Zatrikean Pregeometry - Simple Cosmological Scenarios

ZATRIKEAN PREGEOMETRY: SIMPLE COSMOLOGICAL SCENARIOS T.G. DALLAS and V.S. GEROYANNIS Astronomy Laboratory, Dept. of Physics, University of Patras, Greece (Received: 13 July, 1992)

Abstract. In this paper basic cosmologicalconsequences of zatrikean pregeometry are studied. It is shown that many universal quantities can be calculated with simple applications of this theory. An equation directly linking the angular velocityof the universew, the Hubble constant H and the mean density p of the universe is derived.The relation betweenthe mass M and the radius Æ of the universe is examined.This relationleads to vadous cosmologicalscenarios,includingvariations in physical constantsand/of violationof the mass conservationand/or variable geometricalproperties.

1. Introduction In a recent paper (Geroyannis, 1992; hereafter referred to as ZP1) the foundations of zatrikean (i.e., chess-like) pregeometry were set. According to this theory, space is represented by an abacus A N made up by squares (of, in the three-dimensional case, by cubes; a forthcoming paper will deal with this generalization) called geobits. This simple assumption introduces a transformation to classical mechanics that gives results close to the ones derived by general relativity. It is also shown in ZP1 that rotation can induce such pregeometry. Under this scheme, there is more than ()ne path of minimal length to go from one geobit to another. Also, there is a dilation of all Euclidean distances rT (these topics are covered in Geroyannis and Dallas, 1993; hereafler referred to as ZP2). In the present paper we examine some cosmological consequences of the zatrikean pregeometry. In Sect. 2 we shall apply the method developed in Sect. 9 of ZP1 to the whole universe; this gives an estimate of the angular velocity of the universe. The result leads to a series of formulas that can calculate any of the following quantities of the universe: mass M, radius R, mean density p, angular velocity ~, Hubble constant H, in terms of the other quantities and physical constants. These formulae show that, since R increases with time, other quantities should also vary with time. In Sect. 3 we deal with this result in six different scenarios which have diffenmt cosmological consequences. These scenarios include an increase in the mass M of the universe, a decreasing maximum attainable speed c, a varying gravitational constant G, an increasing pregeometric dilation constant 7, and their combinations. Our conclusions are summarised in Sect. 4. In this paper we deal exclusively with Q premetrics (ZP1, Sect. 3). Therefore, all quantities must be regarded as following such convention; e.g., distances are denoted by 7"instead of r o. For instance, the basic pregeometric transformation is Astrophysics and Space Science 201: 273-280, 1993. @ 1993 KluwerAcademic Publishers. Prh~ted in Belgium.

274

T.G. D A L L A S A N D V.S. G E R O Y A N N I S

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ZATRIKEANPREGEOMETRY:SIMPLECOSMOLOGICALSCENARIOS

r = rT

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275 (1)

We also use S.I. units in all formulae, tables and figures, unless clearly stated otherwise.

2. The Angular Velocity of the Universe and the Hubble Constant Let us assume (following Sect. 9 of ZP1) a mass M and a particle moving with the maximum attainable speed c on the abacus AN. The particle can be captured by the mass in a circular orbit with radius R = 7 G M / c 2. While on this orbit, the partMe moves with angular velocity co = c/tg. If we "homogenize" the mass M to the whole volume V enclosed within the radius R, the resulting mean density is 3w 2

p = 47r7G(1

+ ~to)3,

(2)

where we have used the formula (ZP1, Sect. 9) ¢3 M

-

(3)

7Ga)' and the pregeometric expression for volume v=4rc T

(Ig)3, 1+~0

(4)

in accordance with Eq. (1). Solving Eq. (2) for w, we get /47r G" 1

(5)

Similafly, we can formulate the above equations so that we get an explicit relation for each of M, R, p, w. The relevant formulae are given in Table I. Let us assume now that M refers to the whole universe; then R is the radius of the universe. By substituting in Eq. (5) the value p = 5 × 10 -27 k g m -3, which is a good estimate of the mean density of the universe, we get the upper limit of the angular velocity of the universe, w = 2.15 × 10-is s - 1 . This corresponds to a period of rotation T = 9.27 × 101° years. It is an established observational fact that the universe is expanding. The edge of the universe expands with the maximum attainable speed c. Therefore, the Hubble constant H = c / R results from Eq. (5) as H = 3 × 1Ó19/t/471"

V

T-ya(1 + ~,0)3p,

(6)

276

T.G. DALLAS AND V.S. GEROYANNIS

TABLE II Characteristics of the Universe according to zatrikean pregeometry Mean density, p

