet, the FSC
*R*
_{h}
* = ct *model expansion, by definition, neither decelerates nor accelerates. It may well be that apparent early cosmic deceleration and late cosmic acceleration both ultimately prove to be illusions produced by a constant-velocity, linearly-expanding, FSC universe. Furthermore, as discussed herein, the FSC model would appear to strongly support Freedman
* et al.* in the current Hubble tension debate, if approximately 14 Gyrs can be assumed to be the current cosmic age.

We are currently in a “golden age” of astronomy and cosmology. Astrophysical observations in the coming decade are expected to bring much greater resolution concerning the behavior and fundamental nature of dark matter and dark energy. These are two of the remaining great mysteries of the universe.

With respect to the behavior of dark energy, the expansion history of our universe, going back to the earliest galaxies, should come into greater focus. If all goes well with these observations, we should be able to fill in many details with respect to the velocities of galactic separation going all the way back to the first few hundred million years of cosmic expansion. We should then have a remarkably accurate “moving picture” computer simulation of the history of that portion of the universe we can now observe.

When astrophysicists concern themselves with the velocities of galactic separation on scales greater than those of the local clusters held together by gravity and dark matter, they are studying the Hubble parameter and its tight correlation with cosmological redshift. When the Hubble parameter is characterized as a “snapshot” of the universe at a particular point in cosmic time (at the present time, for instance), it can be referred to as the Hubble constant. On a global scale, making use of cosmic microwave background (CMB) observations, the 2018 Planck Collaboration has arrived at a current Hubble constant H_{0} value of 67.36 +/− 0.54 km∙s^{−}^{1}∙Mpc^{−}^{1} [

The ongoing temporal (i.e., “moving picture”) studies of the universe are expected to show that, over the great span of cosmic time, the Hubble parameter is, in fact, scalar in some way. The first evidence of this became apparent in 1998, with studies of Type Ia supernovae [_{h} = ct cosmological models) or very slightly accelerating (as claimed by ΛCDM concordance model cosmologists). Both types of cosmological models are still viable at the present time [

Flat Space Cosmology (FSC) is perhaps the most successful R_{h} = ct model to date [_{0} value of 66.893 km∙s^{−}^{1}∙Mpc^{−}^{1}, fitting with the 2018 Planck Collaboration consensus. It also predicts the COBE CMB dT/T anisotropy ratio of 0.66 × 10^{−}^{5}. A book chapter summary of FSC is now freely available online [

The purpose of the current report is to show how FSC models the temporal dark energy expansion of the universe. We show in great detail the scalar nature of the FSC Hubble parameter, so that it can be compared to the observations to be made in the coming decade.

Previously-published equations of FSC, relating cosmological (or galactic) redshift z, temporal cosmic temperature T_{t}, temporal cosmic radius R_{t}, the associated temporal Hubble parameter H_{t}, the currently-observed Hubble parameter H_{o}, the currently-observed cosmic temperature T_{o}, and cosmic age t, are brought together in the Results section in order to derive the parameter values given in

The following two FSC equations are useful for deriving the model relationships between a given cosmological (or galactic) redshift z and the associated temporal Hubble parameter H_{t}:

z ≅ ( T t 2 T o 2 − 1 ) 1 / 2 (1)

and

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

The first equation relates the redshift to the temporal cosmic temperature T_{t} and the currently-observed cosmic temperature T_{o} [_{t} to the temporal cosmic radius R_{t} [

Recalling the FSC Hubble parameter definition (H_{t} = c/R_{t}), rearrangement and substitution gives:

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

To convert the H_{t} term from reciprocal seconds (s^{−}^{1}) to the conventional Hubble parameter units of km∙s^{−}^{1}∙Mpc^{−}^{1}, the left-hand term is multiplied by 3.08567758 × 10^{19} km∙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)

Rearrangement of terms gives:

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

Using T_{0} = 2.72548 K, this simplifies to:

H t ≅ ( z 2 + 1 ) 0.014949183831548 (6)

The final useful equation relates cosmic time t (in Gyrs after the Planck epoch) to the current Hubble parameter H_{0} value of 66.893 km∙s^{−}^{1}∙Mpc^{−}^{1}, the temporal Hubble parameter H_{t} value, and the current FSC cosmic age of 14.617 Gyrs:

H t ≅ H 0 ( 14.617 t ) (7)

Equations (5), (6) and (7) can then be used to create

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 | 0.21 | 69.84 | 1.84 |

13.8 | 0.24 | 70.85 | 1.85 |

13 | 0.35 | 75.21 | 1.88 |

12 | 0.47 | 81.48 | 1.91 |

11 | 0.57 | 88.89 | 1.95 |

10 | 0.68 | 97.78 | 1.99 |

9 | 0.79 | 108.64 | 2.04 |

8 | 0.91 | 122.22 | 2.09 |

7 | 1.04 | 139.68 | 2.15 |

6 | 1.20 | 162.96 | 2.21 |

5 | 1.39 | 195.55 | 2.29 |

4 | 1.63 | 244.44 | 2.39 |

3 | 1.97 | 325.92 | 2.51 |

2 | 2.51 | 488.89 | 2.69 |

1 | 3.69 | 977.77 | 2.99 |

0.5 | 5.31 | 1955.55 | 3.29 |

0.25 | 7.58 | 3911.1 | 3.59 |

0.174 | 9.11 | 5618.51 | 3.75 |

0.1179 | 11.09 | 8293.97 | 3.92 |

Notice the sinuous appearance of this graph. Its overall shape greatly resembles what cosmologists have been expecting for a scalar temporal behavior of dark energy!

