Hypersphere Cosmology: A linear relation between Galactic Recession and Arc Angle

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Key features and Hubble's Law: Using the Doppler type redshift in Galactic Recession

The model, named 4-Sphere [*], bases its physics on expansion due to Cosmic Background Radiation, and simplifies its math considering the CMB in absence of matter [**]. Here, the Universe lies on the surface of a hypersphere which expands at a constant rate with its radius stretching as r = ct, and with the Big Bang occurring at its center. 4-Sphere explains why our physics is shown to be that of a system without boundaries, even though our Universe has a finite volume: It proves that if it were not for Relativity, our studies would probably have focused on an infinite and static universe.

Other models hypothesize a hypersphere that expands as r = ct. Reading the main features, you will notice that they are completely different from each other. The novelty of 4-Sphere lies in its definition of the Hubble constant: Its geometry, indeed, suggests a linear relation between the Galactic Recession and the arc angle (not the arc length) [***]. Note that this perspective does not affect the validity of Hubble’s law; it only concerns predictions about the past and the future, which cannot be based purely on observations.

Given the constant expansion speed, it is not necessary to define a new specific type of Redshift (as the standard Cosmological one) to be associated with the Galactic Recession.Here the redshift is Gravitational or Doppler. In fact, for the Galactic Recession the Redshift is of the Doppler type (except for special cases in which the gravity of the star cannot be neglected) while for the Cosmic Background Radiation (CMB) it is exclusively of the gravitational type.

The model is also supported by assumptions that are necessary and which are pure conjecture, but the key to the speculation is contained in Hubble's Law and Star distance validation:

1. Hubble's Law: 4-Sphere geometry suggests a linear relation between the galactic recession and the arc angle. The Galactic Recession is the subject of speculation.
2. Star distance validation: This is the key to the whole speculation. The validation desired is carried out on the Luminosity distance, comparing its value calculated from the Redshift z of a star with that derived by its Distance modulus μ.

HERE ARE THE MAIN QUANTITIES FOR THE 4-SPHERE

Anticipating the results, If r is the radius of the 4-Sphere, θ is the center angle between the star and us, z its Redshift, and t_now the time elapsed from Big Bang (measured by us), the main quantities for the 4-Sphere are:

1. The star Recession velocity v_r = cθ
2. The quantity β equals to angle θ = v_r/c = ((1+z)²−1)/((1+z)²+1)
3. The star Redshift 1+z = (1+β)^1/2(1−β)^−1/2
4. The Time dilation dt_obs/dt_emit = (1−θ²)^−1/2 = γ: the Lorentz factor of Special Relativity
5. The actual radius of the 4-Sphere r_now = ct_now
6. The time from the star’s light beam started t₀ = t_nowe ^−θ
7. The time spent by the star’s light beam to travel the arc θ   Δt = t_now − t₀
8. The Proper distance of the star,  d_P = r_nowθ
9. The Luminosity distance d_L = cΔt   (to be compared with Distance modulus μ)
10. The conceptual equivalent of Comoving distance d_C = θ   (if we prefer the angle to the arc we obtain a dimensionless quantity constant in time with r as the scale factor)
11. The arc corresponding now to 1 Mpc (Megaparsec)   θ_{1 Mpc} = 1 Mpc / r_now = 2.36∗10⁻⁴ rad
12. The equivalent of Hubble’s constant H_Sphere = cθ_{1 Mpc} = 70.9 Km s⁻¹ (per θ_{1 Mpc})

[*] – 4-Sphere is a proper name, but here we also mean the hypersphere embedded in four-dimensional space R4 (someone call it 4-ball too); its surface is named by topologists a S³ sphere.
[**] – Since gravity decreases with the square of the distance, matter clustered in galaxies could be seen as points of discontinuity in the universe, where the distribution of matter is not continuous but concentrated in specific regions. This idea reflects the large-scale structure of the universe, characterized by a web of galaxies and clusters separated by vast voids, a central concept in modern cosmology.
Our hypothesis is that the CMB represents the predominant force in maintaining the 4d-bubble that characterizes, here, the shape of the Universe, and that matter, as a discontinuity, can be neglected.
[***] – See chapter 1.3 in 4-SPHERE FEATURE AND SPECULATION.