5.00 × 10 -30 g cm -3

Angular velocity, w

2.15 × 10 -18 s -1

Period of rotation, T

9.27 x 101° y

Hubble time

1.47 x 101° y

Total mass, M

6.63 x 1055 g

Total number of baryons

3.97 x 1079

Current radius, R

4.51 x 103 Mpc

Hubble constant, H

65 km s - 1 M p c -1

where H is in km S - 1 Mpc -1 and all other quantities are in S.I. units. The above equation shows clearly that H is nothing hut the angular velocity c~ of the universe in somewhat unusual units. By substituting in Eq. (6) the value p = 5 x 10 - 2 7 kg m -3, we ger H = 65. This value is in accordance with the current observational data. By using the formulae of Table I and Eq. (6), we can calculate H by any one of M,/~, p, aJ. A first estimate and an upper limit on the age of the universe is given by the "Hubble time" that is found equal to 5 x 1017 S or 1.6 X 101° years. This means that the universe has completed, at the most, one sixth of a rotation round itself. By using Table I and assuming as an observational result the mean density p of the universe, we can calculate all the relevant quantities concerning the universe. These are listed in Table II.

3. Are Physical Constants Really Constant? From the equation connecting M with R (see Table I) we get c2R

7GM

-

1.

(7)

Since the universe is expanding, R in the above equation increases with time. As a result, other quantities in Eq. (7) should also vary with time. This leads to various scenarios, which will be discussed later. 3.1. VARIATION OF THE M A S S M OF THE UNIVERSE W I T H T1ME

If we assume that c, 7, G a r e constant with time, then as the universe expands its mass should also increase. By differentiating Eq. (7) with respect to time, we get: c2

= ~

k.

(8)

ZATRIKEANPREGEOMETRY:SIMPLECOSMOLOG1CALSCENAR1OS

277

The creation of matter in the universe is linear to the current rate of expansion. Since the value of dR/dt is 3 × 108 m s- l, by substituting in Eq. (8), we estimate the creation of matter at dM~dt = 1 × 10 35 kgs -1. This is equivalent to 3 x 1069 hadrons per year in the whole universe, or to one galaxy like ours every two months. The above result is not consistent with observational data; therefore, we can accept that the mass of the universe remains constant. As a result, G and/or c and/or 7 shou]Ld change with time. 3.2. 'VARIATION OF THE MAXIMUM Aqq'AINABLE SPEED C WITH TIME Accepting M, 7, G a r e constant, then Eq. (7) leads to

_ -7GMÆ 2cÆ2

(9)

,

which gives for the current rate of change ~/e = - 3 . 3 7 × 10 -11 y-1. The variation of c with time will be further examined in Sect. 3.6. 3.3. VARIATION OF THE GRAVlTATIONAL CONSTANT G WITH TIME In this case, by differentiating Eq. (7) with respect to time, we get c2/~ _

(lO)

7M'

G/G

which gives for the current variation of G the value = 6.67 x 10- i l y - 1 Therefore, G should increase with time. By multiplying Eq. (10) with the age of the universe we get a value for G(0) very ctose to zero. In this scenario, gravity is just a reaction to the expansion of the universe and had minor effect in its early stages. Most of the previous theories (Brans-Dicke, Dirac) favor a G decreasing with time, but others accept an increasing G (Barber). In Van Flandern (1981), solar systems measurements are interpreted as a change in G according to the formula

= f(3.2 5z 1.1) × 10 -11

y-1

(11)

where, f is a theory dependent parameter. The results catculated by means of Eq. (t0) are in agreement with Eq. (11) for f ___2. 3.4. VARIATION OF ")'G WITH T1ME

In zatrikean pregeometry the gravitational constant G and the pregeometric dilation constant 7 appear continuously together in the form of their product. We may assurne that this very product changes with time; Eq. (7) leads to c2/~ 76~ + ;~G = M

(12)

The above equation gives for the relative rate ofchange of TG the value 6.67 × 10-11 per year.

278

X.6. DALLAS AND V.S. GEROYANN1S

3.5. VARIATION OF THE PREGEOMETRIC D1LATION CONSTANT ")' W l T H TIME

In this scenario, 7 had a smaller value in the past and the rate of change is calculated by the equation c2R a~_ G M

(13)

which gives for the present time ~//7 = 6.67 x 10-1 ~ y - 1. By multiplying the result with the estimated age of the universe, we get for the initial value 7(0) ~ 0. But what does a variable pregeometric dilation constant mean? In Sect. 9 of ZP1 7 is defined as 7 = (/~/a) 3, where /3 is the diagonal of the geobit and a its edge. Let e be the perimeter of the geobit; then 7 = (4/3/«) 3. Therefore, variation of 7 means an excess perimeter or diagonal for the geobit that leads to its deformation. For example, some time in the past the geobit could have been a circle; then 7 = ( 4 / ~ ) 3 = 2.064. If we assume a linear change for 3', this happened approximately 4 billion years ago. But 3' reltects the effect of matter on the deviation from Euclidean geometry; if 7 is zero, the mass has no effect and space is perfectly Euclidean, As a result, this scenario starts from an Euclidean universe and, as it expands, evolves to a universe where matter has an ever stronger effect on geometry. By Eq. (1) and Eq. (13), we get r=rT

1+

0r--~- ) .

