Since the discovery of Hubble, the universe is expanding in an accelerated fashion according to the current cosmology [1] [2] [3] [4] [5] . The most reliable rationale of this expansion is the redshift of the light emitted from celestial bodies, where the extent of the redshift is larger if the celestial body is farther away [6] - [12] . Recently, the observation precision of the cosmic microwave background (CMB) radiation and the redshift of large galaxies has increased, and the measurement of the expansion velocity of the universe has become more accurate [13] . However, a basic question of what occurred at the beginning of the universe and an ultimate fate of the universe are still unknown [13] . Furthermore, the observation data of more distant celestial bodies are welcome, because they let us know what happened in the past universe. For instance, Chen et al. reported a redshift of z = 6.68 in their observation data of a distant celestial body [14] . However, Stern et al. claim that it may be impossible to observe the redshift of wavelengths less than 9300 Å at such a distance, because the intensity of the light is too weak to observe [15] . Although the observation data of very distant celestial bodies are important, there are questions relative to the data’s reliability [15] .

A question has also been raised against the accelerated expansion of the universe first pointed out by Hubble [1] [4] . Barrow et al. also pointed out a question of whether the expansion of the universe is accelerating or decelerating [16] [17] . Furthermore, although the primary force of the currently believed accelerated expansion of the universe derives from the “repulsive gravity” of dark energy [4] , some researchers doubt even the existence of dark energy [18] . We hereby present the following proposal to construct a clearer theory regarding such a problem. Lehoucq et al. presented the existence of a 3-dimensional spherical universe as a basic structure of the universe [19] . The 3-dimensional sphere is a manifold having a spherical structure that exists in 4-dimensional space time, and the universe is considered to exist on its surface according to Shibata et al. [20] . We think that the universe began explosively from a tiny Big Bang and expanded soon after in an accelerated way. Next, the expansion velocity decreases and ceases, begins to contract in an accelerated fashion, and finally ends with the Big Crunch with contraction to a single point. According to our proposal, it is possible to explain the current structure of the universe without considering the dark energy, and to explain the expansion and contraction of the universe. Focusing on this point, we present a new concept of the current structure of the universe.

The 3-dimensional spherical (S³) universe model we propose in this paper is composed of four dimensions, in which they are shown by 3-dimensional x-, y-, and z-coordinates that represent a space and a w-coordinate that has a dimension of distance by coordinate time t multiplied by light speed c (Equations (1) and (2)). In Equation (2), R represents the maximum radius of the S³ universe.

w = c t , (1)

and

w 2 + x 2 + y 2 + z 2 = R 2 . (2)

In this S³ universe, the space expands and contracts while the coordinate time t advances from the birth of the universe (w = −R) to its end (w = R) in one direction (Figure 1(a)). At arbitrary coordinate time t, the S³ universe can be observed as a 2-dimensional sphere of radius r mapped in a 3-dimensional space when it is observed from one direction outside of the S³ universe fixing one of the space coordinates to zero (Figure 1(a)). On the other hand, for an observer inside the hypersphere S³ universe, the universe is observed as a 3-dimensional observer-centred filled-in sphere, and numerous lights from celestial bodies scattered at each distance are observed as if they are coming straight to us (Figure 1(b)). This observed universe is a map of the light from a light source of a hypersphere of observer-centred space inside of the 3-dimensional sphere (Figure 1(b)).

When the S³ universe is represented in polar coordinates, it is written as shown in Equations (3) - (6). ψ is elevation angle of space coordinate against the w coordinate axis, and it represents the expansion of the observable universe. It changes from zero to π upon the passage of coordinate time t. This ψ will be

called expansion angle hereafter. θ is zenith angle at an arbitrary ψ, when the direction of the z-axis is facing upward, and the zenith top is zero and the zenith bottom is π. Likewise, η shows a deflection angle from the x-axis; it is positive when a right screw rotated clockwise goes ahead. This variable moves from zero to 2π.

w = − R cos ( ψ ) , (3)

x = R sin ( ψ ) sin ( θ ) cos ( η ) , (4)

y = R sin ( ψ ) sin ( θ ) sin ( η ) , (5)

z = R sin ( ψ ) cos ( θ ) (6)

and

ψ = cos − 1 ( − c t R ) , (7)

where 0 ≤ ψ ≤ π, 0 ≤ θ ≤ π, and 0 ≤ η ≤ 2π.

A radius r of the S³ universe is shown in Equation (8), and it becomes the maximal R at a great sphere. The expansion velocity v of the S³ universe at coordinate time t is shown in Equation (9).

r = R sin ( ψ ) (8)

and

d r d t = v = − c cot ( ψ ) . (9)

The minute distance that light runs in the S³ universe becomes cosec(ψ)-times larger than that when it runs on a flat space time, because the radius of the S³ universe changes upon the passage of coordinate time t. This increase of the geodesic can be explained by the increase of the passage of proper time τ on the S³ hypersphere against the passage of proper time τ on the expansion angle ψ, if the concept that light speed is always constant holds (Equation (11)). Therefore, if the ratio cosec(ψ) is integrated by coordinate time t, the ratio of the passage of proper time τ against that of coordinate time t from its origin (−R/c) to the coordinate time t is obtained (Equation (12)).

