Site of the Gleaners who think that the universe is a spherical surface in maybe four-dimensional space
8. An explanation on the variation of the Hubble constant (A second spin-off meeting)
Robot: The subject of this spin-off and urgent meeting is to give some explanation about the variation of the Hubble constant or the Great Bull constant.
First of all, as you know, the Hubble’s law is
where v is a recession velocity of a galaxy, H0 is the Hubble constant, and D is a distance to the galaxy from the earth. If we assume that the expansion speed of the universe (or the maximum recession velocity) is the speed c of light and the age of the universe is A, then about the farthest galaxy v=c and D=cA. In this case Eq. (1) is expressed as
c=H0cA or 1=H0A. …(2)
By substituting a well known value 70(km/s)/Mpc into H0 in Eq. (2), as the value of the age A we obtain 14 billion years: almost the same as the well known age of the universe. The values H0 and A have been measured independently with each other, so we can suppose that Eq. (2) has been holding true since maybe the big bang.
Puzzle Sheep: It should be noted that as for Eq. (2) the speed c can be replaced by any other ‘fixed’ speed v. In our space model the fixed speed v has to be c on S3. The Hubble’s law holds true also on H-axis where the fixed speed v can be 2c/π. If we suppose that Eq. (2) has been holding true, the Hubble constant might have been decreasing in inverse proportion to the age A. Consequently, the value of H0 obtained by observing galaxies near to the earth may be less than that obtained by observing galaxies far from the earth. However, according to the actual observed results, the value of H0 obtained by observing galaxies near to the earth seems to be larger than that obtained by observing galaxies far from the earth or CMB. Now we seem to have met a contradiction.
Robot: In this meeting we are going to present some explanation for the contradiction in terms of our space model.
Stamp Pony: The contradiction seems to be more difficult to solve than the question of who is stronger, Dexter Morgan or Hannibal Lecter. In the first place although when a light was emitted from a galaxy, the universe was much smaller than the present size, when we observe the light on the earth, the size of the universe has become the present one. In the second place let’s suppose that when the universe was much smaller than the present size, a galaxy G1 existing at a point P1 (the distance between the earth and P1: the distance between P1 and the farthest galaxy = a: b) was retreating at a speed S1 from the earth. Then when the universe became larger, a galaxy G2 existing at a point P2 (the distance between the earth and P2: the distance between P2 and the farthest galaxy = a: b) would be retreating at the same speed S1. So theoretically the value of H0 obtained by observing galaxies near to the earth may be the same as that obtained by observing galaxies far from the earth.
Robot: Then we have yet to explain the reason for the variation of the Hubble constant.
Glass Snake: Taking this opportunity, I’d like to add a corrected observable size of a galaxy to the magnification ratio dm/d of the universe derived in the appendix. The magnification ratio dm/d is shown as a curve A in Fig.12. The curve A is expressed as a function of h/R, where h is a coordinate of H-axis (a line passing from the earth to its conjugate point (CP)) and R is the present radius of the universe. Here we suppose that along H-axis the size (radius) or the original size of the universe increases from 0 (at h/R=1) to 1 (at h/R=-1) linearly as shown as a line B in Fig, 12. The size is normalized by the present radius R of the universe. Then by multiplying the magnification ratio dm/d by the original size do/R of the universe, the corrected observable size dco (shown as a curve C in Fig.12) is expressed as follows:
dco = (dm/d)(do/R) …(3)
The corrected observable size dco can be considered as the average observable size of galaxies located at h/R and it is normalized by the average size of galaxies near to the earth. According to the curve C the observable size dco of galaxies near to the earth is larger than that of galaxies far from the earth.
Robot: Since the observable size dco of a galaxy has almost nothing to do with the recession velocity of the galaxy, the curve C may not teach us the reason for the variation of the Hubble constant.
I don’t like your former explanation because the theoretical value of the
Hubble constant in the vicinity of the earth diverges to infinity. So I’d like
to ask you to present another explanation.
Puzzle Sheep: Then how about this? In the new Fig. 13 we show a cross section of the 3-sphere S3 (the universe) and H-axis whose coordinate h is normalized by the present radius R of the universe. Here we plot points A1, A2,…, A4 on S3 where point A3 is on the equator. We then suppose that on the earth the time passes along H-axis, and we draw a normal line L2 from point A2 to H-axis. And when we draw a sphere S3A which was the universe when a light was emitted from point A2, we suppose that the intersection point B2a of line L2 and S3A was a virtual source of the light. Then we suppose that on the earth the light from point A2 seems to come from the point B2 on S3 corresponding to point B2a. Then if we draw a normal line L3 from point A3 to H-axis, the intersection point B3a of line L3 and S3B (the universe when a light was emitted from point A3) corresponds to point B3 on S3 which matches point A3. Then if we draw a normal line L4 from point A4a on S3C (a smaller universe) to H-axis and draw a sphere S3D which was the universe when a light was emitted from point A4a, we suppose that the intersection point B4a of line L4 and S3D was a virtual source of the light. Then we suppose that on the earth the light from point A4 (corresponding to point A4a on S3) seems to come from the point B4 on S3 corresponding to point B4a. It should be noted that the recession velocity of point Ai is cθi/π(θi is the angle of point Ai) and the recession velocity increases linearly from the earth to CP.
