General Relativity and Gravitation, Vol. 9, No. 4 (1978), pp. 329-338

A Plan to Study the "Standard Hot Big-Bang Model" of the Universe

RAMESH CHANDRA

Department of Mathematics, St. Andrew's College, Gorakhpur, Gorakhpur, India

Received April 25, 1977

Abstract

The dynamics of the "standard hot big-bang model" of the universe according to general relativity with modified cosmical constant are discussed.

w Introduction

In 1917 Einstein introduced the cosmological term into his field equations. Later on he abandoned this term because it did not serve the purposes for which it had been introduced. It has been shown that we neither neglect nor accept this term in its present form. As a result, a modification of the cosmical constant has been proposed. The modified field equations thus obtained have been used to study the evolution of the universe according to the "standard hot big-bang model."

w Modification of the Einstein FieM Equations to Study Homogeneous Cosmology

The dynamics of general relativity are expressed in Einstein's field equations

1 R 8nG Ri]--2 g i]+hgi i=- e" Tij ( i ' ] = 1 ' 2 ' 3 ' 4 ) (2.1)

where A is a universal constant which, like R, has the dimensions of a space curvature, namely, (length) -2. It has come to be called the "cosmical constant",

329 0001-7701/78/0900-0329505.00/0 �9 Plenum Publishing Corporation

330 CHANDRA

because only in cosmology does it play a significant role. The reasons that led Einstein to introduce the cosmological term Agii into his field equations are as follows [1].

(i) Einstein idealized the universe as a 3-sphere filled with matter at effec- tively uniform density, its random velocities zero. He was able to show that for positive values of A equations (2.1) admitted of a solution satisfying these conditions.

(ii) Einstein thought that for positive A equations (2.1) had no solutions for Tij = O, that is, for empty space. He was of the opinion that Mach's principle [2] had been incorporated into his theory of gravitation.

The system (2.1) is known as the Einstein static universe, but it has been verified by Hubble [3] that the universe is expanding.

Einstein's result (ii) was shown to be wrong by de Sitter [4], who found a solution of equations (2.1) for empty space. This solution represented an "ex- panding" universe, in which test particles of negligible mass would continually recede from each other with ever-increasing velocity.

Thereupon Einstein abandoned the cosmological team, calling it "the biggest blunder of my life" [5].

On the ground of the two correspondence limits, the Newtonian limit and the special theory limit [6], a modification of the field equations (2.1) has been proposed as [7]

1 8rrG Ri i - -~R g i i+ A(ij)gii- c4 Tij ( i , ] = 1 , 2 , 3 , 4 ) (2.2)

where A(ii) are different functions of the "cosmical time" for i, ] = 1,2, 3 (space components), and the rest of the components (time components) are constants.

The quantities A(i]) are not the components of a tensor because they do not change on coordinate transformations, though some components of A(ii) are functions of "cosmical time." It is due to the fact that the adoption of a com- mon cosmical time, which reminds one of the absolute time of Newton, is made possible by the fact that a spatially homogeneous Universe would have a uniform potential. Thus, we see that the quantities A(i]) are invariant.

In contravariant form and in mixed form, the field equations (2.2) can be written as

1 i" A(i])gii = _ 87rG Ri] - -2 R g l + 7 - Ti] (i, j = 1, 2, 3, 4) (2.3)

i 1 i A(i) i _ 81rG i R i - -~Rgl+~x(])~;] c4 T~ (i,] = 1 ,2 ,3 ,4 ) (2.4)

The symmetric property of the field equations and the invariant nature of the quantities A(i]) suggest that

A PLAN TO STUDY BIG-BANG MODEL 331

a( i i )=a( i i )=A{3=a l i~=a( i J )=A (ji) ( i , j = 1,2, 3, 4) (2.5)

According to the special theory limit [8] the field equations (2.2) yield the diagonal values of A(/]) as [7]

A(/]) = a( t ) > 0 (i = 1, 2, 3)

and (2.6)

A(44) = -/3 = const < 0

Under these conditions the Newtonian limit of the field equations (2.2) is given by [7]

V2r + [~(t) +/3] c 2 = strap (2.7)

This admits a variable density

[ ~ ( t ) + t31 c ~ 0 - (2.8)

8rrG

in the presence of homogeneous matter, namely ~b = const. This is a situation which modifies the concept of Einstein's static universe. For an expanding uni- verse the density decreases with time. Hence the equation (2.8) suggests that a ( t ) is a decreasing function of time.