c Δ τ = cosec ( ψ ) c Δ t , (10)

d τ d t = cosec ( ψ ) (11)

and

c R ∫ − R c t cosec ( ψ ) d t = c R ∫ − R c t [ cos − 1 ( − c t R ) ] d t = cos − 1 ( − c t R ) = ψ . (12)

The result of this integration indicates that the angular distance of the light of the Big Bang (CMB) that ran in the S³ universe is the same as the expansion angle ψ of the S³ universe. Therefore, this expansion angle ψ is the angular distance that CMB runs, and the geodesic distance S_ψ is the same as the arc length ψR of the great sphere of radius R and angle ψ as shown in Figure 2 (Equation (13)). The geodesic distance S_β that light reaches the observation point from a celestial body born after the Big Bang is βR, if the angular distance between an observation point and a space coordinate of the celestial body is β (Equation (14)). As shown in Figure 2, angular distance and the geodesic of a celestial body that remains in the S³ universe are always constant despite the expansion∙contraction of the S³ universe.

S ψ = ψ R (13)

and

S β = β R . (14)

The observable universe that can be observed at the expansion angle ψ in the S³ universe is an image mapped in the 3-dimensional sphere of the radius ψR with its center in the observation point. The radius ψR of the observable universe becomes large soon after the Big Bang, but then the expansion velocity slows down. Thereafter, when the S³ universe enters the contraction phase, the expansion velocity decreases and the radius ψR becomes πR/2 in the great sphere of the S³ universe and the expansion velocity increases again. The radius ψR then reaches maximal πR immediately before the end of the S³ universe (Equation (13) and Figure 3).

The optical path light reaches from an arbitrary celestial body to an observation point is always constant despite the expansion of the observable universe (Figure 2). However, a redshift (or blueshift when minus) is observed due to a difference in the wavelength of light between the time of emission and that of observation, because the passage of the proper time of the light at the time of emission and that of observation differs. The redshift z is shown in Equation (15), when the longitude difference of the space coordinate between the light source and the observation points is β. λ_s is the wavelength of the light source and λ is the wavelength of light at the observation point. The ratio between them is a ratio of the inverse of the elongation of the proper time (Equation (12)).

z = λ λ s − 1 = cosec ( ψ − β ) cosec ( ψ ) − 1 = sin ( ψ ) sin ( ψ − β ) − 1. (15)

The redshift is observed in a flat space due to the receding velocity of the light source, but it is observed due to the difference of the proper times between those of emission and observation, both stationary, in the S³ universe. From Equation (15), it can be concluded that the blueshift begins successively from neighbouring stars of smaller to larger longitudinal difference β when the S³ universe enters the contraction phase (Figure 4).

This result agrees very well with the current observation of the universe that the universe apparently turned from decelerated to accelerated expansion [21] . In fact, the S³ universe is contracting in an accelerated way, and is not expanding. In view of our theory, it would be important to mention the report of Lombriser [22] . He found that the redshift of galaxies can be interpreted to be an evolution of particle masses as a consequence of conformal transformation of the FLRW (Friedmann Lemaître Robertson Walker) metric to a Minkowski space [22] . Although his rationale behind the theory is different from that of our theory, he pointed out that the current universe may not be expanding based upon his theory. The conclusion of his theory is basically same as our conclusion.

Welch et al. recently reported observations of a distant star of redshift 6.2 ± 0.1 and 900 million years after the Big Bang [23] . Based upon their observations we calculated from Equation (13) that R is 13.8/ψ billion light years. The expansion angle of the light source is 9 = ψ_sR from Equation (13). Therefore, ψ_s is 9/R radians and ψ_s = (9/138)ψ = 0.0652ψ. If we assume that the expansion angle of a star at the time of emission is ψ_s, Equation (15) becomes

z = λ λ s − 1 = sin ( ψ ) sin ( ψ s ) − 1. (16)

By substituting z = 6.2 and ψ_s = 0.0652 in the above equation, we obtain

6.2 = sin ( ψ ) sin ( 0.0652 ψ ) − 1. (17)

From this equation we obtain ψ ≈ 113 deg (1.97 rad), indicating that the universe is in a stage after the beginning of the contraction.

In summary, we theoretically demonstrated that the increase in receding velocity of the observable universe is observed when the universe is contracting in an accelerated fashion even though it is not expanding. In our theory, the so-called unknown energy that expands the universe in an accelerated fashion in terms of the energy balance of the entire universe is unnecessary. Our results suggest that this model can be a potential candidate demonstrating the geometrical structure of the current universe.

We proposed in this paper a novel view of redshifts of the lights emitted from celestial bodies based upon the structure of the three-dimensional spherical (S³) universe. Contrary to the common interpretation of the redshift that the universe is expanding in an accelerated fashion, we theoretically demonstrated that the redshift of the celestial bodies can also be observed in a contraction phase of the universe. In our theory, the so-called unknown energy, which is known to expand the universe in an accelerated fashion in terms of the energy balance of the entire universe, is not necessary. Our result suggests that our model can be a potential candidate that clearly explains the geometrical structure of the current universe, and it will open a new concept of the current status of the universe.

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

Shibata, H. and Ogata, N. (2023) Apparent Accelerated Expansion of the Three-Dimensional Spherical Universe Observed during Its Contraction Phase. Journal of Modern Physics, 14, 1693-1702. https://doi.org/10.4236/jmp.2023.1413099