Estimating Galactic Dark Matter and Redshift in Flat Space Cosmology

With reference to known galactic rotation speeds and previous publications on our light-speed expanding Flat Space Cosmology model, a toy model variation is presented herein for the purpose of exploring possible time-dependent relationships between galactic dark matter, visible matter, total matter, redshift, radius and angular velocity. The result of this exploration, in the form of graphs and tables, provides for remarkable correlations with current galactic observations and perhaps moves us closer to understanding the scalar nature and influence of dark matter and Lambda on the expanding universe. With reference to light speed expansion, if one is willing to re-define cosmic red shift as [z/(1+z)], without considering Lambda cosmology inputs, light travel distances can be reproduced with a marginal error of +8.6% at z =1.2, (i.e. traditional light travel distance is 8.6% higher than our estimate), 0% at z = 11.5 and -5.5% at z = 1200.( i.e. traditional light travel distance is 5.5% lower than our estimate).


Introduction
Even though standard model of cosmology is standing on 5 pillars namely, big bang, inflation, super luminal expansion, dark matter and dark energy, we would like to emphasize that, 1) James Peebles, the famous cosmologist and 2019 Nobel laureate, strongly believes that Big bang is an inappropriate concept in understanding the universe. Readers are encouraged to visit, https://phys.org/news/2019-11-cosmologist-lonely-big-theory.html 2) Theoretically big bang, inflation, dark energy and super luminal expansion are no way connected with Planck scale which is having a major role in understanding quantum cosmology having information passing at speed of light. 3) Experimentally so far no one could understand big bang [1,2] inflation [3,4], dark energy, dark matter and super luminal expansion with reference to any underground or ground or satellite based experiment. 4) Big bang, inflation and Super luminal expansion are no way giving a clue for unifying general theory of relativity and quantum mechanics. 5) Even though most of the cosmological observations are being studied and understood with photons that propagate with speed of light, it is very unfortunate to say that, most of the cosmologists are strongly believing in hypothetical 'super luminal expansion' of space. Recently detected gravitational waves that are supposed to originate from massive black holes are also confirmed to be moving at speed of light. If so, superluminal expansion can be considered as a pure human intellectual concept having no experimental support. 6) Big bang, Inflation and Dark energy are inference based intellectual concepts having no proper physical base and probably may misguide the future generation. 7) Compared to Big bang, inflation, dark energy and superluminal expansion, dark matter is having some sort of physical support in terms of an unknown, unidentified and unseen elementary particle having an heuristic gravitational attractive property. In addition to that, ultimately somehow, one should suppose the existence of some kind matter for understanding the unexpected massive nature of trillions of observed galaxies. 8) 'Spin' is a basic property of quantum mechanics and 'rotation' is a very common experience. 9) It is better to understand and develop models of cosmology based on well supported physical concepts rather than extraordinary physical hypothesis that demand super-normal efforts.

Three characteristic galactic applications
Based on the assumptions, 3 characteristic properties of any galaxy can be expressed in the following way [12,13,14]. At present, 1) Radius of any galaxy can be expressed as [19,20], 2) Angular velocity of any galaxy can be expressed as, 3) Mean separation distance of any two neighboring galaxies can be expressed as, represent the flat rotations speeds of galaxies   1 2 , . G G It may be noted that, according to Baryonic acoustic oscillations, current galactic separation distance is around 490 million light years [21,22]. In this context, relation (16) can be recommended for further research.
Replacing MOND's acceleration parameter with cosmic angular acceleration and replacing galactic visible mass with dark mass, relation (18) can be obtained. Considering a simple C++ program, relation (17) can be solved numerically. See Table -1. In most of the cases, estimated total mass of galaxy seems to be in line with galactic halo estimations made with NFW model having marginal errors (http://astroweb.cwru.edu/SPARC/WP50_M200.mrt). It can be confirmed with columns (5), (6) of Table 1 and Figure-1. Red curve is our estimation and green curve is for NFW model [24,25,26]. As total mass of galaxy is assumed to be proportional to 4 th power of rotation speed, a small change in rotation speed will have large effect in galactic total mass. We have prepared figure 1 with respect to increasing galactic rotation speeds for a clear understanding. From the figure it is very clear that, for low rotation speeds, NFW estimated halo mass is on higher side and our estimated total mass is on lower side. Based on relation (14), if one is willing to consider the idea that, galactic radius is proportional to square root of its mass, it seems logical to say that, lower massive galaxies will have small radii and low rotation speeds. Here we would like appeal that, galaxies whose visible mass approaches our reference mass unit of Corresponding flat rotation speed is 18.5 km/sec. It needs further study with respect to NGC1052-DF2 refined data. By means of tidal mass loss [30], if dark matter shifts from satellite galaxy to its mother galaxy, based on our proposed concepts, mother galaxy's flat rotation speed must increase significantly due to increase in total matter. It is for future observational testing.

Understanding cosmic red shift with restricted speed of light
It may be noted that, increased redshifts and increased distances forced Edwin Hubble to propose the Hubble's law [31]. With reference to laboratory, appropriate definition of redshift   z seems to be [32], 1 1.
But not 1 Here, as usual, O  is the wave length of light received from observed galaxy and L  is the wave length of light in laboratory. Even though both relations are ad-hoc definitions, compared to relation (20), relation (19) seems to be appropriate with respect to 'light speed expansion'. Very interesting thing is that, when redshift is very small (up to 0.01 z  ), both relations almost all will give the same result. Important point to be noticed is that, by Hubble's time the maximum redshift noticed was 0.003.  With reference to relation (20), relation (19) can be expressed as, 1 1 universe and receding galaxies. Only thing is that, we are confining to 'light speed expansion' and 'light speed  Planck ball as a sequel of cosmic evolution, it is very simple to understand the cosmic physics. In this context, we appeal that, 1) Even though highly intuitive and impressive, there is no clarity and proper physical support for 'big bang' and 'inflation' concepts.
2) Lambda term can be expressed as a scalar quantity having the form, 2 2 2 3 3 .

Conclusion
The result of this exploration provides for remarkable correlations with current galactic observations and perhaps moves us closer to understanding the scalar nature and influence of dark matter and Lambda on the expanding universe. Our light travel distance versus redshift graphs employ a reasonable and consistent mathematical re-definition of cosmological and galactic redshift. Our ongoing efforts to better understand the observed universe have moved us away from an unquestioning acceptance of cosmology in its standard model form. Problematic issues concerning the questionable physical nature of an infinite singularity, a 'Big Bang' characterized by cosmic inflation and its unproven 'inflaton' field, cosmic acceleration in opposition to steady light-speed expansion, the cosmological constant problem and various cosmic coincidence problems have pointed us in a different direction. The reader is encouraged to further explore Flat Space Cosmology and its model variants, one of which has been presented herein.