The trace of the modified field equations can be obtained from the equa- tions (2.4)

(1 1 (2 2 3 3 A(4/g4) = 8rrG -R + ( A ( l l g l + A ( z l g 2 + A { a / g a + (4 4 - - - T c 4

which yields

87rG R = 7 - T + [3a(t) -/31

in view of the equations (2.5) and (2.6). With the help of this equation, the field equations (2.2), (2.3), and (2.4) can also be written in the following forms:

1 8, G[r 1 0 Rii - -~ [3a(t) - [J]gii + A(ij)gii - -~ \ i! - -~ Tg

R ij - -~ [3a(t) - ~]gij + A(ij)gij = 8rrGc 4 ij _ -21 Zg

i 1 ^(i) i 8rrG(T ~ 1 i) R] - ~- [3a(t) - 5lg~ + ' ~O ' ) s / - c 4 - ~- Tg] (i, j = 1, 2, 3, 4)

Thus, we see that the introduction of a set of invariant quantities A(i/) in place of A is a suitable modification of the Einstein field equations to study homogeneous cosmologies.

332 CHANDRA

w Mach's Principle and the Modified FieM Equations

If the modified field equations satisfy Mach's principle, then we cannot have a homogeneous universe that is empty, that is, the metric of a space-time for a homogeneous universe is not possible when Tii = O. In general the metric of a space-time for a homogeneous universe in a comoving coordinates system (which is always possible in the case of a homogeneous universe) is given by [9]

ds 2 = c 2 dt 2 - gijdxidx j (i, j = 1, 2, 3, 4) (3.1)

where the gij are functions of the coordinates. If we want to substitute the values from the equation (3.1) into the left-hand side of the equation (2.2), to test Mach's principle, it is necessary to know the values of A(/j) for i, j = 1, 2, 3 and for i =] = 4. But we only know the diagonal values of A(ij) given in the equations (2.6). Therefore, it is not possible to check whether the modified field equations satisfy Mach's principle or not in general. To clarify this point let us take the Robertson-Walker (R.W.) metric [10]

R2(t) ds2 = c2 dt2 (1 + �88 2 [dr2 + r2 d02 + r2 sin2 0 dx 2] (3.2)

where K is the curvature index and takes the value +1, -1, or 0, when the uni- verse is closed, open, or flat, respectively. R(t) is the scale factor for measuring distances in the nonstatic universe. This general metric is applicable to all homo- geneous isotropic model universes.

On substituting values from (3.2) and (2.6) into (2.2) with Ti] = 0, we get [11]

2~ R2 K Rc 2 + ~ + - ~ - a(t) = 0 (3.3)

and

/~2 K ~ = 0 (3.4) R2e - - - -~ + ~ + 3

Since/3 is a positive quantity, then for K = +1 or 0, the equation (3.4) does not give a real value of the scale factor. Therefore, a homogeneous and isotropie universe satisfies Mach's principle when it is either closed or flat. Similarly, we can cheek the Machian situation for other homogeneous model universes.

w Equations Governing the "Standard Hot Big-Bang"Model

At the present epoch astronomical observations reveal that the universe is homogeneous and isotropic on scales of ~10 8 fight years and larger [12]. By

A PLAN TO STUDY BIG-BANG MODEL 333

taking such a large-scale viewpoint, one can treat galaxies as "particles" of a "gas" that fills the universe. The energy-momentum tensor for this "cosmic fluid" is given by [13]

T i / = ( p + p ) u i u i - g q p (i,]= 1 , 2 , 3 , 4 ) (4.1)

where u i is the 4-velocity vector of the cosmic fluid. The R.W. Metric (3.2) is applicable to all the homogeneous and isotropic

model universes with the condition that the "cosmic fluid" is at rest relative to the comoving observer [10], whence u i = (0, O, O, c). On substituting (2.6), (3.2), and (4.1) into the field equations (2.2) we get [11]

2R 1~ 2 K 87rGp a c ----i + + U - o (t) - c" (4.2)

and

t~ 2 K /3_ 8~rGp R~e 2 + ~ - "~ 3 3c 2 (4.3)

The equation (4.2) is a dynamic equation that gives the second derivative of the scale factor and thereby governs the dynamic evolution away from the initial moment of time. The equation (4.3) is an "initial value equation" which relates /~ to R and O at any initial moment of time. From these two equations we get

- - 7 - + ~(t) +/3 R2R = ~-;~ ~ 1 (4.4)

Knowing the equation of state for the cosmic fluid and the variation of a(t), the equations (4.3) and (4.4) together will give the dynamic evolution of the universe according to the "standard hot big-bang model" [14].

w Nature o f the Parameters a(t) and/3

The difference of the equations (4.2) and (4.3) gives

4 4rrG [a(t) +/3] c2 R (5.1) = CpR cT pR + 2

in which the right-hand side is essentially the force, producing the acceleration on the left [15]. The force terms

4 47rGpR [a(t) +/3] cZ R -~ lrGpR, e2 , and 2

are due to gravity, pressure, and the cosmological term, respectively. Hence the quantities a(t) and/3 can be treated as the force parameters counteracting the gravity.