Proponents of the ΛCDM concordance model of cosmology, and R_{h} = ct model cosmologists, are currently in a pitched battle to establish which model is more accurate with respect to observations and predictions. As documented in recent publications [_{h} = ct model].

Notice also that this graph correlates a redshift z value of 1.0 with a cosmic scale of 0.5 times the current scale. This is true for FSC as well as ΛCDM, although the two models differ slightly with respect to the current cosmic age.

In ΛCDM cosmology, the post-inflationary cosmological vacuum energy density is assumed to be a constant. This is not an absolute requirement of general relativity, so long as the vacuum energy density is scalar according to Λ = 3 H t 2 / c 2 . In the FSC quintessence model, this scalar relationship holds true and is equivalent to Λ = 3 / R t 2 [

As speculated in the FSC book chapter summary, ongoing cosmological matter creation may be paired with a continual decline in the cosmological vacuum energy density, as a requirement for conservation of energy in such a finite isolated expanding system. It should be remembered that the details of matter creation in all cosmological models are a mystery. In FSC, matter creation is an ongoing process, whereas ΛCDM cosmologists generally assume that all matter was created nearly instantaneously. However, as a result, a major difference between the two models is that only ΛCDM cosmology has a cosmological constant problem, based upon its embedded constant post-inflationary vacuum energy density assumption.

As a consequence of the dark energy observations, in addition to their cosmological constant and instantaneous matter creation assumptions, ΛCDM cosmologists must now also assume certain features of the universal expansion. These features had not been required when it was once thought (i.e., before 1998) that the cosmological vacuum energy density might actually be perpetually zero. They now require that universal expansion decelerated during the first half of the cosmic time span since the Big Bang, and then, almost imperceptibly, began to accelerate approximately 6 billion years ago. This becomes absolutely necessary if one requires a post-inflationary cosmological constant at the currently observed value of about 10^{−}^{9} J∙m^{−}^{3}. Nevertheless, this deceleration-followed-by-acceleration scenario of universal expansion is clearly debatable, especially when one considers the observational statistical error bars in

When one compares the relative luminosity and angular diameter distances between the two competing models, in the form of a ratio, it has recently been shown that the ΛCDM model contention of late cosmic acceleration could be an illusion produced by a R_{h} = ct universe [

Further support that cosmic acceleration could be an illusion is clearly evident in _{h} = ct model expansion, by definition, neither decelerates nor accelerates!

The upward curving portion of our

Regardless, given the overall shape of our

Given the ongoing tension between different research teams considering what current near and deep space observations might be telling us about the H_{0} value as a snapshot in time, it is worth noting the following:

The 2018 Planck Collaboration analysis of the CMB looked at 99.998 percent of the current radius of the universe. Their consensus H_{0} estimate of 67.36 km∙s^{−}^{1}∙Mpc^{−}^{1} appears, in FSC, to fit with a 14.6 Gyr old universe. According to _{0} observation of 69.6 km∙s^{−}^{1}∙Mpc^{−}^{1} [_{0} observations of 74 - 77 km∙s^{−}^{1}∙Mpc^{−}^{1} [_{0} estimate is the most likely outlier.

Equations of FSC have been utilized to characterize the model’s temporal behavior of dark energy. A table relating cosmic age, cosmological redshift, and the temporal FSC Hubble parameter value has been created. The resulting graph of the log of the Hubble parameter as a function of cosmological (or galactic) redshift has a particularly interesting sinuous shape: an upward flexion curve out to a z value of about 1.0 (corresponding to the last 7.3 billion years of the FSC cosmic expansion); a roughly straight line segment for 1.0 < z < 1.7 (corresponding to 3.76 to 7.3 billion years of cosmic age); and an opposite flexion curve for z values greater than about 1.7 (corresponding to the first 3.76 billion years of the FSC cosmic expansion). The overall shape of the graph greatly resembles what ΛCDM proponents have been expecting for a scalar temporal behavior of dark energy. And yet, the FSC R_{h} = ct model expansion, by definition, neither decelerates nor accelerates. It may well be that apparent early cosmic deceleration and late cosmic acceleration both ultimately prove to be illusions produced by a constant-velocity, linearly-expanding, FSC universe.

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. NagaphaniSarma, 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. (2020) How Flat Space Cosmology Models Dark Energy. Journal of Modern Physics, 11, 1493-1501. https://doi.org/10.4236/jmp.2020.1110091