According to Fig, 13 we notice that since point A2 is farther from the earth than the corresponding point B2, the recession velocity of point A2 is faster than that of point B2. This means that observers on the earth are misled to think as if point A2 were on point B2 and observe that the recession velocity of a galaxy near to the earth is faster than its real velocity. On the other hand, since point A4 is nearer to the earth than the corresponding point B4, the recession velocity of point A4 is slower than that of point B4. This means that observers on the earth are misled to think as if point A4 were on point B4 and observe that the recession velocity of a galaxy far from the earth is slower than its real velocity. This is the reason for the variation of the Hubble constant.
Robot: How could you humans possibly fabricate such an acrobatic explanation? If the observed recession velocity were really accelerated, the cosmological principle supported by Isaac Newton would collapse in a large scale. In the light of your explanation we can measure the Hubble constant correctly by observing galaxies on the equator of the universe. Furthermore we can estimate that the difference ratio between the observed recession velocity of a close galaxy and its real velocity may be larger than that between the observed recession velocity of a far galaxy (CMB) and its real velocity, and this estimation seems to match the observed results.
In short, according to the observed results it appears that in the region near to the earth (one hemisphere) the expansion speed of the universe is accelerated while in the region far from the earth (the other hemisphere) the expansion speed is decelerated. And according to your explanation those observed results have been modified from a common speed by one (a kind of time trap) of the spherical effects of the universe.
Glass Snake: Your this time explanation might be right because there seem to be no other contingencies. In other words by observing the variation of the Hubble constant we can measure the curvature of the universe directly. Furthermore a merit of your explanation is that for galaxies nearer and nearer to the earth the Hubble constants don’t become so large.
In the movie ‘The Polar Express’ the Hobo said to the hero boy, “Seeing is believing.” So why don’t you show the theoretical values
of the Hubble constant based on your explanation to the audience?
Puzzle Sheep: I see. Although I don’t know who the audience is, suppose that the age of the universe is 14 billion years and the expansion speed of the universe is the speed c of light. Then in Fig. 14 a theoretical curve of the Hubble constant H(D)( (km/s)/Mpc) of a galaxy (or a star) is shown as a function of the distance D between the galaxy and the earth on S3, and the unit of the distance D is 0.7 billion light years. According to the curve H(D)=70 at D=10 (7 billion light years or on the equator) and H(D)=70 at almost D=20 (14 billion light years or almost on CP). And as D approaches 0, H(D) becomes almost 99 instead of 70.
Robot: We understand that the seeming deceleration effect of the expansion speed is much smaller than the seeming acceleration effect of the expansion speed by Fig. 14.
Glass Snake: Another interpretation of Fig. 13 is this: In the region near to the earth since time flow on S3 is faster than that on the earth, the recession velocity of point A2 seems to be faster than its real value, and consequently the Hubble constant seems to be larger. While in the region far from the earth since time flow on S3 is a little slower than that on the earth, the recession velocity of point A4 seems to be slower than its real value, and consequently the Hubble constant seems to be smaller. A demerit or special feature of this interpretation is that in the vicinity of the earth the Hubble constant seems to diverge to infinity.
comparison we show the former theoretical curve of the Hubble constant in Fig.
14A. Strictly speaking, in the curve of Fig.14A the Hubble constant diverges to
infinity at the earth. The question of which curve (Fig.14 or Fig.14A) is more
correct will be evaluated by the observed results of the Hubble constant.
Stamp Pony: Anyway the importance of the Hubble constant is increasing year by year. The Great Bull was really great.
Robot: Now you have studied much about our closed universe model. So I’d like to ask you a question about the universe to check your comprehension. The question is this:
What effect does the expansion of the universe have on the movement of all the objects in the universe?
Glass Snake: Here is an answer. Under the condition that the average distance among all the objects (including galaxies) has been increasing, what if the expansion of the universe stops? In this case if there are objects A and B between which distance increases, there are objects C and D between which distance decreases. This means that if the universe doesn’t expand the movement of objects C and D will change while the movement of objects A and B will not change. On the contrary if the universe expands objects A-D keep moving the same as before.
Puzzle Sheep: In other words the universe expands to keep being an inertial reference frame. If the universe had not been an inertial reference frame, Isaac Newton would not have discovered His laws of motion.
Stamp Pony: Or if the universe doesn’t expand we cannot move at will.
Your answer may be right. In short the universe is the most generous existence
ever in the world.
Latest update: 2019/10/28
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