334 CHANDRA

w Assumption Regarding the Variation of Pressure

As we have already mentioned, the equation (4.4) is integrable only when we know the equation of state and the dependence of a(t) on t. But both are not known. About o~(t) we know that it is a positive decreasing function of time. Therefore, it is better to take the variation of pressure as a linear function of a(t) to avoid complications. A suitable assumption of this type can be taken as

P = [ a ( t ) - a ( t o ) ] c2 (6.1) 8~rG

where to is the time at the present epoch from the beginning of the universe. The idea behind this assumption about the pressure is that with the help of

this pressure equation we will be able to get a differential equation similar to that of "Friedmann's differential equation" governing uniform model universes, which are best suited with the observations [16].

w Boundary Conditions

Let us suppose that tl is the time when the universe ceased to be radiation dominated and became matter dominated. Then for the radiation-dominated phase of evolution the boundary conditions can be taken as

t= t l , P=Pa, P=Pl, R =R1, or(t) = or(t1)

and (7.1)

T = T1 (absolute temperature).

Also in the radiation dominated phase of evolution (0 ~< t ~< t~) we have [17]

3 _ Gfrr 2(kT) 4 3p = p = 321rt2 c4120itac3 (7.2)

Similarly the boundary conditions for the matter-dominated phase of evolu- tion can be taken as

t = to, p = 0, O = 00, R = Ro, a(t) = (to) (7.3)

Here we have taken zero pressure for the prese/at moment, according to the observations. The pressure equation (6.1) is also consistent with this condition.

Finally, we assume the validity of the pressure equation (6.1) in both phases of evolution. It is due to the fact that if we take two different pressure equa- tions, one for the radiation-dominated phase of evolution and other for the matter-dominated phase of evolution, then we cannot follow the boundary con~ ditions (7.1) strictly.

A PLAN TO STUDY BIG-BANG MODEL 335

w Determination of the Variation of Density

On substituting the expression for the pressure from the equation (6.1) into the equation (4.4) we get

C-~ [a(to) +/3] R3 = 87rGp R3 (8.1) 3 3c 2

where C is a constant of integration, can be determined under the boundary con- ditions (7.5) as

C +-[~(t~ +/31 _ 8~aOo 3 R~ 3c 2 R~ (8"2)

The equations (8.1) and (8.2) imply

8zrGp_ 1 ( ~ [a(to)+/3] _R3) } 3c 2 R 3 R~ 3 (R~ (8.3)

This expresses the variation of density with scale factor. It can also be writ- ten as

( R o a - R 3 ) { [a(to)+/3] } P - Po - R 3 P0 8zrG c2 (8.4)

which shows that for an expanding universe we must have

Po > [a(to) +/31 c2 (8.5) 8zrG

Under this condition the equation (8.2) implies that the constant of integra- tion C must be a positive quantity.

w Determination of the Variation of the Scale Factor

Substituting (8.1) into (4.3) yields an equation identical with "Friedmann's differential equation" [16]

dR t~ t ) r l a ' t o "R2 + C/R - ^]"1/2 = c dt (9.1)

with a slight change that in place of A here we get a positive quantity a(to). The solution of this differential equation gives the variation of the scale factor with time.

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w Determination of or(to) and 3

In terms of the observed parameters H(=I~/R) and q(=-R[~/R 2) the equa- tions (4.2) and (4.3) can be written as

H 2 K 87rGp e2 (1 - 2 q ) + ~ - ~ - a ( t ) - c2

and

H 2 K 3 _ 87rGp c2 + ~ - -r 3 3c 2

These two equations determine the parameters or(to) and/3 under the condi- tions (7.3) as

_ ~ K (10.1) a(to) = (1 - 2qo)+ Ro 2

and

8nGpo 3Ho 2 3K 3 = c2 c2 Ro 2 (10.2)

where Ho and qo are the values of the parameters H and q at the present epoch, respectively.

The parameters Ho, qo, K/Ro 2, and Po are regarded as being determined nu- merically from observations [ 18], hence from the equations (10.1) and (10.2) it is possible to determine the numerical values of a(to) and/3, respectively.

w Procedure to Study the History o f the Evolution o f the Universe

We have seen that the numerical values of the quantities Ho, qo, K/Ro z, Po, or(to), and 3 are known. With the help of these known quantities we can deter- mine some other physical quantities as follows.

Substituting the numerical values of a(to), 3, Po, and R into the equation (8.2) we get the numerical value of the constant of integration C. And then we substitute the values of a (to), K, and C into the equation (9.1), which gives the variation of RI, in terms of tl, after integration between the limit R = 0 to R = R~. This value of Ra gives the density p~ at the time t~ with the help of the equation (8.3). The density thus obtained gives the numerical values of pl, Pt, tl , and T1 from the equation (7.2) under the condition (7.1). Thus, we will be able to study the history of the evolution of the universe in the radiation domi- nated phase of evolution with the help of these known parameters.