(14)

The above equation shows that the departure from the Euclidean distance depends on the premetrics used as well as on the radius R and mass M of the universe. As a result, the large-scale structure of the universe affects the local structure of the abacus. By applying Eq. (14) to the whole universe, we get R = RT(1 + '~0),

(15)

which shows that on the universe as a whole, its mass lacks any apparent effect on the dilation and only the type of premetrics affects its deviation from Euclidean geometry. Since (1 + ~0) is close to unity for both R and Q premetrics (ZP2, Sect. 3), the deviation from Euclidean geometry in a very large scale in the universe has no visible effect. 3.6. THE VARIATION OF c AND G WITH TIME

We introduce the hypothesis that the maximum attainable speed c on the abacus equals the rate of expansion of the universe c-

,

(16)

ZATRIKEANPREGEOMETRY:SIMPLECOSMOLOGICALSCENARIOS

279

due to the fact that no part of the universe can have velocity higher than the rate of expansion. Since the rate of expansion might have been different in the past, c was not always equal to its current value. Therefore, Eq. (7) leads to

7 GM

- 1.

(17)

Let us assume that a particle ra in the expanding universe is moving with the maximum speed e. To conserve its energy, the rate of change of its kinetic and potential energy must be equal. Thus

-~kk

_ raM/~ (1 GM = a---dy+ +o~-~-~) ,

(18)

and by substituting Eq. (17) in Eq. (18) we finally get --R2/~

G - M(1 + ~'0)'

(19)

From Eq. (19) we can estimate the current acceleration of the universe; it is equal to - 2 . 5 4 x 10 - l ° m s -2. The above equation shows that G is the manifestation of the acceleration of the universe; had the maximum attainable speed been constant, G would have been zero and there would be no gravity. Combining Eq. (17) and Eq. (19), we can calculate a formula for 7 to obtain

_/~2 7 -

R/~ (1 + +o).

(20)

Notice that, according to Eq. (20), 7 is inversely proportional to the deceleration parameter q = J~R/R 2. Thus, we get q = 0.39 < 0.50; the deviation from the Euclidean geometry is hyperbolic and the universe will expand forever. Having calculated R, we can estimate the current relative change o f t at - 2 . 5 8 x 10-11 per year, less than the value calculated in Sect. 3.2. By substituting Eq. (20) in Eq. (19) and differentiating it on the assumption that 7 is constant, we get O =

+ 7M

(21)

Substituting the current values in the above equation, we get G/G = - 1 . 4 5 x 10-I1 y - I . Contrary to Sect. 3.3, under the present scenario, G decreases with time; the above result agrees with Van Flandern's formula for f _~ - ~ . Nevertheless, the result is within the limits set in Blake (1977) using paleontological data. A simple approach to the universe's expansion is Æ C< t I/2,

/~ 0< •-1/2

~ Πt -3/2,

(22)

280

T.G. DALLASAND V.S. GEROYANN1S

Under Eq. (22) the universe starts from an ultra dense state with minimum diameter and undergoes a rapid expansion in a very short time; later the expansion gradually slows down (a typical inflationary scenario), but will go on forever at an ever decreasing rate. In the meantime, the maximum attainable speed c starts from very large values close to infinity, but very quickly slows down and will continue to slow down ad infinitum, albeit at a slower rate. Gravitational forces were stronger in the early universe, not only because masses were closer together, but also because G was higher in the past, the value of G decreasing proportionally to that of c. Finally, the pregeometric dilation constant "7 remains constant with time. Since 3' remains constant and G decreases, 7 G will decrease with time. 4. Conclusions In this paper we have examined some consequences of zatrikean pregeometry in cosmology. We have shown that the basic results obtained are in accordance with observational data. We have also verified that zatrikean pregeometry can lead to many cosmological scenarios. Two of these scenarios (Sects. 3.5 and 3.6) were discussed in more detail, because they represent two different interpretations of the pregeometric dilation constant 3'. Under the one scenario, 3' is variable; as a result the structure of the abacus is time-dependent. Then 7 represents the deviation from Euclidean geometry due to mass. However, the large scale structure of the universe looks Euclidean, because only the premetrics have effect on it; matter interacts significantly with space only in shorter distances. Under the second scenario, 7 is constant and represents the deceleration parameter of the universe; c is the rate of expansion of the universe and G is a meter for the acceleration of the universe. The universe will expand forever and both c and G decrease with time. These scenarios will be examined in more detail in the future. References Blake, G.M.: 1977, Monthly Notices Roy. Astron. Soc. 178, 41. Geroyannis V.S.: 1992,Astrophys. Space Sci. 199, 1. Geroyannis V.S. and Dallas T.G.: 1993, Astrophys. Space Sci. 200, 1. Van Flandern, T.C.: 1981, Astrophys. J. 248, 813.