A PLAN TO STUDY BIG-BANG MODEL 337

Again on integrating the equation (9.1) between the limit R = {3 to R = R0 we get the numerical value of the time to at the present epoch which gives the age of the universe.

The physical quantities thus obtained are sufficient to study the history of the evolution of the universe.

w Superiority over the Friedmann Cosmology

The "standard hot big-bang model" of the universe based on Friedmann's differential equation is remarkably powerful and accords well with observations [16]. However, it encounters an apparent difficulty with one item: the variation of pressure. For an expanding universe the pressure must be a decreasing func- tion of time. But this functional relation is not known. For this reason, in Fried- mann cosmology we study the radiation-dominated phase of evolution and the matter-dominated phase of evolution separately. Therefore, it is not possible to know the time when the universe ceased to be radiation dominated and became matter dominated. This difficulty has been removed here. And hence it will be possible to get more accurate information about the evolution of the universe. For example, the prediction of the helium abundance, which has not been pre- dicted accurately by any other theory of gravitation [19].

A ckno wledgm en ts

The author is thankful to Dr. V. B. Johari, Reader in Mathematics, Univer- sity of Gorakhpur, and Shri S. K. Srivastava, Research Scholar, Department of Mathematics, University of Gorakhpur for suggestions during the preparation of this paper.

References

1. Einstein, A. (1917). PressAkad. lCiss. Berlin, Sitzber, 142; English translation available in The Principle o f Relativity, (Lorentz et al., 1923).

2. Rindler, W. (1969). Essential Relativity (Van Nostrand Reinhold Company, New York), pp. 15,271.

3. Hubble, E. (1936).Astrophs. J., 84, 158, 270, 517; (1953).Mon. Not. R. Astron. Soc., 113, 658.

4. De Sitter, W. (1917). Mon. Not. R. Astron. Soc., 78, 3. 5. Einstein, A. quoted by Gamow, G. (1970). inMy World line (Viking Press, New York),

p. 44. 6. Trautmann, A., Pirani, F. A. E., and Bondi, H. (1964). Lectureson GeneralRelativity

(Prentice-Hall, Englewood Cliffs, New Jersey), p. 119. 7. Chandra, Ramesh (1977). Gen. Rel. Gray. 8, 787. 8. McVittic, G. C. (1956). General Relativity and Cosmology (Chapman & Hall Ltd., Lon-

don), p. 62.

338 CHANDRA

9. Misner, W., Thorne, K., and Wheeler, y. (1973). Gravitation (W. H. Freeman, San Fran- cisco), p. 715.

I0. Robertson, H. P. (1935).Astrophys. J., 82, 248; (1936). 83, 187; Walker, A. G. (1936). Proc. London Math. Soc., 42, 90.

11. Tolman, R. C. (1934). Relativity, Thermodynamics and Cosmology (Oxford University Press, Oxford), formulas (98.6).

12. Sandage, A., Tamman, A., and Hardy, E. (1972). Astrophys. J., 172, 253; Boughn, S. P., Fram, D. M., and Partridge, R. B. (1971). Astrophys. J., 165,439.

13. Ehlers, J., Green, P., and Sachs, R. K. (1968). J. Math. Phys., 9, 1344. 14. Sciama, D. W. (1973). Modern Cosmology (Cambridge University Press, Cambridge);

Misner, W., Thorne, K., and Wheeler, Y. (1973). Gravitation (W. H. Freeman and Com- pany, San Francisco), Chap. 28.

15. Milne, E. A. and MeCrea, W. H. (1934); Quart. J. Math., 5, 73; Schluter, A. (1955). Astron. J., 60, 141; Rindler, W. (1969). Essential Relativity (Van Nostrand Reinhold Company, New York), p. 248.

16. Friedmann, A. (1922). Z. Phys., 10, 377; (1924). 21,326; Lemaitre, G. (1927). Ann. Soc. Sei. Bruxelles, A47, 49; Robertson, H. P. (1933). Rev. Mod. Phys., 5, 62.

17. Misner, W., Thorne, K., and Wheeler, Y. (1973). Gravitation (W. H. Freeman and Com- pany, San Francisco) equations (27.30) (27.53) and (27.54).

18. Sandage, A. (1972).Astrophys. J., 173,485; (1972). 178; McVittie, G. C. (1956). Gen- eral Relativity and Cosmology (Chapmann & Hall Ltd., London), equations (9.213), (9.214) and (9.316).

19. Ryan, Michael P., Jr., and Shepley, Lawrence C. (1975). Homogeneous Cosmology (Princeton Series in Physics, Princeton, New Jersey), p. 266.